cvmliftlem2 - Mathbox for Mario Carneiro

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Theorem cvmliftlem2 35351

Description: Lemma for cvmlift 35364. 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 11122	. . 3 ⊢ ((𝜑 ∧ 𝜓) → 0 ∈ ℝ)
3		1red 11120	. . 3 ⊢ ((𝜑 ∧ 𝜓) → 1 ∈ ℝ)
4		cvmliftlem1.m	. . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → 𝑀 ∈ (1...𝑁))
5		elfznn 13455	. . . . . . 7 ⊢ (𝑀 ∈ (1...𝑁) → 𝑀 ∈ ℕ)
6	4, 5	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → 𝑀 ∈ ℕ)
7	6	nnred 12147	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → 𝑀 ∈ ℝ)
8		peano2rem 11435	. . . . 5 ⊢ (𝑀 ∈ ℝ → (𝑀 − 1) ∈ ℝ)
9	7, 8	syl 17	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → (𝑀 − 1) ∈ ℝ)
10		nnm1nn0 12429	. . . . . 6 ⊢ (𝑀 ∈ ℕ → (𝑀 − 1) ∈ ℕ₀)
11	6, 10	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → (𝑀 − 1) ∈ ℕ₀)
12	11	nn0ge0d 12452	. . . 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 12147	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → 𝑁 ∈ ℝ)
16	14	nngt0d 12181	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → 0 < 𝑁)
17		divge0 11998	. . . 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 13430	. . . . . 6 ⊢ (𝑀 ∈ (1...𝑁) → 𝑀 ≤ 𝑁)
20	4, 19	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → 𝑀 ≤ 𝑁)
21	14	nncnd 12148	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → 𝑁 ∈ ℂ)
22	21	mulridd 11136	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → (𝑁 · 1) = 𝑁)
23	20, 22	breqtrrd 5121	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → 𝑀 ≤ (𝑁 · 1))
24		ledivmul 12005	. . . . 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 13316	. . 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 3969	1 ⊢ ((𝜑 ∧ 𝜓) → 𝑊 ⊆ (0[,]1))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1541 ∈ wcel 2113 ∀wral 3048 {crab 3396 ∖ cdif 3895 ∩ cin 3897 ⊆ wss 3898 ∅c0 4282 𝒫 cpw 4549 {csn 4575 ∪ cuni 4858 ∪ ciun 4941 class class class wbr 5093 ↦ cmpt 5174 × cxp 5617 ^◡ccnv 5618 ran crn 5620 ↾ cres 5621 “ cima 5622 ⟶wf 6482 ‘cfv 6486 (class class class)co 7352 1^st c1st 7925 ℝcr 11012 0cc0 11013 1c1 11014 · cmul 11018 < clt 11153 ≤ cle 11154 − cmin 11351 / cdiv 11781 ℕcn 12132 ℕ₀cn0 12388 (,)cioo 13247 [,]cicc 13250 ...cfz 13409 ↾_t crest 17326 topGenctg 17343 Cn ccn 23140 Homeochmeo 23669 IIcii 24796 CovMap ccvm 35320

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 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2705 ax-sep 5236 ax-nul 5246 ax-pow 5305 ax-pr 5372 ax-un 7674 ax-cnex 11069 ax-resscn 11070 ax-1cn 11071 ax-icn 11072 ax-addcl 11073 ax-addrcl 11074 ax-mulcl 11075 ax-mulrcl 11076 ax-mulcom 11077 ax-addass 11078 ax-mulass 11079 ax-distr 11080 ax-i2m1 11081 ax-1ne0 11082 ax-1rid 11083 ax-rnegex 11084 ax-rrecex 11085 ax-cnre 11086 ax-pre-lttri 11087 ax-pre-lttrn 11088 ax-pre-ltadd 11089 ax-pre-mulgt0 11090

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 2537 df-eu 2566 df-clab 2712 df-cleq 2725 df-clel 2808 df-nfc 2882 df-ne 2930 df-nel 3034 df-ral 3049 df-rex 3058 df-rmo 3347 df-reu 3348 df-rab 3397 df-v 3439 df-sbc 3738 df-csb 3847 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3918 df-nul 4283 df-if 4475 df-pw 4551 df-sn 4576 df-pr 4578 df-op 4582 df-uni 4859 df-iun 4943 df-br 5094 df-opab 5156 df-mpt 5175 df-tr 5201 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 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 7309 df-ov 7355 df-oprab 7356 df-mpo 7357 df-om 7803 df-1st 7927 df-2nd 7928 df-frecs 8217 df-wrecs 8248 df-recs 8297 df-rdg 8335 df-er 8628 df-en 8876 df-dom 8877 df-sdom 8878 df-pnf 11155 df-mnf 11156 df-xr 11157 df-ltxr 11158 df-le 11159 df-sub 11353 df-neg 11354 df-div 11782 df-nn 12133 df-n0 12389 df-z 12476 df-uz 12739 df-icc 13254 df-fz 13410

This theorem is referenced by: cvmliftlem3 35352 cvmliftlem6 35355 cvmliftlem8 35357

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