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Mirrors > Home > MPE Home > Th. List > cphclm | Structured version Visualization version GIF version |
Description: A subcomplex pre-Hilbert space is a subcomplex module. (Contributed by Mario Carneiro, 16-Oct-2015.) |
Ref | Expression |
---|---|
cphclm | ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ ℂMod) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cphlmod 22974 | . 2 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ LMod) | |
2 | eqid 2622 | . . 3 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
3 | eqid 2622 | . . 3 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊)) | |
4 | 2, 3 | cphsca 22979 | . 2 ⊢ (𝑊 ∈ ℂPreHil → (Scalar‘𝑊) = (ℂfld ↾s (Base‘(Scalar‘𝑊)))) |
5 | 2, 3 | cphsubrg 22980 | . 2 ⊢ (𝑊 ∈ ℂPreHil → (Base‘(Scalar‘𝑊)) ∈ (SubRing‘ℂfld)) |
6 | 2, 3 | isclm 22864 | . 2 ⊢ (𝑊 ∈ ℂMod ↔ (𝑊 ∈ LMod ∧ (Scalar‘𝑊) = (ℂfld ↾s (Base‘(Scalar‘𝑊))) ∧ (Base‘(Scalar‘𝑊)) ∈ (SubRing‘ℂfld))) |
7 | 1, 4, 5, 6 | syl3anbrc 1246 | 1 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ ℂMod) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ∈ wcel 1990 ‘cfv 5888 (class class class)co 6650 Basecbs 15857 ↾s cress 15858 Scalarcsca 15944 SubRingcsubrg 18776 LModclmod 18863 ℂfldccnfld 19746 ℂModcclm 22862 ℂPreHilccph 22966 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-addf 10015 ax-mulf 10016 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-tpos 7352 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-fz 12327 df-seq 12802 df-exp 12861 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-starv 15956 df-tset 15960 df-ple 15961 df-ds 15964 df-unif 15965 df-0g 16102 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-grp 17425 df-subg 17591 df-cmn 18195 df-mgp 18490 df-ur 18502 df-ring 18549 df-cring 18550 df-oppr 18623 df-dvdsr 18641 df-unit 18642 df-drng 18749 df-subrg 18778 df-lvec 19103 df-cnfld 19747 df-phl 19971 df-nlm 22391 df-clm 22863 df-cph 22968 |
This theorem is referenced by: cphnmvs 22990 cphipcj 22999 cphorthcom 23001 cphip0l 23002 cphip0r 23003 cphipeq0 23004 cphdir 23005 cphdi 23006 cph2di 23007 cphsubdir 23008 cphsubdi 23009 cph2subdi 23010 cphass 23011 cphassr 23012 cph2ass 23013 nmparlem 23038 cphipval 23042 ipcn 23045 csscld 23048 clsocv 23049 minveclem2 23197 pjthlem2 23209 |
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