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Theorem wlkp1lem8 27389
Description: Lemma for wlkp1 27390. (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 27387 . . 3 (𝜑 → ∀𝑘 ∈ (0..^𝑁)((𝑄𝑘) = (𝑃𝑘) ∧ (𝑄‘(𝑘 + 1)) = (𝑃‘(𝑘 + 1)) ∧ ((iEdg‘𝑆)‘(𝐻𝑘)) = (𝐼‘(𝐹𝑘))))
1710elfvexd 6697 . . . . . 6 (𝜑𝐺 ∈ V)
181, 2iswlkg 27322 . . . . . 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 2827 . . . . . . . . 9 (♯‘𝐹) = 𝑁
2120oveq2i 7156 . . . . . . . 8 (0..^(♯‘𝐹)) = (0..^𝑁)
2221raleqi 3411 . . . . . . 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 217 . . . . . 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 1127 . . . . 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 254 . . . 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 2832 . . . . . . 7 (((𝑄𝑘) = (𝑃𝑘) ∧ (𝑄‘(𝑘 + 1)) = (𝑃‘(𝑘 + 1))) → ((𝑄𝑘) = (𝑄‘(𝑘 + 1)) ↔ (𝑃𝑘) = (𝑃‘(𝑘 + 1))))
28273adant3 1124 . . . . . 6 (((𝑄𝑘) = (𝑃𝑘) ∧ (𝑄‘(𝑘 + 1)) = (𝑃‘(𝑘 + 1)) ∧ ((iEdg‘𝑆)‘(𝐻𝑘)) = (𝐼‘(𝐹𝑘))) → ((𝑄𝑘) = (𝑄‘(𝑘 + 1)) ↔ (𝑃𝑘) = (𝑃‘(𝑘 + 1))))
29 simp3 1130 . . . . . . 7 (((𝑄𝑘) = (𝑃𝑘) ∧ (𝑄‘(𝑘 + 1)) = (𝑃‘(𝑘 + 1)) ∧ ((iEdg‘𝑆)‘(𝐻𝑘)) = (𝐼‘(𝐹𝑘))) → ((iEdg‘𝑆)‘(𝐻𝑘)) = (𝐼‘(𝐹𝑘)))
30 simp1 1128 . . . . . . . 8 (((𝑄𝑘) = (𝑃𝑘) ∧ (𝑄‘(𝑘 + 1)) = (𝑃‘(𝑘 + 1)) ∧ ((iEdg‘𝑆)‘(𝐻𝑘)) = (𝐼‘(𝐹𝑘))) → (𝑄𝑘) = (𝑃𝑘))
3130sneqd 4569 . . . . . . 7 (((𝑄𝑘) = (𝑃𝑘) ∧ (𝑄‘(𝑘 + 1)) = (𝑃‘(𝑘 + 1)) ∧ ((iEdg‘𝑆)‘(𝐻𝑘)) = (𝐼‘(𝐹𝑘))) → {(𝑄𝑘)} = {(𝑃𝑘)})
3229, 31eqeq12d 2834 . . . . . 6 (((𝑄𝑘) = (𝑃𝑘) ∧ (𝑄‘(𝑘 + 1)) = (𝑃‘(𝑘 + 1)) ∧ ((iEdg‘𝑆)‘(𝐻𝑘)) = (𝐼‘(𝐹𝑘))) → (((iEdg‘𝑆)‘(𝐻𝑘)) = {(𝑄𝑘)} ↔ (𝐼‘(𝐹𝑘)) = {(𝑃𝑘)}))
33 preq12 4663 . . . . . . . 8 (((𝑄𝑘) = (𝑃𝑘) ∧ (𝑄‘(𝑘 + 1)) = (𝑃‘(𝑘 + 1))) → {(𝑄𝑘), (𝑄‘(𝑘 + 1))} = {(𝑃𝑘), (𝑃‘(𝑘 + 1))})
34333adant3 1124 . . . . . . 7 (((𝑄𝑘) = (𝑃𝑘) ∧ (𝑄‘(𝑘 + 1)) = (𝑃‘(𝑘 + 1)) ∧ ((iEdg‘𝑆)‘(𝐻𝑘)) = (𝐼‘(𝐹𝑘))) → {(𝑄𝑘), (𝑄‘(𝑘 + 1))} = {(𝑃𝑘), (𝑃‘(𝑘 + 1))})
3534, 29sseq12d 3997 . . . . . 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 1069 . . . . 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 249 . . . 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 3153 . . 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 27384 . . . . 5 (𝜑 → ((iEdg‘𝑆)‘(𝐻𝑁)) = ((𝐼 ∪ {⟨𝐵, 𝐸⟩})‘𝐵))
4140adantr 481 . . . 4 ((𝜑 ∧ (𝑄𝑁) = (𝑄‘(𝑁 + 1))) → ((iEdg‘𝑆)‘(𝐻𝑁)) = ((𝐼 ∪ {⟨𝐵, 𝐸⟩})‘𝐵))
425, 10, 73jca 1120 . . . . . 6 (𝜑 → (𝐵 ∈ V ∧ 𝐸 ∈ (Edg‘𝐺) ∧ ¬ 𝐵 ∈ dom 𝐼))
4342adantr 481 . . . . 5 ((𝜑 ∧ (𝑄𝑁) = (𝑄‘(𝑁 + 1))) → (𝐵 ∈ V ∧ 𝐸 ∈ (Edg‘𝐺) ∧ ¬ 𝐵 ∈ dom 𝐼))
44 fsnunfv 6941 . . . . 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 6663 . . . . . . . 8 (𝑥 = 𝑁 → (𝑄𝑥) = (𝑄𝑁))
47 fveq2 6663 . . . . . . . 8 (𝑥 = 𝑁 → (𝑃𝑥) = (𝑃𝑁))
4846, 47eqeq12d 2834 . . . . . . 7 (𝑥 = 𝑁 → ((𝑄𝑥) = (𝑃𝑥) ↔ (𝑄𝑁) = (𝑃𝑁)))
491, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15wlkp1lem5 27386 . . . . . . 7 (𝜑 → ∀𝑥 ∈ (0...𝑁)(𝑄𝑥) = (𝑃𝑥))
502wlkf 27323 . . . . . . . . . 10 (𝐹(Walks‘𝐺)𝑃𝐹 ∈ Word dom 𝐼)
51 lencl 13871 . . . . . . . . . . 11 (𝐹 ∈ Word dom 𝐼 → (♯‘𝐹) ∈ ℕ0)
529eleq1i 2900 . . . . . . . . . . . 12 (𝑁 ∈ ℕ0 ↔ (♯‘𝐹) ∈ ℕ0)
53 elnn0uz 12271 . . . . . . . . . . . 12 (𝑁 ∈ ℕ0𝑁 ∈ (ℤ‘0))
5452, 53sylbb1 238 . . . . . . . . . . 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 235 . . . . . . . 8 (𝜑𝑁 ∈ ℕ0)
58 nn0fz0 12993 . . . . . . . 8 (𝑁 ∈ ℕ0𝑁 ∈ (0...𝑁))
5957, 58sylib 219 . . . . . . 7 (𝜑𝑁 ∈ (0...𝑁))
6048, 49, 59rspcdva 3622 . . . . . 6 (𝜑 → (𝑄𝑁) = (𝑃𝑁))
6114fveq1i 6664 . . . . . . . . . . 11 (𝑄‘(𝑁 + 1)) = ((𝑃 ∪ {⟨(𝑁 + 1), 𝐶⟩})‘(𝑁 + 1))
62 ovex 7178 . . . . . . . . . . . 12 (𝑁 + 1) ∈ V
631, 2, 3, 4, 5, 6, 7, 8, 9wlkp1lem1 27382 . . . . . . . . . . . 12 (𝜑 → ¬ (𝑁 + 1) ∈ dom 𝑃)
64 fsnunfv 6941 . . . . . . . . . . . 12 (((𝑁 + 1) ∈ V ∧ 𝐶𝑉 ∧ ¬ (𝑁 + 1) ∈ dom 𝑃) → ((𝑃 ∪ {⟨(𝑁 + 1), 𝐶⟩})‘(𝑁 + 1)) = 𝐶)
6562, 6, 63, 64mp3an2i 1457 . . . . . . . . . . 11 (𝜑 → ((𝑃 ∪ {⟨(𝑁 + 1), 𝐶⟩})‘(𝑁 + 1)) = 𝐶)
6661, 65syl5eq 2865 . . . . . . . . . 10 (𝜑 → (𝑄‘(𝑁 + 1)) = 𝐶)
6766eqeq2d 2829 . . . . . . . . 9 (𝜑 → ((𝑃𝑁) = (𝑄‘(𝑁 + 1)) ↔ (𝑃𝑁) = 𝐶))
68 eqcom 2825 . . . . . . . . 9 ((𝑃𝑁) = 𝐶𝐶 = (𝑃𝑁))
6967, 68syl6bb 288 . . . . . . . 8 (𝜑 → ((𝑃𝑁) = (𝑄‘(𝑁 + 1)) ↔ 𝐶 = (𝑃𝑁)))
70 wlkp1.l . . . . . . . . . 10 ((𝜑𝐶 = (𝑃𝑁)) → 𝐸 = {𝐶})
71 sneq 4567 . . . . . . . . . . 11 (𝐶 = (𝑃𝑁) → {𝐶} = {(𝑃𝑁)})
7271adantl 482 . . . . . . . . . 10 ((𝜑𝐶 = (𝑃𝑁)) → {𝐶} = {(𝑃𝑁)})
7370, 72eqtrd 2853 . . . . . . . . 9 ((𝜑𝐶 = (𝑃𝑁)) → 𝐸 = {(𝑃𝑁)})
7473ex 413 . . . . . . . 8 (𝜑 → (𝐶 = (𝑃𝑁) → 𝐸 = {(𝑃𝑁)}))
7569, 74sylbid 241 . . . . . . 7 (𝜑 → ((𝑃𝑁) = (𝑄‘(𝑁 + 1)) → 𝐸 = {(𝑃𝑁)}))
76 eqeq1 2822 . . . . . . . 8 ((𝑄𝑁) = (𝑃𝑁) → ((𝑄𝑁) = (𝑄‘(𝑁 + 1)) ↔ (𝑃𝑁) = (𝑄‘(𝑁 + 1))))
77 sneq 4567 . . . . . . . . 9 ((𝑄𝑁) = (𝑃𝑁) → {(𝑄𝑁)} = {(𝑃𝑁)})
7877eqeq2d 2829 . . . . . . . 8 ((𝑄𝑁) = (𝑃𝑁) → (𝐸 = {(𝑄𝑁)} ↔ 𝐸 = {(𝑃𝑁)}))
7976, 78imbi12d 346 . . . . . . 7 ((𝑄𝑁) = (𝑃𝑁) → (((𝑄𝑁) = (𝑄‘(𝑁 + 1)) → 𝐸 = {(𝑄𝑁)}) ↔ ((𝑃𝑁) = (𝑄‘(𝑁 + 1)) → 𝐸 = {(𝑃𝑁)})))
8075, 79syl5ibrcom 248 . . . . . 6 (𝜑 → ((𝑄𝑁) = (𝑃𝑁) → ((𝑄𝑁) = (𝑄‘(𝑁 + 1)) → 𝐸 = {(𝑄𝑁)})))
8160, 80mpd 15 . . . . 5 (𝜑 → ((𝑄𝑁) = (𝑄‘(𝑁 + 1)) → 𝐸 = {(𝑄𝑁)}))
8281imp 407 . . . 4 ((𝜑 ∧ (𝑄𝑁) = (𝑄‘(𝑁 + 1))) → 𝐸 = {(𝑄𝑁)})
8341, 45, 823eqtrd 2857 . . 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 27388 . . . 4 (𝜑 → {(𝑄𝑁), (𝑄‘(𝑁 + 1))} ⊆ ((iEdg‘𝑆)‘(𝐻𝑁)))
8584adantr 481 . . 3 ((𝜑 ∧ ¬ (𝑄𝑁) = (𝑄‘(𝑁 + 1))) → {(𝑄𝑁), (𝑄‘(𝑁 + 1))} ⊆ ((iEdg‘𝑆)‘(𝐻𝑁)))
8683, 85ifpimpda 1070 . 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 27383 . . . . . 6 (𝜑 → (♯‘𝐻) = (𝑁 + 1))
8887oveq2d 7161 . . . . 5 (𝜑 → (0..^(♯‘𝐻)) = (0..^(𝑁 + 1)))
89 fzosplitsn 13133 . . . . . 6 (𝑁 ∈ (ℤ‘0) → (0..^(𝑁 + 1)) = ((0..^𝑁) ∪ {𝑁}))
9056, 89syl 17 . . . . 5 (𝜑 → (0..^(𝑁 + 1)) = ((0..^𝑁) ∪ {𝑁}))
9188, 90eqtrd 2853 . . . 4 (𝜑 → (0..^(♯‘𝐻)) = ((0..^𝑁) ∪ {𝑁}))
9291raleqdv 3413 . . 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 4164 . . . 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 6677 . . . . 5 𝑁 ∈ V
96 wkslem1 27316 . . . . . 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 4605 . . . . 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 628 . . 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 306 . 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 709 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 207  wa 396  if-wif 1054  w3a 1079   = wceq 1528  wcel 2105  wral 3135  Vcvv 3492  cun 3931  wss 3933  {csn 4557  {cpr 4559  cop 4563   class class class wbr 5057  dom cdm 5548  Fun wfun 6342  wf 6344  cfv 6348  (class class class)co 7145  Fincfn 8497  0cc0 10525  1c1 10526   + caddc 10528  0cn0 11885  cuz 12231  ...cfz 12880  ..^cfzo 13021  chash 13678  Word cword 13849  Vtxcvtx 26708  iEdgciedg 26709  Edgcedg 26759  Walkscwlks 27305
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-ifp 1055  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-oadd 8095  df-er 8278  df-map 8397  df-pm 8398  df-en 8498  df-dom 8499  df-sdom 8500  df-fin 8501  df-dju 9318  df-card 9356  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-nn 11627  df-n0 11886  df-z 11970  df-uz 12232  df-fz 12881  df-fzo 13022  df-hash 13679  df-word 13850  df-wlks 27308
This theorem is referenced by:  wlkp1  27390
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