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Theorem wlkeq 30271

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		eqid 2763	. . . . . . 7 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2		eqid 2763	. . . . . . 7 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
3		eqid 2763	. . . . . . 7 ⊢ (1^st ‘𝐴) = (1^st ‘𝐴)
4		eqid 2763	. . . . . . 7 ⊢ (2^nd ‘𝐴) = (2^nd ‘𝐴)
5	1, 2, 3, 4	wlkelwrd 30270	. . . . . 6 ⊢ (𝐴 ∈ (Walks‘𝐺) → ((1^st ‘𝐴) ∈ Word dom (iEdg‘𝐺) ∧ (2^nd ‘𝐴):(0...(♯‘(1^st ‘𝐴)))⟶(Vtx‘𝐺)))
6		eqid 2763	. . . . . . 7 ⊢ (1^st ‘𝐵) = (1^st ‘𝐵)
7		eqid 2763	. . . . . . 7 ⊢ (2^nd ‘𝐵) = (2^nd ‘𝐵)
8	1, 2, 6, 7	wlkelwrd 30270	. . . . . 6 ⊢ (𝐵 ∈ (Walks‘𝐺) → ((1^st ‘𝐵) ∈ Word dom (iEdg‘𝐺) ∧ (2^nd ‘𝐵):(0...(♯‘(1^st ‘𝐵)))⟶(Vtx‘𝐺)))
9	5, 8	anim12i 625	. . . . 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))))
10		wlkop 30265	. . . . . . 7 ⊢ (𝐴 ∈ (Walks‘𝐺) → 𝐴 = ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩)
11		eleq1 2852	. . . . . . . 8 ⊢ (𝐴 = ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ → (𝐴 ∈ (Walks‘𝐺) ↔ ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ (Walks‘𝐺)))
12		df-br 5109	. . . . . . . . 9 ⊢ ((1^st ‘𝐴)(Walks‘𝐺)(2^nd ‘𝐴) ↔ ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ (Walks‘𝐺))
13		wlklenvm1 30259	. . . . . . . . 9 ⊢ ((1^st ‘𝐴)(Walks‘𝐺)(2^nd ‘𝐴) → (♯‘(1^st ‘𝐴)) = ((♯‘(2^nd ‘𝐴)) − 1))
14	12, 13	sylbir 238	. . . . . . . 8 ⊢ (⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ (Walks‘𝐺) → (♯‘(1^st ‘𝐴)) = ((♯‘(2^nd ‘𝐴)) − 1))
15	11, 14	biimtrdi 256	. . . . . . 7 ⊢ (𝐴 = ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ → (𝐴 ∈ (Walks‘𝐺) → (♯‘(1^st ‘𝐴)) = ((♯‘(2^nd ‘𝐴)) − 1)))
16	10, 15	mpcom 39	. . . . . 6 ⊢ (𝐴 ∈ (Walks‘𝐺) → (♯‘(1^st ‘𝐴)) = ((♯‘(2^nd ‘𝐴)) − 1))
17		wlkop 30265	. . . . . . 7 ⊢ (𝐵 ∈ (Walks‘𝐺) → 𝐵 = ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩)
18		eleq1 2852	. . . . . . . 8 ⊢ (𝐵 = ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ → (𝐵 ∈ (Walks‘𝐺) ↔ ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ (Walks‘𝐺)))
19		df-br 5109	. . . . . . . . 9 ⊢ ((1^st ‘𝐵)(Walks‘𝐺)(2^nd ‘𝐵) ↔ ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ (Walks‘𝐺))
20		wlklenvm1 30259	. . . . . . . . 9 ⊢ ((1^st ‘𝐵)(Walks‘𝐺)(2^nd ‘𝐵) → (♯‘(1^st ‘𝐵)) = ((♯‘(2^nd ‘𝐵)) − 1))
21	19, 20	sylbir 238	. . . . . . . 8 ⊢ (⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ (Walks‘𝐺) → (♯‘(1^st ‘𝐵)) = ((♯‘(2^nd ‘𝐵)) − 1))
22	18, 21	biimtrdi 256	. . . . . . 7 ⊢ (𝐵 = ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ → (𝐵 ∈ (Walks‘𝐺) → (♯‘(1^st ‘𝐵)) = ((♯‘(2^nd ‘𝐵)) − 1)))
23	17, 22	mpcom 39	. . . . . 6 ⊢ (𝐵 ∈ (Walks‘𝐺) → (♯‘(1^st ‘𝐵)) = ((♯‘(2^nd ‘𝐵)) − 1))
24	16, 23	anim12i 625	. . . . 5 ⊢ ((𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) → ((♯‘(1^st ‘𝐴)) = ((♯‘(2^nd ‘𝐴)) − 1) ∧ (♯‘(1^st ‘𝐵)) = ((♯‘(2^nd ‘𝐵)) − 1)))
25		eqwrd 14743	. . . . . . . 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))))
26	25	ad2ant2r 760	. . . . . . 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))))
27	26	adantr 486	. . . . . 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))))
28		lencl 14719	. . . . . . . . 9 ⊢ ((1^st ‘𝐴) ∈ Word dom (iEdg‘𝐺) → (♯‘(1^st ‘𝐴)) ∈ ℕ₀)
29	28	adantr 486	. . . . . . . 8 ⊢ (((1^st ‘𝐴) ∈ Word dom (iEdg‘𝐺) ∧ (2^nd ‘𝐴):(0...(♯‘(1^st ‘𝐴)))⟶(Vtx‘𝐺)) → (♯‘(1^st ‘𝐴)) ∈ ℕ₀)
30		simpr 490	. . . . . . . 8 ⊢ (((1^st ‘𝐴) ∈ Word dom (iEdg‘𝐺) ∧ (2^nd ‘𝐴):(0...(♯‘(1^st ‘𝐴)))⟶(Vtx‘𝐺)) → (2^nd ‘𝐴):(0...(♯‘(1^st ‘𝐴)))⟶(Vtx‘𝐺))
31		simpr 490	. . . . . . . 8 ⊢ (((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))
32		2ffzeq 13823	. . . . . . . 8 ⊢ (((♯‘(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))))
33	29, 30, 31, 32	syl2an3an 1450	. . . . . . 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))) → ((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))))
34	33	adantr 486	. . . . . 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))))
35	27, 34	anbi12d 644	. . . . 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)))))
36	9, 24, 35	syl2anc 596	. . . 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)))))
37	36	3adant3 1150	. . 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)))))
38		eqeq1 2767	. . . . . . 7 ⊢ (𝑁 = (♯‘(1^st ‘𝐴)) → (𝑁 = (♯‘(1^st ‘𝐵)) ↔ (♯‘(1^st ‘𝐴)) = (♯‘(1^st ‘𝐵))))
39		oveq2 7459	. . . . . . . 8 ⊢ (𝑁 = (♯‘(1^st ‘𝐴)) → (0..^𝑁) = (0..^(♯‘(1^st ‘𝐴))))
40	39	raleqdv 3323	. . . . . . 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)))
41	38, 40	anbi12d 644	. . . . . 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))))
42		oveq2 7459	. . . . . . . 8 ⊢ (𝑁 = (♯‘(1^st ‘𝐴)) → (0...𝑁) = (0...(♯‘(1^st ‘𝐴))))
43	42	raleqdv 3323	. . . . . . 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)))
44	38, 43	anbi12d 644	. . . . . 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))))
45	41, 44	anbi12d 644	. . . . 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)))))
46	45	bibi2d 345	. . . 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))))))
47	46	3ad2ant3 1153	. . 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))))))
48	37, 47	mpbird 260	. 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)))))
49		1st2ndb 8075	. . . . 5 ⊢ (𝐴 ∈ (V × V) ↔ 𝐴 = ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩)
50	10, 49	sylibr 237	. . . 4 ⊢ (𝐴 ∈ (Walks‘𝐺) → 𝐴 ∈ (V × V))
51		1st2ndb 8075	. . . . 5 ⊢ (𝐵 ∈ (V × V) ↔ 𝐵 = ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩)
52	17, 51	sylibr 237	. . . 4 ⊢ (𝐵 ∈ (Walks‘𝐺) → 𝐵 ∈ (V × V))
53		xpopth 8076	. . . 4 ⊢ ((𝐴 ∈ (V × V) ∧ 𝐵 ∈ (V × V)) → (((1^st ‘𝐴) = (1^st ‘𝐵) ∧ (2^nd ‘𝐴) = (2^nd ‘𝐵)) ↔ 𝐴 = 𝐵))
54	50, 52, 53	syl2an 608	. . 3 ⊢ ((𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) → (((1^st ‘𝐴) = (1^st ‘𝐵) ∧ (2^nd ‘𝐴) = (2^nd ‘𝐵)) ↔ 𝐴 = 𝐵))
55	54	3adant3 1150	. 2 ⊢ ((𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺) ∧ 𝑁 = (♯‘(1^st ‘𝐴))) → (((1^st ‘𝐴) = (1^st ‘𝐵) ∧ (2^nd ‘𝐴) = (2^nd ‘𝐵)) ↔ 𝐴 = 𝐵))
56		3anass 1111	. . . 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))))
57		anandi 689	. . . 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))))
58	56, 57	bitr2i 279	. . 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)))
59	58	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))))
60	48, 55, 59	3bitr3d 312	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

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2147 ∀wral 3080 Vcvv 3454 ⟨cop 4840 class class class wbr 5108 × cxp 5656 dom cdm 5658 ⟶wf 6562 ‘cfv 6566 (class class class)co 7451 1^st c1st 8033 2^nd c2nd 8034 0cc0 11239 1c1 11240 − cmin 11580 ℕ₀cn0 12646 ...cfz 13679 ..^cfzo 13828 ♯chash 14515 Word cword 14699 Vtxcvtx 29631 iEdgciedg 29632 Walkscwlks 30234

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5238 ax-sep 5255 ax-nul 5266 ax-pow 5334 ax-pr 5398 ax-un 7783 ax-cnex 11295 ax-resscn 11296 ax-1cn 11297 ax-icn 11298 ax-addcl 11299 ax-addrcl 11300 ax-mulcl 11301 ax-mulrcl 11302 ax-mulcom 11303 ax-addass 11304 ax-mulass 11305 ax-distr 11306 ax-i2m1 11307 ax-1ne0 11308 ax-1rid 11309 ax-rnegex 11310 ax-rrecex 11311 ax-cnre 11312 ax-pre-lttri 11313 ax-pre-lttrn 11314 ax-pre-ltadd 11315 ax-pre-mulgt0 11316

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-ifp 1079 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-op 4841 df-uni 4873 df-int 4913 df-iun 4958 df-br 5109 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-we 5613 df-xp 5664 df-rel 5665 df-cnv 5666 df-co 5667 df-dm 5668 df-rn 5669 df-res 5670 df-ima 5671 df-pred 6330 df-ord 6391 df-on 6392 df-lim 6393 df-suc 6394 df-iota 6520 df-fun 6568 df-fn 6569 df-f 6570 df-f1 6571 df-fo 6572 df-f1o 6573 df-fv 6574 df-riota 7408 df-ov 7454 df-oprab 7455 df-mpo 7456 df-om 7912 df-1st 8035 df-2nd 8036 df-frecs 8337 df-wrecs 8368 df-recs 8417 df-rdg 8456 df-1o 8514 df-er 8755 df-map 8887 df-pm 8888 df-en 9012 df-dom 9013 df-sdom 9014 df-fin 9015 df-card 10057 df-pnf 11384 df-mnf 11385 df-xr 11386 df-ltxr 11387 df-le 11388 df-sub 11582 df-neg 11583 df-nn 12376 df-n0 12647 df-z 12734 df-uz 13006 df-fz 13680 df-fzo 13829 df-hash 14516 df-word 14700 df-wlks 30237

This theorem is used by: uspgr2wlkeq 30283

W3C validator