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Theorem 1wlkeq 40838
 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
1wlkeq ((𝐴 ∈ (1Walks‘𝐺) ∧ 𝐵 ∈ (1Walks‘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 1wlkeq
StepHypRef Expression
1 1wlkop 40832 . . . . 5 (𝐴 ∈ (1Walks‘𝐺) → 𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩)
2 1st2ndb 7097 . . . . 5 (𝐴 ∈ (V × V) ↔ 𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩)
31, 2sylibr 223 . . . 4 (𝐴 ∈ (1Walks‘𝐺) → 𝐴 ∈ (V × V))
4 1wlkop 40832 . . . . 5 (𝐵 ∈ (1Walks‘𝐺) → 𝐵 = ⟨(1st𝐵), (2nd𝐵)⟩)
5 1st2ndb 7097 . . . . 5 (𝐵 ∈ (V × V) ↔ 𝐵 = ⟨(1st𝐵), (2nd𝐵)⟩)
64, 5sylibr 223 . . . 4 (𝐵 ∈ (1Walks‘𝐺) → 𝐵 ∈ (V × V))
7 xpopth 7098 . . . . 5 ((𝐴 ∈ (V × V) ∧ 𝐵 ∈ (V × V)) → (((1st𝐴) = (1st𝐵) ∧ (2nd𝐴) = (2nd𝐵)) ↔ 𝐴 = 𝐵))
87bicomd 212 . . . 4 ((𝐴 ∈ (V × V) ∧ 𝐵 ∈ (V × V)) → (𝐴 = 𝐵 ↔ ((1st𝐴) = (1st𝐵) ∧ (2nd𝐴) = (2nd𝐵))))
93, 6, 8syl2an 493 . . 3 ((𝐴 ∈ (1Walks‘𝐺) ∧ 𝐵 ∈ (1Walks‘𝐺)) → (𝐴 = 𝐵 ↔ ((1st𝐴) = (1st𝐵) ∧ (2nd𝐴) = (2nd𝐵))))
1093adant3 1074 . 2 ((𝐴 ∈ (1Walks‘𝐺) ∧ 𝐵 ∈ (1Walks‘𝐺) ∧ 𝑁 = (#‘(1st𝐴))) → (𝐴 = 𝐵 ↔ ((1st𝐴) = (1st𝐵) ∧ (2nd𝐴) = (2nd𝐵))))
11 eqid 2610 . . . . . . 7 (Vtx‘𝐺) = (Vtx‘𝐺)
12 eqid 2610 . . . . . . 7 (iEdg‘𝐺) = (iEdg‘𝐺)
13 eqid 2610 . . . . . . 7 (1st𝐴) = (1st𝐴)
14 eqid 2610 . . . . . . 7 (2nd𝐴) = (2nd𝐴)
1511, 12, 13, 141wlkelwrd 40837 . . . . . 6 (𝐴 ∈ (1Walks‘𝐺) → ((1st𝐴) ∈ Word dom (iEdg‘𝐺) ∧ (2nd𝐴):(0...(#‘(1st𝐴)))⟶(Vtx‘𝐺)))
16 eqid 2610 . . . . . . 7 (1st𝐵) = (1st𝐵)
17 eqid 2610 . . . . . . 7 (2nd𝐵) = (2nd𝐵)
1811, 12, 16, 171wlkelwrd 40837 . . . . . 6 (𝐵 ∈ (1Walks‘𝐺) → ((1st𝐵) ∈ Word dom (iEdg‘𝐺) ∧ (2nd𝐵):(0...(#‘(1st𝐵)))⟶(Vtx‘𝐺)))
1915, 18anim12i 588 . . . . 5 ((𝐴 ∈ (1Walks‘𝐺) ∧ 𝐵 ∈ (1Walks‘𝐺)) → (((1st𝐴) ∈ Word dom (iEdg‘𝐺) ∧ (2nd𝐴):(0...(#‘(1st𝐴)))⟶(Vtx‘𝐺)) ∧ ((1st𝐵) ∈ Word dom (iEdg‘𝐺) ∧ (2nd𝐵):(0...(#‘(1st𝐵)))⟶(Vtx‘𝐺))))
20 eleq1 2676 . . . . . . . 8 (𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩ → (𝐴 ∈ (1Walks‘𝐺) ↔ ⟨(1st𝐴), (2nd𝐴)⟩ ∈ (1Walks‘𝐺)))
21 df-br 4584 . . . . . . . . 9 ((1st𝐴)(1Walks‘𝐺)(2nd𝐴) ↔ ⟨(1st𝐴), (2nd𝐴)⟩ ∈ (1Walks‘𝐺))
22 1wlklenvm1 40826 . . . . . . . . 9 ((1st𝐴)(1Walks‘𝐺)(2nd𝐴) → (#‘(1st𝐴)) = ((#‘(2nd𝐴)) − 1))
2321, 22sylbir 224 . . . . . . . 8 (⟨(1st𝐴), (2nd𝐴)⟩ ∈ (1Walks‘𝐺) → (#‘(1st𝐴)) = ((#‘(2nd𝐴)) − 1))
2420, 23syl6bi 242 . . . . . . 7 (𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩ → (𝐴 ∈ (1Walks‘𝐺) → (#‘(1st𝐴)) = ((#‘(2nd𝐴)) − 1)))
251, 24mpcom 37 . . . . . 6 (𝐴 ∈ (1Walks‘𝐺) → (#‘(1st𝐴)) = ((#‘(2nd𝐴)) − 1))
26 eleq1 2676 . . . . . . . 8 (𝐵 = ⟨(1st𝐵), (2nd𝐵)⟩ → (𝐵 ∈ (1Walks‘𝐺) ↔ ⟨(1st𝐵), (2nd𝐵)⟩ ∈ (1Walks‘𝐺)))
27 df-br 4584 . . . . . . . . 9 ((1st𝐵)(1Walks‘𝐺)(2nd𝐵) ↔ ⟨(1st𝐵), (2nd𝐵)⟩ ∈ (1Walks‘𝐺))
28 1wlklenvm1 40826 . . . . . . . . 9 ((1st𝐵)(1Walks‘𝐺)(2nd𝐵) → (#‘(1st𝐵)) = ((#‘(2nd𝐵)) − 1))
2927, 28sylbir 224 . . . . . . . 8 (⟨(1st𝐵), (2nd𝐵)⟩ ∈ (1Walks‘𝐺) → (#‘(1st𝐵)) = ((#‘(2nd𝐵)) − 1))
3026, 29syl6bi 242 . . . . . . 7 (𝐵 = ⟨(1st𝐵), (2nd𝐵)⟩ → (𝐵 ∈ (1Walks‘𝐺) → (#‘(1st𝐵)) = ((#‘(2nd𝐵)) − 1)))
314, 30mpcom 37 . . . . . 6 (𝐵 ∈ (1Walks‘𝐺) → (#‘(1st𝐵)) = ((#‘(2nd𝐵)) − 1))
3225, 31anim12i 588 . . . . 5 ((𝐴 ∈ (1Walks‘𝐺) ∧ 𝐵 ∈ (1Walks‘𝐺)) → ((#‘(1st𝐴)) = ((#‘(2nd𝐴)) − 1) ∧ (#‘(1st𝐵)) = ((#‘(2nd𝐵)) − 1)))
33 eqwrd 13201 . . . . . . . 8 (((1st𝐴) ∈ Word dom (iEdg‘𝐺) ∧ (1st𝐵) ∈ Word dom (iEdg‘𝐺)) → ((1st𝐴) = (1st𝐵) ↔ ((#‘(1st𝐴)) = (#‘(1st𝐵)) ∧ ∀𝑥 ∈ (0..^(#‘(1st𝐴)))((1st𝐴)‘𝑥) = ((1st𝐵)‘𝑥))))
3433ad2ant2r 779 . . . . . . 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 13179 . . . . . . . . . . 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 788 . . . . . . . . 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 792 . . . . . . . . 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 1235 . . . . . . . 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 12329 . . . . . . 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 743 . . . . 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 691 . . . 4 ((𝐴 ∈ (1Walks‘𝐺) ∧ 𝐵 ∈ (1Walks‘𝐺)) → (((1st𝐴) = (1st𝐵) ∧ (2nd𝐴) = (2nd𝐵)) ↔ (((#‘(1st𝐴)) = (#‘(1st𝐵)) ∧ ∀𝑥 ∈ (0..^(#‘(1st𝐴)))((1st𝐴)‘𝑥) = ((1st𝐵)‘𝑥)) ∧ ((#‘(1st𝐴)) = (#‘(1st𝐵)) ∧ ∀𝑥 ∈ (0...(#‘(1st𝐴)))((2nd𝐴)‘𝑥) = ((2nd𝐵)‘𝑥)))))
47463adant3 1074 . . 3 ((𝐴 ∈ (1Walks‘𝐺) ∧ 𝐵 ∈ (1Walks‘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 2614 . . . . . . 7 (𝑁 = (#‘(1st𝐴)) → (𝑁 = (#‘(1st𝐵)) ↔ (#‘(1st𝐴)) = (#‘(1st𝐵))))
49 oveq2 6557 . . . . . . . 8 (𝑁 = (#‘(1st𝐴)) → (0..^𝑁) = (0..^(#‘(1st𝐴))))
5049raleqdv 3121 . . . . . . 7 (𝑁 = (#‘(1st𝐴)) → (∀𝑥 ∈ (0..^𝑁)((1st𝐴)‘𝑥) = ((1st𝐵)‘𝑥) ↔ ∀𝑥 ∈ (0..^(#‘(1st𝐴)))((1st𝐴)‘𝑥) = ((1st𝐵)‘𝑥)))
5148, 50anbi12d 743 . . . . . 6 (𝑁 = (#‘(1st𝐴)) → ((𝑁 = (#‘(1st𝐵)) ∧ ∀𝑥 ∈ (0..^𝑁)((1st𝐴)‘𝑥) = ((1st𝐵)‘𝑥)) ↔ ((#‘(1st𝐴)) = (#‘(1st𝐵)) ∧ ∀𝑥 ∈ (0..^(#‘(1st𝐴)))((1st𝐴)‘𝑥) = ((1st𝐵)‘𝑥))))
52 oveq2 6557 . . . . . . . 8 (𝑁 = (#‘(1st𝐴)) → (0...𝑁) = (0...(#‘(1st𝐴))))
5352raleqdv 3121 . . . . . . 7 (𝑁 = (#‘(1st𝐴)) → (∀𝑥 ∈ (0...𝑁)((2nd𝐴)‘𝑥) = ((2nd𝐵)‘𝑥) ↔ ∀𝑥 ∈ (0...(#‘(1st𝐴)))((2nd𝐴)‘𝑥) = ((2nd𝐵)‘𝑥)))
5448, 53anbi12d 743 . . . . . 6 (𝑁 = (#‘(1st𝐴)) → ((𝑁 = (#‘(1st𝐵)) ∧ ∀𝑥 ∈ (0...𝑁)((2nd𝐴)‘𝑥) = ((2nd𝐵)‘𝑥)) ↔ ((#‘(1st𝐴)) = (#‘(1st𝐵)) ∧ ∀𝑥 ∈ (0...(#‘(1st𝐴)))((2nd𝐴)‘𝑥) = ((2nd𝐵)‘𝑥))))
5551, 54anbi12d 743 . . . . 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 1077 . . 3 ((𝐴 ∈ (1Walks‘𝐺) ∧ 𝐵 ∈ (1Walks‘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 246 . 2 ((𝐴 ∈ (1Walks‘𝐺) ∧ 𝐵 ∈ (1Walks‘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 1035 . . . 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 867 . . . 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 264 . . 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 ∈ (1Walks‘𝐺) ∧ 𝐵 ∈ (1Walks‘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 293 1 ((𝐴 ∈ (1Walks‘𝐺) ∧ 𝐵 ∈ (1Walks‘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 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  Vcvv 3173  ⟨cop 4131   class class class wbr 4583   × cxp 5036  dom cdm 5038  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  1st c1st 7057  2nd c2nd 7058  0cc0 9815  1c1 9816   − cmin 10145  ℕ0cn0 11169  ...cfz 12197  ..^cfzo 12334  #chash 12979  Word cword 13146  Vtxcvtx 25673  iEdgciedg 25674  1Walksc1wlks 40796 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-ifp 1007  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-1wlks 40800 This theorem is referenced by:  uspgr2wlkeq  40854
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