wlkeq - Metamath Proof Explorer

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

Theorem wlkeq 30207

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 2761	. . . . . . 7 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2		eqid 2761	. . . . . . 7 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
3		eqid 2761	. . . . . . 7 ⊢ (1^st ‘𝐴) = (1^st ‘𝐴)
4		eqid 2761	. . . . . . 7 ⊢ (2^nd ‘𝐴) = (2^nd ‘𝐴)
5	1, 2, 3, 4	wlkelwrd 30206	. . . . . 6 ⊢ (𝐴 ∈ (Walks‘𝐺) → ((1^st ‘𝐴) ∈ Word dom (iEdg‘𝐺) ∧ (2^nd ‘𝐴):(0...(♯‘(1^st ‘𝐴)))⟶(Vtx‘𝐺)))
6		eqid 2761	. . . . . . 7 ⊢ (1^st ‘𝐵) = (1^st ‘𝐵)
7		eqid 2761	. . . . . . 7 ⊢ (2^nd ‘𝐵) = (2^nd ‘𝐵)
8	1, 2, 6, 7	wlkelwrd 30206	. . . . . 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 30201	. . . . . . 7 ⊢ (𝐴 ∈ (Walks‘𝐺) → 𝐴 = ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩)
11		eleq1 2849	. . . . . . . 8 ⊢ (𝐴 = ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ → (𝐴 ∈ (Walks‘𝐺) ↔ ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ (Walks‘𝐺)))
12		df-br 5104	. . . . . . . . 9 ⊢ ((1^st ‘𝐴)(Walks‘𝐺)(2^nd ‘𝐴) ↔ ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ (Walks‘𝐺))
13		wlklenvm1 30195	. . . . . . . . 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 30201	. . . . . . 7 ⊢ (𝐵 ∈ (Walks‘𝐺) → 𝐵 = ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩)
18		eleq1 2849	. . . . . . . 8 ⊢ (𝐵 = ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ → (𝐵 ∈ (Walks‘𝐺) ↔ ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ (Walks‘𝐺)))
19		df-br 5104	. . . . . . . . 9 ⊢ ((1^st ‘𝐵)(Walks‘𝐺)(2^nd ‘𝐵) ↔ ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ (Walks‘𝐺))
20		wlklenvm1 30195	. . . . . . . . 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 14695	. . . . . . . 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 14671	. . . . . . . . 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 13776	. . . . . . . 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 1449	. . . . . . 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 2765	. . . . . . 7 ⊢ (𝑁 = (♯‘(1^st ‘𝐴)) → (𝑁 = (♯‘(1^st ‘𝐵)) ↔ (♯‘(1^st ‘𝐴)) = (♯‘(1^st ‘𝐵))))
39		oveq2 7426	. . . . . . . 8 ⊢ (𝑁 = (♯‘(1^st ‘𝐴)) → (0..^𝑁) = (0..^(♯‘(1^st ‘𝐴))))
40	39	raleqdv 3320	. . . . . . 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 7426	. . . . . . . 8 ⊢ (𝑁 = (♯‘(1^st ‘𝐴)) → (0...𝑁) = (0...(♯‘(1^st ‘𝐴))))
43	42	raleqdv 3320	. . . . . . 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 8039	. . . . 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 8039	. . . . 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 8040	. . . 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 2145 ∀wral 3077 Vcvv 3451 ⟨cop 4590 class class class wbr 5103 × cxp 5649 dom cdm 5651 ⟶wf 6533 ‘cfv 6537 (class class class)co 7418 1^st c1st 7997 2^nd c2nd 7998 0cc0 11193 1c1 11194 − cmin 11534 ℕ₀cn0 12599 ...cfz 13632 ..^cfzo 13781 ♯chash 14467 Word cword 14651 Vtxcvtx 29567 iEdgciedg 29568 Walkscwlks 30170

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 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-rep 5232 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7749 ax-cnex 11249 ax-resscn 11250 ax-1cn 11251 ax-icn 11252 ax-addcl 11253 ax-addrcl 11254 ax-mulcl 11255 ax-mulrcl 11256 ax-mulcom 11257 ax-addass 11258 ax-mulass 11259 ax-distr 11260 ax-i2m1 11261 ax-1ne0 11262 ax-1rid 11263 ax-rnegex 11264 ax-rrecex 11265 ax-cnre 11266 ax-pre-lttri 11267 ax-pre-lttrn 11268 ax-pre-ltadd 11269 ax-pre-mulgt0 11270

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 2100 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-reu 3367 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7375 df-ov 7421 df-oprab 7422 df-mpo 7423 df-om 7876 df-1st 7999 df-2nd 8000 df-frecs 8292 df-wrecs 8323 df-recs 8372 df-rdg 8411 df-1o 8469 df-er 8710 df-map 8842 df-pm 8843 df-en 8967 df-dom 8968 df-sdom 8969 df-fin 8970 df-card 10013 df-pnf 11338 df-mnf 11339 df-xr 11340 df-ltxr 11341 df-le 11342 df-sub 11536 df-neg 11537 df-nn 12329 df-n0 12600 df-z 12687 df-uz 12959 df-fz 13633 df-fzo 13782 df-hash 14468 df-word 14652 df-wlks 30173

This theorem is used by: uspgr2wlkeq 30219

W3C validator