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Theorem cvmliftlem9 34272
Description: Lemma for cvmlift 34278. The 𝑄(𝑀) functions are defined on almost disjoint intervals, but they overlap at the edges. Here we show that at these points the 𝑄 functions agree on their common domain. (Contributed by Mario Carneiro, 14-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©}⟩}))
Assertion
Ref Expression
cvmliftlem9 ((πœ‘ ∧ 𝑀 ∈ (1...𝑁)) β†’ ((π‘„β€˜π‘€)β€˜((𝑀 βˆ’ 1) / 𝑁)) = ((π‘„β€˜(𝑀 βˆ’ 1))β€˜((𝑀 βˆ’ 1) / 𝑁)))
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
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)   𝑋(π‘₯,𝑧,𝑣,𝑒,π‘˜,π‘š,𝑠,𝑏)

Proof of Theorem cvmliftlem9
StepHypRef Expression
1 elfznn 13526 . . . 4 (𝑀 ∈ (1...𝑁) β†’ 𝑀 ∈ β„•)
2 cvmliftlem.1 . . . . 5 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 . . . . 5 𝐡 = βˆͺ 𝐢
4 cvmliftlem.x . . . . 5 𝑋 = βˆͺ 𝐽
5 cvmliftlem.f . . . . 5 (πœ‘ β†’ 𝐹 ∈ (𝐢 CovMap 𝐽))
6 cvmliftlem.g . . . . 5 (πœ‘ β†’ 𝐺 ∈ (II Cn 𝐽))
7 cvmliftlem.p . . . . 5 (πœ‘ β†’ 𝑃 ∈ 𝐡)
8 cvmliftlem.e . . . . 5 (πœ‘ β†’ (πΉβ€˜π‘ƒ) = (πΊβ€˜0))
9 cvmliftlem.n . . . . 5 (πœ‘ β†’ 𝑁 ∈ β„•)
10 cvmliftlem.t . . . . 5 (πœ‘ β†’ 𝑇:(1...𝑁)⟢βˆͺ 𝑗 ∈ 𝐽 ({𝑗} Γ— (π‘†β€˜π‘—)))
11 cvmliftlem.a . . . . 5 (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 . . . . 5 𝐿 = (topGenβ€˜ran (,))
13 cvmliftlem.q . . . . 5 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 eqid 2732 . . . . 5 (((𝑀 βˆ’ 1) / 𝑁)[,](𝑀 / 𝑁)) = (((𝑀 βˆ’ 1) / 𝑁)[,](𝑀 / 𝑁))
152, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14cvmliftlem5 34268 . . . 4 ((πœ‘ ∧ 𝑀 ∈ β„•) β†’ (π‘„β€˜π‘€) = (𝑧 ∈ (((𝑀 βˆ’ 1) / 𝑁)[,](𝑀 / 𝑁)) ↦ (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 . . 3 ((πœ‘ ∧ 𝑀 ∈ (1...𝑁)) β†’ (π‘„β€˜π‘€) = (𝑧 ∈ (((𝑀 βˆ’ 1) / 𝑁)[,](𝑀 / 𝑁)) ↦ (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‘§))))
17 simpr 485 . . . . 5 (((πœ‘ ∧ 𝑀 ∈ (1...𝑁)) ∧ 𝑧 = ((𝑀 βˆ’ 1) / 𝑁)) β†’ 𝑧 = ((𝑀 βˆ’ 1) / 𝑁))
1817fveq2d 6892 . . . 4 (((πœ‘ ∧ 𝑀 ∈ (1...𝑁)) ∧ 𝑧 = ((𝑀 βˆ’ 1) / 𝑁)) β†’ (πΊβ€˜π‘§) = (πΊβ€˜((𝑀 βˆ’ 1) / 𝑁)))
1918fveq2d 6892 . . 3 (((πœ‘ ∧ 𝑀 ∈ (1...𝑁)) ∧ 𝑧 = ((𝑀 βˆ’ 1) / 𝑁)) β†’ (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‘§)) = (β—‘(𝐹 β†Ύ (℩𝑏 ∈ (2nd β€˜(π‘‡β€˜π‘€))((π‘„β€˜(𝑀 βˆ’ 1))β€˜((𝑀 βˆ’ 1) / 𝑁)) ∈ 𝑏))β€˜(πΊβ€˜((𝑀 βˆ’ 1) / 𝑁))))
201adantl 482 . . . . . . . 8 ((πœ‘ ∧ 𝑀 ∈ (1...𝑁)) β†’ 𝑀 ∈ β„•)
2120nnred 12223 . . . . . . 7 ((πœ‘ ∧ 𝑀 ∈ (1...𝑁)) β†’ 𝑀 ∈ ℝ)
22 peano2rem 11523 . . . . . . 7 (𝑀 ∈ ℝ β†’ (𝑀 βˆ’ 1) ∈ ℝ)
2321, 22syl 17 . . . . . 6 ((πœ‘ ∧ 𝑀 ∈ (1...𝑁)) β†’ (𝑀 βˆ’ 1) ∈ ℝ)
249adantr 481 . . . . . 6 ((πœ‘ ∧ 𝑀 ∈ (1...𝑁)) β†’ 𝑁 ∈ β„•)
2523, 24nndivred 12262 . . . . 5 ((πœ‘ ∧ 𝑀 ∈ (1...𝑁)) β†’ ((𝑀 βˆ’ 1) / 𝑁) ∈ ℝ)
2625rexrd 11260 . . . 4 ((πœ‘ ∧ 𝑀 ∈ (1...𝑁)) β†’ ((𝑀 βˆ’ 1) / 𝑁) ∈ ℝ*)
2721, 24nndivred 12262 . . . . 5 ((πœ‘ ∧ 𝑀 ∈ (1...𝑁)) β†’ (𝑀 / 𝑁) ∈ ℝ)
2827rexrd 11260 . . . 4 ((πœ‘ ∧ 𝑀 ∈ (1...𝑁)) β†’ (𝑀 / 𝑁) ∈ ℝ*)
2921ltm1d 12142 . . . . . 6 ((πœ‘ ∧ 𝑀 ∈ (1...𝑁)) β†’ (𝑀 βˆ’ 1) < 𝑀)
3024nnred 12223 . . . . . . 7 ((πœ‘ ∧ 𝑀 ∈ (1...𝑁)) β†’ 𝑁 ∈ ℝ)
3124nngt0d 12257 . . . . . . 7 ((πœ‘ ∧ 𝑀 ∈ (1...𝑁)) β†’ 0 < 𝑁)
32 ltdiv1 12074 . . . . . . 7 (((𝑀 βˆ’ 1) ∈ ℝ ∧ 𝑀 ∈ ℝ ∧ (𝑁 ∈ ℝ ∧ 0 < 𝑁)) β†’ ((𝑀 βˆ’ 1) < 𝑀 ↔ ((𝑀 βˆ’ 1) / 𝑁) < (𝑀 / 𝑁)))
3323, 21, 30, 31, 32syl112anc 1374 . . . . . 6 ((πœ‘ ∧ 𝑀 ∈ (1...𝑁)) β†’ ((𝑀 βˆ’ 1) < 𝑀 ↔ ((𝑀 βˆ’ 1) / 𝑁) < (𝑀 / 𝑁)))
3429, 33mpbid 231 . . . . 5 ((πœ‘ ∧ 𝑀 ∈ (1...𝑁)) β†’ ((𝑀 βˆ’ 1) / 𝑁) < (𝑀 / 𝑁))
3525, 27, 34ltled 11358 . . . 4 ((πœ‘ ∧ 𝑀 ∈ (1...𝑁)) β†’ ((𝑀 βˆ’ 1) / 𝑁) ≀ (𝑀 / 𝑁))
36 lbicc2 13437 . . . 4 ((((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)))
3726, 28, 35, 36syl3anc 1371 . . 3 ((πœ‘ ∧ 𝑀 ∈ (1...𝑁)) β†’ ((𝑀 βˆ’ 1) / 𝑁) ∈ (((𝑀 βˆ’ 1) / 𝑁)[,](𝑀 / 𝑁)))
38 fvexd 6903 . . 3 ((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))β€˜(πΊβ€˜((𝑀 βˆ’ 1) / 𝑁))) ∈ V)
3916, 19, 37, 38fvmptd 7002 . 2 ((πœ‘ ∧ 𝑀 ∈ (1...𝑁)) β†’ ((π‘„β€˜π‘€)β€˜((𝑀 βˆ’ 1) / 𝑁)) = (β—‘(𝐹 β†Ύ (℩𝑏 ∈ (2nd β€˜(π‘‡β€˜π‘€))((π‘„β€˜(𝑀 βˆ’ 1))β€˜((𝑀 βˆ’ 1) / 𝑁)) ∈ 𝑏))β€˜(πΊβ€˜((𝑀 βˆ’ 1) / 𝑁))))
405adantr 481 . . . . . . 7 ((πœ‘ ∧ 𝑀 ∈ (1...𝑁)) β†’ 𝐹 ∈ (𝐢 CovMap 𝐽))
41 simpr 485 . . . . . . . 8 ((πœ‘ ∧ 𝑀 ∈ (1...𝑁)) β†’ 𝑀 ∈ (1...𝑁))
422, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 41cvmliftlem1 34264 . . . . . . 7 ((πœ‘ ∧ 𝑀 ∈ (1...𝑁)) β†’ (2nd β€˜(π‘‡β€˜π‘€)) ∈ (π‘†β€˜(1st β€˜(π‘‡β€˜π‘€))))
432, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14cvmliftlem7 34270 . . . . . . . . 9 ((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))}))
44 cvmcn 34241 . . . . . . . . . . 11 (𝐹 ∈ (𝐢 CovMap 𝐽) β†’ 𝐹 ∈ (𝐢 Cn 𝐽))
453, 4cnf 22741 . . . . . . . . . . 11 (𝐹 ∈ (𝐢 Cn 𝐽) β†’ 𝐹:π΅βŸΆπ‘‹)
4640, 44, 453syl 18 . . . . . . . . . 10 ((πœ‘ ∧ 𝑀 ∈ (1...𝑁)) β†’ 𝐹:π΅βŸΆπ‘‹)
47 ffn 6714 . . . . . . . . . 10 (𝐹:π΅βŸΆπ‘‹ β†’ 𝐹 Fn 𝐡)
48 fniniseg 7058 . . . . . . . . . 10 (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)))))
4946, 47, 483syl 18 . . . . . . . . 9 ((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)))))
5043, 49mpbid 231 . . . . . . . 8 ((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))))
5150simpld 495 . . . . . . 7 ((πœ‘ ∧ 𝑀 ∈ (1...𝑁)) β†’ ((π‘„β€˜(𝑀 βˆ’ 1))β€˜((𝑀 βˆ’ 1) / 𝑁)) ∈ 𝐡)
5250simprd 496 . . . . . . . 8 ((πœ‘ ∧ 𝑀 ∈ (1...𝑁)) β†’ (πΉβ€˜((π‘„β€˜(𝑀 βˆ’ 1))β€˜((𝑀 βˆ’ 1) / 𝑁))) = (πΊβ€˜((𝑀 βˆ’ 1) / 𝑁)))
532, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 41, 14, 37cvmliftlem3 34266 . . . . . . . 8 ((πœ‘ ∧ 𝑀 ∈ (1...𝑁)) β†’ (πΊβ€˜((𝑀 βˆ’ 1) / 𝑁)) ∈ (1st β€˜(π‘‡β€˜π‘€)))
5452, 53eqeltrd 2833 . . . . . . 7 ((πœ‘ ∧ 𝑀 ∈ (1...𝑁)) β†’ (πΉβ€˜((π‘„β€˜(𝑀 βˆ’ 1))β€˜((𝑀 βˆ’ 1) / 𝑁))) ∈ (1st β€˜(π‘‡β€˜π‘€)))
55 eqid 2732 . . . . . . . 8 (℩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)
562, 3, 55cvmsiota 34256 . . . . . . 7 ((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)))
5740, 42, 51, 54, 56syl13anc 1372 . . . . . 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‘€)) ∧ ((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)))
5857simprd 496 . . . . 5 ((πœ‘ ∧ 𝑀 ∈ (1...𝑁)) β†’ ((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))
59 fvres 6907 . . . . 5 (((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) β†’ ((𝐹 β†Ύ (℩𝑏 ∈ (2nd β€˜(π‘‡β€˜π‘€))((π‘„β€˜(𝑀 βˆ’ 1))β€˜((𝑀 βˆ’ 1) / 𝑁)) ∈ 𝑏))β€˜((π‘„β€˜(𝑀 βˆ’ 1))β€˜((𝑀 βˆ’ 1) / 𝑁))) = (πΉβ€˜((π‘„β€˜(𝑀 βˆ’ 1))β€˜((𝑀 βˆ’ 1) / 𝑁))))
6058, 59syl 17 . . . 4 ((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))β€˜((π‘„β€˜(𝑀 βˆ’ 1))β€˜((𝑀 βˆ’ 1) / 𝑁))) = (πΉβ€˜((π‘„β€˜(𝑀 βˆ’ 1))β€˜((𝑀 βˆ’ 1) / 𝑁))))
6160, 52eqtrd 2772 . . 3 ((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))β€˜((π‘„β€˜(𝑀 βˆ’ 1))β€˜((𝑀 βˆ’ 1) / 𝑁))) = (πΊβ€˜((𝑀 βˆ’ 1) / 𝑁)))
6257simpld 495 . . . . 5 ((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‘€)))
632cvmsf1o 34251 . . . . 5 ((𝐹 ∈ (𝐢 CovMap 𝐽) ∧ (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)):(℩𝑏 ∈ (2nd β€˜(π‘‡β€˜π‘€))((π‘„β€˜(𝑀 βˆ’ 1))β€˜((𝑀 βˆ’ 1) / 𝑁)) ∈ 𝑏)–1-1-ontoβ†’(1st β€˜(π‘‡β€˜π‘€)))
6440, 42, 62, 63syl3anc 1371 . . . 4 ((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)):(℩𝑏 ∈ (2nd β€˜(π‘‡β€˜π‘€))((π‘„β€˜(𝑀 βˆ’ 1))β€˜((𝑀 βˆ’ 1) / 𝑁)) ∈ 𝑏)–1-1-ontoβ†’(1st β€˜(π‘‡β€˜π‘€)))
65 f1ocnvfv 7272 . . . 4 (((𝐹 β†Ύ (℩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)–1-1-ontoβ†’(1st 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)) β†’ (((𝐹 β†Ύ (℩𝑏 ∈ (2nd β€˜(π‘‡β€˜π‘€))((π‘„β€˜(𝑀 βˆ’ 1))β€˜((𝑀 βˆ’ 1) / 𝑁)) ∈ 𝑏))β€˜((π‘„β€˜(𝑀 βˆ’ 1))β€˜((𝑀 βˆ’ 1) / 𝑁))) = (πΊβ€˜((𝑀 βˆ’ 1) / 𝑁)) β†’ (β—‘(𝐹 β†Ύ (℩𝑏 ∈ (2nd β€˜(π‘‡β€˜π‘€))((π‘„β€˜(𝑀 βˆ’ 1))β€˜((𝑀 βˆ’ 1) / 𝑁)) ∈ 𝑏))β€˜(πΊβ€˜((𝑀 βˆ’ 1) / 𝑁))) = ((π‘„β€˜(𝑀 βˆ’ 1))β€˜((𝑀 βˆ’ 1) / 𝑁))))
6664, 58, 65syl2anc 584 . . 3 ((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))β€˜((π‘„β€˜(𝑀 βˆ’ 1))β€˜((𝑀 βˆ’ 1) / 𝑁))) = (πΊβ€˜((𝑀 βˆ’ 1) / 𝑁)) β†’ (β—‘(𝐹 β†Ύ (℩𝑏 ∈ (2nd β€˜(π‘‡β€˜π‘€))((π‘„β€˜(𝑀 βˆ’ 1))β€˜((𝑀 βˆ’ 1) / 𝑁)) ∈ 𝑏))β€˜(πΊβ€˜((𝑀 βˆ’ 1) / 𝑁))) = ((π‘„β€˜(𝑀 βˆ’ 1))β€˜((𝑀 βˆ’ 1) / 𝑁))))
6761, 66mpd 15 . 2 ((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))β€˜(πΊβ€˜((𝑀 βˆ’ 1) / 𝑁))) = ((π‘„β€˜(𝑀 βˆ’ 1))β€˜((𝑀 βˆ’ 1) / 𝑁)))
6839, 67eqtrd 2772 1 ((πœ‘ ∧ 𝑀 ∈ (1...𝑁)) β†’ ((π‘„β€˜π‘€)β€˜((𝑀 βˆ’ 1) / 𝑁)) = ((π‘„β€˜(𝑀 βˆ’ 1))β€˜((𝑀 βˆ’ 1) / 𝑁)))
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€“1-1-ontoβ†’wf1o 6539  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   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-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  cvmliftlem13  34275
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