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Theorem wlkp1lem8 26558
Description: Lemma for wlkp1 26559. (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 26556 . . 3 (𝜑 → ∀𝑘 ∈ (0..^𝑁)((𝑄𝑘) = (𝑃𝑘) ∧ (𝑄‘(𝑘 + 1)) = (𝑃‘(𝑘 + 1)) ∧ ((iEdg‘𝑆)‘(𝐻𝑘)) = (𝐼‘(𝐹𝑘))))
1710elfvexd 6209 . . . . . 6 (𝜑𝐺 ∈ V)
181, 2iswlkg 26490 . . . . . 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 2629 . . . . . . . . 9 (#‘𝐹) = 𝑁
2120oveq2i 6646 . . . . . . . 8 (0..^(#‘𝐹)) = (0..^𝑁)
2221raleqi 3137 . . . . . . 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 206 . . . . . 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 1082 . . . . 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 243 . . . 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 2633 . . . . . . 7 (((𝑄𝑘) = (𝑃𝑘) ∧ (𝑄‘(𝑘 + 1)) = (𝑃‘(𝑘 + 1))) → ((𝑄𝑘) = (𝑄‘(𝑘 + 1)) ↔ (𝑃𝑘) = (𝑃‘(𝑘 + 1))))
28273adant3 1079 . . . . . 6 (((𝑄𝑘) = (𝑃𝑘) ∧ (𝑄‘(𝑘 + 1)) = (𝑃‘(𝑘 + 1)) ∧ ((iEdg‘𝑆)‘(𝐻𝑘)) = (𝐼‘(𝐹𝑘))) → ((𝑄𝑘) = (𝑄‘(𝑘 + 1)) ↔ (𝑃𝑘) = (𝑃‘(𝑘 + 1))))
29 simp3 1061 . . . . . . 7 (((𝑄𝑘) = (𝑃𝑘) ∧ (𝑄‘(𝑘 + 1)) = (𝑃‘(𝑘 + 1)) ∧ ((iEdg‘𝑆)‘(𝐻𝑘)) = (𝐼‘(𝐹𝑘))) → ((iEdg‘𝑆)‘(𝐻𝑘)) = (𝐼‘(𝐹𝑘)))
30 simp1 1059 . . . . . . . 8 (((𝑄𝑘) = (𝑃𝑘) ∧ (𝑄‘(𝑘 + 1)) = (𝑃‘(𝑘 + 1)) ∧ ((iEdg‘𝑆)‘(𝐻𝑘)) = (𝐼‘(𝐹𝑘))) → (𝑄𝑘) = (𝑃𝑘))
3130sneqd 4180 . . . . . . 7 (((𝑄𝑘) = (𝑃𝑘) ∧ (𝑄‘(𝑘 + 1)) = (𝑃‘(𝑘 + 1)) ∧ ((iEdg‘𝑆)‘(𝐻𝑘)) = (𝐼‘(𝐹𝑘))) → {(𝑄𝑘)} = {(𝑃𝑘)})
3229, 31eqeq12d 2635 . . . . . 6 (((𝑄𝑘) = (𝑃𝑘) ∧ (𝑄‘(𝑘 + 1)) = (𝑃‘(𝑘 + 1)) ∧ ((iEdg‘𝑆)‘(𝐻𝑘)) = (𝐼‘(𝐹𝑘))) → (((iEdg‘𝑆)‘(𝐻𝑘)) = {(𝑄𝑘)} ↔ (𝐼‘(𝐹𝑘)) = {(𝑃𝑘)}))
33 preq12 4261 . . . . . . . 8 (((𝑄𝑘) = (𝑃𝑘) ∧ (𝑄‘(𝑘 + 1)) = (𝑃‘(𝑘 + 1))) → {(𝑄𝑘), (𝑄‘(𝑘 + 1))} = {(𝑃𝑘), (𝑃‘(𝑘 + 1))})
34333adant3 1079 . . . . . . 7 (((𝑄𝑘) = (𝑃𝑘) ∧ (𝑄‘(𝑘 + 1)) = (𝑃‘(𝑘 + 1)) ∧ ((iEdg‘𝑆)‘(𝐻𝑘)) = (𝐼‘(𝐹𝑘))) → {(𝑄𝑘), (𝑄‘(𝑘 + 1))} = {(𝑃𝑘), (𝑃‘(𝑘 + 1))})
3534, 29sseq12d 3626 . . . . . 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 1026 . . . . 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 238 . . . 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 2944 . . 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 26553 . . . . 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 1240 . . . . . 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 6438 . . . . 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 6178 . . . . . . . 8 (𝑥 = 𝑁 → (𝑄𝑥) = (𝑄𝑁))
47 fveq2 6178 . . . . . . . 8 (𝑥 = 𝑁 → (𝑃𝑥) = (𝑃𝑁))
4846, 47eqeq12d 2635 . . . . . . 7 (𝑥 = 𝑁 → ((𝑄𝑥) = (𝑃𝑥) ↔ (𝑄𝑁) = (𝑃𝑁)))
491, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15wlkp1lem5 26555 . . . . . . 7 (𝜑 → ∀𝑥 ∈ (0...𝑁)(𝑄𝑥) = (𝑃𝑥))
502wlkf 26491 . . . . . . . . . 10 (𝐹(Walks‘𝐺)𝑃𝐹 ∈ Word dom 𝐼)
51 lencl 13307 . . . . . . . . . . 11 (𝐹 ∈ Word dom 𝐼 → (#‘𝐹) ∈ ℕ0)
529eleq1i 2690 . . . . . . . . . . . 12 (𝑁 ∈ ℕ0 ↔ (#‘𝐹) ∈ ℕ0)
53 elnn0uz 11710 . . . . . . . . . . . 12 (𝑁 ∈ ℕ0𝑁 ∈ (ℤ‘0))
5452, 53sylbb1 227 . . . . . . . . . . 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 224 . . . . . . . 8 (𝜑𝑁 ∈ ℕ0)
58 nn0fz0 12421 . . . . . . . 8 (𝑁 ∈ ℕ0𝑁 ∈ (0...𝑁))
5957, 58sylib 208 . . . . . . 7 (𝜑𝑁 ∈ (0...𝑁))
6048, 49, 59rspcdva 3311 . . . . . 6 (𝜑 → (𝑄𝑁) = (𝑃𝑁))
6114fveq1i 6179 . . . . . . . . . . 11 (𝑄‘(𝑁 + 1)) = ((𝑃 ∪ {⟨(𝑁 + 1), 𝐶⟩})‘(𝑁 + 1))
62 ovex 6663 . . . . . . . . . . . 12 (𝑁 + 1) ∈ V
631, 2, 3, 4, 5, 6, 7, 8, 9wlkp1lem1 26551 . . . . . . . . . . . 12 (𝜑 → ¬ (𝑁 + 1) ∈ dom 𝑃)
64 fsnunfv 6438 . . . . . . . . . . . 12 (((𝑁 + 1) ∈ V ∧ 𝐶𝑉 ∧ ¬ (𝑁 + 1) ∈ dom 𝑃) → ((𝑃 ∪ {⟨(𝑁 + 1), 𝐶⟩})‘(𝑁 + 1)) = 𝐶)
6562, 6, 63, 64mp3an2i 1427 . . . . . . . . . . 11 (𝜑 → ((𝑃 ∪ {⟨(𝑁 + 1), 𝐶⟩})‘(𝑁 + 1)) = 𝐶)
6661, 65syl5eq 2666 . . . . . . . . . 10 (𝜑 → (𝑄‘(𝑁 + 1)) = 𝐶)
6766eqeq2d 2630 . . . . . . . . 9 (𝜑 → ((𝑃𝑁) = (𝑄‘(𝑁 + 1)) ↔ (𝑃𝑁) = 𝐶))
68 eqcom 2627 . . . . . . . . 9 ((𝑃𝑁) = 𝐶𝐶 = (𝑃𝑁))
6967, 68syl6bb 276 . . . . . . . 8 (𝜑 → ((𝑃𝑁) = (𝑄‘(𝑁 + 1)) ↔ 𝐶 = (𝑃𝑁)))
70 wlkp1.l . . . . . . . . . 10 ((𝜑𝐶 = (𝑃𝑁)) → 𝐸 = {𝐶})
71 sneq 4178 . . . . . . . . . . 11 (𝐶 = (𝑃𝑁) → {𝐶} = {(𝑃𝑁)})
7271adantl 482 . . . . . . . . . 10 ((𝜑𝐶 = (𝑃𝑁)) → {𝐶} = {(𝑃𝑁)})
7370, 72eqtrd 2654 . . . . . . . . 9 ((𝜑𝐶 = (𝑃𝑁)) → 𝐸 = {(𝑃𝑁)})
7473ex 450 . . . . . . . 8 (𝜑 → (𝐶 = (𝑃𝑁) → 𝐸 = {(𝑃𝑁)}))
7569, 74sylbid 230 . . . . . . 7 (𝜑 → ((𝑃𝑁) = (𝑄‘(𝑁 + 1)) → 𝐸 = {(𝑃𝑁)}))
76 eqeq1 2624 . . . . . . . 8 ((𝑄𝑁) = (𝑃𝑁) → ((𝑄𝑁) = (𝑄‘(𝑁 + 1)) ↔ (𝑃𝑁) = (𝑄‘(𝑁 + 1))))
77 sneq 4178 . . . . . . . . 9 ((𝑄𝑁) = (𝑃𝑁) → {(𝑄𝑁)} = {(𝑃𝑁)})
7877eqeq2d 2630 . . . . . . . 8 ((𝑄𝑁) = (𝑃𝑁) → (𝐸 = {(𝑄𝑁)} ↔ 𝐸 = {(𝑃𝑁)}))
7976, 78imbi12d 334 . . . . . . 7 ((𝑄𝑁) = (𝑃𝑁) → (((𝑄𝑁) = (𝑄‘(𝑁 + 1)) → 𝐸 = {(𝑄𝑁)}) ↔ ((𝑃𝑁) = (𝑄‘(𝑁 + 1)) → 𝐸 = {(𝑃𝑁)})))
8075, 79syl5ibrcom 237 . . . . . 6 (𝜑 → ((𝑄𝑁) = (𝑃𝑁) → ((𝑄𝑁) = (𝑄‘(𝑁 + 1)) → 𝐸 = {(𝑄𝑁)})))
8160, 80mpd 15 . . . . 5 (𝜑 → ((𝑄𝑁) = (𝑄‘(𝑁 + 1)) → 𝐸 = {(𝑄𝑁)}))
8281imp 445 . . . 4 ((𝜑 ∧ (𝑄𝑁) = (𝑄‘(𝑁 + 1))) → 𝐸 = {(𝑄𝑁)})
8341, 45, 823eqtrd 2658 . . 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 26557 . . . 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 1027 . 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 26552 . . . . . 6 (𝜑 → (#‘𝐻) = (𝑁 + 1))
8887oveq2d 6651 . . . . 5 (𝜑 → (0..^(#‘𝐻)) = (0..^(𝑁 + 1)))
89 fzosplitsn 12560 . . . . . 6 (𝑁 ∈ (ℤ‘0) → (0..^(𝑁 + 1)) = ((0..^𝑁) ∪ {𝑁}))
9056, 89syl 17 . . . . 5 (𝜑 → (0..^(𝑁 + 1)) = ((0..^𝑁) ∪ {𝑁}))
9188, 90eqtrd 2654 . . . 4 (𝜑 → (0..^(#‘𝐻)) = ((0..^𝑁) ∪ {𝑁}))
9291raleqdv 3139 . . 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 3786 . . . 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))))))
95 fvex 6188 . . . . . 6 (#‘𝐹) ∈ V
969, 95eqeltri 2695 . . . . 5 𝑁 ∈ V
97 fveq2 6178 . . . . . . . 8 (𝑘 = 𝑁 → (𝑄𝑘) = (𝑄𝑁))
98 oveq1 6642 . . . . . . . . 9 (𝑘 = 𝑁 → (𝑘 + 1) = (𝑁 + 1))
9998fveq2d 6182 . . . . . . . 8 (𝑘 = 𝑁 → (𝑄‘(𝑘 + 1)) = (𝑄‘(𝑁 + 1)))
10097, 99eqeq12d 2635 . . . . . . 7 (𝑘 = 𝑁 → ((𝑄𝑘) = (𝑄‘(𝑘 + 1)) ↔ (𝑄𝑁) = (𝑄‘(𝑁 + 1))))
101 fveq2 6178 . . . . . . . . 9 (𝑘 = 𝑁 → (𝐻𝑘) = (𝐻𝑁))
102101fveq2d 6182 . . . . . . . 8 (𝑘 = 𝑁 → ((iEdg‘𝑆)‘(𝐻𝑘)) = ((iEdg‘𝑆)‘(𝐻𝑁)))
10397sneqd 4180 . . . . . . . 8 (𝑘 = 𝑁 → {(𝑄𝑘)} = {(𝑄𝑁)})
104102, 103eqeq12d 2635 . . . . . . 7 (𝑘 = 𝑁 → (((iEdg‘𝑆)‘(𝐻𝑘)) = {(𝑄𝑘)} ↔ ((iEdg‘𝑆)‘(𝐻𝑁)) = {(𝑄𝑁)}))
10597, 99preq12d 4267 . . . . . . . 8 (𝑘 = 𝑁 → {(𝑄𝑘), (𝑄‘(𝑘 + 1))} = {(𝑄𝑁), (𝑄‘(𝑁 + 1))})
106105, 102sseq12d 3626 . . . . . . 7 (𝑘 = 𝑁 → ({(𝑄𝑘), (𝑄‘(𝑘 + 1))} ⊆ ((iEdg‘𝑆)‘(𝐻𝑘)) ↔ {(𝑄𝑁), (𝑄‘(𝑁 + 1))} ⊆ ((iEdg‘𝑆)‘(𝐻𝑁))))
107100, 104, 106ifpbi123d 1026 . . . . . 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)))))
108107ralsng 4209 . . . . 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)))))
10996, 108mp1i 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)))))
110109anbi2d 739 . . 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))))))
11192, 94, 1103bitrd 294 . 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))))))
11239, 86, 111mpbir2and 956 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 196  wa 384  if-wif 1011  w3a 1036   = wceq 1481  wcel 1988  wral 2909  Vcvv 3195  cun 3565  wss 3567  {csn 4168  {cpr 4170  cop 4174   class class class wbr 4644  dom cdm 5104  Fun wfun 5870  wf 5872  cfv 5876  (class class class)co 6635  Fincfn 7940  0cc0 9921  1c1 9922   + caddc 9924  0cn0 11277  cuz 11672  ...cfz 12311  ..^cfzo 12449  #chash 13100  Word cword 13274  Vtxcvtx 25855  iEdgciedg 25856  Edgcedg 25920  Walkscwlks 26473
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934  ax-cnex 9977  ax-resscn 9978  ax-1cn 9979  ax-icn 9980  ax-addcl 9981  ax-addrcl 9982  ax-mulcl 9983  ax-mulrcl 9984  ax-mulcom 9985  ax-addass 9986  ax-mulass 9987  ax-distr 9988  ax-i2m1 9989  ax-1ne0 9990  ax-1rid 9991  ax-rnegex 9992  ax-rrecex 9993  ax-cnre 9994  ax-pre-lttri 9995  ax-pre-lttrn 9996  ax-pre-ltadd 9997  ax-pre-mulgt0 9998
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ifp 1012  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-nel 2895  df-ral 2914  df-rex 2915  df-reu 2916  df-rmo 2917  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-int 4467  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-riota 6596  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-om 7051  df-1st 7153  df-2nd 7154  df-wrecs 7392  df-recs 7453  df-rdg 7491  df-1o 7545  df-oadd 7549  df-er 7727  df-map 7844  df-pm 7845  df-en 7941  df-dom 7942  df-sdom 7943  df-fin 7944  df-card 8750  df-cda 8975  df-pnf 10061  df-mnf 10062  df-xr 10063  df-ltxr 10064  df-le 10065  df-sub 10253  df-neg 10254  df-nn 11006  df-n0 11278  df-z 11363  df-uz 11673  df-fz 12312  df-fzo 12450  df-hash 13101  df-word 13282  df-wlks 26476
This theorem is referenced by:  wlkp1  26559
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