sitgclbn - Mathbox for Thierry Arnoux

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Theorem sitgclbn 32210

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 24413	. . . 4 ⊢ (𝑊 ∈ Ban → 𝑊 ∈ CMetSp)
14	12, 13	syl 17	. . 3 ⊢ (𝜑 → 𝑊 ∈ CMetSp)
15		cmsms 24417	. . 3 ⊢ (𝑊 ∈ CMetSp → 𝑊 ∈ MetSp)
16		mstps 23516	. . 3 ⊢ (𝑊 ∈ MetSp → 𝑊 ∈ TopSp)
17	14, 15, 16	3syl 18	. 2 ⊢ (𝜑 → 𝑊 ∈ TopSp)
18		bnlmod 24412	. . 3 ⊢ (𝑊 ∈ Ban → 𝑊 ∈ LMod)
19		lmodcmn 20086	. . 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 1131	. . 3 ⊢ ((𝜑 ∧ 𝑚 ∈ (𝐻 “ (0[,)+∞)) ∧ 𝑥 ∈ 𝐵) → 𝑊 ∈ LMod)
24		imassrn 5969	. . . . . 6 ⊢ (𝐻 “ (0[,)+∞)) ⊆ ran 𝐻
25	6	rneqi 5835	. . . . . . 7 ⊢ ran 𝐻 = ran (ℝHom‘(Scalar‘𝑊))
26		eqid 2738	. . . . . . . . 9 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
27	26	rrhfe 31862	. . . . . . . 8 ⊢ ((Scalar‘𝑊) ∈ ℝExt → (ℝHom‘(Scalar‘𝑊)):ℝ⟶(Base‘(Scalar‘𝑊)))
28		frn 6591	. . . . . . . 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 3965	. . . . . 6 ⊢ (𝜑 → ran 𝐻 ⊆ (Base‘(Scalar‘𝑊)))
31	24, 30	sstrid 3928	. . . . 5 ⊢ (𝜑 → (𝐻 “ (0[,)+∞)) ⊆ (Base‘(Scalar‘𝑊)))
32	31	sselda 3917	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ (𝐻 “ (0[,)+∞))) → 𝑚 ∈ (Base‘(Scalar‘𝑊)))
33	32	3adant3 1130	. . 3 ⊢ ((𝜑 ∧ 𝑚 ∈ (𝐻 “ (0[,)+∞)) ∧ 𝑥 ∈ 𝐵) → 𝑚 ∈ (Base‘(Scalar‘𝑊)))
34		simp3 1136	. . 3 ⊢ ((𝜑 ∧ 𝑚 ∈ (𝐻 “ (0[,)+∞)) ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵)
35	1, 10, 5, 26	lmodvscl 20055	. . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝑚 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑥 ∈ 𝐵) → (𝑚 · 𝑥) ∈ 𝐵)
36	23, 33, 34, 35	syl3anc 1369	. 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 32209	1 ⊢ (𝜑 → ((𝑊sitg𝑀)‘𝐹) ∈ 𝐵)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ w3a 1085 = wceq 1539 ∈ wcel 2108 ⊆ wss 3883 ∪ cuni 4836 × cxp 5578 dom cdm 5580 ran crn 5581 ↾ cres 5582 “ cima 5583 ⟶wf 6414 ‘cfv 6418 (class class class)co 7255 ℝcr 10801 0cc0 10802 +∞cpnf 10937 [,)cico 13010 Basecbs 16840 Scalarcsca 16891 ·_𝑠 cvsca 16892 distcds 16897 TopOpenctopn 17049 0_gc0g 17067 CMndccmn 19301 LModclmod 20038 TopSpctps 21989 MetSpcms 23379 CMetSpccms 24401 Bancbn 24402 ℝHomcrrh 31843 ℝExt crrext 31844 sigaGencsigagen 32006 measurescmeas 32063 sitgcsitg 32196

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-pre-sup 10880 ax-addf 10881 ax-mulf 10882

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-iin 4924 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-se 5536 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-isom 6427 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-of 7511 df-om 7688 df-1st 7804 df-2nd 7805 df-supp 7949 df-tpos 8013 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-2o 8268 df-er 8456 df-map 8575 df-pm 8576 df-ixp 8644 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-fsupp 9059 df-fi 9100 df-sup 9131 df-inf 9132 df-oi 9199 df-card 9628 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-nn 11904 df-2 11966 df-3 11967 df-4 11968 df-5 11969 df-6 11970 df-7 11971 df-8 11972 df-9 11973 df-n0 12164 df-z 12250 df-dec 12367 df-uz 12512 df-q 12618 df-rp 12660 df-xneg 12777 df-xadd 12778 df-xmul 12779 df-ioo 13012 df-ico 13014 df-icc 13015 df-fz 13169 df-fzo 13312 df-fl 13440 df-mod 13518 df-seq 13650 df-exp 13711 df-hash 13973 df-cj 14738 df-re 14739 df-im 14740 df-sqrt 14874 df-abs 14875 df-dvds 15892 df-gcd 16130 df-numer 16367 df-denom 16368 df-gz 16559 df-struct 16776 df-sets 16793 df-slot 16811 df-ndx 16823 df-base 16841 df-ress 16868 df-plusg 16901 df-mulr 16902 df-starv 16903 df-sca 16904 df-vsca 16905 df-ip 16906 df-tset 16907 df-ple 16908 df-ds 16910 df-unif 16911 df-hom 16912 df-cco 16913 df-rest 17050 df-topn 17051 df-0g 17069 df-gsum 17070 df-topgen 17071 df-pt 17072 df-prds 17075 df-xrs 17130 df-qtop 17135 df-imas 17136 df-xps 17138 df-mre 17212 df-mrc 17213 df-acs 17215 df-mgm 18241 df-sgrp 18290 df-mnd 18301 df-mhm 18345 df-submnd 18346 df-grp 18495 df-minusg 18496 df-sbg 18497 df-mulg 18616 df-subg 18667 df-ghm 18747 df-cntz 18838 df-od 19051 df-cmn 19303 df-abl 19304 df-mgp 19636 df-ur 19653 df-ring 19700 df-cring 19701 df-oppr 19777 df-dvdsr 19798 df-unit 19799 df-invr 19829 df-dvr 19840 df-rnghom 19874 df-drng 19908 df-subrg 19937 df-abv 19992 df-lmod 20040 df-nzr 20442 df-psmet 20502 df-xmet 20503 df-met 20504 df-bl 20505 df-mopn 20506 df-fbas 20507 df-fg 20508 df-metu 20509 df-cnfld 20511 df-zring 20583 df-zrh 20617 df-zlm 20618 df-chr 20619 df-refld 20722 df-top 21951 df-topon 21968 df-topsp 21990 df-bases 22004 df-cld 22078 df-ntr 22079 df-cls 22080 df-nei 22157 df-cn 22286 df-cnp 22287 df-haus 22374 df-reg 22375 df-cmp 22446 df-tx 22621 df-hmeo 22814 df-fil 22905 df-fm 22997 df-flim 22998 df-flf 22999 df-fcls 23000 df-cnext 23119 df-ust 23260 df-utop 23291 df-uss 23316 df-usp 23317 df-ucn 23336 df-cfilu 23347 df-cusp 23358 df-xms 23381 df-ms 23382 df-tms 23383 df-nm 23644 df-ngp 23645 df-nrg 23647 df-nlm 23648 df-nvc 23649 df-cncf 23947 df-cfil 24324 df-cmet 24326 df-cms 24404 df-bn 24405 df-qqh 31823 df-rrh 31845 df-rrext 31849 df-esum 31896 df-siga 31977 df-sigagen 32007 df-meas 32064 df-mbfm 32118 df-sitg 32197

This theorem is referenced by: sitgclcn 32211 sitgclre 32212

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