Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  wlkeq Structured version   Visualization version   GIF version

Theorem wlkeq 27433
 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 2798 . . . . . . 7 (Vtx‘𝐺) = (Vtx‘𝐺)
2 eqid 2798 . . . . . . 7 (iEdg‘𝐺) = (iEdg‘𝐺)
3 eqid 2798 . . . . . . 7 (1st𝐴) = (1st𝐴)
4 eqid 2798 . . . . . . 7 (2nd𝐴) = (2nd𝐴)
51, 2, 3, 4wlkelwrd 27432 . . . . . 6 (𝐴 ∈ (Walks‘𝐺) → ((1st𝐴) ∈ Word dom (iEdg‘𝐺) ∧ (2nd𝐴):(0...(♯‘(1st𝐴)))⟶(Vtx‘𝐺)))
6 eqid 2798 . . . . . . 7 (1st𝐵) = (1st𝐵)
7 eqid 2798 . . . . . . 7 (2nd𝐵) = (2nd𝐵)
81, 2, 6, 7wlkelwrd 27432 . . . . . 6 (𝐵 ∈ (Walks‘𝐺) → ((1st𝐵) ∈ Word dom (iEdg‘𝐺) ∧ (2nd𝐵):(0...(♯‘(1st𝐵)))⟶(Vtx‘𝐺)))
95, 8anim12i 615 . . . . 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 27427 . . . . . . 7 (𝐴 ∈ (Walks‘𝐺) → 𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩)
11 eleq1 2877 . . . . . . . 8 (𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩ → (𝐴 ∈ (Walks‘𝐺) ↔ ⟨(1st𝐴), (2nd𝐴)⟩ ∈ (Walks‘𝐺)))
12 df-br 5032 . . . . . . . . 9 ((1st𝐴)(Walks‘𝐺)(2nd𝐴) ↔ ⟨(1st𝐴), (2nd𝐴)⟩ ∈ (Walks‘𝐺))
13 wlklenvm1 27421 . . . . . . . . 9 ((1st𝐴)(Walks‘𝐺)(2nd𝐴) → (♯‘(1st𝐴)) = ((♯‘(2nd𝐴)) − 1))
1412, 13sylbir 238 . . . . . . . 8 (⟨(1st𝐴), (2nd𝐴)⟩ ∈ (Walks‘𝐺) → (♯‘(1st𝐴)) = ((♯‘(2nd𝐴)) − 1))
1511, 14syl6bi 256 . . . . . . 7 (𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩ → (𝐴 ∈ (Walks‘𝐺) → (♯‘(1st𝐴)) = ((♯‘(2nd𝐴)) − 1)))
1610, 15mpcom 38 . . . . . 6 (𝐴 ∈ (Walks‘𝐺) → (♯‘(1st𝐴)) = ((♯‘(2nd𝐴)) − 1))
17 wlkop 27427 . . . . . . 7 (𝐵 ∈ (Walks‘𝐺) → 𝐵 = ⟨(1st𝐵), (2nd𝐵)⟩)
18 eleq1 2877 . . . . . . . 8 (𝐵 = ⟨(1st𝐵), (2nd𝐵)⟩ → (𝐵 ∈ (Walks‘𝐺) ↔ ⟨(1st𝐵), (2nd𝐵)⟩ ∈ (Walks‘𝐺)))
19 df-br 5032 . . . . . . . . 9 ((1st𝐵)(Walks‘𝐺)(2nd𝐵) ↔ ⟨(1st𝐵), (2nd𝐵)⟩ ∈ (Walks‘𝐺))
20 wlklenvm1 27421 . . . . . . . . 9 ((1st𝐵)(Walks‘𝐺)(2nd𝐵) → (♯‘(1st𝐵)) = ((♯‘(2nd𝐵)) − 1))
2119, 20sylbir 238 . . . . . . . 8 (⟨(1st𝐵), (2nd𝐵)⟩ ∈ (Walks‘𝐺) → (♯‘(1st𝐵)) = ((♯‘(2nd𝐵)) − 1))
2218, 21syl6bi 256 . . . . . . 7 (𝐵 = ⟨(1st𝐵), (2nd𝐵)⟩ → (𝐵 ∈ (Walks‘𝐺) → (♯‘(1st𝐵)) = ((♯‘(2nd𝐵)) − 1)))
2317, 22mpcom 38 . . . . . 6 (𝐵 ∈ (Walks‘𝐺) → (♯‘(1st𝐵)) = ((♯‘(2nd𝐵)) − 1))
2416, 23anim12i 615 . . . . 5 ((𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) → ((♯‘(1st𝐴)) = ((♯‘(2nd𝐴)) − 1) ∧ (♯‘(1st𝐵)) = ((♯‘(2nd𝐵)) − 1)))
25 eqwrd 13903 . . . . . . . 8 (((1st𝐴) ∈ Word dom (iEdg‘𝐺) ∧ (1st𝐵) ∈ Word dom (iEdg‘𝐺)) → ((1st𝐴) = (1st𝐵) ↔ ((♯‘(1st𝐴)) = (♯‘(1st𝐵)) ∧ ∀𝑥 ∈ (0..^(♯‘(1st𝐴)))((1st𝐴)‘𝑥) = ((1st𝐵)‘𝑥))))
2625ad2ant2r 746 . . . . . . 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 484 . . . . . 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 13879 . . . . . . . . 9 ((1st𝐴) ∈ Word dom (iEdg‘𝐺) → (♯‘(1st𝐴)) ∈ ℕ0)
2928adantr 484 . . . . . . . 8 (((1st𝐴) ∈ Word dom (iEdg‘𝐺) ∧ (2nd𝐴):(0...(♯‘(1st𝐴)))⟶(Vtx‘𝐺)) → (♯‘(1st𝐴)) ∈ ℕ0)
30 simpr 488 . . . . . . . 8 (((1st𝐴) ∈ Word dom (iEdg‘𝐺) ∧ (2nd𝐴):(0...(♯‘(1st𝐴)))⟶(Vtx‘𝐺)) → (2nd𝐴):(0...(♯‘(1st𝐴)))⟶(Vtx‘𝐺))
31 simpr 488 . . . . . . . 8 (((1st𝐵) ∈ Word dom (iEdg‘𝐺) ∧ (2nd𝐵):(0...(♯‘(1st𝐵)))⟶(Vtx‘𝐺)) → (2nd𝐵):(0...(♯‘(1st𝐵)))⟶(Vtx‘𝐺))
32 2ffzeq 13026 . . . . . . . 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 1419 . . . . . . 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 484 . . . . . 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 633 . . . . 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 587 . . . 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 1129 . . 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 2802 . . . . . . 7 (𝑁 = (♯‘(1st𝐴)) → (𝑁 = (♯‘(1st𝐵)) ↔ (♯‘(1st𝐴)) = (♯‘(1st𝐵))))
39 oveq2 7144 . . . . . . . 8 (𝑁 = (♯‘(1st𝐴)) → (0..^𝑁) = (0..^(♯‘(1st𝐴))))
4039raleqdv 3364 . . . . . . 7 (𝑁 = (♯‘(1st𝐴)) → (∀𝑥 ∈ (0..^𝑁)((1st𝐴)‘𝑥) = ((1st𝐵)‘𝑥) ↔ ∀𝑥 ∈ (0..^(♯‘(1st𝐴)))((1st𝐴)‘𝑥) = ((1st𝐵)‘𝑥)))
4138, 40anbi12d 633 . . . . . 6 (𝑁 = (♯‘(1st𝐴)) → ((𝑁 = (♯‘(1st𝐵)) ∧ ∀𝑥 ∈ (0..^𝑁)((1st𝐴)‘𝑥) = ((1st𝐵)‘𝑥)) ↔ ((♯‘(1st𝐴)) = (♯‘(1st𝐵)) ∧ ∀𝑥 ∈ (0..^(♯‘(1st𝐴)))((1st𝐴)‘𝑥) = ((1st𝐵)‘𝑥))))
42 oveq2 7144 . . . . . . . 8 (𝑁 = (♯‘(1st𝐴)) → (0...𝑁) = (0...(♯‘(1st𝐴))))
4342raleqdv 3364 . . . . . . 7 (𝑁 = (♯‘(1st𝐴)) → (∀𝑥 ∈ (0...𝑁)((2nd𝐴)‘𝑥) = ((2nd𝐵)‘𝑥) ↔ ∀𝑥 ∈ (0...(♯‘(1st𝐴)))((2nd𝐴)‘𝑥) = ((2nd𝐵)‘𝑥)))
4438, 43anbi12d 633 . . . . . 6 (𝑁 = (♯‘(1st𝐴)) → ((𝑁 = (♯‘(1st𝐵)) ∧ ∀𝑥 ∈ (0...𝑁)((2nd𝐴)‘𝑥) = ((2nd𝐵)‘𝑥)) ↔ ((♯‘(1st𝐴)) = (♯‘(1st𝐵)) ∧ ∀𝑥 ∈ (0...(♯‘(1st𝐴)))((2nd𝐴)‘𝑥) = ((2nd𝐵)‘𝑥))))
4541, 44anbi12d 633 . . . . 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 346 . . . 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 1132 . . 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 260 . 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 7714 . . . . 5 (𝐴 ∈ (V × V) ↔ 𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩)
5010, 49sylibr 237 . . . 4 (𝐴 ∈ (Walks‘𝐺) → 𝐴 ∈ (V × V))
51 1st2ndb 7714 . . . . 5 (𝐵 ∈ (V × V) ↔ 𝐵 = ⟨(1st𝐵), (2nd𝐵)⟩)
5217, 51sylibr 237 . . . 4 (𝐵 ∈ (Walks‘𝐺) → 𝐵 ∈ (V × V))
53 xpopth 7715 . . . 4 ((𝐴 ∈ (V × V) ∧ 𝐵 ∈ (V × V)) → (((1st𝐴) = (1st𝐵) ∧ (2nd𝐴) = (2nd𝐵)) ↔ 𝐴 = 𝐵))
5450, 52, 53syl2an 598 . . 3 ((𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) → (((1st𝐴) = (1st𝐵) ∧ (2nd𝐴) = (2nd𝐵)) ↔ 𝐴 = 𝐵))
55543adant3 1129 . 2 ((𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺) ∧ 𝑁 = (♯‘(1st𝐴))) → (((1st𝐴) = (1st𝐵) ∧ (2nd𝐴) = (2nd𝐵)) ↔ 𝐴 = 𝐵))
56 3anass 1092 . . . 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 675 . . . 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 279 . . 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 312 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 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ∀wral 3106  Vcvv 3441  ⟨cop 4531   class class class wbr 5031   × cxp 5518  dom cdm 5520  ⟶wf 6321  ‘cfv 6325  (class class class)co 7136  1st c1st 7672  2nd c2nd 7673  0cc0 10529  1c1 10530   − cmin 10862  ℕ0cn0 11888  ...cfz 12888  ..^cfzo 13031  ♯chash 13689  Word cword 13860  Vtxcvtx 26799  iEdgciedg 26800  Walkscwlks 27396 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5155  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296  ax-un 7444  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ifp 1059  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-int 4840  df-iun 4884  df-br 5032  df-opab 5094  df-mpt 5112  df-tr 5138  df-id 5426  df-eprel 5431  df-po 5439  df-so 5440  df-fr 5479  df-we 5481  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-res 5532  df-ima 5533  df-pred 6117  df-ord 6163  df-on 6164  df-lim 6165  df-suc 6166  df-iota 6284  df-fun 6327  df-fn 6328  df-f 6329  df-f1 6330  df-fo 6331  df-f1o 6332  df-fv 6333  df-riota 7094  df-ov 7139  df-oprab 7140  df-mpo 7141  df-om 7564  df-1st 7674  df-2nd 7675  df-wrecs 7933  df-recs 7994  df-rdg 8032  df-1o 8088  df-oadd 8092  df-er 8275  df-map 8394  df-pm 8395  df-en 8496  df-dom 8497  df-sdom 8498  df-fin 8499  df-card 9355  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-nn 11629  df-n0 11889  df-z 11973  df-uz 12235  df-fz 12889  df-fzo 13032  df-hash 13690  df-word 13861  df-wlks 27399 This theorem is referenced by:  uspgr2wlkeq  27445
 Copyright terms: Public domain W3C validator