Mathbox for Mario Carneiro |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > cvmliftlem11 | Structured version Visualization version GIF version |
Description: Lemma for cvmlift 32444. (Contributed by Mario Carneiro, 14-Feb-2015.) |
Ref | Expression |
---|---|
cvmliftlem.1 | ⊢ 𝑆 = (𝑘 ∈ 𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (◡𝐹 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘))))}) |
cvmliftlem.b | ⊢ 𝐵 = ∪ 𝐶 |
cvmliftlem.x | ⊢ 𝑋 = ∪ 𝐽 |
cvmliftlem.f | ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽)) |
cvmliftlem.g | ⊢ (𝜑 → 𝐺 ∈ (II Cn 𝐽)) |
cvmliftlem.p | ⊢ (𝜑 → 𝑃 ∈ 𝐵) |
cvmliftlem.e | ⊢ (𝜑 → (𝐹‘𝑃) = (𝐺‘0)) |
cvmliftlem.n | ⊢ (𝜑 → 𝑁 ∈ ℕ) |
cvmliftlem.t | ⊢ (𝜑 → 𝑇:(1...𝑁)⟶∪ 𝑗 ∈ 𝐽 ({𝑗} × (𝑆‘𝑗))) |
cvmliftlem.a | ⊢ (𝜑 → ∀𝑘 ∈ (1...𝑁)(𝐺 “ (((𝑘 − 1) / 𝑁)[,](𝑘 / 𝑁))) ⊆ (1st ‘(𝑇‘𝑘))) |
cvmliftlem.l | ⊢ 𝐿 = (topGen‘ran (,)) |
cvmliftlem.q | ⊢ 𝑄 = seq0((𝑥 ∈ V, 𝑚 ∈ ℕ ↦ (𝑧 ∈ (((𝑚 − 1) / 𝑁)[,](𝑚 / 𝑁)) ↦ (◡(𝐹 ↾ (℩𝑏 ∈ (2nd ‘(𝑇‘𝑚))(𝑥‘((𝑚 − 1) / 𝑁)) ∈ 𝑏))‘(𝐺‘𝑧)))), (( I ↾ ℕ) ∪ {〈0, {〈0, 𝑃〉}〉})) |
cvmliftlem.k | ⊢ 𝐾 = ∪ 𝑘 ∈ (1...𝑁)(𝑄‘𝑘) |
Ref | Expression |
---|---|
cvmliftlem11 | ⊢ (𝜑 → (𝐾 ∈ (II Cn 𝐶) ∧ (𝐹 ∘ 𝐾) = 𝐺)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cvmliftlem.1 | . . . . 5 ⊢ 𝑆 = (𝑘 ∈ 𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (◡𝐹 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘))))}) | |
2 | cvmliftlem.b | . . . . 5 ⊢ 𝐵 = ∪ 𝐶 | |
3 | cvmliftlem.x | . . . . 5 ⊢ 𝑋 = ∪ 𝐽 | |
4 | cvmliftlem.f | . . . . 5 ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽)) | |
5 | cvmliftlem.g | . . . . 5 ⊢ (𝜑 → 𝐺 ∈ (II Cn 𝐽)) | |
6 | cvmliftlem.p | . . . . 5 ⊢ (𝜑 → 𝑃 ∈ 𝐵) | |
7 | cvmliftlem.e | . . . . 5 ⊢ (𝜑 → (𝐹‘𝑃) = (𝐺‘0)) | |
8 | cvmliftlem.n | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℕ) | |
9 | cvmliftlem.t | . . . . 5 ⊢ (𝜑 → 𝑇:(1...𝑁)⟶∪ 𝑗 ∈ 𝐽 ({𝑗} × (𝑆‘𝑗))) | |
10 | cvmliftlem.a | . . . . 5 ⊢ (𝜑 → ∀𝑘 ∈ (1...𝑁)(𝐺 “ (((𝑘 − 1) / 𝑁)[,](𝑘 / 𝑁))) ⊆ (1st ‘(𝑇‘𝑘))) | |
11 | cvmliftlem.l | . . . . 5 ⊢ 𝐿 = (topGen‘ran (,)) | |
12 | cvmliftlem.q | . . . . 5 ⊢ 𝑄 = seq0((𝑥 ∈ V, 𝑚 ∈ ℕ ↦ (𝑧 ∈ (((𝑚 − 1) / 𝑁)[,](𝑚 / 𝑁)) ↦ (◡(𝐹 ↾ (℩𝑏 ∈ (2nd ‘(𝑇‘𝑚))(𝑥‘((𝑚 − 1) / 𝑁)) ∈ 𝑏))‘(𝐺‘𝑧)))), (( I ↾ ℕ) ∪ {〈0, {〈0, 𝑃〉}〉})) | |
13 | cvmliftlem.k | . . . . 5 ⊢ 𝐾 = ∪ 𝑘 ∈ (1...𝑁)(𝑄‘𝑘) | |
14 | biid 262 | . . . . 5 ⊢ (((𝑛 ∈ ℕ ∧ (𝑛 + 1) ∈ (1...𝑁)) ∧ (∪ 𝑘 ∈ (1...𝑛)(𝑄‘𝑘) ∈ ((𝐿 ↾t (0[,](𝑛 / 𝑁))) Cn 𝐶) ∧ (𝐹 ∘ ∪ 𝑘 ∈ (1...𝑛)(𝑄‘𝑘)) = (𝐺 ↾ (0[,](𝑛 / 𝑁))))) ↔ ((𝑛 ∈ ℕ ∧ (𝑛 + 1) ∈ (1...𝑁)) ∧ (∪ 𝑘 ∈ (1...𝑛)(𝑄‘𝑘) ∈ ((𝐿 ↾t (0[,](𝑛 / 𝑁))) Cn 𝐶) ∧ (𝐹 ∘ ∪ 𝑘 ∈ (1...𝑛)(𝑄‘𝑘)) = (𝐺 ↾ (0[,](𝑛 / 𝑁)))))) | |
15 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 | cvmliftlem10 32439 | . . . 4 ⊢ (𝜑 → (𝐾 ∈ ((𝐿 ↾t (0[,](𝑁 / 𝑁))) Cn 𝐶) ∧ (𝐹 ∘ 𝐾) = (𝐺 ↾ (0[,](𝑁 / 𝑁))))) |
16 | 15 | simpld 495 | . . 3 ⊢ (𝜑 → 𝐾 ∈ ((𝐿 ↾t (0[,](𝑁 / 𝑁))) Cn 𝐶)) |
17 | 11 | a1i 11 | . . . . . 6 ⊢ (𝜑 → 𝐿 = (topGen‘ran (,))) |
18 | 8 | nncnd 11643 | . . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℂ) |
19 | 8 | nnne0d 11676 | . . . . . . . 8 ⊢ (𝜑 → 𝑁 ≠ 0) |
20 | 18, 19 | dividd 11403 | . . . . . . 7 ⊢ (𝜑 → (𝑁 / 𝑁) = 1) |
21 | 20 | oveq2d 7161 | . . . . . 6 ⊢ (𝜑 → (0[,](𝑁 / 𝑁)) = (0[,]1)) |
22 | 17, 21 | oveq12d 7163 | . . . . 5 ⊢ (𝜑 → (𝐿 ↾t (0[,](𝑁 / 𝑁))) = ((topGen‘ran (,)) ↾t (0[,]1))) |
23 | dfii2 23419 | . . . . 5 ⊢ II = ((topGen‘ran (,)) ↾t (0[,]1)) | |
24 | 22, 23 | syl6eqr 2874 | . . . 4 ⊢ (𝜑 → (𝐿 ↾t (0[,](𝑁 / 𝑁))) = II) |
25 | 24 | oveq1d 7160 | . . 3 ⊢ (𝜑 → ((𝐿 ↾t (0[,](𝑁 / 𝑁))) Cn 𝐶) = (II Cn 𝐶)) |
26 | 16, 25 | eleqtrd 2915 | . 2 ⊢ (𝜑 → 𝐾 ∈ (II Cn 𝐶)) |
27 | 15 | simprd 496 | . . 3 ⊢ (𝜑 → (𝐹 ∘ 𝐾) = (𝐺 ↾ (0[,](𝑁 / 𝑁)))) |
28 | 21 | reseq2d 5847 | . . 3 ⊢ (𝜑 → (𝐺 ↾ (0[,](𝑁 / 𝑁))) = (𝐺 ↾ (0[,]1))) |
29 | iiuni 23418 | . . . . 5 ⊢ (0[,]1) = ∪ II | |
30 | 29, 3 | cnf 21784 | . . . 4 ⊢ (𝐺 ∈ (II Cn 𝐽) → 𝐺:(0[,]1)⟶𝑋) |
31 | ffn 6508 | . . . 4 ⊢ (𝐺:(0[,]1)⟶𝑋 → 𝐺 Fn (0[,]1)) | |
32 | fnresdm 6460 | . . . 4 ⊢ (𝐺 Fn (0[,]1) → (𝐺 ↾ (0[,]1)) = 𝐺) | |
33 | 5, 30, 31, 32 | 4syl 19 | . . 3 ⊢ (𝜑 → (𝐺 ↾ (0[,]1)) = 𝐺) |
34 | 27, 28, 33 | 3eqtrd 2860 | . 2 ⊢ (𝜑 → (𝐹 ∘ 𝐾) = 𝐺) |
35 | 26, 34 | jca 512 | 1 ⊢ (𝜑 → (𝐾 ∈ (II Cn 𝐶) ∧ (𝐹 ∘ 𝐾) = 𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ∀wral 3138 {crab 3142 Vcvv 3495 ∖ cdif 3932 ∪ cun 3933 ∩ cin 3934 ⊆ wss 3935 ∅c0 4290 𝒫 cpw 4537 {csn 4559 〈cop 4565 ∪ cuni 4832 ∪ ciun 4912 ↦ cmpt 5138 I cid 5453 × cxp 5547 ◡ccnv 5548 ran crn 5550 ↾ cres 5551 “ cima 5552 ∘ ccom 5553 Fn wfn 6344 ⟶wf 6345 ‘cfv 6349 ℩crio 7102 (class class class)co 7145 ∈ cmpo 7147 1st c1st 7678 2nd c2nd 7679 0cc0 10526 1c1 10527 + caddc 10529 − cmin 10859 / cdiv 11286 ℕcn 11627 (,)cioo 12728 [,]cicc 12731 ...cfz 12882 seqcseq 13359 ↾t crest 16684 topGenctg 16701 Cn ccn 21762 Homeochmeo 22291 IIcii 23412 CovMap ccvm 32400 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7450 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4833 df-int 4870 df-iun 4914 df-iin 4915 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7569 df-1st 7680 df-2nd 7681 df-wrecs 7938 df-recs 7999 df-rdg 8037 df-oadd 8097 df-er 8279 df-map 8398 df-en 8499 df-dom 8500 df-sdom 8501 df-fin 8502 df-fi 8864 df-sup 8895 df-inf 8896 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11628 df-2 11689 df-3 11690 df-n0 11887 df-z 11971 df-uz 12233 df-q 12338 df-rp 12380 df-xneg 12497 df-xadd 12498 df-xmul 12499 df-ioo 12732 df-icc 12735 df-fz 12883 df-seq 13360 df-exp 13420 df-cj 14448 df-re 14449 df-im 14450 df-sqrt 14584 df-abs 14585 df-rest 16686 df-topgen 16707 df-psmet 20467 df-xmet 20468 df-met 20469 df-bl 20470 df-mopn 20471 df-top 21432 df-topon 21449 df-bases 21484 df-cld 21557 df-cn 21765 df-hmeo 22293 df-ii 23414 df-cvm 32401 |
This theorem is referenced by: cvmliftlem13 32441 cvmliftlem14 32442 |
Copyright terms: Public domain | W3C validator |