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Mirrors > Home > MPE Home > Th. List > sspnval | Structured version Visualization version GIF version |
Description: The norm on a subspace in terms of the norm on the parent space. (Contributed by NM, 28-Jan-2008.) (New usage is discouraged.) |
Ref | Expression |
---|---|
sspn.y | β’ π = (BaseSetβπ) |
sspn.n | β’ π = (normCVβπ) |
sspn.m | β’ π = (normCVβπ) |
sspn.h | β’ π» = (SubSpβπ) |
Ref | Expression |
---|---|
sspnval | β’ ((π β NrmCVec β§ π β π» β§ π΄ β π) β (πβπ΄) = (πβπ΄)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sspn.y | . . . . 5 β’ π = (BaseSetβπ) | |
2 | sspn.n | . . . . 5 β’ π = (normCVβπ) | |
3 | sspn.m | . . . . 5 β’ π = (normCVβπ) | |
4 | sspn.h | . . . . 5 β’ π» = (SubSpβπ) | |
5 | 1, 2, 3, 4 | sspn 30566 | . . . 4 β’ ((π β NrmCVec β§ π β π») β π = (π βΎ π)) |
6 | 5 | fveq1d 6904 | . . 3 β’ ((π β NrmCVec β§ π β π») β (πβπ΄) = ((π βΎ π)βπ΄)) |
7 | fvres 6921 | . . 3 β’ (π΄ β π β ((π βΎ π)βπ΄) = (πβπ΄)) | |
8 | 6, 7 | sylan9eq 2788 | . 2 β’ (((π β NrmCVec β§ π β π») β§ π΄ β π) β (πβπ΄) = (πβπ΄)) |
9 | 8 | 3impa 1107 | 1 β’ ((π β NrmCVec β§ π β π» β§ π΄ β π) β (πβπ΄) = (πβπ΄)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 β§ w3a 1084 = wceq 1533 β wcel 2098 βΎ cres 5684 βcfv 6553 NrmCVeccnv 30414 BaseSetcba 30416 normCVcnmcv 30420 SubSpcss 30551 |
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 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-ov 7429 df-oprab 7430 df-1st 7999 df-2nd 8000 df-vc 30389 df-nv 30422 df-va 30425 df-ba 30426 df-sm 30427 df-0v 30428 df-nmcv 30430 df-ssp 30552 |
This theorem is referenced by: sspimsval 30568 |
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