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Mirrors > Home > MPE Home > Th. List > Mathboxes > cvmliftlem2 | Structured version Visualization version GIF version |
Description: Lemma for cvmlift 32443. 𝑊 = [(𝑘 − 1) / 𝑁, 𝑘 / 𝑁] is a subset of [0, 1] for each 𝑀 ∈ (1...𝑁). (Contributed by Mario Carneiro, 16-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 (,)) |
cvmliftlem1.m | ⊢ ((𝜑 ∧ 𝜓) → 𝑀 ∈ (1...𝑁)) |
cvmliftlem3.3 | ⊢ 𝑊 = (((𝑀 − 1) / 𝑁)[,](𝑀 / 𝑁)) |
Ref | Expression |
---|---|
cvmliftlem2 | ⊢ ((𝜑 ∧ 𝜓) → 𝑊 ⊆ (0[,]1)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cvmliftlem3.3 | . 2 ⊢ 𝑊 = (((𝑀 − 1) / 𝑁)[,](𝑀 / 𝑁)) | |
2 | 0red 10632 | . . 3 ⊢ ((𝜑 ∧ 𝜓) → 0 ∈ ℝ) | |
3 | 1red 10630 | . . 3 ⊢ ((𝜑 ∧ 𝜓) → 1 ∈ ℝ) | |
4 | cvmliftlem1.m | . . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → 𝑀 ∈ (1...𝑁)) | |
5 | elfznn 12924 | . . . . . . 7 ⊢ (𝑀 ∈ (1...𝑁) → 𝑀 ∈ ℕ) | |
6 | 4, 5 | syl 17 | . . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → 𝑀 ∈ ℕ) |
7 | 6 | nnred 11641 | . . . . 5 ⊢ ((𝜑 ∧ 𝜓) → 𝑀 ∈ ℝ) |
8 | peano2rem 10941 | . . . . 5 ⊢ (𝑀 ∈ ℝ → (𝑀 − 1) ∈ ℝ) | |
9 | 7, 8 | syl 17 | . . . 4 ⊢ ((𝜑 ∧ 𝜓) → (𝑀 − 1) ∈ ℝ) |
10 | nnm1nn0 11926 | . . . . . 6 ⊢ (𝑀 ∈ ℕ → (𝑀 − 1) ∈ ℕ0) | |
11 | 6, 10 | syl 17 | . . . . 5 ⊢ ((𝜑 ∧ 𝜓) → (𝑀 − 1) ∈ ℕ0) |
12 | 11 | nn0ge0d 11946 | . . . 4 ⊢ ((𝜑 ∧ 𝜓) → 0 ≤ (𝑀 − 1)) |
13 | cvmliftlem.n | . . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℕ) | |
14 | 13 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝜓) → 𝑁 ∈ ℕ) |
15 | 14 | nnred 11641 | . . . 4 ⊢ ((𝜑 ∧ 𝜓) → 𝑁 ∈ ℝ) |
16 | 14 | nngt0d 11674 | . . . 4 ⊢ ((𝜑 ∧ 𝜓) → 0 < 𝑁) |
17 | divge0 11497 | . . . 4 ⊢ ((((𝑀 − 1) ∈ ℝ ∧ 0 ≤ (𝑀 − 1)) ∧ (𝑁 ∈ ℝ ∧ 0 < 𝑁)) → 0 ≤ ((𝑀 − 1) / 𝑁)) | |
18 | 9, 12, 15, 16, 17 | syl22anc 834 | . . 3 ⊢ ((𝜑 ∧ 𝜓) → 0 ≤ ((𝑀 − 1) / 𝑁)) |
19 | elfzle2 12899 | . . . . . 6 ⊢ (𝑀 ∈ (1...𝑁) → 𝑀 ≤ 𝑁) | |
20 | 4, 19 | syl 17 | . . . . 5 ⊢ ((𝜑 ∧ 𝜓) → 𝑀 ≤ 𝑁) |
21 | 14 | nncnd 11642 | . . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → 𝑁 ∈ ℂ) |
22 | 21 | mulid1d 10646 | . . . . 5 ⊢ ((𝜑 ∧ 𝜓) → (𝑁 · 1) = 𝑁) |
23 | 20, 22 | breqtrrd 5085 | . . . 4 ⊢ ((𝜑 ∧ 𝜓) → 𝑀 ≤ (𝑁 · 1)) |
24 | ledivmul 11504 | . . . . 5 ⊢ ((𝑀 ∈ ℝ ∧ 1 ∈ ℝ ∧ (𝑁 ∈ ℝ ∧ 0 < 𝑁)) → ((𝑀 / 𝑁) ≤ 1 ↔ 𝑀 ≤ (𝑁 · 1))) | |
25 | 7, 3, 15, 16, 24 | syl112anc 1366 | . . . 4 ⊢ ((𝜑 ∧ 𝜓) → ((𝑀 / 𝑁) ≤ 1 ↔ 𝑀 ≤ (𝑁 · 1))) |
26 | 23, 25 | mpbird 258 | . . 3 ⊢ ((𝜑 ∧ 𝜓) → (𝑀 / 𝑁) ≤ 1) |
27 | iccss 12792 | . . 3 ⊢ (((0 ∈ ℝ ∧ 1 ∈ ℝ) ∧ (0 ≤ ((𝑀 − 1) / 𝑁) ∧ (𝑀 / 𝑁) ≤ 1)) → (((𝑀 − 1) / 𝑁)[,](𝑀 / 𝑁)) ⊆ (0[,]1)) | |
28 | 2, 3, 18, 26, 27 | syl22anc 834 | . 2 ⊢ ((𝜑 ∧ 𝜓) → (((𝑀 − 1) / 𝑁)[,](𝑀 / 𝑁)) ⊆ (0[,]1)) |
29 | 1, 28 | eqsstrid 4012 | 1 ⊢ ((𝜑 ∧ 𝜓) → 𝑊 ⊆ (0[,]1)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ∀wral 3135 {crab 3139 ∖ cdif 3930 ∩ cin 3932 ⊆ wss 3933 ∅c0 4288 𝒫 cpw 4535 {csn 4557 ∪ cuni 4830 ∪ ciun 4910 class class class wbr 5057 ↦ cmpt 5137 × cxp 5546 ◡ccnv 5547 ran crn 5549 ↾ cres 5550 “ cima 5551 ⟶wf 6344 ‘cfv 6348 (class class class)co 7145 1st c1st 7676 ℝcr 10524 0cc0 10525 1c1 10526 · cmul 10530 < clt 10663 ≤ cle 10664 − cmin 10858 / cdiv 11285 ℕcn 11626 ℕ0cn0 11885 (,)cioo 12726 [,]cicc 12729 ...cfz 12880 ↾t crest 16682 topGenctg 16699 Cn ccn 21760 Homeochmeo 22289 IIcii 23410 CovMap ccvm 32399 |
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 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
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 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 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 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-n0 11886 df-z 11970 df-uz 12232 df-icc 12733 df-fz 12881 |
This theorem is referenced by: cvmliftlem3 32431 cvmliftlem6 32434 cvmliftlem8 32436 |
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