![]() |
Mathbox for Mario Carneiro |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > cvmliftlem2 | Structured version Visualization version GIF version |
Description: Lemma for cvmlift 35283. 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 11261 | . . 3 ⊢ ((𝜑 ∧ 𝜓) → 0 ∈ ℝ) | |
3 | 1red 11259 | . . 3 ⊢ ((𝜑 ∧ 𝜓) → 1 ∈ ℝ) | |
4 | cvmliftlem1.m | . . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → 𝑀 ∈ (1...𝑁)) | |
5 | elfznn 13589 | . . . . . . 7 ⊢ (𝑀 ∈ (1...𝑁) → 𝑀 ∈ ℕ) | |
6 | 4, 5 | syl 17 | . . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → 𝑀 ∈ ℕ) |
7 | 6 | nnred 12278 | . . . . 5 ⊢ ((𝜑 ∧ 𝜓) → 𝑀 ∈ ℝ) |
8 | peano2rem 11573 | . . . . 5 ⊢ (𝑀 ∈ ℝ → (𝑀 − 1) ∈ ℝ) | |
9 | 7, 8 | syl 17 | . . . 4 ⊢ ((𝜑 ∧ 𝜓) → (𝑀 − 1) ∈ ℝ) |
10 | nnm1nn0 12564 | . . . . . 6 ⊢ (𝑀 ∈ ℕ → (𝑀 − 1) ∈ ℕ0) | |
11 | 6, 10 | syl 17 | . . . . 5 ⊢ ((𝜑 ∧ 𝜓) → (𝑀 − 1) ∈ ℕ0) |
12 | 11 | nn0ge0d 12587 | . . . 4 ⊢ ((𝜑 ∧ 𝜓) → 0 ≤ (𝑀 − 1)) |
13 | cvmliftlem.n | . . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℕ) | |
14 | 13 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝜓) → 𝑁 ∈ ℕ) |
15 | 14 | nnred 12278 | . . . 4 ⊢ ((𝜑 ∧ 𝜓) → 𝑁 ∈ ℝ) |
16 | 14 | nngt0d 12312 | . . . 4 ⊢ ((𝜑 ∧ 𝜓) → 0 < 𝑁) |
17 | divge0 12134 | . . . 4 ⊢ ((((𝑀 − 1) ∈ ℝ ∧ 0 ≤ (𝑀 − 1)) ∧ (𝑁 ∈ ℝ ∧ 0 < 𝑁)) → 0 ≤ ((𝑀 − 1) / 𝑁)) | |
18 | 9, 12, 15, 16, 17 | syl22anc 839 | . . 3 ⊢ ((𝜑 ∧ 𝜓) → 0 ≤ ((𝑀 − 1) / 𝑁)) |
19 | elfzle2 13564 | . . . . . 6 ⊢ (𝑀 ∈ (1...𝑁) → 𝑀 ≤ 𝑁) | |
20 | 4, 19 | syl 17 | . . . . 5 ⊢ ((𝜑 ∧ 𝜓) → 𝑀 ≤ 𝑁) |
21 | 14 | nncnd 12279 | . . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → 𝑁 ∈ ℂ) |
22 | 21 | mulridd 11275 | . . . . 5 ⊢ ((𝜑 ∧ 𝜓) → (𝑁 · 1) = 𝑁) |
23 | 20, 22 | breqtrrd 5175 | . . . 4 ⊢ ((𝜑 ∧ 𝜓) → 𝑀 ≤ (𝑁 · 1)) |
24 | ledivmul 12141 | . . . . 5 ⊢ ((𝑀 ∈ ℝ ∧ 1 ∈ ℝ ∧ (𝑁 ∈ ℝ ∧ 0 < 𝑁)) → ((𝑀 / 𝑁) ≤ 1 ↔ 𝑀 ≤ (𝑁 · 1))) | |
25 | 7, 3, 15, 16, 24 | syl112anc 1373 | . . . 4 ⊢ ((𝜑 ∧ 𝜓) → ((𝑀 / 𝑁) ≤ 1 ↔ 𝑀 ≤ (𝑁 · 1))) |
26 | 23, 25 | mpbird 257 | . . 3 ⊢ ((𝜑 ∧ 𝜓) → (𝑀 / 𝑁) ≤ 1) |
27 | iccss 13451 | . . 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 839 | . 2 ⊢ ((𝜑 ∧ 𝜓) → (((𝑀 − 1) / 𝑁)[,](𝑀 / 𝑁)) ⊆ (0[,]1)) |
29 | 1, 28 | eqsstrid 4043 | 1 ⊢ ((𝜑 ∧ 𝜓) → 𝑊 ⊆ (0[,]1)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1536 ∈ wcel 2105 ∀wral 3058 {crab 3432 ∖ cdif 3959 ∩ cin 3961 ⊆ wss 3962 ∅c0 4338 𝒫 cpw 4604 {csn 4630 ∪ cuni 4911 ∪ ciun 4995 class class class wbr 5147 ↦ cmpt 5230 × cxp 5686 ◡ccnv 5687 ran crn 5689 ↾ cres 5690 “ cima 5691 ⟶wf 6558 ‘cfv 6562 (class class class)co 7430 1st c1st 8010 ℝcr 11151 0cc0 11152 1c1 11153 · cmul 11157 < clt 11292 ≤ cle 11293 − cmin 11489 / cdiv 11917 ℕcn 12263 ℕ0cn0 12523 (,)cioo 13383 [,]cicc 13386 ...cfz 13543 ↾t crest 17466 topGenctg 17483 Cn ccn 23247 Homeochmeo 23776 IIcii 24914 CovMap ccvm 35239 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-1st 8012 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-er 8743 df-en 8984 df-dom 8985 df-sdom 8986 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-div 11918 df-nn 12264 df-n0 12524 df-z 12611 df-uz 12876 df-icc 13390 df-fz 13544 |
This theorem is referenced by: cvmliftlem3 35271 cvmliftlem6 35274 cvmliftlem8 35276 |
Copyright terms: Public domain | W3C validator |