Type  Label  Description 
Statement 

Theorem  extwwlkfablem1 26601 
Lemma 1 for extwwlkfab 26617. (Contributed by Alexander van der Vekens,
15Sep2018.)

⊢ ((((𝑉 USGrph 𝐸 ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_{≥}‘2))
∧ 𝑤 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁)) ∧ ((𝑤‘0) = 𝑋 ∧ (𝑤‘(𝑁 − 2)) = (𝑤‘0))) → (𝑤‘(𝑁 − 1)) ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑋)) 

Theorem  extwwlkfablem2lem 26602 
Lemma for extwwlkfablem2 26605. (Contributed by Alexander van der Vekens,
17Sep2018.)

⊢ ((𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = 𝑁 ∧ 𝑁 ∈ (ℤ_{≥}‘2))
→ (#‘(𝑤 substr
⟨0, (𝑁 −
2)⟩)) = (𝑁 −
2)) 

Theorem  clwwlkextfrlem1 26603 
Lemma for numclwwlk2lem1 26629. (Contributed by Alexander van der Vekens,
3Oct2018.)

⊢ (((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ∧ 𝑍 ∈ 𝑉) ∧ (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ (𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋)) → (((𝑊 ++ ⟨“𝑍”⟩)‘0) = 𝑋 ∧ ((𝑊 ++ ⟨“𝑍”⟩)‘𝑁) ≠ 𝑋)) 

Theorem  numclwwlkfvc 26604* 
Value of function 𝐶, mapping a nonnegative number n to
the closed
walks having length n. (Contributed by Alexander van der Vekens,
14Sep2018.)

⊢ 𝐶 = (𝑛 ∈ ℕ_{0} ↦ ((𝑉 ClWWalksN 𝐸)‘𝑛)) ⇒ ⊢ (𝑁 ∈ ℕ_{0} → (𝐶‘𝑁) = ((𝑉 ClWWalksN 𝐸)‘𝑁)) 

Theorem  extwwlkfablem2 26605* 
Lemma 2 for extwwlkfab 26617. (Contributed by Alexander van der Vekens,
15Sep2018.)

⊢ 𝐶 = (𝑛 ∈ ℕ_{0} ↦ ((𝑉 ClWWalksN 𝐸)‘𝑛)) ⇒ ⊢ ((((𝑉 USGrph 𝐸 ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_{≥}‘3))
∧ 𝑤 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁)) ∧ ((𝑤‘0) = 𝑋 ∧ (𝑤‘(𝑁 − 2)) = (𝑤‘0))) → (𝑤 substr ⟨0, (𝑁 − 2)⟩) ∈ (𝐶‘(𝑁 − 2))) 

Theorem  numclwwlkun 26606* 
The set of closed walks in an undirected simple graph is the union of
the numbers of closed walks starting at each of the vertices.
(Contributed by Alexander van der Vekens, 7Oct2018.)

⊢ 𝐶 = (𝑛 ∈ ℕ_{0} ↦ ((𝑉 ClWWalksN 𝐸)‘𝑛)) ⇒ ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑁 ∈ ℕ_{0}) → (𝐶‘𝑁) = ∪
𝑥 ∈ 𝑉 {𝑤 ∈ (𝐶‘𝑁) ∣ (𝑤‘0) = 𝑥}) 

Theorem  numclwwlkdisj 26607* 
The sets of closed walks starting at different vertices in an undirected
simple graph are disjunct. (Contributed by Alexander van der Vekens,
7Oct2018.)

⊢ 𝐶 = (𝑛 ∈ ℕ_{0} ↦ ((𝑉 ClWWalksN 𝐸)‘𝑛)) ⇒ ⊢ Disj 𝑥 ∈ 𝑉 {𝑤 ∈ (𝐶‘𝑁) ∣ (𝑤‘0) = 𝑥} 

Theorem  numclwwlkovf 26608* 
Value of operation 𝐹, mapping a vertex v and a
nonnegative integer
n to the "(For a fixed vertex v, let f(n) be the number of) walks
from v
to v of length n" according to definition 5 in [Huneke] p. 2.
(Contributed by Alexander van der Vekens, 14Sep2018.)

⊢ 𝐶 = (𝑛 ∈ ℕ_{0} ↦ ((𝑉 ClWWalksN 𝐸)‘𝑛))
& ⊢ 𝐹 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ_{0} ↦ {𝑤 ∈ (𝐶‘𝑛) ∣ (𝑤‘0) = 𝑣}) ⇒ ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ_{0}) → (𝑋𝐹𝑁) = {𝑤 ∈ (𝐶‘𝑁) ∣ (𝑤‘0) = 𝑋}) 

Theorem  numclwwlkffin 26609* 
In a finite graph, the value of operation 𝐹 is also finite.
(Contributed by Alexander van der Vekens, 26Sep2018.)

⊢ 𝐶 = (𝑛 ∈ ℕ_{0} ↦ ((𝑉 ClWWalksN 𝐸)‘𝑛))
& ⊢ 𝐹 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ_{0} ↦ {𝑤 ∈ (𝐶‘𝑛) ∣ (𝑤‘0) = 𝑣}) ⇒ ⊢ (((𝑉 ∈ Fin ∧ 𝐸 ∈ 𝑈) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ_{0})) → (𝑋𝐹𝑁) ∈ Fin) 

Theorem  numclwwlkovfel2 26610* 
Properties of an element of the value of operation 𝐹. (Contributed
by Alexander van der Vekens, 20Sep2018.)

⊢ 𝐶 = (𝑛 ∈ ℕ_{0} ↦ ((𝑉 ClWWalksN 𝐸)‘𝑛))
& ⊢ 𝐹 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ_{0} ↦ {𝑤 ∈ (𝐶‘𝑛) ∣ (𝑤‘0) = 𝑣}) ⇒ ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑁 ∈ ℕ_{0} ∧ 𝑋 ∈ 𝑉) → (𝐴 ∈ (𝑋𝐹𝑁) ↔ ((𝐴 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝐴) − 1)){(𝐴‘𝑖), (𝐴‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘𝐴), (𝐴‘0)} ∈ ran 𝐸) ∧ (#‘𝐴) = 𝑁 ∧ (𝐴‘0) = 𝑋))) 

Theorem  numclwwlkovf2 26611* 
Value of operation 𝐹 for argument 2. (Contributed by
Alexander van
der Vekens, 19Sep2018.)

⊢ 𝐶 = (𝑛 ∈ ℕ_{0} ↦ ((𝑉 ClWWalksN 𝐸)‘𝑛))
& ⊢ 𝐹 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ_{0} ↦ {𝑤 ∈ (𝐶‘𝑛) ∣ (𝑤‘0) = 𝑣}) ⇒ ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑋 ∈ 𝑉) → (𝑋𝐹2) = {𝑤 ∈ Word 𝑉 ∣ ((#‘𝑤) = 2 ∧ {(𝑤‘0), (𝑤‘1)} ∈ ran 𝐸 ∧ (𝑤‘0) = 𝑋)}) 

Theorem  numclwwlkovf2num 26612* 
In a k regular graph, therere are k closed walks of length 2 starting at
a fixed vertex. (Contributed by Alexander van der Vekens,
19Sep2018.)

⊢ 𝐶 = (𝑛 ∈ ℕ_{0} ↦ ((𝑉 ClWWalksN 𝐸)‘𝑛))
& ⊢ 𝐹 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ_{0} ↦ {𝑤 ∈ (𝐶‘𝑛) ∣ (𝑤‘0) = 𝑣}) ⇒ ⊢ ((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 ∧ 𝑋 ∈ 𝑉) → (#‘(𝑋𝐹2)) = 𝐾) 

Theorem  numclwwlkovf2ex 26613* 
Extending a closed walk starting at a fixed vertex by an additional edge
(forth and back). (Contributed by AV, 22Sep2018.) (Proof shortened
by AV, 23Oct2018.)

⊢ 𝐶 = (𝑛 ∈ ℕ_{0} ↦ ((𝑉 ClWWalksN 𝐸)‘𝑛))
& ⊢ 𝐹 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ_{0} ↦ {𝑤 ∈ (𝐶‘𝑛) ∣ (𝑤‘0) = 𝑣}) ⇒ ⊢ (((𝑉 USGrph 𝐸 ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_{≥}‘3))
∧ 𝑄 ∈
(⟨𝑉, 𝐸⟩ Neighbors 𝑋) ∧ 𝑃 ∈ (𝑋𝐹(𝑁 − 2))) → ((𝑃 ++ ⟨“𝑋”⟩) ++ ⟨“𝑄”⟩) ∈ (𝐶‘𝑁)) 

Theorem  numclwwlkovg 26614* 
Value of operation 𝐺, mapping a vertex v and a
nonnegative
integer n to the "closed nwalks v(0) ... v(n2) v(n1) v(n) from
v =
v(0) = v(n) with v(n2) = v" according to definition 6 in [Huneke]
p. 2. (Contributed by Alexander van der Vekens, 14Sep2018.)

⊢ 𝐶 = (𝑛 ∈ ℕ_{0} ↦ ((𝑉 ClWWalksN 𝐸)‘𝑛))
& ⊢ 𝐹 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ_{0} ↦ {𝑤 ∈ (𝐶‘𝑛) ∣ (𝑤‘0) = 𝑣})
& ⊢ 𝐺 = (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ_{≥}‘2)
↦ {𝑤 ∈ (𝐶‘𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) = (𝑤‘0))}) ⇒ ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_{≥}‘2))
→ (𝑋𝐺𝑁) = {𝑤 ∈ (𝐶‘𝑁) ∣ ((𝑤‘0) = 𝑋 ∧ (𝑤‘(𝑁 − 2)) = (𝑤‘0))}) 

Theorem  numclwwlkovgel 26615* 
Properties of an element of the value of operation 𝐺.
(Contributed by Alexander van der Vekens, 24Sep2018.)

⊢ 𝐶 = (𝑛 ∈ ℕ_{0} ↦ ((𝑉 ClWWalksN 𝐸)‘𝑛))
& ⊢ 𝐹 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ_{0} ↦ {𝑤 ∈ (𝐶‘𝑛) ∣ (𝑤‘0) = 𝑣})
& ⊢ 𝐺 = (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ_{≥}‘2)
↦ {𝑤 ∈ (𝐶‘𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) = (𝑤‘0))}) ⇒ ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_{≥}‘2))
→ (𝑃 ∈ (𝑋𝐺𝑁) ↔ (𝑃 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) ∧ (𝑃‘0) = 𝑋 ∧ (𝑃‘(𝑁 − 2)) = (𝑃‘0)))) 

Theorem  numclwwlkovgelim 26616* 
Properties of an element of the value of operation 𝐺.
(Contributed by Alexander van der Vekens, 24Sep2018.)

⊢ 𝐶 = (𝑛 ∈ ℕ_{0} ↦ ((𝑉 ClWWalksN 𝐸)‘𝑛))
& ⊢ 𝐹 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ_{0} ↦ {𝑤 ∈ (𝐶‘𝑛) ∣ (𝑤‘0) = 𝑣})
& ⊢ 𝐺 = (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ_{≥}‘2)
↦ {𝑤 ∈ (𝐶‘𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) = (𝑤‘0))}) ⇒ ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_{≥}‘2))
→ (𝑃 ∈ (𝑋𝐺𝑁) → ((𝑃 ∈ Word 𝑉 ∧ (#‘𝑃) = 𝑁) ∧ ((𝑃‘0) = 𝑋 ∧ (𝑃‘(𝑁 − 2)) = (𝑃‘0))))) 

Theorem  extwwlkfab 26617* 
The set of closed walks (having a fixed length greater than 1 and
starting at a fixed vertex) with the last but 2 vertex is identical
with the first (and therefore last) vertex can be constructed from the
set of closed walks with length smaller by 2 than the fixed length
appending a neighbor of the last vertex and afterwards the last vertex
(which is the first vertex) itself ("walking forth and back"
from the
last vertex). 3 ≤ 𝑁 is required since for 𝑁 = 2:
(𝑋𝐹(𝑁 − 2)) = (𝑋𝐹0) = ∅, see clwwlkgt0 26299 stating that
a walk of length 0 is not represented as word, at least not for an
undirected simple graph.) (Contributed by Alexander van der Vekens,
18Sep2018.)

⊢ 𝐶 = (𝑛 ∈ ℕ_{0} ↦ ((𝑉 ClWWalksN 𝐸)‘𝑛))
& ⊢ 𝐹 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ_{0} ↦ {𝑤 ∈ (𝐶‘𝑛) ∣ (𝑤‘0) = 𝑣})
& ⊢ 𝐺 = (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ_{≥}‘2)
↦ {𝑤 ∈ (𝐶‘𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) = (𝑤‘0))}) ⇒ ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_{≥}‘3))
→ (𝑋𝐺𝑁) = {𝑤 ∈ (𝐶‘𝑁) ∣ ((𝑤 substr ⟨0, (𝑁 − 2)⟩) ∈ (𝑋𝐹(𝑁 − 2)) ∧ (𝑤‘(𝑁 − 1)) ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑋) ∧ (𝑤‘(𝑁 − 2)) = 𝑋)}) 

Theorem  numclwlk1lem2foa 26618* 
Going forth and back form the end of a (closed) walk. (Contributed by
Alexander van der Vekens, 22Sep2018.)

⊢ 𝐶 = (𝑛 ∈ ℕ_{0} ↦ ((𝑉 ClWWalksN 𝐸)‘𝑛))
& ⊢ 𝐹 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ_{0} ↦ {𝑤 ∈ (𝐶‘𝑛) ∣ (𝑤‘0) = 𝑣})
& ⊢ 𝐺 = (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ_{≥}‘2)
↦ {𝑤 ∈ (𝐶‘𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) = (𝑤‘0))}) ⇒ ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_{≥}‘3))
→ ((𝑃 ∈ (𝑋𝐹(𝑁 − 2)) ∧ 𝑄 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑋)) → ((𝑃 ++ ⟨“𝑋”⟩) ++ ⟨“𝑄”⟩) ∈ (𝑋𝐺𝑁))) 

Theorem  numclwlk1lem2f 26619* 
T is a function. (Contributed by Alexander van der Vekens,
19Sep2018.)

⊢ 𝐶 = (𝑛 ∈ ℕ_{0} ↦ ((𝑉 ClWWalksN 𝐸)‘𝑛))
& ⊢ 𝐹 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ_{0} ↦ {𝑤 ∈ (𝐶‘𝑛) ∣ (𝑤‘0) = 𝑣})
& ⊢ 𝐺 = (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ_{≥}‘2)
↦ {𝑤 ∈ (𝐶‘𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) = (𝑤‘0))}) & ⊢ 𝑇 = (𝑤 ∈ (𝑋𝐺𝑁) ↦ ⟨(𝑤 substr ⟨0, (𝑁 − 2)⟩), (𝑤‘(𝑁 −
1))⟩) ⇒ ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_{≥}‘3))
→ 𝑇:(𝑋𝐺𝑁)⟶((𝑋𝐹(𝑁 − 2)) × (⟨𝑉, 𝐸⟩ Neighbors 𝑋))) 

Theorem  numclwlk1lem2fv 26620* 
Value of the function T. (Contributed by Alexander van der Vekens,
20Sep2018.)

⊢ 𝐶 = (𝑛 ∈ ℕ_{0} ↦ ((𝑉 ClWWalksN 𝐸)‘𝑛))
& ⊢ 𝐹 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ_{0} ↦ {𝑤 ∈ (𝐶‘𝑛) ∣ (𝑤‘0) = 𝑣})
& ⊢ 𝐺 = (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ_{≥}‘2)
↦ {𝑤 ∈ (𝐶‘𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) = (𝑤‘0))}) & ⊢ 𝑇 = (𝑤 ∈ (𝑋𝐺𝑁) ↦ ⟨(𝑤 substr ⟨0, (𝑁 − 2)⟩), (𝑤‘(𝑁 −
1))⟩) ⇒ ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_{≥}‘3))
→ (𝑃 ∈ (𝑋𝐺𝑁) → (𝑇‘𝑃) = ⟨(𝑃 substr ⟨0, (𝑁 − 2)⟩), (𝑃‘(𝑁 − 1))⟩)) 

Theorem  numclwlk1lem2f1 26621* 
T is a 11 function. (Contributed by AV, 26Sep2018.) (Proof
shortened by AV, 23Oct2018.)

⊢ 𝐶 = (𝑛 ∈ ℕ_{0} ↦ ((𝑉 ClWWalksN 𝐸)‘𝑛))
& ⊢ 𝐹 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ_{0} ↦ {𝑤 ∈ (𝐶‘𝑛) ∣ (𝑤‘0) = 𝑣})
& ⊢ 𝐺 = (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ_{≥}‘2)
↦ {𝑤 ∈ (𝐶‘𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) = (𝑤‘0))}) & ⊢ 𝑇 = (𝑤 ∈ (𝑋𝐺𝑁) ↦ ⟨(𝑤 substr ⟨0, (𝑁 − 2)⟩), (𝑤‘(𝑁 −
1))⟩) ⇒ ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_{≥}‘3))
→ 𝑇:(𝑋𝐺𝑁)–11→((𝑋𝐹(𝑁 − 2)) × (⟨𝑉, 𝐸⟩ Neighbors 𝑋))) 

Theorem  numclwlk1lem2fo 26622* 
T is an onto function. (Contributed by Alexander van der Vekens,
20Sep2018.)

⊢ 𝐶 = (𝑛 ∈ ℕ_{0} ↦ ((𝑉 ClWWalksN 𝐸)‘𝑛))
& ⊢ 𝐹 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ_{0} ↦ {𝑤 ∈ (𝐶‘𝑛) ∣ (𝑤‘0) = 𝑣})
& ⊢ 𝐺 = (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ_{≥}‘2)
↦ {𝑤 ∈ (𝐶‘𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) = (𝑤‘0))}) & ⊢ 𝑇 = (𝑤 ∈ (𝑋𝐺𝑁) ↦ ⟨(𝑤 substr ⟨0, (𝑁 − 2)⟩), (𝑤‘(𝑁 −
1))⟩) ⇒ ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_{≥}‘3))
→ 𝑇:(𝑋𝐺𝑁)–onto→((𝑋𝐹(𝑁 − 2)) × (⟨𝑉, 𝐸⟩ Neighbors 𝑋))) 

Theorem  numclwlk1lem2f1o 26623* 
T is a 11 onto function. (Contributed by Alexander van der Vekens,
26Sep2018.)

⊢ 𝐶 = (𝑛 ∈ ℕ_{0} ↦ ((𝑉 ClWWalksN 𝐸)‘𝑛))
& ⊢ 𝐹 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ_{0} ↦ {𝑤 ∈ (𝐶‘𝑛) ∣ (𝑤‘0) = 𝑣})
& ⊢ 𝐺 = (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ_{≥}‘2)
↦ {𝑤 ∈ (𝐶‘𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) = (𝑤‘0))}) & ⊢ 𝑇 = (𝑤 ∈ (𝑋𝐺𝑁) ↦ ⟨(𝑤 substr ⟨0, (𝑁 − 2)⟩), (𝑤‘(𝑁 −
1))⟩) ⇒ ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_{≥}‘3))
→ 𝑇:(𝑋𝐺𝑁)–11onto→((𝑋𝐹(𝑁 − 2)) × (⟨𝑉, 𝐸⟩ Neighbors 𝑋))) 

Theorem  numclwlk1lem2 26624* 
There is a bijection between the set of closed walks (having a fixed
length greater than 2 and starting at a fixed vertex) with the last
but 2 vertex identical with the first (and therefore last) vertex and
the set of closed walks (having a fixed length less by 2 and starting
at the same vertex) and the neighbors of this vertex. (Contributed by
Alexander van der Vekens, 6Jul2018.)

⊢ 𝐶 = (𝑛 ∈ ℕ_{0} ↦ ((𝑉 ClWWalksN 𝐸)‘𝑛))
& ⊢ 𝐹 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ_{0} ↦ {𝑤 ∈ (𝐶‘𝑛) ∣ (𝑤‘0) = 𝑣})
& ⊢ 𝐺 = (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ_{≥}‘2)
↦ {𝑤 ∈ (𝐶‘𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) = (𝑤‘0))}) ⇒ ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_{≥}‘3))
→ ∃𝑓 𝑓:(𝑋𝐺𝑁)–11onto→((𝑋𝐹(𝑁 − 2)) × (⟨𝑉, 𝐸⟩ Neighbors 𝑋))) 

Theorem  numclwwlk1 26625* 
Statement 9 in [Huneke] p. 2: "If n >
1, then the number of closed
nwalks v(0) ... v(n2) v(n1) v(n) from v = v(0) = v(n) with v(n2) =
v is kf(n2)". Since ⟨𝑉, 𝐸⟩ is kregular, the vertex
v(n2)
= v has k neighbors v(n1), so there are k walks from v(n2) = v to
v(n) = v (via each of v's neighbors) completing each of the f(n2)
walks from v=v(0) to v(n2)=v. This theorem holds even for k=0, but
only for finite graphs! (Contributed by Alexander van der Vekens,
26Sep2018.) (Proof shortened by AV, 5May2021.)

⊢ 𝐶 = (𝑛 ∈ ℕ_{0} ↦ ((𝑉 ClWWalksN 𝐸)‘𝑛))
& ⊢ 𝐹 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ_{0} ↦ {𝑤 ∈ (𝐶‘𝑛) ∣ (𝑤‘0) = 𝑣})
& ⊢ 𝐺 = (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ_{≥}‘2)
↦ {𝑤 ∈ (𝐶‘𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) = (𝑤‘0))}) ⇒ ⊢ (((𝑉 ∈ Fin ∧ ⟨𝑉, 𝐸⟩ RegUSGrph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_{≥}‘3)))
→ (#‘(𝑋𝐺𝑁)) = (𝐾 · (#‘(𝑋𝐹(𝑁 − 2))))) 

Theorem  numclwwlkovq 26626* 
Value of operation Q, mapping a vertex v and a nonnegative integer n
to the not closed walks v(0) ... v(n) of length n from a fixed vertex
v = v(0). "Not closed" means v(n) =/= v(0). (Contributed
by
Alexander van der Vekens, 27Sep2018.)

⊢ 𝐶 = (𝑛 ∈ ℕ_{0} ↦ ((𝑉 ClWWalksN 𝐸)‘𝑛))
& ⊢ 𝐹 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ_{0} ↦ {𝑤 ∈ (𝐶‘𝑛) ∣ (𝑤‘0) = 𝑣})
& ⊢ 𝐺 = (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ_{≥}‘2)
↦ {𝑤 ∈ (𝐶‘𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) = (𝑤‘0))}) & ⊢ 𝑄 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ_{0} ↦ {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ ( lastS ‘𝑤) ≠ 𝑣)}) ⇒ ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ_{0}) → (𝑋𝑄𝑁) = {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ ((𝑤‘0) = 𝑋 ∧ ( lastS ‘𝑤) ≠ 𝑋)}) 

Theorem  numclwwlkqhash 26627* 
In a kregular graph, the size of the set of walks of length n
starting with a fixed vertex and ending not at this vertex is the
difference between k to the power of n and the size of the set of
walks of length n starting with this vertex and ending at this vertex.
(Contributed by Alexander van der Vekens, 30Sep2018.) (Proof
shortened by AV, 5May2021.)

⊢ 𝐶 = (𝑛 ∈ ℕ_{0} ↦ ((𝑉 ClWWalksN 𝐸)‘𝑛))
& ⊢ 𝐹 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ_{0} ↦ {𝑤 ∈ (𝐶‘𝑛) ∣ (𝑤‘0) = 𝑣})
& ⊢ 𝐺 = (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ_{≥}‘2)
↦ {𝑤 ∈ (𝐶‘𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) = (𝑤‘0))}) & ⊢ 𝑄 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ_{0} ↦ {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ ( lastS ‘𝑤) ≠ 𝑣)}) ⇒ ⊢ (((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 ∧ 𝑉 ∈ Fin) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ)) → (#‘(𝑋𝑄𝑁)) = ((𝐾↑𝑁) − (#‘(𝑋𝐹𝑁)))) 

Theorem  numclwwlkovh 26628* 
Value of operation H, mapping a vertex v and a nonnegative integer n
to the "closed nwalks v(0) ... v(n2) v(n1) v(n) from v = v(0)
=
v(n) ... with v(n2) =/= v" according to definition 7 in [Huneke]
p. 2. (Contributed by Alexander van der Vekens, 26Aug2018.)

⊢ 𝐶 = (𝑛 ∈ ℕ_{0} ↦ ((𝑉 ClWWalksN 𝐸)‘𝑛))
& ⊢ 𝐹 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ_{0} ↦ {𝑤 ∈ (𝐶‘𝑛) ∣ (𝑤‘0) = 𝑣})
& ⊢ 𝐺 = (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ_{≥}‘2)
↦ {𝑤 ∈ (𝐶‘𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) = (𝑤‘0))}) & ⊢ 𝑄 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ_{0} ↦ {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ ( lastS ‘𝑤) ≠ 𝑣)}) & ⊢ 𝐻 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ_{0} ↦ {𝑤 ∈ (𝐶‘𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) ≠ (𝑤‘0))}) ⇒ ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ_{0}) → (𝑋𝐻𝑁) = {𝑤 ∈ (𝐶‘𝑁) ∣ ((𝑤‘0) = 𝑋 ∧ (𝑤‘(𝑁 − 2)) ≠ (𝑤‘0))}) 

Theorem  numclwwlk2lem1 26629* 
In a friendship graph, for each walk of length n starting with a fixed
vertex and ending not at this vertex, there is a unique vertex so that
the walk extended by an edge to this vertex and an edge from this
vertex to the first vertex of the walk is a value of operation H. If
the walk is represented as a word, it is sufficient to add one vertex
to the word to obtain the closed walk contained in the value of
operation H, since in a word representing a closed walk the starting
vertex is not repeated at the end. This theorem only generally holds
for Friendship Graphs, because these guarantee that for the first and
last vertex there is a third vertex "in between".
(Contributed by
Alexander van der Vekens, 3Oct2018.)

⊢ 𝐶 = (𝑛 ∈ ℕ_{0} ↦ ((𝑉 ClWWalksN 𝐸)‘𝑛))
& ⊢ 𝐹 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ_{0} ↦ {𝑤 ∈ (𝐶‘𝑛) ∣ (𝑤‘0) = 𝑣})
& ⊢ 𝐺 = (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ_{≥}‘2)
↦ {𝑤 ∈ (𝐶‘𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) = (𝑤‘0))}) & ⊢ 𝑄 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ_{0} ↦ {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ ( lastS ‘𝑤) ≠ 𝑣)}) & ⊢ 𝐻 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ_{0} ↦ {𝑤 ∈ (𝐶‘𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) ≠ (𝑤‘0))}) ⇒ ⊢ ((𝑉 FriendGrph 𝐸 ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑊 ∈ (𝑋𝑄𝑁) → ∃!𝑣 ∈ 𝑉 (𝑊 ++ ⟨“𝑣”⟩) ∈ (𝑋𝐻(𝑁 + 2)))) 

Theorem  numclwlk2lem2f 26630* 
R is a function. (Contributed by Alexander van der Vekens,
5Oct2018.)

⊢ 𝐶 = (𝑛 ∈ ℕ_{0} ↦ ((𝑉 ClWWalksN 𝐸)‘𝑛))
& ⊢ 𝐹 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ_{0} ↦ {𝑤 ∈ (𝐶‘𝑛) ∣ (𝑤‘0) = 𝑣})
& ⊢ 𝐺 = (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ_{≥}‘2)
↦ {𝑤 ∈ (𝐶‘𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) = (𝑤‘0))}) & ⊢ 𝑄 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ_{0} ↦ {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ ( lastS ‘𝑤) ≠ 𝑣)}) & ⊢ 𝐻 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ_{0} ↦ {𝑤 ∈ (𝐶‘𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) ≠ (𝑤‘0))}) & ⊢ 𝑅 = (𝑥 ∈ (𝑋𝐻(𝑁 + 2)) ↦ (𝑥 substr ⟨0, (𝑁 + 1)⟩))
⇒ ⊢ ((𝑉 FriendGrph 𝐸 ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → 𝑅:(𝑋𝐻(𝑁 + 2))⟶(𝑋𝑄𝑁)) 

Theorem  numclwlk2lem2fv 26631* 
Value of the function R. (Contributed by Alexander van der Vekens,
6Oct2018.)

⊢ 𝐶 = (𝑛 ∈ ℕ_{0} ↦ ((𝑉 ClWWalksN 𝐸)‘𝑛))
& ⊢ 𝐹 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ_{0} ↦ {𝑤 ∈ (𝐶‘𝑛) ∣ (𝑤‘0) = 𝑣})
& ⊢ 𝐺 = (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ_{≥}‘2)
↦ {𝑤 ∈ (𝐶‘𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) = (𝑤‘0))}) & ⊢ 𝑄 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ_{0} ↦ {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ ( lastS ‘𝑤) ≠ 𝑣)}) & ⊢ 𝐻 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ_{0} ↦ {𝑤 ∈ (𝐶‘𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) ≠ (𝑤‘0))}) & ⊢ 𝑅 = (𝑥 ∈ (𝑋𝐻(𝑁 + 2)) ↦ (𝑥 substr ⟨0, (𝑁 + 1)⟩))
⇒ ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑊 ∈ (𝑋𝐻(𝑁 + 2)) → (𝑅‘𝑊) = (𝑊 substr ⟨0, (𝑁 + 1)⟩))) 

Theorem  numclwlk2lem2f1o 26632* 
R is a 11 onto function. (Contributed by Alexander van der Vekens,
6Oct2018.)

⊢ 𝐶 = (𝑛 ∈ ℕ_{0} ↦ ((𝑉 ClWWalksN 𝐸)‘𝑛))
& ⊢ 𝐹 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ_{0} ↦ {𝑤 ∈ (𝐶‘𝑛) ∣ (𝑤‘0) = 𝑣})
& ⊢ 𝐺 = (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ_{≥}‘2)
↦ {𝑤 ∈ (𝐶‘𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) = (𝑤‘0))}) & ⊢ 𝑄 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ_{0} ↦ {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ ( lastS ‘𝑤) ≠ 𝑣)}) & ⊢ 𝐻 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ_{0} ↦ {𝑤 ∈ (𝐶‘𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) ≠ (𝑤‘0))}) & ⊢ 𝑅 = (𝑥 ∈ (𝑋𝐻(𝑁 + 2)) ↦ (𝑥 substr ⟨0, (𝑁 + 1)⟩))
⇒ ⊢ ((𝑉 FriendGrph 𝐸 ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → 𝑅:(𝑋𝐻(𝑁 + 2))–11onto→(𝑋𝑄𝑁)) 

Theorem  numclwwlk2lem3 26633* 
In a friendship graph, the size of the set of walks of length n
starting with a fixed vertex and ending not at this vertex equals the
size of the set of all closed walks of length (n+2) starting with this
vertex and not having this vertex as last but 2 vertex. (Contributed
by Alexander van der Vekens, 6Oct2018.) (Proof shortened by AV,
5May2021.)

⊢ 𝐶 = (𝑛 ∈ ℕ_{0} ↦ ((𝑉 ClWWalksN 𝐸)‘𝑛))
& ⊢ 𝐹 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ_{0} ↦ {𝑤 ∈ (𝐶‘𝑛) ∣ (𝑤‘0) = 𝑣})
& ⊢ 𝐺 = (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ_{≥}‘2)
↦ {𝑤 ∈ (𝐶‘𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) = (𝑤‘0))}) & ⊢ 𝑄 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ_{0} ↦ {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ ( lastS ‘𝑤) ≠ 𝑣)}) & ⊢ 𝐻 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ_{0} ↦ {𝑤 ∈ (𝐶‘𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) ≠ (𝑤‘0))}) ⇒ ⊢ ((𝑉 FriendGrph 𝐸 ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (#‘(𝑋𝑄𝑁)) = (#‘(𝑋𝐻(𝑁 + 2)))) 

Theorem  numclwwlk2 26634* 
Statement 10 in [Huneke] p. 2: "If n >
1, then the number of closed
nwalks v(0) ... v(n2) v(n1) v(n) from v = v(0) = v(n) ... with
v(n2) =/= v is k^(n2)  f(n2)." According to rusgranumwlkg 26485, we
have k^(n2) different walks of length (n2): v(0) ... v(n2). From
this number, the number of closed walks of length (n2), which is
f(n2) per definition, must be subtracted, because for these walks
v(n2) =/= v(0) = v would hold. Because of the friendship condition,
there is exactly one vertex v(n1) which is a neighbor of v(n2) as
well as of v(n)=v=v(0), because v(n2) and v(n)=v are different, so
the number of walks v(0) ... v(n2) is identical with the number of
walks v(0) ... v(n), that means each (not closed) walk v(0) ... v(n2)
can be extended by two edges to a closed walk v(0) ... v(n)=v=v(0) in
exactly one way. (Contributed by Alexander van der Vekens,
6Oct2018.)

⊢ 𝐶 = (𝑛 ∈ ℕ_{0} ↦ ((𝑉 ClWWalksN 𝐸)‘𝑛))
& ⊢ 𝐹 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ_{0} ↦ {𝑤 ∈ (𝐶‘𝑛) ∣ (𝑤‘0) = 𝑣})
& ⊢ 𝐺 = (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ_{≥}‘2)
↦ {𝑤 ∈ (𝐶‘𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) = (𝑤‘0))}) & ⊢ 𝑄 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ_{0} ↦ {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ ( lastS ‘𝑤) ≠ 𝑣)}) & ⊢ 𝐻 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ_{0} ↦ {𝑤 ∈ (𝐶‘𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) ≠ (𝑤‘0))}) ⇒ ⊢ (((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 ∧ 𝑉 FriendGrph 𝐸) ∧ (𝑉 ∈ Fin ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_{≥}‘3)))
→ (#‘(𝑋𝐻𝑁)) = ((𝐾↑(𝑁 − 2)) − (#‘(𝑋𝐹(𝑁 − 2))))) 

Theorem  numclwwlk3lem 26635* 
Lemma for numclwwlk3 26636. (Contributed by Alexander van der Vekens,
6Oct2018.)

⊢ 𝐶 = (𝑛 ∈ ℕ_{0} ↦ ((𝑉 ClWWalksN 𝐸)‘𝑛))
& ⊢ 𝐹 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ_{0} ↦ {𝑤 ∈ (𝐶‘𝑛) ∣ (𝑤‘0) = 𝑣})
& ⊢ 𝐺 = (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ_{≥}‘2)
↦ {𝑤 ∈ (𝐶‘𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) = (𝑤‘0))}) & ⊢ 𝑄 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ_{0} ↦ {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ ( lastS ‘𝑤) ≠ 𝑣)}) & ⊢ 𝐻 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ_{0} ↦ {𝑤 ∈ (𝐶‘𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) ≠ (𝑤‘0))}) ⇒ ⊢ (((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑋 ∈ 𝑉) ∧ 𝑁 ∈ (ℤ_{≥}‘2))
→ (#‘(𝑋𝐹𝑁)) = ((#‘(𝑋𝐻𝑁)) + (#‘(𝑋𝐺𝑁)))) 

Theorem  numclwwlk3 26636* 
Statement 12 in [Huneke] p. 2: "Thus f(n)
= (k  1)f(n  2) +
k^(n2)."  the number of the closed walks v(0) ... v(n2) v(n1)
v(n)
is the sum of the number of the closed walks v(0) ... v(n2) v(n1)
v(n) with v(n2) = v(n) (see numclwwlk1 26625) and with v(n2) =/= v(n)
( see numclwwlk2 26634): f(n) = kf(n2) + k^(n2)  f(n2) = (k 
1)f(n 
2) + k^(n2). (Contributed by Alexander van der Vekens,
26Aug2018.)

⊢ 𝐶 = (𝑛 ∈ ℕ_{0} ↦ ((𝑉 ClWWalksN 𝐸)‘𝑛))
& ⊢ 𝐹 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ_{0} ↦ {𝑤 ∈ (𝐶‘𝑛) ∣ (𝑤‘0) = 𝑣})
& ⊢ 𝐺 = (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ_{≥}‘2)
↦ {𝑤 ∈ (𝐶‘𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) = (𝑤‘0))}) & ⊢ 𝑄 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ_{0} ↦ {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ ( lastS ‘𝑤) ≠ 𝑣)}) & ⊢ 𝐻 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ_{0} ↦ {𝑤 ∈ (𝐶‘𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) ≠ (𝑤‘0))}) ⇒ ⊢ (((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 ∧ 𝑉 FriendGrph 𝐸) ∧ (𝑉 ∈ Fin ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_{≥}‘3)))
→ (#‘(𝑋𝐹𝑁)) = (((𝐾 − 1) · (#‘(𝑋𝐹(𝑁 − 2)))) + (𝐾↑(𝑁 − 2)))) 

Theorem  numclwwlk4 26637* 
The total number of closed walks in a finite undirected simple graph is
the sum of the numbers of closed walks starting at each of its vertices.
(Contributed by Alexander van der Vekens, 7Oct2018.)

⊢ 𝐶 = (𝑛 ∈ ℕ_{0} ↦ ((𝑉 ClWWalksN 𝐸)‘𝑛))
& ⊢ 𝐹 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ_{0} ↦ {𝑤 ∈ (𝐶‘𝑛) ∣ (𝑤‘0) = 𝑣}) ⇒ ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ ℕ_{0}) →
(#‘(𝐶‘𝑁)) = Σ𝑥 ∈ 𝑉 (#‘(𝑥𝐹𝑁))) 

Theorem  numclwwlk5lem 26638* 
Lemma for numclwwlk5 26639. (Contributed by Alexander van der Vekens,
7Oct2018.)

⊢ 𝐶 = (𝑛 ∈ ℕ_{0} ↦ ((𝑉 ClWWalksN 𝐸)‘𝑛))
& ⊢ 𝐹 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ_{0} ↦ {𝑤 ∈ (𝐶‘𝑛) ∣ (𝑤‘0) = 𝑣}) ⇒ ⊢ ((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 ∧ 2 ∥ (𝐾 − 1) ∧ 𝑋 ∈ 𝑉) → ((#‘(𝑋𝐹2)) mod 2) = 1) 

Theorem  numclwwlk5 26639* 
Statement 13 in [Huneke] p. 2: "Let p be
a prime divisor of k1; then
f(p) = 1 (mod p) [for each vertex v]". (Contributed by Alexander
van
der Vekens, 7Oct2018.)

⊢ 𝐶 = (𝑛 ∈ ℕ_{0} ↦ ((𝑉 ClWWalksN 𝐸)‘𝑛))
& ⊢ 𝐹 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ_{0} ↦ {𝑤 ∈ (𝐶‘𝑛) ∣ (𝑤‘0) = 𝑣}) ⇒ ⊢ (((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 ∧ 𝑉 FriendGrph 𝐸 ∧ 𝑉 ∈ Fin) ∧ (𝑋 ∈ 𝑉 ∧ 𝑃 ∈ ℙ ∧ 𝑃 ∥ (𝐾 − 1))) → ((#‘(𝑋𝐹𝑃)) mod 𝑃) = 1) 

Theorem  numclwwlk6 26640* 
For a prime divisor p of k1, the total number of closed walks of length
p in an undirected simple graph with m vertices mod p is equal to the
number of vertices mod p. (Contributed by Alexander van der Vekens,
7Oct2018.)

⊢ 𝐶 = (𝑛 ∈ ℕ_{0} ↦ ((𝑉 ClWWalksN 𝐸)‘𝑛))
& ⊢ 𝐹 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ_{0} ↦ {𝑤 ∈ (𝐶‘𝑛) ∣ (𝑤‘0) = 𝑣}) ⇒ ⊢ (((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 ∧ 𝑉 FriendGrph 𝐸 ∧ 𝑉 ∈ Fin) ∧ (𝑃 ∈ ℙ ∧ 𝑃 ∥ (𝐾 − 1))) → ((#‘(𝐶‘𝑃)) mod 𝑃) = ((#‘𝑉) mod 𝑃)) 

Theorem  numclwwlk7 26641 
Statement 14 in [Huneke] p. 2: "The total
number of closed walks of
length p [in a friendship graph] is (k(k1)+1)f(p)=1 (mod p)",
since the
number of vertices in a friendship graph is (k(k1)+1), see
frgregordn0 26597 or frrusgraord 26598, and p divides (k1), i.e. (k1) mod p
= 0 => k(k1) mod p = 0 => k(k1)+1 mod p = 1. Since the empty
graph is
a friendship graph, see frgra0 26521, as well as kregular (for any k),
see
0vgrargra 26464, but has no closed walk, see clwlk0 26290, this theorem would
be false: ((#‘(𝐶‘𝑃)) mod 𝑃) = 0 ≠ 1, so this case
must be excluded. ( (Contributed by Alexander van der Vekens,
1Sep2018.)

⊢ (((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 ∧ 𝑉 FriendGrph 𝐸) ∧ (𝑉 ≠ ∅ ∧ 𝑉 ∈ Fin) ∧ (𝑃 ∈ ℙ ∧ 𝑃 ∥ (𝐾 − 1))) → ((#‘((𝑉 ClWWalksN 𝐸)‘𝑃)) mod 𝑃) = 1) 

Theorem  numclwwlk8 26642 
The size of the set of closed walks of length p, p prime, is divisible by
p. This corresponds to statement 9 in [Huneke] p. 2: "It follows that,
if p is a prime number, then the number of closed walks of length p is
divisible by p", see also clwlkndivn 26380. (Contributed by Alexander van
der Vekens, 7Oct2018.)

⊢ ((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑃 ∈ ℙ) → ((#‘((𝑉 ClWWalksN 𝐸)‘𝑃)) mod 𝑃) = 0) 

Theorem  frgrareggt1 26643 
If a finite friendship graph is kregular with k > 1, then k must be 2.
(Contributed by Alexander van der Vekens, 7Oct2018.)

⊢ ((𝑉 FriendGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → ((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 ∧ 1 < 𝐾) → 𝐾 = 2)) 

Theorem  frgrareg 26644 
If a finite friendship graph is kregular, then k must be 2 (or 0).
(Contributed by Alexander van der Vekens, 9Oct2018.)

⊢ ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → ((𝑉 FriendGrph 𝐸 ∧ ⟨𝑉, 𝐸⟩ RegUSGrph 𝐾) → (𝐾 = 0 ∨ 𝐾 = 2))) 

Theorem  frgraregord013 26645 
If a finite friendship graph is kregular, then it must have order 0, 1
or 3. (Contributed by Alexander van der Vekens, 9Oct2018.)

⊢ ((𝑉 FriendGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ ⟨𝑉, 𝐸⟩ RegUSGrph 𝐾) → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3)) 

Theorem  frgraregord13 26646 
If a nonempty finite friendship graph is kregular, then it must have
order 1 or 3. Special case of frgraregord013 26645. (Contributed by
Alexander van der Vekens, 9Oct2018.)

⊢ (((𝑉 FriendGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ∧ ⟨𝑉, 𝐸⟩ RegUSGrph 𝐾) → ((#‘𝑉) = 1 ∨ (#‘𝑉) = 3)) 

Theorem  frgraogt3nreg 26647* 
If a finite friendship graph has an order greater than 3, it cannot be
kregular for any k. (Contributed by Alexander van der Vekens,
9Oct2018.)

⊢ ((𝑉 FriendGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 3 < (#‘𝑉)) → ∀𝑘 ∈ ℕ_{0}
¬ ⟨𝑉, 𝐸⟩ RegUSGrph 𝑘) 

Theorem  friendshipgt3 26648* 
The friendship theorem for big graphs: In every finite friendship graph
with order greater than 3 there is a vertex which is adjacent to all
other vertices. (Contributed by Alexander van der Vekens,
9Oct2018.)

⊢ ((𝑉 FriendGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 3 < (#‘𝑉)) → ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ ran 𝐸) 

Theorem  friendship 26649* 
The friendship theorem: In every finite (nonempty) friendship graph
there is a vertex which is adjacent to all other vertices. This is
Metamath 100 proof #83. (Contributed by Alexander van der Vekens,
9Oct2018.)

⊢ ((𝑉 FriendGrph 𝐸 ∧ 𝑉 ≠ ∅ ∧ 𝑉 ∈ Fin) → ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ ran 𝐸) 

PART 18 GUIDES AND
MISCELLANEA


18.1 Guides (conventions, explanations, and
examples)


18.1.1 Conventions
This section describes the conventions we use.
These conventions often refer to existing mathematical practices, which are
discussed in more detail in other references.
For the general conventions, see conventions 26650, and for conventions related
to labels, see conventionslabel 26651.
Logic and set theory provide a foundation for all of mathematics. To learn
about them, you should study one or more of the references listed below.
We indicate references using square brackets. The textbooks provide a
motivation for what we are doing, whereas Metamath lets you see in detail all
hidden and implicit steps. Most standard theorems are accompanied by
citations. Some closely followed texts include the following:
 Axioms of propositional calculus  [Margaris].
 Axioms of predicate calculus  [Megill] (System S3' in the article
referenced).
 Theorems of propositional calculus  [WhiteheadRussell].
 Theorems of pure predicate calculus  [Margaris].
 Theorems of equality and substitution  [Monk2], [Tarski], [Megill].
 Axioms of set theory  [BellMachover].
 Development of set theory  [TakeutiZaring]. (The first part of [Quine]
has a good explanation of the powerful device of "virtual" or
class abstractions, which is essential to our development.)
 Construction of real and complex numbers  [Gleason]
 Theorems about real numbers  [Apostol]


Theorem  conventions 26650 
Here are some of the conventions we use in the
Metamath Proof Explorer (aka "set.mm"), and how they correspond to
typical textbook language (skipping the many cases
where they are identical).
For conventions related to labels, see conventionslabel 26651.
 Notation.
Where possible, the notation attempts to conform to modern
conventions, with variations due to our choice of the axiom system
or to make proofs shorter. However, our notation is strictly
sequential (lefttoright). For example, summation is written in the
form Σ𝑘 ∈ 𝐴𝐵 (dfsum 14265) which denotes that index
variable 𝑘 ranges over 𝐴 when evaluating 𝐵. Thus,
Σ𝑘 ∈ ℕ (1 / (2↑𝑘)) = 1 means 1/2 + 1/4 + 1/8 + ...
= 1 (geoihalfsum 14453).
The notation is usually explained in more detail when first introduced.
 Axiomatic assertions ($a).
All axiomatic assertions ($a statements)
starting with " ⊢ " have labels starting
with "ax" (axioms) or "df" (definitions). A statement with a
label starting with "ax" corresponds to what is traditionally
called an axiom. A statement with a label starting with "df"
introduces new symbols or a new relationship among symbols
that can be eliminated; they always extend the definition of
a wff or class. Metamath blindly treats $a statements as new
given facts but does not try to justify them. The mmj2 program
will justify the definitions as sound as discussed below,
except for 4 definitions (dfbi 196, dfcleq 2603, dfclel 2606, dfclab 2597)
that require a more complex metalogical justification by hand.
 Proven axioms.
In some cases we wish to treat an expression as an axiom in
later theorems, even though it can be proved. For example,
we derive the postulates or axioms of complex arithmetic as
theorems of ZFC set theory. For convenience, after deriving
the postulates, we reintroduce them as new axioms on
top of set theory. This lets us easily identify which axioms
are needed for a particular complex number proof, without the
obfuscation of the set theory used to derive them. For more, see
mmcomplex.html. When we wish
to use a previouslyproven assertion as an axiom, our convention
is that we use the
regular "axNAME" label naming convention to define the axiom,
but we precede it with a proof of the same statement with the label
"axNAME" . An example is complex arithmetic axiom ax1cn 9873,
proven by the preceding theorem ax1cn 9849.
The metamath.exe program will warn if an axiom does not match the
preceding theorem that justifies it if the names match in this way.
 Definitions (df...).
We encourage definitions to include hypertext links to proven examples.
 Statements with hypotheses. Many theorems and some axioms,
such as axmp 5, have hypotheses that must be satisfied in order for
the conclusion to hold, in this case min and maj. When presented in
summarized form such as in the Theorem List (click on "Nearby theorems"
on the axmp 5 page), the hypotheses are connected with an ampersand and
separated from the conclusion with a big arrow, such as in " ⊢ 𝜑
& ⊢ (𝜑 → 𝜓) => ⊢ 𝜓". These symbols are _not_
part of the Metamath language but are just informal notation meaning
"and" and "implies".
 Discouraged use and modification.
If something should only be used in limited ways, it is marked with
"(New usage is discouraged.)". This is used, for example, when something
can be constructed in more than one way, and we do not want later
theorems to depend on that specific construction.
This marking is also used if we want later proofs to use proven axioms.
For example, we want later proofs to
use ax1cn 9873 (not ax1cn 9849) and ax1ne0 9884 (not ax1ne0 9860), as these
are proven axioms for complex arithmetic. Thus, both
ax1cn 9849 and ax1ne0 9860 are marked as "(New usage is discouraged.)".
In some cases a proof should not normally be changed, e.g., when it
demonstrates some specific technique.
These are marked with "(Proof modification is discouraged.)".
 New definitions infrequent.
Typically, we are minimalist when introducing new definitions; they are
introduced only when a clear advantage becomes apparent for reducing
the number of symbols, shortening proofs, etc. We generally avoid
the introduction of gratuitous definitions because each one requires
associated theorems and additional elimination steps in proofs.
For example, we use < and ≤ for inequality expressions, and
use ((sin‘(i · 𝐴)) / i) instead of (sinh‘𝐴)
for the hyperbolic sine.
 Minimizing axioms and the axiom of choice.
We prefer proofs that depend on fewer and/or weaker axioms,
even if the proofs are longer. In particular, we prefer proofs that do
not use the axiom of choice (dfac 8822) where such proofs can be found.
The axiom of choice is widely accepted, and ZFC is the most
commonlyaccepted fundamental set of axioms for mathematics.
However, there have been and still are some lingering controversies
about the Axiom of Choice. Therefore, where a proof
does not require the axiom of choice, we prefer that proof instead.
E.g., our proof of the SchroederBernstein Theorem (sbth 7965)
does not use the axiom of choice.
In some cases, the weaker axiom of countable choice (axcc 9140)
or axiom of dependent choice (axdc 9151) can be used instead.
Similarly, any theorem in first order logic (FOL) that
contains only set variables that are all mutually distinct,
and has no wff variables, can be proved *without* using
ax10 2006 through ax13 2234, by invoking ax10w 1993 through ax13w 2000.
We encourage proving theorems *without* ax10 2006 through ax13 2234
and moving them up to the ax4 1728 through ax9 1986 section.
 Alternative (ALT) proofs.
If a different proof is significantly shorter or clearer but
uses more or stronger axioms, we prefer to make that proof an
"alternative" proof (marked with an ALT label suffix), even if
this alternative proof was formalized first.
We then make the proof that requires fewer axioms the main proof.
This has the effect of reducing (over time)
the number and strength of axioms used by any particular proof.
There can be multiple alternatives if it makes sense to do so.
Alternative (*ALT) theorems should have "(Proof modification is
discouraged.) (New usage is discouraged.)" in their comment and should
follow the main statement, so that people reading the text in order will
see the main statement first. The alternative and main statement
comments should use hyperlinks to refer to each other (so that a reader
of one will become easily aware of the other).
 Alternative (ALTV) versions.
If a theorem or definition is an alternative/variant of an already
existing theorem resp. definition, its label should have the same name
with suffix ALTV. Such alternatives should be temporary only, until it
is decided which alternative should be used in the future. Alternative
(*ALTV) theorems or definitions are usually contained in mathboxes.
Their comments need not to contain "(Proof modification is discouraged.)
(New usage is discouraged.)". Alternative statements should follow the
main statement, so that people reading the text in order will see the
main statement first.
 Old (OLD) versions or proofs.
If a proof, definition, axiom, or theorem is going to be removed,
we often stage that change by first renaming its
label with an OLD suffix (to make it clear that it is going to
be removed). Old (*OLD) statements should have "(Proof modification is
discouraged.) (New usage is discouraged.)" and "Obsolete version of
~ xxx as of ddmmmyyyy." (not enclosed in parentheses) in the comment.
An old statement should follow the main statement, so that people
reading the text in order will see the main statement first.
This typically happens when a shorter proof to an existing theorem is
found: the existing theorem is kept as an *OLD statement for one year.
When a proof is shortened automatically (using Metamath's minimize_with
command), then it is not necessary to keep the old proof, nor to add
credit for the shortening.
 Variables.
Propositional variables (variables for wellformed formulas or wffs) are
represented with lowercase Greek letters and are normally used
in this order:
𝜑 = phi, 𝜓 = psi, 𝜒 = chi, 𝜃 = theta,
𝜏 = tau, 𝜂 = eta, 𝜁 = zeta, and 𝜎 = sigma.
Individual setvar variables are represented with lowercase Latin letters
and are normally used in this order:
𝑥, 𝑦, 𝑧, 𝑤, 𝑣, 𝑢, and 𝑡.
Variables that represent classes are often represented by
uppercase Latin letters:
𝐴, 𝐵, 𝐶, 𝐷, 𝐸, and so on.
There are other symbols that also represent class variables and suggest
specific purposes, e.g., 0 for poset zero (see p0val 16864) and
connective symbols such as + for some group addition operation.
(See prdsplusgval 15956 for an example of the use of +).
Class variables are selected in alphabetical order starting
from 𝐴 if there is no reason to do otherwise, but many
assertions select different class variables or a different order
to make their intended meaning clearer.
 Turnstile.
"⊢ ", meaning "It is provable that," is the first token
of all assertions
and hypotheses that aren't syntax constructions. This is a standard
convention in logic. For us, it also prevents any ambiguity with
statements that are syntax constructions, such as "wff ¬ 𝜑".
 Biconditional (↔).
There are basically two ways to maximize the effectiveness of
biconditionals (↔):
you can either have onedirectional simplifications of all theorems
that produce biconditionals, or you can have onedirectional
simplifications of theorems that consume biconditionals.
Some tools (like Lean) follow the first approach, but set.mm follows
the second approach. Practically, this means that in set.mm, for
every theorem that uses an implication in the hypothesis, like
axmp 5, there is a corresponding version with a biconditional or a
reversed biconditional, like mpbi 219 or mpbir 220. We prefer this
second approach because the number of duplications in the second
approach is bounded by the size of the propositional calculus section,
which is much smaller than the number of possible theorems in all later
sections that produce biconditionals. So although theorems like
biimpi 205 are available, in most cases there is already a theorem that
combines it with your theorem of choice, like mpbir2an 957, sylbir 224,
or 3imtr4i 280.
 Substitution.
"[𝑦 / 𝑥]𝜑" should be read "the wff that results from the
proper substitution of 𝑦 for 𝑥 in wff 𝜑." See dfsb 1868
and the related dfsbc 3403 and dfcsb 3500.
 Isaset.
"𝐴 ∈ V" should be read "Class 𝐴 is a set (i.e. exists)."
This is a convention based on Definition 2.9 of [Quine] p. 19.
See dfv 3175 and isset 3180.
However, instead of using 𝐼 ∈ V in the antecedent of a theorem for
some variable 𝐼, we now prefer to use 𝐼 ∈ 𝑉 (or another
variable if 𝑉 is not available) to make it more general. That way we
can often avoid needing extra uses of elex 3185 and syl 17 in the common
case where 𝐼 is already a member of something.
For hypotheses ($e statement) of theorems (mostly in inference form),
however, ⊢ 𝐴 ∈ V is used rather than ⊢ 𝐴 ∈ 𝑉 (e.g.
difexi 4736). This is because 𝐴 ∈ V is almost always satisfied using
an existence theorem stating "... ∈ V", and a hardcoded V in
the $e statement saves a couple of syntax building steps that substitute
V into 𝑉. Notice that this does not hold for hypotheses of
theorems in deduction form: Here still ⊢ (𝜑 → 𝐴 ∈ 𝑉) should be
used rather than ⊢ (𝜑 → 𝐴 ∈ V).
 Converse.
"^{◡}𝑅" should be read "converse of (relation) 𝑅"
and is the same as the more standard notation R^{1}
(the standard notation is ambiguous). See dfcnv 5046.
This can be used to define a subset, e.g., dftan 14641 notates
"the set of values whose cosine is a nonzero complex number" as
(^{◡}cos “ (ℂ ∖ {0})).
 Function application.
"(𝐹‘𝑥)" should be read "the value
of function 𝐹 at 𝑥" and has the same meaning as the more
familiar but ambiguous notation F(x). For example,
(cos‘0) = 1 (see cos0 14719). The left apostrophe notation
originated with Peano and was adopted in Definition *30.01 of
[WhiteheadRussell] p. 235, Definition 10.11 of [Quine] p. 68, and
Definition 6.11 of [TakeutiZaring] p. 26. See dffv 5812.
In the ASCII (input) representation there are spaces around the grave
accent; there is a single accent when it is used directly,
and it is doubled within comments.
 Infix and parentheses.
When a function that takes two classes and produces a class
is applied as part of an infix expression, the expression is always
surrounded by parentheses (see dfov 6552).
For example, the + in (2 + 2); see 2p2e4 11021.
Function application is itself an example of this.
Similarly, predicate expressions
in infix form that take two or three wffs and produce a wff
are also always surrounded by parentheses, such as
(𝜑 → 𝜓), (𝜑 ∨ 𝜓), (𝜑 ∧ 𝜓), and
(𝜑 ↔ 𝜓)
(see wi 4, dfor 384, dfan 385, and dfbi 196 respectively).
In contrast, a binary relation (which compares two _classes_ and
produces a _wff_) applied in an infix expression is _not_
surrounded by parentheses.
This includes set membership 𝐴 ∈ 𝐵 (see wel 1978),
equality 𝐴 = 𝐵 (see dfcleq 2603),
subset 𝐴 ⊆ 𝐵 (see dfss 3554), and
lessthan 𝐴 < 𝐵 (see dflt 9828). For the general definition
of a binary relation in the form 𝐴𝑅𝐵, see dfbr 4584.
For example, 0 < 1 (see 0lt1 10429) does not use parentheses.
 Unary minus.
The symbol  is used to indicate a unary minus, e.g., 1.
It is specially defined because it is so commonly used.
See cneg 10146.
 Function definition.
Functions are typically defined by first defining the constant symbol
(using $c) and declaring that its symbol is a class with the
label cNAME (e.g., ccos 14634).
The function is then defined labeled dfNAME; definitions
are typically given using the mapsto notation (e.g., dfcos 14640).
Typically, there are other proofs such as its
closure labeled NAMEcl (e.g., coscl 14696), its
function application form labeled NAMEval (e.g., cosval 14692),
and at least one simple value (e.g., cos0 14719).
 Factorial.
The factorial function is traditionally a postfix operation,
but we treat it as a normal function applied in prefix form, e.g.,
(!‘4) = ;24 (dffac 12923 and fac4 12930).
 Unambiguous symbols.
A given symbol has a single unambiguous meaning in general.
Thus, where the literature might use the same symbol with different
meanings, here we use different (variant) symbols for different
meanings. These variant symbols often have suffixes, subscripts,
or underlines to distinguish them. For example, here
"0" always means the value zero (df0 9822), while
"0_{g}" is the group identity element (df0g 15925),
"0." is the poset zero (dfp0 16862),
"0_{𝑝}" is the zero polynomial (df0p 23243),
"0_{vec}" is the zero vector in a normed complex vector space
(df0v 26837), and
"0" is a class variable for use as a connective symbol
(this is used, for example, in p0val 16864).
There are other class variables used as connective symbols
where traditional notation would use ambiguous symbols, including
"1", "+", "∗", and "∥".
These symbols are very similar to traditional notation, but because
they are different symbols they eliminate ambiguity.
 ASCII representation of symbols.
We must have an ASCII representation for each symbol.
We generally choose short sequences, ideally digraphs, and generally
choose sequences that vaguely resemble the mathematical symbol.
Here are some of the conventions we use when selecting an
ASCII representation.
We generally do not include parentheses inside a symbol because
that confuses text editors (such as emacs).
Greek letters for wff variables always use the first two letters
of their English names, making them easy to type and easy to remember.
Symbols that almost look like letters, such as ∀,
are often represented by that letter followed by a period.
For example, "A." is used to represent ∀,
"e." is used to represent ∈, and
"E." is used to represent ∃.
Single letters are now always variable names, so constants that are
often shown as single letters are now typically preceded with "_"
in their ASCII representation, for example,
"_i" is the ASCII representation for the imaginary unit i.
A script font constant is often the letter
preceded by "~" meaning "curly", such as "~P" to represent
the power class 𝒫.
Originally, all setvar and class variables used only single letters
az and AZ, respectively. A big change in recent years was to
allow the use of certain symbols as variable names to make formulas
more readable, such as a variable representing an additive group
operation. The convention is to take the original constant token
(in this case "+" which means complex number addition) and put
a period in front of it to result in the ASCII representation of the
variable ".+", shown as +, that can
be used instead of say the letter "P" that had to be used before.
Choosing tokens for more advanced concepts that have no standard
symbols but are represented by words in books, is hard. A few are
reasonably obvious, like "Grp" for group and "Top" for topology,
but often they seem to end up being either too long or too
cryptic. It would be nice if the math community came up with
standardized short abbreviations for English math terminology,
like they have more or less done with symbols, but that probably
won't happen any time soon.
Another informal convention that we've somewhat followed, that is also
not uncommon in the literature, is to start tokens with a
capital letter for collectionlike objects and lower case for
functionlike objects. For example, we have the collections On
(ordinal numbers), Fin, Prime, Grp, and we have the functions sin,
tan, log, sup. Predicates like Ord and Lim also tend to start
with upper case, but in a sense they are really collectionlike,
e.g. Lim indirectly represents the collection of limit ordinals,
but it can't be an actual class since not all limit ordinals
are sets.
This initial capital vs. lower case letter convention is sometimes
ambiguous. In the past there's been a debate about whether
domain and range are collectionlike or functionlike, thus whether
we should use Dom, Ran or dom, ran. Both are used in the literature.
In the end dom, ran won out for aesthetic reasons
(Norm Megill simply just felt they looked nicer).
 Typography conventions.
Class symbols for functions (e.g., abs, sin)
should usually not have leading or trailing blanks in their
HTML/Latex representation.
This is in contrast to class symbols for operations
(e.g., gcd, sadd, eval), which usually do
include leading and trailing blanks in their representation.
If a class symbol is used for a function as well as an operation
(according to the definition dfov 6552, each operation value can be
written as function value of an ordered pair), the convention for its
primary usage should be used, e.g. (iEdg‘𝐺) versus
(𝑉iEdg𝐸) for the edges of a graph 𝐺 = ⟨𝑉, 𝐸⟩.
 Number construction independence.
There are many ways to model complex numbers.
After deriving the complex number postulates we
reintroduce them as new axioms on top of set theory.
This lets us easily identify which axioms are needed
for a particular complex number proof, without the obfuscation
of the set theory used to derive them.
This also lets us be independent of the specific construction,
which we believe is valuable.
See mmcomplex.html for details.
Thus, for example, we don't allow the use of ∅ ∉ ℂ,
as handy as that would be, because that would be
constructionspecific. We want proofs about ℂ to be independent
of whether or not ∅ ∈ ℂ.
 Minimize hypotheses
(except for construction independence and number theorem domains).
In most cases we try to minimize hypotheses, that is,
we eliminate or reduce what must be true to prove something, so that
the proof is more general and easier to use.
There are exceptions. For example, we intentionally add hypotheses
if they help make proofs independent of a particular construction
(e.g., the contruction of complex numbers ℂ).
We also intentionally add hypotheses for many real and complex
number theorems to expressly state their domains even when they
aren't strictly needed. For example, we could show that
(𝐴 < 𝐵 → 𝐵 ≠ 𝐴) without any other hypotheses, but in
practice we also require proving at least some domains
(e.g., see ltnei 10040). Here are the reasons as discussed in
https://groups.google.com/g/metamath/c/2AW7T3d2YiQ/m/iSN7g87t3ikJ:
 Having the hypotheses immediately shows the intended domain of
applicability (is it ℝ, ℝ^{*}, ω, or something else?),
without having to trace back to definitions.
 Having the hypotheses forces its use in the intended
domain, which generally is desirable.
 The behavior is dependent on accidental behavior of definitions
outside of their domains, so the theorems are nonportable and
"brittle".
 Only a few theorems can have their hypotheses removed
in this fashion due to happy coincidences for our particular
settheoretical definitions. The poor user (especially a
novice learning real number arithmetic) is going to be
confused not knowing when hypotheses are needed and when
they are not. For someone who hasn't traced back the
settheoretical foundations of the definitions, it is
seemingly random and isn't intuitive at all.
 The consensus of opinion of people on this group seemed to be
against doing this.
 Natural numbers.
There are different definitions of "natural" numbers in the literature.
We use ℕ (dfnn 10898) for the set of positive integers starting
from 1, and ℕ_{0} (dfn0 11170) for the set of nonnegative integers
starting at zero.
 Decimal numbers.
Numbers larger than nine are often expressed in base 10 using the
decimal constructor dfdec 11370, e.g., ;;;4001 (see 4001prm 15690
for a proof that 4001 is prime).
 Theorem forms.
We will use the following descriptive terms to categorize theorems:
 A theorem is in "closed form" if it has no $e hypotheses
(e.g., unss 3749). The term "tautology" is also used, especially in
propositional calculus. This form was formerly called "theorem form"
or "closed theorem form".
 A theorem is in "deduction form" (or is a "deduction") if it
has one or more $e hypotheses, and the hypotheses and the conclusion are
implications that share the same antecedent. More precisely, the
conclusion is an implication with a wff variable as the antecedent
(usually 𝜑), and every hypothesis ($e statement) is either:
 an implication with the same antecedent as the conclusion, or
 a definition. A definition can be for a class variable (this is a
class variable followed by =, e.g. the definition of 𝐷 in
lhop 23583) or a wff variable (this is a wff variable followed by
↔); class variable definitions are more common.
In practice, a proof of a theorem in deduction form will also contain
many steps that are implications where the antecedent is either that
wff variable (usually 𝜑) or is a conjunction (𝜑 ∩ ...)
including that wff variable (𝜑). E.g. a1d 25, unssd 3751.
 A theorem is in "inference form" (or is an "inference") if
it has one or more $e hypotheses, but is not in deduction form,
i.e. there is no common antecedent (e.g., unssi 3750).
Any theorem whose conclusion is an implication has an associated
inference, whose hypotheses are the hypotheses of that theorem
together with the antecedent of its conclusion, and whose conclusion is
the consequent of that conclusion. When both theorems are in set.mm,
then the associated inference is often labeled by adding the suffix "i"
to the label of the original theorem (for instance, con3i 149 is the
inference associated with con3 148). The inference associated with a
theorem is easily derivable from that theorem by a simple use of
axmp 5. The other direction is the subject of the Deduction Theorem
discussed below. We may also use the term "associated inference" when
the above process is iterated. For instance, syl 17 is an
inference associated with imim1 81 because it is the inference
associated with imim1i 61 which is itself the inference
associated with imim1 81.
"Deduction form" is the preferred form for theorems because this form
allows us to easily use the theorem in places where (in traditional
textbook formalizations) the standard Deduction Theorem (see below)
would be used. We call this approach "deduction style".
In contrast, we usually avoid theorems in "inference form" when that
would end up requiring us to use the deduction theorem.
Deductions have a label suffix of "d", especially if there are other
forms of the same theorem (e.g., pm2.43d 51). The labels for inferences
usually have the suffix "i" (e.g., pm2.43i 50). The labels of theorems
in "closed form" would have no special suffix (e.g., pm2.43 54). When
an inference is converted to a theorem by eliminating an "is a set"
hypothesis, we sometimes suffix the closed form with "g" (for "more
general") as in uniex 6851 vs. uniexg 6853.
 Deduction theorem.
The Deduction Theorem is a metalogical theorem that provides an
algorithm for constructing a proof of a theorem from the proof of its
corresponding deduction (its associated inference). See for instance
Theorem 3 in [Margaris] p. 56. In ordinary mathematics, no one actually
carries out the algorithm, because (in its most basic form) it involves
an exponential explosion of the number of proof steps as more hypotheses
are eliminated. Instead, in ordinary mathematics the Deduction Theorem
is invoked simply to claim that something can be done in principle,
without actually doing it. For more details, see mmdeduction.html.
The Deduction Theorem is a metalogical theorem that cannot be applied
directly in metamath, and the explosion of steps would be a problem
anyway, so alternatives are used. One alternative we use sometimes is
the "weak deduction theorem" dedth 4089, which works in certain cases in
set theory. We also sometimes use dedhb 3343. However, the primary
mechanism we use today for emulating the deduction theorem is to write
proofs in deduction form (aka "deduction style") as described earlier;
the prefixed 𝜑 → mimics the context in a deduction proof system.
In practice this mechanism works very well. This approach is described
in the deduction form and natural deduction page mmnatded.html; a
list of translations for common natural deduction rules is given in
natded 26652.
 Recursion.
We define recursive functions using various "recursion constructors".
These allow us to define, with compact direct definitions, functions
that are usually defined in textbooks with indirect selfreferencing
recursive definitions. This produces compact definition and much
simpler proofs, and greatly reduces the risk of creating unsound
definitions. Examples of recursion constructors include
recs(𝐹) in dfrecs 7355, rec(𝐹, 𝐼) in dfrdg 7393,
seq_{𝜔}(𝐹, 𝐼) in dfseqom 7430, and seq𝑀( + , 𝐹) in
dfseq 12664. These have characteristic function 𝐹 and initial value
𝐼. (Σ_{g} in dfgsum 15926 isn't really designed for arbitrary
recursion, but you could do it with the right magma.) The logically
primary one is dfrecs 7355, but for the "average user" the most useful
one is probably dfseq 12664 provided that a countable sequence is
sufficient for the recursion.
 Extensible structures.
Mathematics includes many structures such as ring, group, poset, etc.
We define an "extensible structure" which is then used to define group,
ring, poset, etc. This allows theorems from more general structures
(groups) to be reused for more specialized structures (rings) without
having to reprove them. See dfstruct 15697.
 Undefined results and "junk theorems".
Some expressions are only expected to be meaningful in certain contexts.
For example, consider Russell's definition description binder iota,
where (℩𝑥𝜑) is meant to be "the 𝑥 such that 𝜑"
(where 𝜑 typically depends on x).
What should that expression produce when there is no such 𝑥?
In set.mm we primarily use one of two approaches.
One approach is to make the expression evaluate to the empty set
whenever the expression is being used outside of its expected context.
While not perfect, it makes it a bit more clear when something
is undefined, and it has the advantage that it makes more
things equal outside their domain which can remove hypotheses when
you feel like exploiting these socalled junk theorems.
Note that Quine does this with iota (his definition of iota
evaluates to the empty set when there is no unique value of 𝑥).
Quine has no problem with that and we don't see why we should,
so we define iota exactly the same way that Quine does.
The main place where you see this being systematically exploited is in
"reverse closure" theorems like 𝐴 ∈ (𝐹‘𝐵) → 𝐵 ∈ dom 𝐹,
which is useful when 𝐹 is a family of sets. (by this we
mean it's a set set even in a type theoretic interpretation.)
The second approach uses "(New usage is discouraged.)" to prevent
unintentional uses of certain properties.
For example, you could define some construct dfNAME whose
usage is discouraged, and prove only the specific properties
you wish to use (and add those proofs to the list of permitted uses
of "discouraged" information). From then on, you can only use
those specific properties without a warning.
Other approaches often have hidden problems.
For example, you could try to "not define undefined terms"
by creating definitions like ${ $d 𝑦𝑥 $. $d 𝑦𝜑 $.
dfiota $a ⊢ (∃!𝑥𝜑 → (℩𝑥𝜑) = ∪ {𝑥 ∣ 𝜑}) $. $}.
This will be rejected by the definition checker, but the bigger
theoretical reason to reject this axiom is that it breaks equality 
the metatheorem (𝑥 = 𝑦 → P(x) = P(y) ) fails
to hold if definitions don't unfold without some assumptions.
(That is, iotabidv 5789 is no longer provable and must be added
as an axiom.) It is important for every syntax constructor to
satisfy equality theorems *unconditionally*, e.g., expressions
like (1 / 0) = (1 / 0) should not be rejected.
This is forced on us by the context free term
language, and anything else requires a lot more infrastructure
(e.g., a type checker) to support without making everything else
more painful to use.
Another approach would be to try to make nonsensical
statements syntactically invalid, but that can create its own
complexities; in some cases that would make parsing itself undecidable.
In practice this does not seem to be a serious issue.
No one does these things deliberately in "real" situations,
and some knowledgeable people (such as Mario Carneiro)
have never seen this happen accidentally.
Norman Megill doesn't agree that these "junk" consequences are
necessarily bad anyway, and they can significantly shorten proofs
in some cases. This database would be much larger if, for example,
we had to condition fvex 6113 on the argument being in the domain
of the function. It is impossible to derive a contradiction
from sound definitions (i.e. that pass the definition check),
assuming ZFC is consistent, and he doesn't see the point of all the
extra busy work and huge increase in set.mm size that would result
from restricting *all* definitions.
So instead of implementing a complex system to counter a
problem that does not appear to occur in practice, we use
a significantly simpler set of approaches.
 Organizing proofs.
Humans have trouble understanding long proofs.
It is often preferable to break longer proofs into
smaller parts (just as with traditional proofs). In Metamath
this is done by creating separate proofs of the separate parts.
A proof with the sole purpose of supporting a final proof is a
lemma; the naming convention for a lemma is the final proof's name
followed by "lem", and a number if there is more than one. E.g.,
sbthlem1 7955 is the first lemma for sbth 7965. Also, consider proving
reusable results separately, so that others will be able to easily
reuse that part of your work.
 Limit proof size.
It is often preferable to break longer proofs into
smaller parts, just as you would do with traditional proofs.
One reason is that humans have trouble understanding long proofs.
Another reason is that it's generally best to prove
reusable results separately,
so that others will be able to easily reuse them.
Finally, the "minimize" routine can take much longer with
very long proofs.
We encourage proofs to be no more than 200 essential steps, and
generally no more than 500 essential steps,
though these are simply guidelines and not hardandfast rules.
Much smaller proofs are fine!
We also acknowledge that some proofs, especially autogenerated ones,
should sometimes not be broken up (e.g., because
breaking them up might be useless and inefficient due to many
interconnections and reused terms within the proof).
In Metamath, breaking up longer proofs is done by creating multiple
separate proofs of separate parts.
A proof with the sole purpose of supporting a final proof is a
lemma; the naming convention for a lemma is the final proof's name
followed by "lem", and a number if there is more than one. E.g.,
sbthlem1 7955 is the first lemma for sbth 7965.
 Hypertext links.
We strongly encourage comments to have many links to related material,
with accompanying text that explains the relationship. These can help
readers understand the context. Links to other statements, or to
HTTP/HTTPS URLs, can be inserted in ASCII source text by prepending a
spaceseparated tilde (e.g., " ~ dfprm " results in " dfprm 15224").
When metamath.exe is used to generate HTML it automatically inserts
hypertext links for syntax used (e.g., every symbol used), every axiom
and definition depended on, the justification for each step in a proof,
and to both the next and previous assertion.
 Hypertext links to section headers.
Some section headers have text under them that describes or explains the
section. However, they are not part of the description of axioms or
theorems, and there is no way to link to them directly. To provide for
this, section headers with accompanying text (indicated with "*"
prefixed to mmtheorems.html#mmdtoc entries) have an anchor in
mmtheorems.html whose name is the first $a or $p statement that
follows the header. For example there is a glossary under the section
heading called GRAPH THEORY. The first $a or $p statement that follows
is cuhg 25819, which you can see two lines down. To reference it we link
to the anchor using a spaceseparated tilde followed by the
spaceseparated link mmtheorems.html#cuhg, which will become the
hyperlink mmtheorems.html#cuhg. Note that no theorem in set.mm is
allowed to begin with "mm" (enforced by "verify markup" in the metamath
program).
Whenever the software sees a tilde reference beginning with "http:",
"https:", or "mm", the reference is assumed to be a link to something
other than a statement label, and the tilde reference is used as is.
This can also be useful for relative links to other pages such as
mmcomplex.html.
 Bibliography references.
Please include a bibliographic reference to any external material used.
A name in square brackets in a comment indicates a
bibliographic reference. The full reference must be of the form
KEYWORD IDENTIFIER? NOISEWORD(S)* [AUTHOR(S)] p. NUMBER 
note that this is a very specific form that requires a page number.
There should be no comma between the author reference and the
"p." (a constant indicator).
Whitespace, comma, period, or semicolon should follow NUMBER.
An example is Theorem 3.1 of [Monk1] p. 22,
The KEYWORD, which is not casesensitive,
must be one of the following: Axiom, Chapter, Compare, Condition,
Corollary, Definition, Equation, Example, Exercise, Figure, Item,
Lemma, Lemmas, Line, Lines, Notation, Part, Postulate, Problem,
Property, Proposition, Remark, Rule, Scheme, Section, or Theorem.
The IDENTIFIER is optional, as in for example
"Remark in [Monk1] p. 22".
The NOISEWORDS(S) are zero or more from the list: from, in, of, on.
The AUTHOR(S) must be present in the file identified with the
htmlbibliography assignment (e.g., mmset.html) as a named anchor
(NAME=). If there is more than one document by the same author(s),
add a numeric suffix (as shown here).
The NUMBER is a page number, and may be any alphanumeric string such as
an integer or Roman numeral.
Note that we _require_ page numbers in comments for individual
$a or $p statements. We allow names in square brackets without
page numbers (a reference to an entire document) in
heading comments.
If this is a new reference, please also add it to the
"Bibliography" section of mmset.html.
(The file mmbiblio.html is automatically rebuilt, e.g.,
using the metamath.exe "write bibliography" command.)
 Acceptable shorter proofs
Shorter proofs are welcome, and any shorter proof we accept
will be acknowledged in the theorem's description. However,
in some cases a proof may be "shorter" or not depending on
how it is formatted. This section provides general guidelines.
Usually we automatically accept shorter proofs that (1)
shorten the set.mm file (with compressed proofs), (2) reduce
the size of the HTML file generated with SHOW STATEMENT xx
/ HTML, (3) use only existing, unmodified theorems in the
database (the order of theorems may be changed, though), and
(4) use no additional axioms.
Usually we will also automatically accept a _new_ theorem
that is used to shorten multiple proofs, if the total size
of set.mm (including the comment of the new theorem, not
including the acknowledgment) decreases as a result.
In borderline cases, we typically place more importance on
the number of compressed proof steps and less on the length
of the label section (since the names are in principle
arbitrary). If two proofs have the same number of compressed
proof steps, we will typically give preference to the one
with the smaller number of different labels, or if these
numbers are the same, the proof with the fewest number of
characters that the proofs happen to have by chance when
label lengths are included.
A few theorems have a longer proof than necessary in order
to avoid the use of certain axioms, for pedagogical purposes,
and for other reasons. These theorems will (or should) have
a "(Proof modification is discouraged.)" tag in their
description. For example, idALT 23 shows a proof directly from
axioms. Shorter proofs for such cases won't be accepted,
of course, unless the criteria described continues to be
satisfied.
 Input format.
The input is in ASCII with twospace indents. Tab characters are not
allowed. Use embedded math comments or HTML entities for nonASCII
characters (e.g., "é" for "é").
 Information on syntax, axioms, and definitions.
For a hyperlinked list of syntax, axioms, and definitions, see
mmdefinitions.html.
If you have questions about a specific symbol or axiom, it is best
to go directly to its definition to learn more about it.
The generated HTML for each theorem and axiom includes hypertext
links to each symbol's definition.
 Reserved symbols: 'LETTER.
Some symbols are reserved for potential future use.
Symbols with the pattern 'LETTER are reserved for possibly
representing characters (this is somewhat similar to Lisp).
We would expect '\n to represent newline, 'sp for space, and perhaps
'\x24 for the dollar character.
 Language and spelling.
It is preferred to use American English for comments and symbols, e.g.
we use "neighborhood" instead of the British English "neighbourhood".
An exception is the word "analog", which can be either a noun or an
adjective. Furthermore, "analog" has the confounding meaning "not
digital", whereas "analogue" is often used in the sense something that
bears analogy to something else also in American English. Therefore,
"analogue" is used for the noun and "analogous" for the adjective in
set.mm.
 Comments and layout.
As for formatting of the file set.mm, and in particular formatting and
layout of the comments, the foremost rule is consistency. The first
sections of set.mm, in particular Part 1 "Classical firstorder logic
with equality" can serve as a model for contributors. Some formatting
rules are enforced when using the Metamath program's "WRITE SOURCE"
command with the "REWRAP" option. Here are a few other rules, which are
not enforced, but that we try follow:

The file set.mm should have a double blank line before each section
header, and at no other places. In particular, there are no triple
blank lines. If there is a "@( Begin $[ ... $] @)" comment (where "@"
is actually "$") before the section header, then the double blank line
should go before that comment.

The header comments should be spaced as those of Part 1, namely, with
a blank line before and after the comment, and an indentation of two
spaces.

Header comments are not rewrapped by the Metamath program [as of
24Oct2021], but similar spacing and wrapping should be used as for
other comments: double spaces after a period ending a sentence, line
wrapping with line width of 79, and no trailing spaces at the end of
lines.
The challenge of varying mathematical conventions
We try to follow mathematical conventions, but in many cases
different texts use different conventions.
In those cases we pick some reasonably common convention and stick to
it.
We have already mentioned that the term "natural number" has
varying definitions (some start from 0, others start from 1), but
that is not the only such case.
A useful example is the set of metavariables used to represent
arbitrary wellformed formulas (wffs).
We use an open phi, φ, to represent the first arbitrary wff in an
assertion with one or more wffs; this is a common convention and
this symbol is easily distinguished from the empty set symbol.
That said, it is impossible to please everyone or simply "follow
the literature" because there are many different conventions for
a variable that represents any arbitrary wff.
To demonstrate the point,
here are some conventions for variables that represent an arbitrary
wff and some texts that use each convention:
 open phi φ (and so on): Tarski's papers,
Rasiowa & Sikorski's
The Mathematics of Metamathematics (1963),
Monk's Introduction to Set Theory (1969),
Enderton's Elements of Set Theory (1977),
Bell & Machover's A Course in Mathematical Logic (1977),
Jech's Set Theory (1978),
Takeuti & Zaring's
Introduction to Axiomatic Set Theory (1982).
 closed phi ϕ (and so on):
Levy's Basic Set Theory (1979),
Kunen's Set Theory (1980),
Paulson's Isabelle: A Generic Theorem Prover (1994),
Huth and Ryan's Logic in Computer Science (2004/2006).
 Greek α, β, γ:
Duffy's Principles of Automated Theorem Proving (1991).
 Roman A, B, C:
Kleene's Introduction to Metamathematics (1974),
Smullyan's FirstOrder Logic (1968/1995).
 script A, B, C:
Hamilton's Logic for Mathematicians (1988).
 italic A, B, C:
Mendelson's Introduction to Mathematical Logic (1997).
 italic P, Q, R:
Suppes's Axiomatic Set Theory (1972),
Gries and Schneider's A Logical Approach to Discrete Math
(1993/1994),
Rosser's Logic for Mathematicians (2008).
 italic p, q, r:
Quine's Set Theory and Its Logic (1969),
Kuratowski & Mostowski's Set Theory (1976).
 italic X, Y, Z:
Dijkstra and Scholten's
Predicate Calculus and Program Semantics (1990).
 Fraktur letters:
Fraenkel et. al's Foundations of Set Theory (1973).
Distinctness or freeness
Here are some conventions that address distinctness or freeness of a
variable:
 Ⅎ𝑥𝜑 is read " 𝑥 is not free in (wff) 𝜑";
see dfnf 1701 (whose description has some important technical
details). Similarly, Ⅎ𝑥𝐴 is read 𝑥 is not free in (class)
𝐴, see dfnfc 2740.
 "$d x y $." should be read "Assume x and y are distinct
variables."
 "$d x 𝜑 $." should be read "Assume x does not occur in phi $."
Sometimes a theorem is proved using
Ⅎ𝑥𝜑 (dfnf 1701) in place of
"$d 𝑥𝜑 $." when a more general result is desired;
ax5 1827 can be used to derive the $d version. For an example of
how to get from the $d version back to the $e version, see the
proof of euf 2466 from dfeu 2462.
 "$d x A $." should be read "Assume x is not a variable occurring in
class A."
 "$d x A $. $d x ps $. $e  (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) $."
is an idiom
often used instead of explicit substitution, meaning "Assume psi results
from the proper substitution of A for x in phi."
 " ⊢ (¬ ∀𝑥𝑥 = 𝑦 → ..." occurs early in some cases, and
should be read "If x and y are distinct
variables, then..." This antecedent provides us with a technical
device (called a "distinctor" in Section 7 of [Megill] p. 444)
to avoid the need for the
$d statement early in our development of predicate calculus, permitting
unrestricted substitutions as conceptually simple as those in
propositional calculus. However, the $d eventually becomes a
requirement, and after that this device is rarely used.
There is a general technique to replace a $d x A or
$d x ph condition in a theorem with the corresponding
Ⅎ𝑥𝐴 or Ⅎ𝑥𝜑; here it is.
⊢ T[x, A] where ,
and you wish to prove ⊢ Ⅎ𝑥𝐴 ⇒ ⊢ T[x, A].
You apply the theorem substituting 𝑦 for 𝑥 and 𝐴 for 𝐴,
where 𝑦 is a new dummy variable, so that
$d y A is satisfied.
You obtain ⊢ T[y, A], and apply chvar to obtain ⊢
T[x, A] (or just use mpbir 220 if T[x, A] binds 𝑥).
The side goal is ⊢ (𝑥 = 𝑦 → ( T[y, A] ↔ T[x, A] )),
where you can use equality theorems, except
that when you get to a bound variable you use a nondv bound variable
renamer theorem like cbval 2259. The section
mmtheorems32.html#mm3146s also describes the
metatheorem that underlies this.
Standard Metamath verifiers do not distinguish between axioms and
definitions (both are $a statements).
In practice, we require that definitions (1) be conservative
(a definition should not allow an expression
that previously qualified as a wff but was not provable
to become provable) and be eliminable
(there should exist an algorithmic method for converting any
expression using the definition into
a logically equivalent expression that previously qualified as a wff).
To ensure this, we have additional rules on almost all definitions
($a statements with a label that does not begin with ax).
These additional rules are not applied in a few cases where they
are too strict (dfbi 196, dfclab 2597, dfcleq 2603, and dfclel 2606);
see those definitions for more information.
These additional rules for definitions are checked by at least
mmj2's definition check (see
mmj2 master file mmj2jar/macros/definitionCheck.js).
This definition check relies on the database being very much like
set.mm, down to the names of certain constants and types, so it
cannot apply to all Metamath databases... but it is useful in set.mm.
In this definition check, a $astatement with a given label and
typecode ⊢ passes the test if and only if it
respects the following rules (these rules require that we have
an unambiguous tree parse, which is checked separately):
 The expression must be a biconditional or an equality (i.e. its
rootsymbol must be ↔ or =).
If the proposed definition passes this first rule, we then
define its definiendum as its left hand side (LHS) and
its definiens as its right hand side (RHS).
We define the *defined symbol* as the rootsymbol of the LHS.
We define a *dummy variable* as a variable occurring
in the RHS but not in the LHS.
Note that the "rootsymbol" is the root of the considered tree;
it need not correspond to a single token in the database
(e.g., see w3o 1030 or wsb 1867).
 The defined expression must not appear in any statement
between its syntax axiom () and its definition,
and the defined expression must not be used in its definiens.
See df3an 1033 for an example where the same symbol is used in
different ways (this is allowed).
 No two variables occurring in the LHS may share a
disjoint variable (DV) condition.
 All dummy variables are required to be disjoint from any
other (dummy or not) variable occurring in this labeled expression.
 Either
(a) there must be no nonsetvar dummy variables, or
(b) there must be a justification theorem.
The justification theorem must be of form
⊢ ( definiens rootsymbol definiens' )
where definiens' is definiens but the dummy variables are all
replaced with other unused dummy variables of the same type.
Note that rootsymbol is ↔ or =, and that setvar
variables are simply variables with the setvar typecode.
 One of the following must be true:
(a) there must be no setvar dummy variables,
(b) there must be a justification theorem as described in rule 5, or
(c) if there are setvar dummy variables, every one must not be free.
That is, it must be true that
(𝜑 → ∀𝑥𝜑) for each setvar dummy variable 𝑥
where 𝜑 is the definiens.
We use two different tests for nonfreeness; one must succeed
for each setvar dummy variable 𝑥.
The first test requires that the setvar dummy variable 𝑥
be syntactically bound
(this is sometimes called the "fast" test, and this implies
that we must track binding operators).
The second test requires a successful
search for the directlystated proof of (𝜑 → ∀𝑥𝜑)
Part c of this rule is how most setvar dummy variables
are handled.
Rule 3 may seem unnecessary, but it is needed.
Without this rule, you can define something like
cbar $a wff Foo x y $.
${ $d x y $. dffoo $a  ( Foo x y <> x = y ) $. $}
and now "Foo x x" is not eliminable;
there is no way to prove that it means anything in particular,
because the definitional theorem that is supposed to be
responsible for connecting it to the original language wants
nothing to do with this expression, even though it is well formed.
A justification theorem for a definition (if used this way)
must be proven before the definition that depends on it.
One example of a justification theorem is vjust 3174.
The definition dfv 3175 ⊢ V = {𝑥 ∣ 𝑥 = 𝑥} is justified
by the justification theorem vjust 3174
⊢ {𝑥 ∣ 𝑥 = 𝑥} = {𝑦 ∣ 𝑦 = 𝑦}.
Another example of a justification theorem is trujust 1477;
the definition dftru 1478 ⊢ (⊤ ↔ (∀𝑥𝑥 = 𝑥 → ∀𝑥𝑥 = 𝑥))
is justified by trujust 1477 ⊢ ((∀𝑥𝑥 = 𝑥 → ∀𝑥𝑥 = 𝑥) ↔ (∀𝑦𝑦 = 𝑦 → ∀𝑦𝑦 = 𝑦)).
Here is more information about our processes for checking and
contributing to this work:
 Multiple verifiers.
This entire file is verified by multiple independentlyimplemented
verifiers when it is checked in, giving us extremely high
confidence that all proofs follow from the assumptions.
The checkers also check for various other problems such as
overly long lines.
 Maximum text line length is 79 characters.
You can fix comment line length by running the commands scripts/rewrap
or metamath 'read set.mm' 'save proof */c/f'
'write source set.mm/rewrap' quit .
As a general rule, a math string in a comment should be surrounded
by backquotes on the same line, and if it is too long it should
be broken into multiple adjacent mathstrings on multiple lines.
Those commands don't modify the math content of statements.
In statements we try to break before the outermost important connective
(not including the typecode and perhaps not the antecedent).
For examples, see sqrtmulii 13974 and absmax 13917.
 Discouraged information.
A separate file named "discouraged" lists all
discouraged statements and uses of them, and this file is checked.
If you change the use of discouraged things, you will need to change
this file.
This makes it obvious when there is a change to anything discouraged
(triggering further review).
 LRParser check.
Metamath verifiers ensure that $p statements follow from previous
$a and $p statements.
However, by itself the Metamath language permits certain kinds of
syntactic ambiguity that we choose to avoid in this database.
Thus, we require that this database unambiguously parse
using the "LRParser" check (implemented by at least mmj2).
(For details, see mmj2 master file src/mmj/verify/LRParser.java).
This check
counters, for example, a devious ambiguous construct
developed by saueran at oregonstate dot edu
posted on Mon, 11 Feb 2019 17:32:32 0800 (PST)
based on creating definitions with mismatched parentheses.
 Proposing specific changes.
Please propose specific changes as pull requests (PRs) against the
"develop" branch of set.mm, at:
https://github.com/metamath/set.mm/tree/develop
 Community.
We encourage anyone interested in Metamath to join our mailing list:
https://groups.google.com/forum/#!forum/metamath.
(Contributed by DAW, 27Dec2016.) (New usage is discouraged.)

⊢ 𝜑 ⇒ ⊢ 𝜑 

Theorem  conventionslabel 26651 
The following explains some of the label conventions in use
in the Metamath Proof Explorer ("set.mm").
For the general conventions, see conventions 26650.
Every statement has a unique identifying label, which serves the
same purpose as an equation number in a book.
We use various label naming conventions to provide
easytoremember hints about their contents.
Labels are not a 1to1 mapping, because that would create
long names that would be difficult to remember and tedious to type.
Instead, label names are relatively short while
suggesting their purpose.
Names are occasionally changed to make them more consistent or
as we find better ways to name them.
Here are a few of the label naming conventions:
 Axioms, definitions, and wff syntax.
As noted earlier, axioms are named "axNAME",
proofs of proven axioms are named "axNAME", and
definitions are named "dfNAME".
Wff syntax declarations have labels beginning with "w"
followed by short fragment suggesting its purpose.
 Hypotheses.
Hypotheses have the name of the final axiom or theorem, followed by
".", followed by a unique id (these ids are usually consecutive integers
starting with 1, e.g. for rgen 2906"rgen.1 $e  ( x e. A > ph ) $."
or letters corresponding to the (main) class variable used in the
hypothesis, e.g. for mdet0 20231: "mdet0.d $e  D = ( N maDet R ) $.").
 Common names.
If a theorem has a wellknown name, that name (or a short version of it)
is sometimes used directly. Examples include
barbara 2551 and stirling 38982.
 Principia Mathematica.
Proofs of theorems from Principia Mathematica often use a special
naming convention: "pm" followed by its identifier.
For example, Theorem *2.27 of [WhiteheadRussell] p. 104 is named
pm2.27 41.
 19.x series of theorems.
Similar to the conventions for the theorems from Principia Mathematica,
theorems from Section 19 of [Margaris] p. 90 often use a special naming
convention: "19." resp. "r19." (for corresponding restricted quantifier
versions) followed by its identifier.
For example, Theorem 38 from Section 19 of [Margaris] p. 90 is labeled
19.38 1757, and the restricted quantifier version of Theorem 21 from
Section 19 of [Margaris] p. 90 is labeled r19.21 2939.
 Characters to be used for labels
Although the specification of Metamath allows for dots/periods "." in
any label, it is usually used only in labels for hypotheses (see above).
Exceptions are the labels of theorems from Principia Mathematica and the
19.x series of theorems from Section 19 of [Margaris] p. 90 (see above)
and 0.999... 14451. Furthermore, the underscore "_" should not be used.
 Syntax label fragments.
Most theorems are named using a concatenation of syntax label fragments
(omitting variables) that represent the important part of the theorem's
main conclusion. Almost every syntactic construct has a definition
labeled "dfNAME", and normally NAME is the syntax label fragment. For
example, the class difference construct (𝐴 ∖ 𝐵) is defined in
dfdif 3543, and thus its syntax label fragment is "dif". Similarly, the
subclass relation 𝐴 ⊆ 𝐵 has syntax label fragment "ss"
because it is defined in dfss 3554. Most theorem names follow from
these fragments, for example, the theorem proving (𝐴 ∖ 𝐵) ⊆ 𝐴
involves a class difference ("dif") of a subset ("ss"), and thus is
labeled difss 3699. There are many other syntax label fragments, e.g.,
singleton construct {𝐴} has syntax label fragment "sn" (because it
is defined in dfsn 4126), and the pair construct {𝐴, 𝐵} has
fragment "pr" ( from dfpr 4128). Digits are used to represent
themselves. Suffixes (e.g., with numbers) are sometimes used to
distinguish multiple theorems that would otherwise produce the same
label.
 Phantom definitions.
In some cases there are common label fragments for something that could
be in a definition, but for technical reasons is not. The iselementof
(is member of) construct 𝐴 ∈ 𝐵 does not have a dfNAME definition;
in this case its syntax label fragment is "el". Thus, because the
theorem beginning with (𝐴 ∈ (𝐵 ∖ {𝐶}) uses iselementof
("el") of a class difference ("dif") of a singleton ("sn"), it is
labeled eldifsn 4260. An "n" is often used for negation (¬), e.g.,
nan 602.
 Exceptions.
Sometimes there is a definition dfNAME but the label fragment is not
the NAME part. The definition should note this exception as part of its
definition. In addition, the table below attempts to list all such
cases and marks them in bold. For example, the label fragment "cn"
represents complex numbers ℂ (even though its definition is in
dfc 9821) and "re" represents real numbers ℝ ( definition dfr 9825).
The empty set ∅ often uses fragment 0, even though it is defined
in dfnul 3875. The syntax construct (𝐴 + 𝐵) usually uses the
fragment "add" (which is consistent with dfadd 9826), but "p" is used as
the fragment for constant theorems. Equality (𝐴 = 𝐵) often uses
"e" as the fragment. As a result, "two plus two equals four" is labeled
2p2e4 11021.
 Other markings.
In labels we sometimes use "com" for "commutative", "ass" for
"associative", "rot" for "rotation", and "di" for "distributive".
 Focus on the important part of the conclusion.
Typically the conclusion is the part the user is most interested in.
So, a rough guideline is that a label typically provides a hint
about only the conclusion; a label rarely says anything about the
hypotheses or antecedents.
If there are multiple theorems with the same conclusion
but different hypotheses/antecedents, then the labels will need
to differ; those label differences should emphasize what is different.
There is no need to always fully describe the conclusion; just
identify the important part. For example,
cos0 14719 is the theorem that provides the value for the cosine of 0;
we would need to look at the theorem itself to see what that value is.
The label "cos0" is concise and we use it instead of "cos0eq1".
There is no need to add the "eq1", because there will never be a case
where we have to disambiguate between different values produced by
the cosine of zero, and we generally prefer shorter labels if
they are unambiguous.
 Closures and values.
As noted above, if a function dfNAME is defined, there is typically a
proof of its value labeled "NAMEval" and of its closure labeld "NAMEcl".
E.g., for cosine (dfcos 14640) we have value cosval 14692 and closure
coscl 14696.
 Special cases.
Sometimes, syntax and related markings are insufficient to distinguish
different theorems. For example, there are over a hundred different
implicationonly theorems. They are grouped in a more adhoc way that
attempts to make their distinctions clearer. These often use
abbreviations such as "mp" for "modus ponens", "syl" for syllogism, and
"id" for "identity". It is especially hard to give good names in the
propositional calculus section because there are so few primitives.
However, in most cases this is not a serious problem. There are a few
very common theorems like axmp 5 and syl 17 that you will have no
trouble remembering, a few theorem series like syl*anc and simp* that
you can use parametrically, and a few other useful glue things for
destructuring 'and's and 'or's (see natded 26652 for a list), and that is
about all you need for most things. As for the rest, you can just
assume that if it involves at most three connectives, then it is
probably already proved in set.mm, and searching for it will give you
the label.
 Suffixes.
Suffixes are used to indicate the form of a theorem (see above).
Additionally, we sometimes suffix with "v" the label of a theorem
eliminating a hypothesis such as Ⅎ𝑥𝜑 in 19.21 2062 via the use of
disjoint variable conditions combined with nfv 1830. If two (or three)
such hypotheses are eliminated, the suffix "vv" resp. "vvv" is used,
e.g. exlimivv 1847.
Conversely, we sometimes suffix with "f" the label of a theorem
introducing such a hypothesis to eliminate the need for the disjoint
variable condition; e.g. euf 2466 derived from dfeu 2462. The "f" stands
for "not free in" which is less restrictive than "does not occur in."
The suffix "b" often means "biconditional" (↔, "iff" , "if and
only if"), e.g. sspwb 4844.
We sometimes suffix with "s" the label of an inference that manipulates
an antecedent, leaving the consequent unchanged. The "s" means that the
inference eliminates the need for a syllogism (syl 17) type inference
in a proof. A theorem label is suffixed with "ALT" if it provides an
alternate lesspreferred proof of a theorem (e.g., the proof is
clearer but uses more axioms than the preferred version).
The "ALT" may be further suffixed with a number if there is more
than one alternate theorem.
Furthermore, a theorem label is suffixed with "OLD" if there is a new
version of it and the OLD version is obsolete (and will be removed
within one year).
Finally, it should be mentioned that suffixes can be combined, for
example in cbvaldva 2269 (cbval 2259 in deduction form "d" with a not free
variable replaced by a disjoint variable condition "v" with a
conjunction as antecedent "a").
Here is a nonexhaustive list of common suffixes:
 a : theorem having a conjunction as antecedent
 b : theorem expressing a logical equivalence
 c : contraction (e.g., sylc 63, syl2anc 691), commutes
(e.g., biimpac 502)
 d : theorem in deduction form
 f : theorem with a hypothesis such as Ⅎ𝑥𝜑
 g : theorem in closed form having an "is a set" antecedent
 i : theorem in inference form
 l : theorem concerning something at the left
 r : theorem concerning something at the right
 r : theorem with something reversed (e.g., a biconditional)
 s : inference that manipulates an antecedent ("s" refers to an
application of syl 17 that is eliminated)
 v : theorem with one (main) disjoint variable condition
 vv : theorem with two (main) disjoint variable conditions
 w : weak(er) form of a theorem
 ALT : alternate proof of a theorem
 ALTV : alternate version of a theorem or definition
 OLD : old/obsolete version of a theorem/definition/proof
 Reuse.
When creating a new theorem or axiom, try to reuse abbreviations used
elsewhere. A comment should explain the first use of an abbreviation.
The following table shows some commonly used abbreviations in labels, in
alphabetical order. For each abbreviation we provide a mnenomic, the
source theorem or the assumption defining it, an expression showing what
it looks like, whether or not it is a "syntax fragment" (an abbreviation
that indicates a particular kind of syntax), and hyperlinks to label
examples that use the abbreviation. The abbreviation is bolded if there
is a dfNAME definition but the label fragment is not NAME. This is
not a complete list of abbreviations, though we do want this to
eventually be a complete list of exceptions.
Abbreviation  Mnenomic  Source 
Expression  Syntax?  Example(s) 
a  and (suffix)  
 No  biimpa 500, rexlimiva 3010 
abl  Abelian group  dfabl 18019 
Abel  Yes  ablgrp 18021, zringabl 19641 
abs  absorption    No 
ressabs 15766 
abs  absolute value (of a complex number) 
dfabs 13824  (abs‘𝐴)  Yes 
absval 13826, absneg 13865, abs1 13885 
ad  adding  
 No  adantr 480, ad2antlr 759 
add  add (see "p")  dfadd 9826 
(𝐴 + 𝐵)  Yes 
addcl 9897, addcom 10101, addass 9902 
al  "for all"  
∀𝑥𝜑  No  alim 1729, alex 1743 
ALT  alternative/less preferred (suffix)  
 No  idALT 23 
an  and  dfan 385 
(𝜑 ∧ 𝜓)  Yes 
anor 509, iman 439, imnan 437 
ant  antecedent  
 No  adantr 480 
ass  associative  
 No  biass 373, orass 545, mulass 9903 
asym  asymmetric, antisymmetric  
 No  intasym 5430, asymref 5431, posasymb 16775 
ax  axiom  
 No  ax6dgen 1992, ax1cn 9849 
bas, base 
base (set of an extensible structure)  dfbase 15700 
(Base‘𝑆)  Yes 
baseval 15746, ressbas 15757, cnfldbas 19571 
b, bi  biconditional ("iff", "if and only if")
 dfbi 196  (𝜑 ↔ 𝜓)  Yes 
impbid 201, sspwb 4844 
br  binary relation  dfbr 4584 
𝐴𝑅𝐵  Yes  brab1 4630, brun 4633 
cbv  change bound variable   
No  cbvalivw 1921, cbvrex 3144 
cl  closure    No 
ifclda 4070, ovrcl 6584, zaddcl 11294 
cn  complex numbers  dfc 9821 
ℂ  Yes  nnsscn 10902, nncn 10905 
cnfld  field of complex numbers  dfcnfld 19568 
ℂ_{fld}  Yes  cnfldbas 19571, cnfldinv 19596 
cntz  centralizer  dfcntz 17573 
(Cntz‘𝑀)  Yes 
cntzfval 17576, dprdfcntz 18237 
cnv  converse  dfcnv 5046 
^{◡}𝐴  Yes  opelcnvg 5224, f1ocnv 6062 
co  composition  dfco 5047 
(𝐴 ∘ 𝐵)  Yes  cnvco 5230, fmptco 6303 
com  commutative  
 No  orcom 401, bicomi 213, eqcomi 2619 
con  contradiction, contraposition  
 No  condan 831, con2d 128 
csb  class substitution  dfcsb 3500 
⦋𝐴 / 𝑥⦌𝐵  Yes 
csbid 3507, csbie2g 3530 
cyg  cyclic group  dfcyg 18103 
CycGrp  Yes 
iscyg 18104, zringcyg 19658 
d  deduction form (suffix)  
 No  idd 24, impbid 201 
df  (alternate) definition (prefix)  
 No  dfrel2 5502, dffn2 5960 
di, distr  distributive  
 No 
andi 907, imdi 377, ordi 904, difindi 3840, ndmovdistr 6721 
dif  class difference  dfdif 3543 
(𝐴 ∖ 𝐵)  Yes 
difss 3699, difindi 3840 
div  division  dfdiv 10564 
(𝐴 / 𝐵)  Yes 
divcl 10570, divval 10566, divmul 10567 
dm  domain  dfdm 5048 
dom 𝐴  Yes  dmmpt 5547, iswrddm0 13184 
e, eq, equ  equals  dfcleq 2603 
𝐴 = 𝐵  Yes 
2p2e4 11021, uneqri 3717, equtr 1935 
el  element of  
𝐴 ∈ 𝐵  Yes 
eldif 3550, eldifsn 4260, elssuni 4403 
eu  "there exists exactly one"  dfeu 2462 
∃!𝑥𝜑  Yes  euex 2482, euabsn 4205 
ex  exists (i.e. is a set)  
 No  brrelex 5080, 0ex 4718 
ex  "there exists (at least one)"  dfex 1696 
∃𝑥𝜑  Yes  exim 1751, alex 1743 
exp  export  
 No  expt 167, expcom 450 
f  "not free in" (suffix)  
 No  equs45f 2338, sbf 2368 
f  function  dff 5808 
𝐹:𝐴⟶𝐵  Yes  fssxp 5973, opelf 5978 
fal  false  dffal 1481 
⊥  Yes  bifal 1488, falantru 1499 
fi  finite intersection  dffi 8200 
(fi‘𝐵)  Yes  fival 8201, inelfi 8207 
fi, fin  finite  dffin 7845 
Fin  Yes 
isfi 7865, snfi 7923, onfin 8036 
fld  field (Note: there is an alternative
definition Fld of a field, see dffld 32961)  dffield 18573 
Field  Yes  isfld 18579, fldidom 19126 
fn  function with domain  dffn 5807 
𝐴 Fn 𝐵  Yes  ffn 5958, fndm 5904 
frgp  free group  dffrgp 17946 
(freeGrp‘𝐼)  Yes 
frgpval 17994, frgpadd 17999 
fsupp  finitely supported function 
dffsupp 8159  𝑅 finSupp 𝑍  Yes 
isfsupp 8162, fdmfisuppfi 8167, fsuppco 8190 
fun  function  dffun 5806 
Fun 𝐹  Yes  funrel 5821, ffun 5961 
fv  function value  dffv 5812 
(𝐹‘𝐴)  Yes  fvres 6117, swrdfv 13276 
fz  finite set of sequential integers 
dffz 12198 
(𝑀...𝑁)  Yes  fzval 12199, eluzfz 12208 
fz0  finite set of sequential nonnegative integers 

(0...𝑁)  Yes  nn0fz0 12306, fz0tp 12309 
fzo  halfopen integer range  dffzo 12335 
(𝑀..^𝑁)  Yes 
elfzo 12341, elfzofz 12354 
g  more general (suffix); eliminates "is a set"
hypothsis  
 No  uniexg 6853 
gra  graph  
 No  uhgrav 25825, isumgra 25844, usgrares 25898 
grp  group  dfgrp 17248 
Grp  Yes  isgrp 17251, tgpgrp 21692 
gsum  group sum  dfgsum 15926 
(𝐺 Σ_{g} 𝐹)  Yes 
gsumval 17094, gsumwrev 17619 
hash  size (of a set)  dfhash 12980 
(#‘𝐴)  Yes 
hashgval 12982, hashfz1 12996, hashcl 13009 
hb  hypothesis builder (prefix)  
 No  hbxfrbi 1742, hbald 2028, hbequid 33212 
hm  (monoid, group, ring) homomorphism  
 No  ismhm 17160, isghm 17483, isrhm 18544 
i  inference (suffix)  
 No  eleq1i 2679, tcsni 8502 
i  implication (suffix)  
 No  brwdomi 8356, infeq5i 8416 
id  identity  
 No  biid 250 
idm  idempotent  
 No  anidm 674, tpidm13 4235 
im, imp  implication (label often omitted) 
dfim 13689  (𝐴 → 𝐵)  Yes 
iman 439, imnan 437, impbidd 199 
ima  image  dfima 5051 
(𝐴 “ 𝐵)  Yes  resima 5351, imaundi 5464 
imp  import  
 No  biimpa 500, impcom 445 
in  intersection  dfin 3547 
(𝐴 ∩ 𝐵)  Yes  elin 3758, incom 3767 
inf  infimum  dfinf 8232 
inf(ℝ^{+}, ℝ^{*}, < )  Yes 
fiinfcl 8290, infiso 8296 
is...  is (something a) ...?  
 No  isring 18374 
j  joining, disjoining  
 No  jc 158, jaoi 393 
l  left  
 No  olcd 407, simpl 472 
map  mapping operation or set exponentiation 
dfmap 7746  (𝐴 ↑_{𝑚} 𝐵)  Yes 
mapvalg 7754, elmapex 7764 
mat  matrix  dfmat 20033 
(𝑁 Mat 𝑅)  Yes 
matval 20036, matring 20068 
mdet  determinant (of a square matrix) 
dfmdet 20210  (𝑁 maDet 𝑅)  Yes 
mdetleib 20212, mdetrlin 20227 
mgm  magma  dfmgm 17065 
Magma  Yes 
mgmidmo 17082, mgmlrid 17089, ismgm 17066 
mgp  multiplicative group  dfmgp 18313 
(mulGrp‘𝑅)  Yes 
mgpress 18323, ringmgp 18376 
mnd  monoid  dfmnd 17118 
Mnd  Yes  mndass 17125, mndodcong 17784 
mo  "there exists at most one"  dfmo 2463 
∃*𝑥𝜑  Yes  eumo 2487, moim 2507 
mp  modus ponens  axmp 5 
 No  mpd 15, mpi 20 
mpt  modus ponendo tollens  
 No  mptnan 1684, mptxor 1685 
mpt  mapsto notation for a function 
dfmpt 4645  (𝑥 ∈ 𝐴 ↦ 𝐵)  Yes 
fconstmpt 5085, resmpt 5369 
mpt2  mapsto notation for an operation 
dfmpt2 6554  (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)  Yes 
mpt2mpt 6650, resmpt2 6656 
mul  multiplication (see "t")  dfmul 9827 
(𝐴 · 𝐵)  Yes 
mulcl 9899, divmul 10567, mulcom 9901, mulass 9903 
n, not  not  
¬ 𝜑  Yes 
nan 602, notnotr 124 
ne  not equal  dfne  𝐴 ≠ 𝐵 
Yes  exmidne 2792, neeqtrd 2851 
nel  not element of  dfnel  𝐴 ∉ 𝐵

Yes  neli 2885, nnel 2892 
ne0  not equal to zero (see n0)  
≠ 0  No 
negne0d 10269, ine0 10344, gt0ne0 10372 
nf  "not free in" (prefix)  
 No  nfnd 1769 
ngp  normed group  dfngp 22198 
NrmGrp  Yes  isngp 22210, ngptps 22216 
nm  norm (on a group or ring)  dfnm 22197 
(norm‘𝑊)  Yes 
nmval 22204, subgnm 22247 
nn  positive integers  dfnn 10898 
ℕ  Yes  nnsscn 10902, nncn 10905 
nn0  nonnegative integers  dfn0 11170 
ℕ_{0}  Yes  nnnn0 11176, nn0cn 11179 
n0  not the empty set (see ne0)  
≠ ∅  No  n0i 3879, vn0 3883, ssn0 3928 
OLD  old, obsolete (to be removed soon)  
 No  19.43OLD 1800 
op  ordered pair  dfop 4132 
⟨𝐴, 𝐵⟩  Yes  dfopif 4337, opth 4871 
or  or  dfor 384 
(𝜑 ∨ 𝜓)  Yes 
orcom 401, anor 509 
ot  ordered triple  dfot 4134 
⟨𝐴, 𝐵, 𝐶⟩  Yes 
euotd 4900, fnotovb 6592 
ov  operation value  dfov 6552 
(𝐴𝐹𝐵)  Yes
 fnotovb 6592, fnovrn 6707 
p  plus (see "add"), for allconstant
theorems  dfadd 9826 
(3 + 2) = 5  Yes 
3p2e5 11037 
pfx  prefix  dfpfx 40245 
(𝑊 prefix 𝐿)  Yes 
pfxlen 40254, ccatpfx 40272 
pm  Principia Mathematica  
 No  pm2.27 41 
pm  partial mapping (operation)  dfpm 7747 
(𝐴 ↑_{pm} 𝐵)  Yes  elpmi 7762, pmsspw 7778 
pr  pair  dfpr 4128 
{𝐴, 𝐵}  Yes 
elpr 4146, prcom 4211, prid1g 4239, prnz 4253 
prm, prime  prime (number)  dfprm 15224 
ℙ  Yes  1nprm 15230, dvdsprime 15238 
pss  proper subset  dfpss 3556 
𝐴 ⊊ 𝐵  Yes  pssss 3664, sspsstri 3671 
q  rational numbers ("quotients")  dfq 11665 
ℚ  Yes  elq 11666 
r  right  
 No  orcd 406, simprl 790 
rab  restricted class abstraction 
dfrab 2905  {𝑥 ∈ 𝐴 ∣ 𝜑}  Yes 
rabswap 3098, dfoprab 6553 
ral  restricted universal quantification 
dfral 2901  ∀𝑥 ∈ 𝐴𝜑  Yes 
ralnex 2975, ralrnmpt2 6673 
rcl  reverse closure  
 No  ndmfvrcl 6129, nnarcl 7583 
re  real numbers  dfr 9825 
ℝ  Yes  recn 9905, 0re 9919 
rel  relation  dfrel 5045  Rel 𝐴 
Yes  brrelex 5080, relmpt2opab 7146 
res  restriction  dfres 5050 
(𝐴 ↾ 𝐵)  Yes 
opelres 5322, f1ores 6064 
reu  restricted existential uniqueness 
dfreu 2903  ∃!𝑥 ∈ 𝐴𝜑  Yes 
nfreud 3091, reurex 3137 
rex  restricted existential quantification 
dfrex 2902  ∃𝑥 ∈ 𝐴𝜑  Yes 
rexnal 2978, rexrnmpt2 6674 
rmo  restricted "at most one" 
dfrmo 2904  ∃*𝑥 ∈ 𝐴𝜑  Yes 
nfrmod 3092, nrexrmo 3140 
rn  range  dfrn 5049  ran 𝐴 
Yes  elrng 5236, rncnvcnv 5270 
rng  (unital) ring  dfring 18372 
Ring  Yes 
ringidval 18326, isring 18374, ringgrp 18375 
rot  rotation  
 No  3anrot 1036, 3orrot 1037 
s  eliminates need for syllogism (suffix) 
  No  ancoms 468 
sb  (proper) substitution (of a set) 
dfsb 1868  [𝑦 / 𝑥]𝜑  Yes 
spsbe 1871, sbimi 1873 
sbc  (proper) substitution of a class 
dfsbc 3403  [𝐴 / 𝑥]𝜑  Yes 
sbc2or 3411, sbcth 3417 
sca  scalar  dfsca 15784 
(Scalar‘𝐻)  Yes 
resssca 15854, mgpsca 18319 
simp  simple, simplification  
 No  simpl 472, simp3r3 1164 
sn  singleton  dfsn 4126 
{𝐴}  Yes  eldifsn 4260 
sp  specialization  
 No  spsbe 1871, spei 2249 
ss  subset  dfss 3554 
𝐴 ⊆ 𝐵  Yes  difss 3699 
struct  structure  dfstruct 15697 
Struct  Yes  brstruct 15703, structfn 15708 
sub  subtract  dfsub 10147 
(𝐴 − 𝐵)  Yes 
subval 10151, subaddi 10247 
sup  supremum  dfsup 8231 
sup(𝐴, 𝐵, < )  Yes 
fisupcl 8258, supmo 8241 
supp  support (of a function)  dfsupp 7183 
(𝐹 supp 𝑍)  Yes 
ressuppfi 8184, mptsuppd 7205 
swap  swap (two parts within a theorem) 
  No  rabswap 3098, 2reuswap 3377 
syl  syllogism  syl 17 
 No  3syl 18 
sym  symmetric  
 No  dfsymdif 3806, cnvsym 5429 
symg  symmetric group  dfsymg 17621 
(SymGrp‘𝐴)  Yes 
symghash 17628, pgrpsubgsymg 17651 
t 
times (see "mul"), for allconstant theorems 
dfmul 9827 
(3 · 2) = 6  Yes 
3t2e6 11056 
th  theorem  
 No  nfth 1718, sbcth 3417, weth 9200 
tp  triple  dftp 4130 
{𝐴, 𝐵, 𝐶}  Yes 
eltpi 4176, tpeq1 4221 
tr  transitive  
 No  bitrd 267, biantr 968 
tru  true  dftru 1478 
⊤  Yes  bitru 1487, truanfal 1498 
un  union  dfun 3545 
(𝐴 ∪ 𝐵)  Yes 
uneqri 3717, uncom 3719 
unit  unit (in a ring) 
dfunit 18465  (Unit‘𝑅)  Yes 
isunit 18480, nzrunit 19088 
v  disjoint variable conditions used when
a notfree hypothesis (suffix) 
  No  spimv 2245 
vv  2 disjoint variables (in a notfree hypothesis)
(suffix)    No  19.23vv 1890 
w  weak (version of a theorem) (suffix)  
 No  ax11w 1994, spnfw 1915 
wrd  word 
dfword 13154  Word 𝑆  Yes 
iswrdb 13166, wrdfn 13174, ffz0iswrd 13187 
xp  cross product (Cartesian product) 
dfxp 5044  (𝐴 × 𝐵)  Yes 
elxp 5055, opelxpi 5072, xpundi 5094 
xr  eXtended reals  dfxr 9957 
ℝ^{*}  Yes  ressxr 9962, rexr 9964, 0xr 9965 
z  integers (from German "Zahlen") 
dfz 11255  ℤ  Yes 
elz 11256, zcn 11259 
zn  ring of integers mod 𝑛  dfzn 19674 
(ℤ/nℤ‘𝑁)  Yes 
znval 19702, zncrng 19712, znhash 19726 
zring  ring of integers  dfzring 19638 
ℤ_{ring}  Yes  zringbas 19643, zringcrng 19639

0, z 
slashed zero (empty set) (see n0)  dfnul 3875 
∅  Yes 
n0i 3879, vn0 3883; snnz 4252, prnz 4253 
(Contributed by DAW, 27Dec2016.) (New usage is discouraged.)

⊢ 𝜑 ⇒ ⊢ 𝜑 

18.1.2 Natural deduction


Theorem  natded 26652 
Here are typical natural deduction (ND) rules in the style of Gentzen
and Jaśkowski, along with MPE translations of them. This also
shows the recommended theorems when you find yourself needing these
rules (the recommendations encourage a slightly different proof style
that works more naturally with metamath). A decent list of the standard
rules of natural deduction can be found beginning with definition /\I in
[Pfenning] p. 18. For information about ND and Metamath, see the
page on Deduction Form and Natural Deduction
in Metamath Proof Explorer. Many more citations could be added.
Name  Natural Deduction Rule  Translation 
Recommendation  Comments 
IT 
Γ⊢ 𝜓 => Γ⊢ 𝜓 
idi 2 
nothing  Reiteration is always redundant in Metamath.
Definition "new rule" in [Pfenning] p. 18,
definition IT in [Clemente] p. 10. 
∧I 
Γ⊢ 𝜓 & Γ⊢ 𝜒 => Γ⊢ 𝜓 ∧ 𝜒 
jca 553 
jca 553, pm3.2i 470 
Definition ∧I in [Pfenning] p. 18,
definition I∧m,n in [Clemente] p. 10, and
definition ∧I in [Indrzejczak] p. 34
(representing both Gentzen's system NK and Jaśkowski) 
∧E_{L} 
Γ⊢ 𝜓 ∧ 𝜒 => Γ⊢ 𝜓 
simpld 474 
simpld 474, adantr 480 
Definition ∧E_{L} in [Pfenning] p. 18,
definition E∧(1) in [Clemente] p. 11, and
definition ∧E in [Indrzejczak] p. 34
(representing both Gentzen's system NK and Jaśkowski) 
∧E_{R} 
Γ⊢ 𝜓 ∧ 𝜒 => Γ⊢ 𝜒 
simprd 478 
simpr 476, adantl 481 
Definition ∧E_{R} in [Pfenning] p. 18,
definition E∧(2) in [Clemente] p. 11, and
definition ∧E in [Indrzejczak] p. 34
(representing both Gentzen's system NK and Jaśkowski) 
→I 
Γ, 𝜓⊢ 𝜒 => Γ⊢ 𝜓 → 𝜒 
ex 449  ex 449 
Definition →I in [Pfenning] p. 18,
definition I=>m,n in [Clemente] p. 11, and
definition →I in [Indrzejczak] p. 33. 
→E 
Γ⊢ 𝜓 → 𝜒 & Γ⊢ 𝜓 => Γ⊢ 𝜒 
mpd 15  axmp 5, mpd 15, mpdan 699, imp 444 
Definition →E in [Pfenning] p. 18,
definition E=>m,n in [Clemente] p. 11, and
definition →E in [Indrzejczak] p. 33. 
∨I_{L}  Γ⊢ 𝜓 =>
Γ⊢ 𝜓 ∨ 𝜒 
olcd 407 
olc 398, olci 405, olcd 407 
Definition ∨I in [Pfenning] p. 18,
definition I∨n(1) in [Clemente] p. 12 
∨I_{R}  Γ⊢ 𝜒 =>
Γ⊢ 𝜓 ∨ 𝜒 
orcd 406 
orc 399, orci 404, orcd 406 
Definition ∨I_{R} in [Pfenning] p. 18,
definition I∨n(2) in [Clemente] p. 12. 
∨E  Γ⊢ 𝜓 ∨ 𝜒 & Γ, 𝜓⊢ 𝜃 &
Γ, 𝜒⊢ 𝜃 => Γ⊢ 𝜃 
mpjaodan 823 
mpjaodan 823, jaodan 822, jaod 394 
Definition ∨E in [Pfenning] p. 18,
definition E∨m,n,p in [Clemente] p. 12. 
¬I  Γ, 𝜓⊢ ⊥ => Γ⊢ ¬ 𝜓 
inegd 1494  pm2.01d 180 

¬I  Γ, 𝜓⊢ 𝜃 & Γ⊢ ¬ 𝜃 =>
Γ⊢ ¬ 𝜓 
mtand 689  mtand 689 
definition I¬m,n,p in [Clemente] p. 13. 
¬I  Γ, 𝜓⊢ 𝜒 & Γ, 𝜓⊢ ¬ 𝜒 =>
Γ⊢ ¬ 𝜓 
pm2.65da 598  pm2.65da 598 
Contradiction. 
¬I 
Γ, 𝜓⊢ ¬ 𝜓 => Γ⊢ ¬ 𝜓 
pm2.01da 457  pm2.01d 180, pm2.65da 598, pm2.65d 186 
For an alternative falsumfree natural deduction ruleset 
¬E 
Γ⊢ 𝜓 & Γ⊢ ¬ 𝜓 => Γ⊢ ⊥ 
pm2.21fal 1496 
pm2.21dd 185  
¬E 
Γ, ¬ 𝜓⊢ ⊥ => Γ⊢ 𝜓 

pm2.21dd 185 
definition →E in [Indrzejczak] p. 33. 
¬E 
Γ⊢ 𝜓 & Γ⊢ ¬ 𝜓 => Γ⊢ 𝜃 
pm2.21dd 185  pm2.21dd 185, pm2.21d 117, pm2.21 119 
For an alternative falsumfree natural deduction ruleset.
Definition ¬E in [Pfenning] p. 18. 
⊤I  Γ⊢ ⊤ 
a1tru 1491  tru 1479, a1tru 1491, trud 1484 
Definition ⊤I in [Pfenning] p. 18. 
⊥E  Γ, ⊥⊢ 𝜃 
falimd 1490  falim 1489 
Definition ⊥E in [Pfenning] p. 18. 
∀I 
Γ⊢ [𝑎 / 𝑥]𝜓 => Γ⊢ ∀𝑥𝜓 
alrimiv 1842  alrimiv 1842, ralrimiva 2949 
Definition ∀I^{a} in [Pfenning] p. 18,
definition I∀n in [Clemente] p. 32. 
∀E 
Γ⊢ ∀𝑥𝜓 => Γ⊢ [𝑡 / 𝑥]𝜓 
spsbcd 3416  spcv 3272, rspcv 3278 
Definition ∀E in [Pfenning] p. 18,
definition E∀n,t in [Clemente] p. 32. 
∃I 
Γ⊢ [𝑡 / 𝑥]𝜓 => Γ⊢ ∃𝑥𝜓 
spesbcd 3488  spcev 3273, rspcev 3282 
Definition ∃I in [Pfenning] p. 18,
definition I∃n,t in [Clemente] p. 32. 
∃E 
Γ⊢ ∃𝑥𝜓 & Γ, [𝑎 / 𝑥]𝜓⊢ 𝜃 =>
Γ⊢ 𝜃 
exlimddv 1850  exlimddv 1850, exlimdd 2075,
exlimdv 1848, rexlimdva 3013 
Definition ∃E^{a,u} in [Pfenning] p. 18,
definition E∃m,n,p,a in [Clemente] p. 32. 
⊥C 
Γ, ¬ 𝜓⊢ ⊥ => Γ⊢ 𝜓 
efald 1495  efald 1495 
Proof by contradiction (classical logic),
definition ⊥C in [Pfenning] p. 17. 
⊥C 
Γ, ¬ 𝜓⊢ 𝜓 => Γ⊢ 𝜓 
pm2.18da 458  pm2.18da 458, pm2.18d 123, pm2.18 121 
For an alternative falsumfree natural deduction ruleset 
¬ ¬C 
Γ⊢ ¬ ¬ 𝜓 => Γ⊢ 𝜓 
notnotrd 127  notnotrd 127, notnotr 124 
Double negation rule (classical logic),
definition NNC in [Pfenning] p. 17,
definition E¬n in [Clemente] p. 14. 
EM  Γ⊢ 𝜓 ∨ ¬ 𝜓 
exmidd 431  exmid 430 
Excluded middle (classical logic),
definition XM in [Pfenning] p. 17,
proof 5.11 in [Clemente] p. 14. 
=I  Γ⊢ 𝐴 = 𝐴 
eqidd 2611  eqid 2610, eqidd 2611 
Introduce equality,
definition =I in [Pfenning] p. 127. 
=E  Γ⊢ 𝐴 = 𝐵 & Γ[𝐴 / 𝑥]𝜓 =>
Γ⊢ [𝐵 / 𝑥]𝜓 
sbceq1dd 3408  sbceq1d 3407, equality theorems 
Eliminate equality,
definition =E in [Pfenning] p. 127. (Both E1 and E2.) 
Note that MPE uses classical logic, not intuitionist logic. As is
conventional, the "I" rules are introduction rules, "E" rules are
elimination rules, the "C" rules are conversion rules, and Γ
represents the set of (current) hypotheses. We use wff variable names
beginning with 𝜓 to provide a closer representation
of the Metamath
equivalents (which typically use the antedent 𝜑 to represent the
context Γ).
Most of this information was developed by Mario Carneiro and posted on
3Feb2017. For more information, see the
page on Deduction Form and Natural Deduction
in Metamath Proof Explorer.
For annotated examples where some traditional ND rules
are directly applied in MPE, see exnatded5.2 26653, exnatded5.3 26656,
exnatded5.5 26659, exnatded5.7 26660, exnatded5.8 26662, exnatded5.13 26664,
exnatded9.20 26666, and exnatded9.26 26668.
(Contributed by DAW, 4Feb2017.) (New usage is discouraged.)

⊢ 𝜑 ⇒ ⊢ 𝜑 

18.1.3 Natural deduction examples
These are examples of how natural deduction rules can be applied in Metamath
(both as lineforline translations of ND rules, and as a way to apply
deduction forms without being limited to applying ND rules). For more
information, see natded 26652 and mmnatded.html 26652. Since these examples should
not be used within proofs of other theorems, especially in Mathboxes, they
are marked with "(New usage is discouraged.)".


Theorem  exnatded5.2 26653 
Theorem 5.2 of [Clemente] p. 15, translated line by line using the
interpretation of natural deduction in Metamath.
For information about ND and Metamath, see the
page on Deduction Form and Natural Deduction
in Metamath Proof Explorer.
The original proof, which uses Fitch style, was written as follows:
#  MPE#  ND Expression 
MPE Translation  ND Rationale 
MPE Rationale 
1  5  ((𝜓 ∧ 𝜒) → 𝜃) 
(𝜑 → ((𝜓 ∧ 𝜒) → 𝜃)) 
Given 
$e. 
2  2  (𝜒 → 𝜓) 
(𝜑 → (𝜒 → 𝜓)) 
Given 
$e. 
3  1  𝜒 
(𝜑 → 𝜒) 
Given 
$e. 
4  3  𝜓 
(𝜑 → 𝜓) 
→E 2,3 
mpd 15, the MPE equivalent of →E, 1,2 
5  4  (𝜓 ∧ 𝜒) 
(𝜑 → (𝜓 ∧ 𝜒)) 
∧I 4,3 
jca 553, the MPE equivalent of ∧I, 3,1 
6  6  𝜃 
(𝜑 → 𝜃) 
→E 1,5 
mpd 15, the MPE equivalent of →E, 4,5 
The original used Latin letters for predicates;
we have replaced them with
Greek letters to follow Metamath naming conventions and so that
it is easier to follow the Metamath translation.
The Metamath lineforline translation of this
natural deduction approach precedes every line with an antecedent
including 𝜑 and uses the Metamath equivalents
of the natural deduction rules.
Below is the final metamath proof (which reorders some steps).
A much more efficient proof, using more of Metamath and MPE's
capabilities, is shown in exnatded5.22 26654.
A proof without context is shown in exnatded5.2i 26655.
(Contributed by Mario Carneiro, 9Feb2017.)
(Proof modification is discouraged.) (New usage is discouraged.)

⊢ (𝜑 → ((𝜓 ∧ 𝜒) → 𝜃)) & ⊢ (𝜑 → (𝜒 → 𝜓)) & ⊢ (𝜑 → 𝜒) ⇒ ⊢ (𝜑 → 𝜃) 

Theorem  exnatded5.22 26654 
A more efficient proof of Theorem 5.2 of [Clemente] p. 15. Compare with
exnatded5.2 26653 and exnatded5.2i 26655. (Contributed by Mario Carneiro,
9Feb2017.) (Proof modification is discouraged.)
(New usage is discouraged.)

⊢ (𝜑 → ((𝜓 ∧ 𝜒) → 𝜃)) & ⊢ (𝜑 → (𝜒 → 𝜓)) & ⊢ (𝜑 → 𝜒) ⇒ ⊢ (𝜑 → 𝜃) 

Theorem  exnatded5.2i 26655 
The same as exnatded5.2 26653 and exnatded5.22 26654 but with no context.
(Contributed by Mario Carneiro, 9Feb2017.)
(Proof modification is discouraged.) (New usage is discouraged.)

⊢ ((𝜓 ∧ 𝜒) → 𝜃)
& ⊢ (𝜒 → 𝜓)
& ⊢ 𝜒 ⇒ ⊢ 𝜃 

Theorem  exnatded5.3 26656 
Theorem 5.3 of [Clemente] p. 16, translated line by line using an
interpretation of natural deduction in Metamath.
A much more efficient proof, using more of Metamath and MPE's
capabilities, is shown in exnatded5.32 26657.
A proof without context is shown in exnatded5.3i 26658.
For information about ND and Metamath, see the
page on Deduction Form and Natural Deduction
in Metamath Proof Explorer
.
The original proof, which uses Fitch style, was written as follows:
#  MPE#  ND Expression 
MPE Translation  ND Rationale 
MPE Rationale 
1  2;3  (𝜓 → 𝜒) 
(𝜑 → (𝜓 → 𝜒)) 
Given 
$e; adantr 480 to move it into the ND hypothesis 
2  5;6  (𝜒 → 𝜃) 
(𝜑 → (𝜒 → 𝜃)) 
Given 
$e; adantr 480 to move it into the ND hypothesis 
3  1  ... 𝜓 
((𝜑 ∧ 𝜓) → 𝜓) 
ND hypothesis assumption 
simpr 476, to access the new assumption 
4  4  ... 𝜒 
((𝜑 ∧ 𝜓) → 𝜒) 
→E 1,3 
mpd 15, the MPE equivalent of →E, 1.3.
adantr 480 was used to transform its dependency
(we could also use imp 444 to get this directly from 1)

5  7  ... 𝜃 
((𝜑 ∧ 𝜓) → 𝜃) 
→E 2,4 
mpd 15, the MPE equivalent of →E, 4,6.
adantr 480 was used to transform its dependency 
6  8  ... (𝜒 ∧ 𝜃) 
((𝜑 ∧ 𝜓) → (𝜒 ∧ 𝜃)) 
∧I 4,5 
jca 553, the MPE equivalent of ∧I, 4,7 
7  9  (𝜓 → (𝜒 ∧ 𝜃)) 
(𝜑 → (𝜓 → (𝜒 ∧ 𝜃))) 
→I 3,6 
ex 449, the MPE equivalent of →I, 8 
The original used Latin letters for predicates;
we have replaced them with
Greek letters to follow Metamath naming conventions and so that
it is easier to follow the Metamath translation.
The Metamath lineforline translation of this
natural deduction approach precedes every line with an antecedent
including 𝜑 and uses the Metamath equivalents
of the natural deduction rules.
(Contributed by Mario Carneiro, 9Feb2017.)
(Proof modification is discouraged.) (New usage is discouraged.)

⊢ (𝜑 → (𝜓 → 𝜒)) & ⊢ (𝜑 → (𝜒 → 𝜃)) ⇒ ⊢ (𝜑 → (𝜓 → (𝜒 ∧ 𝜃))) 

Theorem  exnatded5.32 26657 
A more efficient proof of Theorem 5.3 of [Clemente] p. 16. Compare with
exnatded5.3 26656 and exnatded5.3i 26658. (Contributed by Mario Carneiro,
9Feb2017.) (Proof modification is discouraged.)
(New usage is discouraged.)

⊢ (𝜑 → (𝜓 → 𝜒)) & ⊢ (𝜑 → (𝜒 → 𝜃)) ⇒ ⊢ (𝜑 → (𝜓 → (𝜒 ∧ 𝜃))) 

Theorem  exnatded5.3i 26658 
The same as exnatded5.3 26656 and exnatded5.32 26657 but with no context.
Identical to jccir 560, which should be used instead. (Contributed
by
Mario Carneiro, 9Feb2017.) (Proof modification is discouraged.)
(New usage is discouraged.)

⊢ (𝜓 → 𝜒)
& ⊢ (𝜒 → 𝜃) ⇒ ⊢ (𝜓 → (𝜒 ∧ 𝜃)) 

Theorem  exnatded5.5 26659 
Theorem 5.5 of [Clemente] p. 18, translated line by line using the
usual translation of natural deduction (ND) in the
Metamath Proof Explorer (MPE) notation.
For information about ND and Metamath, see the
page on Deduction Form and Natural Deduction
in Metamath Proof Explorer.
The original proof, which uses Fitch style, was written as follows
(the leading "..." shows an embedded ND hypothesis, beginning with
the initial assumption of the ND hypothesis):
#  MPE#  ND Expression 
MPE Translation  ND Rationale 
MPE Rationale 
1  2;3 
(𝜓 → 𝜒) 
(𝜑 → (𝜓 → 𝜒)) 
Given 
$e; adantr 480 to move it into the ND hypothesis 
2  5  ¬ 𝜒 
(𝜑 → ¬ 𝜒)  Given 
$e; we'll use adantr 480 to move it into the ND hypothesis 
3  1 
... 𝜓  (𝜑 → 𝜓) 
ND hypothesis assumption 
simpr 476 
4  4  ... 𝜒 
((𝜑 ∧ 𝜓) → 𝜒) 
→E 1,3 
mpd 15 1,3 
5  6  ... ¬ 𝜒 
((𝜑 ∧ 𝜓) → ¬ 𝜒) 
IT 2 
adantr 480 5 
6  7  ¬ 𝜓 
(𝜑 → ¬ 𝜓) 
∧I 3,4,5 
pm2.65da 598 4,6 
The original used Latin letters; we have replaced them with
Greek letters to follow Metamath naming conventions and so that
it is easier to follow the Metamath translation.
The Metamath lineforline translation of this
natural deduction approach precedes every line with an antecedent
including 𝜑 and uses the Metamath equivalents
of the natural deduction rules.
To add an assumption, the antecedent is modified to include it
(typically by using adantr 480; simpr 476 is useful when you want to
depend directly on the new assumption).
Below is the final metamath proof (which reorders some steps).
A much more efficient proof is mtod 188;
a proof without context is shown in mto 187.
(Contributed by David A. Wheeler, 19Feb2017.)
(Proof modification is discouraged.) (New usage is discouraged.)

⊢ (𝜑 → (𝜓 → 𝜒)) & ⊢ (𝜑 → ¬ 𝜒) ⇒ ⊢ (𝜑 → ¬ 𝜓) 

Theorem  exnatded5.7 26660 
Theorem 5.7 of [Clemente] p. 19, translated line by line using the
interpretation of natural deduction in Metamath.
A much more efficient proof, using more of Metamath and MPE's
capabilities, is shown in exnatded5.72 26661.
For information about ND and Metamath, see the
page on Deduction Form and Natural Deduction
in Metamath Proof Explorer
.
The original proof, which uses Fitch style, was written as follows:
#  MPE#  ND Expression 
MPE Translation  ND Rationale 
MPE Rationale 
1  6 
(𝜓 ∨ (𝜒 ∧ 𝜃)) 
(𝜑 → (𝜓 ∨ (𝜒 ∧ 𝜃))) 
Given 
$e. No need for adantr 480 because we do not move this
into an ND hypothesis 
2  1  ... 𝜓 
((𝜑 ∧ 𝜓) → 𝜓) 
ND hypothesis assumption (new scope) 
simpr 476 
3  2  ... (𝜓 ∨ 𝜒) 
((𝜑 ∧ 𝜓) → (𝜓 ∨ 𝜒)) 
∨I_{L} 2 
orcd 406, the MPE equivalent of ∨I_{L}, 1 
4  3  ... (𝜒 ∧ 𝜃) 
((𝜑 ∧ (𝜒 ∧ 𝜃)) → (𝜒 ∧ 𝜃)) 
ND hypothesis assumption (new scope) 
simpr 476 
5  4  ... 𝜒 
((𝜑 ∧ (𝜒 ∧ 𝜃)) → 𝜒) 
∧E_{L} 4 
simpld 474, the MPE equivalent of ∧E_{L}, 3 
6  6  ... (𝜓 ∨ 𝜒) 
((𝜑 ∧ (𝜒 ∧ 𝜃)) → (𝜓 ∨ 𝜒)) 
∨I_{R} 5 
olcd 407, the MPE equivalent of ∨I_{R}, 4 
7  7  (𝜓 ∨ 𝜒) 
(𝜑 → (𝜓 ∨ 𝜒)) 
∨E 1,3,6 
mpjaodan 823, the MPE equivalent of ∨E, 2,5,6 
The original used Latin letters for predicates;
we have replaced them with
Greek letters to follow Metamath naming conventions and so that
it is easier to follow the Metamath translation.
The Metamath lineforline translation of this
natural deduction approach precedes every line with an antecedent
including 𝜑 and uses the Metamath equivalents
of the natural deduction rules.
(Contributed by Mario Carneiro, 9Feb2017.)
(Proof modification is discouraged.) (New usage is discouraged.)

⊢ (𝜑 → (𝜓 ∨ (𝜒 ∧ 𝜃))) ⇒ ⊢ (𝜑 → (𝜓 ∨ 𝜒)) 

Theorem  exnatded5.72 26661 
A more efficient proof of Theorem 5.7 of [Clemente] p. 19. Compare with
exnatded5.7 26660. (Contributed by Mario Carneiro,
9Feb2017.)
(Proof modification is discouraged.) (New usage is discouraged.)

⊢ (𝜑 → (𝜓 ∨ (𝜒 ∧ 𝜃))) ⇒ ⊢ (𝜑 → (𝜓 ∨ 𝜒)) 

Theorem  exnatded5.8 26662 
Theorem 5.8 of [Clemente] p. 20, translated line by line using the
usual translation of natural deduction (ND) in the
Metamath Proof Explorer (MPE) notation.
For information about ND and Metamath, see the
page on Deduction Form and Natural Deduction
in Metamath Proof Explorer.
The original proof, which uses Fitch style, was written as follows
(the leading "..." shows an embedded ND hypothesis, beginning with
the initial assumption of the ND hypothesis):
#  MPE#  ND Expression 
MPE Translation  ND Rationale 
MPE Rationale 
1  10;11 
((𝜓 ∧ 𝜒) → ¬ 𝜃) 
(𝜑 → ((𝜓 ∧ 𝜒) → ¬ 𝜃)) 
Given 
$e; adantr 480 to move it into the ND hypothesis 
2  3;4  (𝜏 → 𝜃) 
(𝜑 → (𝜏 → 𝜃))  Given 
$e; adantr 480 to move it into the ND hypothesis 
3  7;8 
𝜒  (𝜑 → 𝜒) 
Given 
$e; adantr 480 to move it into the ND hypothesis 
4  1;2  𝜏  (𝜑 → 𝜏) 
Given 
$e. adantr 480 to move it into the ND hypothesis 
5  6  ... 𝜓 
((𝜑 ∧ 𝜓) → 𝜓) 
ND Hypothesis/Assumption 
simpr 476. New ND hypothesis scope, each reference outside
the scope must change antecedent 𝜑 to (𝜑 ∧ 𝜓). 
6  9  ... (𝜓 ∧ 𝜒) 
((𝜑 ∧ 𝜓) → (𝜓 ∧ 𝜒)) 
∧I 5,3 
jca 553 (∧I), 6,8 (adantr 480 to bring in scope) 
7  5  ... ¬ 𝜃 
((𝜑 ∧ 𝜓) → ¬ 𝜃) 
→E 1,6 
mpd 15 (→E), 2,4 
8  12  ... 𝜃 
((𝜑 ∧ 𝜓) → 𝜃) 
→E 2,4 
mpd 15 (→E), 9,11;
note the contradiction with ND line 7 (MPE line 5) 
9  13  ¬ 𝜓 
(𝜑 → ¬ 𝜓) 
¬I 5,7,8 
pm2.65da 598 (¬I), 5,12; proof by contradiction.
MPE step 6 (ND#5) does not need a reference here, because
the assumption is embedded in the antecedents 
The original used Latin letters; we have replaced them with
Greek letters to follow Metamath naming conventions and so that
it is easier to follow the Metamath translation.
The Metamath lineforline translation of this
natural deduction approach precedes every line with an antecedent
including 𝜑 and uses the Metamath equivalents
of the natural deduction rules.
To add an assumption, the antecedent is modified to include it
(typically by using adantr 480; simpr 476 is useful when you want to
depend directly on the new assumption).
Below is the final metamath proof (which reorders some steps).
A much more efficient proof, using more of Metamath and MPE's
capabilities, is shown in exnatded5.82 26663.
(Contributed by Mario Carneiro, 9Feb2017.)
(Proof modification is discouraged.) (New usage is discouraged.)

⊢ (𝜑 → ((𝜓 ∧ 𝜒) → ¬ 𝜃)) & ⊢ (𝜑 → (𝜏 → 𝜃)) & ⊢ (𝜑 → 𝜒)
& ⊢ (𝜑 → 𝜏) ⇒ ⊢ (𝜑 → ¬ 𝜓) 

Theorem  exnatded5.82 26663 
A more efficient proof of Theorem 5.8 of [Clemente] p. 20. For a longer
linebyline translation, see exnatded5.8 26662. (Contributed by Mario
Carneiro, 9Feb2017.) (Proof modification is discouraged.)
(New usage is discouraged.)

⊢ (𝜑 → ((𝜓 ∧ 𝜒) → ¬ 𝜃)) & ⊢ (𝜑 → (𝜏 → 𝜃)) & ⊢ (𝜑 → 𝜒)
& ⊢ (𝜑 → 𝜏) ⇒ ⊢ (𝜑 → ¬ 𝜓) 

Theorem  exnatded5.13 26664 
Theorem 5.13 of [Clemente] p. 20, translated line by line using the
interpretation of natural deduction in Metamath.
For information about ND and Metamath, see the
page on Deduction Form and Natural Deduction
in Metamath Proof Explorer.
A much more efficient proof, using more of Metamath and MPE's
capabilities, is shown in exnatded5.132 26665.
The original proof, which uses Fitch style, was written as follows
(the leading "..." shows an embedded ND hypothesis, beginning with
the initial assumption of the ND hypothesis):
#  MPE#  ND Expression 
MPE Translation  ND Rationale 
MPE Rationale 
1  15  (𝜓 ∨ 𝜒) 
(𝜑 → (𝜓 ∨ 𝜒)) 
Given 
$e. 
2;3  2  (𝜓 → 𝜃) 
(𝜑 → (𝜓 → 𝜃))  Given 
$e. adantr 480 to move it into the ND hypothesis 
3  9  (¬ 𝜏 → ¬ 𝜒) 
(𝜑 → (¬ 𝜏 → ¬ 𝜒)) 
Given 
$e. ad2antrr 758 to move it into the ND subhypothesis 
4  1  ... 𝜓 
((𝜑 ∧ 𝜓) → 𝜓) 
ND hypothesis assumption 
simpr 476 
5  4  ... 𝜃 
((𝜑 ∧ 𝜓) → 𝜃) 
→E 2,4 
mpd 15 1,3 
6  5  ... (𝜃 ∨ 𝜏) 
((𝜑 ∧ 𝜓) → (𝜃 ∨ 𝜏)) 
∨I 5 
orcd 406 4 
7  6  ... 𝜒 
((𝜑 ∧ 𝜒) → 𝜒) 
ND hypothesis assumption 
simpr 476 
8  8  ... ... ¬ 𝜏 
(((𝜑 ∧ 𝜒) ∧ ¬ 𝜏) → ¬ 𝜏) 
(sub) ND hypothesis assumption 
simpr 476 
9  11  ... ... ¬ 𝜒 
(((𝜑 ∧ 𝜒) ∧ ¬ 𝜏) → ¬ 𝜒) 
→E 3,8 
mpd 15 8,10 
10  7  ... ... 𝜒 
(((𝜑 ∧ 𝜒) ∧ ¬ 𝜏) → 𝜒) 
IT 7 
adantr 480 6 
11  12  ... ¬ ¬ 𝜏 
((𝜑 ∧ 𝜒) → ¬ ¬ 𝜏) 
¬I 8,9,10 
pm2.65da 598 7,11 
12  13  ... 𝜏 
((𝜑 ∧ 𝜒) → 𝜏) 
¬E 11 
notnotrd 127 12 
13  14  ... (𝜃 ∨ 𝜏) 
((𝜑 ∧ 𝜒) → (𝜃 ∨ 𝜏)) 
∨I 12 
olcd 407 13 
14  16  (𝜃 ∨ 𝜏) 
(𝜑 → (𝜃 ∨ 𝜏)) 
∨E 1,6,13 
mpjaodan 823 5,14,15 
The original used Latin letters; we have replaced them with
Greek letters to follow Metamath naming conventions and so that
it is easier to follow the Metamath translation.
The Metamath lineforline translation of this
natural deduction approach precedes every line with an antecedent
including 𝜑 and uses the Metamath equivalents
of the natural deduction rules.
To add an assumption, the antecedent is modified to include it
(typically by using adantr 480; simpr 476 is useful when you want to
depend directly on the new assumption).
(Contributed by Mario Carneiro, 9Feb2017.)
(Proof modification is discouraged.) (New usage is discouraged.)

⊢ (𝜑 → (𝜓 ∨ 𝜒)) & ⊢ (𝜑 → (𝜓 → 𝜃)) & ⊢ (𝜑 → (¬ 𝜏 → ¬ 𝜒)) ⇒ ⊢ (𝜑 → (𝜃 ∨ 𝜏)) 

Theorem  exnatded5.132 26665 
A more efficient proof of Theorem 5.13 of [Clemente] p. 20. Compare
with exnatded5.13 26664. (Contributed by Mario Carneiro,
9Feb2017.)
(Proof modification is discouraged.) (New usage is discouraged.)

⊢ (𝜑 → (𝜓 ∨ 𝜒)) & ⊢ (𝜑 → (𝜓 → 𝜃)) & ⊢ (𝜑 → (¬ 𝜏 → ¬ 𝜒)) ⇒ ⊢ (𝜑 → (𝜃 ∨ 𝜏)) 

Theorem  exnatded9.20 26666 
Theorem 9.20 of [Clemente] p. 43, translated line by line using the
usual translation of natural deduction (ND) in the
Metamath Proof Explorer (MPE) notation.
For information about ND and Metamath, see the
page on Deduction Form and Natural Deduction
in Metamath Proof Explorer.
The original proof, which uses Fitch style, was written as follows
(the leading "..." shows an embedded ND hypothesis, beginning with
the initial assumption of the ND hypothesis):
#  MPE#  ND Expression 
MPE Translation  ND Rationale 
MPE Rationale 
1  1 
(𝜓 ∧ (𝜒 ∨ 𝜃)) 
(𝜑 → (𝜓 ∧ (𝜒 ∨ 𝜃))) 
Given 
$e 
2  2  𝜓 
(𝜑 → 𝜓) 
∧E_{L} 1 
simpld 474 1 
3  11 
(𝜒 ∨ 𝜃) 
(𝜑 → (𝜒 ∨ 𝜃)) 
∧E_{R} 1 
simprd 478 1 
4  4 
... 𝜒 
((𝜑 ∧ 𝜒) → 𝜒) 
ND hypothesis assumption 
simpr 476 
5  5 
... (𝜓 ∧ 𝜒) 
((𝜑 ∧ 𝜒) → (𝜓 ∧ 𝜒)) 
∧I 2,4 
jca 553 3,4 
6  6 
... ((𝜓 ∧ 𝜒) ∨ (𝜓 ∧ 𝜃)) 
((𝜑 ∧ 𝜒) → ((𝜓 ∧ 𝜒) ∨ (𝜓 ∧ 𝜃))) 
∨I_{R} 5 
orcd 406 5 
7  8 
... 𝜃 
((𝜑 ∧ 𝜃) → 𝜃) 
ND hypothesis assumption 
simpr 476 
8  9 
... (𝜓 ∧ 𝜃) 
((𝜑 ∧ 𝜃) → (𝜓 ∧ 𝜃)) 
∧I 2,7 
jca 553 7,8 
9  10 
... ((𝜓 ∧ 𝜒) ∨ (𝜓 ∧ 𝜃)) 
((𝜑 ∧ 𝜃) → ((𝜓 ∧ 𝜒) ∨ (𝜓 ∧ 𝜃))) 
∨I_{L} 8 
olcd 407 9 
10  12 
((𝜓 ∧ 𝜒) ∨ (𝜓 ∧ 𝜃)) 
(𝜑 → ((𝜓 ∧ 𝜒) ∨ (𝜓 ∧ 𝜃))) 
∨E 3,6,9 
mpjaodan 823 6,10,11 
The original used Latin letters; we have replaced them with
Greek letters to follow Metamath naming conventions and so that
it is easier to follow the Metamath translation.
The Metamath lineforline translation of this
natural deduction approach precedes every line with an antecedent
including 𝜑 and uses the Metamath equivalents
of the natural deduction rules.
To add an assumption, the antecedent is modified to include it
(typically by using adantr 480; simpr 476 is useful when you want to
depend directly on the new assumption).
Below is the final metamath proof (which reorders some steps).
A much more efficient proof is exnatded9.202 26667.
(Contributed by David A. Wheeler, 19Feb2017.)
(Proof modification is discouraged.) (New usage is discouraged.)

⊢ (𝜑 → (𝜓 ∧ (𝜒 ∨ 𝜃))) ⇒ ⊢ (𝜑 → ((𝜓 ∧ 𝜒) ∨ (𝜓 ∧ 𝜃))) 

Theorem  exnatded9.202 26667 
A more efficient proof of Theorem 9.20 of [Clemente] p. 45. Compare
with exnatded9.20 26666. (Contributed by David A. Wheeler,
19Feb2017.)
(Proof modification is discouraged.) (New usage is discouraged.)

⊢ (𝜑 → (𝜓 ∧ (𝜒 ∨ 𝜃))) ⇒ ⊢ (𝜑 → ((𝜓 ∧ 𝜒) ∨ (𝜓 ∧ 𝜃))) 

Theorem  exnatded9.26 26668* 
Theorem 9.26 of [Clemente] p. 45, translated line by line using an
interpretation of natural deduction in Metamath. This proof has some
additional complications due to the fact that Metamath's existential
elimination rule does not change bound variables, so we need to verify
that 𝑥 is bound in the conclusion.
For information about ND and Metamath, see the
page on Deduction Form and Natural Deduction
in Metamath Proof Explorer.
The original proof, which uses Fitch style, was written as follows
(the leading "..." shows an embedded ND hypothesis, beginning with
the initial assumption of the ND hypothesis):
#  MPE#  ND Expression 
MPE Translation  ND Rationale 
MPE Rationale 
1  3  ∃𝑥∀𝑦𝜓(𝑥, 𝑦) 
(𝜑 → ∃𝑥∀𝑦𝜓) 
Given 
$e. 
2  6  ... ∀𝑦𝜓(𝑥, 𝑦) 
((𝜑 ∧ ∀𝑦𝜓) → ∀𝑦𝜓) 
ND hypothesis assumption 
simpr 476. Later statements will have this scope. 
3  7;5,4  ... 𝜓(𝑥, 𝑦) 
((𝜑 ∧ ∀𝑦𝜓) → 𝜓) 
∀E 2,y 
spsbcd 3416 (∀E), 5,6. To use it we need a1i 11 and vex 3176.
This could be immediately done with 19.21bi 2047, but we want to show
the general approach for substitution.

4  12;8,9,10,11  ... ∃𝑥𝜓(𝑥, 𝑦) 
((𝜑 ∧ ∀𝑦𝜓) → ∃𝑥𝜓) 
∃I 3,a 
spesbcd 3488 (∃I), 11.
To use it we need sylibr 223, which in turn requires sylib 207 and
two uses of sbcid 3419.
This could be more immediately done using 19.8a 2039, but we want to show
the general approach for substitution.

5  13;1,2  ∃𝑥𝜓(𝑥, 𝑦) 
(𝜑 → ∃𝑥𝜓)  ∃E 1,2,4,a 
exlimdd 2075 (∃E), 1,2,3,12.
We'll need supporting
assertions that the variable is free (not bound),
as provided in nfv 1830 and nfe1 2014 (MPE# 1,2) 
6  14  ∀𝑦∃𝑥𝜓(𝑥, 𝑦) 
(𝜑 → ∀𝑦∃𝑥𝜓) 
∀I 5 
alrimiv 1842 (∀I), 13 
The original used Latin letters for predicates;
we have replaced them with
Greek letters to follow Metamath naming conventions and so that
it is easier to follow the Metamath translation.
The Metamath lineforline translation of this
natural deduction approach precedes every line with an antecedent
including 𝜑 and uses the Metamath equivalents
of the natural deduction rules.
Below is the final metamath proof (which reorders some steps).
Note that in the original proof, 𝜓(𝑥, 𝑦) has explicit
parameters. In Metamath, these parameters are always implicit, and the
parameters upon which a wff variable can depend are recorded in the
"allowed substitution hints" below.
A much more efficient proof, using more of Metamath and MPE's
capabilities, is shown in exnatded9.262 26669.
(Contributed by Mario Carneiro, 9Feb2017.)
(Revised by David A. Wheeler, 18Feb2017.)
(Proof modification is discouraged.) (New usage is discouraged.)

⊢ (𝜑 → ∃𝑥∀𝑦𝜓) ⇒ ⊢ (𝜑 → ∀𝑦∃𝑥𝜓) 

Theorem  exnatded9.262 26669* 
A more efficient proof of Theorem 9.26 of [Clemente] p. 45. Compare
with exnatded9.26 26668. (Contributed by Mario Carneiro,
9Feb2017.)
(Proof modification is discouraged.) (New usage is discouraged.)

⊢ (𝜑 → ∃𝑥∀𝑦𝜓) ⇒ ⊢ (𝜑 → ∀𝑦∃𝑥𝜓) 

18.1.4 Definitional examples


Theorem  exor 26670 
Example for dfor 384. Example by David A. Wheeler. (Contributed
by Mario
Carneiro, 9May2015.)

⊢ (2 = 3 ∨ 4 = 4) 

Theorem  exan 26671 
Example for dfan 385. Example by David A. Wheeler. (Contributed
by Mario
Carneiro, 9May2015.)

⊢ (2 = 2 ∧ 3 = 3) 

Theorem  exdif 26672 
Example for dfdif 3543. Example by David A. Wheeler. (Contributed
by
Mario Carneiro, 6May2015.)

⊢ ({1, 3} ∖ {1, 8}) =
{3} 

Theorem  exun 26673 
Example for dfun 3545. Example by David A. Wheeler. (Contributed
by Mario
Carneiro, 6May2015.)

⊢ ({1, 3} ∪ {1, 8}) = {1, 3,
8} 

Theorem  exin 26674 
Example for dfin 3547. Example by David A. Wheeler. (Contributed
by Mario
Carneiro, 6May2015.)

⊢ ({1, 3} ∩ {1, 8}) = {1} 

Theorem  exuni 26675 
Example for dfuni 4373. Example by David A. Wheeler. (Contributed
by
Mario Carneiro, 2Jul2016.)

⊢ ∪ {{1, 3}, {1,
8}} = {1, 3, 8} 

Theorem  exss 26676 
Example for dfss 3554. Example by David A. Wheeler. (Contributed
by Mario
Carneiro, 6May2015.)

⊢ {1, 2} ⊆ {1, 2, 3} 

Theorem  expss 26677 
Example for dfpss 3556. Example by David A. Wheeler. (Contributed
by
Mario Carneiro, 6May2015.)

⊢ {1, 2} ⊊ {1, 2, 3} 

Theorem  expw 26678 
Example for dfpw 4110. Example by David A. Wheeler. (Contributed
by Mario
Carneiro, 2Jul2016.)

⊢ (𝐴 = {3, 5, 7} → 𝒫 𝐴 = (({∅} ∪ {{3}, {5},
{7}}) ∪ ({{3, 5}, {3, 7}, {5, 7}} ∪ {{3, 5, 7}}))) 

Theorem  expr 26679 
Example for dfpr 4128. (Contributed by Mario Carneiro,
7May2015.)

⊢ (𝐴 ∈ {1, 1} → (𝐴↑2) = 1) 

Theorem  exbr 26680 
Example for dfbr 4584. Example by David A. Wheeler. (Contributed
by Mario
Carneiro, 6May2015.)

⊢ (𝑅 = {⟨2, 6⟩, ⟨3, 9⟩}
→ 3𝑅9) 

Theorem  exopab 26681* 
Example for dfopab 4644. Example by David A. Wheeler. (Contributed
by
Mario Carneiro, 18Jun2015.)

⊢ (𝑅 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ (𝑥 + 1) = 𝑦)} → 3𝑅4) 

Theorem  exeprel 26682 
Example for dfeprel 4949. Example by David A. Wheeler. (Contributed
by
Mario Carneiro, 18Jun2015.)

⊢ 5 E {1, 5} 

Theorem  exid 26683 
Example for dfid 4953. Example by David A. Wheeler. (Contributed
by Mario
Carneiro, 18Jun2015.)

⊢ (5 I 5 ∧ ¬ 4 I 5) 

Theorem  expo 26684 
Example for dfpo 4959. Example by David A. Wheeler. (Contributed
by Mario
Carneiro, 18Jun2015.)

⊢ ( < Po ℝ ∧ ¬ ≤ Po
ℝ) 

Theorem  exxp 26685 
Example for dfxp 5044. Example by David A. Wheeler. (Contributed
by Mario
Carneiro, 7May2015.)

⊢ ({1, 5} × {2, 7}) = ({⟨1,
2⟩, ⟨1, 7⟩} ∪ {⟨5, 2⟩, ⟨5,
7⟩}) 

Theorem  excnv 26686 
Example for dfcnv 5046. Example by David A. Wheeler. (Contributed
by
Mario Carneiro, 6May2015.)

⊢ ^{◡}{⟨2, 6⟩, ⟨3, 9⟩} =
{⟨6, 2⟩, ⟨9, 3⟩} 

Theorem  exco 26687 
Example for dfco 5047. Example by David A. Wheeler. (Contributed
by Mario
Carneiro, 7May2015.)

⊢ ((exp ∘ cos)‘0) =
e 

Theorem  exdm 26688 
Example for dfdm 5048. Example by David A. Wheeler. (Contributed
by Mario
Carneiro, 7May2015.)

⊢ (𝐹 = {⟨2, 6⟩, ⟨3, 9⟩}
→ dom 𝐹 = {2,
3}) 

Theorem  exrn 26689 
Example for dfrn 5049. Example by David A. Wheeler. (Contributed
by Mario
Carneiro, 7May2015.)

⊢ (𝐹 = {⟨2, 6⟩, ⟨3, 9⟩}
→ ran 𝐹 = {6,
9}) 

Theorem  exres 26690 
Example for dfres 5050. Example by David A. Wheeler. (Contributed
by
Mario Carneiro, 7May2015.)

⊢ ((𝐹 = {⟨2, 6⟩, ⟨3, 9⟩}
∧ 𝐵 = {1, 2}) →
(𝐹 ↾ 𝐵) = {⟨2,
6⟩}) 

Theorem  exima 26691 
Example for dfima 5051. Example by David A. Wheeler. (Contributed
by
Mario Carneiro, 7May2015.)

⊢ ((𝐹 = {⟨2, 6⟩, ⟨3, 9⟩}
∧ 𝐵 = {1, 2}) →
(𝐹 “ 𝐵) = {6}) 

Theorem  exfv 26692 
Example for dffv 5812. Example by David A. Wheeler. (Contributed
by Mario
Carneiro, 7May2015.)

⊢ (𝐹 = {⟨2, 6⟩, ⟨3, 9⟩}
→ (𝐹‘3) =
9) 

Theorem  ex1st 26693 
Example for df1st 7059. Example by David A. Wheeler. (Contributed
by
Mario Carneiro, 18Jun2015.)

⊢ (1^{st} ‘⟨3, 4⟩) =
3 

Theorem  ex2nd 26694 
Example for df2nd 7060. Example by David A. Wheeler. (Contributed
by
Mario Carneiro, 18Jun2015.)

⊢ (2^{nd} ‘⟨3, 4⟩) =
4 

Theorem  1kp2ke3k 26695 
Example for dfdec 11370, 1000 + 2000 = 3000.
This proof disproves (by counterexample) the assertion of Hao Wang, who
stated, "There is a theorem in the primitive notation of set theory
that
corresponds to the arithmetic theorem 1000 + 2000 = 3000. The formula
would be forbiddingly long... even if (one) knows the definitions and is
asked to simplify the long formula according to them, chances are he will
make errors and arrive at some incorrect result." (Hao Wang,
"Theory and
practice in mathematics" , In Thomas Tymoczko, editor, New
Directions in
the Philosophy of Mathematics, pp 129152, Birkauser Boston, Inc.,
Boston, 1986. (QA8.6.N48). The quote itself is on page 140.)
This is noted in Metamath: A Computer Language for Pure
Mathematics by
Norman Megill (2007) section 1.1.3. Megill then states, "A number of
writers have conveyed the impression that the kind of absolute rigor
provided by Metamath is an impossible dream, suggesting that a complete,
formal verification of a typical theorem would take millions of steps in
untold volumes of books... These writers assume, however, that in order
to achieve the kind of complete formal verification they desire one must
break down a proof into individual primitive steps that make direct
reference to the axioms. This is not necessary. There is no reason not
to make use of previously proved theorems rather than proving them over
and over... A hierarchy of theorems and definitions permits an
exponential growth in the formula sizes and primitive proof steps to be
described with only a linear growth in the number of symbols used. Of
course, this is how ordinary informal mathematics is normally done anyway,
but with Metamath it can be done with absolute rigor and precision."
The proof here starts with (2 + 1) = 3, commutes
it, and repeatedly
multiplies both sides by ten. This is certainly longer than traditional
mathematical proofs, e.g., there are a number of steps explicitly shown
here to show that we're allowed to do operations such as multiplication.
However, while longer, the proof is clearly a manageable size  even
though every step is rigorously derived all the way back to the primitive
notions of set theory and logic. And while there's a risk of making
errors, the many independent verifiers make it much less likely that an
incorrect result will be accepted.
This proof heavily relies on the decimal constructor dfdec 11370 developed by
Mario Carneiro in 2015. The underlying Metamath language has an
intentionally very small set of primitives; it doesn't even have a
builtin construct for numbers. Instead, the digits are defined using
these primitives, and the decimal constructor is used to make it easy to
express larger numbers as combinations of digits.
(Contributed by David A. Wheeler, 29Jun2016.) (Shortened by Mario
Carneiro using the arithmetic algorithm in mmj2, 30Jun2016.)

⊢ (;;;1000 + ;;;2000) = ;;;3000 

Theorem  exfl 26696 
Example for dffl 12455. Example by David A. Wheeler. (Contributed
by Mario
Carneiro, 18Jun2015.)

⊢ ((⌊‘(3 / 2)) = 1 ∧
(⌊‘(3 / 2)) = 2) 

Theorem  exceil 26697 
Example for dfceil 12456. (Contributed by AV, 4Sep2021.)

⊢ ((⌈‘(3 / 2)) = 2 ∧
(⌈‘(3 / 2)) = 1) 

Theorem  exmod 26698 
Example for dfmod 12531. (Contributed by AV, 3Sep2021.)

⊢ ((5 mod 3) = 2 ∧ (7 mod 2) =
1) 

Theorem  exexp 26699 
Example for dfexp 12723. (Contributed by AV, 4Sep2021.)

⊢ ((5↑2) = ;25 ∧ (3↑2) = (1 / 9)) 

Theorem  exfac 26700 
Example for dffac 12923. (Contributed by AV, 4Sep2021.)

⊢ (!‘5) = ;;120 