cvmlift2lem3 - Mathbox for Mario Carneiro

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Theorem cvmlift2lem3 35487

Description: Lemma for cvmlift2 35498. (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 24846	. . . 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 23627	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ (0[,]1)) → (𝑧 ∈ (0[,]1) ↦ 𝑋) ∈ (II Cn II))
9	6	cnmptid 23626	. . 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 23631	. 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 35486	. . . . 5 ⊢ (𝜑 → (𝐻 ∈ (II Cn 𝐶) ∧ (𝐹 ∘ 𝐻) = (𝑧 ∈ (0[,]1) ↦ (𝑧𝐺0)) ∧ (𝐻‘0) = 𝑃))
17	16	simp1d 1143	. . . 4 ⊢ (𝜑 → 𝐻 ∈ (II Cn 𝐶))
18		iiuni 24848	. . . . 5 ⊢ (0[,]1) = ∪ II
19	18, 1	cnf 23211	. . . 4 ⊢ (𝐻 ∈ (II Cn 𝐶) → 𝐻:(0[,]1)⟶𝐵)
20	17, 19	syl 17	. . 3 ⊢ (𝜑 → 𝐻:(0[,]1)⟶𝐵)
21	20	ffvelcdmda 7036	. 2 ⊢ ((𝜑 ∧ 𝑋 ∈ (0[,]1)) → (𝐻‘𝑋) ∈ 𝐵)
22		0elunit 13422	. . . 4 ⊢ 0 ∈ (0[,]1)
23		oveq2 7375	. . . . 5 ⊢ (𝑧 = 0 → (𝑋𝐺𝑧) = (𝑋𝐺0))
24		eqid 2736	. . . . 5 ⊢ (𝑧 ∈ (0[,]1) ↦ (𝑋𝐺𝑧)) = (𝑧 ∈ (0[,]1) ↦ (𝑋𝐺𝑧))
25		ovex 7400	. . . . 5 ⊢ (𝑋𝐺0) ∈ V
26	23, 24, 25	fvmpt 6947	. . . 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 1144	. . . . 5 ⊢ (𝜑 → (𝐹 ∘ 𝐻) = (𝑧 ∈ (0[,]1) ↦ (𝑧𝐺0)))
29	28	fveq1d 6842	. . . 4 ⊢ (𝜑 → ((𝐹 ∘ 𝐻)‘𝑋) = ((𝑧 ∈ (0[,]1) ↦ (𝑧𝐺0))‘𝑋))
30		oveq1 7374	. . . . 5 ⊢ (𝑧 = 𝑋 → (𝑧𝐺0) = (𝑋𝐺0))
31		eqid 2736	. . . . 5 ⊢ (𝑧 ∈ (0[,]1) ↦ (𝑧𝐺0)) = (𝑧 ∈ (0[,]1) ↦ (𝑧𝐺0))
32	30, 31, 25	fvmpt 6947	. . . 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 6939	. . . 4 ⊢ ((𝐻:(0[,]1)⟶𝐵 ∧ 𝑋 ∈ (0[,]1)) → ((𝐹 ∘ 𝐻)‘𝑋) = (𝐹‘(𝐻‘𝑋)))
35	20, 34	sylan 581	. . 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 35483	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 1087 = wceq 1542 ∈ wcel 2114 ∪ cuni 4850 ↦ cmpt 5166 ∘ ccom 5635 ⟶wf 6494 ‘cfv 6498 ℩crio 7323 (class class class)co 7367 0cc0 11038 1c1 11039 [,]cicc 13301 TopOnctopon 22875 Cn ccn 23189 ×_t ctx 23525 IIcii 24842 CovMap ccvm 35437

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 ax-inf2 9562 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116 ax-addf 11117

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3062 df-rmo 3342 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3909 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4851 df-int 4890 df-iun 4935 df-iin 4936 df-br 5086 df-opab 5148 df-mpt 5167 df-tr 5193 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-se 5585 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6265 df-ord 6326 df-on 6327 df-lim 6328 df-suc 6329 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-isom 6507 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-of 7631 df-om 7818 df-1st 7942 df-2nd 7943 df-supp 8111 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-1o 8405 df-2o 8406 df-er 8643 df-ec 8645 df-map 8775 df-ixp 8846 df-en 8894 df-dom 8895 df-sdom 8896 df-fin 8897 df-fsupp 9275 df-fi 9324 df-sup 9355 df-inf 9356 df-oi 9425 df-card 9863 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-div 11808 df-nn 12175 df-2 12244 df-3 12245 df-4 12246 df-5 12247 df-6 12248 df-7 12249 df-8 12250 df-9 12251 df-n0 12438 df-z 12525 df-dec 12645 df-uz 12789 df-q 12899 df-rp 12943 df-xneg 13063 df-xadd 13064 df-xmul 13065 df-ioo 13302 df-ico 13304 df-icc 13305 df-fz 13462 df-fzo 13609 df-fl 13751 df-seq 13964 df-exp 14024 df-hash 14293 df-cj 15061 df-re 15062 df-im 15063 df-sqrt 15197 df-abs 15198 df-clim 15450 df-sum 15649 df-struct 17117 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-mulr 17234 df-starv 17235 df-sca 17236 df-vsca 17237 df-ip 17238 df-tset 17239 df-ple 17240 df-ds 17242 df-unif 17243 df-hom 17244 df-cco 17245 df-rest 17385 df-topn 17386 df-0g 17404 df-gsum 17405 df-topgen 17406 df-pt 17407 df-prds 17410 df-xrs 17466 df-qtop 17471 df-imas 17472 df-xps 17474 df-mre 17548 df-mrc 17549 df-acs 17551 df-mgm 18608 df-sgrp 18687 df-mnd 18703 df-submnd 18752 df-mulg 19044 df-cntz 19292 df-cmn 19757 df-psmet 21344 df-xmet 21345 df-met 21346 df-bl 21347 df-mopn 21348 df-cnfld 21353 df-top 22859 df-topon 22876 df-topsp 22898 df-bases 22911 df-cld 22984 df-ntr 22985 df-cls 22986 df-nei 23063 df-cn 23192 df-cnp 23193 df-cmp 23352 df-conn 23377 df-lly 23431 df-nlly 23432 df-tx 23527 df-hmeo 23720 df-xms 24285 df-ms 24286 df-tms 24287 df-ii 24844 df-cncf 24845 df-htpy 24937 df-phtpy 24938 df-phtpc 24959 df-pconn 35403 df-sconn 35404 df-cvm 35438

This theorem is referenced by: cvmlift2lem5 35489 cvmlift2lem6 35490 cvmlift2lem7 35491 cvmlift2lem8 35492

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