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Theorem wlkp1lem8 26577

Description: Lemma for wlkp1 26578. (Contributed by AV, 6-Mar-2021.)

**Hypotheses**
Ref	Expression
wlkp1.v	⊢ 𝑉 = (Vtx‘𝐺)
wlkp1.i	⊢ 𝐼 = (iEdg‘𝐺)
wlkp1.f	⊢ (𝜑 → Fun 𝐼)
wlkp1.a	⊢ (𝜑 → 𝐼 ∈ Fin)
wlkp1.b	⊢ (𝜑 → 𝐵 ∈ V)
wlkp1.c	⊢ (𝜑 → 𝐶 ∈ 𝑉)
wlkp1.d	⊢ (𝜑 → ¬ 𝐵 ∈ dom 𝐼)
wlkp1.w	⊢ (𝜑 → 𝐹(Walks‘𝐺)𝑃)
wlkp1.n	⊢ 𝑁 = (#‘𝐹)
wlkp1.e	⊢ (𝜑 → 𝐸 ∈ (Edg‘𝐺))
wlkp1.x	⊢ (𝜑 → {(𝑃‘𝑁), 𝐶} ⊆ 𝐸)
wlkp1.u	⊢ (𝜑 → (iEdg‘𝑆) = (𝐼 ∪ {⟨𝐵, 𝐸⟩}))
wlkp1.h	⊢ 𝐻 = (𝐹 ∪ {⟨𝑁, 𝐵⟩})
wlkp1.q	⊢ 𝑄 = (𝑃 ∪ {⟨(𝑁 + 1), 𝐶⟩})
wlkp1.s	⊢ (𝜑 → (Vtx‘𝑆) = 𝑉)
wlkp1.l	⊢ ((𝜑 ∧ 𝐶 = (𝑃‘𝑁)) → 𝐸 = {𝐶})

**Assertion**
Ref	Expression
wlkp1lem8	⊢ (𝜑 → ∀𝑘 ∈ (0..^(#‘𝐻))if-((𝑄‘𝑘) = (𝑄‘(𝑘 + 1)), ((iEdg‘𝑆)‘(𝐻‘𝑘)) = {(𝑄‘𝑘)}, {(𝑄‘𝑘), (𝑄‘(𝑘 + 1))} ⊆ ((iEdg‘𝑆)‘(𝐻‘𝑘))))

Distinct variable groups: 𝜑,𝑘 𝑘,𝑁 𝑃,𝑘 𝑄,𝑘 𝑘,𝐹 𝑘,𝐺 𝑘,𝐻 𝑆,𝑘 Allowed substitution hints: 𝐵(𝑘) 𝐶(𝑘) 𝐸(𝑘) 𝐼(𝑘) 𝑉(𝑘)

Proof of Theorem wlkp1lem8 Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		wlkp1.v	. . . 4 ⊢ 𝑉 = (Vtx‘𝐺)
2		wlkp1.i	. . . 4 ⊢ 𝐼 = (iEdg‘𝐺)
3		wlkp1.f	. . . 4 ⊢ (𝜑 → Fun 𝐼)
4		wlkp1.a	. . . 4 ⊢ (𝜑 → 𝐼 ∈ Fin)
5		wlkp1.b	. . . 4 ⊢ (𝜑 → 𝐵 ∈ V)
6		wlkp1.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑉)
7		wlkp1.d	. . . 4 ⊢ (𝜑 → ¬ 𝐵 ∈ dom 𝐼)
8		wlkp1.w	. . . 4 ⊢ (𝜑 → 𝐹(Walks‘𝐺)𝑃)
9		wlkp1.n	. . . 4 ⊢ 𝑁 = (#‘𝐹)
10		wlkp1.e	. . . 4 ⊢ (𝜑 → 𝐸 ∈ (Edg‘𝐺))
11		wlkp1.x	. . . 4 ⊢ (𝜑 → {(𝑃‘𝑁), 𝐶} ⊆ 𝐸)
12		wlkp1.u	. . . 4 ⊢ (𝜑 → (iEdg‘𝑆) = (𝐼 ∪ {⟨𝐵, 𝐸⟩}))
13		wlkp1.h	. . . 4 ⊢ 𝐻 = (𝐹 ∪ {⟨𝑁, 𝐵⟩})
14		wlkp1.q	. . . 4 ⊢ 𝑄 = (𝑃 ∪ {⟨(𝑁 + 1), 𝐶⟩})
15		wlkp1.s	. . . 4 ⊢ (𝜑 → (Vtx‘𝑆) = 𝑉)
16	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15	wlkp1lem6 26575	. . 3 ⊢ (𝜑 → ∀𝑘 ∈ (0..^𝑁)((𝑄‘𝑘) = (𝑃‘𝑘) ∧ (𝑄‘(𝑘 + 1)) = (𝑃‘(𝑘 + 1)) ∧ ((iEdg‘𝑆)‘(𝐻‘𝑘)) = (𝐼‘(𝐹‘𝑘))))
17	10	elfvexd 6222	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ V)
18	1, 2	iswlkg 26509	. . . . . 6 ⊢ (𝐺 ∈ V → (𝐹(Walks‘𝐺)𝑃 ↔ (𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘))))))
19	17, 18	syl 17	. . . . 5 ⊢ (𝜑 → (𝐹(Walks‘𝐺)𝑃 ↔ (𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘))))))
20	9	eqcomi 2631	. . . . . . . . 9 ⊢ (#‘𝐹) = 𝑁
21	20	oveq2i 6661	. . . . . . . 8 ⊢ (0..^(#‘𝐹)) = (0..^𝑁)
22	21	raleqi 3142	. . . . . . 7 ⊢ (∀𝑘 ∈ (0..^(#‘𝐹))if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘))) ↔ ∀𝑘 ∈ (0..^𝑁)if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘))))
23	22	biimpi 206	. . . . . 6 ⊢ (∀𝑘 ∈ (0..^(#‘𝐹))if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘))) → ∀𝑘 ∈ (0..^𝑁)if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘))))
24	23	3ad2ant3 1084	. . . . 5 ⊢ ((𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘)))) → ∀𝑘 ∈ (0..^𝑁)if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘))))
25	19, 24	syl6bi 243	. . . 4 ⊢ (𝜑 → (𝐹(Walks‘𝐺)𝑃 → ∀𝑘 ∈ (0..^𝑁)if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘)))))
26	8, 25	mpd 15	. . 3 ⊢ (𝜑 → ∀𝑘 ∈ (0..^𝑁)if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘))))
27		eqeq12 2635	. . . . . . 7 ⊢ (((𝑄‘𝑘) = (𝑃‘𝑘) ∧ (𝑄‘(𝑘 + 1)) = (𝑃‘(𝑘 + 1))) → ((𝑄‘𝑘) = (𝑄‘(𝑘 + 1)) ↔ (𝑃‘𝑘) = (𝑃‘(𝑘 + 1))))
28	27	3adant3 1081	. . . . . 6 ⊢ (((𝑄‘𝑘) = (𝑃‘𝑘) ∧ (𝑄‘(𝑘 + 1)) = (𝑃‘(𝑘 + 1)) ∧ ((iEdg‘𝑆)‘(𝐻‘𝑘)) = (𝐼‘(𝐹‘𝑘))) → ((𝑄‘𝑘) = (𝑄‘(𝑘 + 1)) ↔ (𝑃‘𝑘) = (𝑃‘(𝑘 + 1))))
29		simp3 1063	. . . . . . 7 ⊢ (((𝑄‘𝑘) = (𝑃‘𝑘) ∧ (𝑄‘(𝑘 + 1)) = (𝑃‘(𝑘 + 1)) ∧ ((iEdg‘𝑆)‘(𝐻‘𝑘)) = (𝐼‘(𝐹‘𝑘))) → ((iEdg‘𝑆)‘(𝐻‘𝑘)) = (𝐼‘(𝐹‘𝑘)))
30		simp1 1061	. . . . . . . 8 ⊢ (((𝑄‘𝑘) = (𝑃‘𝑘) ∧ (𝑄‘(𝑘 + 1)) = (𝑃‘(𝑘 + 1)) ∧ ((iEdg‘𝑆)‘(𝐻‘𝑘)) = (𝐼‘(𝐹‘𝑘))) → (𝑄‘𝑘) = (𝑃‘𝑘))
31	30	sneqd 4189	. . . . . . 7 ⊢ (((𝑄‘𝑘) = (𝑃‘𝑘) ∧ (𝑄‘(𝑘 + 1)) = (𝑃‘(𝑘 + 1)) ∧ ((iEdg‘𝑆)‘(𝐻‘𝑘)) = (𝐼‘(𝐹‘𝑘))) → {(𝑄‘𝑘)} = {(𝑃‘𝑘)})
32	29, 31	eqeq12d 2637	. . . . . 6 ⊢ (((𝑄‘𝑘) = (𝑃‘𝑘) ∧ (𝑄‘(𝑘 + 1)) = (𝑃‘(𝑘 + 1)) ∧ ((iEdg‘𝑆)‘(𝐻‘𝑘)) = (𝐼‘(𝐹‘𝑘))) → (((iEdg‘𝑆)‘(𝐻‘𝑘)) = {(𝑄‘𝑘)} ↔ (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}))
33		preq12 4270	. . . . . . . 8 ⊢ (((𝑄‘𝑘) = (𝑃‘𝑘) ∧ (𝑄‘(𝑘 + 1)) = (𝑃‘(𝑘 + 1))) → {(𝑄‘𝑘), (𝑄‘(𝑘 + 1))} = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))})
34	33	3adant3 1081	. . . . . . 7 ⊢ (((𝑄‘𝑘) = (𝑃‘𝑘) ∧ (𝑄‘(𝑘 + 1)) = (𝑃‘(𝑘 + 1)) ∧ ((iEdg‘𝑆)‘(𝐻‘𝑘)) = (𝐼‘(𝐹‘𝑘))) → {(𝑄‘𝑘), (𝑄‘(𝑘 + 1))} = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))})
35	34, 29	sseq12d 3634	. . . . . 6 ⊢ (((𝑄‘𝑘) = (𝑃‘𝑘) ∧ (𝑄‘(𝑘 + 1)) = (𝑃‘(𝑘 + 1)) ∧ ((iEdg‘𝑆)‘(𝐻‘𝑘)) = (𝐼‘(𝐹‘𝑘))) → ({(𝑄‘𝑘), (𝑄‘(𝑘 + 1))} ⊆ ((iEdg‘𝑆)‘(𝐻‘𝑘)) ↔ {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘))))
36	28, 32, 35	ifpbi123d 1027	. . . . 5 ⊢ (((𝑄‘𝑘) = (𝑃‘𝑘) ∧ (𝑄‘(𝑘 + 1)) = (𝑃‘(𝑘 + 1)) ∧ ((iEdg‘𝑆)‘(𝐻‘𝑘)) = (𝐼‘(𝐹‘𝑘))) → (if-((𝑄‘𝑘) = (𝑄‘(𝑘 + 1)), ((iEdg‘𝑆)‘(𝐻‘𝑘)) = {(𝑄‘𝑘)}, {(𝑄‘𝑘), (𝑄‘(𝑘 + 1))} ⊆ ((iEdg‘𝑆)‘(𝐻‘𝑘))) ↔ if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘)))))
37	36	biimprd 238	. . . 4 ⊢ (((𝑄‘𝑘) = (𝑃‘𝑘) ∧ (𝑄‘(𝑘 + 1)) = (𝑃‘(𝑘 + 1)) ∧ ((iEdg‘𝑆)‘(𝐻‘𝑘)) = (𝐼‘(𝐹‘𝑘))) → (if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘))) → if-((𝑄‘𝑘) = (𝑄‘(𝑘 + 1)), ((iEdg‘𝑆)‘(𝐻‘𝑘)) = {(𝑄‘𝑘)}, {(𝑄‘𝑘), (𝑄‘(𝑘 + 1))} ⊆ ((iEdg‘𝑆)‘(𝐻‘𝑘)))))
38	37	ral2imi 2947	. . 3 ⊢ (∀𝑘 ∈ (0..^𝑁)((𝑄‘𝑘) = (𝑃‘𝑘) ∧ (𝑄‘(𝑘 + 1)) = (𝑃‘(𝑘 + 1)) ∧ ((iEdg‘𝑆)‘(𝐻‘𝑘)) = (𝐼‘(𝐹‘𝑘))) → (∀𝑘 ∈ (0..^𝑁)if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘))) → ∀𝑘 ∈ (0..^𝑁)if-((𝑄‘𝑘) = (𝑄‘(𝑘 + 1)), ((iEdg‘𝑆)‘(𝐻‘𝑘)) = {(𝑄‘𝑘)}, {(𝑄‘𝑘), (𝑄‘(𝑘 + 1))} ⊆ ((iEdg‘𝑆)‘(𝐻‘𝑘)))))
39	16, 26, 38	sylc 65	. 2 ⊢ (𝜑 → ∀𝑘 ∈ (0..^𝑁)if-((𝑄‘𝑘) = (𝑄‘(𝑘 + 1)), ((iEdg‘𝑆)‘(𝐻‘𝑘)) = {(𝑄‘𝑘)}, {(𝑄‘𝑘), (𝑄‘(𝑘 + 1))} ⊆ ((iEdg‘𝑆)‘(𝐻‘𝑘))))
40	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13	wlkp1lem3 26572	. . . . 5 ⊢ (𝜑 → ((iEdg‘𝑆)‘(𝐻‘𝑁)) = ((𝐼 ∪ {⟨𝐵, 𝐸⟩})‘𝐵))
41	40	adantr 481	. . . 4 ⊢ ((𝜑 ∧ (𝑄‘𝑁) = (𝑄‘(𝑁 + 1))) → ((iEdg‘𝑆)‘(𝐻‘𝑁)) = ((𝐼 ∪ {⟨𝐵, 𝐸⟩})‘𝐵))
42	5, 10, 7	3jca 1242	. . . . . 6 ⊢ (𝜑 → (𝐵 ∈ V ∧ 𝐸 ∈ (Edg‘𝐺) ∧ ¬ 𝐵 ∈ dom 𝐼))
43	42	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ (𝑄‘𝑁) = (𝑄‘(𝑁 + 1))) → (𝐵 ∈ V ∧ 𝐸 ∈ (Edg‘𝐺) ∧ ¬ 𝐵 ∈ dom 𝐼))
44		fsnunfv 6453	. . . . 5 ⊢ ((𝐵 ∈ V ∧ 𝐸 ∈ (Edg‘𝐺) ∧ ¬ 𝐵 ∈ dom 𝐼) → ((𝐼 ∪ {⟨𝐵, 𝐸⟩})‘𝐵) = 𝐸)
45	43, 44	syl 17	. . . 4 ⊢ ((𝜑 ∧ (𝑄‘𝑁) = (𝑄‘(𝑁 + 1))) → ((𝐼 ∪ {⟨𝐵, 𝐸⟩})‘𝐵) = 𝐸)
46		fveq2 6191	. . . . . . . 8 ⊢ (𝑥 = 𝑁 → (𝑄‘𝑥) = (𝑄‘𝑁))
47		fveq2 6191	. . . . . . . 8 ⊢ (𝑥 = 𝑁 → (𝑃‘𝑥) = (𝑃‘𝑁))
48	46, 47	eqeq12d 2637	. . . . . . 7 ⊢ (𝑥 = 𝑁 → ((𝑄‘𝑥) = (𝑃‘𝑥) ↔ (𝑄‘𝑁) = (𝑃‘𝑁)))
49	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15	wlkp1lem5 26574	. . . . . . 7 ⊢ (𝜑 → ∀𝑥 ∈ (0...𝑁)(𝑄‘𝑥) = (𝑃‘𝑥))
50	2	wlkf 26510	. . . . . . . . . 10 ⊢ (𝐹(Walks‘𝐺)𝑃 → 𝐹 ∈ Word dom 𝐼)
51		lencl 13324	. . . . . . . . . . 11 ⊢ (𝐹 ∈ Word dom 𝐼 → (#‘𝐹) ∈ ℕ₀)
52	9	eleq1i 2692	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ₀ ↔ (#‘𝐹) ∈ ℕ₀)
53		elnn0uz 11725	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ₀ ↔ 𝑁 ∈ (ℤ_≥‘0))
54	52, 53	sylbb1 227	. . . . . . . . . . 11 ⊢ ((#‘𝐹) ∈ ℕ₀ → 𝑁 ∈ (ℤ_≥‘0))
55	51, 54	syl 17	. . . . . . . . . 10 ⊢ (𝐹 ∈ Word dom 𝐼 → 𝑁 ∈ (ℤ_≥‘0))
56	8, 50, 55	3syl 18	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘0))
57	56, 53	sylibr 224	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
58		nn0fz0 12437	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ₀ ↔ 𝑁 ∈ (0...𝑁))
59	57, 58	sylib 208	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ (0...𝑁))
60	48, 49, 59	rspcdva 3316	. . . . . 6 ⊢ (𝜑 → (𝑄‘𝑁) = (𝑃‘𝑁))
61	14	fveq1i 6192	. . . . . . . . . . 11 ⊢ (𝑄‘(𝑁 + 1)) = ((𝑃 ∪ {⟨(𝑁 + 1), 𝐶⟩})‘(𝑁 + 1))
62		ovex 6678	. . . . . . . . . . . 12 ⊢ (𝑁 + 1) ∈ V
63	1, 2, 3, 4, 5, 6, 7, 8, 9	wlkp1lem1 26570	. . . . . . . . . . . 12 ⊢ (𝜑 → ¬ (𝑁 + 1) ∈ dom 𝑃)
64		fsnunfv 6453	. . . . . . . . . . . 12 ⊢ (((𝑁 + 1) ∈ V ∧ 𝐶 ∈ 𝑉 ∧ ¬ (𝑁 + 1) ∈ dom 𝑃) → ((𝑃 ∪ {⟨(𝑁 + 1), 𝐶⟩})‘(𝑁 + 1)) = 𝐶)
65	62, 6, 63, 64	mp3an2i 1429	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑃 ∪ {⟨(𝑁 + 1), 𝐶⟩})‘(𝑁 + 1)) = 𝐶)
66	61, 65	syl5eq 2668	. . . . . . . . . 10 ⊢ (𝜑 → (𝑄‘(𝑁 + 1)) = 𝐶)
67	66	eqeq2d 2632	. . . . . . . . 9 ⊢ (𝜑 → ((𝑃‘𝑁) = (𝑄‘(𝑁 + 1)) ↔ (𝑃‘𝑁) = 𝐶))
68		eqcom 2629	. . . . . . . . 9 ⊢ ((𝑃‘𝑁) = 𝐶 ↔ 𝐶 = (𝑃‘𝑁))
69	67, 68	syl6bb 276	. . . . . . . 8 ⊢ (𝜑 → ((𝑃‘𝑁) = (𝑄‘(𝑁 + 1)) ↔ 𝐶 = (𝑃‘𝑁)))
70		wlkp1.l	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐶 = (𝑃‘𝑁)) → 𝐸 = {𝐶})
71		sneq 4187	. . . . . . . . . . 11 ⊢ (𝐶 = (𝑃‘𝑁) → {𝐶} = {(𝑃‘𝑁)})
72	71	adantl 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐶 = (𝑃‘𝑁)) → {𝐶} = {(𝑃‘𝑁)})
73	70, 72	eqtrd 2656	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐶 = (𝑃‘𝑁)) → 𝐸 = {(𝑃‘𝑁)})
74	73	ex 450	. . . . . . . 8 ⊢ (𝜑 → (𝐶 = (𝑃‘𝑁) → 𝐸 = {(𝑃‘𝑁)}))
75	69, 74	sylbid 230	. . . . . . 7 ⊢ (𝜑 → ((𝑃‘𝑁) = (𝑄‘(𝑁 + 1)) → 𝐸 = {(𝑃‘𝑁)}))
76		eqeq1 2626	. . . . . . . 8 ⊢ ((𝑄‘𝑁) = (𝑃‘𝑁) → ((𝑄‘𝑁) = (𝑄‘(𝑁 + 1)) ↔ (𝑃‘𝑁) = (𝑄‘(𝑁 + 1))))
77		sneq 4187	. . . . . . . . 9 ⊢ ((𝑄‘𝑁) = (𝑃‘𝑁) → {(𝑄‘𝑁)} = {(𝑃‘𝑁)})
78	77	eqeq2d 2632	. . . . . . . 8 ⊢ ((𝑄‘𝑁) = (𝑃‘𝑁) → (𝐸 = {(𝑄‘𝑁)} ↔ 𝐸 = {(𝑃‘𝑁)}))
79	76, 78	imbi12d 334	. . . . . . 7 ⊢ ((𝑄‘𝑁) = (𝑃‘𝑁) → (((𝑄‘𝑁) = (𝑄‘(𝑁 + 1)) → 𝐸 = {(𝑄‘𝑁)}) ↔ ((𝑃‘𝑁) = (𝑄‘(𝑁 + 1)) → 𝐸 = {(𝑃‘𝑁)})))
80	75, 79	syl5ibrcom 237	. . . . . 6 ⊢ (𝜑 → ((𝑄‘𝑁) = (𝑃‘𝑁) → ((𝑄‘𝑁) = (𝑄‘(𝑁 + 1)) → 𝐸 = {(𝑄‘𝑁)})))
81	60, 80	mpd 15	. . . . 5 ⊢ (𝜑 → ((𝑄‘𝑁) = (𝑄‘(𝑁 + 1)) → 𝐸 = {(𝑄‘𝑁)}))
82	81	imp 445	. . . 4 ⊢ ((𝜑 ∧ (𝑄‘𝑁) = (𝑄‘(𝑁 + 1))) → 𝐸 = {(𝑄‘𝑁)})
83	41, 45, 82	3eqtrd 2660	. . 3 ⊢ ((𝜑 ∧ (𝑄‘𝑁) = (𝑄‘(𝑁 + 1))) → ((iEdg‘𝑆)‘(𝐻‘𝑁)) = {(𝑄‘𝑁)})
84	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15	wlkp1lem7 26576	. . . 4 ⊢ (𝜑 → {(𝑄‘𝑁), (𝑄‘(𝑁 + 1))} ⊆ ((iEdg‘𝑆)‘(𝐻‘𝑁)))
85	84	adantr 481	. . 3 ⊢ ((𝜑 ∧ ¬ (𝑄‘𝑁) = (𝑄‘(𝑁 + 1))) → {(𝑄‘𝑁), (𝑄‘(𝑁 + 1))} ⊆ ((iEdg‘𝑆)‘(𝐻‘𝑁)))
86	83, 85	ifpimpda 1028	. 2 ⊢ (𝜑 → if-((𝑄‘𝑁) = (𝑄‘(𝑁 + 1)), ((iEdg‘𝑆)‘(𝐻‘𝑁)) = {(𝑄‘𝑁)}, {(𝑄‘𝑁), (𝑄‘(𝑁 + 1))} ⊆ ((iEdg‘𝑆)‘(𝐻‘𝑁))))
87	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13	wlkp1lem2 26571	. . . . . 6 ⊢ (𝜑 → (#‘𝐻) = (𝑁 + 1))
88	87	oveq2d 6666	. . . . 5 ⊢ (𝜑 → (0..^(#‘𝐻)) = (0..^(𝑁 + 1)))
89		fzosplitsn 12576	. . . . . 6 ⊢ (𝑁 ∈ (ℤ_≥‘0) → (0..^(𝑁 + 1)) = ((0..^𝑁) ∪ {𝑁}))
90	56, 89	syl 17	. . . . 5 ⊢ (𝜑 → (0..^(𝑁 + 1)) = ((0..^𝑁) ∪ {𝑁}))
91	88, 90	eqtrd 2656	. . . 4 ⊢ (𝜑 → (0..^(#‘𝐻)) = ((0..^𝑁) ∪ {𝑁}))
92	91	raleqdv 3144	. . 3 ⊢ (𝜑 → (∀𝑘 ∈ (0..^(#‘𝐻))if-((𝑄‘𝑘) = (𝑄‘(𝑘 + 1)), ((iEdg‘𝑆)‘(𝐻‘𝑘)) = {(𝑄‘𝑘)}, {(𝑄‘𝑘), (𝑄‘(𝑘 + 1))} ⊆ ((iEdg‘𝑆)‘(𝐻‘𝑘))) ↔ ∀𝑘 ∈ ((0..^𝑁) ∪ {𝑁})if-((𝑄‘𝑘) = (𝑄‘(𝑘 + 1)), ((iEdg‘𝑆)‘(𝐻‘𝑘)) = {(𝑄‘𝑘)}, {(𝑄‘𝑘), (𝑄‘(𝑘 + 1))} ⊆ ((iEdg‘𝑆)‘(𝐻‘𝑘)))))
93		ralunb 3794	. . . 4 ⊢ (∀𝑘 ∈ ((0..^𝑁) ∪ {𝑁})if-((𝑄‘𝑘) = (𝑄‘(𝑘 + 1)), ((iEdg‘𝑆)‘(𝐻‘𝑘)) = {(𝑄‘𝑘)}, {(𝑄‘𝑘), (𝑄‘(𝑘 + 1))} ⊆ ((iEdg‘𝑆)‘(𝐻‘𝑘))) ↔ (∀𝑘 ∈ (0..^𝑁)if-((𝑄‘𝑘) = (𝑄‘(𝑘 + 1)), ((iEdg‘𝑆)‘(𝐻‘𝑘)) = {(𝑄‘𝑘)}, {(𝑄‘𝑘), (𝑄‘(𝑘 + 1))} ⊆ ((iEdg‘𝑆)‘(𝐻‘𝑘))) ∧ ∀𝑘 ∈ {𝑁}if-((𝑄‘𝑘) = (𝑄‘(𝑘 + 1)), ((iEdg‘𝑆)‘(𝐻‘𝑘)) = {(𝑄‘𝑘)}, {(𝑄‘𝑘), (𝑄‘(𝑘 + 1))} ⊆ ((iEdg‘𝑆)‘(𝐻‘𝑘)))))
94	93	a1i 11	. . 3 ⊢ (𝜑 → (∀𝑘 ∈ ((0..^𝑁) ∪ {𝑁})if-((𝑄‘𝑘) = (𝑄‘(𝑘 + 1)), ((iEdg‘𝑆)‘(𝐻‘𝑘)) = {(𝑄‘𝑘)}, {(𝑄‘𝑘), (𝑄‘(𝑘 + 1))} ⊆ ((iEdg‘𝑆)‘(𝐻‘𝑘))) ↔ (∀𝑘 ∈ (0..^𝑁)if-((𝑄‘𝑘) = (𝑄‘(𝑘 + 1)), ((iEdg‘𝑆)‘(𝐻‘𝑘)) = {(𝑄‘𝑘)}, {(𝑄‘𝑘), (𝑄‘(𝑘 + 1))} ⊆ ((iEdg‘𝑆)‘(𝐻‘𝑘))) ∧ ∀𝑘 ∈ {𝑁}if-((𝑄‘𝑘) = (𝑄‘(𝑘 + 1)), ((iEdg‘𝑆)‘(𝐻‘𝑘)) = {(𝑄‘𝑘)}, {(𝑄‘𝑘), (𝑄‘(𝑘 + 1))} ⊆ ((iEdg‘𝑆)‘(𝐻‘𝑘))))))
95		fvex 6201	. . . . . 6 ⊢ (#‘𝐹) ∈ V
96	9, 95	eqeltri 2697	. . . . 5 ⊢ 𝑁 ∈ V
97		fveq2 6191	. . . . . . . 8 ⊢ (𝑘 = 𝑁 → (𝑄‘𝑘) = (𝑄‘𝑁))
98		oveq1 6657	. . . . . . . . 9 ⊢ (𝑘 = 𝑁 → (𝑘 + 1) = (𝑁 + 1))
99	98	fveq2d 6195	. . . . . . . 8 ⊢ (𝑘 = 𝑁 → (𝑄‘(𝑘 + 1)) = (𝑄‘(𝑁 + 1)))
100	97, 99	eqeq12d 2637	. . . . . . 7 ⊢ (𝑘 = 𝑁 → ((𝑄‘𝑘) = (𝑄‘(𝑘 + 1)) ↔ (𝑄‘𝑁) = (𝑄‘(𝑁 + 1))))
101		fveq2 6191	. . . . . . . . 9 ⊢ (𝑘 = 𝑁 → (𝐻‘𝑘) = (𝐻‘𝑁))
102	101	fveq2d 6195	. . . . . . . 8 ⊢ (𝑘 = 𝑁 → ((iEdg‘𝑆)‘(𝐻‘𝑘)) = ((iEdg‘𝑆)‘(𝐻‘𝑁)))
103	97	sneqd 4189	. . . . . . . 8 ⊢ (𝑘 = 𝑁 → {(𝑄‘𝑘)} = {(𝑄‘𝑁)})
104	102, 103	eqeq12d 2637	. . . . . . 7 ⊢ (𝑘 = 𝑁 → (((iEdg‘𝑆)‘(𝐻‘𝑘)) = {(𝑄‘𝑘)} ↔ ((iEdg‘𝑆)‘(𝐻‘𝑁)) = {(𝑄‘𝑁)}))
105	97, 99	preq12d 4276	. . . . . . . 8 ⊢ (𝑘 = 𝑁 → {(𝑄‘𝑘), (𝑄‘(𝑘 + 1))} = {(𝑄‘𝑁), (𝑄‘(𝑁 + 1))})
106	105, 102	sseq12d 3634	. . . . . . 7 ⊢ (𝑘 = 𝑁 → ({(𝑄‘𝑘), (𝑄‘(𝑘 + 1))} ⊆ ((iEdg‘𝑆)‘(𝐻‘𝑘)) ↔ {(𝑄‘𝑁), (𝑄‘(𝑁 + 1))} ⊆ ((iEdg‘𝑆)‘(𝐻‘𝑁))))
107	100, 104, 106	ifpbi123d 1027	. . . . . 6 ⊢ (𝑘 = 𝑁 → (if-((𝑄‘𝑘) = (𝑄‘(𝑘 + 1)), ((iEdg‘𝑆)‘(𝐻‘𝑘)) = {(𝑄‘𝑘)}, {(𝑄‘𝑘), (𝑄‘(𝑘 + 1))} ⊆ ((iEdg‘𝑆)‘(𝐻‘𝑘))) ↔ if-((𝑄‘𝑁) = (𝑄‘(𝑁 + 1)), ((iEdg‘𝑆)‘(𝐻‘𝑁)) = {(𝑄‘𝑁)}, {(𝑄‘𝑁), (𝑄‘(𝑁 + 1))} ⊆ ((iEdg‘𝑆)‘(𝐻‘𝑁)))))
108	107	ralsng 4218	. . . . 5 ⊢ (𝑁 ∈ V → (∀𝑘 ∈ {𝑁}if-((𝑄‘𝑘) = (𝑄‘(𝑘 + 1)), ((iEdg‘𝑆)‘(𝐻‘𝑘)) = {(𝑄‘𝑘)}, {(𝑄‘𝑘), (𝑄‘(𝑘 + 1))} ⊆ ((iEdg‘𝑆)‘(𝐻‘𝑘))) ↔ if-((𝑄‘𝑁) = (𝑄‘(𝑁 + 1)), ((iEdg‘𝑆)‘(𝐻‘𝑁)) = {(𝑄‘𝑁)}, {(𝑄‘𝑁), (𝑄‘(𝑁 + 1))} ⊆ ((iEdg‘𝑆)‘(𝐻‘𝑁)))))
109	96, 108	mp1i 13	. . . 4 ⊢ (𝜑 → (∀𝑘 ∈ {𝑁}if-((𝑄‘𝑘) = (𝑄‘(𝑘 + 1)), ((iEdg‘𝑆)‘(𝐻‘𝑘)) = {(𝑄‘𝑘)}, {(𝑄‘𝑘), (𝑄‘(𝑘 + 1))} ⊆ ((iEdg‘𝑆)‘(𝐻‘𝑘))) ↔ if-((𝑄‘𝑁) = (𝑄‘(𝑁 + 1)), ((iEdg‘𝑆)‘(𝐻‘𝑁)) = {(𝑄‘𝑁)}, {(𝑄‘𝑁), (𝑄‘(𝑁 + 1))} ⊆ ((iEdg‘𝑆)‘(𝐻‘𝑁)))))
110	109	anbi2d 740	. . 3 ⊢ (𝜑 → ((∀𝑘 ∈ (0..^𝑁)if-((𝑄‘𝑘) = (𝑄‘(𝑘 + 1)), ((iEdg‘𝑆)‘(𝐻‘𝑘)) = {(𝑄‘𝑘)}, {(𝑄‘𝑘), (𝑄‘(𝑘 + 1))} ⊆ ((iEdg‘𝑆)‘(𝐻‘𝑘))) ∧ ∀𝑘 ∈ {𝑁}if-((𝑄‘𝑘) = (𝑄‘(𝑘 + 1)), ((iEdg‘𝑆)‘(𝐻‘𝑘)) = {(𝑄‘𝑘)}, {(𝑄‘𝑘), (𝑄‘(𝑘 + 1))} ⊆ ((iEdg‘𝑆)‘(𝐻‘𝑘)))) ↔ (∀𝑘 ∈ (0..^𝑁)if-((𝑄‘𝑘) = (𝑄‘(𝑘 + 1)), ((iEdg‘𝑆)‘(𝐻‘𝑘)) = {(𝑄‘𝑘)}, {(𝑄‘𝑘), (𝑄‘(𝑘 + 1))} ⊆ ((iEdg‘𝑆)‘(𝐻‘𝑘))) ∧ if-((𝑄‘𝑁) = (𝑄‘(𝑁 + 1)), ((iEdg‘𝑆)‘(𝐻‘𝑁)) = {(𝑄‘𝑁)}, {(𝑄‘𝑁), (𝑄‘(𝑁 + 1))} ⊆ ((iEdg‘𝑆)‘(𝐻‘𝑁))))))
111	92, 94, 110	3bitrd 294	. 2 ⊢ (𝜑 → (∀𝑘 ∈ (0..^(#‘𝐻))if-((𝑄‘𝑘) = (𝑄‘(𝑘 + 1)), ((iEdg‘𝑆)‘(𝐻‘𝑘)) = {(𝑄‘𝑘)}, {(𝑄‘𝑘), (𝑄‘(𝑘 + 1))} ⊆ ((iEdg‘𝑆)‘(𝐻‘𝑘))) ↔ (∀𝑘 ∈ (0..^𝑁)if-((𝑄‘𝑘) = (𝑄‘(𝑘 + 1)), ((iEdg‘𝑆)‘(𝐻‘𝑘)) = {(𝑄‘𝑘)}, {(𝑄‘𝑘), (𝑄‘(𝑘 + 1))} ⊆ ((iEdg‘𝑆)‘(𝐻‘𝑘))) ∧ if-((𝑄‘𝑁) = (𝑄‘(𝑁 + 1)), ((iEdg‘𝑆)‘(𝐻‘𝑁)) = {(𝑄‘𝑁)}, {(𝑄‘𝑁), (𝑄‘(𝑁 + 1))} ⊆ ((iEdg‘𝑆)‘(𝐻‘𝑁))))))
112	39, 86, 111	mpbir2and 957	1 ⊢ (𝜑 → ∀𝑘 ∈ (0..^(#‘𝐻))if-((𝑄‘𝑘) = (𝑄‘(𝑘 + 1)), ((iEdg‘𝑆)‘(𝐻‘𝑘)) = {(𝑄‘𝑘)}, {(𝑄‘𝑘), (𝑄‘(𝑘 + 1))} ⊆ ((iEdg‘𝑆)‘(𝐻‘𝑘))))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∧ wa 384 if-wif 1012 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ∀wral 2912 Vcvv 3200 ∪ cun 3572 ⊆ wss 3574 {csn 4177 {cpr 4179 ⟨cop 4183 class class class wbr 4653 dom cdm 5114 Fun wfun 5882 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 Fincfn 7955 0cc0 9936 1c1 9937 + caddc 9939 ℕ₀cn0 11292 ℤ_≥cuz 11687 ...cfz 12326 ..^cfzo 12465 #chash 13117 Word cword 13291 Vtxcvtx 25874 iEdgciedg 25875 Edgcedg 25939 Walkscwlks 26492

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-ifp 1013 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-card 8765 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-n0 11293 df-z 11378 df-uz 11688 df-fz 12327 df-fzo 12466 df-hash 13118 df-word 13299 df-wlks 26495

This theorem is referenced by: wlkp1 26578

W3C validator