cvmliftlem2 - Mathbox for Mario Carneiro

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Theorem cvmliftlem2 35318

Description: Lemma for cvmlift 35331. 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 11112	. . 3 ⊢ ((𝜑 ∧ 𝜓) → 0 ∈ ℝ)
3		1red 11110	. . 3 ⊢ ((𝜑 ∧ 𝜓) → 1 ∈ ℝ)
4		cvmliftlem1.m	. . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → 𝑀 ∈ (1...𝑁))
5		elfznn 13450	. . . . . . 7 ⊢ (𝑀 ∈ (1...𝑁) → 𝑀 ∈ ℕ)
6	4, 5	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → 𝑀 ∈ ℕ)
7	6	nnred 12137	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → 𝑀 ∈ ℝ)
8		peano2rem 11425	. . . . 5 ⊢ (𝑀 ∈ ℝ → (𝑀 − 1) ∈ ℝ)
9	7, 8	syl 17	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → (𝑀 − 1) ∈ ℝ)
10		nnm1nn0 12419	. . . . . 6 ⊢ (𝑀 ∈ ℕ → (𝑀 − 1) ∈ ℕ₀)
11	6, 10	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → (𝑀 − 1) ∈ ℕ₀)
12	11	nn0ge0d 12442	. . . 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 12137	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → 𝑁 ∈ ℝ)
16	14	nngt0d 12171	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → 0 < 𝑁)
17		divge0 11988	. . . 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 13425	. . . . . 6 ⊢ (𝑀 ∈ (1...𝑁) → 𝑀 ≤ 𝑁)
20	4, 19	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → 𝑀 ≤ 𝑁)
21	14	nncnd 12138	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → 𝑁 ∈ ℂ)
22	21	mulridd 11126	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → (𝑁 · 1) = 𝑁)
23	20, 22	breqtrrd 5119	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → 𝑀 ≤ (𝑁 · 1))
24		ledivmul 11995	. . . . 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 13311	. . 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 3973	1 ⊢ ((𝜑 ∧ 𝜓) → 𝑊 ⊆ (0[,]1))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1541 ∈ wcel 2111 ∀wral 3047 {crab 3395 ∖ cdif 3899 ∩ cin 3901 ⊆ wss 3902 ∅c0 4283 𝒫 cpw 4550 {csn 4576 ∪ cuni 4859 ∪ ciun 4941 class class class wbr 5091 ↦ cmpt 5172 × cxp 5614 ^◡ccnv 5615 ran crn 5617 ↾ cres 5618 “ cima 5619 ⟶wf 6477 ‘cfv 6481 (class class class)co 7346 1^st c1st 7919 ℝcr 11002 0cc0 11003 1c1 11004 · cmul 11008 < clt 11143 ≤ cle 11144 − cmin 11341 / cdiv 11771 ℕcn 12122 ℕ₀cn0 12378 (,)cioo 13242 [,]cicc 13245 ...cfz 13404 ↾_t crest 17321 topGenctg 17338 Cn ccn 23137 Homeochmeo 23666 IIcii 24793 CovMap ccvm 35287

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 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-sep 5234 ax-nul 5244 ax-pow 5303 ax-pr 5370 ax-un 7668 ax-cnex 11059 ax-resscn 11060 ax-1cn 11061 ax-icn 11062 ax-addcl 11063 ax-addrcl 11064 ax-mulcl 11065 ax-mulrcl 11066 ax-mulcom 11067 ax-addass 11068 ax-mulass 11069 ax-distr 11070 ax-i2m1 11071 ax-1ne0 11072 ax-1rid 11073 ax-rnegex 11074 ax-rrecex 11075 ax-cnre 11076 ax-pre-lttri 11077 ax-pre-lttrn 11078 ax-pre-ltadd 11079 ax-pre-mulgt0 11080

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3742 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4284 df-if 4476 df-pw 4552 df-sn 4577 df-pr 4579 df-op 4583 df-uni 4860 df-iun 4943 df-br 5092 df-opab 5154 df-mpt 5173 df-tr 5199 df-id 5511 df-eprel 5516 df-po 5524 df-so 5525 df-fr 5569 df-we 5571 df-xp 5622 df-rel 5623 df-cnv 5624 df-co 5625 df-dm 5626 df-rn 5627 df-res 5628 df-ima 5629 df-pred 6248 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-riota 7303 df-ov 7349 df-oprab 7350 df-mpo 7351 df-om 7797 df-1st 7921 df-2nd 7922 df-frecs 8211 df-wrecs 8242 df-recs 8291 df-rdg 8329 df-er 8622 df-en 8870 df-dom 8871 df-sdom 8872 df-pnf 11145 df-mnf 11146 df-xr 11147 df-ltxr 11148 df-le 11149 df-sub 11343 df-neg 11344 df-div 11772 df-nn 12123 df-n0 12379 df-z 12466 df-uz 12730 df-icc 13249 df-fz 13405

This theorem is referenced by: cvmliftlem3 35319 cvmliftlem6 35322 cvmliftlem8 35324

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