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Theorem cvmliftlem8 32147
Description: Lemma for cvmlift 32154. 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 12786 . . 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 32144 . . 3 ((𝜑𝑀 ∈ ℕ) → (𝑄𝑀) = (𝑧𝑊 ↦ ((𝐹 ↾ (𝑏 ∈ (2nd ‘(𝑇𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏))‘(𝐺𝑧))))
161, 15sylan2 592 . 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 32115 . . . 4 (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐶 ∈ Top)
19 cnrest2r 21579 . . . 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 23055 . . . . . 6 (topGen‘ran (,)) ∈ (TopOn‘ℝ)
2212, 21eqeltri 2879 . . . . 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 32141 . . . . . 6 ((𝜑𝑀 ∈ (1...𝑁)) → 𝑊 ⊆ (0[,]1))
25 unitssre 12735 . . . . . 6 (0[,]1) ⊆ ℝ
2624, 25syl6ss 3901 . . . . 5 ((𝜑𝑀 ∈ (1...𝑁)) → 𝑊 ⊆ ℝ)
27 resttopon 21453 . . . . 5 ((𝐿 ∈ (TopOn‘ℝ) ∧ 𝑊 ⊆ ℝ) → (𝐿t 𝑊) ∈ (TopOn‘𝑊))
2822, 26, 27sylancr 587 . . . 4 ((𝜑𝑀 ∈ (1...𝑁)) → (𝐿t 𝑊) ∈ (TopOn‘𝑊))
29 eqid 2795 . . . . . . 7 (II ↾t 𝑊) = (II ↾t 𝑊)
30 iitopon 23170 . . . . . . . 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 23172 . . . . . . . . . . 11 (0[,]1) = II
3433, 4cnf 21538 . . . . . . . . . 10 (𝐺 ∈ (II Cn 𝐽) → 𝐺:(0[,]1)⟶𝑋)
3532, 34syl 17 . . . . . . . . 9 ((𝜑𝑀 ∈ (1...𝑁)) → 𝐺:(0[,]1)⟶𝑋)
3635feqmptd 6601 . . . . . . . 8 ((𝜑𝑀 ∈ (1...𝑁)) → 𝐺 = (𝑧 ∈ (0[,]1) ↦ (𝐺𝑧)))
3736, 32eqeltrrd 2884 . . . . . . 7 ((𝜑𝑀 ∈ (1...𝑁)) → (𝑧 ∈ (0[,]1) ↦ (𝐺𝑧)) ∈ (II Cn 𝐽))
3829, 31, 24, 37cnmpt1res 21968 . . . . . 6 ((𝜑𝑀 ∈ (1...𝑁)) → (𝑧𝑊 ↦ (𝐺𝑧)) ∈ ((II ↾t 𝑊) Cn 𝐽))
39 dfii2 23173 . . . . . . . . . 10 II = ((topGen‘ran (,)) ↾t (0[,]1))
4012oveq1i 7026 . . . . . . . . . 10 (𝐿t (0[,]1)) = ((topGen‘ran (,)) ↾t (0[,]1))
4139, 40eqtr4i 2822 . . . . . . . . 9 II = (𝐿t (0[,]1))
4241oveq1i 7026 . . . . . . . 8 (II ↾t 𝑊) = ((𝐿t (0[,]1)) ↾t 𝑊)
43 retop 23053 . . . . . . . . . . 11 (topGen‘ran (,)) ∈ Top
4412, 43eqeltri 2879 . . . . . . . . . 10 𝐿 ∈ Top
4544a1i 11 . . . . . . . . 9 ((𝜑𝑀 ∈ (1...𝑁)) → 𝐿 ∈ Top)
46 ovexd 7050 . . . . . . . . 9 ((𝜑𝑀 ∈ (1...𝑁)) → (0[,]1) ∈ V)
47 restabs 21457 . . . . . . . . 9 ((𝐿 ∈ Top ∧ 𝑊 ⊆ (0[,]1) ∧ (0[,]1) ∈ V) → ((𝐿t (0[,]1)) ↾t 𝑊) = (𝐿t 𝑊))
4845, 24, 46, 47syl3anc 1364 . . . . . . . 8 ((𝜑𝑀 ∈ (1...𝑁)) → ((𝐿t (0[,]1)) ↾t 𝑊) = (𝐿t 𝑊))
4942, 48syl5eq 2843 . . . . . . 7 ((𝜑𝑀 ∈ (1...𝑁)) → (II ↾t 𝑊) = (𝐿t 𝑊))
5049oveq1d 7031 . . . . . 6 ((𝜑𝑀 ∈ (1...𝑁)) → ((II ↾t 𝑊) Cn 𝐽) = ((𝐿t 𝑊) Cn 𝐽))
5138, 50eleqtrd 2885 . . . . 5 ((𝜑𝑀 ∈ (1...𝑁)) → (𝑧𝑊 ↦ (𝐺𝑧)) ∈ ((𝐿t 𝑊) Cn 𝐽))
52 cvmtop2 32116 . . . . . . . 8 (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐽 ∈ Top)
5317, 52syl 17 . . . . . . 7 ((𝜑𝑀 ∈ (1...𝑁)) → 𝐽 ∈ Top)
544toptopon 21209 . . . . . . 7 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
5553, 54sylib 219 . . . . . 6 ((𝜑𝑀 ∈ (1...𝑁)) → 𝐽 ∈ (TopOn‘𝑋))
56 simprl 767 . . . . . . . . . 10 ((𝜑 ∧ (𝑀 ∈ (1...𝑁) ∧ 𝑧𝑊)) → 𝑀 ∈ (1...𝑁))
57 simprr 769 . . . . . . . . . 10 ((𝜑 ∧ (𝑀 ∈ (1...𝑁) ∧ 𝑧𝑊)) → 𝑧𝑊)
582, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 56, 14, 57cvmliftlem3 32142 . . . . . . . . 9 ((𝜑 ∧ (𝑀 ∈ (1...𝑁) ∧ 𝑧𝑊)) → (𝐺𝑧) ∈ (1st ‘(𝑇𝑀)))
5958anassrs 468 . . . . . . . 8 (((𝜑𝑀 ∈ (1...𝑁)) ∧ 𝑧𝑊) → (𝐺𝑧) ∈ (1st ‘(𝑇𝑀)))
6059fmpttd 6742 . . . . . . 7 ((𝜑𝑀 ∈ (1...𝑁)) → (𝑧𝑊 ↦ (𝐺𝑧)):𝑊⟶(1st ‘(𝑇𝑀)))
6160frnd 6389 . . . . . 6 ((𝜑𝑀 ∈ (1...𝑁)) → ran (𝑧𝑊 ↦ (𝐺𝑧)) ⊆ (1st ‘(𝑇𝑀)))
622, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 23cvmliftlem1 32140 . . . . . . . 8 ((𝜑𝑀 ∈ (1...𝑁)) → (2nd ‘(𝑇𝑀)) ∈ (𝑆‘(1st ‘(𝑇𝑀))))
632cvmsrcl 32119 . . . . . . . 8 ((2nd ‘(𝑇𝑀)) ∈ (𝑆‘(1st ‘(𝑇𝑀))) → (1st ‘(𝑇𝑀)) ∈ 𝐽)
64 elssuni 4774 . . . . . . . 8 ((1st ‘(𝑇𝑀)) ∈ 𝐽 → (1st ‘(𝑇𝑀)) ⊆ 𝐽)
6562, 63, 643syl 18 . . . . . . 7 ((𝜑𝑀 ∈ (1...𝑁)) → (1st ‘(𝑇𝑀)) ⊆ 𝐽)
6665, 4syl6sseqr 3939 . . . . . 6 ((𝜑𝑀 ∈ (1...𝑁)) → (1st ‘(𝑇𝑀)) ⊆ 𝑋)
67 cnrest2 21578 . . . . . 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 1364 . . . . 5 ((𝜑𝑀 ∈ (1...𝑁)) → ((𝑧𝑊 ↦ (𝐺𝑧)) ∈ ((𝐿t 𝑊) Cn 𝐽) ↔ (𝑧𝑊 ↦ (𝐺𝑧)) ∈ ((𝐿t 𝑊) Cn (𝐽t (1st ‘(𝑇𝑀))))))
6951, 68mpbid 233 . . . 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 32146 . . . . . . . . . 10 ((𝜑𝑀 ∈ (1...𝑁)) → ((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ (𝐹 “ {(𝐺‘((𝑀 − 1) / 𝑁))}))
71 cvmcn 32117 . . . . . . . . . . . 12 (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐹 ∈ (𝐶 Cn 𝐽))
723, 4cnf 21538 . . . . . . . . . . . 12 (𝐹 ∈ (𝐶 Cn 𝐽) → 𝐹:𝐵𝑋)
7317, 71, 723syl 18 . . . . . . . . . . 11 ((𝜑𝑀 ∈ (1...𝑁)) → 𝐹:𝐵𝑋)
74 ffn 6382 . . . . . . . . . . 11 (𝐹:𝐵𝑋𝐹 Fn 𝐵)
75 fniniseg 6695 . . . . . . . . . . 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 233 . . . . . . . . 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 11501 . . . . . . . . . . . . . . 15 ((𝜑𝑀 ∈ (1...𝑁)) → 𝑀 ∈ ℝ)
82 peano2rem 10801 . . . . . . . . . . . . . . 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 11539 . . . . . . . . . . . . 13 ((𝜑𝑀 ∈ (1...𝑁)) → ((𝑀 − 1) / 𝑁) ∈ ℝ)
8685rexrd 10537 . . . . . . . . . . . 12 ((𝜑𝑀 ∈ (1...𝑁)) → ((𝑀 − 1) / 𝑁) ∈ ℝ*)
8781, 84nndivred 11539 . . . . . . . . . . . . 13 ((𝜑𝑀 ∈ (1...𝑁)) → (𝑀 / 𝑁) ∈ ℝ)
8887rexrd 10537 . . . . . . . . . . . 12 ((𝜑𝑀 ∈ (1...𝑁)) → (𝑀 / 𝑁) ∈ ℝ*)
8981ltm1d 11420 . . . . . . . . . . . . . 14 ((𝜑𝑀 ∈ (1...𝑁)) → (𝑀 − 1) < 𝑀)
9084nnred 11501 . . . . . . . . . . . . . . 15 ((𝜑𝑀 ∈ (1...𝑁)) → 𝑁 ∈ ℝ)
9184nngt0d 11534 . . . . . . . . . . . . . . 15 ((𝜑𝑀 ∈ (1...𝑁)) → 0 < 𝑁)
92 ltdiv1 11352 . . . . . . . . . . . . . . 15 (((𝑀 − 1) ∈ ℝ ∧ 𝑀 ∈ ℝ ∧ (𝑁 ∈ ℝ ∧ 0 < 𝑁)) → ((𝑀 − 1) < 𝑀 ↔ ((𝑀 − 1) / 𝑁) < (𝑀 / 𝑁)))
9383, 81, 90, 91, 92syl112anc 1367 . . . . . . . . . . . . . 14 ((𝜑𝑀 ∈ (1...𝑁)) → ((𝑀 − 1) < 𝑀 ↔ ((𝑀 − 1) / 𝑁) < (𝑀 / 𝑁)))
9489, 93mpbid 233 . . . . . . . . . . . . 13 ((𝜑𝑀 ∈ (1...𝑁)) → ((𝑀 − 1) / 𝑁) < (𝑀 / 𝑁))
9585, 87, 94ltled 10635 . . . . . . . . . . . 12 ((𝜑𝑀 ∈ (1...𝑁)) → ((𝑀 − 1) / 𝑁) ≤ (𝑀 / 𝑁))
96 lbicc2 12702 . . . . . . . . . . . 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 1364 . . . . . . . . . . 11 ((𝜑𝑀 ∈ (1...𝑁)) → ((𝑀 − 1) / 𝑁) ∈ (((𝑀 − 1) / 𝑁)[,](𝑀 / 𝑁)))
9897, 14syl6eleqr 2894 . . . . . . . . . 10 ((𝜑𝑀 ∈ (1...𝑁)) → ((𝑀 − 1) / 𝑁) ∈ 𝑊)
992, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 23, 14, 98cvmliftlem3 32142 . . . . . . . . 9 ((𝜑𝑀 ∈ (1...𝑁)) → (𝐺‘((𝑀 − 1) / 𝑁)) ∈ (1st ‘(𝑇𝑀)))
10079, 99eqeltrd 2883 . . . . . . . 8 ((𝜑𝑀 ∈ (1...𝑁)) → (𝐹‘((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁))) ∈ (1st ‘(𝑇𝑀)))
101 eqid 2795 . . . . . . . . 9 (𝑏 ∈ (2nd ‘(𝑇𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏) = (𝑏 ∈ (2nd ‘(𝑇𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏)
1022, 3, 101cvmsiota 32132 . . . . . . . 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 1365 . . . . . . 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 32126 . . . . . 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 22053 . . . . 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 21956 . . 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 3890 . 2 ((𝜑𝑀 ∈ (1...𝑁)) → (𝑧𝑊 ↦ ((𝐹 ↾ (𝑏 ∈ (2nd ‘(𝑇𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏))‘(𝐺𝑧))) ∈ ((𝐿t 𝑊) Cn 𝐶))
11116, 110eqeltrd 2883 1 ((𝜑𝑀 ∈ (1...𝑁)) → (𝑄𝑀) ∈ ((𝐿t 𝑊) Cn 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1522  wcel 2081  wral 3105  {crab 3109  Vcvv 3437  cdif 3856  cun 3857  cin 3858  wss 3859  c0 4211  𝒫 cpw 4453  {csn 4472  cop 4478   cuni 4745   ciun 4825   class class class wbr 4962  cmpt 5041   I cid 5347   × cxp 5441  ccnv 5442  ran crn 5444  cres 5445  cima 5446   Fn wfn 6220  wf 6221  cfv 6225  crio 6976  (class class class)co 7016  cmpo 7018  1st c1st 7543  2nd c2nd 7544  cr 10382  0cc0 10383  1c1 10384  *cxr 10520   < clt 10521  cle 10522  cmin 10717   / cdiv 11145  cn 11486  (,)cioo 12588  [,]cicc 12591  ...cfz 12742  seqcseq 13219  t crest 16523  topGenctg 16540  Topctop 21185  TopOnctopon 21202   Cn ccn 21516  Homeochmeo 22045  IIcii 23166   CovMap ccvm 32110
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-rep 5081  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319  ax-cnex 10439  ax-resscn 10440  ax-1cn 10441  ax-icn 10442  ax-addcl 10443  ax-addrcl 10444  ax-mulcl 10445  ax-mulrcl 10446  ax-mulcom 10447  ax-addass 10448  ax-mulass 10449  ax-distr 10450  ax-i2m1 10451  ax-1ne0 10452  ax-1rid 10453  ax-rnegex 10454  ax-rrecex 10455  ax-cnre 10456  ax-pre-lttri 10457  ax-pre-lttrn 10458  ax-pre-ltadd 10459  ax-pre-mulgt0 10460  ax-pre-sup 10461
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-nel 3091  df-ral 3110  df-rex 3111  df-reu 3112  df-rmo 3113  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-pss 3876  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-tp 4477  df-op 4479  df-uni 4746  df-int 4783  df-iun 4827  df-br 4963  df-opab 5025  df-mpt 5042  df-tr 5064  df-id 5348  df-eprel 5353  df-po 5362  df-so 5363  df-fr 5402  df-we 5404  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-pred 6023  df-ord 6069  df-on 6070  df-lim 6071  df-suc 6072  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-fv 6233  df-riota 6977  df-ov 7019  df-oprab 7020  df-mpo 7021  df-om 7437  df-1st 7545  df-2nd 7546  df-wrecs 7798  df-recs 7860  df-rdg 7898  df-oadd 7957  df-er 8139  df-map 8258  df-en 8358  df-dom 8359  df-sdom 8360  df-fin 8361  df-fi 8721  df-sup 8752  df-inf 8753  df-pnf 10523  df-mnf 10524  df-xr 10525  df-ltxr 10526  df-le 10527  df-sub 10719  df-neg 10720  df-div 11146  df-nn 11487  df-2 11548  df-3 11549  df-n0 11746  df-z 11830  df-uz 12094  df-q 12198  df-rp 12240  df-xneg 12357  df-xadd 12358  df-xmul 12359  df-ioo 12592  df-icc 12595  df-fz 12743  df-seq 13220  df-exp 13280  df-cj 14292  df-re 14293  df-im 14294  df-sqrt 14428  df-abs 14429  df-rest 16525  df-topgen 16546  df-psmet 20219  df-xmet 20220  df-met 20221  df-bl 20222  df-mopn 20223  df-top 21186  df-topon 21203  df-bases 21238  df-cn 21519  df-hmeo 22047  df-ii 23168  df-cvm 32111
This theorem is referenced by:  cvmliftlem10  32149
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