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Mirrors > Home > MPE Home > Th. List > lspprss | Structured version Visualization version GIF version |
Description: The span of a pair of vectors in a subspace belongs to the subspace. (Contributed by NM, 12-Jan-2015.) |
Ref | Expression |
---|---|
lspprss.s | β’ π = (LSubSpβπ) |
lspprss.n | β’ π = (LSpanβπ) |
lspprss.w | β’ (π β π β LMod) |
lspprss.u | β’ (π β π β π) |
lspprss.x | β’ (π β π β π) |
lspprss.y | β’ (π β π β π) |
Ref | Expression |
---|---|
lspprss | β’ (π β (πβ{π, π}) β π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lspprss.w | . 2 β’ (π β π β LMod) | |
2 | lspprss.u | . 2 β’ (π β π β π) | |
3 | lspprss.x | . . 3 β’ (π β π β π) | |
4 | lspprss.y | . . 3 β’ (π β π β π) | |
5 | 3, 4 | prssd 4783 | . 2 β’ (π β {π, π} β π) |
6 | lspprss.s | . . 3 β’ π = (LSubSpβπ) | |
7 | lspprss.n | . . 3 β’ π = (LSpanβπ) | |
8 | 6, 7 | lspssp 20464 | . 2 β’ ((π β LMod β§ π β π β§ {π, π} β π) β (πβ{π, π}) β π) |
9 | 1, 2, 5, 8 | syl3anc 1372 | 1 β’ (π β (πβ{π, π}) β π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1542 β wcel 2107 β wss 3911 {cpr 4589 βcfv 6497 LModclmod 20336 LSubSpclss 20407 LSpanclspn 20447 |
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 |
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-rmo 3352 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-int 4909 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-riota 7314 df-ov 7361 df-0g 17328 df-mgm 18502 df-sgrp 18551 df-mnd 18562 df-grp 18756 df-lmod 20338 df-lss 20408 df-lsp 20448 |
This theorem is referenced by: lsppratlem2 20625 dvh3dim2 39957 dvh3dim3N 39958 lclkrlem2n 40029 |
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