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Theorem cvmliftlem8 33886
Description: Lemma for cvmlift 33893. The functions 𝑄 are continuous functions because they are defined as (𝐹𝐼) ∘ 𝐺 where 𝐺 is continuous and (𝐹𝐼) is a homeomorphism. (Contributed by Mario Carneiro, 16-Feb-2015.)
Hypotheses
Ref Expression
cvmliftlem.1 𝑆 = (𝑘𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑠 = (𝐹𝑘) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘))))})
cvmliftlem.b 𝐵 = 𝐶
cvmliftlem.x 𝑋 = 𝐽
cvmliftlem.f (𝜑𝐹 ∈ (𝐶 CovMap 𝐽))
cvmliftlem.g (𝜑𝐺 ∈ (II Cn 𝐽))
cvmliftlem.p (𝜑𝑃𝐵)
cvmliftlem.e (𝜑 → (𝐹𝑃) = (𝐺‘0))
cvmliftlem.n (𝜑𝑁 ∈ ℕ)
cvmliftlem.t (𝜑𝑇:(1...𝑁)⟶ 𝑗𝐽 ({𝑗} × (𝑆𝑗)))
cvmliftlem.a (𝜑 → ∀𝑘 ∈ (1...𝑁)(𝐺 “ (((𝑘 − 1) / 𝑁)[,](𝑘 / 𝑁))) ⊆ (1st ‘(𝑇𝑘)))
cvmliftlem.l 𝐿 = (topGen‘ran (,))
cvmliftlem.q 𝑄 = seq0((𝑥 ∈ V, 𝑚 ∈ ℕ ↦ (𝑧 ∈ (((𝑚 − 1) / 𝑁)[,](𝑚 / 𝑁)) ↦ ((𝐹 ↾ (𝑏 ∈ (2nd ‘(𝑇𝑚))(𝑥‘((𝑚 − 1) / 𝑁)) ∈ 𝑏))‘(𝐺𝑧)))), (( I ↾ ℕ) ∪ {⟨0, {⟨0, 𝑃⟩}⟩}))
cvmliftlem5.3 𝑊 = (((𝑀 − 1) / 𝑁)[,](𝑀 / 𝑁))
Assertion
Ref Expression
cvmliftlem8 ((𝜑𝑀 ∈ (1...𝑁)) → (𝑄𝑀) ∈ ((𝐿t 𝑊) Cn 𝐶))
Distinct variable groups:   𝑣,𝑏,𝑧,𝐵   𝑗,𝑏,𝑘,𝑚,𝑠,𝑢,𝑥,𝐹,𝑣,𝑧   𝑧,𝐿   𝑀,𝑏,𝑗,𝑘,𝑚,𝑠,𝑢,𝑣,𝑥,𝑧   𝑃,𝑏,𝑘,𝑚,𝑢,𝑣,𝑥,𝑧   𝐶,𝑏,𝑗,𝑘,𝑠,𝑢,𝑣,𝑧   𝜑,𝑗,𝑠,𝑥,𝑧   𝑁,𝑏,𝑘,𝑚,𝑢,𝑣,𝑥,𝑧   𝑆,𝑏,𝑗,𝑘,𝑠,𝑢,𝑣,𝑥,𝑧   𝑗,𝑋   𝐺,𝑏,𝑗,𝑘,𝑚,𝑠,𝑢,𝑣,𝑥,𝑧   𝑇,𝑏,𝑗,𝑘,𝑚,𝑠,𝑢,𝑣,𝑥,𝑧   𝐽,𝑏,𝑗,𝑘,𝑠,𝑢,𝑣,𝑥,𝑧   𝑄,𝑏,𝑘,𝑚,𝑢,𝑣,𝑥,𝑧   𝑘,𝑊,𝑚,𝑥,𝑧
Allowed substitution hints:   𝜑(𝑣,𝑢,𝑘,𝑚,𝑏)   𝐵(𝑥,𝑢,𝑗,𝑘,𝑚,𝑠)   𝐶(𝑥,𝑚)   𝑃(𝑗,𝑠)   𝑄(𝑗,𝑠)   𝑆(𝑚)   𝐽(𝑚)   𝐿(𝑥,𝑣,𝑢,𝑗,𝑘,𝑚,𝑠,𝑏)   𝑁(𝑗,𝑠)   𝑊(𝑣,𝑢,𝑗,𝑠,𝑏)   𝑋(𝑥,𝑧,𝑣,𝑢,𝑘,𝑚,𝑠,𝑏)

Proof of Theorem cvmliftlem8
StepHypRef Expression
1 elfznn 13470 . . 3 (𝑀 ∈ (1...𝑁) → 𝑀 ∈ ℕ)
2 cvmliftlem.1 . . . 4 𝑆 = (𝑘𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑠 = (𝐹𝑘) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘))))})
3 cvmliftlem.b . . . 4 𝐵 = 𝐶
4 cvmliftlem.x . . . 4 𝑋 = 𝐽
5 cvmliftlem.f . . . 4 (𝜑𝐹 ∈ (𝐶 CovMap 𝐽))
6 cvmliftlem.g . . . 4 (𝜑𝐺 ∈ (II Cn 𝐽))
7 cvmliftlem.p . . . 4 (𝜑𝑃𝐵)
8 cvmliftlem.e . . . 4 (𝜑 → (𝐹𝑃) = (𝐺‘0))
9 cvmliftlem.n . . . 4 (𝜑𝑁 ∈ ℕ)
10 cvmliftlem.t . . . 4 (𝜑𝑇:(1...𝑁)⟶ 𝑗𝐽 ({𝑗} × (𝑆𝑗)))
11 cvmliftlem.a . . . 4 (𝜑 → ∀𝑘 ∈ (1...𝑁)(𝐺 “ (((𝑘 − 1) / 𝑁)[,](𝑘 / 𝑁))) ⊆ (1st ‘(𝑇𝑘)))
12 cvmliftlem.l . . . 4 𝐿 = (topGen‘ran (,))
13 cvmliftlem.q . . . 4 𝑄 = seq0((𝑥 ∈ V, 𝑚 ∈ ℕ ↦ (𝑧 ∈ (((𝑚 − 1) / 𝑁)[,](𝑚 / 𝑁)) ↦ ((𝐹 ↾ (𝑏 ∈ (2nd ‘(𝑇𝑚))(𝑥‘((𝑚 − 1) / 𝑁)) ∈ 𝑏))‘(𝐺𝑧)))), (( I ↾ ℕ) ∪ {⟨0, {⟨0, 𝑃⟩}⟩}))
14 cvmliftlem5.3 . . . 4 𝑊 = (((𝑀 − 1) / 𝑁)[,](𝑀 / 𝑁))
152, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14cvmliftlem5 33883 . . 3 ((𝜑𝑀 ∈ ℕ) → (𝑄𝑀) = (𝑧𝑊 ↦ ((𝐹 ↾ (𝑏 ∈ (2nd ‘(𝑇𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏))‘(𝐺𝑧))))
161, 15sylan2 593 . 2 ((𝜑𝑀 ∈ (1...𝑁)) → (𝑄𝑀) = (𝑧𝑊 ↦ ((𝐹 ↾ (𝑏 ∈ (2nd ‘(𝑇𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏))‘(𝐺𝑧))))
175adantr 481 . . . 4 ((𝜑𝑀 ∈ (1...𝑁)) → 𝐹 ∈ (𝐶 CovMap 𝐽))
18 cvmtop1 33854 . . . 4 (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐶 ∈ Top)
19 cnrest2r 22638 . . . 4 (𝐶 ∈ Top → ((𝐿t 𝑊) Cn (𝐶t (𝑏 ∈ (2nd ‘(𝑇𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏))) ⊆ ((𝐿t 𝑊) Cn 𝐶))
2017, 18, 193syl 18 . . 3 ((𝜑𝑀 ∈ (1...𝑁)) → ((𝐿t 𝑊) Cn (𝐶t (𝑏 ∈ (2nd ‘(𝑇𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏))) ⊆ ((𝐿t 𝑊) Cn 𝐶))
21 retopon 24127 . . . . . 6 (topGen‘ran (,)) ∈ (TopOn‘ℝ)
2212, 21eqeltri 2834 . . . . 5 𝐿 ∈ (TopOn‘ℝ)
23 simpr 485 . . . . . . 7 ((𝜑𝑀 ∈ (1...𝑁)) → 𝑀 ∈ (1...𝑁))
242, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 23, 14cvmliftlem2 33880 . . . . . 6 ((𝜑𝑀 ∈ (1...𝑁)) → 𝑊 ⊆ (0[,]1))
25 unitssre 13416 . . . . . 6 (0[,]1) ⊆ ℝ
2624, 25sstrdi 3956 . . . . 5 ((𝜑𝑀 ∈ (1...𝑁)) → 𝑊 ⊆ ℝ)
27 resttopon 22512 . . . . 5 ((𝐿 ∈ (TopOn‘ℝ) ∧ 𝑊 ⊆ ℝ) → (𝐿t 𝑊) ∈ (TopOn‘𝑊))
2822, 26, 27sylancr 587 . . . 4 ((𝜑𝑀 ∈ (1...𝑁)) → (𝐿t 𝑊) ∈ (TopOn‘𝑊))
29 eqid 2736 . . . . . . 7 (II ↾t 𝑊) = (II ↾t 𝑊)
30 iitopon 24242 . . . . . . . 8 II ∈ (TopOn‘(0[,]1))
3130a1i 11 . . . . . . 7 ((𝜑𝑀 ∈ (1...𝑁)) → II ∈ (TopOn‘(0[,]1)))
326adantr 481 . . . . . . . . . 10 ((𝜑𝑀 ∈ (1...𝑁)) → 𝐺 ∈ (II Cn 𝐽))
33 iiuni 24244 . . . . . . . . . . 11 (0[,]1) = II
3433, 4cnf 22597 . . . . . . . . . 10 (𝐺 ∈ (II Cn 𝐽) → 𝐺:(0[,]1)⟶𝑋)
3532, 34syl 17 . . . . . . . . 9 ((𝜑𝑀 ∈ (1...𝑁)) → 𝐺:(0[,]1)⟶𝑋)
3635feqmptd 6910 . . . . . . . 8 ((𝜑𝑀 ∈ (1...𝑁)) → 𝐺 = (𝑧 ∈ (0[,]1) ↦ (𝐺𝑧)))
3736, 32eqeltrrd 2839 . . . . . . 7 ((𝜑𝑀 ∈ (1...𝑁)) → (𝑧 ∈ (0[,]1) ↦ (𝐺𝑧)) ∈ (II Cn 𝐽))
3829, 31, 24, 37cnmpt1res 23027 . . . . . 6 ((𝜑𝑀 ∈ (1...𝑁)) → (𝑧𝑊 ↦ (𝐺𝑧)) ∈ ((II ↾t 𝑊) Cn 𝐽))
39 dfii2 24245 . . . . . . . . . 10 II = ((topGen‘ran (,)) ↾t (0[,]1))
4012oveq1i 7367 . . . . . . . . . 10 (𝐿t (0[,]1)) = ((topGen‘ran (,)) ↾t (0[,]1))
4139, 40eqtr4i 2767 . . . . . . . . 9 II = (𝐿t (0[,]1))
4241oveq1i 7367 . . . . . . . 8 (II ↾t 𝑊) = ((𝐿t (0[,]1)) ↾t 𝑊)
43 retop 24125 . . . . . . . . . . 11 (topGen‘ran (,)) ∈ Top
4412, 43eqeltri 2834 . . . . . . . . . 10 𝐿 ∈ Top
4544a1i 11 . . . . . . . . 9 ((𝜑𝑀 ∈ (1...𝑁)) → 𝐿 ∈ Top)
46 ovexd 7392 . . . . . . . . 9 ((𝜑𝑀 ∈ (1...𝑁)) → (0[,]1) ∈ V)
47 restabs 22516 . . . . . . . . 9 ((𝐿 ∈ Top ∧ 𝑊 ⊆ (0[,]1) ∧ (0[,]1) ∈ V) → ((𝐿t (0[,]1)) ↾t 𝑊) = (𝐿t 𝑊))
4845, 24, 46, 47syl3anc 1371 . . . . . . . 8 ((𝜑𝑀 ∈ (1...𝑁)) → ((𝐿t (0[,]1)) ↾t 𝑊) = (𝐿t 𝑊))
4942, 48eqtrid 2788 . . . . . . 7 ((𝜑𝑀 ∈ (1...𝑁)) → (II ↾t 𝑊) = (𝐿t 𝑊))
5049oveq1d 7372 . . . . . 6 ((𝜑𝑀 ∈ (1...𝑁)) → ((II ↾t 𝑊) Cn 𝐽) = ((𝐿t 𝑊) Cn 𝐽))
5138, 50eleqtrd 2840 . . . . 5 ((𝜑𝑀 ∈ (1...𝑁)) → (𝑧𝑊 ↦ (𝐺𝑧)) ∈ ((𝐿t 𝑊) Cn 𝐽))
52 cvmtop2 33855 . . . . . . . 8 (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐽 ∈ Top)
5317, 52syl 17 . . . . . . 7 ((𝜑𝑀 ∈ (1...𝑁)) → 𝐽 ∈ Top)
544toptopon 22266 . . . . . . 7 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
5553, 54sylib 217 . . . . . 6 ((𝜑𝑀 ∈ (1...𝑁)) → 𝐽 ∈ (TopOn‘𝑋))
56 simprl 769 . . . . . . . . . 10 ((𝜑 ∧ (𝑀 ∈ (1...𝑁) ∧ 𝑧𝑊)) → 𝑀 ∈ (1...𝑁))
57 simprr 771 . . . . . . . . . 10 ((𝜑 ∧ (𝑀 ∈ (1...𝑁) ∧ 𝑧𝑊)) → 𝑧𝑊)
582, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 56, 14, 57cvmliftlem3 33881 . . . . . . . . 9 ((𝜑 ∧ (𝑀 ∈ (1...𝑁) ∧ 𝑧𝑊)) → (𝐺𝑧) ∈ (1st ‘(𝑇𝑀)))
5958anassrs 468 . . . . . . . 8 (((𝜑𝑀 ∈ (1...𝑁)) ∧ 𝑧𝑊) → (𝐺𝑧) ∈ (1st ‘(𝑇𝑀)))
6059fmpttd 7063 . . . . . . 7 ((𝜑𝑀 ∈ (1...𝑁)) → (𝑧𝑊 ↦ (𝐺𝑧)):𝑊⟶(1st ‘(𝑇𝑀)))
6160frnd 6676 . . . . . 6 ((𝜑𝑀 ∈ (1...𝑁)) → ran (𝑧𝑊 ↦ (𝐺𝑧)) ⊆ (1st ‘(𝑇𝑀)))
622, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 23cvmliftlem1 33879 . . . . . . . 8 ((𝜑𝑀 ∈ (1...𝑁)) → (2nd ‘(𝑇𝑀)) ∈ (𝑆‘(1st ‘(𝑇𝑀))))
632cvmsrcl 33858 . . . . . . . 8 ((2nd ‘(𝑇𝑀)) ∈ (𝑆‘(1st ‘(𝑇𝑀))) → (1st ‘(𝑇𝑀)) ∈ 𝐽)
64 elssuni 4898 . . . . . . . 8 ((1st ‘(𝑇𝑀)) ∈ 𝐽 → (1st ‘(𝑇𝑀)) ⊆ 𝐽)
6562, 63, 643syl 18 . . . . . . 7 ((𝜑𝑀 ∈ (1...𝑁)) → (1st ‘(𝑇𝑀)) ⊆ 𝐽)
6665, 4sseqtrrdi 3995 . . . . . 6 ((𝜑𝑀 ∈ (1...𝑁)) → (1st ‘(𝑇𝑀)) ⊆ 𝑋)
67 cnrest2 22637 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ ran (𝑧𝑊 ↦ (𝐺𝑧)) ⊆ (1st ‘(𝑇𝑀)) ∧ (1st ‘(𝑇𝑀)) ⊆ 𝑋) → ((𝑧𝑊 ↦ (𝐺𝑧)) ∈ ((𝐿t 𝑊) Cn 𝐽) ↔ (𝑧𝑊 ↦ (𝐺𝑧)) ∈ ((𝐿t 𝑊) Cn (𝐽t (1st ‘(𝑇𝑀))))))
6855, 61, 66, 67syl3anc 1371 . . . . 5 ((𝜑𝑀 ∈ (1...𝑁)) → ((𝑧𝑊 ↦ (𝐺𝑧)) ∈ ((𝐿t 𝑊) Cn 𝐽) ↔ (𝑧𝑊 ↦ (𝐺𝑧)) ∈ ((𝐿t 𝑊) Cn (𝐽t (1st ‘(𝑇𝑀))))))
6951, 68mpbid 231 . . . 4 ((𝜑𝑀 ∈ (1...𝑁)) → (𝑧𝑊 ↦ (𝐺𝑧)) ∈ ((𝐿t 𝑊) Cn (𝐽t (1st ‘(𝑇𝑀)))))
702, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14cvmliftlem7 33885 . . . . . . . . . 10 ((𝜑𝑀 ∈ (1...𝑁)) → ((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ (𝐹 “ {(𝐺‘((𝑀 − 1) / 𝑁))}))
71 cvmcn 33856 . . . . . . . . . . . 12 (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐹 ∈ (𝐶 Cn 𝐽))
723, 4cnf 22597 . . . . . . . . . . . 12 (𝐹 ∈ (𝐶 Cn 𝐽) → 𝐹:𝐵𝑋)
7317, 71, 723syl 18 . . . . . . . . . . 11 ((𝜑𝑀 ∈ (1...𝑁)) → 𝐹:𝐵𝑋)
74 ffn 6668 . . . . . . . . . . 11 (𝐹:𝐵𝑋𝐹 Fn 𝐵)
75 fniniseg 7010 . . . . . . . . . . 11 (𝐹 Fn 𝐵 → (((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ (𝐹 “ {(𝐺‘((𝑀 − 1) / 𝑁))}) ↔ (((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝐵 ∧ (𝐹‘((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁))) = (𝐺‘((𝑀 − 1) / 𝑁)))))
7673, 74, 753syl 18 . . . . . . . . . 10 ((𝜑𝑀 ∈ (1...𝑁)) → (((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ (𝐹 “ {(𝐺‘((𝑀 − 1) / 𝑁))}) ↔ (((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝐵 ∧ (𝐹‘((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁))) = (𝐺‘((𝑀 − 1) / 𝑁)))))
7770, 76mpbid 231 . . . . . . . . 9 ((𝜑𝑀 ∈ (1...𝑁)) → (((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝐵 ∧ (𝐹‘((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁))) = (𝐺‘((𝑀 − 1) / 𝑁))))
7877simpld 495 . . . . . . . 8 ((𝜑𝑀 ∈ (1...𝑁)) → ((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝐵)
7977simprd 496 . . . . . . . . 9 ((𝜑𝑀 ∈ (1...𝑁)) → (𝐹‘((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁))) = (𝐺‘((𝑀 − 1) / 𝑁)))
801adantl 482 . . . . . . . . . . . . . . . 16 ((𝜑𝑀 ∈ (1...𝑁)) → 𝑀 ∈ ℕ)
8180nnred 12168 . . . . . . . . . . . . . . 15 ((𝜑𝑀 ∈ (1...𝑁)) → 𝑀 ∈ ℝ)
82 peano2rem 11468 . . . . . . . . . . . . . . 15 (𝑀 ∈ ℝ → (𝑀 − 1) ∈ ℝ)
8381, 82syl 17 . . . . . . . . . . . . . 14 ((𝜑𝑀 ∈ (1...𝑁)) → (𝑀 − 1) ∈ ℝ)
849adantr 481 . . . . . . . . . . . . . 14 ((𝜑𝑀 ∈ (1...𝑁)) → 𝑁 ∈ ℕ)
8583, 84nndivred 12207 . . . . . . . . . . . . 13 ((𝜑𝑀 ∈ (1...𝑁)) → ((𝑀 − 1) / 𝑁) ∈ ℝ)
8685rexrd 11205 . . . . . . . . . . . 12 ((𝜑𝑀 ∈ (1...𝑁)) → ((𝑀 − 1) / 𝑁) ∈ ℝ*)
8781, 84nndivred 12207 . . . . . . . . . . . . 13 ((𝜑𝑀 ∈ (1...𝑁)) → (𝑀 / 𝑁) ∈ ℝ)
8887rexrd 11205 . . . . . . . . . . . 12 ((𝜑𝑀 ∈ (1...𝑁)) → (𝑀 / 𝑁) ∈ ℝ*)
8981ltm1d 12087 . . . . . . . . . . . . . 14 ((𝜑𝑀 ∈ (1...𝑁)) → (𝑀 − 1) < 𝑀)
9084nnred 12168 . . . . . . . . . . . . . . 15 ((𝜑𝑀 ∈ (1...𝑁)) → 𝑁 ∈ ℝ)
9184nngt0d 12202 . . . . . . . . . . . . . . 15 ((𝜑𝑀 ∈ (1...𝑁)) → 0 < 𝑁)
92 ltdiv1 12019 . . . . . . . . . . . . . . 15 (((𝑀 − 1) ∈ ℝ ∧ 𝑀 ∈ ℝ ∧ (𝑁 ∈ ℝ ∧ 0 < 𝑁)) → ((𝑀 − 1) < 𝑀 ↔ ((𝑀 − 1) / 𝑁) < (𝑀 / 𝑁)))
9383, 81, 90, 91, 92syl112anc 1374 . . . . . . . . . . . . . 14 ((𝜑𝑀 ∈ (1...𝑁)) → ((𝑀 − 1) < 𝑀 ↔ ((𝑀 − 1) / 𝑁) < (𝑀 / 𝑁)))
9489, 93mpbid 231 . . . . . . . . . . . . 13 ((𝜑𝑀 ∈ (1...𝑁)) → ((𝑀 − 1) / 𝑁) < (𝑀 / 𝑁))
9585, 87, 94ltled 11303 . . . . . . . . . . . 12 ((𝜑𝑀 ∈ (1...𝑁)) → ((𝑀 − 1) / 𝑁) ≤ (𝑀 / 𝑁))
96 lbicc2 13381 . . . . . . . . . . . 12 ((((𝑀 − 1) / 𝑁) ∈ ℝ* ∧ (𝑀 / 𝑁) ∈ ℝ* ∧ ((𝑀 − 1) / 𝑁) ≤ (𝑀 / 𝑁)) → ((𝑀 − 1) / 𝑁) ∈ (((𝑀 − 1) / 𝑁)[,](𝑀 / 𝑁)))
9786, 88, 95, 96syl3anc 1371 . . . . . . . . . . 11 ((𝜑𝑀 ∈ (1...𝑁)) → ((𝑀 − 1) / 𝑁) ∈ (((𝑀 − 1) / 𝑁)[,](𝑀 / 𝑁)))
9897, 14eleqtrrdi 2849 . . . . . . . . . 10 ((𝜑𝑀 ∈ (1...𝑁)) → ((𝑀 − 1) / 𝑁) ∈ 𝑊)
992, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 23, 14, 98cvmliftlem3 33881 . . . . . . . . 9 ((𝜑𝑀 ∈ (1...𝑁)) → (𝐺‘((𝑀 − 1) / 𝑁)) ∈ (1st ‘(𝑇𝑀)))
10079, 99eqeltrd 2838 . . . . . . . 8 ((𝜑𝑀 ∈ (1...𝑁)) → (𝐹‘((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁))) ∈ (1st ‘(𝑇𝑀)))
101 eqid 2736 . . . . . . . . 9 (𝑏 ∈ (2nd ‘(𝑇𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏) = (𝑏 ∈ (2nd ‘(𝑇𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏)
1022, 3, 101cvmsiota 33871 . . . . . . . 8 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ ((2nd ‘(𝑇𝑀)) ∈ (𝑆‘(1st ‘(𝑇𝑀))) ∧ ((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝐵 ∧ (𝐹‘((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁))) ∈ (1st ‘(𝑇𝑀)))) → ((𝑏 ∈ (2nd ‘(𝑇𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏) ∈ (2nd ‘(𝑇𝑀)) ∧ ((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ (𝑏 ∈ (2nd ‘(𝑇𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏)))
10317, 62, 78, 100, 102syl13anc 1372 . . . . . . 7 ((𝜑𝑀 ∈ (1...𝑁)) → ((𝑏 ∈ (2nd ‘(𝑇𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏) ∈ (2nd ‘(𝑇𝑀)) ∧ ((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ (𝑏 ∈ (2nd ‘(𝑇𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏)))
104103simpld 495 . . . . . 6 ((𝜑𝑀 ∈ (1...𝑁)) → (𝑏 ∈ (2nd ‘(𝑇𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏) ∈ (2nd ‘(𝑇𝑀)))
1052cvmshmeo 33865 . . . . . 6 (((2nd ‘(𝑇𝑀)) ∈ (𝑆‘(1st ‘(𝑇𝑀))) ∧ (𝑏 ∈ (2nd ‘(𝑇𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏) ∈ (2nd ‘(𝑇𝑀))) → (𝐹 ↾ (𝑏 ∈ (2nd ‘(𝑇𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏)) ∈ ((𝐶t (𝑏 ∈ (2nd ‘(𝑇𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏))Homeo(𝐽t (1st ‘(𝑇𝑀)))))
10662, 104, 105syl2anc 584 . . . . 5 ((𝜑𝑀 ∈ (1...𝑁)) → (𝐹 ↾ (𝑏 ∈ (2nd ‘(𝑇𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏)) ∈ ((𝐶t (𝑏 ∈ (2nd ‘(𝑇𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏))Homeo(𝐽t (1st ‘(𝑇𝑀)))))
107 hmeocnvcn 23112 . . . . 5 ((𝐹 ↾ (𝑏 ∈ (2nd ‘(𝑇𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏)) ∈ ((𝐶t (𝑏 ∈ (2nd ‘(𝑇𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏))Homeo(𝐽t (1st ‘(𝑇𝑀)))) → (𝐹 ↾ (𝑏 ∈ (2nd ‘(𝑇𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏)) ∈ ((𝐽t (1st ‘(𝑇𝑀))) Cn (𝐶t (𝑏 ∈ (2nd ‘(𝑇𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏))))
108106, 107syl 17 . . . 4 ((𝜑𝑀 ∈ (1...𝑁)) → (𝐹 ↾ (𝑏 ∈ (2nd ‘(𝑇𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏)) ∈ ((𝐽t (1st ‘(𝑇𝑀))) Cn (𝐶t (𝑏 ∈ (2nd ‘(𝑇𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏))))
10928, 69, 108cnmpt11f 23015 . . 3 ((𝜑𝑀 ∈ (1...𝑁)) → (𝑧𝑊 ↦ ((𝐹 ↾ (𝑏 ∈ (2nd ‘(𝑇𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏))‘(𝐺𝑧))) ∈ ((𝐿t 𝑊) Cn (𝐶t (𝑏 ∈ (2nd ‘(𝑇𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏))))
11020, 109sseldd 3945 . 2 ((𝜑𝑀 ∈ (1...𝑁)) → (𝑧𝑊 ↦ ((𝐹 ↾ (𝑏 ∈ (2nd ‘(𝑇𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏))‘(𝐺𝑧))) ∈ ((𝐿t 𝑊) Cn 𝐶))
11116, 110eqeltrd 2838 1 ((𝜑𝑀 ∈ (1...𝑁)) → (𝑄𝑀) ∈ ((𝐿t 𝑊) Cn 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  wral 3064  {crab 3407  Vcvv 3445  cdif 3907  cun 3908  cin 3909  wss 3910  c0 4282  𝒫 cpw 4560  {csn 4586  cop 4592   cuni 4865   ciun 4954   class class class wbr 5105  cmpt 5188   I cid 5530   × cxp 5631  ccnv 5632  ran crn 5634  cres 5635  cima 5636   Fn wfn 6491  wf 6492  cfv 6496  crio 7312  (class class class)co 7357  cmpo 7359  1st c1st 7919  2nd c2nd 7920  cr 11050  0cc0 11051  1c1 11052  *cxr 11188   < clt 11189  cle 11190  cmin 11385   / cdiv 11812  cn 12153  (,)cioo 13264  [,]cicc 13267  ...cfz 13424  seqcseq 13906  t crest 17302  topGenctg 17319  Topctop 22242  TopOnctopon 22259   Cn ccn 22575  Homeochmeo 23104  IIcii 24238   CovMap ccvm 33849
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-er 8648  df-map 8767  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fi 9347  df-sup 9378  df-inf 9379  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-3 12217  df-n0 12414  df-z 12500  df-uz 12764  df-q 12874  df-rp 12916  df-xneg 13033  df-xadd 13034  df-xmul 13035  df-ioo 13268  df-icc 13271  df-fz 13425  df-seq 13907  df-exp 13968  df-cj 14984  df-re 14985  df-im 14986  df-sqrt 15120  df-abs 15121  df-rest 17304  df-topgen 17325  df-psmet 20788  df-xmet 20789  df-met 20790  df-bl 20791  df-mopn 20792  df-top 22243  df-topon 22260  df-bases 22296  df-cn 22578  df-hmeo 23106  df-ii 24240  df-cvm 33850
This theorem is referenced by:  cvmliftlem10  33888
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