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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 33479 | . . 3 ⊢ ℂfld ∈ ℝExt | |
13 | 11, 12 | eqeltrdi 2833 | . 2 ⊢ (𝜑 → (Scalar‘𝑊) ∈ ℝExt ) |
14 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 13 | sitgclbn 33831 | 1 ⊢ (𝜑 → ((𝑊sitg𝑀)‘𝐹) ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ∪ cuni 4899 dom cdm 5666 ran crn 5667 ‘cfv 6533 (class class class)co 7401 Basecbs 17143 Scalarcsca 17199 ·𝑠 cvsca 17200 TopOpenctopn 17366 0gc0g 17384 ℂfldccnfld 21228 Bancbn 25183 ℝHomcrrh 33462 ℝExt crrext 33463 sigaGencsigagen 33625 measurescmeas 33682 sitgcsitg 33817 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 ax-addf 11185 ax-mulf 11186 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-tp 4625 df-op 4627 df-uni 4900 df-int 4941 df-iun 4989 df-iin 4990 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-se 5622 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-isom 6542 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-of 7663 df-om 7849 df-1st 7968 df-2nd 7969 df-supp 8141 df-tpos 8206 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-2o 8462 df-er 8699 df-map 8818 df-pm 8819 df-ixp 8888 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-fsupp 9358 df-fi 9402 df-sup 9433 df-inf 9434 df-oi 9501 df-card 9930 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-div 11869 df-nn 12210 df-2 12272 df-3 12273 df-4 12274 df-5 12275 df-6 12276 df-7 12277 df-8 12278 df-9 12279 df-n0 12470 df-z 12556 df-dec 12675 df-uz 12820 df-q 12930 df-rp 12972 df-xneg 13089 df-xadd 13090 df-xmul 13091 df-ioo 13325 df-ico 13327 df-icc 13328 df-fz 13482 df-fzo 13625 df-fl 13754 df-mod 13832 df-seq 13964 df-exp 14025 df-hash 14288 df-cj 15043 df-re 15044 df-im 15045 df-sqrt 15179 df-abs 15180 df-dvds 16195 df-gcd 16433 df-numer 16670 df-denom 16671 df-gz 16862 df-struct 17079 df-sets 17096 df-slot 17114 df-ndx 17126 df-base 17144 df-ress 17173 df-plusg 17209 df-mulr 17210 df-starv 17211 df-sca 17212 df-vsca 17213 df-ip 17214 df-tset 17215 df-ple 17216 df-ds 17218 df-unif 17219 df-hom 17220 df-cco 17221 df-rest 17367 df-topn 17368 df-0g 17386 df-gsum 17387 df-topgen 17388 df-pt 17389 df-prds 17392 df-xrs 17447 df-qtop 17452 df-imas 17453 df-xps 17455 df-mre 17529 df-mrc 17530 df-acs 17532 df-proset 18250 df-poset 18268 df-plt 18285 df-toset 18372 df-ps 18521 df-tsr 18522 df-mgm 18563 df-sgrp 18642 df-mnd 18658 df-mhm 18703 df-submnd 18704 df-grp 18856 df-minusg 18857 df-sbg 18858 df-mulg 18986 df-subg 19040 df-ghm 19129 df-cntz 19223 df-od 19438 df-cmn 19692 df-abl 19693 df-mgp 20030 df-rng 20048 df-ur 20077 df-ring 20130 df-cring 20131 df-oppr 20226 df-dvdsr 20249 df-unit 20250 df-invr 20280 df-dvr 20293 df-rhm 20364 df-nzr 20405 df-subrng 20436 df-subrg 20461 df-drng 20579 df-field 20580 df-abv 20650 df-lmod 20698 df-psmet 21220 df-xmet 21221 df-met 21222 df-bl 21223 df-mopn 21224 df-fbas 21225 df-fg 21226 df-metu 21227 df-cnfld 21229 df-zring 21302 df-zrh 21358 df-zlm 21359 df-chr 21360 df-refld 21466 df-top 22718 df-topon 22735 df-topsp 22757 df-bases 22771 df-cld 22845 df-ntr 22846 df-cls 22847 df-nei 22924 df-cn 23053 df-cnp 23054 df-haus 23141 df-reg 23142 df-cmp 23213 df-tx 23388 df-hmeo 23581 df-fil 23672 df-fm 23764 df-flim 23765 df-flf 23766 df-fcls 23767 df-cnext 23886 df-ust 24027 df-utop 24058 df-uss 24083 df-usp 24084 df-ucn 24103 df-cfilu 24114 df-cusp 24125 df-xms 24148 df-ms 24149 df-tms 24150 df-nm 24413 df-ngp 24414 df-nrg 24416 df-nlm 24417 df-nvc 24418 df-cncf 24720 df-cfil 25105 df-cmet 25107 df-cms 25185 df-bn 25186 df-omnd 32685 df-ogrp 32686 df-orng 32881 df-ofld 32882 df-qqh 33442 df-rrh 33464 df-rrext 33468 df-esum 33515 df-siga 33596 df-sigagen 33626 df-meas 33683 df-mbfm 33737 df-sitg 33818 |
This theorem is referenced by: (None) |
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