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Mirrors > Home > MPE Home > Th. List > sspgval | Structured version Visualization version GIF version |
Description: Vector addition on a subspace in terms of vector addition on the parent space. (Contributed by NM, 28-Jan-2008.) (New usage is discouraged.) |
Ref | Expression |
---|---|
sspg.y | β’ π = (BaseSetβπ) |
sspg.g | β’ πΊ = ( +π£ βπ) |
sspg.f | β’ πΉ = ( +π£ βπ) |
sspg.h | β’ π» = (SubSpβπ) |
Ref | Expression |
---|---|
sspgval | β’ (((π β NrmCVec β§ π β π») β§ (π΄ β π β§ π΅ β π)) β (π΄πΉπ΅) = (π΄πΊπ΅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sspg.y | . . . 4 β’ π = (BaseSetβπ) | |
2 | sspg.g | . . . 4 β’ πΊ = ( +π£ βπ) | |
3 | sspg.f | . . . 4 β’ πΉ = ( +π£ βπ) | |
4 | sspg.h | . . . 4 β’ π» = (SubSpβπ) | |
5 | 1, 2, 3, 4 | sspg 30486 | . . 3 β’ ((π β NrmCVec β§ π β π») β πΉ = (πΊ βΎ (π Γ π))) |
6 | 5 | oveqd 7421 | . 2 β’ ((π β NrmCVec β§ π β π») β (π΄πΉπ΅) = (π΄(πΊ βΎ (π Γ π))π΅)) |
7 | ovres 7569 | . 2 β’ ((π΄ β π β§ π΅ β π) β (π΄(πΊ βΎ (π Γ π))π΅) = (π΄πΊπ΅)) | |
8 | 6, 7 | sylan9eq 2786 | 1 β’ (((π β NrmCVec β§ π β π») β§ (π΄ β π β§ π΅ β π)) β (π΄πΉπ΅) = (π΄πΊπ΅)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1533 β wcel 2098 Γ cxp 5667 βΎ cres 5671 βcfv 6536 (class class class)co 7404 NrmCVeccnv 30342 +π£ cpv 30343 BaseSetcba 30344 SubSpcss 30479 |
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 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-ov 7407 df-oprab 7408 df-1st 7971 df-2nd 7972 df-grpo 30251 df-ablo 30303 df-vc 30317 df-nv 30350 df-va 30353 df-ba 30354 df-sm 30355 df-0v 30356 df-nmcv 30358 df-ssp 30480 |
This theorem is referenced by: sspmval 30491 minvecolem2 30633 hhshsslem2 31026 |
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