![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > sspm | Structured version Visualization version GIF version |
Description: Vector subtraction on a subspace is a restriction of vector subtraction on the parent space. (Contributed by NM, 28-Jan-2008.) (New usage is discouraged.) |
Ref | Expression |
---|---|
sspm.y | β’ π = (BaseSetβπ) |
sspm.m | β’ π = ( βπ£ βπ) |
sspm.l | β’ πΏ = ( βπ£ βπ) |
sspm.h | β’ π» = (SubSpβπ) |
Ref | Expression |
---|---|
sspm | β’ ((π β NrmCVec β§ π β π») β πΏ = (π βΎ (π Γ π))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sspm.y | . 2 β’ π = (BaseSetβπ) | |
2 | sspm.h | . 2 β’ π» = (SubSpβπ) | |
3 | sspm.m | . . 3 β’ π = ( βπ£ βπ) | |
4 | sspm.l | . . 3 β’ πΏ = ( βπ£ βπ) | |
5 | 1, 3, 4, 2 | sspmval 29717 | . 2 β’ (((π β NrmCVec β§ π β π») β§ (π₯ β π β§ π¦ β π)) β (π₯πΏπ¦) = (π₯ππ¦)) |
6 | 1, 4 | nvmf 29629 | . 2 β’ (π β NrmCVec β πΏ:(π Γ π)βΆπ) |
7 | eqid 2733 | . . 3 β’ (BaseSetβπ) = (BaseSetβπ) | |
8 | 7, 3 | nvmf 29629 | . 2 β’ (π β NrmCVec β π:((BaseSetβπ) Γ (BaseSetβπ))βΆ(BaseSetβπ)) |
9 | 1, 2, 5, 6, 8 | sspmlem 29716 | 1 β’ ((π β NrmCVec β§ π β π») β πΏ = (π βΎ (π Γ π))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 Γ cxp 5632 βΎ cres 5636 βcfv 6497 NrmCVeccnv 29568 BaseSetcba 29570 βπ£ cnsb 29573 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 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 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-nel 3047 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-po 5546 df-so 5547 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-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-1st 7922 df-2nd 7923 df-er 8651 df-en 8887 df-dom 8888 df-sdom 8889 df-pnf 11196 df-mnf 11197 df-ltxr 11199 df-sub 11392 df-neg 11393 df-grpo 29477 df-gid 29478 df-ginv 29479 df-gdiv 29480 df-ablo 29529 df-vc 29543 df-nv 29576 df-va 29579 df-ba 29580 df-sm 29581 df-0v 29582 df-vs 29583 df-nmcv 29584 df-ssp 29706 |
This theorem is referenced by: hhssvs 30256 |
Copyright terms: Public domain | W3C validator |