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Theorem cvmliftlem5 34952
Description: Lemma for cvmlift 34962. Definition of 𝑄 at a successor. This is a function defined on π‘Š as β—‘(𝑇 β†Ύ 𝐼) ∘ 𝐺 where 𝐼 is the unique covering set of 2nd β€˜(π‘‡β€˜π‘€) that contains 𝑄(𝑀 βˆ’ 1) evaluated at the last defined point, namely (𝑀 βˆ’ 1) / 𝑁 (note that for 𝑀 = 1 this is using the seed value 𝑄(0)(0) = 𝑃). (Contributed by Mario Carneiro, 15-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
cvmliftlem5 ((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‘§))))
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 cvmliftlem5
StepHypRef Expression
1 0z 12594 . . . 4 0 ∈ β„€
2 simpr 483 . . . . 5 ((πœ‘ ∧ 𝑀 ∈ β„•) β†’ 𝑀 ∈ β„•)
3 nnuz 12890 . . . . . 6 β„• = (β„€β‰₯β€˜1)
4 1e0p1 12744 . . . . . . 7 1 = (0 + 1)
54fveq2i 6893 . . . . . 6 (β„€β‰₯β€˜1) = (β„€β‰₯β€˜(0 + 1))
63, 5eqtri 2753 . . . . 5 β„• = (β„€β‰₯β€˜(0 + 1))
72, 6eleqtrdi 2835 . . . 4 ((πœ‘ ∧ 𝑀 ∈ β„•) β†’ 𝑀 ∈ (β„€β‰₯β€˜(0 + 1)))
8 seqm1 14011 . . . 4 ((0 ∈ β„€ ∧ 𝑀 ∈ (β„€β‰₯β€˜(0 + 1))) β†’ (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©}⟩}))β€˜π‘€) = ((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©}⟩}))β€˜(𝑀 βˆ’ 1))(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©}⟩})β€˜π‘€)))
91, 7, 8sylancr 585 . . 3 ((πœ‘ ∧ 𝑀 ∈ β„•) β†’ (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©}⟩}))β€˜π‘€) = ((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©}⟩}))β€˜(𝑀 βˆ’ 1))(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©}⟩})β€˜π‘€)))
10 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©}⟩}))
1110fveq1i 6891 . . 3 (π‘„β€˜π‘€) = (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©}⟩}))β€˜π‘€)
1210fveq1i 6891 . . . 4 (π‘„β€˜(𝑀 βˆ’ 1)) = (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©}⟩}))β€˜(𝑀 βˆ’ 1))
1312oveq1i 7423 . . 3 ((π‘„β€˜(𝑀 βˆ’ 1))(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©}⟩})β€˜π‘€)) = ((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©}⟩}))β€˜(𝑀 βˆ’ 1))(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©}⟩})β€˜π‘€))
149, 11, 133eqtr4g 2790 . 2 ((πœ‘ ∧ 𝑀 ∈ β„•) β†’ (π‘„β€˜π‘€) = ((π‘„β€˜(𝑀 βˆ’ 1))(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©}⟩})β€˜π‘€)))
15 0nnn 12273 . . . . . 6 Β¬ 0 ∈ β„•
16 disjsn 4712 . . . . . 6 ((β„• ∩ {0}) = βˆ… ↔ Β¬ 0 ∈ β„•)
1715, 16mpbir 230 . . . . 5 (β„• ∩ {0}) = βˆ…
18 fnresi 6679 . . . . . 6 ( I β†Ύ β„•) Fn β„•
19 c0ex 11233 . . . . . . 7 0 ∈ V
20 snex 5428 . . . . . . 7 {⟨0, π‘ƒβŸ©} ∈ V
2119, 20fnsn 6606 . . . . . 6 {⟨0, {⟨0, π‘ƒβŸ©}⟩} Fn {0}
22 fvun1 6982 . . . . . 6 ((( I β†Ύ β„•) Fn β„• ∧ {⟨0, {⟨0, π‘ƒβŸ©}⟩} Fn {0} ∧ ((β„• ∩ {0}) = βˆ… ∧ 𝑀 ∈ β„•)) β†’ ((( I β†Ύ β„•) βˆͺ {⟨0, {⟨0, π‘ƒβŸ©}⟩})β€˜π‘€) = (( I β†Ύ β„•)β€˜π‘€))
2318, 21, 22mp3an12 1447 . . . . 5 (((β„• ∩ {0}) = βˆ… ∧ 𝑀 ∈ β„•) β†’ ((( I β†Ύ β„•) βˆͺ {⟨0, {⟨0, π‘ƒβŸ©}⟩})β€˜π‘€) = (( I β†Ύ β„•)β€˜π‘€))
2417, 2, 23sylancr 585 . . . 4 ((πœ‘ ∧ 𝑀 ∈ β„•) β†’ ((( I β†Ύ β„•) βˆͺ {⟨0, {⟨0, π‘ƒβŸ©}⟩})β€˜π‘€) = (( I β†Ύ β„•)β€˜π‘€))
25 fvresi 7176 . . . . 5 (𝑀 ∈ β„• β†’ (( I β†Ύ β„•)β€˜π‘€) = 𝑀)
2625adantl 480 . . . 4 ((πœ‘ ∧ 𝑀 ∈ β„•) β†’ (( I β†Ύ β„•)β€˜π‘€) = 𝑀)
2724, 26eqtrd 2765 . . 3 ((πœ‘ ∧ 𝑀 ∈ β„•) β†’ ((( I β†Ύ β„•) βˆͺ {⟨0, {⟨0, π‘ƒβŸ©}⟩})β€˜π‘€) = 𝑀)
2827oveq2d 7429 . 2 ((πœ‘ ∧ 𝑀 ∈ β„•) β†’ ((π‘„β€˜(𝑀 βˆ’ 1))(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©}⟩})β€˜π‘€)) = ((π‘„β€˜(𝑀 βˆ’ 1))(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‘§))))𝑀))
29 fvexd 6905 . . 3 (πœ‘ β†’ (π‘„β€˜(𝑀 βˆ’ 1)) ∈ V)
30 simpr 483 . . . . . . . . 9 ((π‘₯ = (π‘„β€˜(𝑀 βˆ’ 1)) ∧ π‘š = 𝑀) β†’ π‘š = 𝑀)
3130oveq1d 7428 . . . . . . . 8 ((π‘₯ = (π‘„β€˜(𝑀 βˆ’ 1)) ∧ π‘š = 𝑀) β†’ (π‘š βˆ’ 1) = (𝑀 βˆ’ 1))
3231oveq1d 7428 . . . . . . 7 ((π‘₯ = (π‘„β€˜(𝑀 βˆ’ 1)) ∧ π‘š = 𝑀) β†’ ((π‘š βˆ’ 1) / 𝑁) = ((𝑀 βˆ’ 1) / 𝑁))
3330oveq1d 7428 . . . . . . 7 ((π‘₯ = (π‘„β€˜(𝑀 βˆ’ 1)) ∧ π‘š = 𝑀) β†’ (π‘š / 𝑁) = (𝑀 / 𝑁))
3432, 33oveq12d 7431 . . . . . 6 ((π‘₯ = (π‘„β€˜(𝑀 βˆ’ 1)) ∧ π‘š = 𝑀) β†’ (((π‘š βˆ’ 1) / 𝑁)[,](π‘š / 𝑁)) = (((𝑀 βˆ’ 1) / 𝑁)[,](𝑀 / 𝑁)))
35 cvmliftlem5.3 . . . . . 6 π‘Š = (((𝑀 βˆ’ 1) / 𝑁)[,](𝑀 / 𝑁))
3634, 35eqtr4di 2783 . . . . 5 ((π‘₯ = (π‘„β€˜(𝑀 βˆ’ 1)) ∧ π‘š = 𝑀) β†’ (((π‘š βˆ’ 1) / 𝑁)[,](π‘š / 𝑁)) = π‘Š)
3730fveq2d 6894 . . . . . . . . . 10 ((π‘₯ = (π‘„β€˜(𝑀 βˆ’ 1)) ∧ π‘š = 𝑀) β†’ (π‘‡β€˜π‘š) = (π‘‡β€˜π‘€))
3837fveq2d 6894 . . . . . . . . 9 ((π‘₯ = (π‘„β€˜(𝑀 βˆ’ 1)) ∧ π‘š = 𝑀) β†’ (2nd β€˜(π‘‡β€˜π‘š)) = (2nd β€˜(π‘‡β€˜π‘€)))
39 simpl 481 . . . . . . . . . . 11 ((π‘₯ = (π‘„β€˜(𝑀 βˆ’ 1)) ∧ π‘š = 𝑀) β†’ π‘₯ = (π‘„β€˜(𝑀 βˆ’ 1)))
4039, 32fveq12d 6897 . . . . . . . . . 10 ((π‘₯ = (π‘„β€˜(𝑀 βˆ’ 1)) ∧ π‘š = 𝑀) β†’ (π‘₯β€˜((π‘š βˆ’ 1) / 𝑁)) = ((π‘„β€˜(𝑀 βˆ’ 1))β€˜((𝑀 βˆ’ 1) / 𝑁)))
4140eleq1d 2810 . . . . . . . . 9 ((π‘₯ = (π‘„β€˜(𝑀 βˆ’ 1)) ∧ π‘š = 𝑀) β†’ ((π‘₯β€˜((π‘š βˆ’ 1) / 𝑁)) ∈ 𝑏 ↔ ((π‘„β€˜(𝑀 βˆ’ 1))β€˜((𝑀 βˆ’ 1) / 𝑁)) ∈ 𝑏))
4238, 41riotaeqbidv 7372 . . . . . . . 8 ((π‘₯ = (π‘„β€˜(𝑀 βˆ’ 1)) ∧ π‘š = 𝑀) β†’ (℩𝑏 ∈ (2nd β€˜(π‘‡β€˜π‘š))(π‘₯β€˜((π‘š βˆ’ 1) / 𝑁)) ∈ 𝑏) = (℩𝑏 ∈ (2nd β€˜(π‘‡β€˜π‘€))((π‘„β€˜(𝑀 βˆ’ 1))β€˜((𝑀 βˆ’ 1) / 𝑁)) ∈ 𝑏))
4342reseq2d 5980 . . . . . . 7 ((π‘₯ = (π‘„β€˜(𝑀 βˆ’ 1)) ∧ π‘š = 𝑀) β†’ (𝐹 β†Ύ (℩𝑏 ∈ (2nd β€˜(π‘‡β€˜π‘š))(π‘₯β€˜((π‘š βˆ’ 1) / 𝑁)) ∈ 𝑏)) = (𝐹 β†Ύ (℩𝑏 ∈ (2nd β€˜(π‘‡β€˜π‘€))((π‘„β€˜(𝑀 βˆ’ 1))β€˜((𝑀 βˆ’ 1) / 𝑁)) ∈ 𝑏)))
4443cnveqd 5873 . . . . . 6 ((π‘₯ = (π‘„β€˜(𝑀 βˆ’ 1)) ∧ π‘š = 𝑀) β†’ β—‘(𝐹 β†Ύ (℩𝑏 ∈ (2nd β€˜(π‘‡β€˜π‘š))(π‘₯β€˜((π‘š 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)))
4544fveq1d 6892 . . . . 5 ((π‘₯ = (π‘„β€˜(𝑀 βˆ’ 1)) ∧ π‘š = 𝑀) β†’ (β—‘(𝐹 β†Ύ (℩𝑏 ∈ (2nd β€˜(π‘‡β€˜π‘š))(π‘₯β€˜((π‘š 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€˜π‘§)))
4636, 45mpteq12dv 5235 . . . 4 ((π‘₯ = (π‘„β€˜(𝑀 βˆ’ 1)) ∧ π‘š = 𝑀) 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‘§))) = (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‘§))))
47 eqid 2725 . . . 4 (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‘§)))) = (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‘§))))
48 ovex 7446 . . . . . 6 (((𝑀 βˆ’ 1) / 𝑁)[,](𝑀 / 𝑁)) ∈ V
4935, 48eqeltri 2821 . . . . 5 π‘Š ∈ V
5049mptex 7229 . . . 4 (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‘§))) ∈ V
5146, 47, 50ovmpoa 7570 . . 3 (((π‘„β€˜(𝑀 βˆ’ 1)) ∈ V ∧ 𝑀 ∈ β„•) β†’ ((π‘„β€˜(𝑀 βˆ’ 1))(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‘§))))𝑀) = (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‘§))))
5229, 51sylan 578 . 2 ((πœ‘ ∧ 𝑀 ∈ β„•) β†’ ((π‘„β€˜(𝑀 βˆ’ 1))(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‘§))))𝑀) = (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‘§))))
5314, 28, 523eqtrd 2769 1 ((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‘§))))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 394   = wceq 1533   ∈ wcel 2098  βˆ€wral 3051  {crab 3419  Vcvv 3463   βˆ– cdif 3938   βˆͺ cun 3939   ∩ cin 3940   βŠ† wss 3941  βˆ…c0 4319  π’« cpw 4599  {csn 4625  βŸ¨cop 4631  βˆͺ cuni 4904  βˆͺ ciun 4992   ↦ cmpt 5227   I cid 5570   Γ— cxp 5671  β—‘ccnv 5672  ran crn 5674   β†Ύ cres 5675   β€œ cima 5676   Fn wfn 6538  βŸΆwf 6539  β€˜cfv 6543  β„©crio 7368  (class class class)co 7413   ∈ cmpo 7415  1st c1st 7985  2nd c2nd 7986  0cc0 11133  1c1 11134   + caddc 11136   βˆ’ cmin 11469   / cdiv 11896  β„•cn 12237  β„€cz 12583  β„€β‰₯cuz 12847  (,)cioo 13351  [,]cicc 13354  ...cfz 13511  seqcseq 13993   β†Ύt crest 17396  topGenctg 17413   Cn ccn 23141  Homeochmeo 23670  IIcii 24808   CovMap ccvm 34918
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5281  ax-sep 5295  ax-nul 5302  ax-pow 5360  ax-pr 5424  ax-un 7735  ax-cnex 11189  ax-resscn 11190  ax-1cn 11191  ax-icn 11192  ax-addcl 11193  ax-addrcl 11194  ax-mulcl 11195  ax-mulrcl 11196  ax-mulcom 11197  ax-addass 11198  ax-mulass 11199  ax-distr 11200  ax-i2m1 11201  ax-1ne0 11202  ax-1rid 11203  ax-rnegex 11204  ax-rrecex 11205  ax-cnre 11206  ax-pre-lttri 11207  ax-pre-lttrn 11208  ax-pre-ltadd 11209  ax-pre-mulgt0 11210
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-pss 3961  df-nul 4320  df-if 4526  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4905  df-iun 4994  df-br 5145  df-opab 5207  df-mpt 5228  df-tr 5262  df-id 5571  df-eprel 5577  df-po 5585  df-so 5586  df-fr 5628  df-we 5630  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  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 7369  df-ov 7416  df-oprab 7417  df-mpo 7418  df-om 7866  df-2nd 7988  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-er 8718  df-en 8958  df-dom 8959  df-sdom 8960  df-pnf 11275  df-mnf 11276  df-xr 11277  df-ltxr 11278  df-le 11279  df-sub 11471  df-neg 11472  df-nn 12238  df-n0 12498  df-z 12584  df-uz 12848  df-seq 13994
This theorem is referenced by:  cvmliftlem6  34953  cvmliftlem8  34955  cvmliftlem9  34956
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