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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 30458 | . 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 30370 | . 2 β’ (π β NrmCVec β πΏ:(π Γ π)βΆπ) |
7 | eqid 2724 | . . 3 β’ (BaseSetβπ) = (BaseSetβπ) | |
8 | 7, 3 | nvmf 30370 | . 2 β’ (π β NrmCVec β π:((BaseSetβπ) Γ (BaseSetβπ))βΆ(BaseSetβπ)) |
9 | 1, 2, 5, 6, 8 | sspmlem 30457 | 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 395 = wceq 1533 β wcel 2098 Γ cxp 5665 βΎ cres 5669 βcfv 6534 NrmCVeccnv 30309 BaseSetcba 30311 βπ£ cnsb 30314 SubSpcss 30446 |
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 2163 ax-ext 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-id 5565 df-po 5579 df-so 5580 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-1st 7969 df-2nd 7970 df-er 8700 df-en 8937 df-dom 8938 df-sdom 8939 df-pnf 11248 df-mnf 11249 df-ltxr 11251 df-sub 11444 df-neg 11445 df-grpo 30218 df-gid 30219 df-ginv 30220 df-gdiv 30221 df-ablo 30270 df-vc 30284 df-nv 30317 df-va 30320 df-ba 30321 df-sm 30322 df-0v 30323 df-vs 30324 df-nmcv 30325 df-ssp 30447 |
This theorem is referenced by: hhssvs 30997 |
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