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Theorem wlkeq 29667
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 2735 . . . . . . 7 (Vtx‘𝐺) = (Vtx‘𝐺)
2 eqid 2735 . . . . . . 7 (iEdg‘𝐺) = (iEdg‘𝐺)
3 eqid 2735 . . . . . . 7 (1st𝐴) = (1st𝐴)
4 eqid 2735 . . . . . . 7 (2nd𝐴) = (2nd𝐴)
51, 2, 3, 4wlkelwrd 29666 . . . . . 6 (𝐴 ∈ (Walks‘𝐺) → ((1st𝐴) ∈ Word dom (iEdg‘𝐺) ∧ (2nd𝐴):(0...(♯‘(1st𝐴)))⟶(Vtx‘𝐺)))
6 eqid 2735 . . . . . . 7 (1st𝐵) = (1st𝐵)
7 eqid 2735 . . . . . . 7 (2nd𝐵) = (2nd𝐵)
81, 2, 6, 7wlkelwrd 29666 . . . . . 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 29661 . . . . . . 7 (𝐴 ∈ (Walks‘𝐺) → 𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩)
11 eleq1 2827 . . . . . . . 8 (𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩ → (𝐴 ∈ (Walks‘𝐺) ↔ ⟨(1st𝐴), (2nd𝐴)⟩ ∈ (Walks‘𝐺)))
12 df-br 5149 . . . . . . . . 9 ((1st𝐴)(Walks‘𝐺)(2nd𝐴) ↔ ⟨(1st𝐴), (2nd𝐴)⟩ ∈ (Walks‘𝐺))
13 wlklenvm1 29655 . . . . . . . . 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 29661 . . . . . . 7 (𝐵 ∈ (Walks‘𝐺) → 𝐵 = ⟨(1st𝐵), (2nd𝐵)⟩)
18 eleq1 2827 . . . . . . . 8 (𝐵 = ⟨(1st𝐵), (2nd𝐵)⟩ → (𝐵 ∈ (Walks‘𝐺) ↔ ⟨(1st𝐵), (2nd𝐵)⟩ ∈ (Walks‘𝐺)))
19 df-br 5149 . . . . . . . . 9 ((1st𝐵)(Walks‘𝐺)(2nd𝐵) ↔ ⟨(1st𝐵), (2nd𝐵)⟩ ∈ (Walks‘𝐺))
20 wlklenvm1 29655 . . . . . . . . 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 14592 . . . . . . . 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 14568 . . . . . . . . 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 13686 . . . . . . . 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 1421 . . . . . . 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 1131 . . 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 2739 . . . . . . 7 (𝑁 = (♯‘(1st𝐴)) → (𝑁 = (♯‘(1st𝐵)) ↔ (♯‘(1st𝐴)) = (♯‘(1st𝐵))))
39 oveq2 7439 . . . . . . . 8 (𝑁 = (♯‘(1st𝐴)) → (0..^𝑁) = (0..^(♯‘(1st𝐴))))
4039raleqdv 3324 . . . . . . 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 3324 . . . . . . 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 1134 . . 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 8053 . . . . 5 (𝐴 ∈ (V × V) ↔ 𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩)
5010, 49sylibr 234 . . . 4 (𝐴 ∈ (Walks‘𝐺) → 𝐴 ∈ (V × V))
51 1st2ndb 8053 . . . . 5 (𝐵 ∈ (V × V) ↔ 𝐵 = ⟨(1st𝐵), (2nd𝐵)⟩)
5217, 51sylibr 234 . . . 4 (𝐵 ∈ (Walks‘𝐺) → 𝐵 ∈ (V × V))
53 xpopth 8054 . . . 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 1131 . 2 ((𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺) ∧ 𝑁 = (♯‘(1st𝐴))) → (((1st𝐴) = (1st𝐵) ∧ (2nd𝐴) = (2nd𝐵)) ↔ 𝐴 = 𝐵))
56 3anass 1094 . . . 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 1086   = wceq 1537  wcel 2106  wral 3059  Vcvv 3478  cop 4637   class class class wbr 5148   × cxp 5687  dom cdm 5689  wf 6559  cfv 6563  (class class class)co 7431  1st c1st 8011  2nd c2nd 8012  0cc0 11153  1c1 11154  cmin 11490  0cn0 12524  ...cfz 13544  ..^cfzo 13691  chash 14366  Word cword 14549  Vtxcvtx 29028  iEdgciedg 29029  Walkscwlks 29629
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ifp 1063  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-er 8744  df-map 8867  df-pm 8868  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-card 9977  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-n0 12525  df-z 12612  df-uz 12877  df-fz 13545  df-fzo 13692  df-hash 14367  df-word 14550  df-wlks 29632
This theorem is referenced by:  uspgr2wlkeq  29679
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