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| Mirrors > Home > MPE Home > Th. List > nvscl | Structured version Visualization version GIF version | ||
| Description: Closure law for the scalar product operation of a normed complex vector space. (Contributed by NM, 1-Feb-2007.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| nvscl.1 | β’ π = (BaseSetβπ) |
| nvscl.4 | β’ π = ( Β·π OLD βπ) |
| Ref | Expression |
|---|---|
| nvscl | β’ ((π β NrmCVec β§ π΄ β β β§ π΅ β π) β (π΄ππ΅) β π) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | eqid 2733 | . . 3 β’ (1st βπ) = (1st βπ) | |
| 2 | 1 | nvvc 29868 | . 2 β’ (π β NrmCVec β (1st βπ) β CVecOLD) |
| 3 | eqid 2733 | . . . 4 β’ ( +π£ βπ) = ( +π£ βπ) | |
| 4 | 3 | vafval 29856 | . . 3 β’ ( +π£ βπ) = (1st β(1st βπ)) |
| 5 | nvscl.4 | . . . 4 β’ π = ( Β·π OLD βπ) | |
| 6 | 5 | smfval 29858 | . . 3 β’ π = (2nd β(1st βπ)) |
| 7 | nvscl.1 | . . . 4 β’ π = (BaseSetβπ) | |
| 8 | 7, 3 | bafval 29857 | . . 3 β’ π = ran ( +π£ βπ) |
| 9 | 4, 6, 8 | vccl 29816 | . 2 β’ (((1st βπ) β CVecOLD β§ π΄ β β β§ π΅ β π) β (π΄ππ΅) β π) |
| 10 | 2, 9 | syl3an1 1164 | 1 β’ ((π β NrmCVec β§ π΄ β β β§ π΅ β π) β (π΄ππ΅) β π) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β§ w3a 1088 = wceq 1542 β wcel 2107 βcfv 6544 (class class class)co 7409 1st c1st 7973 βcc 11108 CVecOLDcvc 29811 NrmCVeccnv 29837 +π£ cpv 29838 BaseSetcba 29839 Β·π OLD cns 29840 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pr 5428 ax-un 7725 |
| This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-ov 7412 df-oprab 7413 df-1st 7975 df-2nd 7976 df-vc 29812 df-nv 29845 df-va 29848 df-ba 29849 df-sm 29850 df-0v 29851 df-nmcv 29853 |
| This theorem is referenced by: nvmval2 29896 nvmf 29898 nvmdi 29901 nvnegneg 29902 nvpncan2 29906 nvaddsub4 29910 nvdif 29919 nvpi 29920 nvmtri 29924 nvabs 29925 nvge0 29926 imsmetlem 29943 smcnlem 29950 ipval2lem2 29957 4ipval2 29961 ipval3 29962 sspmval 29986 lnocoi 30010 lnomul 30013 0lno 30043 nmlno0lem 30046 nmblolbii 30052 blocnilem 30057 ip0i 30078 ip1ilem 30079 ipdirilem 30082 ipasslem1 30084 ipasslem2 30085 ipasslem4 30087 ipasslem5 30088 ipasslem8 30090 ipasslem9 30091 ipasslem10 30092 ipasslem11 30093 dipassr 30099 dipsubdir 30101 siilem1 30104 ipblnfi 30108 ubthlem2 30124 minvecolem2 30128 hhshsslem2 30521 |
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