Users' Mathboxes Mathbox for Mario Carneiro < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cvmliftlem6 Structured version   Visualization version   GIF version

Theorem cvmliftlem6 34281
Description: Lemma for cvmlift 34290. Induction step for cvmliftlem7 34282. 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 716 . . . . . . . . . . 11 ((πœ‘ ∧ (πœ“ ∧ 𝑧 ∈ π‘Š)) β†’ 𝑀 ∈ (1...𝑁))
141, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13cvmliftlem1 34276 . . . . . . . . . 10 ((πœ‘ ∧ (πœ“ ∧ 𝑧 ∈ π‘Š)) β†’ (2nd β€˜(π‘‡β€˜π‘€)) ∈ (π‘†β€˜(1st β€˜(π‘‡β€˜π‘€))))
151cvmsss 34258 . . . . . . . . . 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 482 . . . . . . . . . . 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 716 . . . . . . . . . . . . 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 34253 . . . . . . . . . . . . . . 15 (𝐹 ∈ (𝐢 CovMap 𝐽) β†’ 𝐹 ∈ (𝐢 Cn 𝐽))
212, 3cnf 22750 . . . . . . . . . . . . . . 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 6718 . . . . . . . . . . . . . 14 (𝐹:π΅βŸΆπ‘‹ β†’ 𝐹 Fn 𝐡)
24 fniniseg 7062 . . . . . . . . . . . . . 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 496 . . . . . . . . . . 11 ((πœ‘ ∧ (πœ“ ∧ 𝑧 ∈ π‘Š)) β†’ ((π‘„β€˜(𝑀 βˆ’ 1))β€˜((𝑀 βˆ’ 1) / 𝑁)) ∈ 𝐡)
2826simprd 497 . . . . . . . . . . . 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 13530 . . . . . . . . . . . . . . . . . . . 20 (𝑀 ∈ (1...𝑁) β†’ 𝑀 ∈ β„•)
3113, 30syl 17 . . . . . . . . . . . . . . . . . . 19 ((πœ‘ ∧ (πœ“ ∧ 𝑧 ∈ π‘Š)) β†’ 𝑀 ∈ β„•)
3231nnred 12227 . . . . . . . . . . . . . . . . . 18 ((πœ‘ ∧ (πœ“ ∧ 𝑧 ∈ π‘Š)) β†’ 𝑀 ∈ ℝ)
33 peano2rem 11527 . . . . . . . . . . . . . . . . . 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 482 . . . . . . . . . . . . . . . . 17 ((πœ‘ ∧ (πœ“ ∧ 𝑧 ∈ π‘Š)) β†’ 𝑁 ∈ β„•)
3634, 35nndivred 12266 . . . . . . . . . . . . . . . 16 ((πœ‘ ∧ (πœ“ ∧ 𝑧 ∈ π‘Š)) β†’ ((𝑀 βˆ’ 1) / 𝑁) ∈ ℝ)
3736rexrd 11264 . . . . . . . . . . . . . . 15 ((πœ‘ ∧ (πœ“ ∧ 𝑧 ∈ π‘Š)) β†’ ((𝑀 βˆ’ 1) / 𝑁) ∈ ℝ*)
3832, 35nndivred 12266 . . . . . . . . . . . . . . . 16 ((πœ‘ ∧ (πœ“ ∧ 𝑧 ∈ π‘Š)) β†’ (𝑀 / 𝑁) ∈ ℝ)
3938rexrd 11264 . . . . . . . . . . . . . . 15 ((πœ‘ ∧ (πœ“ ∧ 𝑧 ∈ π‘Š)) β†’ (𝑀 / 𝑁) ∈ ℝ*)
4032ltm1d 12146 . . . . . . . . . . . . . . . . 17 ((πœ‘ ∧ (πœ“ ∧ 𝑧 ∈ π‘Š)) β†’ (𝑀 βˆ’ 1) < 𝑀)
4135nnred 12227 . . . . . . . . . . . . . . . . . 18 ((πœ‘ ∧ (πœ“ ∧ 𝑧 ∈ π‘Š)) β†’ 𝑁 ∈ ℝ)
4235nngt0d 12261 . . . . . . . . . . . . . . . . . 18 ((πœ‘ ∧ (πœ“ ∧ 𝑧 ∈ π‘Š)) β†’ 0 < 𝑁)
43 ltdiv1 12078 . . . . . . . . . . . . . . . . . 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 1375 . . . . . . . . . . . . . . . . 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 11362 . . . . . . . . . . . . . . 15 ((πœ‘ ∧ (πœ“ ∧ 𝑧 ∈ π‘Š)) β†’ ((𝑀 βˆ’ 1) / 𝑁) ≀ (𝑀 / 𝑁))
47 lbicc2 13441 . . . . . . . . . . . . . . 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 1372 . . . . . . . . . . . . . 14 ((πœ‘ ∧ (πœ“ ∧ 𝑧 ∈ π‘Š)) β†’ ((𝑀 βˆ’ 1) / 𝑁) ∈ (((𝑀 βˆ’ 1) / 𝑁)[,](𝑀 / 𝑁)))
4948, 29eleqtrrdi 2845 . . . . . . . . . . . . 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 34278 . . . . . . . . . . . 12 ((πœ‘ ∧ (πœ“ ∧ 𝑧 ∈ π‘Š)) β†’ (πΊβ€˜((𝑀 βˆ’ 1) / 𝑁)) ∈ (1st β€˜(π‘‡β€˜π‘€)))
5128, 50eqeltrd 2834 . . . . . . . . . . 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 2733 . . . . . . . . . . . 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 34268 . . . . . . . . . . 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 1373 . . . . . . . . . 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 496 . . . . . . . . 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 3984 . . . . . . . 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 4942 . . . . . . . 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 4034 . . . . . 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 34263 . . . . . . . . 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 1372 . . . . . . . 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 6846 . . . . . . . 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 6834 . . . . . . . 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 772 . . . . . . . 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 34278 . . . . . . 7 ((πœ‘ ∧ (πœ“ ∧ 𝑧 ∈ π‘Š)) β†’ (πΊβ€˜π‘§) ∈ (1st β€˜(π‘‡β€˜π‘€)))
6764, 66ffvelcdmd 7088 . . . . . 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 3984 . . . . 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 469 . . . 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 7115 . . 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 34280 . . . . 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 592 . . . 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 6703 . . 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 257 . 2 ((πœ‘ ∧ πœ“) β†’ (π‘„β€˜π‘€):π‘ŠβŸΆπ΅)
77 fvres 6911 . . . . . . 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 7275 . . . . . . 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 585 . . . . . 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 6911 . . . . . . 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 2780 . . . . 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 469 . . . 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 5249 . . 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 482 . . . . 5 ((πœ‘ ∧ πœ“) β†’ 𝐹:π΅βŸΆπ‘‹)
8887feqmptd 6961 . . . 4 ((πœ‘ ∧ πœ“) β†’ 𝐹 = (𝑦 ∈ 𝐡 ↦ (πΉβ€˜π‘¦)))
89 fveq2 6892 . . . 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 7127 . . 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 24397 . . . . . . . 8 (0[,]1) = βˆͺ II
9291, 3cnf 22750 . . . . . . 7 (𝐺 ∈ (II Cn 𝐽) β†’ 𝐺:(0[,]1)βŸΆπ‘‹)
935, 92syl 17 . . . . . 6 (πœ‘ β†’ 𝐺:(0[,]1)βŸΆπ‘‹)
9493adantr 482 . . . . 5 ((πœ‘ ∧ πœ“) β†’ 𝐺:(0[,]1)βŸΆπ‘‹)
951, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 29cvmliftlem2 34277 . . . . 5 ((πœ‘ ∧ πœ“) β†’ π‘Š βŠ† (0[,]1))
9694, 95fssresd 6759 . . . 4 ((πœ‘ ∧ πœ“) β†’ (𝐺 β†Ύ π‘Š):π‘ŠβŸΆπ‘‹)
9796feqmptd 6961 . . 3 ((πœ‘ ∧ πœ“) β†’ (𝐺 β†Ύ π‘Š) = (𝑧 ∈ π‘Š ↦ ((𝐺 β†Ύ π‘Š)β€˜π‘§)))
9885, 90, 973eqtr4d 2783 . 2 ((πœ‘ ∧ πœ“) β†’ (𝐹 ∘ (π‘„β€˜π‘€)) = (𝐺 β†Ύ π‘Š))
9976, 98jca 513 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 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   ∘ ccom 5681   Fn wfn 6539  βŸΆwf 6540  β€“1-1-ontoβ†’wf1o 6543  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   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-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:  cvmliftlem7  34282  cvmliftlem10  34285  cvmliftlem13  34287
  Copyright terms: Public domain W3C validator