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Theorem wlkp1lem8 30028

Description: Lemma for wlkp1 30029. (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 ∈ 𝑊)
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) 𝑊(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 ∈ 𝑊)
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 30026	. . 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 6917	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ V)
18	1, 2	iswlkg 29963	. . . . . 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 18	. . . . 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 2772	. . . . . . . . 9 ⊢ (♯‘𝐹) = 𝑁
21	20	oveq2i 7421	. . . . . . . 8 ⊢ (0..^(♯‘𝐹)) = (0..^𝑁)
22	21	raleqi 3321	. . . . . . 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 219	. . . . . 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 1153	. . . . 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	biimtrdi 256	. . . 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 16	. . 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 2780	. . . . . . 7 ⊢ (((𝑄‘𝑘) = (𝑃‘𝑘) ∧ (𝑄‘(𝑘 + 1)) = (𝑃‘(𝑘 + 1))) → ((𝑄‘𝑘) = (𝑄‘(𝑘 + 1)) ↔ (𝑃‘𝑘) = (𝑃‘(𝑘 + 1))))
28	27	3adant3 1150	. . . . . 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 1156	. . . . . . 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 1154	. . . . . . . 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 4601	. . . . . . 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 2779	. . . . . 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 4701	. . . . . . . 8 ⊢ (((𝑄‘𝑘) = (𝑃‘𝑘) ∧ (𝑄‘(𝑘 + 1)) = (𝑃‘(𝑘 + 1))) → {(𝑄‘𝑘), (𝑄‘(𝑘 + 1))} = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))})
34	33	3adant3 1150	. . . . . . 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 3970	. . . . . 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 1095	. . . . 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 251	. . . 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 3104	. . 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 66	. 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 30023	. . . . 5 ⊢ (𝜑 → ((iEdg‘𝑆)‘(𝐻‘𝑁)) = ((𝐼 ∪ {⟨𝐵, 𝐸⟩})‘𝐵))
41	40	adantr 485	. . . 4 ⊢ ((𝜑 ∧ (𝑄‘𝑁) = (𝑄‘(𝑁 + 1))) → ((iEdg‘𝑆)‘(𝐻‘𝑁)) = ((𝐼 ∪ {⟨𝐵, 𝐸⟩})‘𝐵))
42	5, 10, 7	3jca 1146	. . . . . 6 ⊢ (𝜑 → (𝐵 ∈ 𝑊 ∧ 𝐸 ∈ (Edg‘𝐺) ∧ ¬ 𝐵 ∈ dom 𝐼))
43	42	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ (𝑄‘𝑁) = (𝑄‘(𝑁 + 1))) → (𝐵 ∈ 𝑊 ∧ 𝐸 ∈ (Edg‘𝐺) ∧ ¬ 𝐵 ∈ dom 𝐼))
44		fsnunfv 7185	. . . . 5 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐸 ∈ (Edg‘𝐺) ∧ ¬ 𝐵 ∈ dom 𝐼) → ((𝐼 ∪ {⟨𝐵, 𝐸⟩})‘𝐵) = 𝐸)
45	43, 44	syl 18	. . . 4 ⊢ ((𝜑 ∧ (𝑄‘𝑁) = (𝑄‘(𝑁 + 1))) → ((𝐼 ∪ {⟨𝐵, 𝐸⟩})‘𝐵) = 𝐸)
46		fveq2 6881	. . . . . . . 8 ⊢ (𝑥 = 𝑁 → (𝑄‘𝑥) = (𝑄‘𝑁))
47		fveq2 6881	. . . . . . . 8 ⊢ (𝑥 = 𝑁 → (𝑃‘𝑥) = (𝑃‘𝑁))
48	46, 47	eqeq12d 2779	. . . . . . 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 30025	. . . . . . 7 ⊢ (𝜑 → ∀𝑥 ∈ (0...𝑁)(𝑄‘𝑥) = (𝑃‘𝑥))
50	2	wlkf 29964	. . . . . . . . . 10 ⊢ (𝐹(Walks‘𝐺)𝑃 → 𝐹 ∈ Word dom 𝐼)
51		lencl 14566	. . . . . . . . . . 11 ⊢ (𝐹 ∈ Word dom 𝐼 → (♯‘𝐹) ∈ ℕ₀)
52	9	eleq1i 2854	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ₀ ↔ (♯‘𝐹) ∈ ℕ₀)
53		elnn0uz 12898	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ₀ ↔ 𝑁 ∈ (ℤ_≥‘0))
54	52, 53	sylbb1 240	. . . . . . . . . . 11 ⊢ ((♯‘𝐹) ∈ ℕ₀ → 𝑁 ∈ (ℤ_≥‘0))
55	51, 54	syl 18	. . . . . . . . . 10 ⊢ (𝐹 ∈ Word dom 𝐼 → 𝑁 ∈ (ℤ_≥‘0))
56	8, 50, 55	3syl 19	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘0))
57	56, 53	sylibr 237	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
58		nn0fz0 13649	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ₀ ↔ 𝑁 ∈ (0...𝑁))
59	57, 58	sylib 221	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ (0...𝑁))
60	48, 49, 59	rspcdva 3582	. . . . . 6 ⊢ (𝜑 → (𝑄‘𝑁) = (𝑃‘𝑁))
61	14	fveq1i 6882	. . . . . . . . . . 11 ⊢ (𝑄‘(𝑁 + 1)) = ((𝑃 ∪ {⟨(𝑁 + 1), 𝐶⟩})‘(𝑁 + 1))
62		ovex 7443	. . . . . . . . . . . 12 ⊢ (𝑁 + 1) ∈ V
63	1, 2, 3, 4, 5, 6, 7, 8, 9	wlkp1lem1 30021	. . . . . . . . . . . 12 ⊢ (𝜑 → ¬ (𝑁 + 1) ∈ dom 𝑃)
64		fsnunfv 7185	. . . . . . . . . . . 12 ⊢ (((𝑁 + 1) ∈ V ∧ 𝐶 ∈ 𝑉 ∧ ¬ (𝑁 + 1) ∈ dom 𝑃) → ((𝑃 ∪ {⟨(𝑁 + 1), 𝐶⟩})‘(𝑁 + 1)) = 𝐶)
65	62, 6, 63, 64	mp3an2i 1495	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑃 ∪ {⟨(𝑁 + 1), 𝐶⟩})‘(𝑁 + 1)) = 𝐶)
66	61, 65	eqtrid 2810	. . . . . . . . . 10 ⊢ (𝜑 → (𝑄‘(𝑁 + 1)) = 𝐶)
67	66	eqeq2d 2774	. . . . . . . . 9 ⊢ (𝜑 → ((𝑃‘𝑁) = (𝑄‘(𝑁 + 1)) ↔ (𝑃‘𝑁) = 𝐶))
68		eqcom 2770	. . . . . . . . 9 ⊢ ((𝑃‘𝑁) = 𝐶 ↔ 𝐶 = (𝑃‘𝑁))
69	67, 68	bitrdi 290	. . . . . . . 8 ⊢ (𝜑 → ((𝑃‘𝑁) = (𝑄‘(𝑁 + 1)) ↔ 𝐶 = (𝑃‘𝑁)))
70		wlkp1.l	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐶 = (𝑃‘𝑁)) → 𝐸 = {𝐶})
71		sneq 4599	. . . . . . . . . . 11 ⊢ (𝐶 = (𝑃‘𝑁) → {𝐶} = {(𝑃‘𝑁)})
72	71	adantl 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐶 = (𝑃‘𝑁)) → {𝐶} = {(𝑃‘𝑁)})
73	70, 72	eqtrd 2798	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐶 = (𝑃‘𝑁)) → 𝐸 = {(𝑃‘𝑁)})
74	73	ex 417	. . . . . . . 8 ⊢ (𝜑 → (𝐶 = (𝑃‘𝑁) → 𝐸 = {(𝑃‘𝑁)}))
75	69, 74	sylbid 243	. . . . . . 7 ⊢ (𝜑 → ((𝑃‘𝑁) = (𝑄‘(𝑁 + 1)) → 𝐸 = {(𝑃‘𝑁)}))
76		eqeq1 2767	. . . . . . . 8 ⊢ ((𝑄‘𝑁) = (𝑃‘𝑁) → ((𝑄‘𝑁) = (𝑄‘(𝑁 + 1)) ↔ (𝑃‘𝑁) = (𝑄‘(𝑁 + 1))))
77		sneq 4599	. . . . . . . . 9 ⊢ ((𝑄‘𝑁) = (𝑃‘𝑁) → {(𝑄‘𝑁)} = {(𝑃‘𝑁)})
78	77	eqeq2d 2774	. . . . . . . 8 ⊢ ((𝑄‘𝑁) = (𝑃‘𝑁) → (𝐸 = {(𝑄‘𝑁)} ↔ 𝐸 = {(𝑃‘𝑁)}))
79	76, 78	imbi12d 347	. . . . . . 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 250	. . . . . 6 ⊢ (𝜑 → ((𝑄‘𝑁) = (𝑃‘𝑁) → ((𝑄‘𝑁) = (𝑄‘(𝑁 + 1)) → 𝐸 = {(𝑄‘𝑁)})))
81	60, 80	mpd 16	. . . . 5 ⊢ (𝜑 → ((𝑄‘𝑁) = (𝑄‘(𝑁 + 1)) → 𝐸 = {(𝑄‘𝑁)}))
82	81	imp 411	. . . 4 ⊢ ((𝜑 ∧ (𝑄‘𝑁) = (𝑄‘(𝑁 + 1))) → 𝐸 = {(𝑄‘𝑁)})
83	41, 45, 82	3eqtrd 2802	. . 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 30027	. . . 4 ⊢ (𝜑 → {(𝑄‘𝑁), (𝑄‘(𝑁 + 1))} ⊆ ((iEdg‘𝑆)‘(𝐻‘𝑁)))
85	84	adantr 485	. . 3 ⊢ ((𝜑 ∧ ¬ (𝑄‘𝑁) = (𝑄‘(𝑁 + 1))) → {(𝑄‘𝑁), (𝑄‘(𝑁 + 1))} ⊆ ((iEdg‘𝑆)‘(𝐻‘𝑁)))
86	83, 85	ifpimpda 1097	. 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 30022	. . . . . 6 ⊢ (𝜑 → (♯‘𝐻) = (𝑁 + 1))
88	87	oveq2d 7426	. . . . 5 ⊢ (𝜑 → (0..^(♯‘𝐻)) = (0..^(𝑁 + 1)))
89		fzosplitsn 13801	. . . . . 6 ⊢ (𝑁 ∈ (ℤ_≥‘0) → (0..^(𝑁 + 1)) = ((0..^𝑁) ∪ {𝑁}))
90	56, 89	syl 18	. . . . 5 ⊢ (𝜑 → (0..^(𝑁 + 1)) = ((0..^𝑁) ∪ {𝑁}))
91	88, 90	eqtrd 2798	. . . 4 ⊢ (𝜑 → (0..^(♯‘𝐻)) = ((0..^𝑁) ∪ {𝑁}))
92	91	raleqdv 3323	. . 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 4150	. . . 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	9	fvexi 6895	. . . . 5 ⊢ 𝑁 ∈ V
96		wkslem1 29957	. . . . . 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)))))
97	96	ralsng 4641	. . . . 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)))))
98	95, 97	mp1i 14	. . . 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)))))
99	98	anbi2d 641	. . 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))))))
100	92, 94, 99	3bitrd 308	. 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))))))
101	39, 86, 100	mpbir2and 725	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 209 ∧ wa 400 if-wif 1078 ∧ w3a 1103 = wceq 1570 ∈ wcel 2143 ∀wral 3079 Vcvv 3455 ∪ cun 3903 ⊆ wss 3905 {csn 4589 {cpr 4591 ⟨cop 4595 class class class wbr 5109 dom cdm 5661 Fun wfun 6530 ⟶wf 6532 ‘cfv 6536 (class class class)co 7410 Fincfn 8939 0cc0 11095 1c1 11096 + caddc 11098 ℕ₀cn0 12499 ℤ_≥cuz 12857 ...cfz 13530 ..^cfzo 13678 ♯chash 14362 Word cword 14546 Vtxcvtx 29346 iEdgciedg 29347 Edgcedg 29397 Walkscwlks 29946

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5238 ax-sep 5257 ax-nul 5269 ax-pow 5336 ax-pr 5404 ax-un 7732 ax-cnex 11151 ax-resscn 11152 ax-1cn 11153 ax-icn 11154 ax-addcl 11155 ax-addrcl 11156 ax-mulcl 11157 ax-mulrcl 11158 ax-mulcom 11159 ax-addass 11160 ax-mulass 11161 ax-distr 11162 ax-i2m1 11163 ax-1ne0 11164 ax-1rid 11165 ax-rnegex 11166 ax-rrecex 11167 ax-cnre 11168 ax-pre-lttri 11169 ax-pre-lttrn 11170 ax-pre-ltadd 11171 ax-pre-mulgt0 11172

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-ifp 1079 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4488 df-pw 4564 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4873 df-int 4913 df-iun 4958 df-br 5110 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5556 df-eprel 5561 df-po 5569 df-so 5570 df-fr 5614 df-we 5616 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-om 7859 df-1st 7982 df-2nd 7983 df-frecs 8274 df-wrecs 8305 df-recs 8354 df-rdg 8393 df-1o 8449 df-oadd 8453 df-er 8690 df-map 8822 df-pm 8823 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-dju 9883 df-card 9921 df-pnf 11240 df-mnf 11241 df-xr 11242 df-ltxr 11243 df-le 11244 df-sub 11438 df-neg 11439 df-nn 12229 df-n0 12500 df-z 12587 df-uz 12858 df-fz 13531 df-fzo 13679 df-hash 14363 df-word 14547 df-wlks 29949

This theorem is referenced by: wlkp1 30029

W3C validator