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Theorem eupatrl 26495
 Description: An Eulerian path is a trail. Unfortunately, the edge function 𝐹 of an Eulerian path has the domain (1...(#‘𝐹)), whereas the edge functions of all kinds of walks defined here have the domain (0..^(#‘𝐹)) (i.e. the edge functions are "words of edge indices", see discussion and proposal of Mario Carneiro at https://groups.google.com/d/msg/metamath/KdVXdL3IH3k/2-BYcS_ACQAJ). Therefore, the arguments of the edge function of an Eulerian path must be shifted by 1 to obtain an edge function of a trail in this theorem, using the auxiliary theorems above (fargshiftlem 26162, fargshiftfv 26163, etc.). TODO: The definition of an Eulerian path and all related theorems should be modified to fit to the general definition of a trail. (Contributed by Alexander van der Vekens, 24-Nov-2017.)
Hypothesis
Ref Expression
eupatrl.f 𝐺 = (𝑥 ∈ (0..^(#‘𝐹)) ↦ (𝐹‘(𝑥 + 1)))
Assertion
Ref Expression
eupatrl (𝐹(𝑉 EulPaths 𝐸)𝑃𝐺(𝑉 Trails 𝐸)𝑃)
Distinct variable groups:   𝑥,𝐸   𝑥,𝐹   𝑥,𝑃   𝑥,𝑉
Allowed substitution hint:   𝐺(𝑥)

Proof of Theorem eupatrl
Dummy variables 𝑒 𝑓 𝑘 𝑛 𝑝 𝑣 𝑙 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-eupa 26490 . . . 4 EulPaths = (𝑣 ∈ V, 𝑒 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑣 UMGrph 𝑒 ∧ ∃𝑛 ∈ ℕ0 (𝑓:(1...𝑛)–1-1-onto→dom 𝑒𝑝:(0...𝑛)⟶𝑣 ∧ ∀𝑘 ∈ (1...𝑛)(𝑒‘(𝑓𝑘)) = {(𝑝‘(𝑘 − 1)), (𝑝𝑘)}))})
21brovmpt2ex 7236 . . 3 (𝐹(𝑉 EulPaths 𝐸)𝑃 → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)))
3 eupatrl.f . . . . . . 7 𝐺 = (𝑥 ∈ (0..^(#‘𝐹)) ↦ (𝐹‘(𝑥 + 1)))
4 ovex 6577 . . . . . . . 8 (0..^(#‘𝐹)) ∈ V
54mptex 6390 . . . . . . 7 (𝑥 ∈ (0..^(#‘𝐹)) ↦ (𝐹‘(𝑥 + 1))) ∈ V
63, 5eqeltri 2684 . . . . . 6 𝐺 ∈ V
76a1i 11 . . . . 5 (𝐹 ∈ V → 𝐺 ∈ V)
87anim1i 590 . . . 4 ((𝐹 ∈ V ∧ 𝑃 ∈ V) → (𝐺 ∈ V ∧ 𝑃 ∈ V))
98anim2i 591 . . 3 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐺 ∈ V ∧ 𝑃 ∈ V)))
102, 9syl 17 . 2 (𝐹(𝑉 EulPaths 𝐸)𝑃 → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐺 ∈ V ∧ 𝑃 ∈ V)))
11 f1ofn 6051 . . . . . . . . . . . . 13 (𝐹:(1...𝑛)–1-1-onto→dom 𝐸𝐹 Fn (1...𝑛))
12 fseq1hash 13026 . . . . . . . . . . . . 13 ((𝑛 ∈ ℕ0𝐹 Fn (1...𝑛)) → (#‘𝐹) = 𝑛)
1311, 12sylan2 490 . . . . . . . . . . . 12 ((𝑛 ∈ ℕ0𝐹:(1...𝑛)–1-1-onto→dom 𝐸) → (#‘𝐹) = 𝑛)
1413ancoms 468 . . . . . . . . . . 11 ((𝐹:(1...𝑛)–1-1-onto→dom 𝐸𝑛 ∈ ℕ0) → (#‘𝐹) = 𝑛)
15 f1of1 6049 . . . . . . . . . . . . . . . . . 18 (𝐹:(1...(#‘𝐹))–1-1-onto→dom 𝐸𝐹:(1...(#‘𝐹))–1-1→dom 𝐸)
163fargshiftf1 26165 . . . . . . . . . . . . . . . . . . 19 (((#‘𝐹) ∈ ℕ0𝐹:(1...(#‘𝐹))–1-1→dom 𝐸) → 𝐺:(0..^(#‘𝐹))–1-1→dom 𝐸)
1716expcom 450 . . . . . . . . . . . . . . . . . 18 (𝐹:(1...(#‘𝐹))–1-1→dom 𝐸 → ((#‘𝐹) ∈ ℕ0𝐺:(0..^(#‘𝐹))–1-1→dom 𝐸))
1815, 17syl 17 . . . . . . . . . . . . . . . . 17 (𝐹:(1...(#‘𝐹))–1-1-onto→dom 𝐸 → ((#‘𝐹) ∈ ℕ0𝐺:(0..^(#‘𝐹))–1-1→dom 𝐸))
1918imp 444 . . . . . . . . . . . . . . . 16 ((𝐹:(1...(#‘𝐹))–1-1-onto→dom 𝐸 ∧ (#‘𝐹) ∈ ℕ0) → 𝐺:(0..^(#‘𝐹))–1-1→dom 𝐸)
20 f1ofo 6057 . . . . . . . . . . . . . . . . . 18 (𝐹:(1...(#‘𝐹))–1-1-onto→dom 𝐸𝐹:(1...(#‘𝐹))–onto→dom 𝐸)
213fargshiftfo 26166 . . . . . . . . . . . . . . . . . . 19 (((#‘𝐹) ∈ ℕ0𝐹:(1...(#‘𝐹))–onto→dom 𝐸) → 𝐺:(0..^(#‘𝐹))–onto→dom 𝐸)
2221expcom 450 . . . . . . . . . . . . . . . . . 18 (𝐹:(1...(#‘𝐹))–onto→dom 𝐸 → ((#‘𝐹) ∈ ℕ0𝐺:(0..^(#‘𝐹))–onto→dom 𝐸))
2320, 22syl 17 . . . . . . . . . . . . . . . . 17 (𝐹:(1...(#‘𝐹))–1-1-onto→dom 𝐸 → ((#‘𝐹) ∈ ℕ0𝐺:(0..^(#‘𝐹))–onto→dom 𝐸))
2423imp 444 . . . . . . . . . . . . . . . 16 ((𝐹:(1...(#‘𝐹))–1-1-onto→dom 𝐸 ∧ (#‘𝐹) ∈ ℕ0) → 𝐺:(0..^(#‘𝐹))–onto→dom 𝐸)
25 df-f1o 5811 . . . . . . . . . . . . . . . 16 (𝐺:(0..^(#‘𝐹))–1-1-onto→dom 𝐸 ↔ (𝐺:(0..^(#‘𝐹))–1-1→dom 𝐸𝐺:(0..^(#‘𝐹))–onto→dom 𝐸))
2619, 24, 25sylanbrc 695 . . . . . . . . . . . . . . 15 ((𝐹:(1...(#‘𝐹))–1-1-onto→dom 𝐸 ∧ (#‘𝐹) ∈ ℕ0) → 𝐺:(0..^(#‘𝐹))–1-1-onto→dom 𝐸)
27 f1ofn 6051 . . . . . . . . . . . . . . . . 17 (𝐺:(0..^(#‘𝐹))–1-1-onto→dom 𝐸𝐺 Fn (0..^(#‘𝐹)))
28 hashfn 13025 . . . . . . . . . . . . . . . . . . 19 (𝐺 Fn (0..^(#‘𝐹)) → (#‘𝐺) = (#‘(0..^(#‘𝐹))))
2928anim2i 591 . . . . . . . . . . . . . . . . . 18 (((#‘𝐹) ∈ ℕ0𝐺 Fn (0..^(#‘𝐹))) → ((#‘𝐹) ∈ ℕ0 ∧ (#‘𝐺) = (#‘(0..^(#‘𝐹)))))
3029ancoms 468 . . . . . . . . . . . . . . . . 17 ((𝐺 Fn (0..^(#‘𝐹)) ∧ (#‘𝐹) ∈ ℕ0) → ((#‘𝐹) ∈ ℕ0 ∧ (#‘𝐺) = (#‘(0..^(#‘𝐹)))))
3127, 30sylan 487 . . . . . . . . . . . . . . . 16 ((𝐺:(0..^(#‘𝐹))–1-1-onto→dom 𝐸 ∧ (#‘𝐹) ∈ ℕ0) → ((#‘𝐹) ∈ ℕ0 ∧ (#‘𝐺) = (#‘(0..^(#‘𝐹)))))
32 hashfzo0 13077 . . . . . . . . . . . . . . . . . . 19 ((#‘𝐹) ∈ ℕ0 → (#‘(0..^(#‘𝐹))) = (#‘𝐹))
3332eqeq2d 2620 . . . . . . . . . . . . . . . . . 18 ((#‘𝐹) ∈ ℕ0 → ((#‘𝐺) = (#‘(0..^(#‘𝐹))) ↔ (#‘𝐺) = (#‘𝐹)))
3433biimpa 500 . . . . . . . . . . . . . . . . 17 (((#‘𝐹) ∈ ℕ0 ∧ (#‘𝐺) = (#‘(0..^(#‘𝐹)))) → (#‘𝐺) = (#‘𝐹))
35 pm3.2 462 . . . . . . . . . . . . . . . . . 18 (𝐺:(0..^(#‘𝐹))–1-1-onto→dom 𝐸 → ((#‘𝐺) = (#‘𝐹) → (𝐺:(0..^(#‘𝐹))–1-1-onto→dom 𝐸 ∧ (#‘𝐺) = (#‘𝐹))))
3635adantr 480 . . . . . . . . . . . . . . . . 17 ((𝐺:(0..^(#‘𝐹))–1-1-onto→dom 𝐸 ∧ (#‘𝐹) ∈ ℕ0) → ((#‘𝐺) = (#‘𝐹) → (𝐺:(0..^(#‘𝐹))–1-1-onto→dom 𝐸 ∧ (#‘𝐺) = (#‘𝐹))))
3734, 36syl5com 31 . . . . . . . . . . . . . . . 16 (((#‘𝐹) ∈ ℕ0 ∧ (#‘𝐺) = (#‘(0..^(#‘𝐹)))) → ((𝐺:(0..^(#‘𝐹))–1-1-onto→dom 𝐸 ∧ (#‘𝐹) ∈ ℕ0) → (𝐺:(0..^(#‘𝐹))–1-1-onto→dom 𝐸 ∧ (#‘𝐺) = (#‘𝐹))))
3831, 37mpcom 37 . . . . . . . . . . . . . . 15 ((𝐺:(0..^(#‘𝐹))–1-1-onto→dom 𝐸 ∧ (#‘𝐹) ∈ ℕ0) → (𝐺:(0..^(#‘𝐹))–1-1-onto→dom 𝐸 ∧ (#‘𝐺) = (#‘𝐹)))
3926, 38sylancom 698 . . . . . . . . . . . . . 14 ((𝐹:(1...(#‘𝐹))–1-1-onto→dom 𝐸 ∧ (#‘𝐹) ∈ ℕ0) → (𝐺:(0..^(#‘𝐹))–1-1-onto→dom 𝐸 ∧ (#‘𝐺) = (#‘𝐹)))
40 df-f1 5809 . . . . . . . . . . . . . . . . . . . . 21 (𝐺:(0..^(#‘𝐹))–1-1→dom 𝐸 ↔ (𝐺:(0..^(#‘𝐹))⟶dom 𝐸 ∧ Fun 𝐺))
41 iswrdi 13164 . . . . . . . . . . . . . . . . . . . . . 22 (𝐺:(0..^(#‘𝐹))⟶dom 𝐸𝐺 ∈ Word dom 𝐸)
4241adantr 480 . . . . . . . . . . . . . . . . . . . . 21 ((𝐺:(0..^(#‘𝐹))⟶dom 𝐸 ∧ Fun 𝐺) → 𝐺 ∈ Word dom 𝐸)
4340, 42sylbi 206 . . . . . . . . . . . . . . . . . . . 20 (𝐺:(0..^(#‘𝐹))–1-1→dom 𝐸𝐺 ∈ Word dom 𝐸)
4443adantr 480 . . . . . . . . . . . . . . . . . . 19 ((𝐺:(0..^(#‘𝐹))–1-1→dom 𝐸𝐺:(0..^(#‘𝐹))–onto→dom 𝐸) → 𝐺 ∈ Word dom 𝐸)
4525, 44sylbi 206 . . . . . . . . . . . . . . . . . 18 (𝐺:(0..^(#‘𝐹))–1-1-onto→dom 𝐸𝐺 ∈ Word dom 𝐸)
4645ad4antr 764 . . . . . . . . . . . . . . . . 17 (((((𝐺:(0..^(#‘𝐹))–1-1-onto→dom 𝐸 ∧ (#‘𝐺) = (#‘𝐹)) ∧ (𝐹:(1...(#‘𝐹))–1-1-onto→dom 𝐸 ∧ (#‘𝐹) ∈ ℕ0)) ∧ ∀𝑘 ∈ (1...(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃‘(𝑘 − 1)), (𝑃𝑘)}) ∧ 𝑃:(0...(#‘𝐹))⟶𝑉) → 𝐺 ∈ Word dom 𝐸)
4740simprbi 479 . . . . . . . . . . . . . . . . . . . 20 (𝐺:(0..^(#‘𝐹))–1-1→dom 𝐸 → Fun 𝐺)
4847adantr 480 . . . . . . . . . . . . . . . . . . 19 ((𝐺:(0..^(#‘𝐹))–1-1→dom 𝐸𝐺:(0..^(#‘𝐹))–onto→dom 𝐸) → Fun 𝐺)
4925, 48sylbi 206 . . . . . . . . . . . . . . . . . 18 (𝐺:(0..^(#‘𝐹))–1-1-onto→dom 𝐸 → Fun 𝐺)
5049ad4antr 764 . . . . . . . . . . . . . . . . 17 (((((𝐺:(0..^(#‘𝐹))–1-1-onto→dom 𝐸 ∧ (#‘𝐺) = (#‘𝐹)) ∧ (𝐹:(1...(#‘𝐹))–1-1-onto→dom 𝐸 ∧ (#‘𝐹) ∈ ℕ0)) ∧ ∀𝑘 ∈ (1...(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃‘(𝑘 − 1)), (𝑃𝑘)}) ∧ 𝑃:(0...(#‘𝐹))⟶𝑉) → Fun 𝐺)
5146, 50jca 553 . . . . . . . . . . . . . . . 16 (((((𝐺:(0..^(#‘𝐹))–1-1-onto→dom 𝐸 ∧ (#‘𝐺) = (#‘𝐹)) ∧ (𝐹:(1...(#‘𝐹))–1-1-onto→dom 𝐸 ∧ (#‘𝐹) ∈ ℕ0)) ∧ ∀𝑘 ∈ (1...(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃‘(𝑘 − 1)), (𝑃𝑘)}) ∧ 𝑃:(0...(#‘𝐹))⟶𝑉) → (𝐺 ∈ Word dom 𝐸 ∧ Fun 𝐺))
52 eqcom 2617 . . . . . . . . . . . . . . . . . . . . 21 ((#‘𝐺) = (#‘𝐹) ↔ (#‘𝐹) = (#‘𝐺))
5352biimpi 205 . . . . . . . . . . . . . . . . . . . 20 ((#‘𝐺) = (#‘𝐹) → (#‘𝐹) = (#‘𝐺))
5453ad3antlr 763 . . . . . . . . . . . . . . . . . . 19 ((((𝐺:(0..^(#‘𝐹))–1-1-onto→dom 𝐸 ∧ (#‘𝐺) = (#‘𝐹)) ∧ (𝐹:(1...(#‘𝐹))–1-1-onto→dom 𝐸 ∧ (#‘𝐹) ∈ ℕ0)) ∧ ∀𝑘 ∈ (1...(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃‘(𝑘 − 1)), (𝑃𝑘)}) → (#‘𝐹) = (#‘𝐺))
5554oveq2d 6565 . . . . . . . . . . . . . . . . . 18 ((((𝐺:(0..^(#‘𝐹))–1-1-onto→dom 𝐸 ∧ (#‘𝐺) = (#‘𝐹)) ∧ (𝐹:(1...(#‘𝐹))–1-1-onto→dom 𝐸 ∧ (#‘𝐹) ∈ ℕ0)) ∧ ∀𝑘 ∈ (1...(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃‘(𝑘 − 1)), (𝑃𝑘)}) → (0...(#‘𝐹)) = (0...(#‘𝐺)))
5655feq2d 5944 . . . . . . . . . . . . . . . . 17 ((((𝐺:(0..^(#‘𝐹))–1-1-onto→dom 𝐸 ∧ (#‘𝐺) = (#‘𝐹)) ∧ (𝐹:(1...(#‘𝐹))–1-1-onto→dom 𝐸 ∧ (#‘𝐹) ∈ ℕ0)) ∧ ∀𝑘 ∈ (1...(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃‘(𝑘 − 1)), (𝑃𝑘)}) → (𝑃:(0...(#‘𝐹))⟶𝑉𝑃:(0...(#‘𝐺))⟶𝑉))
5756biimpa 500 . . . . . . . . . . . . . . . 16 (((((𝐺:(0..^(#‘𝐹))–1-1-onto→dom 𝐸 ∧ (#‘𝐺) = (#‘𝐹)) ∧ (𝐹:(1...(#‘𝐹))–1-1-onto→dom 𝐸 ∧ (#‘𝐹) ∈ ℕ0)) ∧ ∀𝑘 ∈ (1...(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃‘(𝑘 − 1)), (𝑃𝑘)}) ∧ 𝑃:(0...(#‘𝐹))⟶𝑉) → 𝑃:(0...(#‘𝐺))⟶𝑉)
58 f1of 6050 . . . . . . . . . . . . . . . . . . . . . 22 (𝐹:(1...(#‘𝐹))–1-1-onto→dom 𝐸𝐹:(1...(#‘𝐹))⟶dom 𝐸)
59 fargshiftlem 26162 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((#‘𝐹) ∈ ℕ0𝑙 ∈ (0..^(#‘𝐹))) → (𝑙 + 1) ∈ (1...(#‘𝐹)))
60 simpll 786 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((((𝑙 + 1) ∈ (1...(#‘𝐹)) ∧ ((#‘𝐹) ∈ ℕ0𝑙 ∈ (0..^(#‘𝐹)))) ∧ 𝐹:(1...(#‘𝐹))⟶dom 𝐸) → (𝑙 + 1) ∈ (1...(#‘𝐹)))
61 fveq2 6103 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 (𝑘 = (𝑙 + 1) → (𝐹𝑘) = (𝐹‘(𝑙 + 1)))
6261fveq2d 6107 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 (𝑘 = (𝑙 + 1) → (𝐸‘(𝐹𝑘)) = (𝐸‘(𝐹‘(𝑙 + 1))))
6362adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((((#‘𝐹) ∈ ℕ0𝑙 ∈ (0..^(#‘𝐹))) ∧ 𝑘 = (𝑙 + 1)) → (𝐸‘(𝐹𝑘)) = (𝐸‘(𝐹‘(𝑙 + 1))))
64 oveq1 6556 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 (𝑘 = (𝑙 + 1) → (𝑘 − 1) = ((𝑙 + 1) − 1))
65 elfzoelz 12339 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 (𝑙 ∈ (0..^(#‘𝐹)) → 𝑙 ∈ ℤ)
6665zcnd 11359 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 (𝑙 ∈ (0..^(#‘𝐹)) → 𝑙 ∈ ℂ)
6766adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 (((#‘𝐹) ∈ ℕ0𝑙 ∈ (0..^(#‘𝐹))) → 𝑙 ∈ ℂ)
68 1cnd 9935 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 (((#‘𝐹) ∈ ℕ0𝑙 ∈ (0..^(#‘𝐹))) → 1 ∈ ℂ)
6967, 68pncand 10272 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 (((#‘𝐹) ∈ ℕ0𝑙 ∈ (0..^(#‘𝐹))) → ((𝑙 + 1) − 1) = 𝑙)
7064, 69sylan9eqr 2666 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ((((#‘𝐹) ∈ ℕ0𝑙 ∈ (0..^(#‘𝐹))) ∧ 𝑘 = (𝑙 + 1)) → (𝑘 − 1) = 𝑙)
7170fveq2d 6107 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ((((#‘𝐹) ∈ ℕ0𝑙 ∈ (0..^(#‘𝐹))) ∧ 𝑘 = (𝑙 + 1)) → (𝑃‘(𝑘 − 1)) = (𝑃𝑙))
72 fveq2 6103 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 (𝑘 = (𝑙 + 1) → (𝑃𝑘) = (𝑃‘(𝑙 + 1)))
7372adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ((((#‘𝐹) ∈ ℕ0𝑙 ∈ (0..^(#‘𝐹))) ∧ 𝑘 = (𝑙 + 1)) → (𝑃𝑘) = (𝑃‘(𝑙 + 1)))
7471, 73preq12d 4220 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((((#‘𝐹) ∈ ℕ0𝑙 ∈ (0..^(#‘𝐹))) ∧ 𝑘 = (𝑙 + 1)) → {(𝑃‘(𝑘 − 1)), (𝑃𝑘)} = {(𝑃𝑙), (𝑃‘(𝑙 + 1))})
7563, 74eqeq12d 2625 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((((#‘𝐹) ∈ ℕ0𝑙 ∈ (0..^(#‘𝐹))) ∧ 𝑘 = (𝑙 + 1)) → ((𝐸‘(𝐹𝑘)) = {(𝑃‘(𝑘 − 1)), (𝑃𝑘)} ↔ (𝐸‘(𝐹‘(𝑙 + 1))) = {(𝑃𝑙), (𝑃‘(𝑙 + 1))}))
7675ex 449 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (((#‘𝐹) ∈ ℕ0𝑙 ∈ (0..^(#‘𝐹))) → (𝑘 = (𝑙 + 1) → ((𝐸‘(𝐹𝑘)) = {(𝑃‘(𝑘 − 1)), (𝑃𝑘)} ↔ (𝐸‘(𝐹‘(𝑙 + 1))) = {(𝑃𝑙), (𝑃‘(𝑙 + 1))})))
7776adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (((𝑙 + 1) ∈ (1...(#‘𝐹)) ∧ ((#‘𝐹) ∈ ℕ0𝑙 ∈ (0..^(#‘𝐹)))) → (𝑘 = (𝑙 + 1) → ((𝐸‘(𝐹𝑘)) = {(𝑃‘(𝑘 − 1)), (𝑃𝑘)} ↔ (𝐸‘(𝐹‘(𝑙 + 1))) = {(𝑃𝑙), (𝑃‘(𝑙 + 1))})))
7877adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((((𝑙 + 1) ∈ (1...(#‘𝐹)) ∧ ((#‘𝐹) ∈ ℕ0𝑙 ∈ (0..^(#‘𝐹)))) ∧ 𝐹:(1...(#‘𝐹))⟶dom 𝐸) → (𝑘 = (𝑙 + 1) → ((𝐸‘(𝐹𝑘)) = {(𝑃‘(𝑘 − 1)), (𝑃𝑘)} ↔ (𝐸‘(𝐹‘(𝑙 + 1))) = {(𝑃𝑙), (𝑃‘(𝑙 + 1))})))
7978imp 444 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (((((𝑙 + 1) ∈ (1...(#‘𝐹)) ∧ ((#‘𝐹) ∈ ℕ0𝑙 ∈ (0..^(#‘𝐹)))) ∧ 𝐹:(1...(#‘𝐹))⟶dom 𝐸) ∧ 𝑘 = (𝑙 + 1)) → ((𝐸‘(𝐹𝑘)) = {(𝑃‘(𝑘 − 1)), (𝑃𝑘)} ↔ (𝐸‘(𝐹‘(𝑙 + 1))) = {(𝑃𝑙), (𝑃‘(𝑙 + 1))}))
80 simprl 790 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (((𝑙 + 1) ∈ (1...(#‘𝐹)) ∧ ((#‘𝐹) ∈ ℕ0𝑙 ∈ (0..^(#‘𝐹)))) → (#‘𝐹) ∈ ℕ0)
8180anim1i 590 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((((𝑙 + 1) ∈ (1...(#‘𝐹)) ∧ ((#‘𝐹) ∈ ℕ0𝑙 ∈ (0..^(#‘𝐹)))) ∧ 𝐹:(1...(#‘𝐹))⟶dom 𝐸) → ((#‘𝐹) ∈ ℕ0𝐹:(1...(#‘𝐹))⟶dom 𝐸))
8281adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (((((𝑙 + 1) ∈ (1...(#‘𝐹)) ∧ ((#‘𝐹) ∈ ℕ0𝑙 ∈ (0..^(#‘𝐹)))) ∧ 𝐹:(1...(#‘𝐹))⟶dom 𝐸) ∧ 𝑘 = (𝑙 + 1)) → ((#‘𝐹) ∈ ℕ0𝐹:(1...(#‘𝐹))⟶dom 𝐸))
83 simpr 476 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (((#‘𝐹) ∈ ℕ0𝑙 ∈ (0..^(#‘𝐹))) → 𝑙 ∈ (0..^(#‘𝐹)))
8483ad3antlr 763 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (((((𝑙 + 1) ∈ (1...(#‘𝐹)) ∧ ((#‘𝐹) ∈ ℕ0𝑙 ∈ (0..^(#‘𝐹)))) ∧ 𝐹:(1...(#‘𝐹))⟶dom 𝐸) ∧ 𝑘 = (𝑙 + 1)) → 𝑙 ∈ (0..^(#‘𝐹)))
853fargshiftfv 26163 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (((#‘𝐹) ∈ ℕ0𝐹:(1...(#‘𝐹))⟶dom 𝐸) → (𝑙 ∈ (0..^(#‘𝐹)) → (𝐺𝑙) = (𝐹‘(𝑙 + 1))))
8685imp 444 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((((#‘𝐹) ∈ ℕ0𝐹:(1...(#‘𝐹))⟶dom 𝐸) ∧ 𝑙 ∈ (0..^(#‘𝐹))) → (𝐺𝑙) = (𝐹‘(𝑙 + 1)))
8786eqcomd 2616 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((((#‘𝐹) ∈ ℕ0𝐹:(1...(#‘𝐹))⟶dom 𝐸) ∧ 𝑙 ∈ (0..^(#‘𝐹))) → (𝐹‘(𝑙 + 1)) = (𝐺𝑙))
8882, 84, 87syl2anc 691 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (((((𝑙 + 1) ∈ (1...(#‘𝐹)) ∧ ((#‘𝐹) ∈ ℕ0𝑙 ∈ (0..^(#‘𝐹)))) ∧ 𝐹:(1...(#‘𝐹))⟶dom 𝐸) ∧ 𝑘 = (𝑙 + 1)) → (𝐹‘(𝑙 + 1)) = (𝐺𝑙))
8988fveq2d 6107 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (((((𝑙 + 1) ∈ (1...(#‘𝐹)) ∧ ((#‘𝐹) ∈ ℕ0𝑙 ∈ (0..^(#‘𝐹)))) ∧ 𝐹:(1...(#‘𝐹))⟶dom 𝐸) ∧ 𝑘 = (𝑙 + 1)) → (𝐸‘(𝐹‘(𝑙 + 1))) = (𝐸‘(𝐺𝑙)))
9089eqeq1d 2612 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (((((𝑙 + 1) ∈ (1...(#‘𝐹)) ∧ ((#‘𝐹) ∈ ℕ0𝑙 ∈ (0..^(#‘𝐹)))) ∧ 𝐹:(1...(#‘𝐹))⟶dom 𝐸) ∧ 𝑘 = (𝑙 + 1)) → ((𝐸‘(𝐹‘(𝑙 + 1))) = {(𝑃𝑙), (𝑃‘(𝑙 + 1))} ↔ (𝐸‘(𝐺𝑙)) = {(𝑃𝑙), (𝑃‘(𝑙 + 1))}))
9179, 90bitrd 267 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (((((𝑙 + 1) ∈ (1...(#‘𝐹)) ∧ ((#‘𝐹) ∈ ℕ0𝑙 ∈ (0..^(#‘𝐹)))) ∧ 𝐹:(1...(#‘𝐹))⟶dom 𝐸) ∧ 𝑘 = (𝑙 + 1)) → ((𝐸‘(𝐹𝑘)) = {(𝑃‘(𝑘 − 1)), (𝑃𝑘)} ↔ (𝐸‘(𝐺𝑙)) = {(𝑃𝑙), (𝑃‘(𝑙 + 1))}))
9260, 91rspcdv 3285 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝑙 + 1) ∈ (1...(#‘𝐹)) ∧ ((#‘𝐹) ∈ ℕ0𝑙 ∈ (0..^(#‘𝐹)))) ∧ 𝐹:(1...(#‘𝐹))⟶dom 𝐸) → (∀𝑘 ∈ (1...(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃‘(𝑘 − 1)), (𝑃𝑘)} → (𝐸‘(𝐺𝑙)) = {(𝑃𝑙), (𝑃‘(𝑙 + 1))}))
9392ex 449 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝑙 + 1) ∈ (1...(#‘𝐹)) ∧ ((#‘𝐹) ∈ ℕ0𝑙 ∈ (0..^(#‘𝐹)))) → (𝐹:(1...(#‘𝐹))⟶dom 𝐸 → (∀𝑘 ∈ (1...(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃‘(𝑘 − 1)), (𝑃𝑘)} → (𝐸‘(𝐺𝑙)) = {(𝑃𝑙), (𝑃‘(𝑙 + 1))})))
9493com23 84 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝑙 + 1) ∈ (1...(#‘𝐹)) ∧ ((#‘𝐹) ∈ ℕ0𝑙 ∈ (0..^(#‘𝐹)))) → (∀𝑘 ∈ (1...(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃‘(𝑘 − 1)), (𝑃𝑘)} → (𝐹:(1...(#‘𝐹))⟶dom 𝐸 → (𝐸‘(𝐺𝑙)) = {(𝑃𝑙), (𝑃‘(𝑙 + 1))})))
9559, 94mpancom 700 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((#‘𝐹) ∈ ℕ0𝑙 ∈ (0..^(#‘𝐹))) → (∀𝑘 ∈ (1...(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃‘(𝑘 − 1)), (𝑃𝑘)} → (𝐹:(1...(#‘𝐹))⟶dom 𝐸 → (𝐸‘(𝐺𝑙)) = {(𝑃𝑙), (𝑃‘(𝑙 + 1))})))
9695ex 449 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((#‘𝐹) ∈ ℕ0 → (𝑙 ∈ (0..^(#‘𝐹)) → (∀𝑘 ∈ (1...(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃‘(𝑘 − 1)), (𝑃𝑘)} → (𝐹:(1...(#‘𝐹))⟶dom 𝐸 → (𝐸‘(𝐺𝑙)) = {(𝑃𝑙), (𝑃‘(𝑙 + 1))}))))
9796com24 93 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((#‘𝐹) ∈ ℕ0 → (𝐹:(1...(#‘𝐹))⟶dom 𝐸 → (∀𝑘 ∈ (1...(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃‘(𝑘 − 1)), (𝑃𝑘)} → (𝑙 ∈ (0..^(#‘𝐹)) → (𝐸‘(𝐺𝑙)) = {(𝑃𝑙), (𝑃‘(𝑙 + 1))}))))
9897imp31 447 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((#‘𝐹) ∈ ℕ0𝐹:(1...(#‘𝐹))⟶dom 𝐸) ∧ ∀𝑘 ∈ (1...(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃‘(𝑘 − 1)), (𝑃𝑘)}) → (𝑙 ∈ (0..^(#‘𝐹)) → (𝐸‘(𝐺𝑙)) = {(𝑃𝑙), (𝑃‘(𝑙 + 1))}))
9998ralrimiv 2948 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((#‘𝐹) ∈ ℕ0𝐹:(1...(#‘𝐹))⟶dom 𝐸) ∧ ∀𝑘 ∈ (1...(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃‘(𝑘 − 1)), (𝑃𝑘)}) → ∀𝑙 ∈ (0..^(#‘𝐹))(𝐸‘(𝐺𝑙)) = {(𝑃𝑙), (𝑃‘(𝑙 + 1))})
10099ex 449 . . . . . . . . . . . . . . . . . . . . . . 23 (((#‘𝐹) ∈ ℕ0𝐹:(1...(#‘𝐹))⟶dom 𝐸) → (∀𝑘 ∈ (1...(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃‘(𝑘 − 1)), (𝑃𝑘)} → ∀𝑙 ∈ (0..^(#‘𝐹))(𝐸‘(𝐺𝑙)) = {(𝑃𝑙), (𝑃‘(𝑙 + 1))}))
101100expcom 450 . . . . . . . . . . . . . . . . . . . . . 22 (𝐹:(1...(#‘𝐹))⟶dom 𝐸 → ((#‘𝐹) ∈ ℕ0 → (∀𝑘 ∈ (1...(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃‘(𝑘 − 1)), (𝑃𝑘)} → ∀𝑙 ∈ (0..^(#‘𝐹))(𝐸‘(𝐺𝑙)) = {(𝑃𝑙), (𝑃‘(𝑙 + 1))})))
10258, 101syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝐹:(1...(#‘𝐹))–1-1-onto→dom 𝐸 → ((#‘𝐹) ∈ ℕ0 → (∀𝑘 ∈ (1...(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃‘(𝑘 − 1)), (𝑃𝑘)} → ∀𝑙 ∈ (0..^(#‘𝐹))(𝐸‘(𝐺𝑙)) = {(𝑃𝑙), (𝑃‘(𝑙 + 1))})))
103102imp 444 . . . . . . . . . . . . . . . . . . . 20 ((𝐹:(1...(#‘𝐹))–1-1-onto→dom 𝐸 ∧ (#‘𝐹) ∈ ℕ0) → (∀𝑘 ∈ (1...(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃‘(𝑘 − 1)), (𝑃𝑘)} → ∀𝑙 ∈ (0..^(#‘𝐹))(𝐸‘(𝐺𝑙)) = {(𝑃𝑙), (𝑃‘(𝑙 + 1))}))
104 oveq2 6557 . . . . . . . . . . . . . . . . . . . . . 22 ((#‘𝐺) = (#‘𝐹) → (0..^(#‘𝐺)) = (0..^(#‘𝐹)))
105104raleqdv 3121 . . . . . . . . . . . . . . . . . . . . 21 ((#‘𝐺) = (#‘𝐹) → (∀𝑙 ∈ (0..^(#‘𝐺))(𝐸‘(𝐺𝑙)) = {(𝑃𝑙), (𝑃‘(𝑙 + 1))} ↔ ∀𝑙 ∈ (0..^(#‘𝐹))(𝐸‘(𝐺𝑙)) = {(𝑃𝑙), (𝑃‘(𝑙 + 1))}))
106105imbi2d 329 . . . . . . . . . . . . . . . . . . . 20 ((#‘𝐺) = (#‘𝐹) → ((∀𝑘 ∈ (1...(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃‘(𝑘 − 1)), (𝑃𝑘)} → ∀𝑙 ∈ (0..^(#‘𝐺))(𝐸‘(𝐺𝑙)) = {(𝑃𝑙), (𝑃‘(𝑙 + 1))}) ↔ (∀𝑘 ∈ (1...(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃‘(𝑘 − 1)), (𝑃𝑘)} → ∀𝑙 ∈ (0..^(#‘𝐹))(𝐸‘(𝐺𝑙)) = {(𝑃𝑙), (𝑃‘(𝑙 + 1))})))
107103, 106syl5ibr 235 . . . . . . . . . . . . . . . . . . 19 ((#‘𝐺) = (#‘𝐹) → ((𝐹:(1...(#‘𝐹))–1-1-onto→dom 𝐸 ∧ (#‘𝐹) ∈ ℕ0) → (∀𝑘 ∈ (1...(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃‘(𝑘 − 1)), (𝑃𝑘)} → ∀𝑙 ∈ (0..^(#‘𝐺))(𝐸‘(𝐺𝑙)) = {(𝑃𝑙), (𝑃‘(𝑙 + 1))})))
108107adantl 481 . . . . . . . . . . . . . . . . . 18 ((𝐺:(0..^(#‘𝐹))–1-1-onto→dom 𝐸 ∧ (#‘𝐺) = (#‘𝐹)) → ((𝐹:(1...(#‘𝐹))–1-1-onto→dom 𝐸 ∧ (#‘𝐹) ∈ ℕ0) → (∀𝑘 ∈ (1...(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃‘(𝑘 − 1)), (𝑃𝑘)} → ∀𝑙 ∈ (0..^(#‘𝐺))(𝐸‘(𝐺𝑙)) = {(𝑃𝑙), (𝑃‘(𝑙 + 1))})))
109108imp31 447 . . . . . . . . . . . . . . . . 17 ((((𝐺:(0..^(#‘𝐹))–1-1-onto→dom 𝐸 ∧ (#‘𝐺) = (#‘𝐹)) ∧ (𝐹:(1...(#‘𝐹))–1-1-onto→dom 𝐸 ∧ (#‘𝐹) ∈ ℕ0)) ∧ ∀𝑘 ∈ (1...(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃‘(𝑘 − 1)), (𝑃𝑘)}) → ∀𝑙 ∈ (0..^(#‘𝐺))(𝐸‘(𝐺𝑙)) = {(𝑃𝑙), (𝑃‘(𝑙 + 1))})
110109adantr 480 . . . . . . . . . . . . . . . 16 (((((𝐺:(0..^(#‘𝐹))–1-1-onto→dom 𝐸 ∧ (#‘𝐺) = (#‘𝐹)) ∧ (𝐹:(1...(#‘𝐹))–1-1-onto→dom 𝐸 ∧ (#‘𝐹) ∈ ℕ0)) ∧ ∀𝑘 ∈ (1...(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃‘(𝑘 − 1)), (𝑃𝑘)}) ∧ 𝑃:(0...(#‘𝐹))⟶𝑉) → ∀𝑙 ∈ (0..^(#‘𝐺))(𝐸‘(𝐺𝑙)) = {(𝑃𝑙), (𝑃‘(𝑙 + 1))})
11151, 57, 1103jca 1235 . . . . . . . . . . . . . . 15 (((((𝐺:(0..^(#‘𝐹))–1-1-onto→dom 𝐸 ∧ (#‘𝐺) = (#‘𝐹)) ∧ (𝐹:(1...(#‘𝐹))–1-1-onto→dom 𝐸 ∧ (#‘𝐹) ∈ ℕ0)) ∧ ∀𝑘 ∈ (1...(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃‘(𝑘 − 1)), (𝑃𝑘)}) ∧ 𝑃:(0...(#‘𝐹))⟶𝑉) → ((𝐺 ∈ Word dom 𝐸 ∧ Fun 𝐺) ∧ 𝑃:(0...(#‘𝐺))⟶𝑉 ∧ ∀𝑙 ∈ (0..^(#‘𝐺))(𝐸‘(𝐺𝑙)) = {(𝑃𝑙), (𝑃‘(𝑙 + 1))}))
112111exp31 628 . . . . . . . . . . . . . 14 (((𝐺:(0..^(#‘𝐹))–1-1-onto→dom 𝐸 ∧ (#‘𝐺) = (#‘𝐹)) ∧ (𝐹:(1...(#‘𝐹))–1-1-onto→dom 𝐸 ∧ (#‘𝐹) ∈ ℕ0)) → (∀𝑘 ∈ (1...(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃‘(𝑘 − 1)), (𝑃𝑘)} → (𝑃:(0...(#‘𝐹))⟶𝑉 → ((𝐺 ∈ Word dom 𝐸 ∧ Fun 𝐺) ∧ 𝑃:(0...(#‘𝐺))⟶𝑉 ∧ ∀𝑙 ∈ (0..^(#‘𝐺))(𝐸‘(𝐺𝑙)) = {(𝑃𝑙), (𝑃‘(𝑙 + 1))}))))
11339, 112mpancom 700 . . . . . . . . . . . . 13 ((𝐹:(1...(#‘𝐹))–1-1-onto→dom 𝐸 ∧ (#‘𝐹) ∈ ℕ0) → (∀𝑘 ∈ (1...(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃‘(𝑘 − 1)), (𝑃𝑘)} → (𝑃:(0...(#‘𝐹))⟶𝑉 → ((𝐺 ∈ Word dom 𝐸 ∧ Fun 𝐺) ∧ 𝑃:(0...(#‘𝐺))⟶𝑉 ∧ ∀𝑙 ∈ (0..^(#‘𝐺))(𝐸‘(𝐺𝑙)) = {(𝑃𝑙), (𝑃‘(𝑙 + 1))}))))
114113a1i 11 . . . . . . . . . . . 12 ((#‘𝐹) = 𝑛 → ((𝐹:(1...(#‘𝐹))–1-1-onto→dom 𝐸 ∧ (#‘𝐹) ∈ ℕ0) → (∀𝑘 ∈ (1...(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃‘(𝑘 − 1)), (𝑃𝑘)} → (𝑃:(0...(#‘𝐹))⟶𝑉 → ((𝐺 ∈ Word dom 𝐸 ∧ Fun 𝐺) ∧ 𝑃:(0...(#‘𝐺))⟶𝑉 ∧ ∀𝑙 ∈ (0..^(#‘𝐺))(𝐸‘(𝐺𝑙)) = {(𝑃𝑙), (𝑃‘(𝑙 + 1))})))))
115 oveq2 6557 . . . . . . . . . . . . . . 15 (𝑛 = (#‘𝐹) → (1...𝑛) = (1...(#‘𝐹)))
116115eqcoms 2618 . . . . . . . . . . . . . 14 ((#‘𝐹) = 𝑛 → (1...𝑛) = (1...(#‘𝐹)))
117 f1oeq2 6041 . . . . . . . . . . . . . 14 ((1...𝑛) = (1...(#‘𝐹)) → (𝐹:(1...𝑛)–1-1-onto→dom 𝐸𝐹:(1...(#‘𝐹))–1-1-onto→dom 𝐸))
118116, 117syl 17 . . . . . . . . . . . . 13 ((#‘𝐹) = 𝑛 → (𝐹:(1...𝑛)–1-1-onto→dom 𝐸𝐹:(1...(#‘𝐹))–1-1-onto→dom 𝐸))
119 eleq1 2676 . . . . . . . . . . . . . 14 (𝑛 = (#‘𝐹) → (𝑛 ∈ ℕ0 ↔ (#‘𝐹) ∈ ℕ0))
120119eqcoms 2618 . . . . . . . . . . . . 13 ((#‘𝐹) = 𝑛 → (𝑛 ∈ ℕ0 ↔ (#‘𝐹) ∈ ℕ0))
121118, 120anbi12d 743 . . . . . . . . . . . 12 ((#‘𝐹) = 𝑛 → ((𝐹:(1...𝑛)–1-1-onto→dom 𝐸𝑛 ∈ ℕ0) ↔ (𝐹:(1...(#‘𝐹))–1-1-onto→dom 𝐸 ∧ (#‘𝐹) ∈ ℕ0)))
122116raleqdv 3121 . . . . . . . . . . . . 13 ((#‘𝐹) = 𝑛 → (∀𝑘 ∈ (1...𝑛)(𝐸‘(𝐹𝑘)) = {(𝑃‘(𝑘 − 1)), (𝑃𝑘)} ↔ ∀𝑘 ∈ (1...(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃‘(𝑘 − 1)), (𝑃𝑘)}))
123 oveq2 6557 . . . . . . . . . . . . . . . 16 (𝑛 = (#‘𝐹) → (0...𝑛) = (0...(#‘𝐹)))
124123eqcoms 2618 . . . . . . . . . . . . . . 15 ((#‘𝐹) = 𝑛 → (0...𝑛) = (0...(#‘𝐹)))
125124feq2d 5944 . . . . . . . . . . . . . 14 ((#‘𝐹) = 𝑛 → (𝑃:(0...𝑛)⟶𝑉𝑃:(0...(#‘𝐹))⟶𝑉))
126125imbi1d 330 . . . . . . . . . . . . 13 ((#‘𝐹) = 𝑛 → ((𝑃:(0...𝑛)⟶𝑉 → ((𝐺 ∈ Word dom 𝐸 ∧ Fun 𝐺) ∧ 𝑃:(0...(#‘𝐺))⟶𝑉 ∧ ∀𝑙 ∈ (0..^(#‘𝐺))(𝐸‘(𝐺𝑙)) = {(𝑃𝑙), (𝑃‘(𝑙 + 1))})) ↔ (𝑃:(0...(#‘𝐹))⟶𝑉 → ((𝐺 ∈ Word dom 𝐸 ∧ Fun 𝐺) ∧ 𝑃:(0...(#‘𝐺))⟶𝑉 ∧ ∀𝑙 ∈ (0..^(#‘𝐺))(𝐸‘(𝐺𝑙)) = {(𝑃𝑙), (𝑃‘(𝑙 + 1))}))))
127122, 126imbi12d 333 . . . . . . . . . . . 12 ((#‘𝐹) = 𝑛 → ((∀𝑘 ∈ (1...𝑛)(𝐸‘(𝐹𝑘)) = {(𝑃‘(𝑘 − 1)), (𝑃𝑘)} → (𝑃:(0...𝑛)⟶𝑉 → ((𝐺 ∈ Word dom 𝐸 ∧ Fun 𝐺) ∧ 𝑃:(0...(#‘𝐺))⟶𝑉 ∧ ∀𝑙 ∈ (0..^(#‘𝐺))(𝐸‘(𝐺𝑙)) = {(𝑃𝑙), (𝑃‘(𝑙 + 1))}))) ↔ (∀𝑘 ∈ (1...(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃‘(𝑘 − 1)), (𝑃𝑘)} → (𝑃:(0...(#‘𝐹))⟶𝑉 → ((𝐺 ∈ Word dom 𝐸 ∧ Fun 𝐺) ∧ 𝑃:(0...(#‘𝐺))⟶𝑉 ∧ ∀𝑙 ∈ (0..^(#‘𝐺))(𝐸‘(𝐺𝑙)) = {(𝑃𝑙), (𝑃‘(𝑙 + 1))})))))
128114, 121, 1273imtr4d 282 . . . . . . . . . . 11 ((#‘𝐹) = 𝑛 → ((𝐹:(1...𝑛)–1-1-onto→dom 𝐸𝑛 ∈ ℕ0) → (∀𝑘 ∈ (1...𝑛)(𝐸‘(𝐹𝑘)) = {(𝑃‘(𝑘 − 1)), (𝑃𝑘)} → (𝑃:(0...𝑛)⟶𝑉 → ((𝐺 ∈ Word dom 𝐸 ∧ Fun 𝐺) ∧ 𝑃:(0...(#‘𝐺))⟶𝑉 ∧ ∀𝑙 ∈ (0..^(#‘𝐺))(𝐸‘(𝐺𝑙)) = {(𝑃𝑙), (𝑃‘(𝑙 + 1))})))))
12914, 128mpcom 37 . . . . . . . . . 10 ((𝐹:(1...𝑛)–1-1-onto→dom 𝐸𝑛 ∈ ℕ0) → (∀𝑘 ∈ (1...𝑛)(𝐸‘(𝐹𝑘)) = {(𝑃‘(𝑘 − 1)), (𝑃𝑘)} → (𝑃:(0...𝑛)⟶𝑉 → ((𝐺 ∈ Word dom 𝐸 ∧ Fun 𝐺) ∧ 𝑃:(0...(#‘𝐺))⟶𝑉 ∧ ∀𝑙 ∈ (0..^(#‘𝐺))(𝐸‘(𝐺𝑙)) = {(𝑃𝑙), (𝑃‘(𝑙 + 1))}))))
130129ex 449 . . . . . . . . 9 (𝐹:(1...𝑛)–1-1-onto→dom 𝐸 → (𝑛 ∈ ℕ0 → (∀𝑘 ∈ (1...𝑛)(𝐸‘(𝐹𝑘)) = {(𝑃‘(𝑘 − 1)), (𝑃𝑘)} → (𝑃:(0...𝑛)⟶𝑉 → ((𝐺 ∈ Word dom 𝐸 ∧ Fun 𝐺) ∧ 𝑃:(0...(#‘𝐺))⟶𝑉 ∧ ∀𝑙 ∈ (0..^(#‘𝐺))(𝐸‘(𝐺𝑙)) = {(𝑃𝑙), (𝑃‘(𝑙 + 1))})))))
131130com24 93 . . . . . . . 8 (𝐹:(1...𝑛)–1-1-onto→dom 𝐸 → (𝑃:(0...𝑛)⟶𝑉 → (∀𝑘 ∈ (1...𝑛)(𝐸‘(𝐹𝑘)) = {(𝑃‘(𝑘 − 1)), (𝑃𝑘)} → (𝑛 ∈ ℕ0 → ((𝐺 ∈ Word dom 𝐸 ∧ Fun 𝐺) ∧ 𝑃:(0...(#‘𝐺))⟶𝑉 ∧ ∀𝑙 ∈ (0..^(#‘𝐺))(𝐸‘(𝐺𝑙)) = {(𝑃𝑙), (𝑃‘(𝑙 + 1))})))))
1321313imp 1249 . . . . . . 7 ((𝐹:(1...𝑛)–1-1-onto→dom 𝐸𝑃:(0...𝑛)⟶𝑉 ∧ ∀𝑘 ∈ (1...𝑛)(𝐸‘(𝐹𝑘)) = {(𝑃‘(𝑘 − 1)), (𝑃𝑘)}) → (𝑛 ∈ ℕ0 → ((𝐺 ∈ Word dom 𝐸 ∧ Fun 𝐺) ∧ 𝑃:(0...(#‘𝐺))⟶𝑉 ∧ ∀𝑙 ∈ (0..^(#‘𝐺))(𝐸‘(𝐺𝑙)) = {(𝑃𝑙), (𝑃‘(𝑙 + 1))})))
133132com12 32 . . . . . 6 (𝑛 ∈ ℕ0 → ((𝐹:(1...𝑛)–1-1-onto→dom 𝐸𝑃:(0...𝑛)⟶𝑉 ∧ ∀𝑘 ∈ (1...𝑛)(𝐸‘(𝐹𝑘)) = {(𝑃‘(𝑘 − 1)), (𝑃𝑘)}) → ((𝐺 ∈ Word dom 𝐸 ∧ Fun 𝐺) ∧ 𝑃:(0...(#‘𝐺))⟶𝑉 ∧ ∀𝑙 ∈ (0..^(#‘𝐺))(𝐸‘(𝐺𝑙)) = {(𝑃𝑙), (𝑃‘(𝑙 + 1))})))
134133rexlimiv 3009 . . . . 5 (∃𝑛 ∈ ℕ0 (𝐹:(1...𝑛)–1-1-onto→dom 𝐸𝑃:(0...𝑛)⟶𝑉 ∧ ∀𝑘 ∈ (1...𝑛)(𝐸‘(𝐹𝑘)) = {(𝑃‘(𝑘 − 1)), (𝑃𝑘)}) → ((𝐺 ∈ Word dom 𝐸 ∧ Fun 𝐺) ∧ 𝑃:(0...(#‘𝐺))⟶𝑉 ∧ ∀𝑙 ∈ (0..^(#‘𝐺))(𝐸‘(𝐺𝑙)) = {(𝑃𝑙), (𝑃‘(𝑙 + 1))}))
135134adantl 481 . . . 4 ((𝑉 UMGrph 𝐸 ∧ ∃𝑛 ∈ ℕ0 (𝐹:(1...𝑛)–1-1-onto→dom 𝐸𝑃:(0...𝑛)⟶𝑉 ∧ ∀𝑘 ∈ (1...𝑛)(𝐸‘(𝐹𝑘)) = {(𝑃‘(𝑘 − 1)), (𝑃𝑘)})) → ((𝐺 ∈ Word dom 𝐸 ∧ Fun 𝐺) ∧ 𝑃:(0...(#‘𝐺))⟶𝑉 ∧ ∀𝑙 ∈ (0..^(#‘𝐺))(𝐸‘(𝐺𝑙)) = {(𝑃𝑙), (𝑃‘(𝑙 + 1))}))
136135a1i 11 . . 3 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐺 ∈ V ∧ 𝑃 ∈ V)) → ((𝑉 UMGrph 𝐸 ∧ ∃𝑛 ∈ ℕ0 (𝐹:(1...𝑛)–1-1-onto→dom 𝐸𝑃:(0...𝑛)⟶𝑉 ∧ ∀𝑘 ∈ (1...𝑛)(𝐸‘(𝐹𝑘)) = {(𝑃‘(𝑘 − 1)), (𝑃𝑘)})) → ((𝐺 ∈ Word dom 𝐸 ∧ Fun 𝐺) ∧ 𝑃:(0...(#‘𝐺))⟶𝑉 ∧ ∀𝑙 ∈ (0..^(#‘𝐺))(𝐸‘(𝐺𝑙)) = {(𝑃𝑙), (𝑃‘(𝑙 + 1))})))
137 eqid 2610 . . . 4 dom 𝐸 = dom 𝐸
138 iseupa 26492 . . . 4 (dom 𝐸 = dom 𝐸 → (𝐹(𝑉 EulPaths 𝐸)𝑃 ↔ (𝑉 UMGrph 𝐸 ∧ ∃𝑛 ∈ ℕ0 (𝐹:(1...𝑛)–1-1-onto→dom 𝐸𝑃:(0...𝑛)⟶𝑉 ∧ ∀𝑘 ∈ (1...𝑛)(𝐸‘(𝐹𝑘)) = {(𝑃‘(𝑘 − 1)), (𝑃𝑘)}))))
139137, 138mp1i 13 . . 3 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐺 ∈ V ∧ 𝑃 ∈ V)) → (𝐹(𝑉 EulPaths 𝐸)𝑃 ↔ (𝑉 UMGrph 𝐸 ∧ ∃𝑛 ∈ ℕ0 (𝐹:(1...𝑛)–1-1-onto→dom 𝐸𝑃:(0...𝑛)⟶𝑉 ∧ ∀𝑘 ∈ (1...𝑛)(𝐸‘(𝐹𝑘)) = {(𝑃‘(𝑘 − 1)), (𝑃𝑘)}))))
140 istrl 26067 . . 3 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐺 ∈ V ∧ 𝑃 ∈ V)) → (𝐺(𝑉 Trails 𝐸)𝑃 ↔ ((𝐺 ∈ Word dom 𝐸 ∧ Fun 𝐺) ∧ 𝑃:(0...(#‘𝐺))⟶𝑉 ∧ ∀𝑙 ∈ (0..^(#‘𝐺))(𝐸‘(𝐺𝑙)) = {(𝑃𝑙), (𝑃‘(𝑙 + 1))})))
141136, 139, 1403imtr4d 282 . 2 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐺 ∈ V ∧ 𝑃 ∈ V)) → (𝐹(𝑉 EulPaths 𝐸)𝑃𝐺(𝑉 Trails 𝐸)𝑃))
14210, 141mpcom 37 1 (𝐹(𝑉 EulPaths 𝐸)𝑃𝐺(𝑉 Trails 𝐸)𝑃)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  Vcvv 3173  {cpr 4127   class class class wbr 4583   ↦ cmpt 4643  ◡ccnv 5037  dom cdm 5038  Fun wfun 5798   Fn wfn 5799  ⟶wf 5800  –1-1→wf1 5801  –onto→wfo 5802  –1-1-onto→wf1o 5803  ‘cfv 5804  (class class class)co 6549  ℂcc 9813  0cc0 9815  1c1 9816   + caddc 9818   − cmin 10145  ℕ0cn0 11169  ...cfz 12197  ..^cfzo 12334  #chash 12979  Word cword 13146   UMGrph cumg 25841   Trails ctrail 26027   EulPaths ceup 26489 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-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-hash 12980  df-word 13154  df-umgra 25842  df-wlk 26036  df-trail 26037  df-eupa 26490 This theorem is referenced by: (None)
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