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Theorem wlkp1lem8 27468
 Description: Lemma for wlkp1 27469. (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 (𝜑𝐵𝑊)
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)   𝑊(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 (𝜑𝐵𝑊)
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 27466 . . 3 (𝜑 → ∀𝑘 ∈ (0..^𝑁)((𝑄𝑘) = (𝑃𝑘) ∧ (𝑄‘(𝑘 + 1)) = (𝑃‘(𝑘 + 1)) ∧ ((iEdg‘𝑆)‘(𝐻𝑘)) = (𝐼‘(𝐹𝑘))))
1710elfvexd 6686 . . . . . 6 (𝜑𝐺 ∈ V)
181, 2iswlkg 27401 . . . . . 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 2831 . . . . . . . . 9 (♯‘𝐹) = 𝑁
2120oveq2i 7151 . . . . . . . 8 (0..^(♯‘𝐹)) = (0..^𝑁)
2221raleqi 3390 . . . . . . 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 219 . . . . . 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 1132 . . . . 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 256 . . . 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 2836 . . . . . . 7 (((𝑄𝑘) = (𝑃𝑘) ∧ (𝑄‘(𝑘 + 1)) = (𝑃‘(𝑘 + 1))) → ((𝑄𝑘) = (𝑄‘(𝑘 + 1)) ↔ (𝑃𝑘) = (𝑃‘(𝑘 + 1))))
28273adant3 1129 . . . . . 6 (((𝑄𝑘) = (𝑃𝑘) ∧ (𝑄‘(𝑘 + 1)) = (𝑃‘(𝑘 + 1)) ∧ ((iEdg‘𝑆)‘(𝐻𝑘)) = (𝐼‘(𝐹𝑘))) → ((𝑄𝑘) = (𝑄‘(𝑘 + 1)) ↔ (𝑃𝑘) = (𝑃‘(𝑘 + 1))))
29 simp3 1135 . . . . . . 7 (((𝑄𝑘) = (𝑃𝑘) ∧ (𝑄‘(𝑘 + 1)) = (𝑃‘(𝑘 + 1)) ∧ ((iEdg‘𝑆)‘(𝐻𝑘)) = (𝐼‘(𝐹𝑘))) → ((iEdg‘𝑆)‘(𝐻𝑘)) = (𝐼‘(𝐹𝑘)))
30 simp1 1133 . . . . . . . 8 (((𝑄𝑘) = (𝑃𝑘) ∧ (𝑄‘(𝑘 + 1)) = (𝑃‘(𝑘 + 1)) ∧ ((iEdg‘𝑆)‘(𝐻𝑘)) = (𝐼‘(𝐹𝑘))) → (𝑄𝑘) = (𝑃𝑘))
3130sneqd 4551 . . . . . . 7 (((𝑄𝑘) = (𝑃𝑘) ∧ (𝑄‘(𝑘 + 1)) = (𝑃‘(𝑘 + 1)) ∧ ((iEdg‘𝑆)‘(𝐻𝑘)) = (𝐼‘(𝐹𝑘))) → {(𝑄𝑘)} = {(𝑃𝑘)})
3229, 31eqeq12d 2838 . . . . . 6 (((𝑄𝑘) = (𝑃𝑘) ∧ (𝑄‘(𝑘 + 1)) = (𝑃‘(𝑘 + 1)) ∧ ((iEdg‘𝑆)‘(𝐻𝑘)) = (𝐼‘(𝐹𝑘))) → (((iEdg‘𝑆)‘(𝐻𝑘)) = {(𝑄𝑘)} ↔ (𝐼‘(𝐹𝑘)) = {(𝑃𝑘)}))
33 preq12 4645 . . . . . . . 8 (((𝑄𝑘) = (𝑃𝑘) ∧ (𝑄‘(𝑘 + 1)) = (𝑃‘(𝑘 + 1))) → {(𝑄𝑘), (𝑄‘(𝑘 + 1))} = {(𝑃𝑘), (𝑃‘(𝑘 + 1))})
34333adant3 1129 . . . . . . 7 (((𝑄𝑘) = (𝑃𝑘) ∧ (𝑄‘(𝑘 + 1)) = (𝑃‘(𝑘 + 1)) ∧ ((iEdg‘𝑆)‘(𝐻𝑘)) = (𝐼‘(𝐹𝑘))) → {(𝑄𝑘), (𝑄‘(𝑘 + 1))} = {(𝑃𝑘), (𝑃‘(𝑘 + 1))})
3534, 29sseq12d 3975 . . . . . 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 1075 . . . . 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 251 . . . 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 3148 . . 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 27463 . . . . 5 (𝜑 → ((iEdg‘𝑆)‘(𝐻𝑁)) = ((𝐼 ∪ {⟨𝐵, 𝐸⟩})‘𝐵))
4140adantr 484 . . . 4 ((𝜑 ∧ (𝑄𝑁) = (𝑄‘(𝑁 + 1))) → ((iEdg‘𝑆)‘(𝐻𝑁)) = ((𝐼 ∪ {⟨𝐵, 𝐸⟩})‘𝐵))
425, 10, 73jca 1125 . . . . . 6 (𝜑 → (𝐵𝑊𝐸 ∈ (Edg‘𝐺) ∧ ¬ 𝐵 ∈ dom 𝐼))
4342adantr 484 . . . . 5 ((𝜑 ∧ (𝑄𝑁) = (𝑄‘(𝑁 + 1))) → (𝐵𝑊𝐸 ∈ (Edg‘𝐺) ∧ ¬ 𝐵 ∈ dom 𝐼))
44 fsnunfv 6931 . . . . 5 ((𝐵𝑊𝐸 ∈ (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 6652 . . . . . . . 8 (𝑥 = 𝑁 → (𝑄𝑥) = (𝑄𝑁))
47 fveq2 6652 . . . . . . . 8 (𝑥 = 𝑁 → (𝑃𝑥) = (𝑃𝑁))
4846, 47eqeq12d 2838 . . . . . . 7 (𝑥 = 𝑁 → ((𝑄𝑥) = (𝑃𝑥) ↔ (𝑄𝑁) = (𝑃𝑁)))
491, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15wlkp1lem5 27465 . . . . . . 7 (𝜑 → ∀𝑥 ∈ (0...𝑁)(𝑄𝑥) = (𝑃𝑥))
502wlkf 27402 . . . . . . . . . 10 (𝐹(Walks‘𝐺)𝑃𝐹 ∈ Word dom 𝐼)
51 lencl 13876 . . . . . . . . . . 11 (𝐹 ∈ Word dom 𝐼 → (♯‘𝐹) ∈ ℕ0)
529eleq1i 2904 . . . . . . . . . . . 12 (𝑁 ∈ ℕ0 ↔ (♯‘𝐹) ∈ ℕ0)
53 elnn0uz 12271 . . . . . . . . . . . 12 (𝑁 ∈ ℕ0𝑁 ∈ (ℤ‘0))
5452, 53sylbb1 240 . . . . . . . . . . 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 237 . . . . . . . 8 (𝜑𝑁 ∈ ℕ0)
58 nn0fz0 13000 . . . . . . . 8 (𝑁 ∈ ℕ0𝑁 ∈ (0...𝑁))
5957, 58sylib 221 . . . . . . 7 (𝜑𝑁 ∈ (0...𝑁))
6048, 49, 59rspcdva 3600 . . . . . 6 (𝜑 → (𝑄𝑁) = (𝑃𝑁))
6114fveq1i 6653 . . . . . . . . . . 11 (𝑄‘(𝑁 + 1)) = ((𝑃 ∪ {⟨(𝑁 + 1), 𝐶⟩})‘(𝑁 + 1))
62 ovex 7173 . . . . . . . . . . . 12 (𝑁 + 1) ∈ V
631, 2, 3, 4, 5, 6, 7, 8, 9wlkp1lem1 27461 . . . . . . . . . . . 12 (𝜑 → ¬ (𝑁 + 1) ∈ dom 𝑃)
64 fsnunfv 6931 . . . . . . . . . . . 12 (((𝑁 + 1) ∈ V ∧ 𝐶𝑉 ∧ ¬ (𝑁 + 1) ∈ dom 𝑃) → ((𝑃 ∪ {⟨(𝑁 + 1), 𝐶⟩})‘(𝑁 + 1)) = 𝐶)
6562, 6, 63, 64mp3an2i 1463 . . . . . . . . . . 11 (𝜑 → ((𝑃 ∪ {⟨(𝑁 + 1), 𝐶⟩})‘(𝑁 + 1)) = 𝐶)
6661, 65syl5eq 2869 . . . . . . . . . 10 (𝜑 → (𝑄‘(𝑁 + 1)) = 𝐶)
6766eqeq2d 2833 . . . . . . . . 9 (𝜑 → ((𝑃𝑁) = (𝑄‘(𝑁 + 1)) ↔ (𝑃𝑁) = 𝐶))
68 eqcom 2829 . . . . . . . . 9 ((𝑃𝑁) = 𝐶𝐶 = (𝑃𝑁))
6967, 68syl6bb 290 . . . . . . . 8 (𝜑 → ((𝑃𝑁) = (𝑄‘(𝑁 + 1)) ↔ 𝐶 = (𝑃𝑁)))
70 wlkp1.l . . . . . . . . . 10 ((𝜑𝐶 = (𝑃𝑁)) → 𝐸 = {𝐶})
71 sneq 4549 . . . . . . . . . . 11 (𝐶 = (𝑃𝑁) → {𝐶} = {(𝑃𝑁)})
7271adantl 485 . . . . . . . . . 10 ((𝜑𝐶 = (𝑃𝑁)) → {𝐶} = {(𝑃𝑁)})
7370, 72eqtrd 2857 . . . . . . . . 9 ((𝜑𝐶 = (𝑃𝑁)) → 𝐸 = {(𝑃𝑁)})
7473ex 416 . . . . . . . 8 (𝜑 → (𝐶 = (𝑃𝑁) → 𝐸 = {(𝑃𝑁)}))
7569, 74sylbid 243 . . . . . . 7 (𝜑 → ((𝑃𝑁) = (𝑄‘(𝑁 + 1)) → 𝐸 = {(𝑃𝑁)}))
76 eqeq1 2826 . . . . . . . 8 ((𝑄𝑁) = (𝑃𝑁) → ((𝑄𝑁) = (𝑄‘(𝑁 + 1)) ↔ (𝑃𝑁) = (𝑄‘(𝑁 + 1))))
77 sneq 4549 . . . . . . . . 9 ((𝑄𝑁) = (𝑃𝑁) → {(𝑄𝑁)} = {(𝑃𝑁)})
7877eqeq2d 2833 . . . . . . . 8 ((𝑄𝑁) = (𝑃𝑁) → (𝐸 = {(𝑄𝑁)} ↔ 𝐸 = {(𝑃𝑁)}))
7976, 78imbi12d 348 . . . . . . 7 ((𝑄𝑁) = (𝑃𝑁) → (((𝑄𝑁) = (𝑄‘(𝑁 + 1)) → 𝐸 = {(𝑄𝑁)}) ↔ ((𝑃𝑁) = (𝑄‘(𝑁 + 1)) → 𝐸 = {(𝑃𝑁)})))
8075, 79syl5ibrcom 250 . . . . . 6 (𝜑 → ((𝑄𝑁) = (𝑃𝑁) → ((𝑄𝑁) = (𝑄‘(𝑁 + 1)) → 𝐸 = {(𝑄𝑁)})))
8160, 80mpd 15 . . . . 5 (𝜑 → ((𝑄𝑁) = (𝑄‘(𝑁 + 1)) → 𝐸 = {(𝑄𝑁)}))
8281imp 410 . . . 4 ((𝜑 ∧ (𝑄𝑁) = (𝑄‘(𝑁 + 1))) → 𝐸 = {(𝑄𝑁)})
8341, 45, 823eqtrd 2861 . . 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 27467 . . . 4 (𝜑 → {(𝑄𝑁), (𝑄‘(𝑁 + 1))} ⊆ ((iEdg‘𝑆)‘(𝐻𝑁)))
8584adantr 484 . . 3 ((𝜑 ∧ ¬ (𝑄𝑁) = (𝑄‘(𝑁 + 1))) → {(𝑄𝑁), (𝑄‘(𝑁 + 1))} ⊆ ((iEdg‘𝑆)‘(𝐻𝑁)))
8683, 85ifpimpda 1078 . 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 27462 . . . . . 6 (𝜑 → (♯‘𝐻) = (𝑁 + 1))
8887oveq2d 7156 . . . . 5 (𝜑 → (0..^(♯‘𝐻)) = (0..^(𝑁 + 1)))
89 fzosplitsn 13140 . . . . . 6 (𝑁 ∈ (ℤ‘0) → (0..^(𝑁 + 1)) = ((0..^𝑁) ∪ {𝑁}))
9056, 89syl 17 . . . . 5 (𝜑 → (0..^(𝑁 + 1)) = ((0..^𝑁) ∪ {𝑁}))
9188, 90eqtrd 2857 . . . 4 (𝜑 → (0..^(♯‘𝐻)) = ((0..^𝑁) ∪ {𝑁}))
9291raleqdv 3392 . . 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 4142 . . . 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 6666 . . . . 5 𝑁 ∈ V
96 wkslem1 27395 . . . . . 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)))))
9796ralsng 4587 . . . . 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)))))
9895, 97mp1i 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)))))
9998anbi2d 631 . . 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))))))
10092, 94, 993bitrd 308 . 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))))))
10139, 86, 100mpbir2and 712 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 209   ∧ wa 399  if-wif 1058   ∧ w3a 1084   = wceq 1538   ∈ wcel 2114  ∀wral 3130  Vcvv 3469   ∪ cun 3906   ⊆ wss 3908  {csn 4539  {cpr 4541  ⟨cop 4545   class class class wbr 5042  dom cdm 5532  Fun wfun 6328  ⟶wf 6330  ‘cfv 6334  (class class class)co 7140  Fincfn 8496  0cc0 10526  1c1 10527   + caddc 10529  ℕ0cn0 11885  ℤ≥cuz 12231  ...cfz 12885  ..^cfzo 13028  ♯chash 13686  Word cword 13857  Vtxcvtx 26787  iEdgciedg 26788  Edgcedg 26838  Walkscwlks 27384 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ifp 1059  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-nel 3116  df-ral 3135  df-rex 3136  df-reu 3137  df-rmo 3138  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-tp 4544  df-op 4546  df-uni 4814  df-int 4852  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-tr 5149  df-id 5437  df-eprel 5442  df-po 5451  df-so 5452  df-fr 5491  df-we 5493  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-pred 6126  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-riota 7098  df-ov 7143  df-oprab 7144  df-mpo 7145  df-om 7566  df-1st 7675  df-2nd 7676  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-oadd 8093  df-er 8276  df-map 8395  df-pm 8396  df-en 8497  df-dom 8498  df-sdom 8499  df-fin 8500  df-dju 9318  df-card 9356  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-nn 11626  df-n0 11886  df-z 11970  df-uz 12232  df-fz 12886  df-fzo 13029  df-hash 13687  df-word 13858  df-wlks 27387 This theorem is referenced by:  wlkp1  27469
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