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Theorem wlkp1lem8 26809
Description: Lemma for wlkp1 26810. (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 26807 . . 3 (𝜑 → ∀𝑘 ∈ (0..^𝑁)((𝑄𝑘) = (𝑃𝑘) ∧ (𝑄‘(𝑘 + 1)) = (𝑃‘(𝑘 + 1)) ∧ ((iEdg‘𝑆)‘(𝐻𝑘)) = (𝐼‘(𝐹𝑘))))
1710elfvexd 6385 . . . . . 6 (𝜑𝐺 ∈ V)
181, 2iswlkg 26741 . . . . . 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 2770 . . . . . . . . 9 (♯‘𝐹) = 𝑁
2120oveq2i 6826 . . . . . . . 8 (0..^(♯‘𝐹)) = (0..^𝑁)
2221raleqi 3282 . . . . . . 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 1130 . . . . 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 2774 . . . . . . 7 (((𝑄𝑘) = (𝑃𝑘) ∧ (𝑄‘(𝑘 + 1)) = (𝑃‘(𝑘 + 1))) → ((𝑄𝑘) = (𝑄‘(𝑘 + 1)) ↔ (𝑃𝑘) = (𝑃‘(𝑘 + 1))))
28273adant3 1127 . . . . . 6 (((𝑄𝑘) = (𝑃𝑘) ∧ (𝑄‘(𝑘 + 1)) = (𝑃‘(𝑘 + 1)) ∧ ((iEdg‘𝑆)‘(𝐻𝑘)) = (𝐼‘(𝐹𝑘))) → ((𝑄𝑘) = (𝑄‘(𝑘 + 1)) ↔ (𝑃𝑘) = (𝑃‘(𝑘 + 1))))
29 simp3 1133 . . . . . . 7 (((𝑄𝑘) = (𝑃𝑘) ∧ (𝑄‘(𝑘 + 1)) = (𝑃‘(𝑘 + 1)) ∧ ((iEdg‘𝑆)‘(𝐻𝑘)) = (𝐼‘(𝐹𝑘))) → ((iEdg‘𝑆)‘(𝐻𝑘)) = (𝐼‘(𝐹𝑘)))
30 simp1 1131 . . . . . . . 8 (((𝑄𝑘) = (𝑃𝑘) ∧ (𝑄‘(𝑘 + 1)) = (𝑃‘(𝑘 + 1)) ∧ ((iEdg‘𝑆)‘(𝐻𝑘)) = (𝐼‘(𝐹𝑘))) → (𝑄𝑘) = (𝑃𝑘))
3130sneqd 4334 . . . . . . 7 (((𝑄𝑘) = (𝑃𝑘) ∧ (𝑄‘(𝑘 + 1)) = (𝑃‘(𝑘 + 1)) ∧ ((iEdg‘𝑆)‘(𝐻𝑘)) = (𝐼‘(𝐹𝑘))) → {(𝑄𝑘)} = {(𝑃𝑘)})
3229, 31eqeq12d 2776 . . . . . 6 (((𝑄𝑘) = (𝑃𝑘) ∧ (𝑄‘(𝑘 + 1)) = (𝑃‘(𝑘 + 1)) ∧ ((iEdg‘𝑆)‘(𝐻𝑘)) = (𝐼‘(𝐹𝑘))) → (((iEdg‘𝑆)‘(𝐻𝑘)) = {(𝑄𝑘)} ↔ (𝐼‘(𝐹𝑘)) = {(𝑃𝑘)}))
33 preq12 4415 . . . . . . . 8 (((𝑄𝑘) = (𝑃𝑘) ∧ (𝑄‘(𝑘 + 1)) = (𝑃‘(𝑘 + 1))) → {(𝑄𝑘), (𝑄‘(𝑘 + 1))} = {(𝑃𝑘), (𝑃‘(𝑘 + 1))})
34333adant3 1127 . . . . . . 7 (((𝑄𝑘) = (𝑃𝑘) ∧ (𝑄‘(𝑘 + 1)) = (𝑃‘(𝑘 + 1)) ∧ ((iEdg‘𝑆)‘(𝐻𝑘)) = (𝐼‘(𝐹𝑘))) → {(𝑄𝑘), (𝑄‘(𝑘 + 1))} = {(𝑃𝑘), (𝑃‘(𝑘 + 1))})
3534, 29sseq12d 3776 . . . . . 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 1065 . . . . 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 3086 . . 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 26804 . . . . 5 (𝜑 → ((iEdg‘𝑆)‘(𝐻𝑁)) = ((𝐼 ∪ {⟨𝐵, 𝐸⟩})‘𝐵))
4140adantr 472 . . . 4 ((𝜑 ∧ (𝑄𝑁) = (𝑄‘(𝑁 + 1))) → ((iEdg‘𝑆)‘(𝐻𝑁)) = ((𝐼 ∪ {⟨𝐵, 𝐸⟩})‘𝐵))
425, 10, 73jca 1123 . . . . . 6 (𝜑 → (𝐵 ∈ V ∧ 𝐸 ∈ (Edg‘𝐺) ∧ ¬ 𝐵 ∈ dom 𝐼))
4342adantr 472 . . . . 5 ((𝜑 ∧ (𝑄𝑁) = (𝑄‘(𝑁 + 1))) → (𝐵 ∈ V ∧ 𝐸 ∈ (Edg‘𝐺) ∧ ¬ 𝐵 ∈ dom 𝐼))
44 fsnunfv 6619 . . . . 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 6354 . . . . . . . 8 (𝑥 = 𝑁 → (𝑄𝑥) = (𝑄𝑁))
47 fveq2 6354 . . . . . . . 8 (𝑥 = 𝑁 → (𝑃𝑥) = (𝑃𝑁))
4846, 47eqeq12d 2776 . . . . . . 7 (𝑥 = 𝑁 → ((𝑄𝑥) = (𝑃𝑥) ↔ (𝑄𝑁) = (𝑃𝑁)))
491, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15wlkp1lem5 26806 . . . . . . 7 (𝜑 → ∀𝑥 ∈ (0...𝑁)(𝑄𝑥) = (𝑃𝑥))
502wlkf 26742 . . . . . . . . . 10 (𝐹(Walks‘𝐺)𝑃𝐹 ∈ Word dom 𝐼)
51 lencl 13531 . . . . . . . . . . 11 (𝐹 ∈ Word dom 𝐼 → (♯‘𝐹) ∈ ℕ0)
529eleq1i 2831 . . . . . . . . . . . 12 (𝑁 ∈ ℕ0 ↔ (♯‘𝐹) ∈ ℕ0)
53 elnn0uz 11939 . . . . . . . . . . . 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 12652 . . . . . . . 8 (𝑁 ∈ ℕ0𝑁 ∈ (0...𝑁))
5957, 58sylib 208 . . . . . . 7 (𝜑𝑁 ∈ (0...𝑁))
6048, 49, 59rspcdva 3456 . . . . . 6 (𝜑 → (𝑄𝑁) = (𝑃𝑁))
6114fveq1i 6355 . . . . . . . . . . 11 (𝑄‘(𝑁 + 1)) = ((𝑃 ∪ {⟨(𝑁 + 1), 𝐶⟩})‘(𝑁 + 1))
62 ovex 6843 . . . . . . . . . . . 12 (𝑁 + 1) ∈ V
631, 2, 3, 4, 5, 6, 7, 8, 9wlkp1lem1 26802 . . . . . . . . . . . 12 (𝜑 → ¬ (𝑁 + 1) ∈ dom 𝑃)
64 fsnunfv 6619 . . . . . . . . . . . 12 (((𝑁 + 1) ∈ V ∧ 𝐶𝑉 ∧ ¬ (𝑁 + 1) ∈ dom 𝑃) → ((𝑃 ∪ {⟨(𝑁 + 1), 𝐶⟩})‘(𝑁 + 1)) = 𝐶)
6562, 6, 63, 64mp3an2i 1578 . . . . . . . . . . 11 (𝜑 → ((𝑃 ∪ {⟨(𝑁 + 1), 𝐶⟩})‘(𝑁 + 1)) = 𝐶)
6661, 65syl5eq 2807 . . . . . . . . . 10 (𝜑 → (𝑄‘(𝑁 + 1)) = 𝐶)
6766eqeq2d 2771 . . . . . . . . 9 (𝜑 → ((𝑃𝑁) = (𝑄‘(𝑁 + 1)) ↔ (𝑃𝑁) = 𝐶))
68 eqcom 2768 . . . . . . . . 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 4332 . . . . . . . . . . 11 (𝐶 = (𝑃𝑁) → {𝐶} = {(𝑃𝑁)})
7271adantl 473 . . . . . . . . . 10 ((𝜑𝐶 = (𝑃𝑁)) → {𝐶} = {(𝑃𝑁)})
7370, 72eqtrd 2795 . . . . . . . . 9 ((𝜑𝐶 = (𝑃𝑁)) → 𝐸 = {(𝑃𝑁)})
7473ex 449 . . . . . . . 8 (𝜑 → (𝐶 = (𝑃𝑁) → 𝐸 = {(𝑃𝑁)}))
7569, 74sylbid 230 . . . . . . 7 (𝜑 → ((𝑃𝑁) = (𝑄‘(𝑁 + 1)) → 𝐸 = {(𝑃𝑁)}))
76 eqeq1 2765 . . . . . . . 8 ((𝑄𝑁) = (𝑃𝑁) → ((𝑄𝑁) = (𝑄‘(𝑁 + 1)) ↔ (𝑃𝑁) = (𝑄‘(𝑁 + 1))))
77 sneq 4332 . . . . . . . . 9 ((𝑄𝑁) = (𝑃𝑁) → {(𝑄𝑁)} = {(𝑃𝑁)})
7877eqeq2d 2771 . . . . . . . 8 ((𝑄𝑁) = (𝑃𝑁) → (𝐸 = {(𝑄𝑁)} ↔ 𝐸 = {(𝑃𝑁)}))
7976, 78imbi12d 333 . . . . . . 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 444 . . . 4 ((𝜑 ∧ (𝑄𝑁) = (𝑄‘(𝑁 + 1))) → 𝐸 = {(𝑄𝑁)})
8341, 45, 823eqtrd 2799 . . 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 26808 . . . 4 (𝜑 → {(𝑄𝑁), (𝑄‘(𝑁 + 1))} ⊆ ((iEdg‘𝑆)‘(𝐻𝑁)))
8584adantr 472 . . 3 ((𝜑 ∧ ¬ (𝑄𝑁) = (𝑄‘(𝑁 + 1))) → {(𝑄𝑁), (𝑄‘(𝑁 + 1))} ⊆ ((iEdg‘𝑆)‘(𝐻𝑁)))
8683, 85ifpimpda 1066 . 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 26803 . . . . . 6 (𝜑 → (♯‘𝐻) = (𝑁 + 1))
8887oveq2d 6831 . . . . 5 (𝜑 → (0..^(♯‘𝐻)) = (0..^(𝑁 + 1)))
89 fzosplitsn 12791 . . . . . 6 (𝑁 ∈ (ℤ‘0) → (0..^(𝑁 + 1)) = ((0..^𝑁) ∪ {𝑁}))
9056, 89syl 17 . . . . 5 (𝜑 → (0..^(𝑁 + 1)) = ((0..^𝑁) ∪ {𝑁}))
9188, 90eqtrd 2795 . . . 4 (𝜑 → (0..^(♯‘𝐻)) = ((0..^𝑁) ∪ {𝑁}))
9291raleqdv 3284 . . 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 3938 . . . 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 6364 . . . . . 6 (♯‘𝐹) ∈ V
969, 95eqeltri 2836 . . . . 5 𝑁 ∈ V
97 fveq2 6354 . . . . . . . 8 (𝑘 = 𝑁 → (𝑄𝑘) = (𝑄𝑁))
98 oveq1 6822 . . . . . . . . 9 (𝑘 = 𝑁 → (𝑘 + 1) = (𝑁 + 1))
9998fveq2d 6358 . . . . . . . 8 (𝑘 = 𝑁 → (𝑄‘(𝑘 + 1)) = (𝑄‘(𝑁 + 1)))
10097, 99eqeq12d 2776 . . . . . . 7 (𝑘 = 𝑁 → ((𝑄𝑘) = (𝑄‘(𝑘 + 1)) ↔ (𝑄𝑁) = (𝑄‘(𝑁 + 1))))
101 fveq2 6354 . . . . . . . . 9 (𝑘 = 𝑁 → (𝐻𝑘) = (𝐻𝑁))
102101fveq2d 6358 . . . . . . . 8 (𝑘 = 𝑁 → ((iEdg‘𝑆)‘(𝐻𝑘)) = ((iEdg‘𝑆)‘(𝐻𝑁)))
10397sneqd 4334 . . . . . . . 8 (𝑘 = 𝑁 → {(𝑄𝑘)} = {(𝑄𝑁)})
104102, 103eqeq12d 2776 . . . . . . 7 (𝑘 = 𝑁 → (((iEdg‘𝑆)‘(𝐻𝑘)) = {(𝑄𝑘)} ↔ ((iEdg‘𝑆)‘(𝐻𝑁)) = {(𝑄𝑁)}))
10597, 99preq12d 4421 . . . . . . . 8 (𝑘 = 𝑁 → {(𝑄𝑘), (𝑄‘(𝑘 + 1))} = {(𝑄𝑁), (𝑄‘(𝑁 + 1))})
106105, 102sseq12d 3776 . . . . . . 7 (𝑘 = 𝑁 → ({(𝑄𝑘), (𝑄‘(𝑘 + 1))} ⊆ ((iEdg‘𝑆)‘(𝐻𝑘)) ↔ {(𝑄𝑁), (𝑄‘(𝑁 + 1))} ⊆ ((iEdg‘𝑆)‘(𝐻𝑁))))
107100, 104, 106ifpbi123d 1065 . . . . . 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 4363 . . . . 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 742 . . 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 995 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 383  if-wif 1050  w3a 1072   = wceq 1632  wcel 2140  wral 3051  Vcvv 3341  cun 3714  wss 3716  {csn 4322  {cpr 4324  cop 4328   class class class wbr 4805  dom cdm 5267  Fun wfun 6044  wf 6046  cfv 6050  (class class class)co 6815  Fincfn 8124  0cc0 10149  1c1 10150   + caddc 10152  0cn0 11505  cuz 11900  ...cfz 12540  ..^cfzo 12680  chash 13332  Word cword 13498  Vtxcvtx 26095  iEdgciedg 26096  Edgcedg 26160  Walkscwlks 26724
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-8 2142  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-rep 4924  ax-sep 4934  ax-nul 4942  ax-pow 4993  ax-pr 5056  ax-un 7116  ax-cnex 10205  ax-resscn 10206  ax-1cn 10207  ax-icn 10208  ax-addcl 10209  ax-addrcl 10210  ax-mulcl 10211  ax-mulrcl 10212  ax-mulcom 10213  ax-addass 10214  ax-mulass 10215  ax-distr 10216  ax-i2m1 10217  ax-1ne0 10218  ax-1rid 10219  ax-rnegex 10220  ax-rrecex 10221  ax-cnre 10222  ax-pre-lttri 10223  ax-pre-lttrn 10224  ax-pre-ltadd 10225  ax-pre-mulgt0 10226
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-ifp 1051  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ne 2934  df-nel 3037  df-ral 3056  df-rex 3057  df-reu 3058  df-rmo 3059  df-rab 3060  df-v 3343  df-sbc 3578  df-csb 3676  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-pss 3732  df-nul 4060  df-if 4232  df-pw 4305  df-sn 4323  df-pr 4325  df-tp 4327  df-op 4329  df-uni 4590  df-int 4629  df-iun 4675  df-br 4806  df-opab 4866  df-mpt 4883  df-tr 4906  df-id 5175  df-eprel 5180  df-po 5188  df-so 5189  df-fr 5226  df-we 5228  df-xp 5273  df-rel 5274  df-cnv 5275  df-co 5276  df-dm 5277  df-rn 5278  df-res 5279  df-ima 5280  df-pred 5842  df-ord 5888  df-on 5889  df-lim 5890  df-suc 5891  df-iota 6013  df-fun 6052  df-fn 6053  df-f 6054  df-f1 6055  df-fo 6056  df-f1o 6057  df-fv 6058  df-riota 6776  df-ov 6818  df-oprab 6819  df-mpt2 6820  df-om 7233  df-1st 7335  df-2nd 7336  df-wrecs 7578  df-recs 7639  df-rdg 7677  df-1o 7731  df-oadd 7735  df-er 7914  df-map 8028  df-pm 8029  df-en 8125  df-dom 8126  df-sdom 8127  df-fin 8128  df-card 8976  df-cda 9203  df-pnf 10289  df-mnf 10290  df-xr 10291  df-ltxr 10292  df-le 10293  df-sub 10481  df-neg 10482  df-nn 11234  df-n0 11506  df-z 11591  df-uz 11901  df-fz 12541  df-fzo 12681  df-hash 13333  df-word 13506  df-wlks 26727
This theorem is referenced by:  wlkp1  26810
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