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Theorem wlkp1lem8 26801
Description: Lemma for wlkp1 26802. (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 (𝜑𝐼 ∈ Fin)
wlkp1.b (𝜑𝐵 ∈ V)
wlkp1.c (𝜑𝐶𝑉)
wlkp1.d (𝜑 → ¬ 𝐵 ∈ dom 𝐼)
wlkp1.w (𝜑𝐹(Walks‘𝐺)𝑃)
wlkp1.n 𝑁 = (♯‘𝐹)
wlkp1.e (𝜑𝐸 ∈ (Edg‘𝐺))
wlkp1.x (𝜑 → {(𝑃𝑁), 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 ((𝜑𝐶 = (𝑃𝑁)) → 𝐸 = {𝐶})
Assertion
Ref Expression
wlkp1lem8 (𝜑 → ∀𝑘 ∈ (0..^(♯‘𝐻))if-((𝑄𝑘) = (𝑄‘(𝑘 + 1)), ((iEdg‘𝑆)‘(𝐻𝑘)) = {(𝑄𝑘)}, {(𝑄𝑘), (𝑄‘(𝑘 + 1))} ⊆ ((iEdg‘𝑆)‘(𝐻𝑘))))
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.
StepHypRef 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 (𝜑𝐼 ∈ Fin)
5 wlkp1.b . . . 4 (𝜑𝐵 ∈ V)
6 wlkp1.c . . . 4 (𝜑𝐶𝑉)
7 wlkp1.d . . . 4 (𝜑 → ¬ 𝐵 ∈ dom 𝐼)
8 wlkp1.w . . . 4 (𝜑𝐹(Walks‘𝐺)𝑃)
9 wlkp1.n . . . 4 𝑁 = (♯‘𝐹)
10 wlkp1.e . . . 4 (𝜑𝐸 ∈ (Edg‘𝐺))
11 wlkp1.x . . . 4 (𝜑 → {(𝑃𝑁), 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)
161, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15wlkp1lem6 26799 . . 3 (𝜑 → ∀𝑘 ∈ (0..^𝑁)((𝑄𝑘) = (𝑃𝑘) ∧ (𝑄‘(𝑘 + 1)) = (𝑃‘(𝑘 + 1)) ∧ ((iEdg‘𝑆)‘(𝐻𝑘)) = (𝐼‘(𝐹𝑘))))
1710elfvexd 6439 . . . . . 6 (𝜑𝐺 ∈ V)
181, 2iswlkg 26733 . . . . . 6 (𝐺 ∈ V → (𝐹(Walks‘𝐺)𝑃 ↔ (𝐹 ∈ Word dom 𝐼𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹𝑘)) = {(𝑃𝑘)}, {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹𝑘))))))
1917, 18syl 17 . . . . 5 (𝜑 → (𝐹(Walks‘𝐺)𝑃 ↔ (𝐹 ∈ Word dom 𝐼𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹𝑘)) = {(𝑃𝑘)}, {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹𝑘))))))
209eqcomi 2814 . . . . . . . . 9 (♯‘𝐹) = 𝑁
2120oveq2i 6882 . . . . . . . 8 (0..^(♯‘𝐹)) = (0..^𝑁)
2221raleqi 3330 . . . . . . 7 (∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹𝑘)) = {(𝑃𝑘)}, {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹𝑘))) ↔ ∀𝑘 ∈ (0..^𝑁)if-((𝑃𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹𝑘)) = {(𝑃𝑘)}, {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹𝑘))))
2322biimpi 207 . . . . . 6 (∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹𝑘)) = {(𝑃𝑘)}, {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹𝑘))) → ∀𝑘 ∈ (0..^𝑁)if-((𝑃𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹𝑘)) = {(𝑃𝑘)}, {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹𝑘))))
24233ad2ant3 1158 . . . . 5 ((𝐹 ∈ Word dom 𝐼𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹𝑘)) = {(𝑃𝑘)}, {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹𝑘)))) → ∀𝑘 ∈ (0..^𝑁)if-((𝑃𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹𝑘)) = {(𝑃𝑘)}, {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹𝑘))))
2519, 24syl6bi 244 . . . 4 (𝜑 → (𝐹(Walks‘𝐺)𝑃 → ∀𝑘 ∈ (0..^𝑁)if-((𝑃𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹𝑘)) = {(𝑃𝑘)}, {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹𝑘)))))
268, 25mpd 15 . . 3 (𝜑 → ∀𝑘 ∈ (0..^𝑁)if-((𝑃𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹𝑘)) = {(𝑃𝑘)}, {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹𝑘))))
27 eqeq12 2818 . . . . . . 7 (((𝑄𝑘) = (𝑃𝑘) ∧ (𝑄‘(𝑘 + 1)) = (𝑃‘(𝑘 + 1))) → ((𝑄𝑘) = (𝑄‘(𝑘 + 1)) ↔ (𝑃𝑘) = (𝑃‘(𝑘 + 1))))
28273adant3 1155 . . . . . 6 (((𝑄𝑘) = (𝑃𝑘) ∧ (𝑄‘(𝑘 + 1)) = (𝑃‘(𝑘 + 1)) ∧ ((iEdg‘𝑆)‘(𝐻𝑘)) = (𝐼‘(𝐹𝑘))) → ((𝑄𝑘) = (𝑄‘(𝑘 + 1)) ↔ (𝑃𝑘) = (𝑃‘(𝑘 + 1))))
29 simp3 1161 . . . . . . 7 (((𝑄𝑘) = (𝑃𝑘) ∧ (𝑄‘(𝑘 + 1)) = (𝑃‘(𝑘 + 1)) ∧ ((iEdg‘𝑆)‘(𝐻𝑘)) = (𝐼‘(𝐹𝑘))) → ((iEdg‘𝑆)‘(𝐻𝑘)) = (𝐼‘(𝐹𝑘)))
30 simp1 1159 . . . . . . . 8 (((𝑄𝑘) = (𝑃𝑘) ∧ (𝑄‘(𝑘 + 1)) = (𝑃‘(𝑘 + 1)) ∧ ((iEdg‘𝑆)‘(𝐻𝑘)) = (𝐼‘(𝐹𝑘))) → (𝑄𝑘) = (𝑃𝑘))
3130sneqd 4379 . . . . . . 7 (((𝑄𝑘) = (𝑃𝑘) ∧ (𝑄‘(𝑘 + 1)) = (𝑃‘(𝑘 + 1)) ∧ ((iEdg‘𝑆)‘(𝐻𝑘)) = (𝐼‘(𝐹𝑘))) → {(𝑄𝑘)} = {(𝑃𝑘)})
3229, 31eqeq12d 2820 . . . . . 6 (((𝑄𝑘) = (𝑃𝑘) ∧ (𝑄‘(𝑘 + 1)) = (𝑃‘(𝑘 + 1)) ∧ ((iEdg‘𝑆)‘(𝐻𝑘)) = (𝐼‘(𝐹𝑘))) → (((iEdg‘𝑆)‘(𝐻𝑘)) = {(𝑄𝑘)} ↔ (𝐼‘(𝐹𝑘)) = {(𝑃𝑘)}))
33 preq12 4458 . . . . . . . 8 (((𝑄𝑘) = (𝑃𝑘) ∧ (𝑄‘(𝑘 + 1)) = (𝑃‘(𝑘 + 1))) → {(𝑄𝑘), (𝑄‘(𝑘 + 1))} = {(𝑃𝑘), (𝑃‘(𝑘 + 1))})
34333adant3 1155 . . . . . . 7 (((𝑄𝑘) = (𝑃𝑘) ∧ (𝑄‘(𝑘 + 1)) = (𝑃‘(𝑘 + 1)) ∧ ((iEdg‘𝑆)‘(𝐻𝑘)) = (𝐼‘(𝐹𝑘))) → {(𝑄𝑘), (𝑄‘(𝑘 + 1))} = {(𝑃𝑘), (𝑃‘(𝑘 + 1))})
3534, 29sseq12d 3828 . . . . . 6 (((𝑄𝑘) = (𝑃𝑘) ∧ (𝑄‘(𝑘 + 1)) = (𝑃‘(𝑘 + 1)) ∧ ((iEdg‘𝑆)‘(𝐻𝑘)) = (𝐼‘(𝐹𝑘))) → ({(𝑄𝑘), (𝑄‘(𝑘 + 1))} ⊆ ((iEdg‘𝑆)‘(𝐻𝑘)) ↔ {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹𝑘))))
3628, 32, 35ifpbi123d 1093 . . . . 5 (((𝑄𝑘) = (𝑃𝑘) ∧ (𝑄‘(𝑘 + 1)) = (𝑃‘(𝑘 + 1)) ∧ ((iEdg‘𝑆)‘(𝐻𝑘)) = (𝐼‘(𝐹𝑘))) → (if-((𝑄𝑘) = (𝑄‘(𝑘 + 1)), ((iEdg‘𝑆)‘(𝐻𝑘)) = {(𝑄𝑘)}, {(𝑄𝑘), (𝑄‘(𝑘 + 1))} ⊆ ((iEdg‘𝑆)‘(𝐻𝑘))) ↔ if-((𝑃𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹𝑘)) = {(𝑃𝑘)}, {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹𝑘)))))
3736biimprd 239 . . . 4 (((𝑄𝑘) = (𝑃𝑘) ∧ (𝑄‘(𝑘 + 1)) = (𝑃‘(𝑘 + 1)) ∧ ((iEdg‘𝑆)‘(𝐻𝑘)) = (𝐼‘(𝐹𝑘))) → (if-((𝑃𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹𝑘)) = {(𝑃𝑘)}, {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹𝑘))) → if-((𝑄𝑘) = (𝑄‘(𝑘 + 1)), ((iEdg‘𝑆)‘(𝐻𝑘)) = {(𝑄𝑘)}, {(𝑄𝑘), (𝑄‘(𝑘 + 1))} ⊆ ((iEdg‘𝑆)‘(𝐻𝑘)))))
3837ral2imi 3134 . . 3 (∀𝑘 ∈ (0..^𝑁)((𝑄𝑘) = (𝑃𝑘) ∧ (𝑄‘(𝑘 + 1)) = (𝑃‘(𝑘 + 1)) ∧ ((iEdg‘𝑆)‘(𝐻𝑘)) = (𝐼‘(𝐹𝑘))) → (∀𝑘 ∈ (0..^𝑁)if-((𝑃𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹𝑘)) = {(𝑃𝑘)}, {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹𝑘))) → ∀𝑘 ∈ (0..^𝑁)if-((𝑄𝑘) = (𝑄‘(𝑘 + 1)), ((iEdg‘𝑆)‘(𝐻𝑘)) = {(𝑄𝑘)}, {(𝑄𝑘), (𝑄‘(𝑘 + 1))} ⊆ ((iEdg‘𝑆)‘(𝐻𝑘)))))
3916, 26, 38sylc 65 . 2 (𝜑 → ∀𝑘 ∈ (0..^𝑁)if-((𝑄𝑘) = (𝑄‘(𝑘 + 1)), ((iEdg‘𝑆)‘(𝐻𝑘)) = {(𝑄𝑘)}, {(𝑄𝑘), (𝑄‘(𝑘 + 1))} ⊆ ((iEdg‘𝑆)‘(𝐻𝑘))))
401, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13wlkp1lem3 26796 . . . . 5 (𝜑 → ((iEdg‘𝑆)‘(𝐻𝑁)) = ((𝐼 ∪ {⟨𝐵, 𝐸⟩})‘𝐵))
4140adantr 468 . . . 4 ((𝜑 ∧ (𝑄𝑁) = (𝑄‘(𝑁 + 1))) → ((iEdg‘𝑆)‘(𝐻𝑁)) = ((𝐼 ∪ {⟨𝐵, 𝐸⟩})‘𝐵))
425, 10, 73jca 1151 . . . . . 6 (𝜑 → (𝐵 ∈ V ∧ 𝐸 ∈ (Edg‘𝐺) ∧ ¬ 𝐵 ∈ dom 𝐼))
4342adantr 468 . . . . 5 ((𝜑 ∧ (𝑄𝑁) = (𝑄‘(𝑁 + 1))) → (𝐵 ∈ V ∧ 𝐸 ∈ (Edg‘𝐺) ∧ ¬ 𝐵 ∈ dom 𝐼))
44 fsnunfv 6675 . . . . 5 ((𝐵 ∈ V ∧ 𝐸 ∈ (Edg‘𝐺) ∧ ¬ 𝐵 ∈ dom 𝐼) → ((𝐼 ∪ {⟨𝐵, 𝐸⟩})‘𝐵) = 𝐸)
4543, 44syl 17 . . . 4 ((𝜑 ∧ (𝑄𝑁) = (𝑄‘(𝑁 + 1))) → ((𝐼 ∪ {⟨𝐵, 𝐸⟩})‘𝐵) = 𝐸)
46 fveq2 6405 . . . . . . . 8 (𝑥 = 𝑁 → (𝑄𝑥) = (𝑄𝑁))
47 fveq2 6405 . . . . . . . 8 (𝑥 = 𝑁 → (𝑃𝑥) = (𝑃𝑁))
4846, 47eqeq12d 2820 . . . . . . 7 (𝑥 = 𝑁 → ((𝑄𝑥) = (𝑃𝑥) ↔ (𝑄𝑁) = (𝑃𝑁)))
491, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15wlkp1lem5 26798 . . . . . . 7 (𝜑 → ∀𝑥 ∈ (0...𝑁)(𝑄𝑥) = (𝑃𝑥))
502wlkf 26734 . . . . . . . . . 10 (𝐹(Walks‘𝐺)𝑃𝐹 ∈ Word dom 𝐼)
51 lencl 13531 . . . . . . . . . . 11 (𝐹 ∈ Word dom 𝐼 → (♯‘𝐹) ∈ ℕ0)
529eleq1i 2875 . . . . . . . . . . . 12 (𝑁 ∈ ℕ0 ↔ (♯‘𝐹) ∈ ℕ0)
53 elnn0uz 11939 . . . . . . . . . . . 12 (𝑁 ∈ ℕ0𝑁 ∈ (ℤ‘0))
5452, 53sylbb1 228 . . . . . . . . . . 11 ((♯‘𝐹) ∈ ℕ0𝑁 ∈ (ℤ‘0))
5551, 54syl 17 . . . . . . . . . 10 (𝐹 ∈ Word dom 𝐼𝑁 ∈ (ℤ‘0))
568, 50, 553syl 18 . . . . . . . . 9 (𝜑𝑁 ∈ (ℤ‘0))
5756, 53sylibr 225 . . . . . . . 8 (𝜑𝑁 ∈ ℕ0)
58 nn0fz0 12657 . . . . . . . 8 (𝑁 ∈ ℕ0𝑁 ∈ (0...𝑁))
5957, 58sylib 209 . . . . . . 7 (𝜑𝑁 ∈ (0...𝑁))
6048, 49, 59rspcdva 3507 . . . . . 6 (𝜑 → (𝑄𝑁) = (𝑃𝑁))
6114fveq1i 6406 . . . . . . . . . . 11 (𝑄‘(𝑁 + 1)) = ((𝑃 ∪ {⟨(𝑁 + 1), 𝐶⟩})‘(𝑁 + 1))
62 ovex 6903 . . . . . . . . . . . 12 (𝑁 + 1) ∈ V
631, 2, 3, 4, 5, 6, 7, 8, 9wlkp1lem1 26794 . . . . . . . . . . . 12 (𝜑 → ¬ (𝑁 + 1) ∈ dom 𝑃)
64 fsnunfv 6675 . . . . . . . . . . . 12 (((𝑁 + 1) ∈ V ∧ 𝐶𝑉 ∧ ¬ (𝑁 + 1) ∈ dom 𝑃) → ((𝑃 ∪ {⟨(𝑁 + 1), 𝐶⟩})‘(𝑁 + 1)) = 𝐶)
6562, 6, 63, 64mp3an2i 1583 . . . . . . . . . . 11 (𝜑 → ((𝑃 ∪ {⟨(𝑁 + 1), 𝐶⟩})‘(𝑁 + 1)) = 𝐶)
6661, 65syl5eq 2851 . . . . . . . . . 10 (𝜑 → (𝑄‘(𝑁 + 1)) = 𝐶)
6766eqeq2d 2815 . . . . . . . . 9 (𝜑 → ((𝑃𝑁) = (𝑄‘(𝑁 + 1)) ↔ (𝑃𝑁) = 𝐶))
68 eqcom 2812 . . . . . . . . 9 ((𝑃𝑁) = 𝐶𝐶 = (𝑃𝑁))
6967, 68syl6bb 278 . . . . . . . 8 (𝜑 → ((𝑃𝑁) = (𝑄‘(𝑁 + 1)) ↔ 𝐶 = (𝑃𝑁)))
70 wlkp1.l . . . . . . . . . 10 ((𝜑𝐶 = (𝑃𝑁)) → 𝐸 = {𝐶})
71 sneq 4377 . . . . . . . . . . 11 (𝐶 = (𝑃𝑁) → {𝐶} = {(𝑃𝑁)})
7271adantl 469 . . . . . . . . . 10 ((𝜑𝐶 = (𝑃𝑁)) → {𝐶} = {(𝑃𝑁)})
7370, 72eqtrd 2839 . . . . . . . . 9 ((𝜑𝐶 = (𝑃𝑁)) → 𝐸 = {(𝑃𝑁)})
7473ex 399 . . . . . . . 8 (𝜑 → (𝐶 = (𝑃𝑁) → 𝐸 = {(𝑃𝑁)}))
7569, 74sylbid 231 . . . . . . 7 (𝜑 → ((𝑃𝑁) = (𝑄‘(𝑁 + 1)) → 𝐸 = {(𝑃𝑁)}))
76 eqeq1 2809 . . . . . . . 8 ((𝑄𝑁) = (𝑃𝑁) → ((𝑄𝑁) = (𝑄‘(𝑁 + 1)) ↔ (𝑃𝑁) = (𝑄‘(𝑁 + 1))))
77 sneq 4377 . . . . . . . . 9 ((𝑄𝑁) = (𝑃𝑁) → {(𝑄𝑁)} = {(𝑃𝑁)})
7877eqeq2d 2815 . . . . . . . 8 ((𝑄𝑁) = (𝑃𝑁) → (𝐸 = {(𝑄𝑁)} ↔ 𝐸 = {(𝑃𝑁)}))
7976, 78imbi12d 335 . . . . . . 7 ((𝑄𝑁) = (𝑃𝑁) → (((𝑄𝑁) = (𝑄‘(𝑁 + 1)) → 𝐸 = {(𝑄𝑁)}) ↔ ((𝑃𝑁) = (𝑄‘(𝑁 + 1)) → 𝐸 = {(𝑃𝑁)})))
8075, 79syl5ibrcom 238 . . . . . 6 (𝜑 → ((𝑄𝑁) = (𝑃𝑁) → ((𝑄𝑁) = (𝑄‘(𝑁 + 1)) → 𝐸 = {(𝑄𝑁)})))
8160, 80mpd 15 . . . . 5 (𝜑 → ((𝑄𝑁) = (𝑄‘(𝑁 + 1)) → 𝐸 = {(𝑄𝑁)}))
8281imp 395 . . . 4 ((𝜑 ∧ (𝑄𝑁) = (𝑄‘(𝑁 + 1))) → 𝐸 = {(𝑄𝑁)})
8341, 45, 823eqtrd 2843 . . 3 ((𝜑 ∧ (𝑄𝑁) = (𝑄‘(𝑁 + 1))) → ((iEdg‘𝑆)‘(𝐻𝑁)) = {(𝑄𝑁)})
841, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15wlkp1lem7 26800 . . . 4 (𝜑 → {(𝑄𝑁), (𝑄‘(𝑁 + 1))} ⊆ ((iEdg‘𝑆)‘(𝐻𝑁)))
8584adantr 468 . . 3 ((𝜑 ∧ ¬ (𝑄𝑁) = (𝑄‘(𝑁 + 1))) → {(𝑄𝑁), (𝑄‘(𝑁 + 1))} ⊆ ((iEdg‘𝑆)‘(𝐻𝑁)))
8683, 85ifpimpda 1094 . 2 (𝜑 → if-((𝑄𝑁) = (𝑄‘(𝑁 + 1)), ((iEdg‘𝑆)‘(𝐻𝑁)) = {(𝑄𝑁)}, {(𝑄𝑁), (𝑄‘(𝑁 + 1))} ⊆ ((iEdg‘𝑆)‘(𝐻𝑁))))
871, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13wlkp1lem2 26795 . . . . . 6 (𝜑 → (♯‘𝐻) = (𝑁 + 1))
8887oveq2d 6887 . . . . 5 (𝜑 → (0..^(♯‘𝐻)) = (0..^(𝑁 + 1)))
89 fzosplitsn 12796 . . . . . 6 (𝑁 ∈ (ℤ‘0) → (0..^(𝑁 + 1)) = ((0..^𝑁) ∪ {𝑁}))
9056, 89syl 17 . . . . 5 (𝜑 → (0..^(𝑁 + 1)) = ((0..^𝑁) ∪ {𝑁}))
9188, 90eqtrd 2839 . . . 4 (𝜑 → (0..^(♯‘𝐻)) = ((0..^𝑁) ∪ {𝑁}))
9291raleqdv 3332 . . 3 (𝜑 → (∀𝑘 ∈ (0..^(♯‘𝐻))if-((𝑄𝑘) = (𝑄‘(𝑘 + 1)), ((iEdg‘𝑆)‘(𝐻𝑘)) = {(𝑄𝑘)}, {(𝑄𝑘), (𝑄‘(𝑘 + 1))} ⊆ ((iEdg‘𝑆)‘(𝐻𝑘))) ↔ ∀𝑘 ∈ ((0..^𝑁) ∪ {𝑁})if-((𝑄𝑘) = (𝑄‘(𝑘 + 1)), ((iEdg‘𝑆)‘(𝐻𝑘)) = {(𝑄𝑘)}, {(𝑄𝑘), (𝑄‘(𝑘 + 1))} ⊆ ((iEdg‘𝑆)‘(𝐻𝑘)))))
93 ralunb 3990 . . . 4 (∀𝑘 ∈ ((0..^𝑁) ∪ {𝑁})if-((𝑄𝑘) = (𝑄‘(𝑘 + 1)), ((iEdg‘𝑆)‘(𝐻𝑘)) = {(𝑄𝑘)}, {(𝑄𝑘), (𝑄‘(𝑘 + 1))} ⊆ ((iEdg‘𝑆)‘(𝐻𝑘))) ↔ (∀𝑘 ∈ (0..^𝑁)if-((𝑄𝑘) = (𝑄‘(𝑘 + 1)), ((iEdg‘𝑆)‘(𝐻𝑘)) = {(𝑄𝑘)}, {(𝑄𝑘), (𝑄‘(𝑘 + 1))} ⊆ ((iEdg‘𝑆)‘(𝐻𝑘))) ∧ ∀𝑘 ∈ {𝑁}if-((𝑄𝑘) = (𝑄‘(𝑘 + 1)), ((iEdg‘𝑆)‘(𝐻𝑘)) = {(𝑄𝑘)}, {(𝑄𝑘), (𝑄‘(𝑘 + 1))} ⊆ ((iEdg‘𝑆)‘(𝐻𝑘)))))
9493a1i 11 . . 3 (𝜑 → (∀𝑘 ∈ ((0..^𝑁) ∪ {𝑁})if-((𝑄𝑘) = (𝑄‘(𝑘 + 1)), ((iEdg‘𝑆)‘(𝐻𝑘)) = {(𝑄𝑘)}, {(𝑄𝑘), (𝑄‘(𝑘 + 1))} ⊆ ((iEdg‘𝑆)‘(𝐻𝑘))) ↔ (∀𝑘 ∈ (0..^𝑁)if-((𝑄𝑘) = (𝑄‘(𝑘 + 1)), ((iEdg‘𝑆)‘(𝐻𝑘)) = {(𝑄𝑘)}, {(𝑄𝑘), (𝑄‘(𝑘 + 1))} ⊆ ((iEdg‘𝑆)‘(𝐻𝑘))) ∧ ∀𝑘 ∈ {𝑁}if-((𝑄𝑘) = (𝑄‘(𝑘 + 1)), ((iEdg‘𝑆)‘(𝐻𝑘)) = {(𝑄𝑘)}, {(𝑄𝑘), (𝑄‘(𝑘 + 1))} ⊆ ((iEdg‘𝑆)‘(𝐻𝑘))))))
959fvexi 6419 . . . . 5 𝑁 ∈ V
96 fveq2 6405 . . . . . . . 8 (𝑘 = 𝑁 → (𝑄𝑘) = (𝑄𝑁))
97 fvoveq1 6894 . . . . . . . 8 (𝑘 = 𝑁 → (𝑄‘(𝑘 + 1)) = (𝑄‘(𝑁 + 1)))
9896, 97eqeq12d 2820 . . . . . . 7 (𝑘 = 𝑁 → ((𝑄𝑘) = (𝑄‘(𝑘 + 1)) ↔ (𝑄𝑁) = (𝑄‘(𝑁 + 1))))
99 2fveq3 6410 . . . . . . . 8 (𝑘 = 𝑁 → ((iEdg‘𝑆)‘(𝐻𝑘)) = ((iEdg‘𝑆)‘(𝐻𝑁)))
10096sneqd 4379 . . . . . . . 8 (𝑘 = 𝑁 → {(𝑄𝑘)} = {(𝑄𝑁)})
10199, 100eqeq12d 2820 . . . . . . 7 (𝑘 = 𝑁 → (((iEdg‘𝑆)‘(𝐻𝑘)) = {(𝑄𝑘)} ↔ ((iEdg‘𝑆)‘(𝐻𝑁)) = {(𝑄𝑁)}))
10296, 97preq12d 4464 . . . . . . . 8 (𝑘 = 𝑁 → {(𝑄𝑘), (𝑄‘(𝑘 + 1))} = {(𝑄𝑁), (𝑄‘(𝑁 + 1))})
103102, 99sseq12d 3828 . . . . . . 7 (𝑘 = 𝑁 → ({(𝑄𝑘), (𝑄‘(𝑘 + 1))} ⊆ ((iEdg‘𝑆)‘(𝐻𝑘)) ↔ {(𝑄𝑁), (𝑄‘(𝑁 + 1))} ⊆ ((iEdg‘𝑆)‘(𝐻𝑁))))
10498, 101, 103ifpbi123d 1093 . . . . . 6 (𝑘 = 𝑁 → (if-((𝑄𝑘) = (𝑄‘(𝑘 + 1)), ((iEdg‘𝑆)‘(𝐻𝑘)) = {(𝑄𝑘)}, {(𝑄𝑘), (𝑄‘(𝑘 + 1))} ⊆ ((iEdg‘𝑆)‘(𝐻𝑘))) ↔ if-((𝑄𝑁) = (𝑄‘(𝑁 + 1)), ((iEdg‘𝑆)‘(𝐻𝑁)) = {(𝑄𝑁)}, {(𝑄𝑁), (𝑄‘(𝑁 + 1))} ⊆ ((iEdg‘𝑆)‘(𝐻𝑁)))))
105104ralsng 4408 . . . . 5 (𝑁 ∈ V → (∀𝑘 ∈ {𝑁}if-((𝑄𝑘) = (𝑄‘(𝑘 + 1)), ((iEdg‘𝑆)‘(𝐻𝑘)) = {(𝑄𝑘)}, {(𝑄𝑘), (𝑄‘(𝑘 + 1))} ⊆ ((iEdg‘𝑆)‘(𝐻𝑘))) ↔ if-((𝑄𝑁) = (𝑄‘(𝑁 + 1)), ((iEdg‘𝑆)‘(𝐻𝑁)) = {(𝑄𝑁)}, {(𝑄𝑁), (𝑄‘(𝑁 + 1))} ⊆ ((iEdg‘𝑆)‘(𝐻𝑁)))))
10695, 105mp1i 13 . . . 4 (𝜑 → (∀𝑘 ∈ {𝑁}if-((𝑄𝑘) = (𝑄‘(𝑘 + 1)), ((iEdg‘𝑆)‘(𝐻𝑘)) = {(𝑄𝑘)}, {(𝑄𝑘), (𝑄‘(𝑘 + 1))} ⊆ ((iEdg‘𝑆)‘(𝐻𝑘))) ↔ if-((𝑄𝑁) = (𝑄‘(𝑁 + 1)), ((iEdg‘𝑆)‘(𝐻𝑁)) = {(𝑄𝑁)}, {(𝑄𝑁), (𝑄‘(𝑁 + 1))} ⊆ ((iEdg‘𝑆)‘(𝐻𝑁)))))
107106anbi2d 616 . . 3 (𝜑 → ((∀𝑘 ∈ (0..^𝑁)if-((𝑄𝑘) = (𝑄‘(𝑘 + 1)), ((iEdg‘𝑆)‘(𝐻𝑘)) = {(𝑄𝑘)}, {(𝑄𝑘), (𝑄‘(𝑘 + 1))} ⊆ ((iEdg‘𝑆)‘(𝐻𝑘))) ∧ ∀𝑘 ∈ {𝑁}if-((𝑄𝑘) = (𝑄‘(𝑘 + 1)), ((iEdg‘𝑆)‘(𝐻𝑘)) = {(𝑄𝑘)}, {(𝑄𝑘), (𝑄‘(𝑘 + 1))} ⊆ ((iEdg‘𝑆)‘(𝐻𝑘)))) ↔ (∀𝑘 ∈ (0..^𝑁)if-((𝑄𝑘) = (𝑄‘(𝑘 + 1)), ((iEdg‘𝑆)‘(𝐻𝑘)) = {(𝑄𝑘)}, {(𝑄𝑘), (𝑄‘(𝑘 + 1))} ⊆ ((iEdg‘𝑆)‘(𝐻𝑘))) ∧ if-((𝑄𝑁) = (𝑄‘(𝑁 + 1)), ((iEdg‘𝑆)‘(𝐻𝑁)) = {(𝑄𝑁)}, {(𝑄𝑁), (𝑄‘(𝑁 + 1))} ⊆ ((iEdg‘𝑆)‘(𝐻𝑁))))))
10892, 94, 1073bitrd 296 . 2 (𝜑 → (∀𝑘 ∈ (0..^(♯‘𝐻))if-((𝑄𝑘) = (𝑄‘(𝑘 + 1)), ((iEdg‘𝑆)‘(𝐻𝑘)) = {(𝑄𝑘)}, {(𝑄𝑘), (𝑄‘(𝑘 + 1))} ⊆ ((iEdg‘𝑆)‘(𝐻𝑘))) ↔ (∀𝑘 ∈ (0..^𝑁)if-((𝑄𝑘) = (𝑄‘(𝑘 + 1)), ((iEdg‘𝑆)‘(𝐻𝑘)) = {(𝑄𝑘)}, {(𝑄𝑘), (𝑄‘(𝑘 + 1))} ⊆ ((iEdg‘𝑆)‘(𝐻𝑘))) ∧ if-((𝑄𝑁) = (𝑄‘(𝑁 + 1)), ((iEdg‘𝑆)‘(𝐻𝑁)) = {(𝑄𝑁)}, {(𝑄𝑁), (𝑄‘(𝑁 + 1))} ⊆ ((iEdg‘𝑆)‘(𝐻𝑁))))))
10939, 86, 108mpbir2and 695 1 (𝜑 → ∀𝑘 ∈ (0..^(♯‘𝐻))if-((𝑄𝑘) = (𝑄‘(𝑘 + 1)), ((iEdg‘𝑆)‘(𝐻𝑘)) = {(𝑄𝑘)}, {(𝑄𝑘), (𝑄‘(𝑘 + 1))} ⊆ ((iEdg‘𝑆)‘(𝐻𝑘))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384  if-wif 1078  w3a 1100   = wceq 1637  wcel 2158  wral 3095  Vcvv 3390  cun 3764  wss 3766  {csn 4367  {cpr 4369  cop 4373   class class class wbr 4840  dom cdm 5308  Fun wfun 6092  wf 6094  cfv 6098  (class class class)co 6871  Fincfn 8189  0cc0 10218  1c1 10219   + caddc 10221  0cn0 11555  cuz 11900  ...cfz 12545  ..^cfzo 12685  chash 13333  Word cword 13498  Vtxcvtx 26084  iEdgciedg 26085  Edgcedg 26149  Walkscwlks 26716
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1880  ax-4 1897  ax-5 2004  ax-6 2070  ax-7 2106  ax-8 2160  ax-9 2167  ax-10 2187  ax-11 2203  ax-12 2216  ax-13 2422  ax-ext 2784  ax-rep 4960  ax-sep 4971  ax-nul 4980  ax-pow 5032  ax-pr 5093  ax-un 7176  ax-cnex 10274  ax-resscn 10275  ax-1cn 10276  ax-icn 10277  ax-addcl 10278  ax-addrcl 10279  ax-mulcl 10280  ax-mulrcl 10281  ax-mulcom 10282  ax-addass 10283  ax-mulass 10284  ax-distr 10285  ax-i2m1 10286  ax-1ne0 10287  ax-1rid 10288  ax-rnegex 10289  ax-rrecex 10290  ax-cnre 10291  ax-pre-lttri 10292  ax-pre-lttrn 10293  ax-pre-ltadd 10294  ax-pre-mulgt0 10295
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-ifp 1079  df-3or 1101  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1865  df-sb 2063  df-eu 2636  df-mo 2637  df-clab 2792  df-cleq 2798  df-clel 2801  df-nfc 2936  df-ne 2978  df-nel 3081  df-ral 3100  df-rex 3101  df-reu 3102  df-rmo 3103  df-rab 3104  df-v 3392  df-sbc 3631  df-csb 3726  df-dif 3769  df-un 3771  df-in 3773  df-ss 3780  df-pss 3782  df-nul 4114  df-if 4277  df-pw 4350  df-sn 4368  df-pr 4370  df-tp 4372  df-op 4374  df-uni 4627  df-int 4666  df-iun 4710  df-br 4841  df-opab 4903  df-mpt 4920  df-tr 4943  df-id 5216  df-eprel 5221  df-po 5229  df-so 5230  df-fr 5267  df-we 5269  df-xp 5314  df-rel 5315  df-cnv 5316  df-co 5317  df-dm 5318  df-rn 5319  df-res 5320  df-ima 5321  df-pred 5890  df-ord 5936  df-on 5937  df-lim 5938  df-suc 5939  df-iota 6061  df-fun 6100  df-fn 6101  df-f 6102  df-f1 6103  df-fo 6104  df-f1o 6105  df-fv 6106  df-riota 6832  df-ov 6874  df-oprab 6875  df-mpt2 6876  df-om 7293  df-1st 7395  df-2nd 7396  df-wrecs 7639  df-recs 7701  df-rdg 7739  df-1o 7793  df-oadd 7797  df-er 7976  df-map 8091  df-pm 8092  df-en 8190  df-dom 8191  df-sdom 8192  df-fin 8193  df-card 9045  df-cda 9272  df-pnf 10358  df-mnf 10359  df-xr 10360  df-ltxr 10361  df-le 10362  df-sub 10550  df-neg 10551  df-nn 11303  df-n0 11556  df-z 11640  df-uz 11901  df-fz 12546  df-fzo 12686  df-hash 13334  df-word 13506  df-wlks 26719
This theorem is referenced by:  wlkp1  26802
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