cvmliftlem2 - Mathbox for Mario Carneiro

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Theorem cvmliftlem2 35258

Description: Lemma for cvmlift 35271. 𝑊 = [(𝑘 − 1) / 𝑁, 𝑘 / 𝑁] is a subset of [0, 1] for each 𝑀 ∈ (1...𝑁). (Contributed by Mario Carneiro, 16-Feb-2015.)

**Hypotheses**
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))) ⊆ (1^st ‘(𝑇‘𝑘)))
cvmliftlem.l	⊢ 𝐿 = (topGen‘ran (,))
cvmliftlem1.m	⊢ ((𝜑 ∧ 𝜓) → 𝑀 ∈ (1...𝑁))
cvmliftlem3.3	⊢ 𝑊 = (((𝑀 − 1) / 𝑁)[,](𝑀 / 𝑁))

**Assertion**
Ref	Expression
cvmliftlem2	⊢ ((𝜑 ∧ 𝜓) → 𝑊 ⊆ (0[,]1))

Distinct variable groups: 𝑣,𝐵 𝑗,𝑘,𝑠,𝑢,𝑣,𝐹 𝑗,𝑀,𝑘,𝑠,𝑢,𝑣 𝑃,𝑘,𝑢,𝑣 𝐶,𝑗,𝑘,𝑠,𝑢,𝑣 𝜑,𝑗,𝑠 𝑘,𝑁,𝑢,𝑣 𝑆,𝑗,𝑘,𝑠,𝑢,𝑣 𝑗,𝑋 𝑗,𝐺,𝑘,𝑠,𝑢,𝑣 𝑇,𝑗,𝑘,𝑠,𝑢,𝑣 𝑗,𝐽,𝑘,𝑠,𝑢,𝑣 𝑘,𝑊 Allowed substitution hints: 𝜑(𝑣,𝑢,𝑘) 𝜓(𝑣,𝑢,𝑗,𝑘,𝑠) 𝐵(𝑢,𝑗,𝑘,𝑠) 𝑃(𝑗,𝑠) 𝐿(𝑣,𝑢,𝑗,𝑘,𝑠) 𝑁(𝑗,𝑠) 𝑊(𝑣,𝑢,𝑗,𝑠) 𝑋(𝑣,𝑢,𝑘,𝑠)

**Proof of Theorem cvmliftlem2**
Step	Hyp	Ref	Expression
1		cvmliftlem3.3	. 2 ⊢ 𝑊 = (((𝑀 − 1) / 𝑁)[,](𝑀 / 𝑁))
2		0red 11137	. . 3 ⊢ ((𝜑 ∧ 𝜓) → 0 ∈ ℝ)
3		1red 11135	. . 3 ⊢ ((𝜑 ∧ 𝜓) → 1 ∈ ℝ)
4		cvmliftlem1.m	. . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → 𝑀 ∈ (1...𝑁))
5		elfznn 13474	. . . . . . 7 ⊢ (𝑀 ∈ (1...𝑁) → 𝑀 ∈ ℕ)
6	4, 5	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → 𝑀 ∈ ℕ)
7	6	nnred 12161	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → 𝑀 ∈ ℝ)
8		peano2rem 11449	. . . . 5 ⊢ (𝑀 ∈ ℝ → (𝑀 − 1) ∈ ℝ)
9	7, 8	syl 17	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → (𝑀 − 1) ∈ ℝ)
10		nnm1nn0 12443	. . . . . 6 ⊢ (𝑀 ∈ ℕ → (𝑀 − 1) ∈ ℕ₀)
11	6, 10	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → (𝑀 − 1) ∈ ℕ₀)
12	11	nn0ge0d 12466	. . . 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 12161	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → 𝑁 ∈ ℝ)
16	14	nngt0d 12195	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → 0 < 𝑁)
17		divge0 12012	. . . 4 ⊢ ((((𝑀 − 1) ∈ ℝ ∧ 0 ≤ (𝑀 − 1)) ∧ (𝑁 ∈ ℝ ∧ 0 < 𝑁)) → 0 ≤ ((𝑀 − 1) / 𝑁))
18	9, 12, 15, 16, 17	syl22anc 838	. . 3 ⊢ ((𝜑 ∧ 𝜓) → 0 ≤ ((𝑀 − 1) / 𝑁))
19		elfzle2 13449	. . . . . 6 ⊢ (𝑀 ∈ (1...𝑁) → 𝑀 ≤ 𝑁)
20	4, 19	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → 𝑀 ≤ 𝑁)
21	14	nncnd 12162	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → 𝑁 ∈ ℂ)
22	21	mulridd 11151	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → (𝑁 · 1) = 𝑁)
23	20, 22	breqtrrd 5123	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → 𝑀 ≤ (𝑁 · 1))
24		ledivmul 12019	. . . . 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 13335	. . 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 838	. 2 ⊢ ((𝜑 ∧ 𝜓) → (((𝑀 − 1) / 𝑁)[,](𝑀 / 𝑁)) ⊆ (0[,]1))
29	1, 28	eqsstrid 3976	1 ⊢ ((𝜑 ∧ 𝜓) → 𝑊 ⊆ (0[,]1))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ∀wral 3044 {crab 3396 ∖ cdif 3902 ∩ cin 3904 ⊆ wss 3905 ∅c0 4286 𝒫 cpw 4553 {csn 4579 ∪ cuni 4861 ∪ ciun 4944 class class class wbr 5095 ↦ cmpt 5176 × cxp 5621 ^◡ccnv 5622 ran crn 5624 ↾ cres 5625 “ cima 5626 ⟶wf 6482 ‘cfv 6486 (class class class)co 7353 1^st c1st 7929 ℝcr 11027 0cc0 11028 1c1 11029 · cmul 11033 < clt 11168 ≤ cle 11169 − cmin 11365 / cdiv 11795 ℕcn 12146 ℕ₀cn0 12402 (,)cioo 13266 [,]cicc 13269 ...cfz 13428 ↾_t crest 17342 topGenctg 17359 Cn ccn 23127 Homeochmeo 23656 IIcii 24784 CovMap ccvm 35227

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 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7675 ax-cnex 11084 ax-resscn 11085 ax-1cn 11086 ax-icn 11087 ax-addcl 11088 ax-addrcl 11089 ax-mulcl 11090 ax-mulrcl 11091 ax-mulcom 11092 ax-addass 11093 ax-mulass 11094 ax-distr 11095 ax-i2m1 11096 ax-1ne0 11097 ax-1rid 11098 ax-rnegex 11099 ax-rrecex 11100 ax-cnre 11101 ax-pre-lttri 11102 ax-pre-lttrn 11103 ax-pre-ltadd 11104 ax-pre-mulgt0 11105

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3345 df-reu 3346 df-rab 3397 df-v 3440 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4479 df-pw 4555 df-sn 4580 df-pr 4582 df-op 4586 df-uni 4862 df-iun 4946 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 df-id 5518 df-eprel 5523 df-po 5531 df-so 5532 df-fr 5576 df-we 5578 df-xp 5629 df-rel 5630 df-cnv 5631 df-co 5632 df-dm 5633 df-rn 5634 df-res 5635 df-ima 5636 df-pred 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-riota 7310 df-ov 7356 df-oprab 7357 df-mpo 7358 df-om 7807 df-1st 7931 df-2nd 7932 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-er 8632 df-en 8880 df-dom 8881 df-sdom 8882 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11367 df-neg 11368 df-div 11796 df-nn 12147 df-n0 12403 df-z 12490 df-uz 12754 df-icc 13273 df-fz 13429

This theorem is referenced by: cvmliftlem3 35259 cvmliftlem6 35262 cvmliftlem8 35264

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