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Mirrors > Home > MPE Home > Th. List > nvsass | Structured version Visualization version GIF version |
Description: Associative law for the scalar product of a normed complex vector space. (Contributed by NM, 17-Nov-2007.) (New usage is discouraged.) |
Ref | Expression |
---|---|
nvscl.1 | β’ π = (BaseSetβπ) |
nvscl.4 | β’ π = ( Β·π OLD βπ) |
Ref | Expression |
---|---|
nvsass | β’ ((π β NrmCVec β§ (π΄ β β β§ π΅ β β β§ πΆ β π)) β ((π΄ Β· π΅)ππΆ) = (π΄π(π΅ππΆ))) |
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 | vcass 29820 | . 2 β’ (((1st βπ) β CVecOLD β§ (π΄ β β β§ π΅ β β β§ πΆ β π)) β ((π΄ Β· π΅)ππΆ) = (π΄π(π΅ππΆ))) |
10 | 2, 9 | sylan 581 | 1 β’ ((π β NrmCVec β§ (π΄ β β β§ π΅ β β β§ πΆ β π)) β ((π΄ Β· π΅)ππΆ) = (π΄π(π΅ππΆ))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 β§ w3a 1088 = wceq 1542 β wcel 2107 βcfv 6544 (class class class)co 7409 1st c1st 7973 βcc 11108 Β· cmul 11115 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: nvscom 29882 nvmul0or 29903 nvpi 29920 smcnlem 29950 ipval3 29962 ipdirilem 30082 ipasslem2 30085 ipasslem4 30087 ipasslem5 30088 ipasslem10 30092 ipasslem11 30093 minvecolem2 30128 hlmulass 30159 |
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