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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 29720 | . . . 4 β’ ((π β NrmCVec β§ π β π») β π = (π βΎ π)) |
6 | 5 | fveq1d 6845 | . . 3 β’ ((π β NrmCVec β§ π β π») β (πβπ΄) = ((π βΎ π)βπ΄)) |
7 | fvres 6862 | . . 3 β’ (π΄ β π β ((π βΎ π)βπ΄) = (πβπ΄)) | |
8 | 6, 7 | sylan9eq 2793 | . 2 β’ (((π β NrmCVec β§ π β π») β§ π΄ β π) β (πβπ΄) = (πβπ΄)) |
9 | 8 | 3impa 1111 | 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 397 β§ w3a 1088 = wceq 1542 β wcel 2107 βΎ cres 5636 βcfv 6497 NrmCVeccnv 29568 BaseSetcba 29570 normCVcnmcv 29574 SubSpcss 29705 |
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 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 |
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 2941 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-ov 7361 df-oprab 7362 df-1st 7922 df-2nd 7923 df-vc 29543 df-nv 29576 df-va 29579 df-ba 29580 df-sm 29581 df-0v 29582 df-nmcv 29584 df-ssp 29706 |
This theorem is referenced by: sspimsval 29722 |
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