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Mirrors > Home > MPE Home > Th. List > Mathboxes > cvmliftlem11 | Structured version Visualization version GIF version |
Description: Lemma for cvmlift 33269. (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 260 | . . . . 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 33264 | . . . 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 11999 | . . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℂ) |
19 | 8 | nnne0d 12033 | . . . . . . . 8 ⊢ (𝜑 → 𝑁 ≠ 0) |
20 | 18, 19 | dividd 11759 | . . . . . . 7 ⊢ (𝜑 → (𝑁 / 𝑁) = 1) |
21 | 20 | oveq2d 7283 | . . . . . 6 ⊢ (𝜑 → (0[,](𝑁 / 𝑁)) = (0[,]1)) |
22 | 17, 21 | oveq12d 7285 | . . . . 5 ⊢ (𝜑 → (𝐿 ↾t (0[,](𝑁 / 𝑁))) = ((topGen‘ran (,)) ↾t (0[,]1))) |
23 | dfii2 24055 | . . . . 5 ⊢ II = ((topGen‘ran (,)) ↾t (0[,]1)) | |
24 | 22, 23 | eqtr4di 2796 | . . . 4 ⊢ (𝜑 → (𝐿 ↾t (0[,](𝑁 / 𝑁))) = II) |
25 | 24 | oveq1d 7282 | . . 3 ⊢ (𝜑 → ((𝐿 ↾t (0[,](𝑁 / 𝑁))) Cn 𝐶) = (II Cn 𝐶)) |
26 | 16, 25 | eleqtrd 2841 | . 2 ⊢ (𝜑 → 𝐾 ∈ (II Cn 𝐶)) |
27 | 15 | simprd 496 | . . 3 ⊢ (𝜑 → (𝐹 ∘ 𝐾) = (𝐺 ↾ (0[,](𝑁 / 𝑁)))) |
28 | 21 | reseq2d 5884 | . . 3 ⊢ (𝜑 → (𝐺 ↾ (0[,](𝑁 / 𝑁))) = (𝐺 ↾ (0[,]1))) |
29 | iiuni 24054 | . . . . 5 ⊢ (0[,]1) = ∪ II | |
30 | 29, 3 | cnf 22407 | . . . 4 ⊢ (𝐺 ∈ (II Cn 𝐽) → 𝐺:(0[,]1)⟶𝑋) |
31 | ffn 6592 | . . . 4 ⊢ (𝐺:(0[,]1)⟶𝑋 → 𝐺 Fn (0[,]1)) | |
32 | fnresdm 6543 | . . . 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 2782 | . 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 1539 ∈ wcel 2106 ∀wral 3064 {crab 3068 Vcvv 3429 ∖ cdif 3883 ∪ cun 3884 ∩ cin 3885 ⊆ wss 3886 ∅c0 4256 𝒫 cpw 4533 {csn 4561 〈cop 4567 ∪ cuni 4839 ∪ ciun 4924 ↦ cmpt 5156 I cid 5483 × cxp 5582 ◡ccnv 5583 ran crn 5585 ↾ cres 5586 “ cima 5587 ∘ ccom 5588 Fn wfn 6421 ⟶wf 6422 ‘cfv 6426 ℩crio 7223 (class class class)co 7267 ∈ cmpo 7269 1st c1st 7818 2nd c2nd 7819 0cc0 10881 1c1 10882 + caddc 10884 − cmin 11215 / cdiv 11642 ℕcn 11983 (,)cioo 13089 [,]cicc 13092 ...cfz 13249 seqcseq 13731 ↾t crest 17141 topGenctg 17158 Cn ccn 22385 Homeochmeo 22914 IIcii 24048 CovMap ccvm 33225 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5208 ax-sep 5221 ax-nul 5228 ax-pow 5286 ax-pr 5350 ax-un 7578 ax-cnex 10937 ax-resscn 10938 ax-1cn 10939 ax-icn 10940 ax-addcl 10941 ax-addrcl 10942 ax-mulcl 10943 ax-mulrcl 10944 ax-mulcom 10945 ax-addass 10946 ax-mulass 10947 ax-distr 10948 ax-i2m1 10949 ax-1ne0 10950 ax-1rid 10951 ax-rnegex 10952 ax-rrecex 10953 ax-cnre 10954 ax-pre-lttri 10955 ax-pre-lttrn 10956 ax-pre-ltadd 10957 ax-pre-mulgt0 10958 ax-pre-sup 10959 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3071 df-rmo 3072 df-rab 3073 df-v 3431 df-sbc 3716 df-csb 3832 df-dif 3889 df-un 3891 df-in 3893 df-ss 3903 df-pss 3905 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-iin 4927 df-br 5074 df-opab 5136 df-mpt 5157 df-tr 5191 df-id 5484 df-eprel 5490 df-po 5498 df-so 5499 df-fr 5539 df-we 5541 df-xp 5590 df-rel 5591 df-cnv 5592 df-co 5593 df-dm 5594 df-rn 5595 df-res 5596 df-ima 5597 df-pred 6195 df-ord 6262 df-on 6263 df-lim 6264 df-suc 6265 df-iota 6384 df-fun 6428 df-fn 6429 df-f 6430 df-f1 6431 df-fo 6432 df-f1o 6433 df-fv 6434 df-riota 7224 df-ov 7270 df-oprab 7271 df-mpo 7272 df-om 7703 df-1st 7820 df-2nd 7821 df-frecs 8084 df-wrecs 8115 df-recs 8189 df-rdg 8228 df-er 8485 df-map 8604 df-en 8721 df-dom 8722 df-sdom 8723 df-fin 8724 df-fi 9157 df-sup 9188 df-inf 9189 df-pnf 11021 df-mnf 11022 df-xr 11023 df-ltxr 11024 df-le 11025 df-sub 11217 df-neg 11218 df-div 11643 df-nn 11984 df-2 12046 df-3 12047 df-n0 12244 df-z 12330 df-uz 12593 df-q 12699 df-rp 12741 df-xneg 12858 df-xadd 12859 df-xmul 12860 df-ioo 13093 df-icc 13096 df-fz 13250 df-seq 13732 df-exp 13793 df-cj 14820 df-re 14821 df-im 14822 df-sqrt 14956 df-abs 14957 df-rest 17143 df-topgen 17164 df-psmet 20599 df-xmet 20600 df-met 20601 df-bl 20602 df-mopn 20603 df-top 22053 df-topon 22070 df-bases 22106 df-cld 22180 df-cn 22388 df-hmeo 22916 df-ii 24050 df-cvm 33226 |
This theorem is referenced by: cvmliftlem13 33266 cvmliftlem14 33267 |
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