Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > sitgclcn | Structured version Visualization version GIF version |
Description: Closure of the Bochner integral on a simple function. This version is specific to Banach spaces on the complex numbers. (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)) |
sitgclcn.1 | ⊢ (𝜑 → 𝑊 ∈ Ban) |
sitgclcn.2 | ⊢ (𝜑 → (Scalar‘𝑊) = ℂfld) |
Ref | Expression |
---|---|
sitgclcn | ⊢ (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 | sitgclcn.1 | . 2 ⊢ (𝜑 → 𝑊 ∈ Ban) | |
11 | sitgclcn.2 | . . 3 ⊢ (𝜑 → (Scalar‘𝑊) = ℂfld) | |
12 | cnrrext 31543 | . . 3 ⊢ ℂfld ∈ ℝExt | |
13 | 11, 12 | eqeltrdi 2842 | . 2 ⊢ (𝜑 → (Scalar‘𝑊) ∈ ℝExt ) |
14 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 13 | sitgclbn 31893 | 1 ⊢ (𝜑 → ((𝑊sitg𝑀)‘𝐹) ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2114 ∪ cuni 4806 dom cdm 5535 ran crn 5536 ‘cfv 6350 (class class class)co 7183 Basecbs 16599 Scalarcsca 16684 ·𝑠 cvsca 16685 TopOpenctopn 16811 0gc0g 16829 ℂfldccnfld 20230 Bancbn 24098 ℝHomcrrh 31526 ℝExt crrext 31527 sigaGencsigagen 31689 measurescmeas 31746 sitgcsitg 31879 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5242 ax-pr 5306 ax-un 7492 ax-cnex 10684 ax-resscn 10685 ax-1cn 10686 ax-icn 10687 ax-addcl 10688 ax-addrcl 10689 ax-mulcl 10690 ax-mulrcl 10691 ax-mulcom 10692 ax-addass 10693 ax-mulass 10694 ax-distr 10695 ax-i2m1 10696 ax-1ne0 10697 ax-1rid 10698 ax-rnegex 10699 ax-rrecex 10700 ax-cnre 10701 ax-pre-lttri 10702 ax-pre-lttrn 10703 ax-pre-ltadd 10704 ax-pre-mulgt0 10705 ax-pre-sup 10706 ax-addf 10707 ax-mulf 10708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3402 df-sbc 3686 df-csb 3801 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4222 df-if 4425 df-pw 4500 df-sn 4527 df-pr 4529 df-tp 4531 df-op 4533 df-uni 4807 df-int 4847 df-iun 4893 df-iin 4894 df-br 5041 df-opab 5103 df-mpt 5121 df-tr 5147 df-id 5439 df-eprel 5444 df-po 5452 df-so 5453 df-fr 5493 df-se 5494 df-we 5495 df-xp 5541 df-rel 5542 df-cnv 5543 df-co 5544 df-dm 5545 df-rn 5546 df-res 5547 df-ima 5548 df-pred 6139 df-ord 6186 df-on 6187 df-lim 6188 df-suc 6189 df-iota 6308 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-isom 6359 df-riota 7140 df-ov 7186 df-oprab 7187 df-mpo 7188 df-of 7438 df-om 7613 df-1st 7727 df-2nd 7728 df-supp 7870 df-tpos 7934 df-wrecs 7989 df-recs 8050 df-rdg 8088 df-1o 8144 df-2o 8145 df-er 8333 df-map 8452 df-pm 8453 df-ixp 8521 df-en 8569 df-dom 8570 df-sdom 8571 df-fin 8572 df-fsupp 8920 df-fi 8961 df-sup 8992 df-inf 8993 df-oi 9060 df-card 9454 df-pnf 10768 df-mnf 10769 df-xr 10770 df-ltxr 10771 df-le 10772 df-sub 10963 df-neg 10964 df-div 11389 df-nn 11730 df-2 11792 df-3 11793 df-4 11794 df-5 11795 df-6 11796 df-7 11797 df-8 11798 df-9 11799 df-n0 11990 df-z 12076 df-dec 12193 df-uz 12338 df-q 12444 df-rp 12486 df-xneg 12603 df-xadd 12604 df-xmul 12605 df-ioo 12838 df-ico 12840 df-icc 12841 df-fz 12995 df-fzo 13138 df-fl 13266 df-mod 13342 df-seq 13474 df-exp 13535 df-hash 13796 df-cj 14561 df-re 14562 df-im 14563 df-sqrt 14697 df-abs 14698 df-dvds 15713 df-gcd 15951 df-numer 16188 df-denom 16189 df-gz 16379 df-struct 16601 df-ndx 16602 df-slot 16603 df-base 16605 df-sets 16606 df-ress 16607 df-plusg 16694 df-mulr 16695 df-starv 16696 df-sca 16697 df-vsca 16698 df-ip 16699 df-tset 16700 df-ple 16701 df-ds 16703 df-unif 16704 df-hom 16705 df-cco 16706 df-rest 16812 df-topn 16813 df-0g 16831 df-gsum 16832 df-topgen 16833 df-pt 16834 df-prds 16837 df-xrs 16891 df-qtop 16896 df-imas 16897 df-xps 16899 df-mre 16973 df-mrc 16974 df-acs 16976 df-proset 17667 df-poset 17685 df-plt 17697 df-toset 17773 df-ps 17939 df-tsr 17940 df-mgm 17981 df-sgrp 18030 df-mnd 18041 df-mhm 18085 df-submnd 18086 df-grp 18235 df-minusg 18236 df-sbg 18237 df-mulg 18356 df-subg 18407 df-ghm 18487 df-cntz 18578 df-od 18787 df-cmn 19039 df-abl 19040 df-mgp 19372 df-ur 19384 df-ring 19431 df-cring 19432 df-oppr 19508 df-dvdsr 19526 df-unit 19527 df-invr 19557 df-dvr 19568 df-rnghom 19602 df-drng 19636 df-field 19637 df-subrg 19665 df-abv 19720 df-lmod 19768 df-nzr 20163 df-psmet 20222 df-xmet 20223 df-met 20224 df-bl 20225 df-mopn 20226 df-fbas 20227 df-fg 20228 df-metu 20229 df-cnfld 20231 df-zring 20303 df-zrh 20337 df-zlm 20338 df-chr 20339 df-refld 20434 df-top 21658 df-topon 21675 df-topsp 21697 df-bases 21710 df-cld 21783 df-ntr 21784 df-cls 21785 df-nei 21862 df-cn 21991 df-cnp 21992 df-haus 22079 df-reg 22080 df-cmp 22151 df-tx 22326 df-hmeo 22519 df-fil 22610 df-fm 22702 df-flim 22703 df-flf 22704 df-fcls 22705 df-cnext 22824 df-ust 22965 df-utop 22996 df-uss 23021 df-usp 23022 df-ucn 23041 df-cfilu 23052 df-cusp 23063 df-xms 23086 df-ms 23087 df-tms 23088 df-nm 23348 df-ngp 23349 df-nrg 23351 df-nlm 23352 df-nvc 23353 df-cncf 23643 df-cfil 24020 df-cmet 24022 df-cms 24100 df-bn 24101 df-omnd 30915 df-ogrp 30916 df-orng 31086 df-ofld 31087 df-qqh 31506 df-rrh 31528 df-rrext 31532 df-esum 31579 df-siga 31660 df-sigagen 31690 df-meas 31747 df-mbfm 31801 df-sitg 31880 |
This theorem is referenced by: (None) |
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