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Mirrors > Home > MPE Home > Th. List > Mathboxes > cvmliftlem2 | Structured version Visualization version GIF version |
Description: Lemma for cvmlift 32548. 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 10646 | . . 3 ⊢ ((𝜑 ∧ 𝜓) → 0 ∈ ℝ) | |
3 | 1red 10644 | . . 3 ⊢ ((𝜑 ∧ 𝜓) → 1 ∈ ℝ) | |
4 | cvmliftlem1.m | . . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → 𝑀 ∈ (1...𝑁)) | |
5 | elfznn 12939 | . . . . . . 7 ⊢ (𝑀 ∈ (1...𝑁) → 𝑀 ∈ ℕ) | |
6 | 4, 5 | syl 17 | . . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → 𝑀 ∈ ℕ) |
7 | 6 | nnred 11655 | . . . . 5 ⊢ ((𝜑 ∧ 𝜓) → 𝑀 ∈ ℝ) |
8 | peano2rem 10955 | . . . . 5 ⊢ (𝑀 ∈ ℝ → (𝑀 − 1) ∈ ℝ) | |
9 | 7, 8 | syl 17 | . . . 4 ⊢ ((𝜑 ∧ 𝜓) → (𝑀 − 1) ∈ ℝ) |
10 | nnm1nn0 11941 | . . . . . 6 ⊢ (𝑀 ∈ ℕ → (𝑀 − 1) ∈ ℕ0) | |
11 | 6, 10 | syl 17 | . . . . 5 ⊢ ((𝜑 ∧ 𝜓) → (𝑀 − 1) ∈ ℕ0) |
12 | 11 | nn0ge0d 11961 | . . . 4 ⊢ ((𝜑 ∧ 𝜓) → 0 ≤ (𝑀 − 1)) |
13 | cvmliftlem.n | . . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℕ) | |
14 | 13 | adantr 483 | . . . . 5 ⊢ ((𝜑 ∧ 𝜓) → 𝑁 ∈ ℕ) |
15 | 14 | nnred 11655 | . . . 4 ⊢ ((𝜑 ∧ 𝜓) → 𝑁 ∈ ℝ) |
16 | 14 | nngt0d 11689 | . . . 4 ⊢ ((𝜑 ∧ 𝜓) → 0 < 𝑁) |
17 | divge0 11511 | . . . 4 ⊢ ((((𝑀 − 1) ∈ ℝ ∧ 0 ≤ (𝑀 − 1)) ∧ (𝑁 ∈ ℝ ∧ 0 < 𝑁)) → 0 ≤ ((𝑀 − 1) / 𝑁)) | |
18 | 9, 12, 15, 16, 17 | syl22anc 836 | . . 3 ⊢ ((𝜑 ∧ 𝜓) → 0 ≤ ((𝑀 − 1) / 𝑁)) |
19 | elfzle2 12914 | . . . . . 6 ⊢ (𝑀 ∈ (1...𝑁) → 𝑀 ≤ 𝑁) | |
20 | 4, 19 | syl 17 | . . . . 5 ⊢ ((𝜑 ∧ 𝜓) → 𝑀 ≤ 𝑁) |
21 | 14 | nncnd 11656 | . . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → 𝑁 ∈ ℂ) |
22 | 21 | mulid1d 10660 | . . . . 5 ⊢ ((𝜑 ∧ 𝜓) → (𝑁 · 1) = 𝑁) |
23 | 20, 22 | breqtrrd 5096 | . . . 4 ⊢ ((𝜑 ∧ 𝜓) → 𝑀 ≤ (𝑁 · 1)) |
24 | ledivmul 11518 | . . . . 5 ⊢ ((𝑀 ∈ ℝ ∧ 1 ∈ ℝ ∧ (𝑁 ∈ ℝ ∧ 0 < 𝑁)) → ((𝑀 / 𝑁) ≤ 1 ↔ 𝑀 ≤ (𝑁 · 1))) | |
25 | 7, 3, 15, 16, 24 | syl112anc 1370 | . . . 4 ⊢ ((𝜑 ∧ 𝜓) → ((𝑀 / 𝑁) ≤ 1 ↔ 𝑀 ≤ (𝑁 · 1))) |
26 | 23, 25 | mpbird 259 | . . 3 ⊢ ((𝜑 ∧ 𝜓) → (𝑀 / 𝑁) ≤ 1) |
27 | iccss 12807 | . . 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 836 | . 2 ⊢ ((𝜑 ∧ 𝜓) → (((𝑀 − 1) / 𝑁)[,](𝑀 / 𝑁)) ⊆ (0[,]1)) |
29 | 1, 28 | eqsstrid 4017 | 1 ⊢ ((𝜑 ∧ 𝜓) → 𝑊 ⊆ (0[,]1)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∀wral 3140 {crab 3144 ∖ cdif 3935 ∩ cin 3937 ⊆ wss 3938 ∅c0 4293 𝒫 cpw 4541 {csn 4569 ∪ cuni 4840 ∪ ciun 4921 class class class wbr 5068 ↦ cmpt 5148 × cxp 5555 ◡ccnv 5556 ran crn 5558 ↾ cres 5559 “ cima 5560 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 1st c1st 7689 ℝcr 10538 0cc0 10539 1c1 10540 · cmul 10544 < clt 10677 ≤ cle 10678 − cmin 10872 / cdiv 11299 ℕcn 11640 ℕ0cn0 11900 (,)cioo 12741 [,]cicc 12744 ...cfz 12895 ↾t crest 16696 topGenctg 16713 Cn ccn 21834 Homeochmeo 22363 IIcii 23485 CovMap ccvm 32504 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-n0 11901 df-z 11985 df-uz 12247 df-icc 12748 df-fz 12896 |
This theorem is referenced by: cvmliftlem3 32536 cvmliftlem6 32539 cvmliftlem8 32541 |
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