![]() |
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 29981 | . 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 29893 | . 2 β’ (π β NrmCVec β πΏ:(π Γ π)βΆπ) |
7 | eqid 2732 | . . 3 β’ (BaseSetβπ) = (BaseSetβπ) | |
8 | 7, 3 | nvmf 29893 | . 2 β’ (π β NrmCVec β π:((BaseSetβπ) Γ (BaseSetβπ))βΆ(BaseSetβπ)) |
9 | 1, 2, 5, 6, 8 | sspmlem 29980 | 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 396 = wceq 1541 β wcel 2106 Γ cxp 5674 βΎ cres 5678 βcfv 6543 NrmCVeccnv 29832 BaseSetcba 29834 βπ£ cnsb 29837 SubSpcss 29969 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-po 5588 df-so 5589 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7364 df-ov 7411 df-oprab 7412 df-mpo 7413 df-1st 7974 df-2nd 7975 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-pnf 11249 df-mnf 11250 df-ltxr 11252 df-sub 11445 df-neg 11446 df-grpo 29741 df-gid 29742 df-ginv 29743 df-gdiv 29744 df-ablo 29793 df-vc 29807 df-nv 29840 df-va 29843 df-ba 29844 df-sm 29845 df-0v 29846 df-vs 29847 df-nmcv 29848 df-ssp 29970 |
This theorem is referenced by: hhssvs 30520 |
Copyright terms: Public domain | W3C validator |