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Theorem wlkeq 26529

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 = (#‘(1^st ‘𝐴))) → (𝐴 = 𝐵 ↔ (𝑁 = (#‘(1^st ‘𝐵)) ∧ ∀𝑥 ∈ (0..^𝑁)((1^st ‘𝐴)‘𝑥) = ((1^st ‘𝐵)‘𝑥) ∧ ∀𝑥 ∈ (0...𝑁)((2^nd ‘𝐴)‘𝑥) = ((2^nd ‘𝐵)‘𝑥))))

Distinct variable groups: 𝑥,𝐴 𝑥,𝐵 𝑥,𝑁 Allowed substitution hint: 𝐺(𝑥)

**Proof of Theorem wlkeq**
Step	Hyp	Ref	Expression
1		wlkop 26523	. . . . 5 ⊢ (𝐴 ∈ (Walks‘𝐺) → 𝐴 = ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩)
2		1st2ndb 7206	. . . . 5 ⊢ (𝐴 ∈ (V × V) ↔ 𝐴 = ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩)
3	1, 2	sylibr 224	. . . 4 ⊢ (𝐴 ∈ (Walks‘𝐺) → 𝐴 ∈ (V × V))
4		wlkop 26523	. . . . 5 ⊢ (𝐵 ∈ (Walks‘𝐺) → 𝐵 = ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩)
5		1st2ndb 7206	. . . . 5 ⊢ (𝐵 ∈ (V × V) ↔ 𝐵 = ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩)
6	4, 5	sylibr 224	. . . 4 ⊢ (𝐵 ∈ (Walks‘𝐺) → 𝐵 ∈ (V × V))
7		xpopth 7207	. . . . 5 ⊢ ((𝐴 ∈ (V × V) ∧ 𝐵 ∈ (V × V)) → (((1^st ‘𝐴) = (1^st ‘𝐵) ∧ (2^nd ‘𝐴) = (2^nd ‘𝐵)) ↔ 𝐴 = 𝐵))
8	7	bicomd 213	. . . 4 ⊢ ((𝐴 ∈ (V × V) ∧ 𝐵 ∈ (V × V)) → (𝐴 = 𝐵 ↔ ((1^st ‘𝐴) = (1^st ‘𝐵) ∧ (2^nd ‘𝐴) = (2^nd ‘𝐵))))
9	3, 6, 8	syl2an 494	. . 3 ⊢ ((𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) → (𝐴 = 𝐵 ↔ ((1^st ‘𝐴) = (1^st ‘𝐵) ∧ (2^nd ‘𝐴) = (2^nd ‘𝐵))))
10	9	3adant3 1081	. 2 ⊢ ((𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺) ∧ 𝑁 = (#‘(1^st ‘𝐴))) → (𝐴 = 𝐵 ↔ ((1^st ‘𝐴) = (1^st ‘𝐵) ∧ (2^nd ‘𝐴) = (2^nd ‘𝐵))))
11		eqid 2622	. . . . . . 7 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
12		eqid 2622	. . . . . . 7 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
13		eqid 2622	. . . . . . 7 ⊢ (1^st ‘𝐴) = (1^st ‘𝐴)
14		eqid 2622	. . . . . . 7 ⊢ (2^nd ‘𝐴) = (2^nd ‘𝐴)
15	11, 12, 13, 14	wlkelwrd 26528	. . . . . 6 ⊢ (𝐴 ∈ (Walks‘𝐺) → ((1^st ‘𝐴) ∈ Word dom (iEdg‘𝐺) ∧ (2^nd ‘𝐴):(0...(#‘(1^st ‘𝐴)))⟶(Vtx‘𝐺)))
16		eqid 2622	. . . . . . 7 ⊢ (1^st ‘𝐵) = (1^st ‘𝐵)
17		eqid 2622	. . . . . . 7 ⊢ (2^nd ‘𝐵) = (2^nd ‘𝐵)
18	11, 12, 16, 17	wlkelwrd 26528	. . . . . 6 ⊢ (𝐵 ∈ (Walks‘𝐺) → ((1^st ‘𝐵) ∈ Word dom (iEdg‘𝐺) ∧ (2^nd ‘𝐵):(0...(#‘(1^st ‘𝐵)))⟶(Vtx‘𝐺)))
19	15, 18	anim12i 590	. . . . 5 ⊢ ((𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) → (((1^st ‘𝐴) ∈ Word dom (iEdg‘𝐺) ∧ (2^nd ‘𝐴):(0...(#‘(1^st ‘𝐴)))⟶(Vtx‘𝐺)) ∧ ((1^st ‘𝐵) ∈ Word dom (iEdg‘𝐺) ∧ (2^nd ‘𝐵):(0...(#‘(1^st ‘𝐵)))⟶(Vtx‘𝐺))))
20		eleq1 2689	. . . . . . . 8 ⊢ (𝐴 = ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ → (𝐴 ∈ (Walks‘𝐺) ↔ ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ (Walks‘𝐺)))
21		df-br 4654	. . . . . . . . 9 ⊢ ((1^st ‘𝐴)(Walks‘𝐺)(2^nd ‘𝐴) ↔ ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ (Walks‘𝐺))
22		wlklenvm1 26517	. . . . . . . . 9 ⊢ ((1^st ‘𝐴)(Walks‘𝐺)(2^nd ‘𝐴) → (#‘(1^st ‘𝐴)) = ((#‘(2^nd ‘𝐴)) − 1))
23	21, 22	sylbir 225	. . . . . . . 8 ⊢ (⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ (Walks‘𝐺) → (#‘(1^st ‘𝐴)) = ((#‘(2^nd ‘𝐴)) − 1))
24	20, 23	syl6bi 243	. . . . . . 7 ⊢ (𝐴 = ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ → (𝐴 ∈ (Walks‘𝐺) → (#‘(1^st ‘𝐴)) = ((#‘(2^nd ‘𝐴)) − 1)))
25	1, 24	mpcom 38	. . . . . 6 ⊢ (𝐴 ∈ (Walks‘𝐺) → (#‘(1^st ‘𝐴)) = ((#‘(2^nd ‘𝐴)) − 1))
26		eleq1 2689	. . . . . . . 8 ⊢ (𝐵 = ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ → (𝐵 ∈ (Walks‘𝐺) ↔ ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ (Walks‘𝐺)))
27		df-br 4654	. . . . . . . . 9 ⊢ ((1^st ‘𝐵)(Walks‘𝐺)(2^nd ‘𝐵) ↔ ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ (Walks‘𝐺))
28		wlklenvm1 26517	. . . . . . . . 9 ⊢ ((1^st ‘𝐵)(Walks‘𝐺)(2^nd ‘𝐵) → (#‘(1^st ‘𝐵)) = ((#‘(2^nd ‘𝐵)) − 1))
29	27, 28	sylbir 225	. . . . . . . 8 ⊢ (⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ (Walks‘𝐺) → (#‘(1^st ‘𝐵)) = ((#‘(2^nd ‘𝐵)) − 1))
30	26, 29	syl6bi 243	. . . . . . 7 ⊢ (𝐵 = ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ → (𝐵 ∈ (Walks‘𝐺) → (#‘(1^st ‘𝐵)) = ((#‘(2^nd ‘𝐵)) − 1)))
31	4, 30	mpcom 38	. . . . . 6 ⊢ (𝐵 ∈ (Walks‘𝐺) → (#‘(1^st ‘𝐵)) = ((#‘(2^nd ‘𝐵)) − 1))
32	25, 31	anim12i 590	. . . . 5 ⊢ ((𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) → ((#‘(1^st ‘𝐴)) = ((#‘(2^nd ‘𝐴)) − 1) ∧ (#‘(1^st ‘𝐵)) = ((#‘(2^nd ‘𝐵)) − 1)))
33		eqwrd 13346	. . . . . . . 8 ⊢ (((1^st ‘𝐴) ∈ Word dom (iEdg‘𝐺) ∧ (1^st ‘𝐵) ∈ Word dom (iEdg‘𝐺)) → ((1^st ‘𝐴) = (1^st ‘𝐵) ↔ ((#‘(1^st ‘𝐴)) = (#‘(1^st ‘𝐵)) ∧ ∀𝑥 ∈ (0..^(#‘(1^st ‘𝐴)))((1^st ‘𝐴)‘𝑥) = ((1^st ‘𝐵)‘𝑥))))
34	33	ad2ant2r 783	. . . . . . 7 ⊢ ((((1^st ‘𝐴) ∈ Word dom (iEdg‘𝐺) ∧ (2^nd ‘𝐴):(0...(#‘(1^st ‘𝐴)))⟶(Vtx‘𝐺)) ∧ ((1^st ‘𝐵) ∈ Word dom (iEdg‘𝐺) ∧ (2^nd ‘𝐵):(0...(#‘(1^st ‘𝐵)))⟶(Vtx‘𝐺))) → ((1^st ‘𝐴) = (1^st ‘𝐵) ↔ ((#‘(1^st ‘𝐴)) = (#‘(1^st ‘𝐵)) ∧ ∀𝑥 ∈ (0..^(#‘(1^st ‘𝐴)))((1^st ‘𝐴)‘𝑥) = ((1^st ‘𝐵)‘𝑥))))
35	34	adantr 481	. . . . . 6 ⊢ (((((1^st ‘𝐴) ∈ Word dom (iEdg‘𝐺) ∧ (2^nd ‘𝐴):(0...(#‘(1^st ‘𝐴)))⟶(Vtx‘𝐺)) ∧ ((1^st ‘𝐵) ∈ Word dom (iEdg‘𝐺) ∧ (2^nd ‘𝐵):(0...(#‘(1^st ‘𝐵)))⟶(Vtx‘𝐺))) ∧ ((#‘(1^st ‘𝐴)) = ((#‘(2^nd ‘𝐴)) − 1) ∧ (#‘(1^st ‘𝐵)) = ((#‘(2^nd ‘𝐵)) − 1))) → ((1^st ‘𝐴) = (1^st ‘𝐵) ↔ ((#‘(1^st ‘𝐴)) = (#‘(1^st ‘𝐵)) ∧ ∀𝑥 ∈ (0..^(#‘(1^st ‘𝐴)))((1^st ‘𝐴)‘𝑥) = ((1^st ‘𝐵)‘𝑥))))
36		lencl 13324	. . . . . . . . . . 11 ⊢ ((1^st ‘𝐴) ∈ Word dom (iEdg‘𝐺) → (#‘(1^st ‘𝐴)) ∈ ℕ₀)
37	36	adantr 481	. . . . . . . . . 10 ⊢ (((1^st ‘𝐴) ∈ Word dom (iEdg‘𝐺) ∧ (2^nd ‘𝐴):(0...(#‘(1^st ‘𝐴)))⟶(Vtx‘𝐺)) → (#‘(1^st ‘𝐴)) ∈ ℕ₀)
38	37	adantr 481	. . . . . . . . 9 ⊢ ((((1^st ‘𝐴) ∈ Word dom (iEdg‘𝐺) ∧ (2^nd ‘𝐴):(0...(#‘(1^st ‘𝐴)))⟶(Vtx‘𝐺)) ∧ ((1^st ‘𝐵) ∈ Word dom (iEdg‘𝐺) ∧ (2^nd ‘𝐵):(0...(#‘(1^st ‘𝐵)))⟶(Vtx‘𝐺))) → (#‘(1^st ‘𝐴)) ∈ ℕ₀)
39		simplr 792	. . . . . . . . 9 ⊢ ((((1^st ‘𝐴) ∈ Word dom (iEdg‘𝐺) ∧ (2^nd ‘𝐴):(0...(#‘(1^st ‘𝐴)))⟶(Vtx‘𝐺)) ∧ ((1^st ‘𝐵) ∈ Word dom (iEdg‘𝐺) ∧ (2^nd ‘𝐵):(0...(#‘(1^st ‘𝐵)))⟶(Vtx‘𝐺))) → (2^nd ‘𝐴):(0...(#‘(1^st ‘𝐴)))⟶(Vtx‘𝐺))
40		simprr 796	. . . . . . . . 9 ⊢ ((((1^st ‘𝐴) ∈ Word dom (iEdg‘𝐺) ∧ (2^nd ‘𝐴):(0...(#‘(1^st ‘𝐴)))⟶(Vtx‘𝐺)) ∧ ((1^st ‘𝐵) ∈ Word dom (iEdg‘𝐺) ∧ (2^nd ‘𝐵):(0...(#‘(1^st ‘𝐵)))⟶(Vtx‘𝐺))) → (2^nd ‘𝐵):(0...(#‘(1^st ‘𝐵)))⟶(Vtx‘𝐺))
41	38, 39, 40	3jca 1242	. . . . . . . 8 ⊢ ((((1^st ‘𝐴) ∈ Word dom (iEdg‘𝐺) ∧ (2^nd ‘𝐴):(0...(#‘(1^st ‘𝐴)))⟶(Vtx‘𝐺)) ∧ ((1^st ‘𝐵) ∈ Word dom (iEdg‘𝐺) ∧ (2^nd ‘𝐵):(0...(#‘(1^st ‘𝐵)))⟶(Vtx‘𝐺))) → ((#‘(1^st ‘𝐴)) ∈ ℕ₀ ∧ (2^nd ‘𝐴):(0...(#‘(1^st ‘𝐴)))⟶(Vtx‘𝐺) ∧ (2^nd ‘𝐵):(0...(#‘(1^st ‘𝐵)))⟶(Vtx‘𝐺)))
42	41	adantr 481	. . . . . . 7 ⊢ (((((1^st ‘𝐴) ∈ Word dom (iEdg‘𝐺) ∧ (2^nd ‘𝐴):(0...(#‘(1^st ‘𝐴)))⟶(Vtx‘𝐺)) ∧ ((1^st ‘𝐵) ∈ Word dom (iEdg‘𝐺) ∧ (2^nd ‘𝐵):(0...(#‘(1^st ‘𝐵)))⟶(Vtx‘𝐺))) ∧ ((#‘(1^st ‘𝐴)) = ((#‘(2^nd ‘𝐴)) − 1) ∧ (#‘(1^st ‘𝐵)) = ((#‘(2^nd ‘𝐵)) − 1))) → ((#‘(1^st ‘𝐴)) ∈ ℕ₀ ∧ (2^nd ‘𝐴):(0...(#‘(1^st ‘𝐴)))⟶(Vtx‘𝐺) ∧ (2^nd ‘𝐵):(0...(#‘(1^st ‘𝐵)))⟶(Vtx‘𝐺)))
43		2ffzeq 12460	. . . . . . 7 ⊢ (((#‘(1^st ‘𝐴)) ∈ ℕ₀ ∧ (2^nd ‘𝐴):(0...(#‘(1^st ‘𝐴)))⟶(Vtx‘𝐺) ∧ (2^nd ‘𝐵):(0...(#‘(1^st ‘𝐵)))⟶(Vtx‘𝐺)) → ((2^nd ‘𝐴) = (2^nd ‘𝐵) ↔ ((#‘(1^st ‘𝐴)) = (#‘(1^st ‘𝐵)) ∧ ∀𝑥 ∈ (0...(#‘(1^st ‘𝐴)))((2^nd ‘𝐴)‘𝑥) = ((2^nd ‘𝐵)‘𝑥))))
44	42, 43	syl 17	. . . . . 6 ⊢ (((((1^st ‘𝐴) ∈ Word dom (iEdg‘𝐺) ∧ (2^nd ‘𝐴):(0...(#‘(1^st ‘𝐴)))⟶(Vtx‘𝐺)) ∧ ((1^st ‘𝐵) ∈ Word dom (iEdg‘𝐺) ∧ (2^nd ‘𝐵):(0...(#‘(1^st ‘𝐵)))⟶(Vtx‘𝐺))) ∧ ((#‘(1^st ‘𝐴)) = ((#‘(2^nd ‘𝐴)) − 1) ∧ (#‘(1^st ‘𝐵)) = ((#‘(2^nd ‘𝐵)) − 1))) → ((2^nd ‘𝐴) = (2^nd ‘𝐵) ↔ ((#‘(1^st ‘𝐴)) = (#‘(1^st ‘𝐵)) ∧ ∀𝑥 ∈ (0...(#‘(1^st ‘𝐴)))((2^nd ‘𝐴)‘𝑥) = ((2^nd ‘𝐵)‘𝑥))))
45	35, 44	anbi12d 747	. . . . 5 ⊢ (((((1^st ‘𝐴) ∈ Word dom (iEdg‘𝐺) ∧ (2^nd ‘𝐴):(0...(#‘(1^st ‘𝐴)))⟶(Vtx‘𝐺)) ∧ ((1^st ‘𝐵) ∈ Word dom (iEdg‘𝐺) ∧ (2^nd ‘𝐵):(0...(#‘(1^st ‘𝐵)))⟶(Vtx‘𝐺))) ∧ ((#‘(1^st ‘𝐴)) = ((#‘(2^nd ‘𝐴)) − 1) ∧ (#‘(1^st ‘𝐵)) = ((#‘(2^nd ‘𝐵)) − 1))) → (((1^st ‘𝐴) = (1^st ‘𝐵) ∧ (2^nd ‘𝐴) = (2^nd ‘𝐵)) ↔ (((#‘(1^st ‘𝐴)) = (#‘(1^st ‘𝐵)) ∧ ∀𝑥 ∈ (0..^(#‘(1^st ‘𝐴)))((1^st ‘𝐴)‘𝑥) = ((1^st ‘𝐵)‘𝑥)) ∧ ((#‘(1^st ‘𝐴)) = (#‘(1^st ‘𝐵)) ∧ ∀𝑥 ∈ (0...(#‘(1^st ‘𝐴)))((2^nd ‘𝐴)‘𝑥) = ((2^nd ‘𝐵)‘𝑥)))))
46	19, 32, 45	syl2anc 693	. . . 4 ⊢ ((𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) → (((1^st ‘𝐴) = (1^st ‘𝐵) ∧ (2^nd ‘𝐴) = (2^nd ‘𝐵)) ↔ (((#‘(1^st ‘𝐴)) = (#‘(1^st ‘𝐵)) ∧ ∀𝑥 ∈ (0..^(#‘(1^st ‘𝐴)))((1^st ‘𝐴)‘𝑥) = ((1^st ‘𝐵)‘𝑥)) ∧ ((#‘(1^st ‘𝐴)) = (#‘(1^st ‘𝐵)) ∧ ∀𝑥 ∈ (0...(#‘(1^st ‘𝐴)))((2^nd ‘𝐴)‘𝑥) = ((2^nd ‘𝐵)‘𝑥)))))
47	46	3adant3 1081	. . 3 ⊢ ((𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺) ∧ 𝑁 = (#‘(1^st ‘𝐴))) → (((1^st ‘𝐴) = (1^st ‘𝐵) ∧ (2^nd ‘𝐴) = (2^nd ‘𝐵)) ↔ (((#‘(1^st ‘𝐴)) = (#‘(1^st ‘𝐵)) ∧ ∀𝑥 ∈ (0..^(#‘(1^st ‘𝐴)))((1^st ‘𝐴)‘𝑥) = ((1^st ‘𝐵)‘𝑥)) ∧ ((#‘(1^st ‘𝐴)) = (#‘(1^st ‘𝐵)) ∧ ∀𝑥 ∈ (0...(#‘(1^st ‘𝐴)))((2^nd ‘𝐴)‘𝑥) = ((2^nd ‘𝐵)‘𝑥)))))
48		eqeq1 2626	. . . . . . 7 ⊢ (𝑁 = (#‘(1^st ‘𝐴)) → (𝑁 = (#‘(1^st ‘𝐵)) ↔ (#‘(1^st ‘𝐴)) = (#‘(1^st ‘𝐵))))
49		oveq2 6658	. . . . . . . 8 ⊢ (𝑁 = (#‘(1^st ‘𝐴)) → (0..^𝑁) = (0..^(#‘(1^st ‘𝐴))))
50	49	raleqdv 3144	. . . . . . 7 ⊢ (𝑁 = (#‘(1^st ‘𝐴)) → (∀𝑥 ∈ (0..^𝑁)((1^st ‘𝐴)‘𝑥) = ((1^st ‘𝐵)‘𝑥) ↔ ∀𝑥 ∈ (0..^(#‘(1^st ‘𝐴)))((1^st ‘𝐴)‘𝑥) = ((1^st ‘𝐵)‘𝑥)))
51	48, 50	anbi12d 747	. . . . . 6 ⊢ (𝑁 = (#‘(1^st ‘𝐴)) → ((𝑁 = (#‘(1^st ‘𝐵)) ∧ ∀𝑥 ∈ (0..^𝑁)((1^st ‘𝐴)‘𝑥) = ((1^st ‘𝐵)‘𝑥)) ↔ ((#‘(1^st ‘𝐴)) = (#‘(1^st ‘𝐵)) ∧ ∀𝑥 ∈ (0..^(#‘(1^st ‘𝐴)))((1^st ‘𝐴)‘𝑥) = ((1^st ‘𝐵)‘𝑥))))
52		oveq2 6658	. . . . . . . 8 ⊢ (𝑁 = (#‘(1^st ‘𝐴)) → (0...𝑁) = (0...(#‘(1^st ‘𝐴))))
53	52	raleqdv 3144	. . . . . . 7 ⊢ (𝑁 = (#‘(1^st ‘𝐴)) → (∀𝑥 ∈ (0...𝑁)((2^nd ‘𝐴)‘𝑥) = ((2^nd ‘𝐵)‘𝑥) ↔ ∀𝑥 ∈ (0...(#‘(1^st ‘𝐴)))((2^nd ‘𝐴)‘𝑥) = ((2^nd ‘𝐵)‘𝑥)))
54	48, 53	anbi12d 747	. . . . . 6 ⊢ (𝑁 = (#‘(1^st ‘𝐴)) → ((𝑁 = (#‘(1^st ‘𝐵)) ∧ ∀𝑥 ∈ (0...𝑁)((2^nd ‘𝐴)‘𝑥) = ((2^nd ‘𝐵)‘𝑥)) ↔ ((#‘(1^st ‘𝐴)) = (#‘(1^st ‘𝐵)) ∧ ∀𝑥 ∈ (0...(#‘(1^st ‘𝐴)))((2^nd ‘𝐴)‘𝑥) = ((2^nd ‘𝐵)‘𝑥))))
55	51, 54	anbi12d 747	. . . . 5 ⊢ (𝑁 = (#‘(1^st ‘𝐴)) → (((𝑁 = (#‘(1^st ‘𝐵)) ∧ ∀𝑥 ∈ (0..^𝑁)((1^st ‘𝐴)‘𝑥) = ((1^st ‘𝐵)‘𝑥)) ∧ (𝑁 = (#‘(1^st ‘𝐵)) ∧ ∀𝑥 ∈ (0...𝑁)((2^nd ‘𝐴)‘𝑥) = ((2^nd ‘𝐵)‘𝑥))) ↔ (((#‘(1^st ‘𝐴)) = (#‘(1^st ‘𝐵)) ∧ ∀𝑥 ∈ (0..^(#‘(1^st ‘𝐴)))((1^st ‘𝐴)‘𝑥) = ((1^st ‘𝐵)‘𝑥)) ∧ ((#‘(1^st ‘𝐴)) = (#‘(1^st ‘𝐵)) ∧ ∀𝑥 ∈ (0...(#‘(1^st ‘𝐴)))((2^nd ‘𝐴)‘𝑥) = ((2^nd ‘𝐵)‘𝑥)))))
56	55	bibi2d 332	. . . 4 ⊢ (𝑁 = (#‘(1^st ‘𝐴)) → ((((1^st ‘𝐴) = (1^st ‘𝐵) ∧ (2^nd ‘𝐴) = (2^nd ‘𝐵)) ↔ ((𝑁 = (#‘(1^st ‘𝐵)) ∧ ∀𝑥 ∈ (0..^𝑁)((1^st ‘𝐴)‘𝑥) = ((1^st ‘𝐵)‘𝑥)) ∧ (𝑁 = (#‘(1^st ‘𝐵)) ∧ ∀𝑥 ∈ (0...𝑁)((2^nd ‘𝐴)‘𝑥) = ((2^nd ‘𝐵)‘𝑥)))) ↔ (((1^st ‘𝐴) = (1^st ‘𝐵) ∧ (2^nd ‘𝐴) = (2^nd ‘𝐵)) ↔ (((#‘(1^st ‘𝐴)) = (#‘(1^st ‘𝐵)) ∧ ∀𝑥 ∈ (0..^(#‘(1^st ‘𝐴)))((1^st ‘𝐴)‘𝑥) = ((1^st ‘𝐵)‘𝑥)) ∧ ((#‘(1^st ‘𝐴)) = (#‘(1^st ‘𝐵)) ∧ ∀𝑥 ∈ (0...(#‘(1^st ‘𝐴)))((2^nd ‘𝐴)‘𝑥) = ((2^nd ‘𝐵)‘𝑥))))))
57	56	3ad2ant3 1084	. . 3 ⊢ ((𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺) ∧ 𝑁 = (#‘(1^st ‘𝐴))) → ((((1^st ‘𝐴) = (1^st ‘𝐵) ∧ (2^nd ‘𝐴) = (2^nd ‘𝐵)) ↔ ((𝑁 = (#‘(1^st ‘𝐵)) ∧ ∀𝑥 ∈ (0..^𝑁)((1^st ‘𝐴)‘𝑥) = ((1^st ‘𝐵)‘𝑥)) ∧ (𝑁 = (#‘(1^st ‘𝐵)) ∧ ∀𝑥 ∈ (0...𝑁)((2^nd ‘𝐴)‘𝑥) = ((2^nd ‘𝐵)‘𝑥)))) ↔ (((1^st ‘𝐴) = (1^st ‘𝐵) ∧ (2^nd ‘𝐴) = (2^nd ‘𝐵)) ↔ (((#‘(1^st ‘𝐴)) = (#‘(1^st ‘𝐵)) ∧ ∀𝑥 ∈ (0..^(#‘(1^st ‘𝐴)))((1^st ‘𝐴)‘𝑥) = ((1^st ‘𝐵)‘𝑥)) ∧ ((#‘(1^st ‘𝐴)) = (#‘(1^st ‘𝐵)) ∧ ∀𝑥 ∈ (0...(#‘(1^st ‘𝐴)))((2^nd ‘𝐴)‘𝑥) = ((2^nd ‘𝐵)‘𝑥))))))
58	47, 57	mpbird 247	. 2 ⊢ ((𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺) ∧ 𝑁 = (#‘(1^st ‘𝐴))) → (((1^st ‘𝐴) = (1^st ‘𝐵) ∧ (2^nd ‘𝐴) = (2^nd ‘𝐵)) ↔ ((𝑁 = (#‘(1^st ‘𝐵)) ∧ ∀𝑥 ∈ (0..^𝑁)((1^st ‘𝐴)‘𝑥) = ((1^st ‘𝐵)‘𝑥)) ∧ (𝑁 = (#‘(1^st ‘𝐵)) ∧ ∀𝑥 ∈ (0...𝑁)((2^nd ‘𝐴)‘𝑥) = ((2^nd ‘𝐵)‘𝑥)))))
59		3anass 1042	. . . 4 ⊢ ((𝑁 = (#‘(1^st ‘𝐵)) ∧ ∀𝑥 ∈ (0..^𝑁)((1^st ‘𝐴)‘𝑥) = ((1^st ‘𝐵)‘𝑥) ∧ ∀𝑥 ∈ (0...𝑁)((2^nd ‘𝐴)‘𝑥) = ((2^nd ‘𝐵)‘𝑥)) ↔ (𝑁 = (#‘(1^st ‘𝐵)) ∧ (∀𝑥 ∈ (0..^𝑁)((1^st ‘𝐴)‘𝑥) = ((1^st ‘𝐵)‘𝑥) ∧ ∀𝑥 ∈ (0...𝑁)((2^nd ‘𝐴)‘𝑥) = ((2^nd ‘𝐵)‘𝑥))))
60		anandi 871	. . . 4 ⊢ ((𝑁 = (#‘(1^st ‘𝐵)) ∧ (∀𝑥 ∈ (0..^𝑁)((1^st ‘𝐴)‘𝑥) = ((1^st ‘𝐵)‘𝑥) ∧ ∀𝑥 ∈ (0...𝑁)((2^nd ‘𝐴)‘𝑥) = ((2^nd ‘𝐵)‘𝑥))) ↔ ((𝑁 = (#‘(1^st ‘𝐵)) ∧ ∀𝑥 ∈ (0..^𝑁)((1^st ‘𝐴)‘𝑥) = ((1^st ‘𝐵)‘𝑥)) ∧ (𝑁 = (#‘(1^st ‘𝐵)) ∧ ∀𝑥 ∈ (0...𝑁)((2^nd ‘𝐴)‘𝑥) = ((2^nd ‘𝐵)‘𝑥))))
61	59, 60	bitr2i 265	. . 3 ⊢ (((𝑁 = (#‘(1^st ‘𝐵)) ∧ ∀𝑥 ∈ (0..^𝑁)((1^st ‘𝐴)‘𝑥) = ((1^st ‘𝐵)‘𝑥)) ∧ (𝑁 = (#‘(1^st ‘𝐵)) ∧ ∀𝑥 ∈ (0...𝑁)((2^nd ‘𝐴)‘𝑥) = ((2^nd ‘𝐵)‘𝑥))) ↔ (𝑁 = (#‘(1^st ‘𝐵)) ∧ ∀𝑥 ∈ (0..^𝑁)((1^st ‘𝐴)‘𝑥) = ((1^st ‘𝐵)‘𝑥) ∧ ∀𝑥 ∈ (0...𝑁)((2^nd ‘𝐴)‘𝑥) = ((2^nd ‘𝐵)‘𝑥)))
62	61	a1i 11	. 2 ⊢ ((𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺) ∧ 𝑁 = (#‘(1^st ‘𝐴))) → (((𝑁 = (#‘(1^st ‘𝐵)) ∧ ∀𝑥 ∈ (0..^𝑁)((1^st ‘𝐴)‘𝑥) = ((1^st ‘𝐵)‘𝑥)) ∧ (𝑁 = (#‘(1^st ‘𝐵)) ∧ ∀𝑥 ∈ (0...𝑁)((2^nd ‘𝐴)‘𝑥) = ((2^nd ‘𝐵)‘𝑥))) ↔ (𝑁 = (#‘(1^st ‘𝐵)) ∧ ∀𝑥 ∈ (0..^𝑁)((1^st ‘𝐴)‘𝑥) = ((1^st ‘𝐵)‘𝑥) ∧ ∀𝑥 ∈ (0...𝑁)((2^nd ‘𝐴)‘𝑥) = ((2^nd ‘𝐵)‘𝑥))))
63	10, 58, 62	3bitrd 294	1 ⊢ ((𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺) ∧ 𝑁 = (#‘(1^st ‘𝐴))) → (𝐴 = 𝐵 ↔ (𝑁 = (#‘(1^st ‘𝐵)) ∧ ∀𝑥 ∈ (0..^𝑁)((1^st ‘𝐴)‘𝑥) = ((1^st ‘𝐵)‘𝑥) ∧ ∀𝑥 ∈ (0...𝑁)((2^nd ‘𝐴)‘𝑥) = ((2^nd ‘𝐵)‘𝑥))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ∀wral 2912 Vcvv 3200 ⟨cop 4183 class class class wbr 4653 × cxp 5112 dom cdm 5114 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 1^st c1st 7166 2^nd c2nd 7167 0cc0 9936 1c1 9937 − cmin 10266 ℕ₀cn0 11292 ...cfz 12326 ..^cfzo 12465 #chash 13117 Word cword 13291 Vtxcvtx 25874 iEdgciedg 25875 Walkscwlks 26492

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-ifp 1013 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-n0 11293 df-z 11378 df-uz 11688 df-fz 12327 df-fzo 12466 df-hash 13118 df-word 13299 df-wlks 26495

This theorem is referenced by: uspgr2wlkeq 26542

W3C validator