Users' Mathboxes Mathbox for Mario Carneiro < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cvmliftlem8 Structured version   Visualization version   GIF version
Theorem cvmliftlem8 34271
Description: Lemma for cvmlift 34278. 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 13526 . . 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 34268 . . 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 593 . 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 481 . . . 4 ((πœ‘ ∧ 𝑀 ∈ (1...𝑁)) β†’ 𝐹 ∈ (𝐢 CovMap 𝐽))
18 cvmtop1 34239 . . . 4 (𝐹 ∈ (𝐢 CovMap 𝐽) β†’ 𝐢 ∈ Top)
19 cnrest2r 22782 . . . 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 24271 . . . . . 6 (topGenβ€˜ran (,)) ∈ (TopOnβ€˜β„)
2212, 21eqeltri 2829 . . . . 5 𝐿 ∈ (TopOnβ€˜β„)
23 simpr 485 . . . . . . 7 ((πœ‘ ∧ 𝑀 ∈ (1...𝑁)) β†’ 𝑀 ∈ (1...𝑁))
242, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 23, 14cvmliftlem2 34265 . . . . . 6 ((πœ‘ ∧ 𝑀 ∈ (1...𝑁)) β†’ π‘Š βŠ† (0[,]1))
25 unitssre 13472 . . . . . 6 (0[,]1) βŠ† ℝ
2624, 25sstrdi 3993 . . . . 5 ((πœ‘ ∧ 𝑀 ∈ (1...𝑁)) β†’ π‘Š βŠ† ℝ)
27 resttopon 22656 . . . . 5 ((𝐿 ∈ (TopOnβ€˜β„) ∧ π‘Š βŠ† ℝ) β†’ (𝐿 β†Ύt π‘Š) ∈ (TopOnβ€˜π‘Š))
2822, 26, 27sylancr 587 . . . 4 ((πœ‘ ∧ 𝑀 ∈ (1...𝑁)) β†’ (𝐿 β†Ύt π‘Š) ∈ (TopOnβ€˜π‘Š))
29 eqid 2732 . . . . . . 7 (II β†Ύt π‘Š) = (II β†Ύt π‘Š)
30 iitopon 24386 . . . . . . . 8 II ∈ (TopOnβ€˜(0[,]1))
3130a1i 11 . . . . . . 7 ((πœ‘ ∧ 𝑀 ∈ (1...𝑁)) β†’ II ∈ (TopOnβ€˜(0[,]1)))
326adantr 481 . . . . . . . . . 10 ((πœ‘ ∧ 𝑀 ∈ (1...𝑁)) β†’ 𝐺 ∈ (II Cn 𝐽))
33 iiuni 24388 . . . . . . . . . . 11 (0[,]1) = βˆͺ II
3433, 4cnf 22741 . . . . . . . . . 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 6957 . . . . . . . 8 ((πœ‘ ∧ 𝑀 ∈ (1...𝑁)) β†’ 𝐺 = (𝑧 ∈ (0[,]1) ↦ (πΊβ€˜π‘§)))
3736, 32eqeltrrd 2834 . . . . . . 7 ((πœ‘ ∧ 𝑀 ∈ (1...𝑁)) β†’ (𝑧 ∈ (0[,]1) ↦ (πΊβ€˜π‘§)) ∈ (II Cn 𝐽))
3829, 31, 24, 37cnmpt1res 23171 . . . . . 6 ((πœ‘ ∧ 𝑀 ∈ (1...𝑁)) β†’ (𝑧 ∈ π‘Š ↦ (πΊβ€˜π‘§)) ∈ ((II β†Ύt π‘Š) Cn 𝐽))
39 dfii2 24389 . . . . . . . . . 10 II = ((topGenβ€˜ran (,)) β†Ύt (0[,]1))
4012oveq1i 7415 . . . . . . . . . 10 (𝐿 β†Ύt (0[,]1)) = ((topGenβ€˜ran (,)) β†Ύt (0[,]1))
4139, 40eqtr4i 2763 . . . . . . . . 9 II = (𝐿 β†Ύt (0[,]1))
4241oveq1i 7415 . . . . . . . 8 (II β†Ύt π‘Š) = ((𝐿 β†Ύt (0[,]1)) β†Ύt π‘Š)
43 retop 24269 . . . . . . . . . . 11 (topGenβ€˜ran (,)) ∈ Top
4412, 43eqeltri 2829 . . . . . . . . . 10 𝐿 ∈ Top
4544a1i 11 . . . . . . . . 9 ((πœ‘ ∧ 𝑀 ∈ (1...𝑁)) β†’ 𝐿 ∈ Top)
46 ovexd 7440 . . . . . . . . 9 ((πœ‘ ∧ 𝑀 ∈ (1...𝑁)) β†’ (0[,]1) ∈ V)
47 restabs 22660 . . . . . . . . 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 1371 . . . . . . . 8 ((πœ‘ ∧ 𝑀 ∈ (1...𝑁)) β†’ ((𝐿 β†Ύt (0[,]1)) β†Ύt π‘Š) = (𝐿 β†Ύt π‘Š))
4942, 48eqtrid 2784 . . . . . . 7 ((πœ‘ ∧ 𝑀 ∈ (1...𝑁)) β†’ (II β†Ύt π‘Š) = (𝐿 β†Ύt π‘Š))
5049oveq1d 7420 . . . . . 6 ((πœ‘ ∧ 𝑀 ∈ (1...𝑁)) β†’ ((II β†Ύt π‘Š) Cn 𝐽) = ((𝐿 β†Ύt π‘Š) Cn 𝐽))
5138, 50eleqtrd 2835 . . . . 5 ((πœ‘ ∧ 𝑀 ∈ (1...𝑁)) β†’ (𝑧 ∈ π‘Š ↦ (πΊβ€˜π‘§)) ∈ ((𝐿 β†Ύt π‘Š) Cn 𝐽))
52 cvmtop2 34240 . . . . . . . 8 (𝐹 ∈ (𝐢 CovMap 𝐽) β†’ 𝐽 ∈ Top)
5317, 52syl 17 . . . . . . 7 ((πœ‘ ∧ 𝑀 ∈ (1...𝑁)) β†’ 𝐽 ∈ Top)
544toptopon 22410 . . . . . . 7 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOnβ€˜π‘‹))
5553, 54sylib 217 . . . . . 6 ((πœ‘ ∧ 𝑀 ∈ (1...𝑁)) β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
56 simprl 769 . . . . . . . . . 10 ((πœ‘ ∧ (𝑀 ∈ (1...𝑁) ∧ 𝑧 ∈ π‘Š)) β†’ 𝑀 ∈ (1...𝑁))
57 simprr 771 . . . . . . . . . 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 34266 . . . . . . . . 9 ((πœ‘ ∧ (𝑀 ∈ (1...𝑁) ∧ 𝑧 ∈ π‘Š)) β†’ (πΊβ€˜π‘§) ∈ (1st β€˜(π‘‡β€˜π‘€)))
5958anassrs 468 . . . . . . . 8 (((πœ‘ ∧ 𝑀 ∈ (1...𝑁)) ∧ 𝑧 ∈ π‘Š) β†’ (πΊβ€˜π‘§) ∈ (1st β€˜(π‘‡β€˜π‘€)))
6059fmpttd 7111 . . . . . . 7 ((πœ‘ ∧ 𝑀 ∈ (1...𝑁)) β†’ (𝑧 ∈ π‘Š ↦ (πΊβ€˜π‘§)):π‘ŠβŸΆ(1st β€˜(π‘‡β€˜π‘€)))
6160frnd 6722 . . . . . 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 34264 . . . . . . . 8 ((πœ‘ ∧ 𝑀 ∈ (1...𝑁)) β†’ (2nd β€˜(π‘‡β€˜π‘€)) ∈ (π‘†β€˜(1st β€˜(π‘‡β€˜π‘€))))
632cvmsrcl 34243 . . . . . . . 8 ((2nd β€˜(π‘‡β€˜π‘€)) ∈ (π‘†β€˜(1st β€˜(π‘‡β€˜π‘€))) β†’ (1st β€˜(π‘‡β€˜π‘€)) ∈ 𝐽)
64 elssuni 4940 . . . . . . . 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 4032 . . . . . 6 ((πœ‘ ∧ 𝑀 ∈ (1...𝑁)) β†’ (1st β€˜(π‘‡β€˜π‘€)) βŠ† 𝑋)
67 cnrest2 22781 . . . . . 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 1371 . . . . 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 34270 . . . . . . . . . 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 34241 . . . . . . . . . . . 12 (𝐹 ∈ (𝐢 CovMap 𝐽) β†’ 𝐹 ∈ (𝐢 Cn 𝐽))
723, 4cnf 22741 . . . . . . . . . . . 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 6714 . . . . . . . . . . 11 (𝐹:π΅βŸΆπ‘‹ β†’ 𝐹 Fn 𝐡)
75 fniniseg 7058 . . . . . . . . . . 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 495 . . . . . . . 8 ((πœ‘ ∧ 𝑀 ∈ (1...𝑁)) β†’ ((π‘„β€˜(𝑀 βˆ’ 1))β€˜((𝑀 βˆ’ 1) / 𝑁)) ∈ 𝐡)
7977simprd 496 . . . . . . . . 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 482 . . . . . . . . . . . . . . . 16 ((πœ‘ ∧ 𝑀 ∈ (1...𝑁)) β†’ 𝑀 ∈ β„•)
8180nnred 12223 . . . . . . . . . . . . . . 15 ((πœ‘ ∧ 𝑀 ∈ (1...𝑁)) β†’ 𝑀 ∈ ℝ)
82 peano2rem 11523 . . . . . . . . . . . . . . 15 (𝑀 ∈ ℝ β†’ (𝑀 βˆ’ 1) ∈ ℝ)
8381, 82syl 17 . . . . . . . . . . . . . 14 ((πœ‘ ∧ 𝑀 ∈ (1...𝑁)) β†’ (𝑀 βˆ’ 1) ∈ ℝ)
849adantr 481 . . . . . . . . . . . . . 14 ((πœ‘ ∧ 𝑀 ∈ (1...𝑁)) β†’ 𝑁 ∈ β„•)
8583, 84nndivred 12262 . . . . . . . . . . . . 13 ((πœ‘ ∧ 𝑀 ∈ (1...𝑁)) β†’ ((𝑀 βˆ’ 1) / 𝑁) ∈ ℝ)
8685rexrd 11260 . . . . . . . . . . . 12 ((πœ‘ ∧ 𝑀 ∈ (1...𝑁)) β†’ ((𝑀 βˆ’ 1) / 𝑁) ∈ ℝ*)
8781, 84nndivred 12262 . . . . . . . . . . . . 13 ((πœ‘ ∧ 𝑀 ∈ (1...𝑁)) β†’ (𝑀 / 𝑁) ∈ ℝ)
8887rexrd 11260 . . . . . . . . . . . 12 ((πœ‘ ∧ 𝑀 ∈ (1...𝑁)) β†’ (𝑀 / 𝑁) ∈ ℝ*)
8981ltm1d 12142 . . . . . . . . . . . . . 14 ((πœ‘ ∧ 𝑀 ∈ (1...𝑁)) β†’ (𝑀 βˆ’ 1) < 𝑀)
9084nnred 12223 . . . . . . . . . . . . . . 15 ((πœ‘ ∧ 𝑀 ∈ (1...𝑁)) β†’ 𝑁 ∈ ℝ)
9184nngt0d 12257 . . . . . . . . . . . . . . 15 ((πœ‘ ∧ 𝑀 ∈ (1...𝑁)) β†’ 0 < 𝑁)
92 ltdiv1 12074 . . . . . . . . . . . . . . 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 1374 . . . . . . . . . . . . . 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 11358 . . . . . . . . . . . 12 ((πœ‘ ∧ 𝑀 ∈ (1...𝑁)) β†’ ((𝑀 βˆ’ 1) / 𝑁) ≀ (𝑀 / 𝑁))
96 lbicc2 13437 . . . . . . . . . . . 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 1371 . . . . . . . . . . 11 ((πœ‘ ∧ 𝑀 ∈ (1...𝑁)) β†’ ((𝑀 βˆ’ 1) / 𝑁) ∈ (((𝑀 βˆ’ 1) / 𝑁)[,](𝑀 / 𝑁)))
9897, 14eleqtrrdi 2844 . . . . . . . . . 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 34266 . . . . . . . . 9 ((πœ‘ ∧ 𝑀 ∈ (1...𝑁)) β†’ (πΊβ€˜((𝑀 βˆ’ 1) / 𝑁)) ∈ (1st β€˜(π‘‡β€˜π‘€)))
10079, 99eqeltrd 2833 . . . . . . . 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 2732 . . . . . . . . 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 34256 . . . . . . . 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 1372 . . . . . . 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 495 . . . . . 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 34250 . . . . . 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 584 . . . . 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 23256 . . . . 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 23159 . . 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 3982 . 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 2833 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 396   = wceq 1541   ∈ wcel 2106  βˆ€wral 3061  {crab 3432  Vcvv 3474   βˆ– cdif 3944   βˆͺ cun 3945   ∩ cin 3946   βŠ† wss 3947  βˆ…c0 4321  π’« cpw 4601  {csn 4627  βŸ¨cop 4633  βˆͺ cuni 4907  βˆͺ ciun 4996   class class class wbr 5147   ↦ cmpt 5230   I cid 5572   Γ— cxp 5673  β—‘ccnv 5674  ran crn 5676   β†Ύ cres 5677   β€œ cima 5678   Fn wfn 6535  βŸΆwf 6536  β€˜cfv 6540  β„©crio 7360  (class class class)co 7405   ∈ cmpo 7407  1st c1st 7969  2nd c2nd 7970  β„cr 11105  0cc0 11106  1c1 11107  β„*cxr 11243   < clt 11244   ≀ cle 11245   βˆ’ cmin 11440   / cdiv 11867  β„•cn 12208  (,)cioo 13320  [,]cicc 13323  ...cfz 13480  seqcseq 13962   β†Ύt crest 17362  topGenctg 17379  Topctop 22386  TopOnctopon 22403   Cn ccn 22719  Homeochmeo 23248  IIcii 24382   CovMap ccvm 34234
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 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183  ax-pre-sup 11184
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 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7852  df-1st 7971  df-2nd 7972  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-er 8699  df-map 8818  df-en 8936  df-dom 8937  df-sdom 8938  df-fin 8939  df-fi 9402  df-sup 9433  df-inf 9434  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-div 11868  df-nn 12209  df-2 12271  df-3 12272  df-n0 12469  df-z 12555  df-uz 12819  df-q 12929  df-rp 12971  df-xneg 13088  df-xadd 13089  df-xmul 13090  df-ioo 13324  df-icc 13327  df-fz 13481  df-seq 13963  df-exp 14024  df-cj 15042  df-re 15043  df-im 15044  df-sqrt 15178  df-abs 15179  df-rest 17364  df-topgen 17385  df-psmet 20928  df-xmet 20929  df-met 20930  df-bl 20931  df-mopn 20932  df-top 22387  df-topon 22404  df-bases 22440  df-cn 22722  df-hmeo 23250  df-ii 24384  df-cvm 34235
This theorem is referenced by:  cvmliftlem10  34273
  Copyright terms: Public domain W3C validator