numclwlk1lem2fo - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > numclwlk1lem2fo		Structured version Visualization version GIF version

Theorem numclwlk1lem2fo 26622

Description: T is an onto function. (Contributed by Alexander van der Vekens, 20-Sep-2018.)

**Hypotheses**
Ref	Expression
numclwwlk.c	⊢ 𝐶 = (𝑛 ∈ ℕ₀ ↦ ((𝑉 ClWWalksN 𝐸)‘𝑛))
numclwwlk.f	⊢ 𝐹 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ₀ ↦ {𝑤 ∈ (𝐶‘𝑛) ∣ (𝑤‘0) = 𝑣})
numclwwlk.g	⊢ 𝐺 = (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ_≥‘2) ↦ {𝑤 ∈ (𝐶‘𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) = (𝑤‘0))})
numclwwlk.t	⊢ 𝑇 = (𝑤 ∈ (𝑋𝐺𝑁) ↦ ⟨(𝑤 substr ⟨0, (𝑁 − 2)⟩), (𝑤‘(𝑁 − 1))⟩)

**Assertion**
Ref	Expression
numclwlk1lem2fo	⊢ ((𝑉 USGrph 𝐸 ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3)) → 𝑇:(𝑋𝐺𝑁)–onto→((𝑋𝐹(𝑁 − 2)) × (⟨𝑉, 𝐸⟩ Neighbors 𝑋)))

Distinct variable groups: 𝑛,𝐸 𝑛,𝑁 𝑛,𝑉 𝑤,𝐶 𝑤,𝑁 𝐶,𝑛,𝑣,𝑤 𝑣,𝑁 𝑛,𝑋,𝑣,𝑤 𝑣,𝑉 𝑤,𝐸 𝑤,𝑉 𝑤,𝐹 𝑤,𝐺 Allowed substitution hints: 𝑇(𝑤,𝑣,𝑛) 𝐸(𝑣) 𝐹(𝑣,𝑛) 𝐺(𝑣,𝑛)

Proof of Theorem numclwlk1lem2fo Dummy variables 𝑖 𝑥 𝑝 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		numclwwlk.c	. . 3 ⊢ 𝐶 = (𝑛 ∈ ℕ₀ ↦ ((𝑉 ClWWalksN 𝐸)‘𝑛))
2		numclwwlk.f	. . 3 ⊢ 𝐹 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ₀ ↦ {𝑤 ∈ (𝐶‘𝑛) ∣ (𝑤‘0) = 𝑣})
3		numclwwlk.g	. . 3 ⊢ 𝐺 = (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ_≥‘2) ↦ {𝑤 ∈ (𝐶‘𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) = (𝑤‘0))})
4		numclwwlk.t	. . 3 ⊢ 𝑇 = (𝑤 ∈ (𝑋𝐺𝑁) ↦ ⟨(𝑤 substr ⟨0, (𝑁 − 2)⟩), (𝑤‘(𝑁 − 1))⟩)
5	1, 2, 3, 4	numclwlk1lem2f 26619	. 2 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3)) → 𝑇:(𝑋𝐺𝑁)⟶((𝑋𝐹(𝑁 − 2)) × (⟨𝑉, 𝐸⟩ Neighbors 𝑋)))
6		elxp 5055	. . . . 5 ⊢ (𝑝 ∈ ((𝑋𝐹(𝑁 − 2)) × (⟨𝑉, 𝐸⟩ Neighbors 𝑋)) ↔ ∃𝑎∃𝑏(𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑎 ∈ (𝑋𝐹(𝑁 − 2)) ∧ 𝑏 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑋))))
7	1, 2, 3	numclwlk1lem2foa 26618	. . . . . . . . . . 11 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3)) → ((𝑎 ∈ (𝑋𝐹(𝑁 − 2)) ∧ 𝑏 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑋)) → ((𝑎 ++ ⟨“𝑋”⟩) ++ ⟨“𝑏”⟩) ∈ (𝑋𝐺𝑁)))
8	7	com12 32	. . . . . . . . . 10 ⊢ ((𝑎 ∈ (𝑋𝐹(𝑁 − 2)) ∧ 𝑏 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑋)) → ((𝑉 USGrph 𝐸 ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3)) → ((𝑎 ++ ⟨“𝑋”⟩) ++ ⟨“𝑏”⟩) ∈ (𝑋𝐺𝑁)))
9	8	adantl 481	. . . . . . . . 9 ⊢ ((𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑎 ∈ (𝑋𝐹(𝑁 − 2)) ∧ 𝑏 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑋))) → ((𝑉 USGrph 𝐸 ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3)) → ((𝑎 ++ ⟨“𝑋”⟩) ++ ⟨“𝑏”⟩) ∈ (𝑋𝐺𝑁)))
10	9	imp 444	. . . . . . . 8 ⊢ (((𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑎 ∈ (𝑋𝐹(𝑁 − 2)) ∧ 𝑏 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑋))) ∧ (𝑉 USGrph 𝐸 ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3))) → ((𝑎 ++ ⟨“𝑋”⟩) ++ ⟨“𝑏”⟩) ∈ (𝑋𝐺𝑁))
11		simpl 472	. . . . . . . . 9 ⊢ ((((𝑎 ++ ⟨“𝑋”⟩) ++ ⟨“𝑏”⟩) ∈ (𝑋𝐺𝑁) ∧ ((𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑎 ∈ (𝑋𝐹(𝑁 − 2)) ∧ 𝑏 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑋))) ∧ (𝑉 USGrph 𝐸 ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3)))) → ((𝑎 ++ ⟨“𝑋”⟩) ++ ⟨“𝑏”⟩) ∈ (𝑋𝐺𝑁))
12		fveq2 6103	. . . . . . . . . . 11 ⊢ (𝑥 = ((𝑎 ++ ⟨“𝑋”⟩) ++ ⟨“𝑏”⟩) → (𝑇‘𝑥) = (𝑇‘((𝑎 ++ ⟨“𝑋”⟩) ++ ⟨“𝑏”⟩)))
13	12	eqeq2d 2620	. . . . . . . . . 10 ⊢ (𝑥 = ((𝑎 ++ ⟨“𝑋”⟩) ++ ⟨“𝑏”⟩) → (𝑝 = (𝑇‘𝑥) ↔ 𝑝 = (𝑇‘((𝑎 ++ ⟨“𝑋”⟩) ++ ⟨“𝑏”⟩))))
14		simprr 792	. . . . . . . . . . . 12 ⊢ ((((𝑎 ++ ⟨“𝑋”⟩) ++ ⟨“𝑏”⟩) ∈ (𝑋𝐺𝑁) ∧ ((𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑎 ∈ (𝑋𝐹(𝑁 − 2)) ∧ 𝑏 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑋))) ∧ (𝑉 USGrph 𝐸 ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3)))) → (𝑉 USGrph 𝐸 ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3)))
15	1, 2, 3, 4	numclwlk1lem2fv 26620	. . . . . . . . . . . 12 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3)) → (((𝑎 ++ ⟨“𝑋”⟩) ++ ⟨“𝑏”⟩) ∈ (𝑋𝐺𝑁) → (𝑇‘((𝑎 ++ ⟨“𝑋”⟩) ++ ⟨“𝑏”⟩)) = ⟨(((𝑎 ++ ⟨“𝑋”⟩) ++ ⟨“𝑏”⟩) substr ⟨0, (𝑁 − 2)⟩), (((𝑎 ++ ⟨“𝑋”⟩) ++ ⟨“𝑏”⟩)‘(𝑁 − 1))⟩))
16	14, 11, 15	sylc 63	. . . . . . . . . . 11 ⊢ ((((𝑎 ++ ⟨“𝑋”⟩) ++ ⟨“𝑏”⟩) ∈ (𝑋𝐺𝑁) ∧ ((𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑎 ∈ (𝑋𝐹(𝑁 − 2)) ∧ 𝑏 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑋))) ∧ (𝑉 USGrph 𝐸 ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3)))) → (𝑇‘((𝑎 ++ ⟨“𝑋”⟩) ++ ⟨“𝑏”⟩)) = ⟨(((𝑎 ++ ⟨“𝑋”⟩) ++ ⟨“𝑏”⟩) substr ⟨0, (𝑁 − 2)⟩), (((𝑎 ++ ⟨“𝑋”⟩) ++ ⟨“𝑏”⟩)‘(𝑁 − 1))⟩)
17	16	eqeq2d 2620	. . . . . . . . . 10 ⊢ ((((𝑎 ++ ⟨“𝑋”⟩) ++ ⟨“𝑏”⟩) ∈ (𝑋𝐺𝑁) ∧ ((𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑎 ∈ (𝑋𝐹(𝑁 − 2)) ∧ 𝑏 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑋))) ∧ (𝑉 USGrph 𝐸 ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3)))) → (𝑝 = (𝑇‘((𝑎 ++ ⟨“𝑋”⟩) ++ ⟨“𝑏”⟩)) ↔ 𝑝 = ⟨(((𝑎 ++ ⟨“𝑋”⟩) ++ ⟨“𝑏”⟩) substr ⟨0, (𝑁 − 2)⟩), (((𝑎 ++ ⟨“𝑋”⟩) ++ ⟨“𝑏”⟩)‘(𝑁 − 1))⟩))
18	13, 17	sylan9bbr 733	. . . . . . . . 9 ⊢ (((((𝑎 ++ ⟨“𝑋”⟩) ++ ⟨“𝑏”⟩) ∈ (𝑋𝐺𝑁) ∧ ((𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑎 ∈ (𝑋𝐹(𝑁 − 2)) ∧ 𝑏 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑋))) ∧ (𝑉 USGrph 𝐸 ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3)))) ∧ 𝑥 = ((𝑎 ++ ⟨“𝑋”⟩) ++ ⟨“𝑏”⟩)) → (𝑝 = (𝑇‘𝑥) ↔ 𝑝 = ⟨(((𝑎 ++ ⟨“𝑋”⟩) ++ ⟨“𝑏”⟩) substr ⟨0, (𝑁 − 2)⟩), (((𝑎 ++ ⟨“𝑋”⟩) ++ ⟨“𝑏”⟩)‘(𝑁 − 1))⟩))
19		simprll 798	. . . . . . . . . 10 ⊢ ((((𝑎 ++ ⟨“𝑋”⟩) ++ ⟨“𝑏”⟩) ∈ (𝑋𝐺𝑁) ∧ ((𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑎 ∈ (𝑋𝐹(𝑁 − 2)) ∧ 𝑏 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑋))) ∧ (𝑉 USGrph 𝐸 ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3)))) → 𝑝 = ⟨𝑎, 𝑏⟩)
20		nbgraisvtx 25960	. . . . . . . . . . . . . . . . 17 ⊢ (𝑉 USGrph 𝐸 → (𝑏 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑋) → 𝑏 ∈ 𝑉))
21	20	3ad2ant1 1075	. . . . . . . . . . . . . . . 16 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3)) → (𝑏 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑋) → 𝑏 ∈ 𝑉))
22		simp1 1054	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3)) → 𝑉 USGrph 𝐸)
23		uzuzle23 11605	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑁 ∈ (ℤ_≥‘3) → 𝑁 ∈ (ℤ_≥‘2))
24		uznn0sub 11595	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑁 ∈ (ℤ_≥‘2) → (𝑁 − 2) ∈ ℕ₀)
25	23, 24	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑁 ∈ (ℤ_≥‘3) → (𝑁 − 2) ∈ ℕ₀)
26	25	3ad2ant3 1077	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3)) → (𝑁 − 2) ∈ ℕ₀)
27		simp2 1055	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3)) → 𝑋 ∈ 𝑉)
28	1, 2	numclwwlkovfel2 26610	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑉 USGrph 𝐸 ∧ (𝑁 − 2) ∈ ℕ₀ ∧ 𝑋 ∈ 𝑉) → (𝑎 ∈ (𝑋𝐹(𝑁 − 2)) ↔ ((𝑎 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑎) − 1)){(𝑎‘𝑖), (𝑎‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘𝑎), (𝑎‘0)} ∈ ran 𝐸) ∧ (#‘𝑎) = (𝑁 − 2) ∧ (𝑎‘0) = 𝑋)))
29	22, 26, 27, 28	syl3anc 1318	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3)) → (𝑎 ∈ (𝑋𝐹(𝑁 − 2)) ↔ ((𝑎 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑎) − 1)){(𝑎‘𝑖), (𝑎‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘𝑎), (𝑎‘0)} ∈ ran 𝐸) ∧ (#‘𝑎) = (𝑁 − 2) ∧ (𝑎‘0) = 𝑋)))
30		df-3an 1033	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑎 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑎) − 1)){(𝑎‘𝑖), (𝑎‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘𝑎), (𝑎‘0)} ∈ ran 𝐸) ∧ (#‘𝑎) = (𝑁 − 2) ∧ (𝑎‘0) = 𝑋) ↔ (((𝑎 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑎) − 1)){(𝑎‘𝑖), (𝑎‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘𝑎), (𝑎‘0)} ∈ ran 𝐸) ∧ (#‘𝑎) = (𝑁 − 2)) ∧ (𝑎‘0) = 𝑋))
31	29, 30	syl6bb 275	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3)) → (𝑎 ∈ (𝑋𝐹(𝑁 − 2)) ↔ (((𝑎 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑎) − 1)){(𝑎‘𝑖), (𝑎‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘𝑎), (𝑎‘0)} ∈ ran 𝐸) ∧ (#‘𝑎) = (𝑁 − 2)) ∧ (𝑎‘0) = 𝑋)))
32		simplll 794	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝑎 ∈ Word 𝑉 ∧ (#‘𝑎) = (𝑁 − 2)) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3))) ∧ 𝑏 ∈ 𝑉) → 𝑎 ∈ Word 𝑉)
33		s1cl 13235	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑋 ∈ 𝑉 → ⟨“𝑋”⟩ ∈ Word 𝑉)
34	33	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3)) → ⟨“𝑋”⟩ ∈ Word 𝑉)
35	34	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑎 ∈ Word 𝑉 ∧ (#‘𝑎) = (𝑁 − 2)) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3))) → ⟨“𝑋”⟩ ∈ Word 𝑉)
36	35	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝑎 ∈ Word 𝑉 ∧ (#‘𝑎) = (𝑁 − 2)) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3))) ∧ 𝑏 ∈ 𝑉) → ⟨“𝑋”⟩ ∈ Word 𝑉)
37		s1cl 13235	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑏 ∈ 𝑉 → ⟨“𝑏”⟩ ∈ Word 𝑉)
38	37	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝑎 ∈ Word 𝑉 ∧ (#‘𝑎) = (𝑁 − 2)) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3))) ∧ 𝑏 ∈ 𝑉) → ⟨“𝑏”⟩ ∈ Word 𝑉)
39		ccatass 13224	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑎 ∈ Word 𝑉 ∧ ⟨“𝑋”⟩ ∈ Word 𝑉 ∧ ⟨“𝑏”⟩ ∈ Word 𝑉) → ((𝑎 ++ ⟨“𝑋”⟩) ++ ⟨“𝑏”⟩) = (𝑎 ++ (⟨“𝑋”⟩ ++ ⟨“𝑏”⟩)))
40	39	oveq1d 6564	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑎 ∈ Word 𝑉 ∧ ⟨“𝑋”⟩ ∈ Word 𝑉 ∧ ⟨“𝑏”⟩ ∈ Word 𝑉) → (((𝑎 ++ ⟨“𝑋”⟩) ++ ⟨“𝑏”⟩) substr ⟨0, (𝑁 − 2)⟩) = ((𝑎 ++ (⟨“𝑋”⟩ ++ ⟨“𝑏”⟩)) substr ⟨0, (𝑁 − 2)⟩))
41	32, 36, 38, 40	syl3anc 1318	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝑎 ∈ Word 𝑉 ∧ (#‘𝑎) = (𝑁 − 2)) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3))) ∧ 𝑏 ∈ 𝑉) → (((𝑎 ++ ⟨“𝑋”⟩) ++ ⟨“𝑏”⟩) substr ⟨0, (𝑁 − 2)⟩) = ((𝑎 ++ (⟨“𝑋”⟩ ++ ⟨“𝑏”⟩)) substr ⟨0, (𝑁 − 2)⟩))
42		ccatcl 13212	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((⟨“𝑋”⟩ ∈ Word 𝑉 ∧ ⟨“𝑏”⟩ ∈ Word 𝑉) → (⟨“𝑋”⟩ ++ ⟨“𝑏”⟩) ∈ Word 𝑉)
43	35, 37, 42	syl2an 493	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝑎 ∈ Word 𝑉 ∧ (#‘𝑎) = (𝑁 − 2)) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3))) ∧ 𝑏 ∈ 𝑉) → (⟨“𝑋”⟩ ++ ⟨“𝑏”⟩) ∈ Word 𝑉)
44		simpr 476	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑎 ∈ Word 𝑉 ∧ (#‘𝑎) = (𝑁 − 2)) → (#‘𝑎) = (𝑁 − 2))
45	44	eqcomd 2616	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑎 ∈ Word 𝑉 ∧ (#‘𝑎) = (𝑁 − 2)) → (𝑁 − 2) = (#‘𝑎))
46	45	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑎 ∈ Word 𝑉 ∧ (#‘𝑎) = (𝑁 − 2)) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3))) → (𝑁 − 2) = (#‘𝑎))
47	46	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝑎 ∈ Word 𝑉 ∧ (#‘𝑎) = (𝑁 − 2)) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3))) ∧ 𝑏 ∈ 𝑉) → (𝑁 − 2) = (#‘𝑎))
48		swrdccatid 13348	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑎 ∈ Word 𝑉 ∧ (⟨“𝑋”⟩ ++ ⟨“𝑏”⟩) ∈ Word 𝑉 ∧ (𝑁 − 2) = (#‘𝑎)) → ((𝑎 ++ (⟨“𝑋”⟩ ++ ⟨“𝑏”⟩)) substr ⟨0, (𝑁 − 2)⟩) = 𝑎)
49	32, 43, 47, 48	syl3anc 1318	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝑎 ∈ Word 𝑉 ∧ (#‘𝑎) = (𝑁 − 2)) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3))) ∧ 𝑏 ∈ 𝑉) → ((𝑎 ++ (⟨“𝑋”⟩ ++ ⟨“𝑏”⟩)) substr ⟨0, (𝑁 − 2)⟩) = 𝑎)
50	41, 49	eqtr2d 2645	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝑎 ∈ Word 𝑉 ∧ (#‘𝑎) = (𝑁 − 2)) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3))) ∧ 𝑏 ∈ 𝑉) → 𝑎 = (((𝑎 ++ ⟨“𝑋”⟩) ++ ⟨“𝑏”⟩) substr ⟨0, (𝑁 − 2)⟩))
51		ovex 6577	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑎 ++ ⟨“𝑋”⟩) ++ ⟨“𝑏”⟩) ∈ V
52		lsw 13204	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑎 ++ ⟨“𝑋”⟩) ++ ⟨“𝑏”⟩) ∈ V → ( lastS ‘((𝑎 ++ ⟨“𝑋”⟩) ++ ⟨“𝑏”⟩)) = (((𝑎 ++ ⟨“𝑋”⟩) ++ ⟨“𝑏”⟩)‘((#‘((𝑎 ++ ⟨“𝑋”⟩) ++ ⟨“𝑏”⟩)) − 1)))
53	51, 52	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ( lastS ‘((𝑎 ++ ⟨“𝑋”⟩) ++ ⟨“𝑏”⟩)) = (((𝑎 ++ ⟨“𝑋”⟩) ++ ⟨“𝑏”⟩)‘((#‘((𝑎 ++ ⟨“𝑋”⟩) ++ ⟨“𝑏”⟩)) − 1))
54		simpl 472	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑎 ∈ Word 𝑉 ∧ (#‘𝑎) = (𝑁 − 2)) → 𝑎 ∈ Word 𝑉)
55		ccatcl 13212	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑎 ∈ Word 𝑉 ∧ ⟨“𝑋”⟩ ∈ Word 𝑉) → (𝑎 ++ ⟨“𝑋”⟩) ∈ Word 𝑉)
56	54, 34, 55	syl2an 493	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑎 ∈ Word 𝑉 ∧ (#‘𝑎) = (𝑁 − 2)) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3))) → (𝑎 ++ ⟨“𝑋”⟩) ∈ Word 𝑉)
57		lswccats1 13263	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑎 ++ ⟨“𝑋”⟩) ∈ Word 𝑉 ∧ 𝑏 ∈ 𝑉) → ( lastS ‘((𝑎 ++ ⟨“𝑋”⟩) ++ ⟨“𝑏”⟩)) = 𝑏)
58	56, 57	sylan 487	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝑎 ∈ Word 𝑉 ∧ (#‘𝑎) = (𝑁 − 2)) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3))) ∧ 𝑏 ∈ 𝑉) → ( lastS ‘((𝑎 ++ ⟨“𝑋”⟩) ++ ⟨“𝑏”⟩)) = 𝑏)
59		ccatlen 13213	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝑎 ++ ⟨“𝑋”⟩) ∈ Word 𝑉 ∧ ⟨“𝑏”⟩ ∈ Word 𝑉) → (#‘((𝑎 ++ ⟨“𝑋”⟩) ++ ⟨“𝑏”⟩)) = ((#‘(𝑎 ++ ⟨“𝑋”⟩)) + (#‘⟨“𝑏”⟩)))
60	56, 37, 59	syl2an 493	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝑎 ∈ Word 𝑉 ∧ (#‘𝑎) = (𝑁 − 2)) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3))) ∧ 𝑏 ∈ 𝑉) → (#‘((𝑎 ++ ⟨“𝑋”⟩) ++ ⟨“𝑏”⟩)) = ((#‘(𝑎 ++ ⟨“𝑋”⟩)) + (#‘⟨“𝑏”⟩)))
61	54, 34	anim12i 588	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝑎 ∈ Word 𝑉 ∧ (#‘𝑎) = (𝑁 − 2)) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3))) → (𝑎 ∈ Word 𝑉 ∧ ⟨“𝑋”⟩ ∈ Word 𝑉))
62	61	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((𝑎 ∈ Word 𝑉 ∧ (#‘𝑎) = (𝑁 − 2)) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3))) ∧ 𝑏 ∈ 𝑉) → (𝑎 ∈ Word 𝑉 ∧ ⟨“𝑋”⟩ ∈ Word 𝑉))
63		ccatlen 13213	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑎 ∈ Word 𝑉 ∧ ⟨“𝑋”⟩ ∈ Word 𝑉) → (#‘(𝑎 ++ ⟨“𝑋”⟩)) = ((#‘𝑎) + (#‘⟨“𝑋”⟩)))
64	62, 63	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((𝑎 ∈ Word 𝑉 ∧ (#‘𝑎) = (𝑁 − 2)) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3))) ∧ 𝑏 ∈ 𝑉) → (#‘(𝑎 ++ ⟨“𝑋”⟩)) = ((#‘𝑎) + (#‘⟨“𝑋”⟩)))
65		s1len 13238	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (#‘⟨“𝑏”⟩) = 1
66	65	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((𝑎 ∈ Word 𝑉 ∧ (#‘𝑎) = (𝑁 − 2)) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3))) ∧ 𝑏 ∈ 𝑉) → (#‘⟨“𝑏”⟩) = 1)
67	64, 66	oveq12d 6567	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝑎 ∈ Word 𝑉 ∧ (#‘𝑎) = (𝑁 − 2)) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3))) ∧ 𝑏 ∈ 𝑉) → ((#‘(𝑎 ++ ⟨“𝑋”⟩)) + (#‘⟨“𝑏”⟩)) = (((#‘𝑎) + (#‘⟨“𝑋”⟩)) + 1))
68		s1len 13238	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (#‘⟨“𝑋”⟩) = 1
69	68	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3)) → (#‘⟨“𝑋”⟩) = 1)
70	44, 69	oveqan12d 6568	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝑎 ∈ Word 𝑉 ∧ (#‘𝑎) = (𝑁 − 2)) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3))) → ((#‘𝑎) + (#‘⟨“𝑋”⟩)) = ((𝑁 − 2) + 1))
71	70	oveq1d 6564	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝑎 ∈ Word 𝑉 ∧ (#‘𝑎) = (𝑁 − 2)) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3))) → (((#‘𝑎) + (#‘⟨“𝑋”⟩)) + 1) = (((𝑁 − 2) + 1) + 1))
72		eluzelcn 11575	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑁 ∈ (ℤ_≥‘3) → 𝑁 ∈ ℂ)
73		id 22	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (𝑁 ∈ ℂ → 𝑁 ∈ ℂ)
74		2cnd 10970	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (𝑁 ∈ ℂ → 2 ∈ ℂ)
75	73, 74	subcld 10271	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝑁 ∈ ℂ → (𝑁 − 2) ∈ ℂ)
76		1cnd 9935	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝑁 ∈ ℂ → 1 ∈ ℂ)
77	75, 76, 76	addassd 9941	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑁 ∈ ℂ → (((𝑁 − 2) + 1) + 1) = ((𝑁 − 2) + (1 + 1)))
78		1p1e2 11011	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (1 + 1) = 2
79	78	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝑁 ∈ ℂ → (1 + 1) = 2)
80	79	oveq2d 6565	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑁 ∈ ℂ → ((𝑁 − 2) + (1 + 1)) = ((𝑁 − 2) + 2))
81	77, 80	eqtrd 2644	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑁 ∈ ℂ → (((𝑁 − 2) + 1) + 1) = ((𝑁 − 2) + 2))
82	72, 81	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑁 ∈ (ℤ_≥‘3) → (((𝑁 − 2) + 1) + 1) = ((𝑁 − 2) + 2))
83		2cnd 10970	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑁 ∈ (ℤ_≥‘3) → 2 ∈ ℂ)
84	72, 83	npcand 10275	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑁 ∈ (ℤ_≥‘3) → ((𝑁 − 2) + 2) = 𝑁)
85	82, 84	eqtrd 2644	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑁 ∈ (ℤ_≥‘3) → (((𝑁 − 2) + 1) + 1) = 𝑁)
86	85	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3)) → (((𝑁 − 2) + 1) + 1) = 𝑁)
87	86	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝑎 ∈ Word 𝑉 ∧ (#‘𝑎) = (𝑁 − 2)) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3))) → (((𝑁 − 2) + 1) + 1) = 𝑁)
88	71, 87	eqtrd 2644	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝑎 ∈ Word 𝑉 ∧ (#‘𝑎) = (𝑁 − 2)) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3))) → (((#‘𝑎) + (#‘⟨“𝑋”⟩)) + 1) = 𝑁)
89	88	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝑎 ∈ Word 𝑉 ∧ (#‘𝑎) = (𝑁 − 2)) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3))) ∧ 𝑏 ∈ 𝑉) → (((#‘𝑎) + (#‘⟨“𝑋”⟩)) + 1) = 𝑁)
90	60, 67, 89	3eqtrd 2648	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝑎 ∈ Word 𝑉 ∧ (#‘𝑎) = (𝑁 − 2)) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3))) ∧ 𝑏 ∈ 𝑉) → (#‘((𝑎 ++ ⟨“𝑋”⟩) ++ ⟨“𝑏”⟩)) = 𝑁)
91	90	oveq1d 6564	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝑎 ∈ Word 𝑉 ∧ (#‘𝑎) = (𝑁 − 2)) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3))) ∧ 𝑏 ∈ 𝑉) → ((#‘((𝑎 ++ ⟨“𝑋”⟩) ++ ⟨“𝑏”⟩)) − 1) = (𝑁 − 1))
92	91	fveq2d 6107	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝑎 ∈ Word 𝑉 ∧ (#‘𝑎) = (𝑁 − 2)) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3))) ∧ 𝑏 ∈ 𝑉) → (((𝑎 ++ ⟨“𝑋”⟩) ++ ⟨“𝑏”⟩)‘((#‘((𝑎 ++ ⟨“𝑋”⟩) ++ ⟨“𝑏”⟩)) − 1)) = (((𝑎 ++ ⟨“𝑋”⟩) ++ ⟨“𝑏”⟩)‘(𝑁 − 1)))
93	53, 58, 92	3eqtr3a 2668	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝑎 ∈ Word 𝑉 ∧ (#‘𝑎) = (𝑁 − 2)) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3))) ∧ 𝑏 ∈ 𝑉) → 𝑏 = (((𝑎 ++ ⟨“𝑋”⟩) ++ ⟨“𝑏”⟩)‘(𝑁 − 1)))
94	50, 93	opeq12d 4348	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝑎 ∈ Word 𝑉 ∧ (#‘𝑎) = (𝑁 − 2)) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3))) ∧ 𝑏 ∈ 𝑉) → ⟨𝑎, 𝑏⟩ = ⟨(((𝑎 ++ ⟨“𝑋”⟩) ++ ⟨“𝑏”⟩) substr ⟨0, (𝑁 − 2)⟩), (((𝑎 ++ ⟨“𝑋”⟩) ++ ⟨“𝑏”⟩)‘(𝑁 − 1))⟩)
95	94	exp31 628	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑎 ∈ Word 𝑉 ∧ (#‘𝑎) = (𝑁 − 2)) → ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3)) → (𝑏 ∈ 𝑉 → ⟨𝑎, 𝑏⟩ = ⟨(((𝑎 ++ ⟨“𝑋”⟩) ++ ⟨“𝑏”⟩) substr ⟨0, (𝑁 − 2)⟩), (((𝑎 ++ ⟨“𝑋”⟩) ++ ⟨“𝑏”⟩)‘(𝑁 − 1))⟩)))
96	95	3ad2antl1 1216	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑎 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑎) − 1)){(𝑎‘𝑖), (𝑎‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘𝑎), (𝑎‘0)} ∈ ran 𝐸) ∧ (#‘𝑎) = (𝑁 − 2)) → ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3)) → (𝑏 ∈ 𝑉 → ⟨𝑎, 𝑏⟩ = ⟨(((𝑎 ++ ⟨“𝑋”⟩) ++ ⟨“𝑏”⟩) substr ⟨0, (𝑁 − 2)⟩), (((𝑎 ++ ⟨“𝑋”⟩) ++ ⟨“𝑏”⟩)‘(𝑁 − 1))⟩)))
97	96	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑎 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑎) − 1)){(𝑎‘𝑖), (𝑎‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘𝑎), (𝑎‘0)} ∈ ran 𝐸) ∧ (#‘𝑎) = (𝑁 − 2)) ∧ (𝑎‘0) = 𝑋) → ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3)) → (𝑏 ∈ 𝑉 → ⟨𝑎, 𝑏⟩ = ⟨(((𝑎 ++ ⟨“𝑋”⟩) ++ ⟨“𝑏”⟩) substr ⟨0, (𝑁 − 2)⟩), (((𝑎 ++ ⟨“𝑋”⟩) ++ ⟨“𝑏”⟩)‘(𝑁 − 1))⟩)))
98	97	com12 32	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3)) → ((((𝑎 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑎) − 1)){(𝑎‘𝑖), (𝑎‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘𝑎), (𝑎‘0)} ∈ ran 𝐸) ∧ (#‘𝑎) = (𝑁 − 2)) ∧ (𝑎‘0) = 𝑋) → (𝑏 ∈ 𝑉 → ⟨𝑎, 𝑏⟩ = ⟨(((𝑎 ++ ⟨“𝑋”⟩) ++ ⟨“𝑏”⟩) substr ⟨0, (𝑁 − 2)⟩), (((𝑎 ++ ⟨“𝑋”⟩) ++ ⟨“𝑏”⟩)‘(𝑁 − 1))⟩)))
99	98	3adant1 1072	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3)) → ((((𝑎 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑎) − 1)){(𝑎‘𝑖), (𝑎‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘𝑎), (𝑎‘0)} ∈ ran 𝐸) ∧ (#‘𝑎) = (𝑁 − 2)) ∧ (𝑎‘0) = 𝑋) → (𝑏 ∈ 𝑉 → ⟨𝑎, 𝑏⟩ = ⟨(((𝑎 ++ ⟨“𝑋”⟩) ++ ⟨“𝑏”⟩) substr ⟨0, (𝑁 − 2)⟩), (((𝑎 ++ ⟨“𝑋”⟩) ++ ⟨“𝑏”⟩)‘(𝑁 − 1))⟩)))
100	31, 99	sylbid 229	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3)) → (𝑎 ∈ (𝑋𝐹(𝑁 − 2)) → (𝑏 ∈ 𝑉 → ⟨𝑎, 𝑏⟩ = ⟨(((𝑎 ++ ⟨“𝑋”⟩) ++ ⟨“𝑏”⟩) substr ⟨0, (𝑁 − 2)⟩), (((𝑎 ++ ⟨“𝑋”⟩) ++ ⟨“𝑏”⟩)‘(𝑁 − 1))⟩)))
101	100	com23 84	. . . . . . . . . . . . . . . 16 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3)) → (𝑏 ∈ 𝑉 → (𝑎 ∈ (𝑋𝐹(𝑁 − 2)) → ⟨𝑎, 𝑏⟩ = ⟨(((𝑎 ++ ⟨“𝑋”⟩) ++ ⟨“𝑏”⟩) substr ⟨0, (𝑁 − 2)⟩), (((𝑎 ++ ⟨“𝑋”⟩) ++ ⟨“𝑏”⟩)‘(𝑁 − 1))⟩)))
102	21, 101	syld 46	. . . . . . . . . . . . . . 15 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3)) → (𝑏 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑋) → (𝑎 ∈ (𝑋𝐹(𝑁 − 2)) → ⟨𝑎, 𝑏⟩ = ⟨(((𝑎 ++ ⟨“𝑋”⟩) ++ ⟨“𝑏”⟩) substr ⟨0, (𝑁 − 2)⟩), (((𝑎 ++ ⟨“𝑋”⟩) ++ ⟨“𝑏”⟩)‘(𝑁 − 1))⟩)))
103	102	com13 86	. . . . . . . . . . . . . 14 ⊢ (𝑎 ∈ (𝑋𝐹(𝑁 − 2)) → (𝑏 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑋) → ((𝑉 USGrph 𝐸 ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3)) → ⟨𝑎, 𝑏⟩ = ⟨(((𝑎 ++ ⟨“𝑋”⟩) ++ ⟨“𝑏”⟩) substr ⟨0, (𝑁 − 2)⟩), (((𝑎 ++ ⟨“𝑋”⟩) ++ ⟨“𝑏”⟩)‘(𝑁 − 1))⟩)))
104	103	imp 444	. . . . . . . . . . . . 13 ⊢ ((𝑎 ∈ (𝑋𝐹(𝑁 − 2)) ∧ 𝑏 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑋)) → ((𝑉 USGrph 𝐸 ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3)) → ⟨𝑎, 𝑏⟩ = ⟨(((𝑎 ++ ⟨“𝑋”⟩) ++ ⟨“𝑏”⟩) substr ⟨0, (𝑁 − 2)⟩), (((𝑎 ++ ⟨“𝑋”⟩) ++ ⟨“𝑏”⟩)‘(𝑁 − 1))⟩))
105	104	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑎 ∈ (𝑋𝐹(𝑁 − 2)) ∧ 𝑏 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑋))) → ((𝑉 USGrph 𝐸 ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3)) → ⟨𝑎, 𝑏⟩ = ⟨(((𝑎 ++ ⟨“𝑋”⟩) ++ ⟨“𝑏”⟩) substr ⟨0, (𝑁 − 2)⟩), (((𝑎 ++ ⟨“𝑋”⟩) ++ ⟨“𝑏”⟩)‘(𝑁 − 1))⟩))
106	105	imp 444	. . . . . . . . . . 11 ⊢ (((𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑎 ∈ (𝑋𝐹(𝑁 − 2)) ∧ 𝑏 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑋))) ∧ (𝑉 USGrph 𝐸 ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3))) → ⟨𝑎, 𝑏⟩ = ⟨(((𝑎 ++ ⟨“𝑋”⟩) ++ ⟨“𝑏”⟩) substr ⟨0, (𝑁 − 2)⟩), (((𝑎 ++ ⟨“𝑋”⟩) ++ ⟨“𝑏”⟩)‘(𝑁 − 1))⟩)
107	106	adantl 481	. . . . . . . . . 10 ⊢ ((((𝑎 ++ ⟨“𝑋”⟩) ++ ⟨“𝑏”⟩) ∈ (𝑋𝐺𝑁) ∧ ((𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑎 ∈ (𝑋𝐹(𝑁 − 2)) ∧ 𝑏 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑋))) ∧ (𝑉 USGrph 𝐸 ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3)))) → ⟨𝑎, 𝑏⟩ = ⟨(((𝑎 ++ ⟨“𝑋”⟩) ++ ⟨“𝑏”⟩) substr ⟨0, (𝑁 − 2)⟩), (((𝑎 ++ ⟨“𝑋”⟩) ++ ⟨“𝑏”⟩)‘(𝑁 − 1))⟩)
108	19, 107	eqtrd 2644	. . . . . . . . 9 ⊢ ((((𝑎 ++ ⟨“𝑋”⟩) ++ ⟨“𝑏”⟩) ∈ (𝑋𝐺𝑁) ∧ ((𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑎 ∈ (𝑋𝐹(𝑁 − 2)) ∧ 𝑏 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑋))) ∧ (𝑉 USGrph 𝐸 ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3)))) → 𝑝 = ⟨(((𝑎 ++ ⟨“𝑋”⟩) ++ ⟨“𝑏”⟩) substr ⟨0, (𝑁 − 2)⟩), (((𝑎 ++ ⟨“𝑋”⟩) ++ ⟨“𝑏”⟩)‘(𝑁 − 1))⟩)
109	11, 18, 108	rspcedvd 3289	. . . . . . . 8 ⊢ ((((𝑎 ++ ⟨“𝑋”⟩) ++ ⟨“𝑏”⟩) ∈ (𝑋𝐺𝑁) ∧ ((𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑎 ∈ (𝑋𝐹(𝑁 − 2)) ∧ 𝑏 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑋))) ∧ (𝑉 USGrph 𝐸 ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3)))) → ∃𝑥 ∈ (𝑋𝐺𝑁)𝑝 = (𝑇‘𝑥))
110	10, 109	mpancom 700	. . . . . . 7 ⊢ (((𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑎 ∈ (𝑋𝐹(𝑁 − 2)) ∧ 𝑏 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑋))) ∧ (𝑉 USGrph 𝐸 ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3))) → ∃𝑥 ∈ (𝑋𝐺𝑁)𝑝 = (𝑇‘𝑥))
111	110	ex 449	. . . . . 6 ⊢ ((𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑎 ∈ (𝑋𝐹(𝑁 − 2)) ∧ 𝑏 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑋))) → ((𝑉 USGrph 𝐸 ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3)) → ∃𝑥 ∈ (𝑋𝐺𝑁)𝑝 = (𝑇‘𝑥)))
112	111	exlimivv 1847	. . . . 5 ⊢ (∃𝑎∃𝑏(𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑎 ∈ (𝑋𝐹(𝑁 − 2)) ∧ 𝑏 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑋))) → ((𝑉 USGrph 𝐸 ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3)) → ∃𝑥 ∈ (𝑋𝐺𝑁)𝑝 = (𝑇‘𝑥)))
113	6, 112	sylbi 206	. . . 4 ⊢ (𝑝 ∈ ((𝑋𝐹(𝑁 − 2)) × (⟨𝑉, 𝐸⟩ Neighbors 𝑋)) → ((𝑉 USGrph 𝐸 ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3)) → ∃𝑥 ∈ (𝑋𝐺𝑁)𝑝 = (𝑇‘𝑥)))
114	113	impcom 445	. . 3 ⊢ (((𝑉 USGrph 𝐸 ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3)) ∧ 𝑝 ∈ ((𝑋𝐹(𝑁 − 2)) × (⟨𝑉, 𝐸⟩ Neighbors 𝑋))) → ∃𝑥 ∈ (𝑋𝐺𝑁)𝑝 = (𝑇‘𝑥))
115	114	ralrimiva 2949	. 2 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3)) → ∀𝑝 ∈ ((𝑋𝐹(𝑁 − 2)) × (⟨𝑉, 𝐸⟩ Neighbors 𝑋))∃𝑥 ∈ (𝑋𝐺𝑁)𝑝 = (𝑇‘𝑥))
116		dffo3 6282	. 2 ⊢ (𝑇:(𝑋𝐺𝑁)–onto→((𝑋𝐹(𝑁 − 2)) × (⟨𝑉, 𝐸⟩ Neighbors 𝑋)) ↔ (𝑇:(𝑋𝐺𝑁)⟶((𝑋𝐹(𝑁 − 2)) × (⟨𝑉, 𝐸⟩ Neighbors 𝑋)) ∧ ∀𝑝 ∈ ((𝑋𝐹(𝑁 − 2)) × (⟨𝑉, 𝐸⟩ Neighbors 𝑋))∃𝑥 ∈ (𝑋𝐺𝑁)𝑝 = (𝑇‘𝑥)))
117	5, 115, 116	sylanbrc 695	1 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3)) → 𝑇:(𝑋𝐺𝑁)–onto→((𝑋𝐹(𝑁 − 2)) × (⟨𝑉, 𝐸⟩ Neighbors 𝑋)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∃wex 1695 ∈ wcel 1977 ∀wral 2896 ∃wrex 2897 {crab 2900 Vcvv 3173 {cpr 4127 ⟨cop 4131 class class class wbr 4583 ↦ cmpt 4643 × cxp 5036 ran crn 5039 ⟶wf 5800 –onto→wfo 5802 ‘cfv 5804 (class class class)co 6549 ↦ cmpt2 6551 ℂcc 9813 0cc0 9815 1c1 9816 + caddc 9818 − cmin 10145 2c2 10947 3c3 10948 ℕ₀cn0 11169 ℤ_≥cuz 11563 ..^cfzo 12334 #chash 12979 Word cword 13146 lastS clsw 13147 ++ cconcat 13148 ⟨“cs1 13149 substr csubstr 13150 USGrph cusg 25859 Neighbors cnbgra 25946 ClWWalksN cclwwlkn 26277

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-map 7746 df-pm 7747 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-n0 11170 df-z 11255 df-uz 11564 df-rp 11709 df-fz 12198 df-fzo 12335 df-hash 12980 df-word 13154 df-lsw 13155 df-concat 13156 df-s1 13157 df-substr 13158 df-usgra 25862 df-nbgra 25949 df-clwwlk 26279 df-clwwlkn 26280

This theorem is referenced by: numclwlk1lem2f1o 26623

W3C validator