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Theorem cvmliftlem8 34283
Description: Lemma for cvmlift 34290. The functions 𝑄 are continuous functions because they are defined as β—‘(𝐹 β†Ύ 𝐼) ∘ 𝐺 where 𝐺 is continuous and (𝐹 β†Ύ 𝐼) is a homeomorphism. (Contributed by Mario Carneiro, 16-Feb-2015.)
Hypotheses
Ref Expression
cvmliftlem.1 𝑆 = (π‘˜ ∈ 𝐽 ↦ {𝑠 ∈ (𝒫 𝐢 βˆ– {βˆ…}) ∣ (βˆͺ 𝑠 = (◑𝐹 β€œ π‘˜) ∧ βˆ€π‘’ ∈ 𝑠 (βˆ€π‘£ ∈ (𝑠 βˆ– {𝑒})(𝑒 ∩ 𝑣) = βˆ… ∧ (𝐹 β†Ύ 𝑒) ∈ ((𝐢 β†Ύt 𝑒)Homeo(𝐽 β†Ύt π‘˜))))})
cvmliftlem.b 𝐡 = βˆͺ 𝐢
cvmliftlem.x 𝑋 = βˆͺ 𝐽
cvmliftlem.f (πœ‘ β†’ 𝐹 ∈ (𝐢 CovMap 𝐽))
cvmliftlem.g (πœ‘ β†’ 𝐺 ∈ (II Cn 𝐽))
cvmliftlem.p (πœ‘ β†’ 𝑃 ∈ 𝐡)
cvmliftlem.e (πœ‘ β†’ (πΉβ€˜π‘ƒ) = (πΊβ€˜0))
cvmliftlem.n (πœ‘ β†’ 𝑁 ∈ β„•)
cvmliftlem.t (πœ‘ β†’ 𝑇:(1...𝑁)⟢βˆͺ 𝑗 ∈ 𝐽 ({𝑗} Γ— (π‘†β€˜π‘—)))
cvmliftlem.a (πœ‘ β†’ βˆ€π‘˜ ∈ (1...𝑁)(𝐺 β€œ (((π‘˜ βˆ’ 1) / 𝑁)[,](π‘˜ / 𝑁))) βŠ† (1st β€˜(π‘‡β€˜π‘˜)))
cvmliftlem.l 𝐿 = (topGenβ€˜ran (,))
cvmliftlem.q 𝑄 = seq0((π‘₯ ∈ V, π‘š ∈ β„• ↦ (𝑧 ∈ (((π‘š βˆ’ 1) / 𝑁)[,](π‘š / 𝑁)) ↦ (β—‘(𝐹 β†Ύ (℩𝑏 ∈ (2nd β€˜(π‘‡β€˜π‘š))(π‘₯β€˜((π‘š βˆ’ 1) / 𝑁)) ∈ 𝑏))β€˜(πΊβ€˜π‘§)))), (( I β†Ύ β„•) βˆͺ {⟨0, {⟨0, π‘ƒβŸ©}⟩}))
cvmliftlem5.3 π‘Š = (((𝑀 βˆ’ 1) / 𝑁)[,](𝑀 / 𝑁))
Assertion
Ref Expression
cvmliftlem8 ((πœ‘ ∧ 𝑀 ∈ (1...𝑁)) β†’ (π‘„β€˜π‘€) ∈ ((𝐿 β†Ύt π‘Š) Cn 𝐢))
Distinct variable groups:   𝑣,𝑏,𝑧,𝐡   𝑗,𝑏,π‘˜,π‘š,𝑠,𝑒,π‘₯,𝐹,𝑣,𝑧   𝑧,𝐿   𝑀,𝑏,𝑗,π‘˜,π‘š,𝑠,𝑒,𝑣,π‘₯,𝑧   𝑃,𝑏,π‘˜,π‘š,𝑒,𝑣,π‘₯,𝑧   𝐢,𝑏,𝑗,π‘˜,𝑠,𝑒,𝑣,𝑧   πœ‘,𝑗,𝑠,π‘₯,𝑧   𝑁,𝑏,π‘˜,π‘š,𝑒,𝑣,π‘₯,𝑧   𝑆,𝑏,𝑗,π‘˜,𝑠,𝑒,𝑣,π‘₯,𝑧   𝑗,𝑋   𝐺,𝑏,𝑗,π‘˜,π‘š,𝑠,𝑒,𝑣,π‘₯,𝑧   𝑇,𝑏,𝑗,π‘˜,π‘š,𝑠,𝑒,𝑣,π‘₯,𝑧   𝐽,𝑏,𝑗,π‘˜,𝑠,𝑒,𝑣,π‘₯,𝑧   𝑄,𝑏,π‘˜,π‘š,𝑒,𝑣,π‘₯,𝑧   π‘˜,π‘Š,π‘š,π‘₯,𝑧
Allowed substitution hints:   πœ‘(𝑣,𝑒,π‘˜,π‘š,𝑏)   𝐡(π‘₯,𝑒,𝑗,π‘˜,π‘š,𝑠)   𝐢(π‘₯,π‘š)   𝑃(𝑗,𝑠)   𝑄(𝑗,𝑠)   𝑆(π‘š)   𝐽(π‘š)   𝐿(π‘₯,𝑣,𝑒,𝑗,π‘˜,π‘š,𝑠,𝑏)   𝑁(𝑗,𝑠)   π‘Š(𝑣,𝑒,𝑗,𝑠,𝑏)   𝑋(π‘₯,𝑧,𝑣,𝑒,π‘˜,π‘š,𝑠,𝑏)
Proof of Theorem cvmliftlem8
StepHypRef Expression
1 elfznn 13530 . . 3 (𝑀 ∈ (1...𝑁) β†’ 𝑀 ∈ β„•)
2 cvmliftlem.1 . . . 4 𝑆 = (π‘˜ ∈ 𝐽 ↦ {𝑠 ∈ (𝒫 𝐢 βˆ– {βˆ…}) ∣ (βˆͺ 𝑠 = (◑𝐹 β€œ π‘˜) ∧ βˆ€π‘’ ∈ 𝑠 (βˆ€π‘£ ∈ (𝑠 βˆ– {𝑒})(𝑒 ∩ 𝑣) = βˆ… ∧ (𝐹 β†Ύ 𝑒) ∈ ((𝐢 β†Ύt 𝑒)Homeo(𝐽 β†Ύt π‘˜))))})
3 cvmliftlem.b . . . 4 𝐡 = βˆͺ 𝐢
4 cvmliftlem.x . . . 4 𝑋 = βˆͺ 𝐽
5 cvmliftlem.f . . . 4 (πœ‘ β†’ 𝐹 ∈ (𝐢 CovMap 𝐽))
6 cvmliftlem.g . . . 4 (πœ‘ β†’ 𝐺 ∈ (II Cn 𝐽))
7 cvmliftlem.p . . . 4 (πœ‘ β†’ 𝑃 ∈ 𝐡)
8 cvmliftlem.e . . . 4 (πœ‘ β†’ (πΉβ€˜π‘ƒ) = (πΊβ€˜0))
9 cvmliftlem.n . . . 4 (πœ‘ β†’ 𝑁 ∈ β„•)
10 cvmliftlem.t . . . 4 (πœ‘ β†’ 𝑇:(1...𝑁)⟢βˆͺ 𝑗 ∈ 𝐽 ({𝑗} Γ— (π‘†β€˜π‘—)))
11 cvmliftlem.a . . . 4 (πœ‘ β†’ βˆ€π‘˜ ∈ (1...𝑁)(𝐺 β€œ (((π‘˜ βˆ’ 1) / 𝑁)[,](π‘˜ / 𝑁))) βŠ† (1st β€˜(π‘‡β€˜π‘˜)))
12 cvmliftlem.l . . . 4 𝐿 = (topGenβ€˜ran (,))
13 cvmliftlem.q . . . 4 𝑄 = seq0((π‘₯ ∈ V, π‘š ∈ β„• ↦ (𝑧 ∈ (((π‘š βˆ’ 1) / 𝑁)[,](π‘š / 𝑁)) ↦ (β—‘(𝐹 β†Ύ (℩𝑏 ∈ (2nd β€˜(π‘‡β€˜π‘š))(π‘₯β€˜((π‘š βˆ’ 1) / 𝑁)) ∈ 𝑏))β€˜(πΊβ€˜π‘§)))), (( I β†Ύ β„•) βˆͺ {⟨0, {⟨0, π‘ƒβŸ©}⟩}))
14 cvmliftlem5.3 . . . 4 π‘Š = (((𝑀 βˆ’ 1) / 𝑁)[,](𝑀 / 𝑁))
152, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14cvmliftlem5 34280 . . 3 ((πœ‘ ∧ 𝑀 ∈ β„•) β†’ (π‘„β€˜π‘€) = (𝑧 ∈ π‘Š ↦ (β—‘(𝐹 β†Ύ (℩𝑏 ∈ (2nd β€˜(π‘‡β€˜π‘€))((π‘„β€˜(𝑀 βˆ’ 1))β€˜((𝑀 βˆ’ 1) / 𝑁)) ∈ 𝑏))β€˜(πΊβ€˜π‘§))))
161, 15sylan2 594 . 2 ((πœ‘ ∧ 𝑀 ∈ (1...𝑁)) β†’ (π‘„β€˜π‘€) = (𝑧 ∈ π‘Š ↦ (β—‘(𝐹 β†Ύ (℩𝑏 ∈ (2nd β€˜(π‘‡β€˜π‘€))((π‘„β€˜(𝑀 βˆ’ 1))β€˜((𝑀 βˆ’ 1) / 𝑁)) ∈ 𝑏))β€˜(πΊβ€˜π‘§))))
175adantr 482 . . . 4 ((πœ‘ ∧ 𝑀 ∈ (1...𝑁)) β†’ 𝐹 ∈ (𝐢 CovMap 𝐽))
18 cvmtop1 34251 . . . 4 (𝐹 ∈ (𝐢 CovMap 𝐽) β†’ 𝐢 ∈ Top)
19 cnrest2r 22791 . . . 4 (𝐢 ∈ Top β†’ ((𝐿 β†Ύt π‘Š) Cn (𝐢 β†Ύt (℩𝑏 ∈ (2nd β€˜(π‘‡β€˜π‘€))((π‘„β€˜(𝑀 βˆ’ 1))β€˜((𝑀 βˆ’ 1) / 𝑁)) ∈ 𝑏))) βŠ† ((𝐿 β†Ύt π‘Š) Cn 𝐢))
2017, 18, 193syl 18 . . 3 ((πœ‘ ∧ 𝑀 ∈ (1...𝑁)) β†’ ((𝐿 β†Ύt π‘Š) Cn (𝐢 β†Ύt (℩𝑏 ∈ (2nd β€˜(π‘‡β€˜π‘€))((π‘„β€˜(𝑀 βˆ’ 1))β€˜((𝑀 βˆ’ 1) / 𝑁)) ∈ 𝑏))) βŠ† ((𝐿 β†Ύt π‘Š) Cn 𝐢))
21 retopon 24280 . . . . . 6 (topGenβ€˜ran (,)) ∈ (TopOnβ€˜β„)
2212, 21eqeltri 2830 . . . . 5 𝐿 ∈ (TopOnβ€˜β„)
23 simpr 486 . . . . . . 7 ((πœ‘ ∧ 𝑀 ∈ (1...𝑁)) β†’ 𝑀 ∈ (1...𝑁))
242, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 23, 14cvmliftlem2 34277 . . . . . 6 ((πœ‘ ∧ 𝑀 ∈ (1...𝑁)) β†’ π‘Š βŠ† (0[,]1))
25 unitssre 13476 . . . . . 6 (0[,]1) βŠ† ℝ
2624, 25sstrdi 3995 . . . . 5 ((πœ‘ ∧ 𝑀 ∈ (1...𝑁)) β†’ π‘Š βŠ† ℝ)
27 resttopon 22665 . . . . 5 ((𝐿 ∈ (TopOnβ€˜β„) ∧ π‘Š βŠ† ℝ) β†’ (𝐿 β†Ύt π‘Š) ∈ (TopOnβ€˜π‘Š))
2822, 26, 27sylancr 588 . . . 4 ((πœ‘ ∧ 𝑀 ∈ (1...𝑁)) β†’ (𝐿 β†Ύt π‘Š) ∈ (TopOnβ€˜π‘Š))
29 eqid 2733 . . . . . . 7 (II β†Ύt π‘Š) = (II β†Ύt π‘Š)
30 iitopon 24395 . . . . . . . 8 II ∈ (TopOnβ€˜(0[,]1))
3130a1i 11 . . . . . . 7 ((πœ‘ ∧ 𝑀 ∈ (1...𝑁)) β†’ II ∈ (TopOnβ€˜(0[,]1)))
326adantr 482 . . . . . . . . . 10 ((πœ‘ ∧ 𝑀 ∈ (1...𝑁)) β†’ 𝐺 ∈ (II Cn 𝐽))
33 iiuni 24397 . . . . . . . . . . 11 (0[,]1) = βˆͺ II
3433, 4cnf 22750 . . . . . . . . . 10 (𝐺 ∈ (II Cn 𝐽) β†’ 𝐺:(0[,]1)βŸΆπ‘‹)
3532, 34syl 17 . . . . . . . . 9 ((πœ‘ ∧ 𝑀 ∈ (1...𝑁)) β†’ 𝐺:(0[,]1)βŸΆπ‘‹)
3635feqmptd 6961 . . . . . . . 8 ((πœ‘ ∧ 𝑀 ∈ (1...𝑁)) β†’ 𝐺 = (𝑧 ∈ (0[,]1) ↦ (πΊβ€˜π‘§)))
3736, 32eqeltrrd 2835 . . . . . . 7 ((πœ‘ ∧ 𝑀 ∈ (1...𝑁)) β†’ (𝑧 ∈ (0[,]1) ↦ (πΊβ€˜π‘§)) ∈ (II Cn 𝐽))
3829, 31, 24, 37cnmpt1res 23180 . . . . . 6 ((πœ‘ ∧ 𝑀 ∈ (1...𝑁)) β†’ (𝑧 ∈ π‘Š ↦ (πΊβ€˜π‘§)) ∈ ((II β†Ύt π‘Š) Cn 𝐽))
39 dfii2 24398 . . . . . . . . . 10 II = ((topGenβ€˜ran (,)) β†Ύt (0[,]1))
4012oveq1i 7419 . . . . . . . . . 10 (𝐿 β†Ύt (0[,]1)) = ((topGenβ€˜ran (,)) β†Ύt (0[,]1))
4139, 40eqtr4i 2764 . . . . . . . . 9 II = (𝐿 β†Ύt (0[,]1))
4241oveq1i 7419 . . . . . . . 8 (II β†Ύt π‘Š) = ((𝐿 β†Ύt (0[,]1)) β†Ύt π‘Š)
43 retop 24278 . . . . . . . . . . 11 (topGenβ€˜ran (,)) ∈ Top
4412, 43eqeltri 2830 . . . . . . . . . 10 𝐿 ∈ Top
4544a1i 11 . . . . . . . . 9 ((πœ‘ ∧ 𝑀 ∈ (1...𝑁)) β†’ 𝐿 ∈ Top)
46 ovexd 7444 . . . . . . . . 9 ((πœ‘ ∧ 𝑀 ∈ (1...𝑁)) β†’ (0[,]1) ∈ V)
47 restabs 22669 . . . . . . . . 9 ((𝐿 ∈ Top ∧ π‘Š βŠ† (0[,]1) ∧ (0[,]1) ∈ V) β†’ ((𝐿 β†Ύt (0[,]1)) β†Ύt π‘Š) = (𝐿 β†Ύt π‘Š))
4845, 24, 46, 47syl3anc 1372 . . . . . . . 8 ((πœ‘ ∧ 𝑀 ∈ (1...𝑁)) β†’ ((𝐿 β†Ύt (0[,]1)) β†Ύt π‘Š) = (𝐿 β†Ύt π‘Š))
4942, 48eqtrid 2785 . . . . . . 7 ((πœ‘ ∧ 𝑀 ∈ (1...𝑁)) β†’ (II β†Ύt π‘Š) = (𝐿 β†Ύt π‘Š))
5049oveq1d 7424 . . . . . 6 ((πœ‘ ∧ 𝑀 ∈ (1...𝑁)) β†’ ((II β†Ύt π‘Š) Cn 𝐽) = ((𝐿 β†Ύt π‘Š) Cn 𝐽))
5138, 50eleqtrd 2836 . . . . 5 ((πœ‘ ∧ 𝑀 ∈ (1...𝑁)) β†’ (𝑧 ∈ π‘Š ↦ (πΊβ€˜π‘§)) ∈ ((𝐿 β†Ύt π‘Š) Cn 𝐽))
52 cvmtop2 34252 . . . . . . . 8 (𝐹 ∈ (𝐢 CovMap 𝐽) β†’ 𝐽 ∈ Top)
5317, 52syl 17 . . . . . . 7 ((πœ‘ ∧ 𝑀 ∈ (1...𝑁)) β†’ 𝐽 ∈ Top)
544toptopon 22419 . . . . . . 7 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOnβ€˜π‘‹))
5553, 54sylib 217 . . . . . 6 ((πœ‘ ∧ 𝑀 ∈ (1...𝑁)) β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
56 simprl 770 . . . . . . . . . 10 ((πœ‘ ∧ (𝑀 ∈ (1...𝑁) ∧ 𝑧 ∈ π‘Š)) β†’ 𝑀 ∈ (1...𝑁))
57 simprr 772 . . . . . . . . . 10 ((πœ‘ ∧ (𝑀 ∈ (1...𝑁) ∧ 𝑧 ∈ π‘Š)) β†’ 𝑧 ∈ π‘Š)
582, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 56, 14, 57cvmliftlem3 34278 . . . . . . . . 9 ((πœ‘ ∧ (𝑀 ∈ (1...𝑁) ∧ 𝑧 ∈ π‘Š)) β†’ (πΊβ€˜π‘§) ∈ (1st β€˜(π‘‡β€˜π‘€)))
5958anassrs 469 . . . . . . . 8 (((πœ‘ ∧ 𝑀 ∈ (1...𝑁)) ∧ 𝑧 ∈ π‘Š) β†’ (πΊβ€˜π‘§) ∈ (1st β€˜(π‘‡β€˜π‘€)))
6059fmpttd 7115 . . . . . . 7 ((πœ‘ ∧ 𝑀 ∈ (1...𝑁)) β†’ (𝑧 ∈ π‘Š ↦ (πΊβ€˜π‘§)):π‘ŠβŸΆ(1st β€˜(π‘‡β€˜π‘€)))
6160frnd 6726 . . . . . 6 ((πœ‘ ∧ 𝑀 ∈ (1...𝑁)) β†’ ran (𝑧 ∈ π‘Š ↦ (πΊβ€˜π‘§)) βŠ† (1st β€˜(π‘‡β€˜π‘€)))
622, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 23cvmliftlem1 34276 . . . . . . . 8 ((πœ‘ ∧ 𝑀 ∈ (1...𝑁)) β†’ (2nd β€˜(π‘‡β€˜π‘€)) ∈ (π‘†β€˜(1st β€˜(π‘‡β€˜π‘€))))
632cvmsrcl 34255 . . . . . . . 8 ((2nd β€˜(π‘‡β€˜π‘€)) ∈ (π‘†β€˜(1st β€˜(π‘‡β€˜π‘€))) β†’ (1st β€˜(π‘‡β€˜π‘€)) ∈ 𝐽)
64 elssuni 4942 . . . . . . . 8 ((1st β€˜(π‘‡β€˜π‘€)) ∈ 𝐽 β†’ (1st β€˜(π‘‡β€˜π‘€)) βŠ† βˆͺ 𝐽)
6562, 63, 643syl 18 . . . . . . 7 ((πœ‘ ∧ 𝑀 ∈ (1...𝑁)) β†’ (1st β€˜(π‘‡β€˜π‘€)) βŠ† βˆͺ 𝐽)
6665, 4sseqtrrdi 4034 . . . . . 6 ((πœ‘ ∧ 𝑀 ∈ (1...𝑁)) β†’ (1st β€˜(π‘‡β€˜π‘€)) βŠ† 𝑋)
67 cnrest2 22790 . . . . . 6 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ ran (𝑧 ∈ π‘Š ↦ (πΊβ€˜π‘§)) βŠ† (1st β€˜(π‘‡β€˜π‘€)) ∧ (1st β€˜(π‘‡β€˜π‘€)) βŠ† 𝑋) β†’ ((𝑧 ∈ π‘Š ↦ (πΊβ€˜π‘§)) ∈ ((𝐿 β†Ύt π‘Š) Cn 𝐽) ↔ (𝑧 ∈ π‘Š ↦ (πΊβ€˜π‘§)) ∈ ((𝐿 β†Ύt π‘Š) Cn (𝐽 β†Ύt (1st β€˜(π‘‡β€˜π‘€))))))
6855, 61, 66, 67syl3anc 1372 . . . . 5 ((πœ‘ ∧ 𝑀 ∈ (1...𝑁)) β†’ ((𝑧 ∈ π‘Š ↦ (πΊβ€˜π‘§)) ∈ ((𝐿 β†Ύt π‘Š) Cn 𝐽) ↔ (𝑧 ∈ π‘Š ↦ (πΊβ€˜π‘§)) ∈ ((𝐿 β†Ύt π‘Š) Cn (𝐽 β†Ύt (1st β€˜(π‘‡β€˜π‘€))))))
6951, 68mpbid 231 . . . 4 ((πœ‘ ∧ 𝑀 ∈ (1...𝑁)) β†’ (𝑧 ∈ π‘Š ↦ (πΊβ€˜π‘§)) ∈ ((𝐿 β†Ύt π‘Š) Cn (𝐽 β†Ύt (1st β€˜(π‘‡β€˜π‘€)))))
702, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14cvmliftlem7 34282 . . . . . . . . . 10 ((πœ‘ ∧ 𝑀 ∈ (1...𝑁)) β†’ ((π‘„β€˜(𝑀 βˆ’ 1))β€˜((𝑀 βˆ’ 1) / 𝑁)) ∈ (◑𝐹 β€œ {(πΊβ€˜((𝑀 βˆ’ 1) / 𝑁))}))
71 cvmcn 34253 . . . . . . . . . . . 12 (𝐹 ∈ (𝐢 CovMap 𝐽) β†’ 𝐹 ∈ (𝐢 Cn 𝐽))
723, 4cnf 22750 . . . . . . . . . . . 12 (𝐹 ∈ (𝐢 Cn 𝐽) β†’ 𝐹:π΅βŸΆπ‘‹)
7317, 71, 723syl 18 . . . . . . . . . . 11 ((πœ‘ ∧ 𝑀 ∈ (1...𝑁)) β†’ 𝐹:π΅βŸΆπ‘‹)
74 ffn 6718 . . . . . . . . . . 11 (𝐹:π΅βŸΆπ‘‹ β†’ 𝐹 Fn 𝐡)
75 fniniseg 7062 . . . . . . . . . . 11 (𝐹 Fn 𝐡 β†’ (((π‘„β€˜(𝑀 βˆ’ 1))β€˜((𝑀 βˆ’ 1) / 𝑁)) ∈ (◑𝐹 β€œ {(πΊβ€˜((𝑀 βˆ’ 1) / 𝑁))}) ↔ (((π‘„β€˜(𝑀 βˆ’ 1))β€˜((𝑀 βˆ’ 1) / 𝑁)) ∈ 𝐡 ∧ (πΉβ€˜((π‘„β€˜(𝑀 βˆ’ 1))β€˜((𝑀 βˆ’ 1) / 𝑁))) = (πΊβ€˜((𝑀 βˆ’ 1) / 𝑁)))))
7673, 74, 753syl 18 . . . . . . . . . 10 ((πœ‘ ∧ 𝑀 ∈ (1...𝑁)) β†’ (((π‘„β€˜(𝑀 βˆ’ 1))β€˜((𝑀 βˆ’ 1) / 𝑁)) ∈ (◑𝐹 β€œ {(πΊβ€˜((𝑀 βˆ’ 1) / 𝑁))}) ↔ (((π‘„β€˜(𝑀 βˆ’ 1))β€˜((𝑀 βˆ’ 1) / 𝑁)) ∈ 𝐡 ∧ (πΉβ€˜((π‘„β€˜(𝑀 βˆ’ 1))β€˜((𝑀 βˆ’ 1) / 𝑁))) = (πΊβ€˜((𝑀 βˆ’ 1) / 𝑁)))))
7770, 76mpbid 231 . . . . . . . . 9 ((πœ‘ ∧ 𝑀 ∈ (1...𝑁)) β†’ (((π‘„β€˜(𝑀 βˆ’ 1))β€˜((𝑀 βˆ’ 1) / 𝑁)) ∈ 𝐡 ∧ (πΉβ€˜((π‘„β€˜(𝑀 βˆ’ 1))β€˜((𝑀 βˆ’ 1) / 𝑁))) = (πΊβ€˜((𝑀 βˆ’ 1) / 𝑁))))
7877simpld 496 . . . . . . . 8 ((πœ‘ ∧ 𝑀 ∈ (1...𝑁)) β†’ ((π‘„β€˜(𝑀 βˆ’ 1))β€˜((𝑀 βˆ’ 1) / 𝑁)) ∈ 𝐡)
7977simprd 497 . . . . . . . . 9 ((πœ‘ ∧ 𝑀 ∈ (1...𝑁)) β†’ (πΉβ€˜((π‘„β€˜(𝑀 βˆ’ 1))β€˜((𝑀 βˆ’ 1) / 𝑁))) = (πΊβ€˜((𝑀 βˆ’ 1) / 𝑁)))
801adantl 483 . . . . . . . . . . . . . . . 16 ((πœ‘ ∧ 𝑀 ∈ (1...𝑁)) β†’ 𝑀 ∈ β„•)
8180nnred 12227 . . . . . . . . . . . . . . 15 ((πœ‘ ∧ 𝑀 ∈ (1...𝑁)) β†’ 𝑀 ∈ ℝ)
82 peano2rem 11527 . . . . . . . . . . . . . . 15 (𝑀 ∈ ℝ β†’ (𝑀 βˆ’ 1) ∈ ℝ)
8381, 82syl 17 . . . . . . . . . . . . . 14 ((πœ‘ ∧ 𝑀 ∈ (1...𝑁)) β†’ (𝑀 βˆ’ 1) ∈ ℝ)
849adantr 482 . . . . . . . . . . . . . 14 ((πœ‘ ∧ 𝑀 ∈ (1...𝑁)) β†’ 𝑁 ∈ β„•)
8583, 84nndivred 12266 . . . . . . . . . . . . 13 ((πœ‘ ∧ 𝑀 ∈ (1...𝑁)) β†’ ((𝑀 βˆ’ 1) / 𝑁) ∈ ℝ)
8685rexrd 11264 . . . . . . . . . . . 12 ((πœ‘ ∧ 𝑀 ∈ (1...𝑁)) β†’ ((𝑀 βˆ’ 1) / 𝑁) ∈ ℝ*)
8781, 84nndivred 12266 . . . . . . . . . . . . 13 ((πœ‘ ∧ 𝑀 ∈ (1...𝑁)) β†’ (𝑀 / 𝑁) ∈ ℝ)
8887rexrd 11264 . . . . . . . . . . . 12 ((πœ‘ ∧ 𝑀 ∈ (1...𝑁)) β†’ (𝑀 / 𝑁) ∈ ℝ*)
8981ltm1d 12146 . . . . . . . . . . . . . 14 ((πœ‘ ∧ 𝑀 ∈ (1...𝑁)) β†’ (𝑀 βˆ’ 1) < 𝑀)
9084nnred 12227 . . . . . . . . . . . . . . 15 ((πœ‘ ∧ 𝑀 ∈ (1...𝑁)) β†’ 𝑁 ∈ ℝ)
9184nngt0d 12261 . . . . . . . . . . . . . . 15 ((πœ‘ ∧ 𝑀 ∈ (1...𝑁)) β†’ 0 < 𝑁)
92 ltdiv1 12078 . . . . . . . . . . . . . . 15 (((𝑀 βˆ’ 1) ∈ ℝ ∧ 𝑀 ∈ ℝ ∧ (𝑁 ∈ ℝ ∧ 0 < 𝑁)) β†’ ((𝑀 βˆ’ 1) < 𝑀 ↔ ((𝑀 βˆ’ 1) / 𝑁) < (𝑀 / 𝑁)))
9383, 81, 90, 91, 92syl112anc 1375 . . . . . . . . . . . . . 14 ((πœ‘ ∧ 𝑀 ∈ (1...𝑁)) β†’ ((𝑀 βˆ’ 1) < 𝑀 ↔ ((𝑀 βˆ’ 1) / 𝑁) < (𝑀 / 𝑁)))
9489, 93mpbid 231 . . . . . . . . . . . . 13 ((πœ‘ ∧ 𝑀 ∈ (1...𝑁)) β†’ ((𝑀 βˆ’ 1) / 𝑁) < (𝑀 / 𝑁))
9585, 87, 94ltled 11362 . . . . . . . . . . . 12 ((πœ‘ ∧ 𝑀 ∈ (1...𝑁)) β†’ ((𝑀 βˆ’ 1) / 𝑁) ≀ (𝑀 / 𝑁))
96 lbicc2 13441 . . . . . . . . . . . 12 ((((𝑀 βˆ’ 1) / 𝑁) ∈ ℝ* ∧ (𝑀 / 𝑁) ∈ ℝ* ∧ ((𝑀 βˆ’ 1) / 𝑁) ≀ (𝑀 / 𝑁)) β†’ ((𝑀 βˆ’ 1) / 𝑁) ∈ (((𝑀 βˆ’ 1) / 𝑁)[,](𝑀 / 𝑁)))
9786, 88, 95, 96syl3anc 1372 . . . . . . . . . . 11 ((πœ‘ ∧ 𝑀 ∈ (1...𝑁)) β†’ ((𝑀 βˆ’ 1) / 𝑁) ∈ (((𝑀 βˆ’ 1) / 𝑁)[,](𝑀 / 𝑁)))
9897, 14eleqtrrdi 2845 . . . . . . . . . 10 ((πœ‘ ∧ 𝑀 ∈ (1...𝑁)) β†’ ((𝑀 βˆ’ 1) / 𝑁) ∈ π‘Š)
992, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 23, 14, 98cvmliftlem3 34278 . . . . . . . . 9 ((πœ‘ ∧ 𝑀 ∈ (1...𝑁)) β†’ (πΊβ€˜((𝑀 βˆ’ 1) / 𝑁)) ∈ (1st β€˜(π‘‡β€˜π‘€)))
10079, 99eqeltrd 2834 . . . . . . . 8 ((πœ‘ ∧ 𝑀 ∈ (1...𝑁)) β†’ (πΉβ€˜((π‘„β€˜(𝑀 βˆ’ 1))β€˜((𝑀 βˆ’ 1) / 𝑁))) ∈ (1st β€˜(π‘‡β€˜π‘€)))
101 eqid 2733 . . . . . . . . 9 (℩𝑏 ∈ (2nd β€˜(π‘‡β€˜π‘€))((π‘„β€˜(𝑀 βˆ’ 1))β€˜((𝑀 βˆ’ 1) / 𝑁)) ∈ 𝑏) = (℩𝑏 ∈ (2nd β€˜(π‘‡β€˜π‘€))((π‘„β€˜(𝑀 βˆ’ 1))β€˜((𝑀 βˆ’ 1) / 𝑁)) ∈ 𝑏)
1022, 3, 101cvmsiota 34268 . . . . . . . 8 ((𝐹 ∈ (𝐢 CovMap 𝐽) ∧ ((2nd β€˜(π‘‡β€˜π‘€)) ∈ (π‘†β€˜(1st β€˜(π‘‡β€˜π‘€))) ∧ ((π‘„β€˜(𝑀 βˆ’ 1))β€˜((𝑀 βˆ’ 1) / 𝑁)) ∈ 𝐡 ∧ (πΉβ€˜((π‘„β€˜(𝑀 βˆ’ 1))β€˜((𝑀 βˆ’ 1) / 𝑁))) ∈ (1st β€˜(π‘‡β€˜π‘€)))) β†’ ((℩𝑏 ∈ (2nd β€˜(π‘‡β€˜π‘€))((π‘„β€˜(𝑀 βˆ’ 1))β€˜((𝑀 βˆ’ 1) / 𝑁)) ∈ 𝑏) ∈ (2nd β€˜(π‘‡β€˜π‘€)) ∧ ((π‘„β€˜(𝑀 βˆ’ 1))β€˜((𝑀 βˆ’ 1) / 𝑁)) ∈ (℩𝑏 ∈ (2nd β€˜(π‘‡β€˜π‘€))((π‘„β€˜(𝑀 βˆ’ 1))β€˜((𝑀 βˆ’ 1) / 𝑁)) ∈ 𝑏)))
10317, 62, 78, 100, 102syl13anc 1373 . . . . . . 7 ((πœ‘ ∧ 𝑀 ∈ (1...𝑁)) β†’ ((℩𝑏 ∈ (2nd β€˜(π‘‡β€˜π‘€))((π‘„β€˜(𝑀 βˆ’ 1))β€˜((𝑀 βˆ’ 1) / 𝑁)) ∈ 𝑏) ∈ (2nd β€˜(π‘‡β€˜π‘€)) ∧ ((π‘„β€˜(𝑀 βˆ’ 1))β€˜((𝑀 βˆ’ 1) / 𝑁)) ∈ (℩𝑏 ∈ (2nd β€˜(π‘‡β€˜π‘€))((π‘„β€˜(𝑀 βˆ’ 1))β€˜((𝑀 βˆ’ 1) / 𝑁)) ∈ 𝑏)))
104103simpld 496 . . . . . 6 ((πœ‘ ∧ 𝑀 ∈ (1...𝑁)) β†’ (℩𝑏 ∈ (2nd β€˜(π‘‡β€˜π‘€))((π‘„β€˜(𝑀 βˆ’ 1))β€˜((𝑀 βˆ’ 1) / 𝑁)) ∈ 𝑏) ∈ (2nd β€˜(π‘‡β€˜π‘€)))
1052cvmshmeo 34262 . . . . . 6 (((2nd β€˜(π‘‡β€˜π‘€)) ∈ (π‘†β€˜(1st β€˜(π‘‡β€˜π‘€))) ∧ (℩𝑏 ∈ (2nd β€˜(π‘‡β€˜π‘€))((π‘„β€˜(𝑀 βˆ’ 1))β€˜((𝑀 βˆ’ 1) / 𝑁)) ∈ 𝑏) ∈ (2nd β€˜(π‘‡β€˜π‘€))) β†’ (𝐹 β†Ύ (℩𝑏 ∈ (2nd β€˜(π‘‡β€˜π‘€))((π‘„β€˜(𝑀 βˆ’ 1))β€˜((𝑀 βˆ’ 1) / 𝑁)) ∈ 𝑏)) ∈ ((𝐢 β†Ύt (℩𝑏 ∈ (2nd β€˜(π‘‡β€˜π‘€))((π‘„β€˜(𝑀 βˆ’ 1))β€˜((𝑀 βˆ’ 1) / 𝑁)) ∈ 𝑏))Homeo(𝐽 β†Ύt (1st β€˜(π‘‡β€˜π‘€)))))
10662, 104, 105syl2anc 585 . . . . 5 ((πœ‘ ∧ 𝑀 ∈ (1...𝑁)) β†’ (𝐹 β†Ύ (℩𝑏 ∈ (2nd β€˜(π‘‡β€˜π‘€))((π‘„β€˜(𝑀 βˆ’ 1))β€˜((𝑀 βˆ’ 1) / 𝑁)) ∈ 𝑏)) ∈ ((𝐢 β†Ύt (℩𝑏 ∈ (2nd β€˜(π‘‡β€˜π‘€))((π‘„β€˜(𝑀 βˆ’ 1))β€˜((𝑀 βˆ’ 1) / 𝑁)) ∈ 𝑏))Homeo(𝐽 β†Ύt (1st β€˜(π‘‡β€˜π‘€)))))
107 hmeocnvcn 23265 . . . . 5 ((𝐹 β†Ύ (℩𝑏 ∈ (2nd β€˜(π‘‡β€˜π‘€))((π‘„β€˜(𝑀 βˆ’ 1))β€˜((𝑀 βˆ’ 1) / 𝑁)) ∈ 𝑏)) ∈ ((𝐢 β†Ύt (℩𝑏 ∈ (2nd β€˜(π‘‡β€˜π‘€))((π‘„β€˜(𝑀 βˆ’ 1))β€˜((𝑀 βˆ’ 1) / 𝑁)) ∈ 𝑏))Homeo(𝐽 β†Ύt (1st β€˜(π‘‡β€˜π‘€)))) β†’ β—‘(𝐹 β†Ύ (℩𝑏 ∈ (2nd β€˜(π‘‡β€˜π‘€))((π‘„β€˜(𝑀 βˆ’ 1))β€˜((𝑀 βˆ’ 1) / 𝑁)) ∈ 𝑏)) ∈ ((𝐽 β†Ύt (1st β€˜(π‘‡β€˜π‘€))) Cn (𝐢 β†Ύt (℩𝑏 ∈ (2nd β€˜(π‘‡β€˜π‘€))((π‘„β€˜(𝑀 βˆ’ 1))β€˜((𝑀 βˆ’ 1) / 𝑁)) ∈ 𝑏))))
108106, 107syl 17 . . . 4 ((πœ‘ ∧ 𝑀 ∈ (1...𝑁)) β†’ β—‘(𝐹 β†Ύ (℩𝑏 ∈ (2nd β€˜(π‘‡β€˜π‘€))((π‘„β€˜(𝑀 βˆ’ 1))β€˜((𝑀 βˆ’ 1) / 𝑁)) ∈ 𝑏)) ∈ ((𝐽 β†Ύt (1st β€˜(π‘‡β€˜π‘€))) Cn (𝐢 β†Ύt (℩𝑏 ∈ (2nd β€˜(π‘‡β€˜π‘€))((π‘„β€˜(𝑀 βˆ’ 1))β€˜((𝑀 βˆ’ 1) / 𝑁)) ∈ 𝑏))))
10928, 69, 108cnmpt11f 23168 . . 3 ((πœ‘ ∧ 𝑀 ∈ (1...𝑁)) β†’ (𝑧 ∈ π‘Š ↦ (β—‘(𝐹 β†Ύ (℩𝑏 ∈ (2nd β€˜(π‘‡β€˜π‘€))((π‘„β€˜(𝑀 βˆ’ 1))β€˜((𝑀 βˆ’ 1) / 𝑁)) ∈ 𝑏))β€˜(πΊβ€˜π‘§))) ∈ ((𝐿 β†Ύt π‘Š) Cn (𝐢 β†Ύt (℩𝑏 ∈ (2nd β€˜(π‘‡β€˜π‘€))((π‘„β€˜(𝑀 βˆ’ 1))β€˜((𝑀 βˆ’ 1) / 𝑁)) ∈ 𝑏))))
11020, 109sseldd 3984 . 2 ((πœ‘ ∧ 𝑀 ∈ (1...𝑁)) β†’ (𝑧 ∈ π‘Š ↦ (β—‘(𝐹 β†Ύ (℩𝑏 ∈ (2nd β€˜(π‘‡β€˜π‘€))((π‘„β€˜(𝑀 βˆ’ 1))β€˜((𝑀 βˆ’ 1) / 𝑁)) ∈ 𝑏))β€˜(πΊβ€˜π‘§))) ∈ ((𝐿 β†Ύt π‘Š) Cn 𝐢))
11116, 110eqeltrd 2834 1 ((πœ‘ ∧ 𝑀 ∈ (1...𝑁)) β†’ (π‘„β€˜π‘€) ∈ ((𝐿 β†Ύt π‘Š) Cn 𝐢))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107  βˆ€wral 3062  {crab 3433  Vcvv 3475   βˆ– cdif 3946   βˆͺ cun 3947   ∩ cin 3948   βŠ† wss 3949  βˆ…c0 4323  π’« cpw 4603  {csn 4629  βŸ¨cop 4635  βˆͺ cuni 4909  βˆͺ ciun 4998   class class class wbr 5149   ↦ cmpt 5232   I cid 5574   Γ— cxp 5675  β—‘ccnv 5676  ran crn 5678   β†Ύ cres 5679   β€œ cima 5680   Fn wfn 6539  βŸΆwf 6540  β€˜cfv 6544  β„©crio 7364  (class class class)co 7409   ∈ cmpo 7411  1st c1st 7973  2nd c2nd 7974  β„cr 11109  0cc0 11110  1c1 11111  β„*cxr 11247   < clt 11248   ≀ cle 11249   βˆ’ cmin 11444   / cdiv 11871  β„•cn 12212  (,)cioo 13324  [,]cicc 13327  ...cfz 13484  seqcseq 13966   β†Ύt crest 17366  topGenctg 17383  Topctop 22395  TopOnctopon 22412   Cn ccn 22728  Homeochmeo 23257  IIcii 24391   CovMap ccvm 34246
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187  ax-pre-sup 11188
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-er 8703  df-map 8822  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-fi 9406  df-sup 9437  df-inf 9438  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-div 11872  df-nn 12213  df-2 12275  df-3 12276  df-n0 12473  df-z 12559  df-uz 12823  df-q 12933  df-rp 12975  df-xneg 13092  df-xadd 13093  df-xmul 13094  df-ioo 13328  df-icc 13331  df-fz 13485  df-seq 13967  df-exp 14028  df-cj 15046  df-re 15047  df-im 15048  df-sqrt 15182  df-abs 15183  df-rest 17368  df-topgen 17389  df-psmet 20936  df-xmet 20937  df-met 20938  df-bl 20939  df-mopn 20940  df-top 22396  df-topon 22413  df-bases 22449  df-cn 22731  df-hmeo 23259  df-ii 24393  df-cvm 34247
This theorem is referenced by:  cvmliftlem10  34285
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