cvmliftlem11 - Mathbox for Mario Carneiro

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Theorem cvmliftlem11 35327

Description: Lemma for cvmlift 35331. (Contributed by Mario Carneiro, 14-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 (,))
cvmliftlem.q	⊢ 𝑄 = seq0((𝑥 ∈ V, 𝑚 ∈ ℕ ↦ (𝑧 ∈ (((𝑚 − 1) / 𝑁)[,](𝑚 / 𝑁)) ↦ (^◡(𝐹 ↾ (℩𝑏 ∈ (2^nd ‘(𝑇‘𝑚))(𝑥‘((𝑚 − 1) / 𝑁)) ∈ 𝑏))‘(𝐺‘𝑧)))), (( I ↾ ℕ) ∪ {⟨0, {⟨0, 𝑃⟩}⟩}))
cvmliftlem.k	⊢ 𝐾 = ∪ 𝑘 ∈ (1...𝑁)(𝑄‘𝑘)

**Assertion**
Ref	Expression
cvmliftlem11	⊢ (𝜑 → (𝐾 ∈ (II Cn 𝐶) ∧ (𝐹 ∘ 𝐾) = 𝐺))

Distinct variable groups: 𝑣,𝑏,𝑧,𝐵 𝑗,𝑏,𝑘,𝑚,𝑠,𝑢,𝑥,𝐹,𝑣,𝑧 𝑧,𝐿 𝑃,𝑏,𝑘,𝑚,𝑢,𝑣,𝑥,𝑧 𝐶,𝑏,𝑗,𝑘,𝑠,𝑢,𝑣,𝑧 𝜑,𝑗,𝑠,𝑥,𝑧 𝑁,𝑏,𝑘,𝑚,𝑢,𝑣,𝑥,𝑧 𝑆,𝑏,𝑗,𝑘,𝑠,𝑢,𝑣,𝑥,𝑧 𝑗,𝑋 𝐺,𝑏,𝑗,𝑘,𝑚,𝑠,𝑢,𝑣,𝑥,𝑧 𝑇,𝑏,𝑗,𝑘,𝑚,𝑠,𝑢,𝑣,𝑥,𝑧 𝐽,𝑏,𝑗,𝑘,𝑠,𝑢,𝑣,𝑥,𝑧 𝑄,𝑏,𝑘,𝑚,𝑢,𝑣,𝑥,𝑧 Allowed substitution hints: 𝜑(𝑣,𝑢,𝑘,𝑚,𝑏) 𝐵(𝑥,𝑢,𝑗,𝑘,𝑚,𝑠) 𝐶(𝑥,𝑚) 𝑃(𝑗,𝑠) 𝑄(𝑗,𝑠) 𝑆(𝑚) 𝐽(𝑚) 𝐾(𝑥,𝑧,𝑣,𝑢,𝑗,𝑘,𝑚,𝑠,𝑏) 𝐿(𝑥,𝑣,𝑢,𝑗,𝑘,𝑚,𝑠,𝑏) 𝑁(𝑗,𝑠) 𝑋(𝑥,𝑧,𝑣,𝑢,𝑘,𝑚,𝑠,𝑏)

Proof of Theorem cvmliftlem11 Dummy variable 𝑛 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cvmliftlem.1	. . . . 5 ⊢ 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))))})
2		cvmliftlem.b	. . . . 5 ⊢ 𝐵 = ∪ 𝐶
3		cvmliftlem.x	. . . . 5 ⊢ 𝑋 = ∪ 𝐽
4		cvmliftlem.f	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽))
5		cvmliftlem.g	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ (II Cn 𝐽))
6		cvmliftlem.p	. . . . 5 ⊢ (𝜑 → 𝑃 ∈ 𝐵)
7		cvmliftlem.e	. . . . 5 ⊢ (𝜑 → (𝐹‘𝑃) = (𝐺‘0))
8		cvmliftlem.n	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℕ)
9		cvmliftlem.t	. . . . 5 ⊢ (𝜑 → 𝑇:(1...𝑁)⟶∪ 𝑗 ∈ 𝐽 ({𝑗} × (𝑆‘𝑗)))
10		cvmliftlem.a	. . . . 5 ⊢ (𝜑 → ∀𝑘 ∈ (1...𝑁)(𝐺 “ (((𝑘 − 1) / 𝑁)[,](𝑘 / 𝑁))) ⊆ (1^st ‘(𝑇‘𝑘)))
11		cvmliftlem.l	. . . . 5 ⊢ 𝐿 = (topGen‘ran (,))
12		cvmliftlem.q	. . . . 5 ⊢ 𝑄 = seq0((𝑥 ∈ V, 𝑚 ∈ ℕ ↦ (𝑧 ∈ (((𝑚 − 1) / 𝑁)[,](𝑚 / 𝑁)) ↦ (^◡(𝐹 ↾ (℩𝑏 ∈ (2^nd ‘(𝑇‘𝑚))(𝑥‘((𝑚 − 1) / 𝑁)) ∈ 𝑏))‘(𝐺‘𝑧)))), (( I ↾ ℕ) ∪ {⟨0, {⟨0, 𝑃⟩}⟩}))
13		cvmliftlem.k	. . . . 5 ⊢ 𝐾 = ∪ 𝑘 ∈ (1...𝑁)(𝑄‘𝑘)
14		biid 261	. . . . 5 ⊢ (((𝑛 ∈ ℕ ∧ (𝑛 + 1) ∈ (1...𝑁)) ∧ (∪ 𝑘 ∈ (1...𝑛)(𝑄‘𝑘) ∈ ((𝐿 ↾_t (0[,](𝑛 / 𝑁))) Cn 𝐶) ∧ (𝐹 ∘ ∪ 𝑘 ∈ (1...𝑛)(𝑄‘𝑘)) = (𝐺 ↾ (0[,](𝑛 / 𝑁))))) ↔ ((𝑛 ∈ ℕ ∧ (𝑛 + 1) ∈ (1...𝑁)) ∧ (∪ 𝑘 ∈ (1...𝑛)(𝑄‘𝑘) ∈ ((𝐿 ↾_t (0[,](𝑛 / 𝑁))) Cn 𝐶) ∧ (𝐹 ∘ ∪ 𝑘 ∈ (1...𝑛)(𝑄‘𝑘)) = (𝐺 ↾ (0[,](𝑛 / 𝑁))))))
15	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14	cvmliftlem10 35326	. . . 4 ⊢ (𝜑 → (𝐾 ∈ ((𝐿 ↾_t (0[,](𝑁 / 𝑁))) Cn 𝐶) ∧ (𝐹 ∘ 𝐾) = (𝐺 ↾ (0[,](𝑁 / 𝑁)))))
16	15	simpld 494	. . 3 ⊢ (𝜑 → 𝐾 ∈ ((𝐿 ↾_t (0[,](𝑁 / 𝑁))) Cn 𝐶))
17	11	a1i 11	. . . . . 6 ⊢ (𝜑 → 𝐿 = (topGen‘ran (,)))
18	8	nncnd 12138	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℂ)
19	8	nnne0d 12172	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ≠ 0)
20	18, 19	dividd 11892	. . . . . . 7 ⊢ (𝜑 → (𝑁 / 𝑁) = 1)
21	20	oveq2d 7362	. . . . . 6 ⊢ (𝜑 → (0[,](𝑁 / 𝑁)) = (0[,]1))
22	17, 21	oveq12d 7364	. . . . 5 ⊢ (𝜑 → (𝐿 ↾_t (0[,](𝑁 / 𝑁))) = ((topGen‘ran (,)) ↾_t (0[,]1)))
23		dfii2 24800	. . . . 5 ⊢ II = ((topGen‘ran (,)) ↾_t (0[,]1))
24	22, 23	eqtr4di 2784	. . . 4 ⊢ (𝜑 → (𝐿 ↾_t (0[,](𝑁 / 𝑁))) = II)
25	24	oveq1d 7361	. . 3 ⊢ (𝜑 → ((𝐿 ↾_t (0[,](𝑁 / 𝑁))) Cn 𝐶) = (II Cn 𝐶))
26	16, 25	eleqtrd 2833	. 2 ⊢ (𝜑 → 𝐾 ∈ (II Cn 𝐶))
27	15	simprd 495	. . 3 ⊢ (𝜑 → (𝐹 ∘ 𝐾) = (𝐺 ↾ (0[,](𝑁 / 𝑁))))
28	21	reseq2d 5928	. . 3 ⊢ (𝜑 → (𝐺 ↾ (0[,](𝑁 / 𝑁))) = (𝐺 ↾ (0[,]1)))
29		iiuni 24799	. . . . 5 ⊢ (0[,]1) = ∪ II
30	29, 3	cnf 23159	. . . 4 ⊢ (𝐺 ∈ (II Cn 𝐽) → 𝐺:(0[,]1)⟶𝑋)
31		ffn 6651	. . . 4 ⊢ (𝐺:(0[,]1)⟶𝑋 → 𝐺 Fn (0[,]1))
32		fnresdm 6600	. . . 4 ⊢ (𝐺 Fn (0[,]1) → (𝐺 ↾ (0[,]1)) = 𝐺)
33	5, 30, 31, 32	4syl 19	. . 3 ⊢ (𝜑 → (𝐺 ↾ (0[,]1)) = 𝐺)
34	27, 28, 33	3eqtrd 2770	. 2 ⊢ (𝜑 → (𝐹 ∘ 𝐾) = 𝐺)
35	26, 34	jca 511	1 ⊢ (𝜑 → (𝐾 ∈ (II Cn 𝐶) ∧ (𝐹 ∘ 𝐾) = 𝐺))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1541 ∈ wcel 2111 ∀wral 3047 {crab 3395 Vcvv 3436 ∖ cdif 3899 ∪ cun 3900 ∩ cin 3901 ⊆ wss 3902 ∅c0 4283 𝒫 cpw 4550 {csn 4576 ⟨cop 4582 ∪ cuni 4859 ∪ ciun 4941 ↦ cmpt 5172 I cid 5510 × cxp 5614 ^◡ccnv 5615 ran crn 5617 ↾ cres 5618 “ cima 5619 ∘ ccom 5620 Fn wfn 6476 ⟶wf 6477 ‘cfv 6481 ℩crio 7302 (class class class)co 7346 ∈ cmpo 7348 1^st c1st 7919 2^nd c2nd 7920 0cc0 11003 1c1 11004 + caddc 11006 − cmin 11341 / cdiv 11771 ℕcn 12122 (,)cioo 13242 [,]cicc 13245 ...cfz 13404 seqcseq 13905 ↾_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-rep 5217 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 ax-pre-sup 11081

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-int 4898 df-iun 4943 df-iin 4944 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-map 8752 df-en 8870 df-dom 8871 df-sdom 8872 df-fin 8873 df-fi 9295 df-sup 9326 df-inf 9327 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-2 12185 df-3 12186 df-n0 12379 df-z 12466 df-uz 12730 df-q 12844 df-rp 12888 df-xneg 13008 df-xadd 13009 df-xmul 13010 df-ioo 13246 df-icc 13249 df-fz 13405 df-seq 13906 df-exp 13966 df-cj 15003 df-re 15004 df-im 15005 df-sqrt 15139 df-abs 15140 df-rest 17323 df-topgen 17344 df-psmet 21281 df-xmet 21282 df-met 21283 df-bl 21284 df-mopn 21285 df-top 22807 df-topon 22824 df-bases 22859 df-cld 22932 df-cn 23140 df-hmeo 23668 df-ii 24795 df-cvm 35288

This theorem is referenced by: cvmliftlem13 35328 cvmliftlem14 35329

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