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| Mirrors > Home > MPE Home > Th. List > Mathboxes > cvmliftlem2 | Structured version Visualization version GIF version | ||
| Description: Lemma for cvmlift 35304. 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 11264 | . . 3 ⊢ ((𝜑 ∧ 𝜓) → 0 ∈ ℝ) | |
| 3 | 1red 11262 | . . 3 ⊢ ((𝜑 ∧ 𝜓) → 1 ∈ ℝ) | |
| 4 | cvmliftlem1.m | . . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → 𝑀 ∈ (1...𝑁)) | |
| 5 | elfznn 13593 | . . . . . . 7 ⊢ (𝑀 ∈ (1...𝑁) → 𝑀 ∈ ℕ) | |
| 6 | 4, 5 | syl 17 | . . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → 𝑀 ∈ ℕ) | 
| 7 | 6 | nnred 12281 | . . . . 5 ⊢ ((𝜑 ∧ 𝜓) → 𝑀 ∈ ℝ) | 
| 8 | peano2rem 11576 | . . . . 5 ⊢ (𝑀 ∈ ℝ → (𝑀 − 1) ∈ ℝ) | |
| 9 | 7, 8 | syl 17 | . . . 4 ⊢ ((𝜑 ∧ 𝜓) → (𝑀 − 1) ∈ ℝ) | 
| 10 | nnm1nn0 12567 | . . . . . 6 ⊢ (𝑀 ∈ ℕ → (𝑀 − 1) ∈ ℕ0) | |
| 11 | 6, 10 | syl 17 | . . . . 5 ⊢ ((𝜑 ∧ 𝜓) → (𝑀 − 1) ∈ ℕ0) | 
| 12 | 11 | nn0ge0d 12590 | . . . 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 12281 | . . . 4 ⊢ ((𝜑 ∧ 𝜓) → 𝑁 ∈ ℝ) | 
| 16 | 14 | nngt0d 12315 | . . . 4 ⊢ ((𝜑 ∧ 𝜓) → 0 < 𝑁) | 
| 17 | divge0 12137 | . . . 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 13568 | . . . . . 6 ⊢ (𝑀 ∈ (1...𝑁) → 𝑀 ≤ 𝑁) | |
| 20 | 4, 19 | syl 17 | . . . . 5 ⊢ ((𝜑 ∧ 𝜓) → 𝑀 ≤ 𝑁) | 
| 21 | 14 | nncnd 12282 | . . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → 𝑁 ∈ ℂ) | 
| 22 | 21 | mulridd 11278 | . . . . 5 ⊢ ((𝜑 ∧ 𝜓) → (𝑁 · 1) = 𝑁) | 
| 23 | 20, 22 | breqtrrd 5171 | . . . 4 ⊢ ((𝜑 ∧ 𝜓) → 𝑀 ≤ (𝑁 · 1)) | 
| 24 | ledivmul 12144 | . . . . 5 ⊢ ((𝑀 ∈ ℝ ∧ 1 ∈ ℝ ∧ (𝑁 ∈ ℝ ∧ 0 < 𝑁)) → ((𝑀 / 𝑁) ≤ 1 ↔ 𝑀 ≤ (𝑁 · 1))) | |
| 25 | 7, 3, 15, 16, 24 | syl112anc 1376 | . . . 4 ⊢ ((𝜑 ∧ 𝜓) → ((𝑀 / 𝑁) ≤ 1 ↔ 𝑀 ≤ (𝑁 · 1))) | 
| 26 | 23, 25 | mpbird 257 | . . 3 ⊢ ((𝜑 ∧ 𝜓) → (𝑀 / 𝑁) ≤ 1) | 
| 27 | iccss 13455 | . . 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 4022 | 1 ⊢ ((𝜑 ∧ 𝜓) → 𝑊 ⊆ (0[,]1)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∀wral 3061 {crab 3436 ∖ cdif 3948 ∩ cin 3950 ⊆ wss 3951 ∅c0 4333 𝒫 cpw 4600 {csn 4626 ∪ cuni 4907 ∪ ciun 4991 class class class wbr 5143 ↦ cmpt 5225 × cxp 5683 ◡ccnv 5684 ran crn 5686 ↾ cres 5687 “ cima 5688 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 1st c1st 8012 ℝcr 11154 0cc0 11155 1c1 11156 · cmul 11160 < clt 11295 ≤ cle 11296 − cmin 11492 / cdiv 11920 ℕcn 12266 ℕ0cn0 12526 (,)cioo 13387 [,]cicc 13390 ...cfz 13547 ↾t crest 17465 topGenctg 17482 Cn ccn 23232 Homeochmeo 23761 IIcii 24901 CovMap ccvm 35260 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-n0 12527 df-z 12614 df-uz 12879 df-icc 13394 df-fz 13548 | 
| This theorem is referenced by: cvmliftlem3 35292 cvmliftlem6 35295 cvmliftlem8 35297 | 
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