sitgclbn - Mathbox for Thierry Arnoux

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Theorem sitgclbn 32310

Description: Closure of the Bochner integral on a simple function. This version is specific to Banach spaces, with additional conditions on its scalar field. (Contributed by Thierry Arnoux, 24-Feb-2018.)

**Hypotheses**
Ref	Expression
sitgval.b	⊢ 𝐵 = (Base‘𝑊)
sitgval.j	⊢ 𝐽 = (TopOpen‘𝑊)
sitgval.s	⊢ 𝑆 = (sigaGen‘𝐽)
sitgval.0	⊢ 0 = (0_g‘𝑊)
sitgval.x	⊢ · = ( ·_𝑠 ‘𝑊)
sitgval.h	⊢ 𝐻 = (ℝHom‘(Scalar‘𝑊))
sitgval.1	⊢ (𝜑 → 𝑊 ∈ 𝑉)
sitgval.2	⊢ (𝜑 → 𝑀 ∈ ∪ ran measures)
sibfmbl.1	⊢ (𝜑 → 𝐹 ∈ dom (𝑊sitg𝑀))
sitgclbn.1	⊢ (𝜑 → 𝑊 ∈ Ban)
sitgclbn.2	⊢ (𝜑 → (Scalar‘𝑊) ∈ ℝExt )

**Assertion**
Ref	Expression
sitgclbn	⊢ (𝜑 → ((𝑊sitg𝑀)‘𝐹) ∈ 𝐵)

Proof of Theorem sitgclbn Dummy variables 𝑚 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		sitgval.b	. 2 ⊢ 𝐵 = (Base‘𝑊)
2		sitgval.j	. 2 ⊢ 𝐽 = (TopOpen‘𝑊)
3		sitgval.s	. 2 ⊢ 𝑆 = (sigaGen‘𝐽)
4		sitgval.0	. 2 ⊢ 0 = (0_g‘𝑊)
5		sitgval.x	. 2 ⊢ · = ( ·_𝑠 ‘𝑊)
6		sitgval.h	. 2 ⊢ 𝐻 = (ℝHom‘(Scalar‘𝑊))
7		sitgval.1	. 2 ⊢ (𝜑 → 𝑊 ∈ 𝑉)
8		sitgval.2	. 2 ⊢ (𝜑 → 𝑀 ∈ ∪ ran measures)
9		sibfmbl.1	. 2 ⊢ (𝜑 → 𝐹 ∈ dom (𝑊sitg𝑀))
10		eqid 2738	. 2 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
11		eqid 2738	. 2 ⊢ ((dist‘(Scalar‘𝑊)) ↾ ((Base‘(Scalar‘𝑊)) × (Base‘(Scalar‘𝑊)))) = ((dist‘(Scalar‘𝑊)) ↾ ((Base‘(Scalar‘𝑊)) × (Base‘(Scalar‘𝑊))))
12		sitgclbn.1	. . . 4 ⊢ (𝜑 → 𝑊 ∈ Ban)
13		bncms 24508	. . . 4 ⊢ (𝑊 ∈ Ban → 𝑊 ∈ CMetSp)
14	12, 13	syl 17	. . 3 ⊢ (𝜑 → 𝑊 ∈ CMetSp)
15		cmsms 24512	. . 3 ⊢ (𝑊 ∈ CMetSp → 𝑊 ∈ MetSp)
16		mstps 23608	. . 3 ⊢ (𝑊 ∈ MetSp → 𝑊 ∈ TopSp)
17	14, 15, 16	3syl 18	. 2 ⊢ (𝜑 → 𝑊 ∈ TopSp)
18		bnlmod 24507	. . 3 ⊢ (𝑊 ∈ Ban → 𝑊 ∈ LMod)
19		lmodcmn 20171	. . 3 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ CMnd)
20	12, 18, 19	3syl 18	. 2 ⊢ (𝜑 → 𝑊 ∈ CMnd)
21		sitgclbn.2	. 2 ⊢ (𝜑 → (Scalar‘𝑊) ∈ ℝExt )
22	12, 18	syl 17	. . . 4 ⊢ (𝜑 → 𝑊 ∈ LMod)
23	22	3ad2ant1 1132	. . 3 ⊢ ((𝜑 ∧ 𝑚 ∈ (𝐻 “ (0[,)+∞)) ∧ 𝑥 ∈ 𝐵) → 𝑊 ∈ LMod)
24		imassrn 5980	. . . . . 6 ⊢ (𝐻 “ (0[,)+∞)) ⊆ ran 𝐻
25	6	rneqi 5846	. . . . . . 7 ⊢ ran 𝐻 = ran (ℝHom‘(Scalar‘𝑊))
26		eqid 2738	. . . . . . . . 9 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
27	26	rrhfe 31962	. . . . . . . 8 ⊢ ((Scalar‘𝑊) ∈ ℝExt → (ℝHom‘(Scalar‘𝑊)):ℝ⟶(Base‘(Scalar‘𝑊)))
28		frn 6607	. . . . . . . 8 ⊢ ((ℝHom‘(Scalar‘𝑊)):ℝ⟶(Base‘(Scalar‘𝑊)) → ran (ℝHom‘(Scalar‘𝑊)) ⊆ (Base‘(Scalar‘𝑊)))
29	21, 27, 28	3syl 18	. . . . . . 7 ⊢ (𝜑 → ran (ℝHom‘(Scalar‘𝑊)) ⊆ (Base‘(Scalar‘𝑊)))
30	25, 29	eqsstrid 3969	. . . . . 6 ⊢ (𝜑 → ran 𝐻 ⊆ (Base‘(Scalar‘𝑊)))
31	24, 30	sstrid 3932	. . . . 5 ⊢ (𝜑 → (𝐻 “ (0[,)+∞)) ⊆ (Base‘(Scalar‘𝑊)))
32	31	sselda 3921	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ (𝐻 “ (0[,)+∞))) → 𝑚 ∈ (Base‘(Scalar‘𝑊)))
33	32	3adant3 1131	. . 3 ⊢ ((𝜑 ∧ 𝑚 ∈ (𝐻 “ (0[,)+∞)) ∧ 𝑥 ∈ 𝐵) → 𝑚 ∈ (Base‘(Scalar‘𝑊)))
34		simp3 1137	. . 3 ⊢ ((𝜑 ∧ 𝑚 ∈ (𝐻 “ (0[,)+∞)) ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵)
35	1, 10, 5, 26	lmodvscl 20140	. . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝑚 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑥 ∈ 𝐵) → (𝑚 · 𝑥) ∈ 𝐵)
36	23, 33, 34, 35	syl3anc 1370	. 2 ⊢ ((𝜑 ∧ 𝑚 ∈ (𝐻 “ (0[,)+∞)) ∧ 𝑥 ∈ 𝐵) → (𝑚 · 𝑥) ∈ 𝐵)
37	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 17, 20, 21, 36	sitgclg 32309	1 ⊢ (𝜑 → ((𝑊sitg𝑀)‘𝐹) ∈ 𝐵)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 ⊆ wss 3887 ∪ cuni 4839 × cxp 5587 dom cdm 5589 ran crn 5590 ↾ cres 5591 “ cima 5592 ⟶wf 6429 ‘cfv 6433 (class class class)co 7275 ℝcr 10870 0cc0 10871 +∞cpnf 11006 [,)cico 13081 Basecbs 16912 Scalarcsca 16965 ·_𝑠 cvsca 16966 distcds 16971 TopOpenctopn 17132 0_gc0g 17150 CMndccmn 19386 LModclmod 20123 TopSpctps 22081 MetSpcms 23471 CMetSpccms 24496 Bancbn 24497 ℝHomcrrh 31943 ℝExt crrext 31944 sigaGencsigagen 32106 measurescmeas 32163 sitgcsitg 32296

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949 ax-addf 10950 ax-mulf 10951

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-iin 4927 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-isom 6442 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-of 7533 df-om 7713 df-1st 7831 df-2nd 7832 df-supp 7978 df-tpos 8042 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-2o 8298 df-er 8498 df-map 8617 df-pm 8618 df-ixp 8686 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-fsupp 9129 df-fi 9170 df-sup 9201 df-inf 9202 df-oi 9269 df-card 9697 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-z 12320 df-dec 12438 df-uz 12583 df-q 12689 df-rp 12731 df-xneg 12848 df-xadd 12849 df-xmul 12850 df-ioo 13083 df-ico 13085 df-icc 13086 df-fz 13240 df-fzo 13383 df-fl 13512 df-mod 13590 df-seq 13722 df-exp 13783 df-hash 14045 df-cj 14810 df-re 14811 df-im 14812 df-sqrt 14946 df-abs 14947 df-dvds 15964 df-gcd 16202 df-numer 16439 df-denom 16440 df-gz 16631 df-struct 16848 df-sets 16865 df-slot 16883 df-ndx 16895 df-base 16913 df-ress 16942 df-plusg 16975 df-mulr 16976 df-starv 16977 df-sca 16978 df-vsca 16979 df-ip 16980 df-tset 16981 df-ple 16982 df-ds 16984 df-unif 16985 df-hom 16986 df-cco 16987 df-rest 17133 df-topn 17134 df-0g 17152 df-gsum 17153 df-topgen 17154 df-pt 17155 df-prds 17158 df-xrs 17213 df-qtop 17218 df-imas 17219 df-xps 17221 df-mre 17295 df-mrc 17296 df-acs 17298 df-mgm 18326 df-sgrp 18375 df-mnd 18386 df-mhm 18430 df-submnd 18431 df-grp 18580 df-minusg 18581 df-sbg 18582 df-mulg 18701 df-subg 18752 df-ghm 18832 df-cntz 18923 df-od 19136 df-cmn 19388 df-abl 19389 df-mgp 19721 df-ur 19738 df-ring 19785 df-cring 19786 df-oppr 19862 df-dvdsr 19883 df-unit 19884 df-invr 19914 df-dvr 19925 df-rnghom 19959 df-drng 19993 df-subrg 20022 df-abv 20077 df-lmod 20125 df-nzr 20529 df-psmet 20589 df-xmet 20590 df-met 20591 df-bl 20592 df-mopn 20593 df-fbas 20594 df-fg 20595 df-metu 20596 df-cnfld 20598 df-zring 20671 df-zrh 20705 df-zlm 20706 df-chr 20707 df-refld 20810 df-top 22043 df-topon 22060 df-topsp 22082 df-bases 22096 df-cld 22170 df-ntr 22171 df-cls 22172 df-nei 22249 df-cn 22378 df-cnp 22379 df-haus 22466 df-reg 22467 df-cmp 22538 df-tx 22713 df-hmeo 22906 df-fil 22997 df-fm 23089 df-flim 23090 df-flf 23091 df-fcls 23092 df-cnext 23211 df-ust 23352 df-utop 23383 df-uss 23408 df-usp 23409 df-ucn 23428 df-cfilu 23439 df-cusp 23450 df-xms 23473 df-ms 23474 df-tms 23475 df-nm 23738 df-ngp 23739 df-nrg 23741 df-nlm 23742 df-nvc 23743 df-cncf 24041 df-cfil 24419 df-cmet 24421 df-cms 24499 df-bn 24500 df-qqh 31923 df-rrh 31945 df-rrext 31949 df-esum 31996 df-siga 32077 df-sigagen 32107 df-meas 32164 df-mbfm 32218 df-sitg 32297

This theorem is referenced by: sitgclcn 32311 sitgclre 32312

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