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Theorem cvmliftlem6 34350
Description: Lemma for cvmlift 34359. Induction step for cvmliftlem7 34351. Assuming that 𝑄(𝑀 βˆ’ 1) is defined at (𝑀 βˆ’ 1) / 𝑁 and is a preimage of 𝐺((𝑀 βˆ’ 1) / 𝑁), the next segment 𝑄(𝑀) is also defined and is a function on π‘Š which is a lift 𝐺 for this segment. This follows explicitly from the definition 𝑄(𝑀) = β—‘(𝐹 β†Ύ 𝐼) ∘ 𝐺 since 𝐺 is in 1st β€˜(πΉβ€˜π‘€) for the entire interval so that β—‘(𝐹 β†Ύ 𝐼) maps this into 𝐼 and 𝐹 ∘ 𝑄 maps back to 𝐺. (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))
cvmliftlem6.1 ((πœ‘ ∧ πœ“) β†’ 𝑀 ∈ (1...𝑁))
cvmliftlem6.2 ((πœ‘ ∧ πœ“) β†’ ((π‘„β€˜(𝑀 βˆ’ 1))β€˜((𝑀 βˆ’ 1) / 𝑁)) ∈ (◑𝐹 β€œ {(πΊβ€˜((𝑀 βˆ’ 1) / 𝑁))}))
Assertion
Ref Expression
cvmliftlem6 ((πœ‘ ∧ πœ“) β†’ ((π‘„β€˜π‘€):π‘ŠβŸΆπ΅ ∧ (𝐹 ∘ (π‘„β€˜π‘€)) = (𝐺 β†Ύ π‘Š)))
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   πœ“,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)   πœ“(π‘₯,𝑣,𝑒,𝑗,π‘˜,π‘š,𝑠,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 cvmliftlem6
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 cvmliftlem.1 . . . . . . . . . . 11 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‘˜))))})
2 cvmliftlem.b . . . . . . . . . . 11 𝐡 = βˆͺ 𝐢
3 cvmliftlem.x . . . . . . . . . . 11 𝑋 = βˆͺ 𝐽
4 cvmliftlem.f . . . . . . . . . . 11 (πœ‘ β†’ 𝐹 ∈ (𝐢 CovMap 𝐽))
5 cvmliftlem.g . . . . . . . . . . 11 (πœ‘ β†’ 𝐺 ∈ (II Cn 𝐽))
6 cvmliftlem.p . . . . . . . . . . 11 (πœ‘ β†’ 𝑃 ∈ 𝐡)
7 cvmliftlem.e . . . . . . . . . . 11 (πœ‘ β†’ (πΉβ€˜π‘ƒ) = (πΊβ€˜0))
8 cvmliftlem.n . . . . . . . . . . 11 (πœ‘ β†’ 𝑁 ∈ β„•)
9 cvmliftlem.t . . . . . . . . . . 11 (πœ‘ β†’ 𝑇:(1...𝑁)⟢βˆͺ 𝑗 ∈ 𝐽 ({𝑗} Γ— (π‘†β€˜π‘—)))
10 cvmliftlem.a . . . . . . . . . . 11 (πœ‘ β†’ βˆ€π‘˜ ∈ (1...𝑁)(𝐺 β€œ (((π‘˜ βˆ’ 1) / 𝑁)[,](π‘˜ / 𝑁))) βŠ† (1st β€˜(π‘‡β€˜π‘˜)))
11 cvmliftlem.l . . . . . . . . . . 11 𝐿 = (topGenβ€˜ran (,))
12 cvmliftlem6.1 . . . . . . . . . . . 12 ((πœ‘ ∧ πœ“) β†’ 𝑀 ∈ (1...𝑁))
1312adantrr 715 . . . . . . . . . . 11 ((πœ‘ ∧ (πœ“ ∧ 𝑧 ∈ π‘Š)) β†’ 𝑀 ∈ (1...𝑁))
141, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13cvmliftlem1 34345 . . . . . . . . . 10 ((πœ‘ ∧ (πœ“ ∧ 𝑧 ∈ π‘Š)) β†’ (2nd β€˜(π‘‡β€˜π‘€)) ∈ (π‘†β€˜(1st β€˜(π‘‡β€˜π‘€))))
151cvmsss 34327 . . . . . . . . . 10 ((2nd β€˜(π‘‡β€˜π‘€)) ∈ (π‘†β€˜(1st β€˜(π‘‡β€˜π‘€))) β†’ (2nd β€˜(π‘‡β€˜π‘€)) βŠ† 𝐢)
1614, 15syl 17 . . . . . . . . 9 ((πœ‘ ∧ (πœ“ ∧ 𝑧 ∈ π‘Š)) β†’ (2nd β€˜(π‘‡β€˜π‘€)) βŠ† 𝐢)
174adantr 481 . . . . . . . . . . 11 ((πœ‘ ∧ (πœ“ ∧ 𝑧 ∈ π‘Š)) β†’ 𝐹 ∈ (𝐢 CovMap 𝐽))
18 cvmliftlem6.2 . . . . . . . . . . . . . 14 ((πœ‘ ∧ πœ“) β†’ ((π‘„β€˜(𝑀 βˆ’ 1))β€˜((𝑀 βˆ’ 1) / 𝑁)) ∈ (◑𝐹 β€œ {(πΊβ€˜((𝑀 βˆ’ 1) / 𝑁))}))
1918adantrr 715 . . . . . . . . . . . . 13 ((πœ‘ ∧ (πœ“ ∧ 𝑧 ∈ π‘Š)) β†’ ((π‘„β€˜(𝑀 βˆ’ 1))β€˜((𝑀 βˆ’ 1) / 𝑁)) ∈ (◑𝐹 β€œ {(πΊβ€˜((𝑀 βˆ’ 1) / 𝑁))}))
20 cvmcn 34322 . . . . . . . . . . . . . . 15 (𝐹 ∈ (𝐢 CovMap 𝐽) β†’ 𝐹 ∈ (𝐢 Cn 𝐽))
212, 3cnf 22757 . . . . . . . . . . . . . . 15 (𝐹 ∈ (𝐢 Cn 𝐽) β†’ 𝐹:π΅βŸΆπ‘‹)
2217, 20, 213syl 18 . . . . . . . . . . . . . 14 ((πœ‘ ∧ (πœ“ ∧ 𝑧 ∈ π‘Š)) β†’ 𝐹:π΅βŸΆπ‘‹)
23 ffn 6717 . . . . . . . . . . . . . 14 (𝐹:π΅βŸΆπ‘‹ β†’ 𝐹 Fn 𝐡)
24 fniniseg 7061 . . . . . . . . . . . . . 14 (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)))))
2522, 23, 243syl 18 . . . . . . . . . . . . 13 ((πœ‘ ∧ (πœ“ ∧ 𝑧 ∈ π‘Š)) 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)))))
2619, 25mpbid 231 . . . . . . . . . . . 12 ((πœ‘ ∧ (πœ“ ∧ 𝑧 ∈ π‘Š)) 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))))
2726simpld 495 . . . . . . . . . . 11 ((πœ‘ ∧ (πœ“ ∧ 𝑧 ∈ π‘Š)) β†’ ((π‘„β€˜(𝑀 βˆ’ 1))β€˜((𝑀 βˆ’ 1) / 𝑁)) ∈ 𝐡)
2826simprd 496 . . . . . . . . . . . 12 ((πœ‘ ∧ (πœ“ ∧ 𝑧 ∈ π‘Š)) β†’ (πΉβ€˜((π‘„β€˜(𝑀 βˆ’ 1))β€˜((𝑀 βˆ’ 1) / 𝑁))) = (πΊβ€˜((𝑀 βˆ’ 1) / 𝑁)))
29 cvmliftlem5.3 . . . . . . . . . . . . 13 π‘Š = (((𝑀 βˆ’ 1) / 𝑁)[,](𝑀 / 𝑁))
30 elfznn 13532 . . . . . . . . . . . . . . . . . . . 20 (𝑀 ∈ (1...𝑁) β†’ 𝑀 ∈ β„•)
3113, 30syl 17 . . . . . . . . . . . . . . . . . . 19 ((πœ‘ ∧ (πœ“ ∧ 𝑧 ∈ π‘Š)) β†’ 𝑀 ∈ β„•)
3231nnred 12229 . . . . . . . . . . . . . . . . . 18 ((πœ‘ ∧ (πœ“ ∧ 𝑧 ∈ π‘Š)) β†’ 𝑀 ∈ ℝ)
33 peano2rem 11529 . . . . . . . . . . . . . . . . . 18 (𝑀 ∈ ℝ β†’ (𝑀 βˆ’ 1) ∈ ℝ)
3432, 33syl 17 . . . . . . . . . . . . . . . . 17 ((πœ‘ ∧ (πœ“ ∧ 𝑧 ∈ π‘Š)) β†’ (𝑀 βˆ’ 1) ∈ ℝ)
358adantr 481 . . . . . . . . . . . . . . . . 17 ((πœ‘ ∧ (πœ“ ∧ 𝑧 ∈ π‘Š)) β†’ 𝑁 ∈ β„•)
3634, 35nndivred 12268 . . . . . . . . . . . . . . . 16 ((πœ‘ ∧ (πœ“ ∧ 𝑧 ∈ π‘Š)) β†’ ((𝑀 βˆ’ 1) / 𝑁) ∈ ℝ)
3736rexrd 11266 . . . . . . . . . . . . . . 15 ((πœ‘ ∧ (πœ“ ∧ 𝑧 ∈ π‘Š)) β†’ ((𝑀 βˆ’ 1) / 𝑁) ∈ ℝ*)
3832, 35nndivred 12268 . . . . . . . . . . . . . . . 16 ((πœ‘ ∧ (πœ“ ∧ 𝑧 ∈ π‘Š)) β†’ (𝑀 / 𝑁) ∈ ℝ)
3938rexrd 11266 . . . . . . . . . . . . . . 15 ((πœ‘ ∧ (πœ“ ∧ 𝑧 ∈ π‘Š)) β†’ (𝑀 / 𝑁) ∈ ℝ*)
4032ltm1d 12148 . . . . . . . . . . . . . . . . 17 ((πœ‘ ∧ (πœ“ ∧ 𝑧 ∈ π‘Š)) β†’ (𝑀 βˆ’ 1) < 𝑀)
4135nnred 12229 . . . . . . . . . . . . . . . . . 18 ((πœ‘ ∧ (πœ“ ∧ 𝑧 ∈ π‘Š)) β†’ 𝑁 ∈ ℝ)
4235nngt0d 12263 . . . . . . . . . . . . . . . . . 18 ((πœ‘ ∧ (πœ“ ∧ 𝑧 ∈ π‘Š)) β†’ 0 < 𝑁)
43 ltdiv1 12080 . . . . . . . . . . . . . . . . . 18 (((𝑀 βˆ’ 1) ∈ ℝ ∧ 𝑀 ∈ ℝ ∧ (𝑁 ∈ ℝ ∧ 0 < 𝑁)) β†’ ((𝑀 βˆ’ 1) < 𝑀 ↔ ((𝑀 βˆ’ 1) / 𝑁) < (𝑀 / 𝑁)))
4434, 32, 41, 42, 43syl112anc 1374 . . . . . . . . . . . . . . . . 17 ((πœ‘ ∧ (πœ“ ∧ 𝑧 ∈ π‘Š)) β†’ ((𝑀 βˆ’ 1) < 𝑀 ↔ ((𝑀 βˆ’ 1) / 𝑁) < (𝑀 / 𝑁)))
4540, 44mpbid 231 . . . . . . . . . . . . . . . 16 ((πœ‘ ∧ (πœ“ ∧ 𝑧 ∈ π‘Š)) β†’ ((𝑀 βˆ’ 1) / 𝑁) < (𝑀 / 𝑁))
4636, 38, 45ltled 11364 . . . . . . . . . . . . . . 15 ((πœ‘ ∧ (πœ“ ∧ 𝑧 ∈ π‘Š)) β†’ ((𝑀 βˆ’ 1) / 𝑁) ≀ (𝑀 / 𝑁))
47 lbicc2 13443 . . . . . . . . . . . . . . 15 ((((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)))
4837, 39, 46, 47syl3anc 1371 . . . . . . . . . . . . . 14 ((πœ‘ ∧ (πœ“ ∧ 𝑧 ∈ π‘Š)) β†’ ((𝑀 βˆ’ 1) / 𝑁) ∈ (((𝑀 βˆ’ 1) / 𝑁)[,](𝑀 / 𝑁)))
4948, 29eleqtrrdi 2844 . . . . . . . . . . . . 13 ((πœ‘ ∧ (πœ“ ∧ 𝑧 ∈ π‘Š)) β†’ ((𝑀 βˆ’ 1) / 𝑁) ∈ π‘Š)
501, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 29, 49cvmliftlem3 34347 . . . . . . . . . . . 12 ((πœ‘ ∧ (πœ“ ∧ 𝑧 ∈ π‘Š)) β†’ (πΊβ€˜((𝑀 βˆ’ 1) / 𝑁)) ∈ (1st β€˜(π‘‡β€˜π‘€)))
5128, 50eqeltrd 2833 . . . . . . . . . . 11 ((πœ‘ ∧ (πœ“ ∧ 𝑧 ∈ π‘Š)) β†’ (πΉβ€˜((π‘„β€˜(𝑀 βˆ’ 1))β€˜((𝑀 βˆ’ 1) / 𝑁))) ∈ (1st β€˜(π‘‡β€˜π‘€)))
52 eqid 2732 . . . . . . . . . . . 12 (℩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)
531, 2, 52cvmsiota 34337 . . . . . . . . . . 11 ((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)))
5417, 14, 27, 51, 53syl13anc 1372 . . . . . . . . . 10 ((πœ‘ ∧ (πœ“ ∧ 𝑧 ∈ π‘Š)) 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)))
5554simpld 495 . . . . . . . . 9 ((πœ‘ ∧ (πœ“ ∧ 𝑧 ∈ π‘Š)) β†’ (℩𝑏 ∈ (2nd β€˜(π‘‡β€˜π‘€))((π‘„β€˜(𝑀 βˆ’ 1))β€˜((𝑀 βˆ’ 1) / 𝑁)) ∈ 𝑏) ∈ (2nd β€˜(π‘‡β€˜π‘€)))
5616, 55sseldd 3983 . . . . . . . 8 ((πœ‘ ∧ (πœ“ ∧ 𝑧 ∈ π‘Š)) β†’ (℩𝑏 ∈ (2nd β€˜(π‘‡β€˜π‘€))((π‘„β€˜(𝑀 βˆ’ 1))β€˜((𝑀 βˆ’ 1) / 𝑁)) ∈ 𝑏) ∈ 𝐢)
57 elssuni 4941 . . . . . . . 8 ((℩𝑏 ∈ (2nd β€˜(π‘‡β€˜π‘€))((π‘„β€˜(𝑀 βˆ’ 1))β€˜((𝑀 βˆ’ 1) / 𝑁)) ∈ 𝑏) ∈ 𝐢 β†’ (℩𝑏 ∈ (2nd β€˜(π‘‡β€˜π‘€))((π‘„β€˜(𝑀 βˆ’ 1))β€˜((𝑀 βˆ’ 1) / 𝑁)) ∈ 𝑏) βŠ† βˆͺ 𝐢)
5856, 57syl 17 . . . . . . 7 ((πœ‘ ∧ (πœ“ ∧ 𝑧 ∈ π‘Š)) β†’ (℩𝑏 ∈ (2nd β€˜(π‘‡β€˜π‘€))((π‘„β€˜(𝑀 βˆ’ 1))β€˜((𝑀 βˆ’ 1) / 𝑁)) ∈ 𝑏) βŠ† βˆͺ 𝐢)
5958, 2sseqtrrdi 4033 . . . . . 6 ((πœ‘ ∧ (πœ“ ∧ 𝑧 ∈ π‘Š)) β†’ (℩𝑏 ∈ (2nd β€˜(π‘‡β€˜π‘€))((π‘„β€˜(𝑀 βˆ’ 1))β€˜((𝑀 βˆ’ 1) / 𝑁)) ∈ 𝑏) βŠ† 𝐡)
601cvmsf1o 34332 . . . . . . . . 9 ((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‘€)))
6117, 14, 55, 60syl3anc 1371 . . . . . . . 8 ((πœ‘ ∧ (πœ“ ∧ 𝑧 ∈ π‘Š)) 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‘€)))
62 f1ocnv 6845 . . . . . . . 8 ((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‘€)) β†’ β—‘(𝐹 β†Ύ (℩𝑏 ∈ (2nd β€˜(π‘‡β€˜π‘€))((π‘„β€˜(𝑀 βˆ’ 1))β€˜((𝑀 βˆ’ 1) / 𝑁)) ∈ 𝑏)):(1st β€˜(π‘‡β€˜π‘€))–1-1-ontoβ†’(℩𝑏 ∈ (2nd β€˜(π‘‡β€˜π‘€))((π‘„β€˜(𝑀 βˆ’ 1))β€˜((𝑀 βˆ’ 1) / 𝑁)) ∈ 𝑏))
63 f1of 6833 . . . . . . . 8 (β—‘(𝐹 β†Ύ (℩𝑏 ∈ (2nd β€˜(π‘‡β€˜π‘€))((π‘„β€˜(𝑀 βˆ’ 1))β€˜((𝑀 βˆ’ 1) / 𝑁)) ∈ 𝑏)):(1st β€˜(π‘‡β€˜π‘€))–1-1-ontoβ†’(℩𝑏 ∈ (2nd β€˜(π‘‡β€˜π‘€))((π‘„β€˜(𝑀 βˆ’ 1))β€˜((𝑀 βˆ’ 1) / 𝑁)) ∈ 𝑏) β†’ β—‘(𝐹 β†Ύ (℩𝑏 ∈ (2nd β€˜(π‘‡β€˜π‘€))((π‘„β€˜(𝑀 βˆ’ 1))β€˜((𝑀 βˆ’ 1) / 𝑁)) ∈ 𝑏)):(1st β€˜(π‘‡β€˜π‘€))⟢(℩𝑏 ∈ (2nd β€˜(π‘‡β€˜π‘€))((π‘„β€˜(𝑀 βˆ’ 1))β€˜((𝑀 βˆ’ 1) / 𝑁)) ∈ 𝑏))
6461, 62, 633syl 18 . . . . . . 7 ((πœ‘ ∧ (πœ“ ∧ 𝑧 ∈ π‘Š)) β†’ β—‘(𝐹 β†Ύ (℩𝑏 ∈ (2nd β€˜(π‘‡β€˜π‘€))((π‘„β€˜(𝑀 βˆ’ 1))β€˜((𝑀 βˆ’ 1) / 𝑁)) ∈ 𝑏)):(1st β€˜(π‘‡β€˜π‘€))⟢(℩𝑏 ∈ (2nd β€˜(π‘‡β€˜π‘€))((π‘„β€˜(𝑀 βˆ’ 1))β€˜((𝑀 βˆ’ 1) / 𝑁)) ∈ 𝑏))
65 simprr 771 . . . . . . . 8 ((πœ‘ ∧ (πœ“ ∧ 𝑧 ∈ π‘Š)) β†’ 𝑧 ∈ π‘Š)
661, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 29, 65cvmliftlem3 34347 . . . . . . 7 ((πœ‘ ∧ (πœ“ ∧ 𝑧 ∈ π‘Š)) β†’ (πΊβ€˜π‘§) ∈ (1st β€˜(π‘‡β€˜π‘€)))
6764, 66ffvelcdmd 7087 . . . . . 6 ((πœ‘ ∧ (πœ“ ∧ 𝑧 ∈ π‘Š)) 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‘§)) ∈ (℩𝑏 ∈ (2nd β€˜(π‘‡β€˜π‘€))((π‘„β€˜(𝑀 βˆ’ 1))β€˜((𝑀 βˆ’ 1) / 𝑁)) ∈ 𝑏))
6859, 67sseldd 3983 . . . . 5 ((πœ‘ ∧ (πœ“ ∧ 𝑧 ∈ π‘Š)) 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‘§)) ∈ 𝐡)
6968anassrs 468 . . . 4 (((πœ‘ ∧ πœ“) ∧ 𝑧 ∈ π‘Š) 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‘§)) ∈ 𝐡)
7069fmpttd 7116 . . 3 ((πœ‘ ∧ πœ“) 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‘§))):π‘ŠβŸΆπ΅)
7112, 30syl 17 . . . . 5 ((πœ‘ ∧ πœ“) β†’ 𝑀 ∈ β„•)
72 cvmliftlem.q . . . . . 6 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©}⟩}))
731, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 72, 29cvmliftlem5 34349 . . . . 5 ((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‘§))))
7471, 73syldan 591 . . . 4 ((πœ‘ ∧ πœ“) 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‘§))))
7574feq1d 6702 . . 3 ((πœ‘ ∧ πœ“) β†’ ((π‘„β€˜π‘€):π‘ŠβŸΆπ΅ ↔ (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‘§))):π‘ŠβŸΆπ΅))
7670, 75mpbird 256 . 2 ((πœ‘ ∧ πœ“) β†’ (π‘„β€˜π‘€):π‘ŠβŸΆπ΅)
77 fvres 6910 . . . . . . 7 (𝑧 ∈ π‘Š β†’ ((𝐺 β†Ύ π‘Š)β€˜π‘§) = (πΊβ€˜π‘§))
7865, 77syl 17 . . . . . 6 ((πœ‘ ∧ (πœ“ ∧ 𝑧 ∈ π‘Š)) β†’ ((𝐺 β†Ύ π‘Š)β€˜π‘§) = (πΊβ€˜π‘§))
79 f1ocnvfv2 7277 . . . . . . 7 (((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Ίβ€˜π‘§) ∈ (1st 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))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‘§))) = (πΊβ€˜π‘§))
8061, 66, 79syl2anc 584 . . . . . 6 ((πœ‘ ∧ (πœ“ ∧ 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€˜(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‘§))) = (πΊβ€˜π‘§))
81 fvres 6910 . . . . . . 7 ((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‘§)) ∈ (℩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€˜(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‘§))) = (πΉβ€˜(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‘§))))
8267, 81syl 17 . . . . . 6 ((πœ‘ ∧ (πœ“ ∧ 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€˜(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‘§))) = (πΉβ€˜(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‘§))))
8378, 80, 823eqtr2rd 2779 . . . . 5 ((πœ‘ ∧ (πœ“ ∧ 𝑧 ∈ π‘Š)) β†’ (πΉβ€˜(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‘§))) = ((𝐺 β†Ύ π‘Š)β€˜π‘§))
8483anassrs 468 . . . 4 (((πœ‘ ∧ πœ“) ∧ 𝑧 ∈ π‘Š) β†’ (πΉβ€˜(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‘§))) = ((𝐺 β†Ύ π‘Š)β€˜π‘§))
8584mpteq2dva 5248 . . 3 ((πœ‘ ∧ πœ“) β†’ (𝑧 ∈ π‘Š ↦ (πΉβ€˜(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‘§)))) = (𝑧 ∈ π‘Š ↦ ((𝐺 β†Ύ π‘Š)β€˜π‘§)))
864, 20, 213syl 18 . . . . . 6 (πœ‘ β†’ 𝐹:π΅βŸΆπ‘‹)
8786adantr 481 . . . . 5 ((πœ‘ ∧ πœ“) β†’ 𝐹:π΅βŸΆπ‘‹)
8887feqmptd 6960 . . . 4 ((πœ‘ ∧ πœ“) β†’ 𝐹 = (𝑦 ∈ 𝐡 ↦ (πΉβ€˜π‘¦)))
89 fveq2 6891 . . . 4 (𝑦 = (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†’ (πΉβ€˜π‘¦) = (πΉβ€˜(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‘§))))
9069, 74, 88, 89fmptco 7129 . . 3 ((πœ‘ ∧ πœ“) β†’ (𝐹 ∘ (π‘„β€˜π‘€)) = (𝑧 ∈ π‘Š ↦ (πΉβ€˜(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‘§)))))
91 iiuni 24404 . . . . . . . 8 (0[,]1) = βˆͺ II
9291, 3cnf 22757 . . . . . . 7 (𝐺 ∈ (II Cn 𝐽) β†’ 𝐺:(0[,]1)βŸΆπ‘‹)
935, 92syl 17 . . . . . 6 (πœ‘ β†’ 𝐺:(0[,]1)βŸΆπ‘‹)
9493adantr 481 . . . . 5 ((πœ‘ ∧ πœ“) β†’ 𝐺:(0[,]1)βŸΆπ‘‹)
951, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 29cvmliftlem2 34346 . . . . 5 ((πœ‘ ∧ πœ“) β†’ π‘Š βŠ† (0[,]1))
9694, 95fssresd 6758 . . . 4 ((πœ‘ ∧ πœ“) β†’ (𝐺 β†Ύ π‘Š):π‘ŠβŸΆπ‘‹)
9796feqmptd 6960 . . 3 ((πœ‘ ∧ πœ“) β†’ (𝐺 β†Ύ π‘Š) = (𝑧 ∈ π‘Š ↦ ((𝐺 β†Ύ π‘Š)β€˜π‘§)))
9885, 90, 973eqtr4d 2782 . 2 ((πœ‘ ∧ πœ“) β†’ (𝐹 ∘ (π‘„β€˜π‘€)) = (𝐺 β†Ύ π‘Š))
9976, 98jca 512 1 ((πœ‘ ∧ πœ“) β†’ ((π‘„β€˜π‘€):π‘ŠβŸΆπ΅ ∧ (𝐹 ∘ (π‘„β€˜π‘€)) = (𝐺 β†Ύ π‘Š)))
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 3945   βˆͺ cun 3946   ∩ cin 3947   βŠ† wss 3948  βˆ…c0 4322  π’« cpw 4602  {csn 4628  βŸ¨cop 4634  βˆͺ cuni 4908  βˆͺ ciun 4997   class class class wbr 5148   ↦ cmpt 5231   I cid 5573   Γ— cxp 5674  β—‘ccnv 5675  ran crn 5677   β†Ύ cres 5678   β€œ cima 5679   ∘ ccom 5680   Fn wfn 6538  βŸΆwf 6539  β€“1-1-ontoβ†’wf1o 6542  β€˜cfv 6543  β„©crio 7366  (class class class)co 7411   ∈ cmpo 7413  1st c1st 7975  2nd c2nd 7976  β„cr 11111  0cc0 11112  1c1 11113  β„*cxr 11249   < clt 11250   ≀ cle 11251   βˆ’ cmin 11446   / cdiv 11873  β„•cn 12214  (,)cioo 13326  [,]cicc 13329  ...cfz 13486  seqcseq 13968   β†Ύt crest 17368  topGenctg 17385   Cn ccn 22735  Homeochmeo 23264  IIcii 24398   CovMap ccvm 34315
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 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189  ax-pre-sup 11190
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 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 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-er 8705  df-map 8824  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-fi 9408  df-sup 9439  df-inf 9440  df-pnf 11252  df-mnf 11253  df-xr 11254  df-ltxr 11255  df-le 11256  df-sub 11448  df-neg 11449  df-div 11874  df-nn 12215  df-2 12277  df-3 12278  df-n0 12475  df-z 12561  df-uz 12825  df-q 12935  df-rp 12977  df-xneg 13094  df-xadd 13095  df-xmul 13096  df-icc 13333  df-fz 13487  df-seq 13969  df-exp 14030  df-cj 15048  df-re 15049  df-im 15050  df-sqrt 15184  df-abs 15185  df-rest 17370  df-topgen 17391  df-psmet 20942  df-xmet 20943  df-met 20944  df-bl 20945  df-mopn 20946  df-top 22403  df-topon 22420  df-bases 22456  df-cn 22738  df-hmeo 23266  df-ii 24400  df-cvm 34316
This theorem is referenced by:  cvmliftlem7  34351  cvmliftlem10  34354  cvmliftlem13  34356
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