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Theorem wlkeq 26764
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 26758 . . . . 5 (𝐴 ∈ (Walks‘𝐺) → 𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩)
2 1st2ndb 7355 . . . . 5 (𝐴 ∈ (V × V) ↔ 𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩)
31, 2sylibr 224 . . . 4 (𝐴 ∈ (Walks‘𝐺) → 𝐴 ∈ (V × V))
4 wlkop 26758 . . . . 5 (𝐵 ∈ (Walks‘𝐺) → 𝐵 = ⟨(1st𝐵), (2nd𝐵)⟩)
5 1st2ndb 7355 . . . . 5 (𝐵 ∈ (V × V) ↔ 𝐵 = ⟨(1st𝐵), (2nd𝐵)⟩)
64, 5sylibr 224 . . . 4 (𝐵 ∈ (Walks‘𝐺) → 𝐵 ∈ (V × V))
7 xpopth 7356 . . . . 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 583 . . 3 ((𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) → (𝐴 = 𝐵 ↔ ((1st𝐴) = (1st𝐵) ∧ (2nd𝐴) = (2nd𝐵))))
1093adant3 1126 . 2 ((𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺) ∧ 𝑁 = (♯‘(1st𝐴))) → (𝐴 = 𝐵 ↔ ((1st𝐴) = (1st𝐵) ∧ (2nd𝐴) = (2nd𝐵))))
11 eqid 2771 . . . . . . 7 (Vtx‘𝐺) = (Vtx‘𝐺)
12 eqid 2771 . . . . . . 7 (iEdg‘𝐺) = (iEdg‘𝐺)
13 eqid 2771 . . . . . . 7 (1st𝐴) = (1st𝐴)
14 eqid 2771 . . . . . . 7 (2nd𝐴) = (2nd𝐴)
1511, 12, 13, 14wlkelwrd 26763 . . . . . 6 (𝐴 ∈ (Walks‘𝐺) → ((1st𝐴) ∈ Word dom (iEdg‘𝐺) ∧ (2nd𝐴):(0...(♯‘(1st𝐴)))⟶(Vtx‘𝐺)))
16 eqid 2771 . . . . . . 7 (1st𝐵) = (1st𝐵)
17 eqid 2771 . . . . . . 7 (2nd𝐵) = (2nd𝐵)
1811, 12, 16, 17wlkelwrd 26763 . . . . . 6 (𝐵 ∈ (Walks‘𝐺) → ((1st𝐵) ∈ Word dom (iEdg‘𝐺) ∧ (2nd𝐵):(0...(♯‘(1st𝐵)))⟶(Vtx‘𝐺)))
1915, 18anim12i 600 . . . . 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 2838 . . . . . . . 8 (𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩ → (𝐴 ∈ (Walks‘𝐺) ↔ ⟨(1st𝐴), (2nd𝐴)⟩ ∈ (Walks‘𝐺)))
21 df-br 4787 . . . . . . . . 9 ((1st𝐴)(Walks‘𝐺)(2nd𝐴) ↔ ⟨(1st𝐴), (2nd𝐴)⟩ ∈ (Walks‘𝐺))
22 wlklenvm1 26752 . . . . . . . . 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 2838 . . . . . . . 8 (𝐵 = ⟨(1st𝐵), (2nd𝐵)⟩ → (𝐵 ∈ (Walks‘𝐺) ↔ ⟨(1st𝐵), (2nd𝐵)⟩ ∈ (Walks‘𝐺)))
27 df-br 4787 . . . . . . . . 9 ((1st𝐵)(Walks‘𝐺)(2nd𝐵) ↔ ⟨(1st𝐵), (2nd𝐵)⟩ ∈ (Walks‘𝐺))
28 wlklenvm1 26752 . . . . . . . . 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 600 . . . . 5 ((𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) → ((♯‘(1st𝐴)) = ((♯‘(2nd𝐴)) − 1) ∧ (♯‘(1st𝐵)) = ((♯‘(2nd𝐵)) − 1)))
33 eqwrd 13543 . . . . . . . 8 (((1st𝐴) ∈ Word dom (iEdg‘𝐺) ∧ (1st𝐵) ∈ Word dom (iEdg‘𝐺)) → ((1st𝐴) = (1st𝐵) ↔ ((♯‘(1st𝐴)) = (♯‘(1st𝐵)) ∧ ∀𝑥 ∈ (0..^(♯‘(1st𝐴)))((1st𝐴)‘𝑥) = ((1st𝐵)‘𝑥))))
3433ad2ant2r 741 . . . . . . 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 466 . . . . . 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 13520 . . . . . . . . . . 11 ((1st𝐴) ∈ Word dom (iEdg‘𝐺) → (♯‘(1st𝐴)) ∈ ℕ0)
3736adantr 466 . . . . . . . . . 10 (((1st𝐴) ∈ Word dom (iEdg‘𝐺) ∧ (2nd𝐴):(0...(♯‘(1st𝐴)))⟶(Vtx‘𝐺)) → (♯‘(1st𝐴)) ∈ ℕ0)
3837adantr 466 . . . . . . . . 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 752 . . . . . . . . 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 756 . . . . . . . . 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 1122 . . . . . . . 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 466 . . . . . . 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 12668 . . . . . . 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 616 . . . . 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 573 . . . 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 1126 . . 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 2775 . . . . . . 7 (𝑁 = (♯‘(1st𝐴)) → (𝑁 = (♯‘(1st𝐵)) ↔ (♯‘(1st𝐴)) = (♯‘(1st𝐵))))
49 oveq2 6801 . . . . . . . 8 (𝑁 = (♯‘(1st𝐴)) → (0..^𝑁) = (0..^(♯‘(1st𝐴))))
5049raleqdv 3293 . . . . . . 7 (𝑁 = (♯‘(1st𝐴)) → (∀𝑥 ∈ (0..^𝑁)((1st𝐴)‘𝑥) = ((1st𝐵)‘𝑥) ↔ ∀𝑥 ∈ (0..^(♯‘(1st𝐴)))((1st𝐴)‘𝑥) = ((1st𝐵)‘𝑥)))
5148, 50anbi12d 616 . . . . . 6 (𝑁 = (♯‘(1st𝐴)) → ((𝑁 = (♯‘(1st𝐵)) ∧ ∀𝑥 ∈ (0..^𝑁)((1st𝐴)‘𝑥) = ((1st𝐵)‘𝑥)) ↔ ((♯‘(1st𝐴)) = (♯‘(1st𝐵)) ∧ ∀𝑥 ∈ (0..^(♯‘(1st𝐴)))((1st𝐴)‘𝑥) = ((1st𝐵)‘𝑥))))
52 oveq2 6801 . . . . . . . 8 (𝑁 = (♯‘(1st𝐴)) → (0...𝑁) = (0...(♯‘(1st𝐴))))
5352raleqdv 3293 . . . . . . 7 (𝑁 = (♯‘(1st𝐴)) → (∀𝑥 ∈ (0...𝑁)((2nd𝐴)‘𝑥) = ((2nd𝐵)‘𝑥) ↔ ∀𝑥 ∈ (0...(♯‘(1st𝐴)))((2nd𝐴)‘𝑥) = ((2nd𝐵)‘𝑥)))
5448, 53anbi12d 616 . . . . . 6 (𝑁 = (♯‘(1st𝐴)) → ((𝑁 = (♯‘(1st𝐵)) ∧ ∀𝑥 ∈ (0...𝑁)((2nd𝐴)‘𝑥) = ((2nd𝐵)‘𝑥)) ↔ ((♯‘(1st𝐴)) = (♯‘(1st𝐵)) ∧ ∀𝑥 ∈ (0...(♯‘(1st𝐴)))((2nd𝐴)‘𝑥) = ((2nd𝐵)‘𝑥))))
5551, 54anbi12d 616 . . . . 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 1129 . . 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 1080 . . . 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 655 . . . 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 382  w3a 1071   = wceq 1631  wcel 2145  wral 3061  Vcvv 3351  cop 4322   class class class wbr 4786   × cxp 5247  dom cdm 5249  wf 6027  cfv 6031  (class class class)co 6793  1st c1st 7313  2nd c2nd 7314  0cc0 10138  1c1 10139  cmin 10468  0cn0 11494  ...cfz 12533  ..^cfzo 12673  chash 13321  Word cword 13487  Vtxcvtx 26095  iEdgciedg 26096  Walkscwlks 26727
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-cnex 10194  ax-resscn 10195  ax-1cn 10196  ax-icn 10197  ax-addcl 10198  ax-addrcl 10199  ax-mulcl 10200  ax-mulrcl 10201  ax-mulcom 10202  ax-addass 10203  ax-mulass 10204  ax-distr 10205  ax-i2m1 10206  ax-1ne0 10207  ax-1rid 10208  ax-rnegex 10209  ax-rrecex 10210  ax-cnre 10211  ax-pre-lttri 10212  ax-pre-lttrn 10213  ax-pre-ltadd 10214  ax-pre-mulgt0 10215
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-ifp 1050  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6754  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-om 7213  df-1st 7315  df-2nd 7316  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-1o 7713  df-oadd 7717  df-er 7896  df-map 8011  df-pm 8012  df-en 8110  df-dom 8111  df-sdom 8112  df-fin 8113  df-card 8965  df-pnf 10278  df-mnf 10279  df-xr 10280  df-ltxr 10281  df-le 10282  df-sub 10470  df-neg 10471  df-nn 11223  df-n0 11495  df-z 11580  df-uz 11889  df-fz 12534  df-fzo 12674  df-hash 13322  df-word 13495  df-wlks 26730
This theorem is referenced by:  uspgr2wlkeq  26777
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