1wlkeq - Mathbox for Alexander van der Vekens

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Theorem 1wlkeq 40838

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
1wlkeq	⊢ ((𝐴 ∈ (1Walks‘𝐺) ∧ 𝐵 ∈ (1Walks‘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 1wlkeq**
Step	Hyp	Ref	Expression
1		1wlkop 40832	. . . . 5 ⊢ (𝐴 ∈ (1Walks‘𝐺) → 𝐴 = ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩)
2		1st2ndb 7097	. . . . 5 ⊢ (𝐴 ∈ (V × V) ↔ 𝐴 = ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩)
3	1, 2	sylibr 223	. . . 4 ⊢ (𝐴 ∈ (1Walks‘𝐺) → 𝐴 ∈ (V × V))
4		1wlkop 40832	. . . . 5 ⊢ (𝐵 ∈ (1Walks‘𝐺) → 𝐵 = ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩)
5		1st2ndb 7097	. . . . 5 ⊢ (𝐵 ∈ (V × V) ↔ 𝐵 = ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩)
6	4, 5	sylibr 223	. . . 4 ⊢ (𝐵 ∈ (1Walks‘𝐺) → 𝐵 ∈ (V × V))
7		xpopth 7098	. . . . 5 ⊢ ((𝐴 ∈ (V × V) ∧ 𝐵 ∈ (V × V)) → (((1^st ‘𝐴) = (1^st ‘𝐵) ∧ (2^nd ‘𝐴) = (2^nd ‘𝐵)) ↔ 𝐴 = 𝐵))
8	7	bicomd 212	. . . 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 493	. . 3 ⊢ ((𝐴 ∈ (1Walks‘𝐺) ∧ 𝐵 ∈ (1Walks‘𝐺)) → (𝐴 = 𝐵 ↔ ((1^st ‘𝐴) = (1^st ‘𝐵) ∧ (2^nd ‘𝐴) = (2^nd ‘𝐵))))
10	9	3adant3 1074	. 2 ⊢ ((𝐴 ∈ (1Walks‘𝐺) ∧ 𝐵 ∈ (1Walks‘𝐺) ∧ 𝑁 = (#‘(1^st ‘𝐴))) → (𝐴 = 𝐵 ↔ ((1^st ‘𝐴) = (1^st ‘𝐵) ∧ (2^nd ‘𝐴) = (2^nd ‘𝐵))))
11		eqid 2610	. . . . . . 7 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
12		eqid 2610	. . . . . . 7 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
13		eqid 2610	. . . . . . 7 ⊢ (1^st ‘𝐴) = (1^st ‘𝐴)
14		eqid 2610	. . . . . . 7 ⊢ (2^nd ‘𝐴) = (2^nd ‘𝐴)
15	11, 12, 13, 14	1wlkelwrd 40837	. . . . . 6 ⊢ (𝐴 ∈ (1Walks‘𝐺) → ((1^st ‘𝐴) ∈ Word dom (iEdg‘𝐺) ∧ (2^nd ‘𝐴):(0...(#‘(1^st ‘𝐴)))⟶(Vtx‘𝐺)))
16		eqid 2610	. . . . . . 7 ⊢ (1^st ‘𝐵) = (1^st ‘𝐵)
17		eqid 2610	. . . . . . 7 ⊢ (2^nd ‘𝐵) = (2^nd ‘𝐵)
18	11, 12, 16, 17	1wlkelwrd 40837	. . . . . 6 ⊢ (𝐵 ∈ (1Walks‘𝐺) → ((1^st ‘𝐵) ∈ Word dom (iEdg‘𝐺) ∧ (2^nd ‘𝐵):(0...(#‘(1^st ‘𝐵)))⟶(Vtx‘𝐺)))
19	15, 18	anim12i 588	. . . . 5 ⊢ ((𝐴 ∈ (1Walks‘𝐺) ∧ 𝐵 ∈ (1Walks‘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 2676	. . . . . . . 8 ⊢ (𝐴 = ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ → (𝐴 ∈ (1Walks‘𝐺) ↔ ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ (1Walks‘𝐺)))
21		df-br 4584	. . . . . . . . 9 ⊢ ((1^st ‘𝐴)(1Walks‘𝐺)(2^nd ‘𝐴) ↔ ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ (1Walks‘𝐺))
22		1wlklenvm1 40826	. . . . . . . . 9 ⊢ ((1^st ‘𝐴)(1Walks‘𝐺)(2^nd ‘𝐴) → (#‘(1^st ‘𝐴)) = ((#‘(2^nd ‘𝐴)) − 1))
23	21, 22	sylbir 224	. . . . . . . 8 ⊢ (⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ (1Walks‘𝐺) → (#‘(1^st ‘𝐴)) = ((#‘(2^nd ‘𝐴)) − 1))
24	20, 23	syl6bi 242	. . . . . . 7 ⊢ (𝐴 = ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ → (𝐴 ∈ (1Walks‘𝐺) → (#‘(1^st ‘𝐴)) = ((#‘(2^nd ‘𝐴)) − 1)))
25	1, 24	mpcom 37	. . . . . 6 ⊢ (𝐴 ∈ (1Walks‘𝐺) → (#‘(1^st ‘𝐴)) = ((#‘(2^nd ‘𝐴)) − 1))
26		eleq1 2676	. . . . . . . 8 ⊢ (𝐵 = ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ → (𝐵 ∈ (1Walks‘𝐺) ↔ ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ (1Walks‘𝐺)))
27		df-br 4584	. . . . . . . . 9 ⊢ ((1^st ‘𝐵)(1Walks‘𝐺)(2^nd ‘𝐵) ↔ ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ (1Walks‘𝐺))
28		1wlklenvm1 40826	. . . . . . . . 9 ⊢ ((1^st ‘𝐵)(1Walks‘𝐺)(2^nd ‘𝐵) → (#‘(1^st ‘𝐵)) = ((#‘(2^nd ‘𝐵)) − 1))
29	27, 28	sylbir 224	. . . . . . . 8 ⊢ (⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ (1Walks‘𝐺) → (#‘(1^st ‘𝐵)) = ((#‘(2^nd ‘𝐵)) − 1))
30	26, 29	syl6bi 242	. . . . . . 7 ⊢ (𝐵 = ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ → (𝐵 ∈ (1Walks‘𝐺) → (#‘(1^st ‘𝐵)) = ((#‘(2^nd ‘𝐵)) − 1)))
31	4, 30	mpcom 37	. . . . . 6 ⊢ (𝐵 ∈ (1Walks‘𝐺) → (#‘(1^st ‘𝐵)) = ((#‘(2^nd ‘𝐵)) − 1))
32	25, 31	anim12i 588	. . . . 5 ⊢ ((𝐴 ∈ (1Walks‘𝐺) ∧ 𝐵 ∈ (1Walks‘𝐺)) → ((#‘(1^st ‘𝐴)) = ((#‘(2^nd ‘𝐴)) − 1) ∧ (#‘(1^st ‘𝐵)) = ((#‘(2^nd ‘𝐵)) − 1)))
33		eqwrd 13201	. . . . . . . 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 779	. . . . . . 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 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))))
36		lencl 13179	. . . . . . . . . . 11 ⊢ ((1^st ‘𝐴) ∈ Word dom (iEdg‘𝐺) → (#‘(1^st ‘𝐴)) ∈ ℕ₀)
37	36	adantr 480	. . . . . . . . . 10 ⊢ (((1^st ‘𝐴) ∈ Word dom (iEdg‘𝐺) ∧ (2^nd ‘𝐴):(0...(#‘(1^st ‘𝐴)))⟶(Vtx‘𝐺)) → (#‘(1^st ‘𝐴)) ∈ ℕ₀)
38	37	adantr 480	. . . . . . . . 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 788	. . . . . . . . 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 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 ‘𝐵):(0...(#‘(1^st ‘𝐵)))⟶(Vtx‘𝐺))
41	38, 39, 40	3jca 1235	. . . . . . . 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 480	. . . . . . 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 12329	. . . . . . 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 743	. . . . 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 691	. . . 4 ⊢ ((𝐴 ∈ (1Walks‘𝐺) ∧ 𝐵 ∈ (1Walks‘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 1074	. . 3 ⊢ ((𝐴 ∈ (1Walks‘𝐺) ∧ 𝐵 ∈ (1Walks‘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 2614	. . . . . . 7 ⊢ (𝑁 = (#‘(1^st ‘𝐴)) → (𝑁 = (#‘(1^st ‘𝐵)) ↔ (#‘(1^st ‘𝐴)) = (#‘(1^st ‘𝐵))))
49		oveq2 6557	. . . . . . . 8 ⊢ (𝑁 = (#‘(1^st ‘𝐴)) → (0..^𝑁) = (0..^(#‘(1^st ‘𝐴))))
50	49	raleqdv 3121	. . . . . . 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 743	. . . . . 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 6557	. . . . . . . 8 ⊢ (𝑁 = (#‘(1^st ‘𝐴)) → (0...𝑁) = (0...(#‘(1^st ‘𝐴))))
53	52	raleqdv 3121	. . . . . . 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 743	. . . . . 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 743	. . . . 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 331	. . . 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 1077	. . 3 ⊢ ((𝐴 ∈ (1Walks‘𝐺) ∧ 𝐵 ∈ (1Walks‘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 246	. 2 ⊢ ((𝐴 ∈ (1Walks‘𝐺) ∧ 𝐵 ∈ (1Walks‘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 1035	. . . 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 867	. . . 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 264	. . 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 ∈ (1Walks‘𝐺) ∧ 𝐵 ∈ (1Walks‘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 293	1 ⊢ ((𝐴 ∈ (1Walks‘𝐺) ∧ 𝐵 ∈ (1Walks‘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 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ∀wral 2896 Vcvv 3173 ⟨cop 4131 class class class wbr 4583 × cxp 5036 dom cdm 5038 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 1^st c1st 7057 2^nd c2nd 7058 0cc0 9815 1c1 9816 − cmin 10145 ℕ₀cn0 11169 ...cfz 12197 ..^cfzo 12334 #chash 12979 Word cword 13146 Vtxcvtx 25673 iEdgciedg 25674 1Walksc1wlks 40796

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-ifp 1007 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-map 7746 df-pm 7747 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-fzo 12335 df-hash 12980 df-word 13154 df-1wlks 40800

This theorem is referenced by: uspgr2wlkeq 40854

W3C validator