sitgclbn - Mathbox for Thierry Arnoux

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Theorem sitgclbn 34300

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 2734	. 2 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
11		eqid 2734	. 2 ⊢ ((dist‘(Scalar‘𝑊)) ↾ ((Base‘(Scalar‘𝑊)) × (Base‘(Scalar‘𝑊)))) = ((dist‘(Scalar‘𝑊)) ↾ ((Base‘(Scalar‘𝑊)) × (Base‘(Scalar‘𝑊))))
12		sitgclbn.1	. . 3 ⊢ (𝜑 → 𝑊 ∈ Ban)
13		bncms 25390	. . 3 ⊢ (𝑊 ∈ Ban → 𝑊 ∈ CMetSp)
14		cmsms 25394	. . 3 ⊢ (𝑊 ∈ CMetSp → 𝑊 ∈ MetSp)
15		mstps 24479	. . 3 ⊢ (𝑊 ∈ MetSp → 𝑊 ∈ TopSp)
16	12, 13, 14, 15	4syl 19	. 2 ⊢ (𝜑 → 𝑊 ∈ TopSp)
17		bnlmod 25389	. . 3 ⊢ (𝑊 ∈ Ban → 𝑊 ∈ LMod)
18		lmodcmn 20925	. . 3 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ CMnd)
19	12, 17, 18	3syl 18	. 2 ⊢ (𝜑 → 𝑊 ∈ CMnd)
20		sitgclbn.2	. 2 ⊢ (𝜑 → (Scalar‘𝑊) ∈ ℝExt )
21	12, 17	syl 17	. . . 4 ⊢ (𝜑 → 𝑊 ∈ LMod)
22	21	3ad2ant1 1133	. . 3 ⊢ ((𝜑 ∧ 𝑚 ∈ (𝐻 “ (0[,)+∞)) ∧ 𝑥 ∈ 𝐵) → 𝑊 ∈ LMod)
23		imassrn 6099	. . . . . 6 ⊢ (𝐻 “ (0[,)+∞)) ⊆ ran 𝐻
24	6	rneqi 5961	. . . . . . 7 ⊢ ran 𝐻 = ran (ℝHom‘(Scalar‘𝑊))
25		eqid 2734	. . . . . . . . 9 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
26	25	rrhfe 33950	. . . . . . . 8 ⊢ ((Scalar‘𝑊) ∈ ℝExt → (ℝHom‘(Scalar‘𝑊)):ℝ⟶(Base‘(Scalar‘𝑊)))
27		frn 6753	. . . . . . . 8 ⊢ ((ℝHom‘(Scalar‘𝑊)):ℝ⟶(Base‘(Scalar‘𝑊)) → ran (ℝHom‘(Scalar‘𝑊)) ⊆ (Base‘(Scalar‘𝑊)))
28	20, 26, 27	3syl 18	. . . . . . 7 ⊢ (𝜑 → ran (ℝHom‘(Scalar‘𝑊)) ⊆ (Base‘(Scalar‘𝑊)))
29	24, 28	eqsstrid 4051	. . . . . 6 ⊢ (𝜑 → ran 𝐻 ⊆ (Base‘(Scalar‘𝑊)))
30	23, 29	sstrid 4014	. . . . 5 ⊢ (𝜑 → (𝐻 “ (0[,)+∞)) ⊆ (Base‘(Scalar‘𝑊)))
31	30	sselda 4002	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ (𝐻 “ (0[,)+∞))) → 𝑚 ∈ (Base‘(Scalar‘𝑊)))
32	31	3adant3 1132	. . 3 ⊢ ((𝜑 ∧ 𝑚 ∈ (𝐻 “ (0[,)+∞)) ∧ 𝑥 ∈ 𝐵) → 𝑚 ∈ (Base‘(Scalar‘𝑊)))
33		simp3 1138	. . 3 ⊢ ((𝜑 ∧ 𝑚 ∈ (𝐻 “ (0[,)+∞)) ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵)
34	1, 10, 5, 25	lmodvscl 20893	. . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝑚 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑥 ∈ 𝐵) → (𝑚 · 𝑥) ∈ 𝐵)
35	22, 32, 33, 34	syl3anc 1371	. 2 ⊢ ((𝜑 ∧ 𝑚 ∈ (𝐻 “ (0[,)+∞)) ∧ 𝑥 ∈ 𝐵) → (𝑚 · 𝑥) ∈ 𝐵)
36	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 16, 19, 20, 35	sitgclg 34299	1 ⊢ (𝜑 → ((𝑊sitg𝑀)‘𝐹) ∈ 𝐵)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ w3a 1087 = wceq 1537 ∈ wcel 2103 ⊆ wss 3970 ∪ cuni 4931 × cxp 5697 dom cdm 5699 ran crn 5700 ↾ cres 5701 “ cima 5702 ⟶wf 6568 ‘cfv 6572 (class class class)co 7445 ℝcr 11179 0cc0 11180 +∞cpnf 11317 [,)cico 13405 Basecbs 17253 Scalarcsca 17309 ·_𝑠 cvsca 17310 distcds 17315 TopOpenctopn 17476 0_gc0g 17494 CMndccmn 19817 LModclmod 20875 TopSpctps 22952 MetSpcms 24342 CMetSpccms 25378 Bancbn 25379 ℝHomcrrh 33931 ℝExt crrext 33932 sigaGencsigagen 34094 measurescmeas 34151 sitgcsitg 34286

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2105 ax-9 2113 ax-10 2136 ax-11 2153 ax-12 2173 ax-ext 2705 ax-rep 5306 ax-sep 5320 ax-nul 5327 ax-pow 5386 ax-pr 5450 ax-un 7766 ax-cnex 11236 ax-resscn 11237 ax-1cn 11238 ax-icn 11239 ax-addcl 11240 ax-addrcl 11241 ax-mulcl 11242 ax-mulrcl 11243 ax-mulcom 11244 ax-addass 11245 ax-mulass 11246 ax-distr 11247 ax-i2m1 11248 ax-1ne0 11249 ax-1rid 11250 ax-rnegex 11251 ax-rrecex 11252 ax-cnre 11253 ax-pre-lttri 11254 ax-pre-lttrn 11255 ax-pre-ltadd 11256 ax-pre-mulgt0 11257 ax-pre-sup 11258 ax-addf 11259 ax-mulf 11260

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2890 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-rmo 3383 df-reu 3384 df-rab 3439 df-v 3484 df-sbc 3799 df-csb 3916 df-dif 3973 df-un 3975 df-in 3977 df-ss 3987 df-pss 3990 df-nul 4348 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-int 4973 df-iun 5021 df-iin 5022 df-br 5170 df-opab 5232 df-mpt 5253 df-tr 5287 df-id 5597 df-eprel 5603 df-po 5611 df-so 5612 df-fr 5654 df-se 5655 df-we 5656 df-xp 5705 df-rel 5706 df-cnv 5707 df-co 5708 df-dm 5709 df-rn 5710 df-res 5711 df-ima 5712 df-pred 6331 df-ord 6397 df-on 6398 df-lim 6399 df-suc 6400 df-iota 6524 df-fun 6574 df-fn 6575 df-f 6576 df-f1 6577 df-fo 6578 df-f1o 6579 df-fv 6580 df-isom 6581 df-riota 7401 df-ov 7448 df-oprab 7449 df-mpo 7450 df-of 7710 df-om 7900 df-1st 8026 df-2nd 8027 df-supp 8198 df-tpos 8263 df-frecs 8318 df-wrecs 8349 df-recs 8423 df-rdg 8462 df-1o 8518 df-2o 8519 df-er 8759 df-map 8882 df-pm 8883 df-ixp 8952 df-en 9000 df-dom 9001 df-sdom 9002 df-fin 9003 df-fsupp 9428 df-fi 9476 df-sup 9507 df-inf 9508 df-oi 9575 df-card 10004 df-pnf 11322 df-mnf 11323 df-xr 11324 df-ltxr 11325 df-le 11326 df-sub 11518 df-neg 11519 df-div 11944 df-nn 12290 df-2 12352 df-3 12353 df-4 12354 df-5 12355 df-6 12356 df-7 12357 df-8 12358 df-9 12359 df-n0 12550 df-z 12636 df-dec 12755 df-uz 12900 df-q 13010 df-rp 13054 df-xneg 13171 df-xadd 13172 df-xmul 13173 df-ioo 13407 df-ico 13409 df-icc 13410 df-fz 13564 df-fzo 13708 df-fl 13839 df-mod 13917 df-seq 14049 df-exp 14109 df-hash 14376 df-cj 15144 df-re 15145 df-im 15146 df-sqrt 15280 df-abs 15281 df-dvds 16297 df-gcd 16535 df-numer 16777 df-denom 16778 df-gz 16972 df-struct 17189 df-sets 17206 df-slot 17224 df-ndx 17236 df-base 17254 df-ress 17283 df-plusg 17319 df-mulr 17320 df-starv 17321 df-sca 17322 df-vsca 17323 df-ip 17324 df-tset 17325 df-ple 17326 df-ds 17328 df-unif 17329 df-hom 17330 df-cco 17331 df-rest 17477 df-topn 17478 df-0g 17496 df-gsum 17497 df-topgen 17498 df-pt 17499 df-prds 17502 df-xrs 17557 df-qtop 17562 df-imas 17563 df-xps 17565 df-mre 17639 df-mrc 17640 df-acs 17642 df-mgm 18673 df-sgrp 18752 df-mnd 18768 df-mhm 18813 df-submnd 18814 df-grp 18971 df-minusg 18972 df-sbg 18973 df-mulg 19103 df-subg 19158 df-ghm 19248 df-cntz 19352 df-od 19565 df-cmn 19819 df-abl 19820 df-mgp 20157 df-rng 20175 df-ur 20204 df-ring 20257 df-cring 20258 df-oppr 20355 df-dvdsr 20378 df-unit 20379 df-invr 20409 df-dvr 20422 df-rhm 20493 df-nzr 20534 df-subrng 20567 df-subrg 20592 df-drng 20748 df-abv 20827 df-lmod 20877 df-psmet 21374 df-xmet 21375 df-met 21376 df-bl 21377 df-mopn 21378 df-fbas 21379 df-fg 21380 df-metu 21381 df-cnfld 21383 df-zring 21476 df-zrh 21532 df-zlm 21533 df-chr 21534 df-refld 21641 df-top 22914 df-topon 22931 df-topsp 22953 df-bases 22967 df-cld 23041 df-ntr 23042 df-cls 23043 df-nei 23120 df-cn 23249 df-cnp 23250 df-haus 23337 df-reg 23338 df-cmp 23409 df-tx 23584 df-hmeo 23777 df-fil 23868 df-fm 23960 df-flim 23961 df-flf 23962 df-fcls 23963 df-cnext 24082 df-ust 24223 df-utop 24254 df-uss 24279 df-usp 24280 df-ucn 24299 df-cfilu 24310 df-cusp 24321 df-xms 24344 df-ms 24345 df-tms 24346 df-nm 24609 df-ngp 24610 df-nrg 24612 df-nlm 24613 df-nvc 24614 df-cncf 24916 df-cfil 25301 df-cmet 25303 df-cms 25381 df-bn 25382 df-qqh 33911 df-rrh 33933 df-rrext 33937 df-esum 33984 df-siga 34065 df-sigagen 34095 df-meas 34152 df-mbfm 34206 df-sitg 34287

This theorem is referenced by: sitgclcn 34301 sitgclre 34302

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