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Theorem cvmliftlem8 31581
Description: Lemma for cvmlift 31588. 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 12563 . . 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 31578 . . 3 ((𝜑𝑀 ∈ ℕ) → (𝑄𝑀) = (𝑧𝑊 ↦ ((𝐹 ↾ (𝑏 ∈ (2nd ‘(𝑇𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏))‘(𝐺𝑧))))
161, 15sylan2 492 . 2 ((𝜑𝑀 ∈ (1...𝑁)) → (𝑄𝑀) = (𝑧𝑊 ↦ ((𝐹 ↾ (𝑏 ∈ (2nd ‘(𝑇𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏))‘(𝐺𝑧))))
175adantr 472 . . . 4 ((𝜑𝑀 ∈ (1...𝑁)) → 𝐹 ∈ (𝐶 CovMap 𝐽))
18 cvmtop1 31549 . . . 4 (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐶 ∈ Top)
19 cnrest2r 21293 . . . 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 22768 . . . . . 6 (topGen‘ran (,)) ∈ (TopOn‘ℝ)
2212, 21eqeltri 2835 . . . . 5 𝐿 ∈ (TopOn‘ℝ)
23 simpr 479 . . . . . . 7 ((𝜑𝑀 ∈ (1...𝑁)) → 𝑀 ∈ (1...𝑁))
242, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 23, 14cvmliftlem2 31575 . . . . . 6 ((𝜑𝑀 ∈ (1...𝑁)) → 𝑊 ⊆ (0[,]1))
25 unitssre 12512 . . . . . 6 (0[,]1) ⊆ ℝ
2624, 25syl6ss 3756 . . . . 5 ((𝜑𝑀 ∈ (1...𝑁)) → 𝑊 ⊆ ℝ)
27 resttopon 21167 . . . . 5 ((𝐿 ∈ (TopOn‘ℝ) ∧ 𝑊 ⊆ ℝ) → (𝐿t 𝑊) ∈ (TopOn‘𝑊))
2822, 26, 27sylancr 698 . . . 4 ((𝜑𝑀 ∈ (1...𝑁)) → (𝐿t 𝑊) ∈ (TopOn‘𝑊))
29 eqid 2760 . . . . . . 7 (II ↾t 𝑊) = (II ↾t 𝑊)
30 iitopon 22883 . . . . . . . 8 II ∈ (TopOn‘(0[,]1))
3130a1i 11 . . . . . . 7 ((𝜑𝑀 ∈ (1...𝑁)) → II ∈ (TopOn‘(0[,]1)))
326adantr 472 . . . . . . . . . 10 ((𝜑𝑀 ∈ (1...𝑁)) → 𝐺 ∈ (II Cn 𝐽))
33 iiuni 22885 . . . . . . . . . . 11 (0[,]1) = II
3433, 4cnf 21252 . . . . . . . . . 10 (𝐺 ∈ (II Cn 𝐽) → 𝐺:(0[,]1)⟶𝑋)
3532, 34syl 17 . . . . . . . . 9 ((𝜑𝑀 ∈ (1...𝑁)) → 𝐺:(0[,]1)⟶𝑋)
3635feqmptd 6411 . . . . . . . 8 ((𝜑𝑀 ∈ (1...𝑁)) → 𝐺 = (𝑧 ∈ (0[,]1) ↦ (𝐺𝑧)))
3736, 32eqeltrrd 2840 . . . . . . 7 ((𝜑𝑀 ∈ (1...𝑁)) → (𝑧 ∈ (0[,]1) ↦ (𝐺𝑧)) ∈ (II Cn 𝐽))
3829, 31, 24, 37cnmpt1res 21681 . . . . . 6 ((𝜑𝑀 ∈ (1...𝑁)) → (𝑧𝑊 ↦ (𝐺𝑧)) ∈ ((II ↾t 𝑊) Cn 𝐽))
39 dfii2 22886 . . . . . . . . . 10 II = ((topGen‘ran (,)) ↾t (0[,]1))
4012oveq1i 6823 . . . . . . . . . 10 (𝐿t (0[,]1)) = ((topGen‘ran (,)) ↾t (0[,]1))
4139, 40eqtr4i 2785 . . . . . . . . 9 II = (𝐿t (0[,]1))
4241oveq1i 6823 . . . . . . . 8 (II ↾t 𝑊) = ((𝐿t (0[,]1)) ↾t 𝑊)
43 retop 22766 . . . . . . . . . . 11 (topGen‘ran (,)) ∈ Top
4412, 43eqeltri 2835 . . . . . . . . . 10 𝐿 ∈ Top
4544a1i 11 . . . . . . . . 9 ((𝜑𝑀 ∈ (1...𝑁)) → 𝐿 ∈ Top)
46 ovexd 6843 . . . . . . . . 9 ((𝜑𝑀 ∈ (1...𝑁)) → (0[,]1) ∈ V)
47 restabs 21171 . . . . . . . . 9 ((𝐿 ∈ Top ∧ 𝑊 ⊆ (0[,]1) ∧ (0[,]1) ∈ V) → ((𝐿t (0[,]1)) ↾t 𝑊) = (𝐿t 𝑊))
4845, 24, 46, 47syl3anc 1477 . . . . . . . 8 ((𝜑𝑀 ∈ (1...𝑁)) → ((𝐿t (0[,]1)) ↾t 𝑊) = (𝐿t 𝑊))
4942, 48syl5eq 2806 . . . . . . 7 ((𝜑𝑀 ∈ (1...𝑁)) → (II ↾t 𝑊) = (𝐿t 𝑊))
5049oveq1d 6828 . . . . . 6 ((𝜑𝑀 ∈ (1...𝑁)) → ((II ↾t 𝑊) Cn 𝐽) = ((𝐿t 𝑊) Cn 𝐽))
5138, 50eleqtrd 2841 . . . . 5 ((𝜑𝑀 ∈ (1...𝑁)) → (𝑧𝑊 ↦ (𝐺𝑧)) ∈ ((𝐿t 𝑊) Cn 𝐽))
52 cvmtop2 31550 . . . . . . . 8 (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐽 ∈ Top)
5317, 52syl 17 . . . . . . 7 ((𝜑𝑀 ∈ (1...𝑁)) → 𝐽 ∈ Top)
544toptopon 20924 . . . . . . 7 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
5553, 54sylib 208 . . . . . 6 ((𝜑𝑀 ∈ (1...𝑁)) → 𝐽 ∈ (TopOn‘𝑋))
56 simprl 811 . . . . . . . . . 10 ((𝜑 ∧ (𝑀 ∈ (1...𝑁) ∧ 𝑧𝑊)) → 𝑀 ∈ (1...𝑁))
57 simprr 813 . . . . . . . . . 10 ((𝜑 ∧ (𝑀 ∈ (1...𝑁) ∧ 𝑧𝑊)) → 𝑧𝑊)
582, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 56, 14, 57cvmliftlem3 31576 . . . . . . . . 9 ((𝜑 ∧ (𝑀 ∈ (1...𝑁) ∧ 𝑧𝑊)) → (𝐺𝑧) ∈ (1st ‘(𝑇𝑀)))
5958anassrs 683 . . . . . . . 8 (((𝜑𝑀 ∈ (1...𝑁)) ∧ 𝑧𝑊) → (𝐺𝑧) ∈ (1st ‘(𝑇𝑀)))
60 eqid 2760 . . . . . . . 8 (𝑧𝑊 ↦ (𝐺𝑧)) = (𝑧𝑊 ↦ (𝐺𝑧))
6159, 60fmptd 6548 . . . . . . 7 ((𝜑𝑀 ∈ (1...𝑁)) → (𝑧𝑊 ↦ (𝐺𝑧)):𝑊⟶(1st ‘(𝑇𝑀)))
62 frn 6214 . . . . . . 7 ((𝑧𝑊 ↦ (𝐺𝑧)):𝑊⟶(1st ‘(𝑇𝑀)) → ran (𝑧𝑊 ↦ (𝐺𝑧)) ⊆ (1st ‘(𝑇𝑀)))
6361, 62syl 17 . . . . . 6 ((𝜑𝑀 ∈ (1...𝑁)) → ran (𝑧𝑊 ↦ (𝐺𝑧)) ⊆ (1st ‘(𝑇𝑀)))
642, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 23cvmliftlem1 31574 . . . . . . . 8 ((𝜑𝑀 ∈ (1...𝑁)) → (2nd ‘(𝑇𝑀)) ∈ (𝑆‘(1st ‘(𝑇𝑀))))
652cvmsrcl 31553 . . . . . . . 8 ((2nd ‘(𝑇𝑀)) ∈ (𝑆‘(1st ‘(𝑇𝑀))) → (1st ‘(𝑇𝑀)) ∈ 𝐽)
66 elssuni 4619 . . . . . . . 8 ((1st ‘(𝑇𝑀)) ∈ 𝐽 → (1st ‘(𝑇𝑀)) ⊆ 𝐽)
6764, 65, 663syl 18 . . . . . . 7 ((𝜑𝑀 ∈ (1...𝑁)) → (1st ‘(𝑇𝑀)) ⊆ 𝐽)
6867, 4syl6sseqr 3793 . . . . . 6 ((𝜑𝑀 ∈ (1...𝑁)) → (1st ‘(𝑇𝑀)) ⊆ 𝑋)
69 cnrest2 21292 . . . . . 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))))))
7055, 63, 68, 69syl3anc 1477 . . . . 5 ((𝜑𝑀 ∈ (1...𝑁)) → ((𝑧𝑊 ↦ (𝐺𝑧)) ∈ ((𝐿t 𝑊) Cn 𝐽) ↔ (𝑧𝑊 ↦ (𝐺𝑧)) ∈ ((𝐿t 𝑊) Cn (𝐽t (1st ‘(𝑇𝑀))))))
7151, 70mpbid 222 . . . 4 ((𝜑𝑀 ∈ (1...𝑁)) → (𝑧𝑊 ↦ (𝐺𝑧)) ∈ ((𝐿t 𝑊) Cn (𝐽t (1st ‘(𝑇𝑀)))))
722, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14cvmliftlem7 31580 . . . . . . . . . 10 ((𝜑𝑀 ∈ (1...𝑁)) → ((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ (𝐹 “ {(𝐺‘((𝑀 − 1) / 𝑁))}))
73 cvmcn 31551 . . . . . . . . . . . 12 (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐹 ∈ (𝐶 Cn 𝐽))
743, 4cnf 21252 . . . . . . . . . . . 12 (𝐹 ∈ (𝐶 Cn 𝐽) → 𝐹:𝐵𝑋)
7517, 73, 743syl 18 . . . . . . . . . . 11 ((𝜑𝑀 ∈ (1...𝑁)) → 𝐹:𝐵𝑋)
76 ffn 6206 . . . . . . . . . . 11 (𝐹:𝐵𝑋𝐹 Fn 𝐵)
77 fniniseg 6501 . . . . . . . . . . 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)))))
7875, 76, 773syl 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)))))
7972, 78mpbid 222 . . . . . . . . 9 ((𝜑𝑀 ∈ (1...𝑁)) → (((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝐵 ∧ (𝐹‘((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁))) = (𝐺‘((𝑀 − 1) / 𝑁))))
8079simpld 477 . . . . . . . 8 ((𝜑𝑀 ∈ (1...𝑁)) → ((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝐵)
8179simprd 482 . . . . . . . . 9 ((𝜑𝑀 ∈ (1...𝑁)) → (𝐹‘((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁))) = (𝐺‘((𝑀 − 1) / 𝑁)))
821adantl 473 . . . . . . . . . . . . . . . 16 ((𝜑𝑀 ∈ (1...𝑁)) → 𝑀 ∈ ℕ)
8382nnred 11227 . . . . . . . . . . . . . . 15 ((𝜑𝑀 ∈ (1...𝑁)) → 𝑀 ∈ ℝ)
84 peano2rem 10540 . . . . . . . . . . . . . . 15 (𝑀 ∈ ℝ → (𝑀 − 1) ∈ ℝ)
8583, 84syl 17 . . . . . . . . . . . . . 14 ((𝜑𝑀 ∈ (1...𝑁)) → (𝑀 − 1) ∈ ℝ)
869adantr 472 . . . . . . . . . . . . . 14 ((𝜑𝑀 ∈ (1...𝑁)) → 𝑁 ∈ ℕ)
8785, 86nndivred 11261 . . . . . . . . . . . . 13 ((𝜑𝑀 ∈ (1...𝑁)) → ((𝑀 − 1) / 𝑁) ∈ ℝ)
8887rexrd 10281 . . . . . . . . . . . 12 ((𝜑𝑀 ∈ (1...𝑁)) → ((𝑀 − 1) / 𝑁) ∈ ℝ*)
8983, 86nndivred 11261 . . . . . . . . . . . . 13 ((𝜑𝑀 ∈ (1...𝑁)) → (𝑀 / 𝑁) ∈ ℝ)
9089rexrd 10281 . . . . . . . . . . . 12 ((𝜑𝑀 ∈ (1...𝑁)) → (𝑀 / 𝑁) ∈ ℝ*)
9183ltm1d 11148 . . . . . . . . . . . . . 14 ((𝜑𝑀 ∈ (1...𝑁)) → (𝑀 − 1) < 𝑀)
9286nnred 11227 . . . . . . . . . . . . . . 15 ((𝜑𝑀 ∈ (1...𝑁)) → 𝑁 ∈ ℝ)
9386nngt0d 11256 . . . . . . . . . . . . . . 15 ((𝜑𝑀 ∈ (1...𝑁)) → 0 < 𝑁)
94 ltdiv1 11079 . . . . . . . . . . . . . . 15 (((𝑀 − 1) ∈ ℝ ∧ 𝑀 ∈ ℝ ∧ (𝑁 ∈ ℝ ∧ 0 < 𝑁)) → ((𝑀 − 1) < 𝑀 ↔ ((𝑀 − 1) / 𝑁) < (𝑀 / 𝑁)))
9585, 83, 92, 93, 94syl112anc 1481 . . . . . . . . . . . . . 14 ((𝜑𝑀 ∈ (1...𝑁)) → ((𝑀 − 1) < 𝑀 ↔ ((𝑀 − 1) / 𝑁) < (𝑀 / 𝑁)))
9691, 95mpbid 222 . . . . . . . . . . . . 13 ((𝜑𝑀 ∈ (1...𝑁)) → ((𝑀 − 1) / 𝑁) < (𝑀 / 𝑁))
9787, 89, 96ltled 10377 . . . . . . . . . . . 12 ((𝜑𝑀 ∈ (1...𝑁)) → ((𝑀 − 1) / 𝑁) ≤ (𝑀 / 𝑁))
98 lbicc2 12481 . . . . . . . . . . . 12 ((((𝑀 − 1) / 𝑁) ∈ ℝ* ∧ (𝑀 / 𝑁) ∈ ℝ* ∧ ((𝑀 − 1) / 𝑁) ≤ (𝑀 / 𝑁)) → ((𝑀 − 1) / 𝑁) ∈ (((𝑀 − 1) / 𝑁)[,](𝑀 / 𝑁)))
9988, 90, 97, 98syl3anc 1477 . . . . . . . . . . 11 ((𝜑𝑀 ∈ (1...𝑁)) → ((𝑀 − 1) / 𝑁) ∈ (((𝑀 − 1) / 𝑁)[,](𝑀 / 𝑁)))
10099, 14syl6eleqr 2850 . . . . . . . . . 10 ((𝜑𝑀 ∈ (1...𝑁)) → ((𝑀 − 1) / 𝑁) ∈ 𝑊)
1012, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 23, 14, 100cvmliftlem3 31576 . . . . . . . . 9 ((𝜑𝑀 ∈ (1...𝑁)) → (𝐺‘((𝑀 − 1) / 𝑁)) ∈ (1st ‘(𝑇𝑀)))
10281, 101eqeltrd 2839 . . . . . . . 8 ((𝜑𝑀 ∈ (1...𝑁)) → (𝐹‘((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁))) ∈ (1st ‘(𝑇𝑀)))
103 eqid 2760 . . . . . . . . 9 (𝑏 ∈ (2nd ‘(𝑇𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏) = (𝑏 ∈ (2nd ‘(𝑇𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏)
1042, 3, 103cvmsiota 31566 . . . . . . . 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)))
10517, 64, 80, 102, 104syl13anc 1479 . . . . . . 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)))
106105simpld 477 . . . . . 6 ((𝜑𝑀 ∈ (1...𝑁)) → (𝑏 ∈ (2nd ‘(𝑇𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏) ∈ (2nd ‘(𝑇𝑀)))
1072cvmshmeo 31560 . . . . . 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)))))
10864, 106, 107syl2anc 696 . . . . 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)))))
109 hmeocnvcn 21766 . . . . 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))))
110108, 109syl 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))))
11128, 71, 110cnmpt11f 21669 . . 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))))
11220, 111sseldd 3745 . 2 ((𝜑𝑀 ∈ (1...𝑁)) → (𝑧𝑊 ↦ ((𝐹 ↾ (𝑏 ∈ (2nd ‘(𝑇𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏))‘(𝐺𝑧))) ∈ ((𝐿t 𝑊) Cn 𝐶))
11316, 112eqeltrd 2839 1 ((𝜑𝑀 ∈ (1...𝑁)) → (𝑄𝑀) ∈ ((𝐿t 𝑊) Cn 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1632  wcel 2139  wral 3050  {crab 3054  Vcvv 3340  cdif 3712  cun 3713  cin 3714  wss 3715  c0 4058  𝒫 cpw 4302  {csn 4321  cop 4327   cuni 4588   ciun 4672   class class class wbr 4804  cmpt 4881   I cid 5173   × cxp 5264  ccnv 5265  ran crn 5267  cres 5268  cima 5269   Fn wfn 6044  wf 6045  cfv 6049  crio 6773  (class class class)co 6813  cmpt2 6815  1st c1st 7331  2nd c2nd 7332  cr 10127  0cc0 10128  1c1 10129  *cxr 10265   < clt 10266  cle 10267  cmin 10458   / cdiv 10876  cn 11212  (,)cioo 12368  [,]cicc 12371  ...cfz 12519  seqcseq 12995  t crest 16283  topGenctg 16300  Topctop 20900  TopOnctopon 20917   Cn ccn 21230  Homeochmeo 21758  IIcii 22879   CovMap ccvm 31544
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 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-cnex 10184  ax-resscn 10185  ax-1cn 10186  ax-icn 10187  ax-addcl 10188  ax-addrcl 10189  ax-mulcl 10190  ax-mulrcl 10191  ax-mulcom 10192  ax-addass 10193  ax-mulass 10194  ax-distr 10195  ax-i2m1 10196  ax-1ne0 10197  ax-1rid 10198  ax-rnegex 10199  ax-rrecex 10200  ax-cnre 10201  ax-pre-lttri 10202  ax-pre-lttrn 10203  ax-pre-ltadd 10204  ax-pre-mulgt0 10205  ax-pre-sup 10206
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6774  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-om 7231  df-1st 7333  df-2nd 7334  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-oadd 7733  df-er 7911  df-map 8025  df-en 8122  df-dom 8123  df-sdom 8124  df-fin 8125  df-fi 8482  df-sup 8513  df-inf 8514  df-pnf 10268  df-mnf 10269  df-xr 10270  df-ltxr 10271  df-le 10272  df-sub 10460  df-neg 10461  df-div 10877  df-nn 11213  df-2 11271  df-3 11272  df-n0 11485  df-z 11570  df-uz 11880  df-q 11982  df-rp 12026  df-xneg 12139  df-xadd 12140  df-xmul 12141  df-ioo 12372  df-icc 12375  df-fz 12520  df-seq 12996  df-exp 13055  df-cj 14038  df-re 14039  df-im 14040  df-sqrt 14174  df-abs 14175  df-rest 16285  df-topgen 16306  df-psmet 19940  df-xmet 19941  df-met 19942  df-bl 19943  df-mopn 19944  df-top 20901  df-topon 20918  df-bases 20952  df-cn 21233  df-hmeo 21760  df-ii 22881  df-cvm 31545
This theorem is referenced by:  cvmliftlem10  31583
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