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Theorem wlkeq 26585
Description: Conditions for two walks (within the same graph) being the same. (Contributed by AV, 1-Jul-2018.) (Revised by AV, 16-May-2019.) (Revised by AV, 14-Apr-2021.)
Assertion
Ref Expression
wlkeq ((𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺) ∧ 𝑁 = (#‘(1st𝐴))) → (𝐴 = 𝐵 ↔ (𝑁 = (#‘(1st𝐵)) ∧ ∀𝑥 ∈ (0..^𝑁)((1st𝐴)‘𝑥) = ((1st𝐵)‘𝑥) ∧ ∀𝑥 ∈ (0...𝑁)((2nd𝐴)‘𝑥) = ((2nd𝐵)‘𝑥))))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝑁
Allowed substitution hint:   𝐺(𝑥)

Proof of Theorem wlkeq
StepHypRef Expression
1 wlkop 26579 . . . . 5 (𝐴 ∈ (Walks‘𝐺) → 𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩)
2 1st2ndb 7250 . . . . 5 (𝐴 ∈ (V × V) ↔ 𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩)
31, 2sylibr 224 . . . 4 (𝐴 ∈ (Walks‘𝐺) → 𝐴 ∈ (V × V))
4 wlkop 26579 . . . . 5 (𝐵 ∈ (Walks‘𝐺) → 𝐵 = ⟨(1st𝐵), (2nd𝐵)⟩)
5 1st2ndb 7250 . . . . 5 (𝐵 ∈ (V × V) ↔ 𝐵 = ⟨(1st𝐵), (2nd𝐵)⟩)
64, 5sylibr 224 . . . 4 (𝐵 ∈ (Walks‘𝐺) → 𝐵 ∈ (V × V))
7 xpopth 7251 . . . . 5 ((𝐴 ∈ (V × V) ∧ 𝐵 ∈ (V × V)) → (((1st𝐴) = (1st𝐵) ∧ (2nd𝐴) = (2nd𝐵)) ↔ 𝐴 = 𝐵))
87bicomd 213 . . . 4 ((𝐴 ∈ (V × V) ∧ 𝐵 ∈ (V × V)) → (𝐴 = 𝐵 ↔ ((1st𝐴) = (1st𝐵) ∧ (2nd𝐴) = (2nd𝐵))))
93, 6, 8syl2an 493 . . 3 ((𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) → (𝐴 = 𝐵 ↔ ((1st𝐴) = (1st𝐵) ∧ (2nd𝐴) = (2nd𝐵))))
1093adant3 1101 . 2 ((𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺) ∧ 𝑁 = (#‘(1st𝐴))) → (𝐴 = 𝐵 ↔ ((1st𝐴) = (1st𝐵) ∧ (2nd𝐴) = (2nd𝐵))))
11 eqid 2651 . . . . . . 7 (Vtx‘𝐺) = (Vtx‘𝐺)
12 eqid 2651 . . . . . . 7 (iEdg‘𝐺) = (iEdg‘𝐺)
13 eqid 2651 . . . . . . 7 (1st𝐴) = (1st𝐴)
14 eqid 2651 . . . . . . 7 (2nd𝐴) = (2nd𝐴)
1511, 12, 13, 14wlkelwrd 26584 . . . . . 6 (𝐴 ∈ (Walks‘𝐺) → ((1st𝐴) ∈ Word dom (iEdg‘𝐺) ∧ (2nd𝐴):(0...(#‘(1st𝐴)))⟶(Vtx‘𝐺)))
16 eqid 2651 . . . . . . 7 (1st𝐵) = (1st𝐵)
17 eqid 2651 . . . . . . 7 (2nd𝐵) = (2nd𝐵)
1811, 12, 16, 17wlkelwrd 26584 . . . . . 6 (𝐵 ∈ (Walks‘𝐺) → ((1st𝐵) ∈ Word dom (iEdg‘𝐺) ∧ (2nd𝐵):(0...(#‘(1st𝐵)))⟶(Vtx‘𝐺)))
1915, 18anim12i 589 . . . . 5 ((𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) → (((1st𝐴) ∈ Word dom (iEdg‘𝐺) ∧ (2nd𝐴):(0...(#‘(1st𝐴)))⟶(Vtx‘𝐺)) ∧ ((1st𝐵) ∈ Word dom (iEdg‘𝐺) ∧ (2nd𝐵):(0...(#‘(1st𝐵)))⟶(Vtx‘𝐺))))
20 eleq1 2718 . . . . . . . 8 (𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩ → (𝐴 ∈ (Walks‘𝐺) ↔ ⟨(1st𝐴), (2nd𝐴)⟩ ∈ (Walks‘𝐺)))
21 df-br 4686 . . . . . . . . 9 ((1st𝐴)(Walks‘𝐺)(2nd𝐴) ↔ ⟨(1st𝐴), (2nd𝐴)⟩ ∈ (Walks‘𝐺))
22 wlklenvm1 26573 . . . . . . . . 9 ((1st𝐴)(Walks‘𝐺)(2nd𝐴) → (#‘(1st𝐴)) = ((#‘(2nd𝐴)) − 1))
2321, 22sylbir 225 . . . . . . . 8 (⟨(1st𝐴), (2nd𝐴)⟩ ∈ (Walks‘𝐺) → (#‘(1st𝐴)) = ((#‘(2nd𝐴)) − 1))
2420, 23syl6bi 243 . . . . . . 7 (𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩ → (𝐴 ∈ (Walks‘𝐺) → (#‘(1st𝐴)) = ((#‘(2nd𝐴)) − 1)))
251, 24mpcom 38 . . . . . 6 (𝐴 ∈ (Walks‘𝐺) → (#‘(1st𝐴)) = ((#‘(2nd𝐴)) − 1))
26 eleq1 2718 . . . . . . . 8 (𝐵 = ⟨(1st𝐵), (2nd𝐵)⟩ → (𝐵 ∈ (Walks‘𝐺) ↔ ⟨(1st𝐵), (2nd𝐵)⟩ ∈ (Walks‘𝐺)))
27 df-br 4686 . . . . . . . . 9 ((1st𝐵)(Walks‘𝐺)(2nd𝐵) ↔ ⟨(1st𝐵), (2nd𝐵)⟩ ∈ (Walks‘𝐺))
28 wlklenvm1 26573 . . . . . . . . 9 ((1st𝐵)(Walks‘𝐺)(2nd𝐵) → (#‘(1st𝐵)) = ((#‘(2nd𝐵)) − 1))
2927, 28sylbir 225 . . . . . . . 8 (⟨(1st𝐵), (2nd𝐵)⟩ ∈ (Walks‘𝐺) → (#‘(1st𝐵)) = ((#‘(2nd𝐵)) − 1))
3026, 29syl6bi 243 . . . . . . 7 (𝐵 = ⟨(1st𝐵), (2nd𝐵)⟩ → (𝐵 ∈ (Walks‘𝐺) → (#‘(1st𝐵)) = ((#‘(2nd𝐵)) − 1)))
314, 30mpcom 38 . . . . . 6 (𝐵 ∈ (Walks‘𝐺) → (#‘(1st𝐵)) = ((#‘(2nd𝐵)) − 1))
3225, 31anim12i 589 . . . . 5 ((𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) → ((#‘(1st𝐴)) = ((#‘(2nd𝐴)) − 1) ∧ (#‘(1st𝐵)) = ((#‘(2nd𝐵)) − 1)))
33 eqwrd 13379 . . . . . . . 8 (((1st𝐴) ∈ Word dom (iEdg‘𝐺) ∧ (1st𝐵) ∈ Word dom (iEdg‘𝐺)) → ((1st𝐴) = (1st𝐵) ↔ ((#‘(1st𝐴)) = (#‘(1st𝐵)) ∧ ∀𝑥 ∈ (0..^(#‘(1st𝐴)))((1st𝐴)‘𝑥) = ((1st𝐵)‘𝑥))))
3433ad2ant2r 798 . . . . . . 7 ((((1st𝐴) ∈ Word dom (iEdg‘𝐺) ∧ (2nd𝐴):(0...(#‘(1st𝐴)))⟶(Vtx‘𝐺)) ∧ ((1st𝐵) ∈ Word dom (iEdg‘𝐺) ∧ (2nd𝐵):(0...(#‘(1st𝐵)))⟶(Vtx‘𝐺))) → ((1st𝐴) = (1st𝐵) ↔ ((#‘(1st𝐴)) = (#‘(1st𝐵)) ∧ ∀𝑥 ∈ (0..^(#‘(1st𝐴)))((1st𝐴)‘𝑥) = ((1st𝐵)‘𝑥))))
3534adantr 480 . . . . . 6 (((((1st𝐴) ∈ Word dom (iEdg‘𝐺) ∧ (2nd𝐴):(0...(#‘(1st𝐴)))⟶(Vtx‘𝐺)) ∧ ((1st𝐵) ∈ Word dom (iEdg‘𝐺) ∧ (2nd𝐵):(0...(#‘(1st𝐵)))⟶(Vtx‘𝐺))) ∧ ((#‘(1st𝐴)) = ((#‘(2nd𝐴)) − 1) ∧ (#‘(1st𝐵)) = ((#‘(2nd𝐵)) − 1))) → ((1st𝐴) = (1st𝐵) ↔ ((#‘(1st𝐴)) = (#‘(1st𝐵)) ∧ ∀𝑥 ∈ (0..^(#‘(1st𝐴)))((1st𝐴)‘𝑥) = ((1st𝐵)‘𝑥))))
36 lencl 13356 . . . . . . . . . . 11 ((1st𝐴) ∈ Word dom (iEdg‘𝐺) → (#‘(1st𝐴)) ∈ ℕ0)
3736adantr 480 . . . . . . . . . 10 (((1st𝐴) ∈ Word dom (iEdg‘𝐺) ∧ (2nd𝐴):(0...(#‘(1st𝐴)))⟶(Vtx‘𝐺)) → (#‘(1st𝐴)) ∈ ℕ0)
3837adantr 480 . . . . . . . . 9 ((((1st𝐴) ∈ Word dom (iEdg‘𝐺) ∧ (2nd𝐴):(0...(#‘(1st𝐴)))⟶(Vtx‘𝐺)) ∧ ((1st𝐵) ∈ Word dom (iEdg‘𝐺) ∧ (2nd𝐵):(0...(#‘(1st𝐵)))⟶(Vtx‘𝐺))) → (#‘(1st𝐴)) ∈ ℕ0)
39 simplr 807 . . . . . . . . 9 ((((1st𝐴) ∈ Word dom (iEdg‘𝐺) ∧ (2nd𝐴):(0...(#‘(1st𝐴)))⟶(Vtx‘𝐺)) ∧ ((1st𝐵) ∈ Word dom (iEdg‘𝐺) ∧ (2nd𝐵):(0...(#‘(1st𝐵)))⟶(Vtx‘𝐺))) → (2nd𝐴):(0...(#‘(1st𝐴)))⟶(Vtx‘𝐺))
40 simprr 811 . . . . . . . . 9 ((((1st𝐴) ∈ Word dom (iEdg‘𝐺) ∧ (2nd𝐴):(0...(#‘(1st𝐴)))⟶(Vtx‘𝐺)) ∧ ((1st𝐵) ∈ Word dom (iEdg‘𝐺) ∧ (2nd𝐵):(0...(#‘(1st𝐵)))⟶(Vtx‘𝐺))) → (2nd𝐵):(0...(#‘(1st𝐵)))⟶(Vtx‘𝐺))
4138, 39, 403jca 1261 . . . . . . . 8 ((((1st𝐴) ∈ Word dom (iEdg‘𝐺) ∧ (2nd𝐴):(0...(#‘(1st𝐴)))⟶(Vtx‘𝐺)) ∧ ((1st𝐵) ∈ Word dom (iEdg‘𝐺) ∧ (2nd𝐵):(0...(#‘(1st𝐵)))⟶(Vtx‘𝐺))) → ((#‘(1st𝐴)) ∈ ℕ0 ∧ (2nd𝐴):(0...(#‘(1st𝐴)))⟶(Vtx‘𝐺) ∧ (2nd𝐵):(0...(#‘(1st𝐵)))⟶(Vtx‘𝐺)))
4241adantr 480 . . . . . . 7 (((((1st𝐴) ∈ Word dom (iEdg‘𝐺) ∧ (2nd𝐴):(0...(#‘(1st𝐴)))⟶(Vtx‘𝐺)) ∧ ((1st𝐵) ∈ Word dom (iEdg‘𝐺) ∧ (2nd𝐵):(0...(#‘(1st𝐵)))⟶(Vtx‘𝐺))) ∧ ((#‘(1st𝐴)) = ((#‘(2nd𝐴)) − 1) ∧ (#‘(1st𝐵)) = ((#‘(2nd𝐵)) − 1))) → ((#‘(1st𝐴)) ∈ ℕ0 ∧ (2nd𝐴):(0...(#‘(1st𝐴)))⟶(Vtx‘𝐺) ∧ (2nd𝐵):(0...(#‘(1st𝐵)))⟶(Vtx‘𝐺)))
43 2ffzeq 12499 . . . . . . 7 (((#‘(1st𝐴)) ∈ ℕ0 ∧ (2nd𝐴):(0...(#‘(1st𝐴)))⟶(Vtx‘𝐺) ∧ (2nd𝐵):(0...(#‘(1st𝐵)))⟶(Vtx‘𝐺)) → ((2nd𝐴) = (2nd𝐵) ↔ ((#‘(1st𝐴)) = (#‘(1st𝐵)) ∧ ∀𝑥 ∈ (0...(#‘(1st𝐴)))((2nd𝐴)‘𝑥) = ((2nd𝐵)‘𝑥))))
4442, 43syl 17 . . . . . 6 (((((1st𝐴) ∈ Word dom (iEdg‘𝐺) ∧ (2nd𝐴):(0...(#‘(1st𝐴)))⟶(Vtx‘𝐺)) ∧ ((1st𝐵) ∈ Word dom (iEdg‘𝐺) ∧ (2nd𝐵):(0...(#‘(1st𝐵)))⟶(Vtx‘𝐺))) ∧ ((#‘(1st𝐴)) = ((#‘(2nd𝐴)) − 1) ∧ (#‘(1st𝐵)) = ((#‘(2nd𝐵)) − 1))) → ((2nd𝐴) = (2nd𝐵) ↔ ((#‘(1st𝐴)) = (#‘(1st𝐵)) ∧ ∀𝑥 ∈ (0...(#‘(1st𝐴)))((2nd𝐴)‘𝑥) = ((2nd𝐵)‘𝑥))))
4535, 44anbi12d 747 . . . . 5 (((((1st𝐴) ∈ Word dom (iEdg‘𝐺) ∧ (2nd𝐴):(0...(#‘(1st𝐴)))⟶(Vtx‘𝐺)) ∧ ((1st𝐵) ∈ Word dom (iEdg‘𝐺) ∧ (2nd𝐵):(0...(#‘(1st𝐵)))⟶(Vtx‘𝐺))) ∧ ((#‘(1st𝐴)) = ((#‘(2nd𝐴)) − 1) ∧ (#‘(1st𝐵)) = ((#‘(2nd𝐵)) − 1))) → (((1st𝐴) = (1st𝐵) ∧ (2nd𝐴) = (2nd𝐵)) ↔ (((#‘(1st𝐴)) = (#‘(1st𝐵)) ∧ ∀𝑥 ∈ (0..^(#‘(1st𝐴)))((1st𝐴)‘𝑥) = ((1st𝐵)‘𝑥)) ∧ ((#‘(1st𝐴)) = (#‘(1st𝐵)) ∧ ∀𝑥 ∈ (0...(#‘(1st𝐴)))((2nd𝐴)‘𝑥) = ((2nd𝐵)‘𝑥)))))
4619, 32, 45syl2anc 694 . . . 4 ((𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) → (((1st𝐴) = (1st𝐵) ∧ (2nd𝐴) = (2nd𝐵)) ↔ (((#‘(1st𝐴)) = (#‘(1st𝐵)) ∧ ∀𝑥 ∈ (0..^(#‘(1st𝐴)))((1st𝐴)‘𝑥) = ((1st𝐵)‘𝑥)) ∧ ((#‘(1st𝐴)) = (#‘(1st𝐵)) ∧ ∀𝑥 ∈ (0...(#‘(1st𝐴)))((2nd𝐴)‘𝑥) = ((2nd𝐵)‘𝑥)))))
47463adant3 1101 . . 3 ((𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺) ∧ 𝑁 = (#‘(1st𝐴))) → (((1st𝐴) = (1st𝐵) ∧ (2nd𝐴) = (2nd𝐵)) ↔ (((#‘(1st𝐴)) = (#‘(1st𝐵)) ∧ ∀𝑥 ∈ (0..^(#‘(1st𝐴)))((1st𝐴)‘𝑥) = ((1st𝐵)‘𝑥)) ∧ ((#‘(1st𝐴)) = (#‘(1st𝐵)) ∧ ∀𝑥 ∈ (0...(#‘(1st𝐴)))((2nd𝐴)‘𝑥) = ((2nd𝐵)‘𝑥)))))
48 eqeq1 2655 . . . . . . 7 (𝑁 = (#‘(1st𝐴)) → (𝑁 = (#‘(1st𝐵)) ↔ (#‘(1st𝐴)) = (#‘(1st𝐵))))
49 oveq2 6698 . . . . . . . 8 (𝑁 = (#‘(1st𝐴)) → (0..^𝑁) = (0..^(#‘(1st𝐴))))
5049raleqdv 3174 . . . . . . 7 (𝑁 = (#‘(1st𝐴)) → (∀𝑥 ∈ (0..^𝑁)((1st𝐴)‘𝑥) = ((1st𝐵)‘𝑥) ↔ ∀𝑥 ∈ (0..^(#‘(1st𝐴)))((1st𝐴)‘𝑥) = ((1st𝐵)‘𝑥)))
5148, 50anbi12d 747 . . . . . 6 (𝑁 = (#‘(1st𝐴)) → ((𝑁 = (#‘(1st𝐵)) ∧ ∀𝑥 ∈ (0..^𝑁)((1st𝐴)‘𝑥) = ((1st𝐵)‘𝑥)) ↔ ((#‘(1st𝐴)) = (#‘(1st𝐵)) ∧ ∀𝑥 ∈ (0..^(#‘(1st𝐴)))((1st𝐴)‘𝑥) = ((1st𝐵)‘𝑥))))
52 oveq2 6698 . . . . . . . 8 (𝑁 = (#‘(1st𝐴)) → (0...𝑁) = (0...(#‘(1st𝐴))))
5352raleqdv 3174 . . . . . . 7 (𝑁 = (#‘(1st𝐴)) → (∀𝑥 ∈ (0...𝑁)((2nd𝐴)‘𝑥) = ((2nd𝐵)‘𝑥) ↔ ∀𝑥 ∈ (0...(#‘(1st𝐴)))((2nd𝐴)‘𝑥) = ((2nd𝐵)‘𝑥)))
5448, 53anbi12d 747 . . . . . 6 (𝑁 = (#‘(1st𝐴)) → ((𝑁 = (#‘(1st𝐵)) ∧ ∀𝑥 ∈ (0...𝑁)((2nd𝐴)‘𝑥) = ((2nd𝐵)‘𝑥)) ↔ ((#‘(1st𝐴)) = (#‘(1st𝐵)) ∧ ∀𝑥 ∈ (0...(#‘(1st𝐴)))((2nd𝐴)‘𝑥) = ((2nd𝐵)‘𝑥))))
5551, 54anbi12d 747 . . . . 5 (𝑁 = (#‘(1st𝐴)) → (((𝑁 = (#‘(1st𝐵)) ∧ ∀𝑥 ∈ (0..^𝑁)((1st𝐴)‘𝑥) = ((1st𝐵)‘𝑥)) ∧ (𝑁 = (#‘(1st𝐵)) ∧ ∀𝑥 ∈ (0...𝑁)((2nd𝐴)‘𝑥) = ((2nd𝐵)‘𝑥))) ↔ (((#‘(1st𝐴)) = (#‘(1st𝐵)) ∧ ∀𝑥 ∈ (0..^(#‘(1st𝐴)))((1st𝐴)‘𝑥) = ((1st𝐵)‘𝑥)) ∧ ((#‘(1st𝐴)) = (#‘(1st𝐵)) ∧ ∀𝑥 ∈ (0...(#‘(1st𝐴)))((2nd𝐴)‘𝑥) = ((2nd𝐵)‘𝑥)))))
5655bibi2d 331 . . . 4 (𝑁 = (#‘(1st𝐴)) → ((((1st𝐴) = (1st𝐵) ∧ (2nd𝐴) = (2nd𝐵)) ↔ ((𝑁 = (#‘(1st𝐵)) ∧ ∀𝑥 ∈ (0..^𝑁)((1st𝐴)‘𝑥) = ((1st𝐵)‘𝑥)) ∧ (𝑁 = (#‘(1st𝐵)) ∧ ∀𝑥 ∈ (0...𝑁)((2nd𝐴)‘𝑥) = ((2nd𝐵)‘𝑥)))) ↔ (((1st𝐴) = (1st𝐵) ∧ (2nd𝐴) = (2nd𝐵)) ↔ (((#‘(1st𝐴)) = (#‘(1st𝐵)) ∧ ∀𝑥 ∈ (0..^(#‘(1st𝐴)))((1st𝐴)‘𝑥) = ((1st𝐵)‘𝑥)) ∧ ((#‘(1st𝐴)) = (#‘(1st𝐵)) ∧ ∀𝑥 ∈ (0...(#‘(1st𝐴)))((2nd𝐴)‘𝑥) = ((2nd𝐵)‘𝑥))))))
57563ad2ant3 1104 . . 3 ((𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺) ∧ 𝑁 = (#‘(1st𝐴))) → ((((1st𝐴) = (1st𝐵) ∧ (2nd𝐴) = (2nd𝐵)) ↔ ((𝑁 = (#‘(1st𝐵)) ∧ ∀𝑥 ∈ (0..^𝑁)((1st𝐴)‘𝑥) = ((1st𝐵)‘𝑥)) ∧ (𝑁 = (#‘(1st𝐵)) ∧ ∀𝑥 ∈ (0...𝑁)((2nd𝐴)‘𝑥) = ((2nd𝐵)‘𝑥)))) ↔ (((1st𝐴) = (1st𝐵) ∧ (2nd𝐴) = (2nd𝐵)) ↔ (((#‘(1st𝐴)) = (#‘(1st𝐵)) ∧ ∀𝑥 ∈ (0..^(#‘(1st𝐴)))((1st𝐴)‘𝑥) = ((1st𝐵)‘𝑥)) ∧ ((#‘(1st𝐴)) = (#‘(1st𝐵)) ∧ ∀𝑥 ∈ (0...(#‘(1st𝐴)))((2nd𝐴)‘𝑥) = ((2nd𝐵)‘𝑥))))))
5847, 57mpbird 247 . 2 ((𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺) ∧ 𝑁 = (#‘(1st𝐴))) → (((1st𝐴) = (1st𝐵) ∧ (2nd𝐴) = (2nd𝐵)) ↔ ((𝑁 = (#‘(1st𝐵)) ∧ ∀𝑥 ∈ (0..^𝑁)((1st𝐴)‘𝑥) = ((1st𝐵)‘𝑥)) ∧ (𝑁 = (#‘(1st𝐵)) ∧ ∀𝑥 ∈ (0...𝑁)((2nd𝐴)‘𝑥) = ((2nd𝐵)‘𝑥)))))
59 3anass 1059 . . . 4 ((𝑁 = (#‘(1st𝐵)) ∧ ∀𝑥 ∈ (0..^𝑁)((1st𝐴)‘𝑥) = ((1st𝐵)‘𝑥) ∧ ∀𝑥 ∈ (0...𝑁)((2nd𝐴)‘𝑥) = ((2nd𝐵)‘𝑥)) ↔ (𝑁 = (#‘(1st𝐵)) ∧ (∀𝑥 ∈ (0..^𝑁)((1st𝐴)‘𝑥) = ((1st𝐵)‘𝑥) ∧ ∀𝑥 ∈ (0...𝑁)((2nd𝐴)‘𝑥) = ((2nd𝐵)‘𝑥))))
60 anandi 888 . . . 4 ((𝑁 = (#‘(1st𝐵)) ∧ (∀𝑥 ∈ (0..^𝑁)((1st𝐴)‘𝑥) = ((1st𝐵)‘𝑥) ∧ ∀𝑥 ∈ (0...𝑁)((2nd𝐴)‘𝑥) = ((2nd𝐵)‘𝑥))) ↔ ((𝑁 = (#‘(1st𝐵)) ∧ ∀𝑥 ∈ (0..^𝑁)((1st𝐴)‘𝑥) = ((1st𝐵)‘𝑥)) ∧ (𝑁 = (#‘(1st𝐵)) ∧ ∀𝑥 ∈ (0...𝑁)((2nd𝐴)‘𝑥) = ((2nd𝐵)‘𝑥))))
6159, 60bitr2i 265 . . 3 (((𝑁 = (#‘(1st𝐵)) ∧ ∀𝑥 ∈ (0..^𝑁)((1st𝐴)‘𝑥) = ((1st𝐵)‘𝑥)) ∧ (𝑁 = (#‘(1st𝐵)) ∧ ∀𝑥 ∈ (0...𝑁)((2nd𝐴)‘𝑥) = ((2nd𝐵)‘𝑥))) ↔ (𝑁 = (#‘(1st𝐵)) ∧ ∀𝑥 ∈ (0..^𝑁)((1st𝐴)‘𝑥) = ((1st𝐵)‘𝑥) ∧ ∀𝑥 ∈ (0...𝑁)((2nd𝐴)‘𝑥) = ((2nd𝐵)‘𝑥)))
6261a1i 11 . 2 ((𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺) ∧ 𝑁 = (#‘(1st𝐴))) → (((𝑁 = (#‘(1st𝐵)) ∧ ∀𝑥 ∈ (0..^𝑁)((1st𝐴)‘𝑥) = ((1st𝐵)‘𝑥)) ∧ (𝑁 = (#‘(1st𝐵)) ∧ ∀𝑥 ∈ (0...𝑁)((2nd𝐴)‘𝑥) = ((2nd𝐵)‘𝑥))) ↔ (𝑁 = (#‘(1st𝐵)) ∧ ∀𝑥 ∈ (0..^𝑁)((1st𝐴)‘𝑥) = ((1st𝐵)‘𝑥) ∧ ∀𝑥 ∈ (0...𝑁)((2nd𝐴)‘𝑥) = ((2nd𝐵)‘𝑥))))
6310, 58, 623bitrd 294 1 ((𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺) ∧ 𝑁 = (#‘(1st𝐴))) → (𝐴 = 𝐵 ↔ (𝑁 = (#‘(1st𝐵)) ∧ ∀𝑥 ∈ (0..^𝑁)((1st𝐴)‘𝑥) = ((1st𝐵)‘𝑥) ∧ ∀𝑥 ∈ (0...𝑁)((2nd𝐴)‘𝑥) = ((2nd𝐵)‘𝑥))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  wral 2941  Vcvv 3231  cop 4216   class class class wbr 4685   × cxp 5141  dom cdm 5143  wf 5922  cfv 5926  (class class class)co 6690  1st c1st 7208  2nd c2nd 7209  0cc0 9974  1c1 9975  cmin 10304  0cn0 11330  ...cfz 12364  ..^cfzo 12504  #chash 13157  Word cword 13323  Vtxcvtx 25919  iEdgciedg 25920  Walkscwlks 26548
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-ifp 1033  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-map 7901  df-pm 7902  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-card 8803  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-n0 11331  df-z 11416  df-uz 11726  df-fz 12365  df-fzo 12505  df-hash 13158  df-word 13331  df-wlks 26551
This theorem is referenced by:  uspgr2wlkeq  26598
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