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Theorem wlkp1lem8 30259

Description: Lemma for wlkp1 30260. (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 30257	. . 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 6921	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ V)
18	1, 2	iswlkg 30194	. . . . . 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 2769	. . . . . . . . 9 ⊢ (♯‘𝐹) = 𝑁
21	20	oveq2i 7431	. . . . . . . 8 ⊢ (0..^(♯‘𝐹)) = (0..^𝑁)
22	21	raleqi 3317	. . . . . . 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 2777	. . . . . . 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 4595	. . . . . . 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 2776	. . . . . 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 4695	. . . . . . . 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 3941	. . . . . 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 3101	. . 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 30254	. . . . 5 ⊢ (𝜑 → ((iEdg‘𝑆)‘(𝐻‘𝑁)) = ((𝐼 ∪ {⟨𝐵, 𝐸⟩})‘𝐵))
41	40	adantr 486	. . . 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 486	. . . . 5 ⊢ ((𝜑 ∧ (𝑄‘𝑁) = (𝑄‘(𝑁 + 1))) → (𝐵 ∈ 𝑊 ∧ 𝐸 ∈ (Edg‘𝐺) ∧ ¬ 𝐵 ∈ dom 𝐼))
44		fsnunfv 7192	. . . . 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 6885	. . . . . . . 8 ⊢ (𝑥 = 𝑁 → (𝑄‘𝑥) = (𝑄‘𝑁))
47		fveq2 6885	. . . . . . . 8 ⊢ (𝑥 = 𝑁 → (𝑃‘𝑥) = (𝑃‘𝑁))
48	46, 47	eqeq12d 2776	. . . . . . 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 30256	. . . . . . 7 ⊢ (𝜑 → ∀𝑥 ∈ (0...𝑁)(𝑄‘𝑥) = (𝑃‘𝑥))
50	2	wlkf 30195	. . . . . . . . . 10 ⊢ (𝐹(Walks‘𝐺)𝑃 → 𝐹 ∈ Word dom 𝐼)
51		lencl 14678	. . . . . . . . . . 11 ⊢ (𝐹 ∈ Word dom 𝐼 → (♯‘𝐹) ∈ ℕ₀)
52	9	eleq1i 2851	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ₀ ↔ (♯‘𝐹) ∈ ℕ₀)
53		elnn0uz 13006	. . . . . . . . . . . 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 13759	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ₀ ↔ 𝑁 ∈ (0...𝑁))
59	57, 58	sylib 221	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ (0...𝑁))
60	48, 49, 59	rspcdva 3577	. . . . . 6 ⊢ (𝜑 → (𝑄‘𝑁) = (𝑃‘𝑁))
61	14	fveq1i 6886	. . . . . . . . . . 11 ⊢ (𝑄‘(𝑁 + 1)) = ((𝑃 ∪ {⟨(𝑁 + 1), 𝐶⟩})‘(𝑁 + 1))
62		ovex 7453	. . . . . . . . . . . 12 ⊢ (𝑁 + 1) ∈ V
63	1, 2, 3, 4, 5, 6, 7, 8, 9	wlkp1lem1 30252	. . . . . . . . . . . 12 ⊢ (𝜑 → ¬ (𝑁 + 1) ∈ dom 𝑃)
64		fsnunfv 7192	. . . . . . . . . . . 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 2807	. . . . . . . . . 10 ⊢ (𝜑 → (𝑄‘(𝑁 + 1)) = 𝐶)
67	66	eqeq2d 2771	. . . . . . . . 9 ⊢ (𝜑 → ((𝑃‘𝑁) = (𝑄‘(𝑁 + 1)) ↔ (𝑃‘𝑁) = 𝐶))
68		eqcom 2767	. . . . . . . . 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 4593	. . . . . . . . . . 11 ⊢ (𝐶 = (𝑃‘𝑁) → {𝐶} = {(𝑃‘𝑁)})
72	71	adantl 487	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐶 = (𝑃‘𝑁)) → {𝐶} = {(𝑃‘𝑁)})
73	70, 72	eqtrd 2795	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐶 = (𝑃‘𝑁)) → 𝐸 = {(𝑃‘𝑁)})
74	73	ex 418	. . . . . . . 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 2764	. . . . . . . 8 ⊢ ((𝑄‘𝑁) = (𝑃‘𝑁) → ((𝑄‘𝑁) = (𝑄‘(𝑁 + 1)) ↔ (𝑃‘𝑁) = (𝑄‘(𝑁 + 1))))
77		sneq 4593	. . . . . . . . 9 ⊢ ((𝑄‘𝑁) = (𝑃‘𝑁) → {(𝑄‘𝑁)} = {(𝑃‘𝑁)})
78	77	eqeq2d 2771	. . . . . . . 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 412	. . . 4 ⊢ ((𝜑 ∧ (𝑄‘𝑁) = (𝑄‘(𝑁 + 1))) → 𝐸 = {(𝑄‘𝑁)})
83	41, 45, 82	3eqtrd 2799	. . 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 30258	. . . 4 ⊢ (𝜑 → {(𝑄‘𝑁), (𝑄‘(𝑁 + 1))} ⊆ ((iEdg‘𝑆)‘(𝐻‘𝑁)))
85	84	adantr 486	. . 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 30253	. . . . . 6 ⊢ (𝜑 → (♯‘𝐻) = (𝑁 + 1))
88	87	oveq2d 7436	. . . . 5 ⊢ (𝜑 → (0..^(♯‘𝐻)) = (0..^(𝑁 + 1)))
89		fzosplitsn 13911	. . . . . 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 2795	. . . 4 ⊢ (𝜑 → (0..^(♯‘𝐻)) = ((0..^𝑁) ∪ {𝑁}))
92	91	raleqdv 3319	. . 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 4195	. . . 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 6899	. . . . 5 ⊢ 𝑁 ∈ V
96		wkslem1 30188	. . . . . 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 4635	. . . . 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 642	. . 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 726	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

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 if-wif 1078 ∧ w3a 1103 = wceq 1569 ∈ wcel 2144 ∀wral 3076 Vcvv 3450 ⊆ wss 3895 ∪ cun 4149 {csn 4583 {cpr 4585 ⟨cop 4589 class class class wbr 5102 dom cdm 5650 Fun wfun 6532 ⟶wf 6534 ‘cfv 6538 (class class class)co 7420 Fincfn 8973 0cc0 11200 1c1 11201 + caddc 11203 ℕ₀cn0 12606 ℤ_≥cuz 12965 ...cfz 13639 ..^cfzo 13788 ♯chash 14474 Word cword 14658 Vtxcvtx 29574 iEdgciedg 29575 Edgcedg 29625 Walkscwlks 30177

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7751 ax-cnex 11256 ax-resscn 11257 ax-1cn 11258 ax-icn 11259 ax-addcl 11260 ax-addrcl 11261 ax-mulcl 11262 ax-mulrcl 11263 ax-mulcom 11264 ax-addass 11265 ax-mulass 11266 ax-distr 11267 ax-i2m1 11268 ax-1ne0 11269 ax-1rid 11270 ax-rnegex 11271 ax-rrecex 11272 ax-cnre 11273 ax-pre-lttri 11274 ax-pre-lttrn 11275 ax-pre-ltadd 11276 ax-pre-mulgt0 11277

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 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-ss 3896 df-pss 4014 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4584 df-pr 4586 df-op 4590 df-uni 4867 df-int 4907 df-iun 4952 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5545 df-eprel 5550 df-po 5558 df-so 5559 df-fr 5603 df-we 5605 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-riota 7377 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7878 df-1st 8001 df-2nd 8002 df-frecs 8299 df-wrecs 8330 df-recs 8379 df-rdg 8418 df-1o 8476 df-oadd 8480 df-er 8717 df-map 8849 df-pm 8850 df-en 8974 df-dom 8975 df-sdom 8976 df-fin 8977 df-dju 9982 df-card 10020 df-pnf 11345 df-mnf 11346 df-xr 11347 df-ltxr 11348 df-le 11349 df-sub 11543 df-neg 11544 df-nn 12336 df-n0 12607 df-z 12694 df-uz 12966 df-fz 13640 df-fzo 13789 df-hash 14475 df-word 14659 df-wlks 30180

This theorem is used by: wlkp1 30260

W3C validator