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Theorem wlkeq 29670

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 2740	. . . . . . 7 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2		eqid 2740	. . . . . . 7 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
3		eqid 2740	. . . . . . 7 ⊢ (1^st ‘𝐴) = (1^st ‘𝐴)
4		eqid 2740	. . . . . . 7 ⊢ (2^nd ‘𝐴) = (2^nd ‘𝐴)
5	1, 2, 3, 4	wlkelwrd 29669	. . . . . 6 ⊢ (𝐴 ∈ (Walks‘𝐺) → ((1^st ‘𝐴) ∈ Word dom (iEdg‘𝐺) ∧ (2^nd ‘𝐴):(0...(♯‘(1^st ‘𝐴)))⟶(Vtx‘𝐺)))
6		eqid 2740	. . . . . . 7 ⊢ (1^st ‘𝐵) = (1^st ‘𝐵)
7		eqid 2740	. . . . . . 7 ⊢ (2^nd ‘𝐵) = (2^nd ‘𝐵)
8	1, 2, 6, 7	wlkelwrd 29669	. . . . . 6 ⊢ (𝐵 ∈ (Walks‘𝐺) → ((1^st ‘𝐵) ∈ Word dom (iEdg‘𝐺) ∧ (2^nd ‘𝐵):(0...(♯‘(1^st ‘𝐵)))⟶(Vtx‘𝐺)))
9	5, 8	anim12i 612	. . . . 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 29664	. . . . . . 7 ⊢ (𝐴 ∈ (Walks‘𝐺) → 𝐴 = ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩)
11		eleq1 2832	. . . . . . . 8 ⊢ (𝐴 = ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ → (𝐴 ∈ (Walks‘𝐺) ↔ ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ (Walks‘𝐺)))
12		df-br 5167	. . . . . . . . 9 ⊢ ((1^st ‘𝐴)(Walks‘𝐺)(2^nd ‘𝐴) ↔ ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ (Walks‘𝐺))
13		wlklenvm1 29658	. . . . . . . . 9 ⊢ ((1^st ‘𝐴)(Walks‘𝐺)(2^nd ‘𝐴) → (♯‘(1^st ‘𝐴)) = ((♯‘(2^nd ‘𝐴)) − 1))
14	12, 13	sylbir 235	. . . . . . . 8 ⊢ (⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ (Walks‘𝐺) → (♯‘(1^st ‘𝐴)) = ((♯‘(2^nd ‘𝐴)) − 1))
15	11, 14	biimtrdi 253	. . . . . . 7 ⊢ (𝐴 = ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ → (𝐴 ∈ (Walks‘𝐺) → (♯‘(1^st ‘𝐴)) = ((♯‘(2^nd ‘𝐴)) − 1)))
16	10, 15	mpcom 38	. . . . . 6 ⊢ (𝐴 ∈ (Walks‘𝐺) → (♯‘(1^st ‘𝐴)) = ((♯‘(2^nd ‘𝐴)) − 1))
17		wlkop 29664	. . . . . . 7 ⊢ (𝐵 ∈ (Walks‘𝐺) → 𝐵 = ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩)
18		eleq1 2832	. . . . . . . 8 ⊢ (𝐵 = ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ → (𝐵 ∈ (Walks‘𝐺) ↔ ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ (Walks‘𝐺)))
19		df-br 5167	. . . . . . . . 9 ⊢ ((1^st ‘𝐵)(Walks‘𝐺)(2^nd ‘𝐵) ↔ ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ (Walks‘𝐺))
20		wlklenvm1 29658	. . . . . . . . 9 ⊢ ((1^st ‘𝐵)(Walks‘𝐺)(2^nd ‘𝐵) → (♯‘(1^st ‘𝐵)) = ((♯‘(2^nd ‘𝐵)) − 1))
21	19, 20	sylbir 235	. . . . . . . 8 ⊢ (⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ (Walks‘𝐺) → (♯‘(1^st ‘𝐵)) = ((♯‘(2^nd ‘𝐵)) − 1))
22	18, 21	biimtrdi 253	. . . . . . 7 ⊢ (𝐵 = ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ → (𝐵 ∈ (Walks‘𝐺) → (♯‘(1^st ‘𝐵)) = ((♯‘(2^nd ‘𝐵)) − 1)))
23	17, 22	mpcom 38	. . . . . 6 ⊢ (𝐵 ∈ (Walks‘𝐺) → (♯‘(1^st ‘𝐵)) = ((♯‘(2^nd ‘𝐵)) − 1))
24	16, 23	anim12i 612	. . . . 5 ⊢ ((𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) → ((♯‘(1^st ‘𝐴)) = ((♯‘(2^nd ‘𝐴)) − 1) ∧ (♯‘(1^st ‘𝐵)) = ((♯‘(2^nd ‘𝐵)) − 1)))
25		eqwrd 14605	. . . . . . . 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 746	. . . . . . 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 480	. . . . . 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 14581	. . . . . . . . 9 ⊢ ((1^st ‘𝐴) ∈ Word dom (iEdg‘𝐺) → (♯‘(1^st ‘𝐴)) ∈ ℕ₀)
29	28	adantr 480	. . . . . . . 8 ⊢ (((1^st ‘𝐴) ∈ Word dom (iEdg‘𝐺) ∧ (2^nd ‘𝐴):(0...(♯‘(1^st ‘𝐴)))⟶(Vtx‘𝐺)) → (♯‘(1^st ‘𝐴)) ∈ ℕ₀)
30		simpr 484	. . . . . . . 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 484	. . . . . . . 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 13706	. . . . . . . 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 1422	. . . . . . 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 480	. . . . . 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 631	. . . . 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 583	. . . 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 1132	. . 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 2744	. . . . . . 7 ⊢ (𝑁 = (♯‘(1^st ‘𝐴)) → (𝑁 = (♯‘(1^st ‘𝐵)) ↔ (♯‘(1^st ‘𝐴)) = (♯‘(1^st ‘𝐵))))
39		oveq2 7456	. . . . . . . 8 ⊢ (𝑁 = (♯‘(1^st ‘𝐴)) → (0..^𝑁) = (0..^(♯‘(1^st ‘𝐴))))
40	39	raleqdv 3334	. . . . . . 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 631	. . . . . 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 7456	. . . . . . . 8 ⊢ (𝑁 = (♯‘(1^st ‘𝐴)) → (0...𝑁) = (0...(♯‘(1^st ‘𝐴))))
43	42	raleqdv 3334	. . . . . . 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 631	. . . . . 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 631	. . . . 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 342	. . . 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 1135	. . 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 257	. 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 8070	. . . . 5 ⊢ (𝐴 ∈ (V × V) ↔ 𝐴 = ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩)
50	10, 49	sylibr 234	. . . 4 ⊢ (𝐴 ∈ (Walks‘𝐺) → 𝐴 ∈ (V × V))
51		1st2ndb 8070	. . . . 5 ⊢ (𝐵 ∈ (V × V) ↔ 𝐵 = ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩)
52	17, 51	sylibr 234	. . . 4 ⊢ (𝐵 ∈ (Walks‘𝐺) → 𝐵 ∈ (V × V))
53		xpopth 8071	. . . 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 595	. . 3 ⊢ ((𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) → (((1^st ‘𝐴) = (1^st ‘𝐵) ∧ (2^nd ‘𝐴) = (2^nd ‘𝐵)) ↔ 𝐴 = 𝐵))
55	54	3adant3 1132	. 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 1095	. . . 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 675	. . . 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 276	. . 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 309	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 206 ∧ wa 395 ∧ w3a 1087 = wceq 1537 ∈ wcel 2108 ∀wral 3067 Vcvv 3488 ⟨cop 4654 class class class wbr 5166 × cxp 5698 dom cdm 5700 ⟶wf 6569 ‘cfv 6573 (class class class)co 7448 1^st c1st 8028 2^nd c2nd 8029 0cc0 11184 1c1 11185 − cmin 11520 ℕ₀cn0 12553 ...cfz 13567 ..^cfzo 13711 ♯chash 14379 Word cword 14562 Vtxcvtx 29031 iEdgciedg 29032 Walkscwlks 29632

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-ifp 1064 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-om 7904 df-1st 8030 df-2nd 8031 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-er 8763 df-map 8886 df-pm 8887 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-card 10008 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-nn 12294 df-n0 12554 df-z 12640 df-uz 12904 df-fz 13568 df-fzo 13712 df-hash 14380 df-word 14563 df-wlks 29635

This theorem is referenced by: uspgr2wlkeq 29682

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