cvmliftlem11 - Mathbox for Mario Carneiro

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Theorem cvmliftlem11 35289

Description: Lemma for cvmlift 35293. (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 35288	. . . 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 12209	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℂ)
19	8	nnne0d 12243	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ≠ 0)
20	18, 19	dividd 11963	. . . . . . 7 ⊢ (𝜑 → (𝑁 / 𝑁) = 1)
21	20	oveq2d 7406	. . . . . 6 ⊢ (𝜑 → (0[,](𝑁 / 𝑁)) = (0[,]1))
22	17, 21	oveq12d 7408	. . . . 5 ⊢ (𝜑 → (𝐿 ↾_t (0[,](𝑁 / 𝑁))) = ((topGen‘ran (,)) ↾_t (0[,]1)))
23		dfii2 24782	. . . . 5 ⊢ II = ((topGen‘ran (,)) ↾_t (0[,]1))
24	22, 23	eqtr4di 2783	. . . 4 ⊢ (𝜑 → (𝐿 ↾_t (0[,](𝑁 / 𝑁))) = II)
25	24	oveq1d 7405	. . 3 ⊢ (𝜑 → ((𝐿 ↾_t (0[,](𝑁 / 𝑁))) Cn 𝐶) = (II Cn 𝐶))
26	16, 25	eleqtrd 2831	. 2 ⊢ (𝜑 → 𝐾 ∈ (II Cn 𝐶))
27	15	simprd 495	. . 3 ⊢ (𝜑 → (𝐹 ∘ 𝐾) = (𝐺 ↾ (0[,](𝑁 / 𝑁))))
28	21	reseq2d 5953	. . 3 ⊢ (𝜑 → (𝐺 ↾ (0[,](𝑁 / 𝑁))) = (𝐺 ↾ (0[,]1)))
29		iiuni 24781	. . . . 5 ⊢ (0[,]1) = ∪ II
30	29, 3	cnf 23140	. . . 4 ⊢ (𝐺 ∈ (II Cn 𝐽) → 𝐺:(0[,]1)⟶𝑋)
31		ffn 6691	. . . 4 ⊢ (𝐺:(0[,]1)⟶𝑋 → 𝐺 Fn (0[,]1))
32		fnresdm 6640	. . . 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 2769	. 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 1540 ∈ wcel 2109 ∀wral 3045 {crab 3408 Vcvv 3450 ∖ cdif 3914 ∪ cun 3915 ∩ cin 3916 ⊆ wss 3917 ∅c0 4299 𝒫 cpw 4566 {csn 4592 ⟨cop 4598 ∪ cuni 4874 ∪ ciun 4958 ↦ cmpt 5191 I cid 5535 × cxp 5639 ^◡ccnv 5640 ran crn 5642 ↾ cres 5643 “ cima 5644 ∘ ccom 5645 Fn wfn 6509 ⟶wf 6510 ‘cfv 6514 ℩crio 7346 (class class class)co 7390 ∈ cmpo 7392 1^st c1st 7969 2^nd c2nd 7970 0cc0 11075 1c1 11076 + caddc 11078 − cmin 11412 / cdiv 11842 ℕcn 12193 (,)cioo 13313 [,]cicc 13316 ...cfz 13475 seqcseq 13973 ↾_t crest 17390 topGenctg 17407 Cn ccn 23118 Homeochmeo 23647 IIcii 24775 CovMap ccvm 35249

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 2702 ax-rep 5237 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152 ax-pre-sup 11153

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 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-pss 3937 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4875 df-int 4914 df-iun 4960 df-iin 4961 df-br 5111 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5536 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5594 df-we 5596 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-pred 6277 df-ord 6338 df-on 6339 df-lim 6340 df-suc 6341 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-om 7846 df-1st 7971 df-2nd 7972 df-frecs 8263 df-wrecs 8294 df-recs 8343 df-rdg 8381 df-er 8674 df-map 8804 df-en 8922 df-dom 8923 df-sdom 8924 df-fin 8925 df-fi 9369 df-sup 9400 df-inf 9401 df-pnf 11217 df-mnf 11218 df-xr 11219 df-ltxr 11220 df-le 11221 df-sub 11414 df-neg 11415 df-div 11843 df-nn 12194 df-2 12256 df-3 12257 df-n0 12450 df-z 12537 df-uz 12801 df-q 12915 df-rp 12959 df-xneg 13079 df-xadd 13080 df-xmul 13081 df-ioo 13317 df-icc 13320 df-fz 13476 df-seq 13974 df-exp 14034 df-cj 15072 df-re 15073 df-im 15074 df-sqrt 15208 df-abs 15209 df-rest 17392 df-topgen 17413 df-psmet 21263 df-xmet 21264 df-met 21265 df-bl 21266 df-mopn 21267 df-top 22788 df-topon 22805 df-bases 22840 df-cld 22913 df-cn 23121 df-hmeo 23649 df-ii 24777 df-cvm 35250

This theorem is referenced by: cvmliftlem13 35290 cvmliftlem14 35291

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