Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > sitgclbn | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
sitgval.b | ⊢ 𝐵 = (Base‘𝑊) |
sitgval.j | ⊢ 𝐽 = (TopOpen‘𝑊) |
sitgval.s | ⊢ 𝑆 = (sigaGen‘𝐽) |
sitgval.0 | ⊢ 0 = (0g‘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 ) |
Ref | Expression |
---|---|
sitgclbn | ⊢ (𝜑 → ((𝑊sitg𝑀)‘𝐹) ∈ 𝐵) |
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 = (0g‘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 2824 | . 2 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
11 | eqid 2824 | . 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 23950 | . . . 4 ⊢ (𝑊 ∈ Ban → 𝑊 ∈ CMetSp) | |
14 | 12, 13 | syl 17 | . . 3 ⊢ (𝜑 → 𝑊 ∈ CMetSp) |
15 | cmsms 23954 | . . 3 ⊢ (𝑊 ∈ CMetSp → 𝑊 ∈ MetSp) | |
16 | mstps 23068 | . . 3 ⊢ (𝑊 ∈ MetSp → 𝑊 ∈ TopSp) | |
17 | 14, 15, 16 | 3syl 18 | . 2 ⊢ (𝜑 → 𝑊 ∈ TopSp) |
18 | bnlmod 23949 | . . 3 ⊢ (𝑊 ∈ Ban → 𝑊 ∈ LMod) | |
19 | lmodcmn 19685 | . . 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 1129 | . . 3 ⊢ ((𝜑 ∧ 𝑚 ∈ (𝐻 “ (0[,)+∞)) ∧ 𝑥 ∈ 𝐵) → 𝑊 ∈ LMod) |
24 | imassrn 5943 | . . . . . 6 ⊢ (𝐻 “ (0[,)+∞)) ⊆ ran 𝐻 | |
25 | 6 | rneqi 5810 | . . . . . . 7 ⊢ ran 𝐻 = ran (ℝHom‘(Scalar‘𝑊)) |
26 | eqid 2824 | . . . . . . . . 9 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊)) | |
27 | 26 | rrhfe 31257 | . . . . . . . 8 ⊢ ((Scalar‘𝑊) ∈ ℝExt → (ℝHom‘(Scalar‘𝑊)):ℝ⟶(Base‘(Scalar‘𝑊))) |
28 | frn 6523 | . . . . . . . 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 4018 | . . . . . 6 ⊢ (𝜑 → ran 𝐻 ⊆ (Base‘(Scalar‘𝑊))) |
31 | 24, 30 | sstrid 3981 | . . . . 5 ⊢ (𝜑 → (𝐻 “ (0[,)+∞)) ⊆ (Base‘(Scalar‘𝑊))) |
32 | 31 | sselda 3970 | . . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ (𝐻 “ (0[,)+∞))) → 𝑚 ∈ (Base‘(Scalar‘𝑊))) |
33 | 32 | 3adant3 1128 | . . 3 ⊢ ((𝜑 ∧ 𝑚 ∈ (𝐻 “ (0[,)+∞)) ∧ 𝑥 ∈ 𝐵) → 𝑚 ∈ (Base‘(Scalar‘𝑊))) |
34 | simp3 1134 | . . 3 ⊢ ((𝜑 ∧ 𝑚 ∈ (𝐻 “ (0[,)+∞)) ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵) | |
35 | 1, 10, 5, 26 | lmodvscl 19654 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝑚 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑥 ∈ 𝐵) → (𝑚 · 𝑥) ∈ 𝐵) |
36 | 23, 33, 34, 35 | syl3anc 1367 | . 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 31604 | 1 ⊢ (𝜑 → ((𝑊sitg𝑀)‘𝐹) ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1083 = wceq 1536 ∈ wcel 2113 ⊆ wss 3939 ∪ cuni 4841 × cxp 5556 dom cdm 5558 ran crn 5559 ↾ cres 5560 “ cima 5561 ⟶wf 6354 ‘cfv 6358 (class class class)co 7159 ℝcr 10539 0cc0 10540 +∞cpnf 10675 [,)cico 12743 Basecbs 16486 Scalarcsca 16571 ·𝑠 cvsca 16572 distcds 16577 TopOpenctopn 16698 0gc0g 16716 CMndccmn 18909 LModclmod 19637 TopSpctps 21543 MetSpcms 22931 CMetSpccms 23938 Bancbn 23939 ℝHomcrrh 31238 ℝExt crrext 31239 sigaGencsigagen 31401 measurescmeas 31458 sitgcsitg 31591 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 ax-pre-sup 10618 ax-addf 10619 ax-mulf 10620 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-fal 1549 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-int 4880 df-iun 4924 df-iin 4925 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-se 5518 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-isom 6367 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-of 7412 df-om 7584 df-1st 7692 df-2nd 7693 df-supp 7834 df-tpos 7895 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-1o 8105 df-2o 8106 df-oadd 8109 df-er 8292 df-map 8411 df-pm 8412 df-ixp 8465 df-en 8513 df-dom 8514 df-sdom 8515 df-fin 8516 df-fsupp 8837 df-fi 8878 df-sup 8909 df-inf 8910 df-oi 8977 df-card 9371 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-div 11301 df-nn 11642 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-q 12352 df-rp 12393 df-xneg 12510 df-xadd 12511 df-xmul 12512 df-ioo 12745 df-ico 12747 df-icc 12748 df-fz 12896 df-fzo 13037 df-fl 13165 df-mod 13241 df-seq 13373 df-exp 13433 df-hash 13694 df-cj 14461 df-re 14462 df-im 14463 df-sqrt 14597 df-abs 14598 df-dvds 15611 df-gcd 15847 df-numer 16078 df-denom 16079 df-gz 16269 df-struct 16488 df-ndx 16489 df-slot 16490 df-base 16492 df-sets 16493 df-ress 16494 df-plusg 16581 df-mulr 16582 df-starv 16583 df-sca 16584 df-vsca 16585 df-ip 16586 df-tset 16587 df-ple 16588 df-ds 16590 df-unif 16591 df-hom 16592 df-cco 16593 df-rest 16699 df-topn 16700 df-0g 16718 df-gsum 16719 df-topgen 16720 df-pt 16721 df-prds 16724 df-xrs 16778 df-qtop 16783 df-imas 16784 df-xps 16786 df-mre 16860 df-mrc 16861 df-acs 16863 df-mgm 17855 df-sgrp 17904 df-mnd 17915 df-mhm 17959 df-submnd 17960 df-grp 18109 df-minusg 18110 df-sbg 18111 df-mulg 18228 df-subg 18279 df-ghm 18359 df-cntz 18450 df-od 18659 df-cmn 18911 df-abl 18912 df-mgp 19243 df-ur 19255 df-ring 19302 df-cring 19303 df-oppr 19376 df-dvdsr 19394 df-unit 19395 df-invr 19425 df-dvr 19436 df-rnghom 19470 df-drng 19507 df-subrg 19536 df-abv 19591 df-lmod 19639 df-nzr 20034 df-psmet 20540 df-xmet 20541 df-met 20542 df-bl 20543 df-mopn 20544 df-fbas 20545 df-fg 20546 df-metu 20547 df-cnfld 20549 df-zring 20621 df-zrh 20654 df-zlm 20655 df-chr 20656 df-refld 20752 df-top 21505 df-topon 21522 df-topsp 21544 df-bases 21557 df-cld 21630 df-ntr 21631 df-cls 21632 df-nei 21709 df-cn 21838 df-cnp 21839 df-haus 21926 df-reg 21927 df-cmp 21998 df-tx 22173 df-hmeo 22366 df-fil 22457 df-fm 22549 df-flim 22550 df-flf 22551 df-fcls 22552 df-cnext 22671 df-ust 22812 df-utop 22843 df-uss 22868 df-usp 22869 df-ucn 22888 df-cfilu 22899 df-cusp 22910 df-xms 22933 df-ms 22934 df-tms 22935 df-nm 23195 df-ngp 23196 df-nrg 23198 df-nlm 23199 df-nvc 23200 df-cncf 23489 df-cfil 23861 df-cmet 23863 df-cms 23941 df-bn 23942 df-qqh 31218 df-rrh 31240 df-rrext 31244 df-esum 31291 df-siga 31372 df-sigagen 31402 df-meas 31459 df-mbfm 31513 df-sitg 31592 |
This theorem is referenced by: sitgclcn 31606 sitgclre 31607 |
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