cvmlift2lem3 - Mathbox for Mario Carneiro

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Theorem cvmlift2lem3 35311

Description: Lemma for cvmlift2 35322. (Contributed by Mario Carneiro, 7-May-2015.)

**Hypotheses**
Ref	Expression
cvmlift2.b	⊢ 𝐵 = ∪ 𝐶
cvmlift2.f	⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽))
cvmlift2.g	⊢ (𝜑 → 𝐺 ∈ ((II ×_t II) Cn 𝐽))
cvmlift2.p	⊢ (𝜑 → 𝑃 ∈ 𝐵)
cvmlift2.i	⊢ (𝜑 → (𝐹‘𝑃) = (0𝐺0))
cvmlift2.h	⊢ 𝐻 = (℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = (𝑧 ∈ (0[,]1) ↦ (𝑧𝐺0)) ∧ (𝑓‘0) = 𝑃))
cvmlift2lem3.1	⊢ 𝐾 = (℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = (𝑧 ∈ (0[,]1) ↦ (𝑋𝐺𝑧)) ∧ (𝑓‘0) = (𝐻‘𝑋)))

**Assertion**
Ref	Expression
cvmlift2lem3	⊢ ((𝜑 ∧ 𝑋 ∈ (0[,]1)) → (𝐾 ∈ (II Cn 𝐶) ∧ (𝐹 ∘ 𝐾) = (𝑧 ∈ (0[,]1) ↦ (𝑋𝐺𝑧)) ∧ (𝐾‘0) = (𝐻‘𝑋)))

Distinct variable groups: 𝑧,𝑓,𝐹 𝜑,𝑓,𝑧 𝑓,𝐽,𝑧 𝑓,𝐺,𝑧 𝑓,𝐻,𝑧 𝑓,𝑋,𝑧 𝐶,𝑓,𝑧 𝑃,𝑓,𝑧 𝑧,𝐵 Allowed substitution hints: 𝐵(𝑓) 𝐾(𝑧,𝑓)

**Proof of Theorem cvmlift2lem3**
Step	Hyp	Ref	Expression
1		cvmlift2.b	. 2 ⊢ 𝐵 = ∪ 𝐶
2		cvmlift2lem3.1	. 2 ⊢ 𝐾 = (℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = (𝑧 ∈ (0[,]1) ↦ (𝑋𝐺𝑧)) ∧ (𝑓‘0) = (𝐻‘𝑋)))
3		cvmlift2.f	. . 3 ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽))
4	3	adantr 480	. 2 ⊢ ((𝜑 ∧ 𝑋 ∈ (0[,]1)) → 𝐹 ∈ (𝐶 CovMap 𝐽))
5		iitopon 24906	. . . 4 ⊢ II ∈ (TopOn‘(0[,]1))
6	5	a1i 11	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ (0[,]1)) → II ∈ (TopOn‘(0[,]1)))
7		simpr 484	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ (0[,]1)) → 𝑋 ∈ (0[,]1))
8	6, 6, 7	cnmptc 23671	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ (0[,]1)) → (𝑧 ∈ (0[,]1) ↦ 𝑋) ∈ (II Cn II))
9	6	cnmptid 23670	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ (0[,]1)) → (𝑧 ∈ (0[,]1) ↦ 𝑧) ∈ (II Cn II))
10		cvmlift2.g	. . . 4 ⊢ (𝜑 → 𝐺 ∈ ((II ×_t II) Cn 𝐽))
11	10	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ (0[,]1)) → 𝐺 ∈ ((II ×_t II) Cn 𝐽))
12	6, 8, 9, 11	cnmpt12f 23675	. 2 ⊢ ((𝜑 ∧ 𝑋 ∈ (0[,]1)) → (𝑧 ∈ (0[,]1) ↦ (𝑋𝐺𝑧)) ∈ (II Cn 𝐽))
13		cvmlift2.p	. . . . . 6 ⊢ (𝜑 → 𝑃 ∈ 𝐵)
14		cvmlift2.i	. . . . . 6 ⊢ (𝜑 → (𝐹‘𝑃) = (0𝐺0))
15		cvmlift2.h	. . . . . 6 ⊢ 𝐻 = (℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = (𝑧 ∈ (0[,]1) ↦ (𝑧𝐺0)) ∧ (𝑓‘0) = 𝑃))
16	1, 3, 10, 13, 14, 15	cvmlift2lem2 35310	. . . . 5 ⊢ (𝜑 → (𝐻 ∈ (II Cn 𝐶) ∧ (𝐹 ∘ 𝐻) = (𝑧 ∈ (0[,]1) ↦ (𝑧𝐺0)) ∧ (𝐻‘0) = 𝑃))
17	16	simp1d 1142	. . . 4 ⊢ (𝜑 → 𝐻 ∈ (II Cn 𝐶))
18		iiuni 24908	. . . . 5 ⊢ (0[,]1) = ∪ II
19	18, 1	cnf 23255	. . . 4 ⊢ (𝐻 ∈ (II Cn 𝐶) → 𝐻:(0[,]1)⟶𝐵)
20	17, 19	syl 17	. . 3 ⊢ (𝜑 → 𝐻:(0[,]1)⟶𝐵)
21	20	ffvelcdmda 7103	. 2 ⊢ ((𝜑 ∧ 𝑋 ∈ (0[,]1)) → (𝐻‘𝑋) ∈ 𝐵)
22		0elunit 13510	. . . 4 ⊢ 0 ∈ (0[,]1)
23		oveq2 7440	. . . . 5 ⊢ (𝑧 = 0 → (𝑋𝐺𝑧) = (𝑋𝐺0))
24		eqid 2736	. . . . 5 ⊢ (𝑧 ∈ (0[,]1) ↦ (𝑋𝐺𝑧)) = (𝑧 ∈ (0[,]1) ↦ (𝑋𝐺𝑧))
25		ovex 7465	. . . . 5 ⊢ (𝑋𝐺0) ∈ V
26	23, 24, 25	fvmpt 7015	. . . 4 ⊢ (0 ∈ (0[,]1) → ((𝑧 ∈ (0[,]1) ↦ (𝑋𝐺𝑧))‘0) = (𝑋𝐺0))
27	22, 26	mp1i 13	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ (0[,]1)) → ((𝑧 ∈ (0[,]1) ↦ (𝑋𝐺𝑧))‘0) = (𝑋𝐺0))
28	16	simp2d 1143	. . . . 5 ⊢ (𝜑 → (𝐹 ∘ 𝐻) = (𝑧 ∈ (0[,]1) ↦ (𝑧𝐺0)))
29	28	fveq1d 6907	. . . 4 ⊢ (𝜑 → ((𝐹 ∘ 𝐻)‘𝑋) = ((𝑧 ∈ (0[,]1) ↦ (𝑧𝐺0))‘𝑋))
30		oveq1 7439	. . . . 5 ⊢ (𝑧 = 𝑋 → (𝑧𝐺0) = (𝑋𝐺0))
31		eqid 2736	. . . . 5 ⊢ (𝑧 ∈ (0[,]1) ↦ (𝑧𝐺0)) = (𝑧 ∈ (0[,]1) ↦ (𝑧𝐺0))
32	30, 31, 25	fvmpt 7015	. . . 4 ⊢ (𝑋 ∈ (0[,]1) → ((𝑧 ∈ (0[,]1) ↦ (𝑧𝐺0))‘𝑋) = (𝑋𝐺0))
33	29, 32	sylan9eq 2796	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ (0[,]1)) → ((𝐹 ∘ 𝐻)‘𝑋) = (𝑋𝐺0))
34		fvco3 7007	. . . 4 ⊢ ((𝐻:(0[,]1)⟶𝐵 ∧ 𝑋 ∈ (0[,]1)) → ((𝐹 ∘ 𝐻)‘𝑋) = (𝐹‘(𝐻‘𝑋)))
35	20, 34	sylan 580	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ (0[,]1)) → ((𝐹 ∘ 𝐻)‘𝑋) = (𝐹‘(𝐻‘𝑋)))
36	27, 33, 35	3eqtr2rd 2783	. 2 ⊢ ((𝜑 ∧ 𝑋 ∈ (0[,]1)) → (𝐹‘(𝐻‘𝑋)) = ((𝑧 ∈ (0[,]1) ↦ (𝑋𝐺𝑧))‘0))
37	1, 2, 4, 12, 21, 36	cvmliftiota 35307	1 ⊢ ((𝜑 ∧ 𝑋 ∈ (0[,]1)) → (𝐾 ∈ (II Cn 𝐶) ∧ (𝐹 ∘ 𝐾) = (𝑧 ∈ (0[,]1) ↦ (𝑋𝐺𝑧)) ∧ (𝐾‘0) = (𝐻‘𝑋)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1539 ∈ wcel 2107 ∪ cuni 4906 ↦ cmpt 5224 ∘ ccom 5688 ⟶wf 6556 ‘cfv 6560 ℩crio 7388 (class class class)co 7432 0cc0 11156 1c1 11157 [,]cicc 13391 TopOnctopon 22917 Cn ccn 23233 ×_t ctx 23569 IIcii 24902 CovMap ccvm 35261

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-inf2 9682 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 ax-pre-sup 11234 ax-addf 11235

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-uni 4907 df-int 4946 df-iun 4992 df-iin 4993 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-se 5637 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-isom 6569 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-of 7698 df-om 7889 df-1st 8015 df-2nd 8016 df-supp 8187 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-1o 8507 df-2o 8508 df-er 8746 df-ec 8748 df-map 8869 df-ixp 8939 df-en 8987 df-dom 8988 df-sdom 8989 df-fin 8990 df-fsupp 9403 df-fi 9452 df-sup 9483 df-inf 9484 df-oi 9551 df-card 9980 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-div 11922 df-nn 12268 df-2 12330 df-3 12331 df-4 12332 df-5 12333 df-6 12334 df-7 12335 df-8 12336 df-9 12337 df-n0 12529 df-z 12616 df-dec 12736 df-uz 12880 df-q 12992 df-rp 13036 df-xneg 13155 df-xadd 13156 df-xmul 13157 df-ioo 13392 df-ico 13394 df-icc 13395 df-fz 13549 df-fzo 13696 df-fl 13833 df-seq 14044 df-exp 14104 df-hash 14371 df-cj 15139 df-re 15140 df-im 15141 df-sqrt 15275 df-abs 15276 df-clim 15525 df-sum 15724 df-struct 17185 df-sets 17202 df-slot 17220 df-ndx 17232 df-base 17249 df-ress 17276 df-plusg 17311 df-mulr 17312 df-starv 17313 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-hom 17322 df-cco 17323 df-rest 17468 df-topn 17469 df-0g 17487 df-gsum 17488 df-topgen 17489 df-pt 17490 df-prds 17493 df-xrs 17548 df-qtop 17553 df-imas 17554 df-xps 17556 df-mre 17630 df-mrc 17631 df-acs 17633 df-mgm 18654 df-sgrp 18733 df-mnd 18749 df-submnd 18798 df-mulg 19087 df-cntz 19336 df-cmn 19801 df-psmet 21357 df-xmet 21358 df-met 21359 df-bl 21360 df-mopn 21361 df-cnfld 21366 df-top 22901 df-topon 22918 df-topsp 22940 df-bases 22954 df-cld 23028 df-ntr 23029 df-cls 23030 df-nei 23107 df-cn 23236 df-cnp 23237 df-cmp 23396 df-conn 23421 df-lly 23475 df-nlly 23476 df-tx 23571 df-hmeo 23764 df-xms 24331 df-ms 24332 df-tms 24333 df-ii 24904 df-cncf 24905 df-htpy 25003 df-phtpy 25004 df-phtpc 25025 df-pconn 35227 df-sconn 35228 df-cvm 35262

This theorem is referenced by: cvmlift2lem5 35313 cvmlift2lem6 35314 cvmlift2lem7 35315 cvmlift2lem8 35316

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