cvmlift2lem3 - Mathbox for Mario Carneiro

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Theorem cvmlift2lem3 35332

Description: Lemma for cvmlift2 35343. (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 24828	. . . 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 23605	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ (0[,]1)) → (𝑧 ∈ (0[,]1) ↦ 𝑋) ∈ (II Cn II))
9	6	cnmptid 23604	. . 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 23609	. 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 35331	. . . . 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 24830	. . . . 5 ⊢ (0[,]1) = ∪ II
19	18, 1	cnf 23189	. . . 4 ⊢ (𝐻 ∈ (II Cn 𝐶) → 𝐻:(0[,]1)⟶𝐵)
20	17, 19	syl 17	. . 3 ⊢ (𝜑 → 𝐻:(0[,]1)⟶𝐵)
21	20	ffvelcdmda 7079	. 2 ⊢ ((𝜑 ∧ 𝑋 ∈ (0[,]1)) → (𝐻‘𝑋) ∈ 𝐵)
22		0elunit 13491	. . . 4 ⊢ 0 ∈ (0[,]1)
23		oveq2 7418	. . . . 5 ⊢ (𝑧 = 0 → (𝑋𝐺𝑧) = (𝑋𝐺0))
24		eqid 2736	. . . . 5 ⊢ (𝑧 ∈ (0[,]1) ↦ (𝑋𝐺𝑧)) = (𝑧 ∈ (0[,]1) ↦ (𝑋𝐺𝑧))
25		ovex 7443	. . . . 5 ⊢ (𝑋𝐺0) ∈ V
26	23, 24, 25	fvmpt 6991	. . . 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 6883	. . . 4 ⊢ (𝜑 → ((𝐹 ∘ 𝐻)‘𝑋) = ((𝑧 ∈ (0[,]1) ↦ (𝑧𝐺0))‘𝑋))
30		oveq1 7417	. . . . 5 ⊢ (𝑧 = 𝑋 → (𝑧𝐺0) = (𝑋𝐺0))
31		eqid 2736	. . . . 5 ⊢ (𝑧 ∈ (0[,]1) ↦ (𝑧𝐺0)) = (𝑧 ∈ (0[,]1) ↦ (𝑧𝐺0))
32	30, 31, 25	fvmpt 6991	. . . 4 ⊢ (𝑋 ∈ (0[,]1) → ((𝑧 ∈ (0[,]1) ↦ (𝑧𝐺0))‘𝑋) = (𝑋𝐺0))
33	29, 32	sylan9eq 2791	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ (0[,]1)) → ((𝐹 ∘ 𝐻)‘𝑋) = (𝑋𝐺0))
34		fvco3 6983	. . . 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 2778	. 2 ⊢ ((𝜑 ∧ 𝑋 ∈ (0[,]1)) → (𝐹‘(𝐻‘𝑋)) = ((𝑧 ∈ (0[,]1) ↦ (𝑋𝐺𝑧))‘0))
37	1, 2, 4, 12, 21, 36	cvmliftiota 35328	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 1540 ∈ wcel 2109 ∪ cuni 4888 ↦ cmpt 5206 ∘ ccom 5663 ⟶wf 6532 ‘cfv 6536 ℩crio 7366 (class class class)co 7410 0cc0 11134 1c1 11135 [,]cicc 13370 TopOnctopon 22853 Cn ccn 23167 ×_t ctx 23503 IIcii 24824 CovMap ccvm 35282

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 2708 ax-rep 5254 ax-sep 5271 ax-nul 5281 ax-pow 5340 ax-pr 5407 ax-un 7734 ax-inf2 9660 ax-cnex 11190 ax-resscn 11191 ax-1cn 11192 ax-icn 11193 ax-addcl 11194 ax-addrcl 11195 ax-mulcl 11196 ax-mulrcl 11197 ax-mulcom 11198 ax-addass 11199 ax-mulass 11200 ax-distr 11201 ax-i2m1 11202 ax-1ne0 11203 ax-1rid 11204 ax-rnegex 11205 ax-rrecex 11206 ax-cnre 11207 ax-pre-lttri 11208 ax-pre-lttrn 11209 ax-pre-ltadd 11210 ax-pre-mulgt0 11211 ax-pre-sup 11212 ax-addf 11213

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 2540 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2810 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3062 df-rmo 3364 df-reu 3365 df-rab 3421 df-v 3466 df-sbc 3771 df-csb 3880 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-pss 3951 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-tp 4611 df-op 4613 df-uni 4889 df-int 4928 df-iun 4974 df-iin 4975 df-br 5125 df-opab 5187 df-mpt 5207 df-tr 5235 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-se 5612 df-we 5613 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6295 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-of 7676 df-om 7867 df-1st 7993 df-2nd 7994 df-supp 8165 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-2o 8486 df-er 8724 df-ec 8726 df-map 8847 df-ixp 8917 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fsupp 9379 df-fi 9428 df-sup 9459 df-inf 9460 df-oi 9529 df-card 9958 df-pnf 11276 df-mnf 11277 df-xr 11278 df-ltxr 11279 df-le 11280 df-sub 11473 df-neg 11474 df-div 11900 df-nn 12246 df-2 12308 df-3 12309 df-4 12310 df-5 12311 df-6 12312 df-7 12313 df-8 12314 df-9 12315 df-n0 12507 df-z 12594 df-dec 12714 df-uz 12858 df-q 12970 df-rp 13014 df-xneg 13133 df-xadd 13134 df-xmul 13135 df-ioo 13371 df-ico 13373 df-icc 13374 df-fz 13530 df-fzo 13677 df-fl 13814 df-seq 14025 df-exp 14085 df-hash 14354 df-cj 15123 df-re 15124 df-im 15125 df-sqrt 15259 df-abs 15260 df-clim 15509 df-sum 15708 df-struct 17171 df-sets 17188 df-slot 17206 df-ndx 17218 df-base 17234 df-ress 17257 df-plusg 17289 df-mulr 17290 df-starv 17291 df-sca 17292 df-vsca 17293 df-ip 17294 df-tset 17295 df-ple 17296 df-ds 17298 df-unif 17299 df-hom 17300 df-cco 17301 df-rest 17441 df-topn 17442 df-0g 17460 df-gsum 17461 df-topgen 17462 df-pt 17463 df-prds 17466 df-xrs 17521 df-qtop 17526 df-imas 17527 df-xps 17529 df-mre 17603 df-mrc 17604 df-acs 17606 df-mgm 18623 df-sgrp 18702 df-mnd 18718 df-submnd 18767 df-mulg 19056 df-cntz 19305 df-cmn 19768 df-psmet 21312 df-xmet 21313 df-met 21314 df-bl 21315 df-mopn 21316 df-cnfld 21321 df-top 22837 df-topon 22854 df-topsp 22876 df-bases 22889 df-cld 22962 df-ntr 22963 df-cls 22964 df-nei 23041 df-cn 23170 df-cnp 23171 df-cmp 23330 df-conn 23355 df-lly 23409 df-nlly 23410 df-tx 23505 df-hmeo 23698 df-xms 24264 df-ms 24265 df-tms 24266 df-ii 24826 df-cncf 24827 df-htpy 24925 df-phtpy 24926 df-phtpc 24947 df-pconn 35248 df-sconn 35249 df-cvm 35283

This theorem is referenced by: cvmlift2lem5 35334 cvmlift2lem6 35335 cvmlift2lem7 35336 cvmlift2lem8 35337

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