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 32539. (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 263 | . . . . 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 32534 | . . . 4 ⊢ (𝜑 → (𝐾 ∈ ((𝐿 ↾t (0[,](𝑁 / 𝑁))) Cn 𝐶) ∧ (𝐹 ∘ 𝐾) = (𝐺 ↾ (0[,](𝑁 / 𝑁))))) |
16 | 15 | simpld 497 | . . 3 ⊢ (𝜑 → 𝐾 ∈ ((𝐿 ↾t (0[,](𝑁 / 𝑁))) Cn 𝐶)) |
17 | 11 | a1i 11 | . . . . . 6 ⊢ (𝜑 → 𝐿 = (topGen‘ran (,))) |
18 | 8 | nncnd 11646 | . . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℂ) |
19 | 8 | nnne0d 11679 | . . . . . . . 8 ⊢ (𝜑 → 𝑁 ≠ 0) |
20 | 18, 19 | dividd 11406 | . . . . . . 7 ⊢ (𝜑 → (𝑁 / 𝑁) = 1) |
21 | 20 | oveq2d 7164 | . . . . . 6 ⊢ (𝜑 → (0[,](𝑁 / 𝑁)) = (0[,]1)) |
22 | 17, 21 | oveq12d 7166 | . . . . 5 ⊢ (𝜑 → (𝐿 ↾t (0[,](𝑁 / 𝑁))) = ((topGen‘ran (,)) ↾t (0[,]1))) |
23 | dfii2 23482 | . . . . 5 ⊢ II = ((topGen‘ran (,)) ↾t (0[,]1)) | |
24 | 22, 23 | syl6eqr 2872 | . . . 4 ⊢ (𝜑 → (𝐿 ↾t (0[,](𝑁 / 𝑁))) = II) |
25 | 24 | oveq1d 7163 | . . 3 ⊢ (𝜑 → ((𝐿 ↾t (0[,](𝑁 / 𝑁))) Cn 𝐶) = (II Cn 𝐶)) |
26 | 16, 25 | eleqtrd 2913 | . 2 ⊢ (𝜑 → 𝐾 ∈ (II Cn 𝐶)) |
27 | 15 | simprd 498 | . . 3 ⊢ (𝜑 → (𝐹 ∘ 𝐾) = (𝐺 ↾ (0[,](𝑁 / 𝑁)))) |
28 | 21 | reseq2d 5846 | . . 3 ⊢ (𝜑 → (𝐺 ↾ (0[,](𝑁 / 𝑁))) = (𝐺 ↾ (0[,]1))) |
29 | iiuni 23481 | . . . . 5 ⊢ (0[,]1) = ∪ II | |
30 | 29, 3 | cnf 21846 | . . . 4 ⊢ (𝐺 ∈ (II Cn 𝐽) → 𝐺:(0[,]1)⟶𝑋) |
31 | ffn 6507 | . . . 4 ⊢ (𝐺:(0[,]1)⟶𝑋 → 𝐺 Fn (0[,]1)) | |
32 | fnresdm 6459 | . . . 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 2858 | . 2 ⊢ (𝜑 → (𝐹 ∘ 𝐾) = 𝐺) |
35 | 26, 34 | jca 514 | 1 ⊢ (𝜑 → (𝐾 ∈ (II Cn 𝐶) ∧ (𝐹 ∘ 𝐾) = 𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1531 ∈ wcel 2108 ∀wral 3136 {crab 3140 Vcvv 3493 ∖ cdif 3931 ∪ cun 3932 ∩ cin 3933 ⊆ wss 3934 ∅c0 4289 𝒫 cpw 4537 {csn 4559 〈cop 4565 ∪ cuni 4830 ∪ ciun 4910 ↦ cmpt 5137 I cid 5452 × cxp 5546 ◡ccnv 5547 ran crn 5549 ↾ cres 5550 “ cima 5551 ∘ ccom 5552 Fn wfn 6343 ⟶wf 6344 ‘cfv 6348 ℩crio 7105 (class class class)co 7148 ∈ cmpo 7150 1st c1st 7679 2nd c2nd 7680 0cc0 10529 1c1 10530 + caddc 10532 − cmin 10862 / cdiv 11289 ℕcn 11630 (,)cioo 12730 [,]cicc 12733 ...cfz 12884 seqcseq 13361 ↾t crest 16686 topGenctg 16703 Cn ccn 21824 Homeochmeo 22353 IIcii 23475 CovMap ccvm 32495 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 ax-cnex 10585 ax-resscn 10586 ax-1cn 10587 ax-icn 10588 ax-addcl 10589 ax-addrcl 10590 ax-mulcl 10591 ax-mulrcl 10592 ax-mulcom 10593 ax-addass 10594 ax-mulass 10595 ax-distr 10596 ax-i2m1 10597 ax-1ne0 10598 ax-1rid 10599 ax-rnegex 10600 ax-rrecex 10601 ax-cnre 10602 ax-pre-lttri 10603 ax-pre-lttrn 10604 ax-pre-ltadd 10605 ax-pre-mulgt0 10606 ax-pre-sup 10607 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-nel 3122 df-ral 3141 df-rex 3142 df-reu 3143 df-rmo 3144 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-pss 3952 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4831 df-int 4868 df-iun 4912 df-iin 4913 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7106 df-ov 7151 df-oprab 7152 df-mpo 7153 df-om 7573 df-1st 7681 df-2nd 7682 df-wrecs 7939 df-recs 8000 df-rdg 8038 df-oadd 8098 df-er 8281 df-map 8400 df-en 8502 df-dom 8503 df-sdom 8504 df-fin 8505 df-fi 8867 df-sup 8898 df-inf 8899 df-pnf 10669 df-mnf 10670 df-xr 10671 df-ltxr 10672 df-le 10673 df-sub 10864 df-neg 10865 df-div 11290 df-nn 11631 df-2 11692 df-3 11693 df-n0 11890 df-z 11974 df-uz 12236 df-q 12341 df-rp 12382 df-xneg 12499 df-xadd 12500 df-xmul 12501 df-ioo 12734 df-icc 12737 df-fz 12885 df-seq 13362 df-exp 13422 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-rest 16688 df-topgen 16709 df-psmet 20529 df-xmet 20530 df-met 20531 df-bl 20532 df-mopn 20533 df-top 21494 df-topon 21511 df-bases 21546 df-cld 21619 df-cn 21827 df-hmeo 22355 df-ii 23477 df-cvm 32496 |
This theorem is referenced by: cvmliftlem13 32536 cvmliftlem14 32537 |
Copyright terms: Public domain | W3C validator |