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Theorem wlkeq 29652
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 eqid 2737 . . . . . . 7 (Vtx‘𝐺) = (Vtx‘𝐺)
2 eqid 2737 . . . . . . 7 (iEdg‘𝐺) = (iEdg‘𝐺)
3 eqid 2737 . . . . . . 7 (1st𝐴) = (1st𝐴)
4 eqid 2737 . . . . . . 7 (2nd𝐴) = (2nd𝐴)
51, 2, 3, 4wlkelwrd 29651 . . . . . 6 (𝐴 ∈ (Walks‘𝐺) → ((1st𝐴) ∈ Word dom (iEdg‘𝐺) ∧ (2nd𝐴):(0...(♯‘(1st𝐴)))⟶(Vtx‘𝐺)))
6 eqid 2737 . . . . . . 7 (1st𝐵) = (1st𝐵)
7 eqid 2737 . . . . . . 7 (2nd𝐵) = (2nd𝐵)
81, 2, 6, 7wlkelwrd 29651 . . . . . 6 (𝐵 ∈ (Walks‘𝐺) → ((1st𝐵) ∈ Word dom (iEdg‘𝐺) ∧ (2nd𝐵):(0...(♯‘(1st𝐵)))⟶(Vtx‘𝐺)))
95, 8anim12i 613 . . . . 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))))
10 wlkop 29646 . . . . . . 7 (𝐴 ∈ (Walks‘𝐺) → 𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩)
11 eleq1 2829 . . . . . . . 8 (𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩ → (𝐴 ∈ (Walks‘𝐺) ↔ ⟨(1st𝐴), (2nd𝐴)⟩ ∈ (Walks‘𝐺)))
12 df-br 5144 . . . . . . . . 9 ((1st𝐴)(Walks‘𝐺)(2nd𝐴) ↔ ⟨(1st𝐴), (2nd𝐴)⟩ ∈ (Walks‘𝐺))
13 wlklenvm1 29640 . . . . . . . . 9 ((1st𝐴)(Walks‘𝐺)(2nd𝐴) → (♯‘(1st𝐴)) = ((♯‘(2nd𝐴)) − 1))
1412, 13sylbir 235 . . . . . . . 8 (⟨(1st𝐴), (2nd𝐴)⟩ ∈ (Walks‘𝐺) → (♯‘(1st𝐴)) = ((♯‘(2nd𝐴)) − 1))
1511, 14biimtrdi 253 . . . . . . 7 (𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩ → (𝐴 ∈ (Walks‘𝐺) → (♯‘(1st𝐴)) = ((♯‘(2nd𝐴)) − 1)))
1610, 15mpcom 38 . . . . . 6 (𝐴 ∈ (Walks‘𝐺) → (♯‘(1st𝐴)) = ((♯‘(2nd𝐴)) − 1))
17 wlkop 29646 . . . . . . 7 (𝐵 ∈ (Walks‘𝐺) → 𝐵 = ⟨(1st𝐵), (2nd𝐵)⟩)
18 eleq1 2829 . . . . . . . 8 (𝐵 = ⟨(1st𝐵), (2nd𝐵)⟩ → (𝐵 ∈ (Walks‘𝐺) ↔ ⟨(1st𝐵), (2nd𝐵)⟩ ∈ (Walks‘𝐺)))
19 df-br 5144 . . . . . . . . 9 ((1st𝐵)(Walks‘𝐺)(2nd𝐵) ↔ ⟨(1st𝐵), (2nd𝐵)⟩ ∈ (Walks‘𝐺))
20 wlklenvm1 29640 . . . . . . . . 9 ((1st𝐵)(Walks‘𝐺)(2nd𝐵) → (♯‘(1st𝐵)) = ((♯‘(2nd𝐵)) − 1))
2119, 20sylbir 235 . . . . . . . 8 (⟨(1st𝐵), (2nd𝐵)⟩ ∈ (Walks‘𝐺) → (♯‘(1st𝐵)) = ((♯‘(2nd𝐵)) − 1))
2218, 21biimtrdi 253 . . . . . . 7 (𝐵 = ⟨(1st𝐵), (2nd𝐵)⟩ → (𝐵 ∈ (Walks‘𝐺) → (♯‘(1st𝐵)) = ((♯‘(2nd𝐵)) − 1)))
2317, 22mpcom 38 . . . . . 6 (𝐵 ∈ (Walks‘𝐺) → (♯‘(1st𝐵)) = ((♯‘(2nd𝐵)) − 1))
2416, 23anim12i 613 . . . . 5 ((𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) → ((♯‘(1st𝐴)) = ((♯‘(2nd𝐴)) − 1) ∧ (♯‘(1st𝐵)) = ((♯‘(2nd𝐵)) − 1)))
25 eqwrd 14595 . . . . . . . 8 (((1st𝐴) ∈ Word dom (iEdg‘𝐺) ∧ (1st𝐵) ∈ Word dom (iEdg‘𝐺)) → ((1st𝐴) = (1st𝐵) ↔ ((♯‘(1st𝐴)) = (♯‘(1st𝐵)) ∧ ∀𝑥 ∈ (0..^(♯‘(1st𝐴)))((1st𝐴)‘𝑥) = ((1st𝐵)‘𝑥))))
2625ad2ant2r 747 . . . . . . 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))))
2726adantr 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))))
28 lencl 14571 . . . . . . . . 9 ((1st𝐴) ∈ Word dom (iEdg‘𝐺) → (♯‘(1st𝐴)) ∈ ℕ0)
2928adantr 480 . . . . . . . 8 (((1st𝐴) ∈ Word dom (iEdg‘𝐺) ∧ (2nd𝐴):(0...(♯‘(1st𝐴)))⟶(Vtx‘𝐺)) → (♯‘(1st𝐴)) ∈ ℕ0)
30 simpr 484 . . . . . . . 8 (((1st𝐴) ∈ Word dom (iEdg‘𝐺) ∧ (2nd𝐴):(0...(♯‘(1st𝐴)))⟶(Vtx‘𝐺)) → (2nd𝐴):(0...(♯‘(1st𝐴)))⟶(Vtx‘𝐺))
31 simpr 484 . . . . . . . 8 (((1st𝐵) ∈ Word dom (iEdg‘𝐺) ∧ (2nd𝐵):(0...(♯‘(1st𝐵)))⟶(Vtx‘𝐺)) → (2nd𝐵):(0...(♯‘(1st𝐵)))⟶(Vtx‘𝐺))
32 2ffzeq 13689 . . . . . . . 8 (((♯‘(1st𝐴)) ∈ ℕ0 ∧ (2nd𝐴):(0...(♯‘(1st𝐴)))⟶(Vtx‘𝐺) ∧ (2nd𝐵):(0...(♯‘(1st𝐵)))⟶(Vtx‘𝐺)) → ((2nd𝐴) = (2nd𝐵) ↔ ((♯‘(1st𝐴)) = (♯‘(1st𝐵)) ∧ ∀𝑥 ∈ (0...(♯‘(1st𝐴)))((2nd𝐴)‘𝑥) = ((2nd𝐵)‘𝑥))))
3329, 30, 31, 32syl2an3an 1424 . . . . . . 7 ((((1st𝐴) ∈ Word dom (iEdg‘𝐺) ∧ (2nd𝐴):(0...(♯‘(1st𝐴)))⟶(Vtx‘𝐺)) ∧ ((1st𝐵) ∈ Word dom (iEdg‘𝐺) ∧ (2nd𝐵):(0...(♯‘(1st𝐵)))⟶(Vtx‘𝐺))) → ((2nd𝐴) = (2nd𝐵) ↔ ((♯‘(1st𝐴)) = (♯‘(1st𝐵)) ∧ ∀𝑥 ∈ (0...(♯‘(1st𝐴)))((2nd𝐴)‘𝑥) = ((2nd𝐵)‘𝑥))))
3433adantr 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))) → ((2nd𝐴) = (2nd𝐵) ↔ ((♯‘(1st𝐴)) = (♯‘(1st𝐵)) ∧ ∀𝑥 ∈ (0...(♯‘(1st𝐴)))((2nd𝐴)‘𝑥) = ((2nd𝐵)‘𝑥))))
3527, 34anbi12d 632 . . . . 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)))))
369, 24, 35syl2anc 584 . . . 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)))))
37363adant3 1133 . . 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)))))
38 eqeq1 2741 . . . . . . 7 (𝑁 = (♯‘(1st𝐴)) → (𝑁 = (♯‘(1st𝐵)) ↔ (♯‘(1st𝐴)) = (♯‘(1st𝐵))))
39 oveq2 7439 . . . . . . . 8 (𝑁 = (♯‘(1st𝐴)) → (0..^𝑁) = (0..^(♯‘(1st𝐴))))
4039raleqdv 3326 . . . . . . 7 (𝑁 = (♯‘(1st𝐴)) → (∀𝑥 ∈ (0..^𝑁)((1st𝐴)‘𝑥) = ((1st𝐵)‘𝑥) ↔ ∀𝑥 ∈ (0..^(♯‘(1st𝐴)))((1st𝐴)‘𝑥) = ((1st𝐵)‘𝑥)))
4138, 40anbi12d 632 . . . . . 6 (𝑁 = (♯‘(1st𝐴)) → ((𝑁 = (♯‘(1st𝐵)) ∧ ∀𝑥 ∈ (0..^𝑁)((1st𝐴)‘𝑥) = ((1st𝐵)‘𝑥)) ↔ ((♯‘(1st𝐴)) = (♯‘(1st𝐵)) ∧ ∀𝑥 ∈ (0..^(♯‘(1st𝐴)))((1st𝐴)‘𝑥) = ((1st𝐵)‘𝑥))))
42 oveq2 7439 . . . . . . . 8 (𝑁 = (♯‘(1st𝐴)) → (0...𝑁) = (0...(♯‘(1st𝐴))))
4342raleqdv 3326 . . . . . . 7 (𝑁 = (♯‘(1st𝐴)) → (∀𝑥 ∈ (0...𝑁)((2nd𝐴)‘𝑥) = ((2nd𝐵)‘𝑥) ↔ ∀𝑥 ∈ (0...(♯‘(1st𝐴)))((2nd𝐴)‘𝑥) = ((2nd𝐵)‘𝑥)))
4438, 43anbi12d 632 . . . . . 6 (𝑁 = (♯‘(1st𝐴)) → ((𝑁 = (♯‘(1st𝐵)) ∧ ∀𝑥 ∈ (0...𝑁)((2nd𝐴)‘𝑥) = ((2nd𝐵)‘𝑥)) ↔ ((♯‘(1st𝐴)) = (♯‘(1st𝐵)) ∧ ∀𝑥 ∈ (0...(♯‘(1st𝐴)))((2nd𝐴)‘𝑥) = ((2nd𝐵)‘𝑥))))
4541, 44anbi12d 632 . . . . 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)))))
4645bibi2d 342 . . . 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))))))
47463ad2ant3 1136 . . 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))))))
4837, 47mpbird 257 . 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)))))
49 1st2ndb 8054 . . . . 5 (𝐴 ∈ (V × V) ↔ 𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩)
5010, 49sylibr 234 . . . 4 (𝐴 ∈ (Walks‘𝐺) → 𝐴 ∈ (V × V))
51 1st2ndb 8054 . . . . 5 (𝐵 ∈ (V × V) ↔ 𝐵 = ⟨(1st𝐵), (2nd𝐵)⟩)
5217, 51sylibr 234 . . . 4 (𝐵 ∈ (Walks‘𝐺) → 𝐵 ∈ (V × V))
53 xpopth 8055 . . . 4 ((𝐴 ∈ (V × V) ∧ 𝐵 ∈ (V × V)) → (((1st𝐴) = (1st𝐵) ∧ (2nd𝐴) = (2nd𝐵)) ↔ 𝐴 = 𝐵))
5450, 52, 53syl2an 596 . . 3 ((𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) → (((1st𝐴) = (1st𝐵) ∧ (2nd𝐴) = (2nd𝐵)) ↔ 𝐴 = 𝐵))
55543adant3 1133 . 2 ((𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺) ∧ 𝑁 = (♯‘(1st𝐴))) → (((1st𝐴) = (1st𝐵) ∧ (2nd𝐴) = (2nd𝐵)) ↔ 𝐴 = 𝐵))
56 3anass 1095 . . . 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))))
57 anandi 676 . . . 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))))
5856, 57bitr2i 276 . . 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)))
5958a1i 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))))
6048, 55, 593bitr3d 309 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 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  wral 3061  Vcvv 3480  cop 4632   class class class wbr 5143   × cxp 5683  dom cdm 5685  wf 6557  cfv 6561  (class class class)co 7431  1st c1st 8012  2nd c2nd 8013  0cc0 11155  1c1 11156  cmin 11492  0cn0 12526  ...cfz 13547  ..^cfzo 13694  chash 14369  Word cword 14552  Vtxcvtx 29013  iEdgciedg 29014  Walkscwlks 29614
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-ifp 1064  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-er 8745  df-map 8868  df-pm 8869  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-n0 12527  df-z 12614  df-uz 12879  df-fz 13548  df-fzo 13695  df-hash 14370  df-word 14553  df-wlks 29617
This theorem is referenced by:  uspgr2wlkeq  29664
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