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Mirrors > Home > MPE Home > Th. List > nvf | Structured version Visualization version GIF version |
Description: Mapping for the norm function. (Contributed by NM, 11-Nov-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
nvf.1 | β’ π = (BaseSetβπ) |
nvf.6 | β’ π = (normCVβπ) |
Ref | Expression |
---|---|
nvf | β’ (π β NrmCVec β π:πβΆβ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nvf.1 | . . 3 β’ π = (BaseSetβπ) | |
2 | eqid 2733 | . . 3 β’ ( +π£ βπ) = ( +π£ βπ) | |
3 | eqid 2733 | . . 3 β’ ( Β·π OLD βπ) = ( Β·π OLD βπ) | |
4 | eqid 2733 | . . 3 β’ (0vecβπ) = (0vecβπ) | |
5 | nvf.6 | . . 3 β’ π = (normCVβπ) | |
6 | 1, 2, 3, 4, 5 | nvi 29867 | . 2 β’ (π β NrmCVec β (β¨( +π£ βπ), ( Β·π OLD βπ)β© β CVecOLD β§ π:πβΆβ β§ βπ₯ β π (((πβπ₯) = 0 β π₯ = (0vecβπ)) β§ βπ¦ β β (πβ(π¦( Β·π OLD βπ)π₯)) = ((absβπ¦) Β· (πβπ₯)) β§ βπ¦ β π (πβ(π₯( +π£ βπ)π¦)) β€ ((πβπ₯) + (πβπ¦))))) |
7 | 6 | simp2d 1144 | 1 β’ (π β NrmCVec β π:πβΆβ) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ w3a 1088 = wceq 1542 β wcel 2107 βwral 3062 β¨cop 4635 class class class wbr 5149 βΆwf 6540 βcfv 6544 (class class class)co 7409 βcc 11108 βcr 11109 0cc0 11110 + caddc 11113 Β· cmul 11115 β€ cle 11249 abscabs 15181 CVecOLDcvc 29811 NrmCVeccnv 29837 +π£ cpv 29838 BaseSetcba 29839 Β·π OLD cns 29840 0veccn0v 29841 normCVcnmcv 29843 |
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: nvcl 29914 imsdf 29942 nmcvcn 29948 sspn 29989 hilnormi 30416 |
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