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Theorem cvmliftlem8 34746
Description: Lemma for cvmlift 34753. 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 13537 . . 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 34743 . . 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 480 . . . 4 ((𝜑𝑀 ∈ (1...𝑁)) → 𝐹 ∈ (𝐶 CovMap 𝐽))
18 cvmtop1 34714 . . . 4 (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐶 ∈ Top)
19 cnrest2r 23110 . . . 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 24599 . . . . . 6 (topGen‘ran (,)) ∈ (TopOn‘ℝ)
2212, 21eqeltri 2828 . . . . 5 𝐿 ∈ (TopOn‘ℝ)
23 simpr 484 . . . . . . 7 ((𝜑𝑀 ∈ (1...𝑁)) → 𝑀 ∈ (1...𝑁))
242, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 23, 14cvmliftlem2 34740 . . . . . 6 ((𝜑𝑀 ∈ (1...𝑁)) → 𝑊 ⊆ (0[,]1))
25 unitssre 13483 . . . . . 6 (0[,]1) ⊆ ℝ
2624, 25sstrdi 3994 . . . . 5 ((𝜑𝑀 ∈ (1...𝑁)) → 𝑊 ⊆ ℝ)
27 resttopon 22984 . . . . 5 ((𝐿 ∈ (TopOn‘ℝ) ∧ 𝑊 ⊆ ℝ) → (𝐿t 𝑊) ∈ (TopOn‘𝑊))
2822, 26, 27sylancr 586 . . . 4 ((𝜑𝑀 ∈ (1...𝑁)) → (𝐿t 𝑊) ∈ (TopOn‘𝑊))
29 eqid 2731 . . . . . . 7 (II ↾t 𝑊) = (II ↾t 𝑊)
30 iitopon 24718 . . . . . . . 8 II ∈ (TopOn‘(0[,]1))
3130a1i 11 . . . . . . 7 ((𝜑𝑀 ∈ (1...𝑁)) → II ∈ (TopOn‘(0[,]1)))
326adantr 480 . . . . . . . . . 10 ((𝜑𝑀 ∈ (1...𝑁)) → 𝐺 ∈ (II Cn 𝐽))
33 iiuni 24720 . . . . . . . . . . 11 (0[,]1) = II
3433, 4cnf 23069 . . . . . . . . . 10 (𝐺 ∈ (II Cn 𝐽) → 𝐺:(0[,]1)⟶𝑋)
3532, 34syl 17 . . . . . . . . 9 ((𝜑𝑀 ∈ (1...𝑁)) → 𝐺:(0[,]1)⟶𝑋)
3635feqmptd 6960 . . . . . . . 8 ((𝜑𝑀 ∈ (1...𝑁)) → 𝐺 = (𝑧 ∈ (0[,]1) ↦ (𝐺𝑧)))
3736, 32eqeltrrd 2833 . . . . . . 7 ((𝜑𝑀 ∈ (1...𝑁)) → (𝑧 ∈ (0[,]1) ↦ (𝐺𝑧)) ∈ (II Cn 𝐽))
3829, 31, 24, 37cnmpt1res 23499 . . . . . 6 ((𝜑𝑀 ∈ (1...𝑁)) → (𝑧𝑊 ↦ (𝐺𝑧)) ∈ ((II ↾t 𝑊) Cn 𝐽))
39 dfii2 24721 . . . . . . . . . 10 II = ((topGen‘ran (,)) ↾t (0[,]1))
4012oveq1i 7422 . . . . . . . . . 10 (𝐿t (0[,]1)) = ((topGen‘ran (,)) ↾t (0[,]1))
4139, 40eqtr4i 2762 . . . . . . . . 9 II = (𝐿t (0[,]1))
4241oveq1i 7422 . . . . . . . 8 (II ↾t 𝑊) = ((𝐿t (0[,]1)) ↾t 𝑊)
43 retop 24597 . . . . . . . . . . 11 (topGen‘ran (,)) ∈ Top
4412, 43eqeltri 2828 . . . . . . . . . 10 𝐿 ∈ Top
4544a1i 11 . . . . . . . . 9 ((𝜑𝑀 ∈ (1...𝑁)) → 𝐿 ∈ Top)
46 ovexd 7447 . . . . . . . . 9 ((𝜑𝑀 ∈ (1...𝑁)) → (0[,]1) ∈ V)
47 restabs 22988 . . . . . . . . 9 ((𝐿 ∈ Top ∧ 𝑊 ⊆ (0[,]1) ∧ (0[,]1) ∈ V) → ((𝐿t (0[,]1)) ↾t 𝑊) = (𝐿t 𝑊))
4845, 24, 46, 47syl3anc 1370 . . . . . . . 8 ((𝜑𝑀 ∈ (1...𝑁)) → ((𝐿t (0[,]1)) ↾t 𝑊) = (𝐿t 𝑊))
4942, 48eqtrid 2783 . . . . . . 7 ((𝜑𝑀 ∈ (1...𝑁)) → (II ↾t 𝑊) = (𝐿t 𝑊))
5049oveq1d 7427 . . . . . 6 ((𝜑𝑀 ∈ (1...𝑁)) → ((II ↾t 𝑊) Cn 𝐽) = ((𝐿t 𝑊) Cn 𝐽))
5138, 50eleqtrd 2834 . . . . 5 ((𝜑𝑀 ∈ (1...𝑁)) → (𝑧𝑊 ↦ (𝐺𝑧)) ∈ ((𝐿t 𝑊) Cn 𝐽))
52 cvmtop2 34715 . . . . . . . 8 (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐽 ∈ Top)
5317, 52syl 17 . . . . . . 7 ((𝜑𝑀 ∈ (1...𝑁)) → 𝐽 ∈ Top)
544toptopon 22738 . . . . . . 7 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
5553, 54sylib 217 . . . . . 6 ((𝜑𝑀 ∈ (1...𝑁)) → 𝐽 ∈ (TopOn‘𝑋))
56 simprl 768 . . . . . . . . . 10 ((𝜑 ∧ (𝑀 ∈ (1...𝑁) ∧ 𝑧𝑊)) → 𝑀 ∈ (1...𝑁))
57 simprr 770 . . . . . . . . . 10 ((𝜑 ∧ (𝑀 ∈ (1...𝑁) ∧ 𝑧𝑊)) → 𝑧𝑊)
582, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 56, 14, 57cvmliftlem3 34741 . . . . . . . . 9 ((𝜑 ∧ (𝑀 ∈ (1...𝑁) ∧ 𝑧𝑊)) → (𝐺𝑧) ∈ (1st ‘(𝑇𝑀)))
5958anassrs 467 . . . . . . . 8 (((𝜑𝑀 ∈ (1...𝑁)) ∧ 𝑧𝑊) → (𝐺𝑧) ∈ (1st ‘(𝑇𝑀)))
6059fmpttd 7116 . . . . . . 7 ((𝜑𝑀 ∈ (1...𝑁)) → (𝑧𝑊 ↦ (𝐺𝑧)):𝑊⟶(1st ‘(𝑇𝑀)))
6160frnd 6725 . . . . . 6 ((𝜑𝑀 ∈ (1...𝑁)) → ran (𝑧𝑊 ↦ (𝐺𝑧)) ⊆ (1st ‘(𝑇𝑀)))
622, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 23cvmliftlem1 34739 . . . . . . . 8 ((𝜑𝑀 ∈ (1...𝑁)) → (2nd ‘(𝑇𝑀)) ∈ (𝑆‘(1st ‘(𝑇𝑀))))
632cvmsrcl 34718 . . . . . . . 8 ((2nd ‘(𝑇𝑀)) ∈ (𝑆‘(1st ‘(𝑇𝑀))) → (1st ‘(𝑇𝑀)) ∈ 𝐽)
64 elssuni 4941 . . . . . . . 8 ((1st ‘(𝑇𝑀)) ∈ 𝐽 → (1st ‘(𝑇𝑀)) ⊆ 𝐽)
6562, 63, 643syl 18 . . . . . . 7 ((𝜑𝑀 ∈ (1...𝑁)) → (1st ‘(𝑇𝑀)) ⊆ 𝐽)
6665, 4sseqtrrdi 4033 . . . . . 6 ((𝜑𝑀 ∈ (1...𝑁)) → (1st ‘(𝑇𝑀)) ⊆ 𝑋)
67 cnrest2 23109 . . . . . 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 1370 . . . . 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 34745 . . . . . . . . . 10 ((𝜑𝑀 ∈ (1...𝑁)) → ((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ (𝐹 “ {(𝐺‘((𝑀 − 1) / 𝑁))}))
71 cvmcn 34716 . . . . . . . . . . . 12 (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐹 ∈ (𝐶 Cn 𝐽))
723, 4cnf 23069 . . . . . . . . . . . 12 (𝐹 ∈ (𝐶 Cn 𝐽) → 𝐹:𝐵𝑋)
7317, 71, 723syl 18 . . . . . . . . . . 11 ((𝜑𝑀 ∈ (1...𝑁)) → 𝐹:𝐵𝑋)
74 ffn 6717 . . . . . . . . . . 11 (𝐹:𝐵𝑋𝐹 Fn 𝐵)
75 fniniseg 7061 . . . . . . . . . . 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 494 . . . . . . . 8 ((𝜑𝑀 ∈ (1...𝑁)) → ((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝐵)
7977simprd 495 . . . . . . . . 9 ((𝜑𝑀 ∈ (1...𝑁)) → (𝐹‘((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁))) = (𝐺‘((𝑀 − 1) / 𝑁)))
801adantl 481 . . . . . . . . . . . . . . . 16 ((𝜑𝑀 ∈ (1...𝑁)) → 𝑀 ∈ ℕ)
8180nnred 12234 . . . . . . . . . . . . . . 15 ((𝜑𝑀 ∈ (1...𝑁)) → 𝑀 ∈ ℝ)
82 peano2rem 11534 . . . . . . . . . . . . . . 15 (𝑀 ∈ ℝ → (𝑀 − 1) ∈ ℝ)
8381, 82syl 17 . . . . . . . . . . . . . 14 ((𝜑𝑀 ∈ (1...𝑁)) → (𝑀 − 1) ∈ ℝ)
849adantr 480 . . . . . . . . . . . . . 14 ((𝜑𝑀 ∈ (1...𝑁)) → 𝑁 ∈ ℕ)
8583, 84nndivred 12273 . . . . . . . . . . . . 13 ((𝜑𝑀 ∈ (1...𝑁)) → ((𝑀 − 1) / 𝑁) ∈ ℝ)
8685rexrd 11271 . . . . . . . . . . . 12 ((𝜑𝑀 ∈ (1...𝑁)) → ((𝑀 − 1) / 𝑁) ∈ ℝ*)
8781, 84nndivred 12273 . . . . . . . . . . . . 13 ((𝜑𝑀 ∈ (1...𝑁)) → (𝑀 / 𝑁) ∈ ℝ)
8887rexrd 11271 . . . . . . . . . . . 12 ((𝜑𝑀 ∈ (1...𝑁)) → (𝑀 / 𝑁) ∈ ℝ*)
8981ltm1d 12153 . . . . . . . . . . . . . 14 ((𝜑𝑀 ∈ (1...𝑁)) → (𝑀 − 1) < 𝑀)
9084nnred 12234 . . . . . . . . . . . . . . 15 ((𝜑𝑀 ∈ (1...𝑁)) → 𝑁 ∈ ℝ)
9184nngt0d 12268 . . . . . . . . . . . . . . 15 ((𝜑𝑀 ∈ (1...𝑁)) → 0 < 𝑁)
92 ltdiv1 12085 . . . . . . . . . . . . . . 15 (((𝑀 − 1) ∈ ℝ ∧ 𝑀 ∈ ℝ ∧ (𝑁 ∈ ℝ ∧ 0 < 𝑁)) → ((𝑀 − 1) < 𝑀 ↔ ((𝑀 − 1) / 𝑁) < (𝑀 / 𝑁)))
9383, 81, 90, 91, 92syl112anc 1373 . . . . . . . . . . . . . 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 11369 . . . . . . . . . . . 12 ((𝜑𝑀 ∈ (1...𝑁)) → ((𝑀 − 1) / 𝑁) ≤ (𝑀 / 𝑁))
96 lbicc2 13448 . . . . . . . . . . . 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 1370 . . . . . . . . . . 11 ((𝜑𝑀 ∈ (1...𝑁)) → ((𝑀 − 1) / 𝑁) ∈ (((𝑀 − 1) / 𝑁)[,](𝑀 / 𝑁)))
9897, 14eleqtrrdi 2843 . . . . . . . . . 10 ((𝜑𝑀 ∈ (1...𝑁)) → ((𝑀 − 1) / 𝑁) ∈ 𝑊)
992, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 23, 14, 98cvmliftlem3 34741 . . . . . . . . 9 ((𝜑𝑀 ∈ (1...𝑁)) → (𝐺‘((𝑀 − 1) / 𝑁)) ∈ (1st ‘(𝑇𝑀)))
10079, 99eqeltrd 2832 . . . . . . . 8 ((𝜑𝑀 ∈ (1...𝑁)) → (𝐹‘((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁))) ∈ (1st ‘(𝑇𝑀)))
101 eqid 2731 . . . . . . . . 9 (𝑏 ∈ (2nd ‘(𝑇𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏) = (𝑏 ∈ (2nd ‘(𝑇𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏)
1022, 3, 101cvmsiota 34731 . . . . . . . 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 1371 . . . . . . 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 494 . . . . . 6 ((𝜑𝑀 ∈ (1...𝑁)) → (𝑏 ∈ (2nd ‘(𝑇𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏) ∈ (2nd ‘(𝑇𝑀)))
1052cvmshmeo 34725 . . . . . 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 583 . . . . 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 23584 . . . . 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 23487 . . 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 3983 . 2 ((𝜑𝑀 ∈ (1...𝑁)) → (𝑧𝑊 ↦ ((𝐹 ↾ (𝑏 ∈ (2nd ‘(𝑇𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏))‘(𝐺𝑧))) ∈ ((𝐿t 𝑊) Cn 𝐶))
11116, 110eqeltrd 2832 1 ((𝜑𝑀 ∈ (1...𝑁)) → (𝑄𝑀) ∈ ((𝐿t 𝑊) Cn 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1540  wcel 2105  wral 3060  {crab 3431  Vcvv 3473  cdif 3945  cun 3946  cin 3947  wss 3948  c0 4322  𝒫 cpw 4602  {csn 4628  cop 4634   cuni 4908   ciun 4997   class class class wbr 5148  cmpt 5231   I cid 5573   × cxp 5674  ccnv 5675  ran crn 5677  cres 5678  cima 5679   Fn wfn 6538  wf 6539  cfv 6543  crio 7367  (class class class)co 7412  cmpo 7414  1st c1st 7977  2nd c2nd 7978  cr 11115  0cc0 11116  1c1 11117  *cxr 11254   < clt 11255  cle 11256  cmin 11451   / cdiv 11878  cn 12219  (,)cioo 13331  [,]cicc 13334  ...cfz 13491  seqcseq 13973  t crest 17373  topGenctg 17390  Topctop 22714  TopOnctopon 22731   Cn ccn 23047  Homeochmeo 23576  IIcii 24714   CovMap ccvm 34709
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729  ax-cnex 11172  ax-resscn 11173  ax-1cn 11174  ax-icn 11175  ax-addcl 11176  ax-addrcl 11177  ax-mulcl 11178  ax-mulrcl 11179  ax-mulcom 11180  ax-addass 11181  ax-mulass 11182  ax-distr 11183  ax-i2m1 11184  ax-1ne0 11185  ax-1rid 11186  ax-rnegex 11187  ax-rrecex 11188  ax-cnre 11189  ax-pre-lttri 11190  ax-pre-lttrn 11191  ax-pre-ltadd 11192  ax-pre-mulgt0 11193  ax-pre-sup 11194
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7860  df-1st 7979  df-2nd 7980  df-frecs 8272  df-wrecs 8303  df-recs 8377  df-rdg 8416  df-er 8709  df-map 8828  df-en 8946  df-dom 8947  df-sdom 8948  df-fin 8949  df-fi 9412  df-sup 9443  df-inf 9444  df-pnf 11257  df-mnf 11258  df-xr 11259  df-ltxr 11260  df-le 11261  df-sub 11453  df-neg 11454  df-div 11879  df-nn 12220  df-2 12282  df-3 12283  df-n0 12480  df-z 12566  df-uz 12830  df-q 12940  df-rp 12982  df-xneg 13099  df-xadd 13100  df-xmul 13101  df-ioo 13335  df-icc 13338  df-fz 13492  df-seq 13974  df-exp 14035  df-cj 15053  df-re 15054  df-im 15055  df-sqrt 15189  df-abs 15190  df-rest 17375  df-topgen 17396  df-psmet 21224  df-xmet 21225  df-met 21226  df-bl 21227  df-mopn 21228  df-top 22715  df-topon 22732  df-bases 22768  df-cn 23050  df-hmeo 23578  df-ii 24716  df-cvm 34710
This theorem is referenced by:  cvmliftlem10  34748
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