revwlk - Mathbox for BTernaryTau

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Theorem revwlk 34115

Description: The reverse of a walk is a walk. (Contributed by BTernaryTau, 30-Nov-2023.)

**Assertion**
Ref	Expression
revwlk	⊢ (𝐹(Walks‘𝐺)𝑃 → (reverse‘𝐹)(Walks‘𝐺)(reverse‘𝑃))

Proof of Theorem revwlk Dummy variables 𝑘 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2733	. . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
2	1	wlkf 28871	. . 3 ⊢ (𝐹(Walks‘𝐺)𝑃 → 𝐹 ∈ Word dom (iEdg‘𝐺))
3		revcl 14711	. . 3 ⊢ (𝐹 ∈ Word dom (iEdg‘𝐺) → (reverse‘𝐹) ∈ Word dom (iEdg‘𝐺))
4	2, 3	syl 17	. 2 ⊢ (𝐹(Walks‘𝐺)𝑃 → (reverse‘𝐹) ∈ Word dom (iEdg‘𝐺))
5		eqid 2733	. . . . 5 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
6	5	wlkpwrd 28874	. . . 4 ⊢ (𝐹(Walks‘𝐺)𝑃 → 𝑃 ∈ Word (Vtx‘𝐺))
7		revcl 14711	. . . 4 ⊢ (𝑃 ∈ Word (Vtx‘𝐺) → (reverse‘𝑃) ∈ Word (Vtx‘𝐺))
8		wrdf 14469	. . . 4 ⊢ ((reverse‘𝑃) ∈ Word (Vtx‘𝐺) → (reverse‘𝑃):(0..^(♯‘(reverse‘𝑃)))⟶(Vtx‘𝐺))
9	6, 7, 8	3syl 18	. . 3 ⊢ (𝐹(Walks‘𝐺)𝑃 → (reverse‘𝑃):(0..^(♯‘(reverse‘𝑃)))⟶(Vtx‘𝐺))
10		revlen 14712	. . . . . . 7 ⊢ (𝐹 ∈ Word dom (iEdg‘𝐺) → (♯‘(reverse‘𝐹)) = (♯‘𝐹))
11	2, 10	syl 17	. . . . . 6 ⊢ (𝐹(Walks‘𝐺)𝑃 → (♯‘(reverse‘𝐹)) = (♯‘𝐹))
12	11	oveq2d 7425	. . . . 5 ⊢ (𝐹(Walks‘𝐺)𝑃 → (0...(♯‘(reverse‘𝐹))) = (0...(♯‘𝐹)))
13		wlklenvp1 28875	. . . . . . 7 ⊢ (𝐹(Walks‘𝐺)𝑃 → (♯‘𝑃) = ((♯‘𝐹) + 1))
14	13	oveq2d 7425	. . . . . 6 ⊢ (𝐹(Walks‘𝐺)𝑃 → (0..^(♯‘𝑃)) = (0..^((♯‘𝐹) + 1)))
15		revlen 14712	. . . . . . . 8 ⊢ (𝑃 ∈ Word (Vtx‘𝐺) → (♯‘(reverse‘𝑃)) = (♯‘𝑃))
16	6, 15	syl 17	. . . . . . 7 ⊢ (𝐹(Walks‘𝐺)𝑃 → (♯‘(reverse‘𝑃)) = (♯‘𝑃))
17	16	oveq2d 7425	. . . . . 6 ⊢ (𝐹(Walks‘𝐺)𝑃 → (0..^(♯‘(reverse‘𝑃))) = (0..^(♯‘𝑃)))
18		wlkcl 28872	. . . . . . . 8 ⊢ (𝐹(Walks‘𝐺)𝑃 → (♯‘𝐹) ∈ ℕ₀)
19	18	nn0zd 12584	. . . . . . 7 ⊢ (𝐹(Walks‘𝐺)𝑃 → (♯‘𝐹) ∈ ℤ)
20		fzval3 13701	. . . . . . 7 ⊢ ((♯‘𝐹) ∈ ℤ → (0...(♯‘𝐹)) = (0..^((♯‘𝐹) + 1)))
21	19, 20	syl 17	. . . . . 6 ⊢ (𝐹(Walks‘𝐺)𝑃 → (0...(♯‘𝐹)) = (0..^((♯‘𝐹) + 1)))
22	14, 17, 21	3eqtr4rd 2784	. . . . 5 ⊢ (𝐹(Walks‘𝐺)𝑃 → (0...(♯‘𝐹)) = (0..^(♯‘(reverse‘𝑃))))
23	12, 22	eqtrd 2773	. . . 4 ⊢ (𝐹(Walks‘𝐺)𝑃 → (0...(♯‘(reverse‘𝐹))) = (0..^(♯‘(reverse‘𝑃))))
24	23	feq2d 6704	. . 3 ⊢ (𝐹(Walks‘𝐺)𝑃 → ((reverse‘𝑃):(0...(♯‘(reverse‘𝐹)))⟶(Vtx‘𝐺) ↔ (reverse‘𝑃):(0..^(♯‘(reverse‘𝑃)))⟶(Vtx‘𝐺)))
25	9, 24	mpbird 257	. 2 ⊢ (𝐹(Walks‘𝐺)𝑃 → (reverse‘𝑃):(0...(♯‘(reverse‘𝐹)))⟶(Vtx‘𝐺))
26	11	oveq2d 7425	. . . . . 6 ⊢ (𝐹(Walks‘𝐺)𝑃 → (0..^(♯‘(reverse‘𝐹))) = (0..^(♯‘𝐹)))
27	26	eleq2d 2820	. . . . 5 ⊢ (𝐹(Walks‘𝐺)𝑃 → (𝑘 ∈ (0..^(♯‘(reverse‘𝐹))) ↔ 𝑘 ∈ (0..^(♯‘𝐹))))
28	27	biimpa 478	. . . 4 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝑘 ∈ (0..^(♯‘(reverse‘𝐹)))) → 𝑘 ∈ (0..^(♯‘𝐹)))
29		revfv 14713	. . . . . . . . . 10 ⊢ ((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑘 ∈ (0..^(♯‘𝐹))) → ((reverse‘𝐹)‘𝑘) = (𝐹‘(((♯‘𝐹) − 1) − 𝑘)))
30	2, 29	sylan 581	. . . . . . . . 9 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝑘 ∈ (0..^(♯‘𝐹))) → ((reverse‘𝐹)‘𝑘) = (𝐹‘(((♯‘𝐹) − 1) − 𝑘)))
31		wlklenvm1 28879	. . . . . . . . . . . . 13 ⊢ (𝐹(Walks‘𝐺)𝑃 → (♯‘𝐹) = ((♯‘𝑃) − 1))
32	31	oveq1d 7424	. . . . . . . . . . . 12 ⊢ (𝐹(Walks‘𝐺)𝑃 → ((♯‘𝐹) − 1) = (((♯‘𝑃) − 1) − 1))
33		lencl 14483	. . . . . . . . . . . . . 14 ⊢ (𝑃 ∈ Word (Vtx‘𝐺) → (♯‘𝑃) ∈ ℕ₀)
34	33	nn0cnd 12534	. . . . . . . . . . . . 13 ⊢ (𝑃 ∈ Word (Vtx‘𝐺) → (♯‘𝑃) ∈ ℂ)
35		sub1m1 12464	. . . . . . . . . . . . 13 ⊢ ((♯‘𝑃) ∈ ℂ → (((♯‘𝑃) − 1) − 1) = ((♯‘𝑃) − 2))
36	6, 34, 35	3syl 18	. . . . . . . . . . . 12 ⊢ (𝐹(Walks‘𝐺)𝑃 → (((♯‘𝑃) − 1) − 1) = ((♯‘𝑃) − 2))
37	32, 36	eqtrd 2773	. . . . . . . . . . 11 ⊢ (𝐹(Walks‘𝐺)𝑃 → ((♯‘𝐹) − 1) = ((♯‘𝑃) − 2))
38	37	fvoveq1d 7431	. . . . . . . . . 10 ⊢ (𝐹(Walks‘𝐺)𝑃 → (𝐹‘(((♯‘𝐹) − 1) − 𝑘)) = (𝐹‘(((♯‘𝑃) − 2) − 𝑘)))
39	38	adantr 482	. . . . . . . . 9 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝑘 ∈ (0..^(♯‘𝐹))) → (𝐹‘(((♯‘𝐹) − 1) − 𝑘)) = (𝐹‘(((♯‘𝑃) − 2) − 𝑘)))
40	30, 39	eqtrd 2773	. . . . . . . 8 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝑘 ∈ (0..^(♯‘𝐹))) → ((reverse‘𝐹)‘𝑘) = (𝐹‘(((♯‘𝑃) − 2) − 𝑘)))
41	40	fveq2d 6896	. . . . . . 7 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝑘 ∈ (0..^(♯‘𝐹))) → ((iEdg‘𝐺)‘((reverse‘𝐹)‘𝑘)) = ((iEdg‘𝐺)‘(𝐹‘(((♯‘𝑃) − 2) − 𝑘))))
42	41	adantr 482	. . . . . 6 ⊢ (((𝐹(Walks‘𝐺)𝑃 ∧ 𝑘 ∈ (0..^(♯‘𝐹))) ∧ ((reverse‘𝑃)‘𝑘) = ((reverse‘𝑃)‘(𝑘 + 1))) → ((iEdg‘𝐺)‘((reverse‘𝐹)‘𝑘)) = ((iEdg‘𝐺)‘(𝐹‘(((♯‘𝑃) − 2) − 𝑘))))
43		fzonn0p1p1 13711	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (0..^(♯‘𝐹)) → (𝑘 + 1) ∈ (0..^((♯‘𝐹) + 1)))
44	43	adantl 483	. . . . . . . . . . . 12 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝑘 ∈ (0..^(♯‘𝐹))) → (𝑘 + 1) ∈ (0..^((♯‘𝐹) + 1)))
45	14	adantr 482	. . . . . . . . . . . 12 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝑘 ∈ (0..^(♯‘𝐹))) → (0..^(♯‘𝑃)) = (0..^((♯‘𝐹) + 1)))
46	44, 45	eleqtrrd 2837	. . . . . . . . . . 11 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝑘 ∈ (0..^(♯‘𝐹))) → (𝑘 + 1) ∈ (0..^(♯‘𝑃)))
47		revfv 14713	. . . . . . . . . . 11 ⊢ ((𝑃 ∈ Word (Vtx‘𝐺) ∧ (𝑘 + 1) ∈ (0..^(♯‘𝑃))) → ((reverse‘𝑃)‘(𝑘 + 1)) = (𝑃‘(((♯‘𝑃) − 1) − (𝑘 + 1))))
48	6, 46, 47	syl2an2r 684	. . . . . . . . . 10 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝑘 ∈ (0..^(♯‘𝐹))) → ((reverse‘𝑃)‘(𝑘 + 1)) = (𝑃‘(((♯‘𝑃) − 1) − (𝑘 + 1))))
49		elfzoelz 13632	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ (0..^(♯‘𝐹)) → 𝑘 ∈ ℤ)
50	49	zcnd 12667	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (0..^(♯‘𝐹)) → 𝑘 ∈ ℂ)
51	50	adantl 483	. . . . . . . . . . . . . 14 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝑘 ∈ (0..^(♯‘𝐹))) → 𝑘 ∈ ℂ)
52		1cnd 11209	. . . . . . . . . . . . . 14 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝑘 ∈ (0..^(♯‘𝐹))) → 1 ∈ ℂ)
53	51, 52	addcomd 11416	. . . . . . . . . . . . 13 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝑘 ∈ (0..^(♯‘𝐹))) → (𝑘 + 1) = (1 + 𝑘))
54	53	oveq2d 7425	. . . . . . . . . . . 12 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝑘 ∈ (0..^(♯‘𝐹))) → (((♯‘𝑃) − 1) − (𝑘 + 1)) = (((♯‘𝑃) − 1) − (1 + 𝑘)))
55	6, 34	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝐹(Walks‘𝐺)𝑃 → (♯‘𝑃) ∈ ℂ)
56	55	adantr 482	. . . . . . . . . . . . . 14 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝑘 ∈ (0..^(♯‘𝐹))) → (♯‘𝑃) ∈ ℂ)
57	56, 52	subcld 11571	. . . . . . . . . . . . 13 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝑘 ∈ (0..^(♯‘𝐹))) → ((♯‘𝑃) − 1) ∈ ℂ)
58	57, 52, 51	subsub4d 11602	. . . . . . . . . . . 12 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝑘 ∈ (0..^(♯‘𝐹))) → ((((♯‘𝑃) − 1) − 1) − 𝑘) = (((♯‘𝑃) − 1) − (1 + 𝑘)))
59	36	oveq1d 7424	. . . . . . . . . . . . 13 ⊢ (𝐹(Walks‘𝐺)𝑃 → ((((♯‘𝑃) − 1) − 1) − 𝑘) = (((♯‘𝑃) − 2) − 𝑘))
60	59	adantr 482	. . . . . . . . . . . 12 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝑘 ∈ (0..^(♯‘𝐹))) → ((((♯‘𝑃) − 1) − 1) − 𝑘) = (((♯‘𝑃) − 2) − 𝑘))
61	54, 58, 60	3eqtr2d 2779	. . . . . . . . . . 11 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝑘 ∈ (0..^(♯‘𝐹))) → (((♯‘𝑃) − 1) − (𝑘 + 1)) = (((♯‘𝑃) − 2) − 𝑘))
62	61	fveq2d 6896	. . . . . . . . . 10 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝑘 ∈ (0..^(♯‘𝐹))) → (𝑃‘(((♯‘𝑃) − 1) − (𝑘 + 1))) = (𝑃‘(((♯‘𝑃) − 2) − 𝑘)))
63	48, 62	eqtrd 2773	. . . . . . . . 9 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝑘 ∈ (0..^(♯‘𝐹))) → ((reverse‘𝑃)‘(𝑘 + 1)) = (𝑃‘(((♯‘𝑃) − 2) − 𝑘)))
64	63	sneqd 4641	. . . . . . . 8 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝑘 ∈ (0..^(♯‘𝐹))) → {((reverse‘𝑃)‘(𝑘 + 1))} = {(𝑃‘(((♯‘𝑃) − 2) − 𝑘))})
65	64	adantr 482	. . . . . . 7 ⊢ (((𝐹(Walks‘𝐺)𝑃 ∧ 𝑘 ∈ (0..^(♯‘𝐹))) ∧ ((reverse‘𝑃)‘𝑘) = ((reverse‘𝑃)‘(𝑘 + 1))) → {((reverse‘𝑃)‘(𝑘 + 1))} = {(𝑃‘(((♯‘𝑃) − 2) − 𝑘))})
66		sneq 4639	. . . . . . . 8 ⊢ (((reverse‘𝑃)‘𝑘) = ((reverse‘𝑃)‘(𝑘 + 1)) → {((reverse‘𝑃)‘𝑘)} = {((reverse‘𝑃)‘(𝑘 + 1))})
67	66	adantl 483	. . . . . . 7 ⊢ (((𝐹(Walks‘𝐺)𝑃 ∧ 𝑘 ∈ (0..^(♯‘𝐹))) ∧ ((reverse‘𝑃)‘𝑘) = ((reverse‘𝑃)‘(𝑘 + 1))) → {((reverse‘𝑃)‘𝑘)} = {((reverse‘𝑃)‘(𝑘 + 1))})
68		eqcom 2740	. . . . . . . . . 10 ⊢ (((reverse‘𝑃)‘𝑘) = ((reverse‘𝑃)‘(𝑘 + 1)) ↔ ((reverse‘𝑃)‘(𝑘 + 1)) = ((reverse‘𝑃)‘𝑘))
69		fzossfzop1 13710	. . . . . . . . . . . . . . . 16 ⊢ ((♯‘𝐹) ∈ ℕ₀ → (0..^(♯‘𝐹)) ⊆ (0..^((♯‘𝐹) + 1)))
70	18, 69	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝐹(Walks‘𝐺)𝑃 → (0..^(♯‘𝐹)) ⊆ (0..^((♯‘𝐹) + 1)))
71	70, 14	sseqtrrd 4024	. . . . . . . . . . . . . 14 ⊢ (𝐹(Walks‘𝐺)𝑃 → (0..^(♯‘𝐹)) ⊆ (0..^(♯‘𝑃)))
72	71	sselda 3983	. . . . . . . . . . . . 13 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝑘 ∈ (0..^(♯‘𝐹))) → 𝑘 ∈ (0..^(♯‘𝑃)))
73		revfv 14713	. . . . . . . . . . . . 13 ⊢ ((𝑃 ∈ Word (Vtx‘𝐺) ∧ 𝑘 ∈ (0..^(♯‘𝑃))) → ((reverse‘𝑃)‘𝑘) = (𝑃‘(((♯‘𝑃) − 1) − 𝑘)))
74	6, 72, 73	syl2an2r 684	. . . . . . . . . . . 12 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝑘 ∈ (0..^(♯‘𝐹))) → ((reverse‘𝑃)‘𝑘) = (𝑃‘(((♯‘𝑃) − 1) − 𝑘)))
75	57, 51, 52	sub32d 11603	. . . . . . . . . . . . . . 15 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝑘 ∈ (0..^(♯‘𝐹))) → ((((♯‘𝑃) − 1) − 𝑘) − 1) = ((((♯‘𝑃) − 1) − 1) − 𝑘))
76	75	oveq1d 7424	. . . . . . . . . . . . . 14 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝑘 ∈ (0..^(♯‘𝐹))) → (((((♯‘𝑃) − 1) − 𝑘) − 1) + 1) = (((((♯‘𝑃) − 1) − 1) − 𝑘) + 1))
77	57, 51	subcld 11571	. . . . . . . . . . . . . . 15 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝑘 ∈ (0..^(♯‘𝐹))) → (((♯‘𝑃) − 1) − 𝑘) ∈ ℂ)
78	77, 52	npcand 11575	. . . . . . . . . . . . . 14 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝑘 ∈ (0..^(♯‘𝐹))) → (((((♯‘𝑃) − 1) − 𝑘) − 1) + 1) = (((♯‘𝑃) − 1) − 𝑘))
79	59	oveq1d 7424	. . . . . . . . . . . . . . 15 ⊢ (𝐹(Walks‘𝐺)𝑃 → (((((♯‘𝑃) − 1) − 1) − 𝑘) + 1) = ((((♯‘𝑃) − 2) − 𝑘) + 1))
80	79	adantr 482	. . . . . . . . . . . . . 14 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝑘 ∈ (0..^(♯‘𝐹))) → (((((♯‘𝑃) − 1) − 1) − 𝑘) + 1) = ((((♯‘𝑃) − 2) − 𝑘) + 1))
81	76, 78, 80	3eqtr3d 2781	. . . . . . . . . . . . 13 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝑘 ∈ (0..^(♯‘𝐹))) → (((♯‘𝑃) − 1) − 𝑘) = ((((♯‘𝑃) − 2) − 𝑘) + 1))
82	81	fveq2d 6896	. . . . . . . . . . . 12 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝑘 ∈ (0..^(♯‘𝐹))) → (𝑃‘(((♯‘𝑃) − 1) − 𝑘)) = (𝑃‘((((♯‘𝑃) − 2) − 𝑘) + 1)))
83	74, 82	eqtrd 2773	. . . . . . . . . . 11 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝑘 ∈ (0..^(♯‘𝐹))) → ((reverse‘𝑃)‘𝑘) = (𝑃‘((((♯‘𝑃) − 2) − 𝑘) + 1)))
84	63, 83	eqeq12d 2749	. . . . . . . . . 10 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝑘 ∈ (0..^(♯‘𝐹))) → (((reverse‘𝑃)‘(𝑘 + 1)) = ((reverse‘𝑃)‘𝑘) ↔ (𝑃‘(((♯‘𝑃) − 2) − 𝑘)) = (𝑃‘((((♯‘𝑃) − 2) − 𝑘) + 1))))
85	68, 84	bitrid 283	. . . . . . . . 9 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝑘 ∈ (0..^(♯‘𝐹))) → (((reverse‘𝑃)‘𝑘) = ((reverse‘𝑃)‘(𝑘 + 1)) ↔ (𝑃‘(((♯‘𝑃) − 2) − 𝑘)) = (𝑃‘((((♯‘𝑃) − 2) − 𝑘) + 1))))
86		wkslem1 28864	. . . . . . . . . . . 12 ⊢ (𝑥 = (((♯‘𝑃) − 2) − 𝑘) → (if-((𝑃‘𝑥) = (𝑃‘(𝑥 + 1)), ((iEdg‘𝐺)‘(𝐹‘𝑥)) = {(𝑃‘𝑥)}, {(𝑃‘𝑥), (𝑃‘(𝑥 + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹‘𝑥))) ↔ if-((𝑃‘(((♯‘𝑃) − 2) − 𝑘)) = (𝑃‘((((♯‘𝑃) − 2) − 𝑘) + 1)), ((iEdg‘𝐺)‘(𝐹‘(((♯‘𝑃) − 2) − 𝑘))) = {(𝑃‘(((♯‘𝑃) − 2) − 𝑘))}, {(𝑃‘(((♯‘𝑃) − 2) − 𝑘)), (𝑃‘((((♯‘𝑃) − 2) − 𝑘) + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹‘(((♯‘𝑃) − 2) − 𝑘))))))
87	5, 1	wlkprop 28868	. . . . . . . . . . . . . 14 ⊢ (𝐹(Walks‘𝐺)𝑃 → (𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ ∀𝑥 ∈ (0..^(♯‘𝐹))if-((𝑃‘𝑥) = (𝑃‘(𝑥 + 1)), ((iEdg‘𝐺)‘(𝐹‘𝑥)) = {(𝑃‘𝑥)}, {(𝑃‘𝑥), (𝑃‘(𝑥 + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹‘𝑥)))))
88	87	simp3d 1145	. . . . . . . . . . . . 13 ⊢ (𝐹(Walks‘𝐺)𝑃 → ∀𝑥 ∈ (0..^(♯‘𝐹))if-((𝑃‘𝑥) = (𝑃‘(𝑥 + 1)), ((iEdg‘𝐺)‘(𝐹‘𝑥)) = {(𝑃‘𝑥)}, {(𝑃‘𝑥), (𝑃‘(𝑥 + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹‘𝑥))))
89	88	adantr 482	. . . . . . . . . . . 12 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝑘 ∈ (0..^(♯‘𝐹))) → ∀𝑥 ∈ (0..^(♯‘𝐹))if-((𝑃‘𝑥) = (𝑃‘(𝑥 + 1)), ((iEdg‘𝐺)‘(𝐹‘𝑥)) = {(𝑃‘𝑥)}, {(𝑃‘𝑥), (𝑃‘(𝑥 + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹‘𝑥))))
90	18	nn0cnd 12534	. . . . . . . . . . . . . . . 16 ⊢ (𝐹(Walks‘𝐺)𝑃 → (♯‘𝐹) ∈ ℂ)
91	90	adantr 482	. . . . . . . . . . . . . . 15 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝑘 ∈ (0..^(♯‘𝐹))) → (♯‘𝐹) ∈ ℂ)
92	91, 51, 52	sub32d 11603	. . . . . . . . . . . . . 14 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝑘 ∈ (0..^(♯‘𝐹))) → (((♯‘𝐹) − 𝑘) − 1) = (((♯‘𝐹) − 1) − 𝑘))
93	37	adantr 482	. . . . . . . . . . . . . . 15 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝑘 ∈ (0..^(♯‘𝐹))) → ((♯‘𝐹) − 1) = ((♯‘𝑃) − 2))
94	93	oveq1d 7424	. . . . . . . . . . . . . 14 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝑘 ∈ (0..^(♯‘𝐹))) → (((♯‘𝐹) − 1) − 𝑘) = (((♯‘𝑃) − 2) − 𝑘))
95	92, 94	eqtrd 2773	. . . . . . . . . . . . 13 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝑘 ∈ (0..^(♯‘𝐹))) → (((♯‘𝐹) − 𝑘) − 1) = (((♯‘𝑃) − 2) − 𝑘))
96		ubmelm1fzo 13728	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (0..^(♯‘𝐹)) → (((♯‘𝐹) − 𝑘) − 1) ∈ (0..^(♯‘𝐹)))
97	96	adantl 483	. . . . . . . . . . . . 13 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝑘 ∈ (0..^(♯‘𝐹))) → (((♯‘𝐹) − 𝑘) − 1) ∈ (0..^(♯‘𝐹)))
98	95, 97	eqeltrrd 2835	. . . . . . . . . . . 12 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝑘 ∈ (0..^(♯‘𝐹))) → (((♯‘𝑃) − 2) − 𝑘) ∈ (0..^(♯‘𝐹)))
99	86, 89, 98	rspcdva 3614	. . . . . . . . . . 11 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝑘 ∈ (0..^(♯‘𝐹))) → if-((𝑃‘(((♯‘𝑃) − 2) − 𝑘)) = (𝑃‘((((♯‘𝑃) − 2) − 𝑘) + 1)), ((iEdg‘𝐺)‘(𝐹‘(((♯‘𝑃) − 2) − 𝑘))) = {(𝑃‘(((♯‘𝑃) − 2) − 𝑘))}, {(𝑃‘(((♯‘𝑃) − 2) − 𝑘)), (𝑃‘((((♯‘𝑃) − 2) − 𝑘) + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹‘(((♯‘𝑃) − 2) − 𝑘)))))
100		dfifp2 1064	. . . . . . . . . . 11 ⊢ (if-((𝑃‘(((♯‘𝑃) − 2) − 𝑘)) = (𝑃‘((((♯‘𝑃) − 2) − 𝑘) + 1)), ((iEdg‘𝐺)‘(𝐹‘(((♯‘𝑃) − 2) − 𝑘))) = {(𝑃‘(((♯‘𝑃) − 2) − 𝑘))}, {(𝑃‘(((♯‘𝑃) − 2) − 𝑘)), (𝑃‘((((♯‘𝑃) − 2) − 𝑘) + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹‘(((♯‘𝑃) − 2) − 𝑘)))) ↔ (((𝑃‘(((♯‘𝑃) − 2) − 𝑘)) = (𝑃‘((((♯‘𝑃) − 2) − 𝑘) + 1)) → ((iEdg‘𝐺)‘(𝐹‘(((♯‘𝑃) − 2) − 𝑘))) = {(𝑃‘(((♯‘𝑃) − 2) − 𝑘))}) ∧ (¬ (𝑃‘(((♯‘𝑃) − 2) − 𝑘)) = (𝑃‘((((♯‘𝑃) − 2) − 𝑘) + 1)) → {(𝑃‘(((♯‘𝑃) − 2) − 𝑘)), (𝑃‘((((♯‘𝑃) − 2) − 𝑘) + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹‘(((♯‘𝑃) − 2) − 𝑘))))))
101	99, 100	sylib 217	. . . . . . . . . 10 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝑘 ∈ (0..^(♯‘𝐹))) → (((𝑃‘(((♯‘𝑃) − 2) − 𝑘)) = (𝑃‘((((♯‘𝑃) − 2) − 𝑘) + 1)) → ((iEdg‘𝐺)‘(𝐹‘(((♯‘𝑃) − 2) − 𝑘))) = {(𝑃‘(((♯‘𝑃) − 2) − 𝑘))}) ∧ (¬ (𝑃‘(((♯‘𝑃) − 2) − 𝑘)) = (𝑃‘((((♯‘𝑃) − 2) − 𝑘) + 1)) → {(𝑃‘(((♯‘𝑃) − 2) − 𝑘)), (𝑃‘((((♯‘𝑃) − 2) − 𝑘) + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹‘(((♯‘𝑃) − 2) − 𝑘))))))
102	101	simpld 496	. . . . . . . . 9 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝑘 ∈ (0..^(♯‘𝐹))) → ((𝑃‘(((♯‘𝑃) − 2) − 𝑘)) = (𝑃‘((((♯‘𝑃) − 2) − 𝑘) + 1)) → ((iEdg‘𝐺)‘(𝐹‘(((♯‘𝑃) − 2) − 𝑘))) = {(𝑃‘(((♯‘𝑃) − 2) − 𝑘))}))
103	85, 102	sylbid 239	. . . . . . . 8 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝑘 ∈ (0..^(♯‘𝐹))) → (((reverse‘𝑃)‘𝑘) = ((reverse‘𝑃)‘(𝑘 + 1)) → ((iEdg‘𝐺)‘(𝐹‘(((♯‘𝑃) − 2) − 𝑘))) = {(𝑃‘(((♯‘𝑃) − 2) − 𝑘))}))
104	103	imp 408	. . . . . . 7 ⊢ (((𝐹(Walks‘𝐺)𝑃 ∧ 𝑘 ∈ (0..^(♯‘𝐹))) ∧ ((reverse‘𝑃)‘𝑘) = ((reverse‘𝑃)‘(𝑘 + 1))) → ((iEdg‘𝐺)‘(𝐹‘(((♯‘𝑃) − 2) − 𝑘))) = {(𝑃‘(((♯‘𝑃) − 2) − 𝑘))})
105	65, 67, 104	3eqtr4rd 2784	. . . . . 6 ⊢ (((𝐹(Walks‘𝐺)𝑃 ∧ 𝑘 ∈ (0..^(♯‘𝐹))) ∧ ((reverse‘𝑃)‘𝑘) = ((reverse‘𝑃)‘(𝑘 + 1))) → ((iEdg‘𝐺)‘(𝐹‘(((♯‘𝑃) − 2) − 𝑘))) = {((reverse‘𝑃)‘𝑘)})
106	42, 105	eqtrd 2773	. . . . 5 ⊢ (((𝐹(Walks‘𝐺)𝑃 ∧ 𝑘 ∈ (0..^(♯‘𝐹))) ∧ ((reverse‘𝑃)‘𝑘) = ((reverse‘𝑃)‘(𝑘 + 1))) → ((iEdg‘𝐺)‘((reverse‘𝐹)‘𝑘)) = {((reverse‘𝑃)‘𝑘)})
107	85	notbid 318	. . . . . . . 8 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝑘 ∈ (0..^(♯‘𝐹))) → (¬ ((reverse‘𝑃)‘𝑘) = ((reverse‘𝑃)‘(𝑘 + 1)) ↔ ¬ (𝑃‘(((♯‘𝑃) − 2) − 𝑘)) = (𝑃‘((((♯‘𝑃) − 2) − 𝑘) + 1))))
108	101	simprd 497	. . . . . . . 8 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝑘 ∈ (0..^(♯‘𝐹))) → (¬ (𝑃‘(((♯‘𝑃) − 2) − 𝑘)) = (𝑃‘((((♯‘𝑃) − 2) − 𝑘) + 1)) → {(𝑃‘(((♯‘𝑃) − 2) − 𝑘)), (𝑃‘((((♯‘𝑃) − 2) − 𝑘) + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹‘(((♯‘𝑃) − 2) − 𝑘)))))
109	107, 108	sylbid 239	. . . . . . 7 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝑘 ∈ (0..^(♯‘𝐹))) → (¬ ((reverse‘𝑃)‘𝑘) = ((reverse‘𝑃)‘(𝑘 + 1)) → {(𝑃‘(((♯‘𝑃) − 2) − 𝑘)), (𝑃‘((((♯‘𝑃) − 2) − 𝑘) + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹‘(((♯‘𝑃) − 2) − 𝑘)))))
110	109	imp 408	. . . . . 6 ⊢ (((𝐹(Walks‘𝐺)𝑃 ∧ 𝑘 ∈ (0..^(♯‘𝐹))) ∧ ¬ ((reverse‘𝑃)‘𝑘) = ((reverse‘𝑃)‘(𝑘 + 1))) → {(𝑃‘(((♯‘𝑃) − 2) − 𝑘)), (𝑃‘((((♯‘𝑃) − 2) − 𝑘) + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹‘(((♯‘𝑃) − 2) − 𝑘))))
111		prcom 4737	. . . . . . . 8 ⊢ {((reverse‘𝑃)‘(𝑘 + 1)), ((reverse‘𝑃)‘𝑘)} = {((reverse‘𝑃)‘𝑘), ((reverse‘𝑃)‘(𝑘 + 1))}
112	63, 83	preq12d 4746	. . . . . . . 8 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝑘 ∈ (0..^(♯‘𝐹))) → {((reverse‘𝑃)‘(𝑘 + 1)), ((reverse‘𝑃)‘𝑘)} = {(𝑃‘(((♯‘𝑃) − 2) − 𝑘)), (𝑃‘((((♯‘𝑃) − 2) − 𝑘) + 1))})
113	111, 112	eqtr3id 2787	. . . . . . 7 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝑘 ∈ (0..^(♯‘𝐹))) → {((reverse‘𝑃)‘𝑘), ((reverse‘𝑃)‘(𝑘 + 1))} = {(𝑃‘(((♯‘𝑃) − 2) − 𝑘)), (𝑃‘((((♯‘𝑃) − 2) − 𝑘) + 1))})
114	113	adantr 482	. . . . . 6 ⊢ (((𝐹(Walks‘𝐺)𝑃 ∧ 𝑘 ∈ (0..^(♯‘𝐹))) ∧ ¬ ((reverse‘𝑃)‘𝑘) = ((reverse‘𝑃)‘(𝑘 + 1))) → {((reverse‘𝑃)‘𝑘), ((reverse‘𝑃)‘(𝑘 + 1))} = {(𝑃‘(((♯‘𝑃) − 2) − 𝑘)), (𝑃‘((((♯‘𝑃) − 2) − 𝑘) + 1))})
115	41	adantr 482	. . . . . 6 ⊢ (((𝐹(Walks‘𝐺)𝑃 ∧ 𝑘 ∈ (0..^(♯‘𝐹))) ∧ ¬ ((reverse‘𝑃)‘𝑘) = ((reverse‘𝑃)‘(𝑘 + 1))) → ((iEdg‘𝐺)‘((reverse‘𝐹)‘𝑘)) = ((iEdg‘𝐺)‘(𝐹‘(((♯‘𝑃) − 2) − 𝑘))))
116	110, 114, 115	3sstr4d 4030	. . . . 5 ⊢ (((𝐹(Walks‘𝐺)𝑃 ∧ 𝑘 ∈ (0..^(♯‘𝐹))) ∧ ¬ ((reverse‘𝑃)‘𝑘) = ((reverse‘𝑃)‘(𝑘 + 1))) → {((reverse‘𝑃)‘𝑘), ((reverse‘𝑃)‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘((reverse‘𝐹)‘𝑘)))
117	106, 116	ifpimpda 1082	. . . 4 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝑘 ∈ (0..^(♯‘𝐹))) → if-(((reverse‘𝑃)‘𝑘) = ((reverse‘𝑃)‘(𝑘 + 1)), ((iEdg‘𝐺)‘((reverse‘𝐹)‘𝑘)) = {((reverse‘𝑃)‘𝑘)}, {((reverse‘𝑃)‘𝑘), ((reverse‘𝑃)‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘((reverse‘𝐹)‘𝑘))))
118	28, 117	syldan 592	. . 3 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝑘 ∈ (0..^(♯‘(reverse‘𝐹)))) → if-(((reverse‘𝑃)‘𝑘) = ((reverse‘𝑃)‘(𝑘 + 1)), ((iEdg‘𝐺)‘((reverse‘𝐹)‘𝑘)) = {((reverse‘𝑃)‘𝑘)}, {((reverse‘𝑃)‘𝑘), ((reverse‘𝑃)‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘((reverse‘𝐹)‘𝑘))))
119	118	ralrimiva 3147	. 2 ⊢ (𝐹(Walks‘𝐺)𝑃 → ∀𝑘 ∈ (0..^(♯‘(reverse‘𝐹)))if-(((reverse‘𝑃)‘𝑘) = ((reverse‘𝑃)‘(𝑘 + 1)), ((iEdg‘𝐺)‘((reverse‘𝐹)‘𝑘)) = {((reverse‘𝑃)‘𝑘)}, {((reverse‘𝑃)‘𝑘), ((reverse‘𝑃)‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘((reverse‘𝐹)‘𝑘))))
120		wlkv 28869	. . . 4 ⊢ (𝐹(Walks‘𝐺)𝑃 → (𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V))
121	120	simp1d 1143	. . 3 ⊢ (𝐹(Walks‘𝐺)𝑃 → 𝐺 ∈ V)
122	5, 1	iswlkg 28870	. . 3 ⊢ (𝐺 ∈ V → ((reverse‘𝐹)(Walks‘𝐺)(reverse‘𝑃) ↔ ((reverse‘𝐹) ∈ Word dom (iEdg‘𝐺) ∧ (reverse‘𝑃):(0...(♯‘(reverse‘𝐹)))⟶(Vtx‘𝐺) ∧ ∀𝑘 ∈ (0..^(♯‘(reverse‘𝐹)))if-(((reverse‘𝑃)‘𝑘) = ((reverse‘𝑃)‘(𝑘 + 1)), ((iEdg‘𝐺)‘((reverse‘𝐹)‘𝑘)) = {((reverse‘𝑃)‘𝑘)}, {((reverse‘𝑃)‘𝑘), ((reverse‘𝑃)‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘((reverse‘𝐹)‘𝑘))))))
123	121, 122	syl 17	. 2 ⊢ (𝐹(Walks‘𝐺)𝑃 → ((reverse‘𝐹)(Walks‘𝐺)(reverse‘𝑃) ↔ ((reverse‘𝐹) ∈ Word dom (iEdg‘𝐺) ∧ (reverse‘𝑃):(0...(♯‘(reverse‘𝐹)))⟶(Vtx‘𝐺) ∧ ∀𝑘 ∈ (0..^(♯‘(reverse‘𝐹)))if-(((reverse‘𝑃)‘𝑘) = ((reverse‘𝑃)‘(𝑘 + 1)), ((iEdg‘𝐺)‘((reverse‘𝐹)‘𝑘)) = {((reverse‘𝑃)‘𝑘)}, {((reverse‘𝑃)‘𝑘), ((reverse‘𝑃)‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘((reverse‘𝐹)‘𝑘))))))
124	4, 25, 119, 123	mpbir3and 1343	1 ⊢ (𝐹(Walks‘𝐺)𝑃 → (reverse‘𝐹)(Walks‘𝐺)(reverse‘𝑃))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 397 if-wif 1062 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ∀wral 3062 Vcvv 3475 ⊆ wss 3949 {csn 4629 {cpr 4631 class class class wbr 5149 dom cdm 5677 ⟶wf 6540 ‘cfv 6544 (class class class)co 7409 ℂcc 11108 0cc0 11110 1c1 11111 + caddc 11113 − cmin 11444 2c2 12267 ℕ₀cn0 12472 ℤcz 12558 ...cfz 13484 ..^cfzo 13627 ♯chash 14290 Word cword 14464 reversecreverse 14708 Vtxcvtx 28256 iEdgciedg 28257 Walkscwlks 28853

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-ifp 1063 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-1st 7975 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-er 8703 df-map 8822 df-pm 8823 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-card 9934 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-nn 12213 df-2 12275 df-n0 12473 df-z 12559 df-uz 12823 df-fz 13485 df-fzo 13628 df-hash 14291 df-word 14465 df-reverse 14709 df-wlks 28856

This theorem is referenced by: revwlkb 34116 swrdwlk 34117

W3C validator