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Mirrors > Home > MPE Home > Th. List > lsscl | Structured version Visualization version GIF version |
Description: Closure property of a subspace. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 8-Jan-2015.) |
Ref | Expression |
---|---|
lsscl.f | β’ πΉ = (Scalarβπ) |
lsscl.b | β’ π΅ = (BaseβπΉ) |
lsscl.p | β’ + = (+gβπ) |
lsscl.t | β’ Β· = ( Β·π βπ) |
lsscl.s | β’ π = (LSubSpβπ) |
Ref | Expression |
---|---|
lsscl | β’ ((π β π β§ (π β π΅ β§ π β π β§ π β π)) β ((π Β· π) + π) β π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lsscl.f | . . . 4 β’ πΉ = (Scalarβπ) | |
2 | lsscl.b | . . . 4 β’ π΅ = (BaseβπΉ) | |
3 | eqid 2733 | . . . 4 β’ (Baseβπ) = (Baseβπ) | |
4 | lsscl.p | . . . 4 β’ + = (+gβπ) | |
5 | lsscl.t | . . . 4 β’ Β· = ( Β·π βπ) | |
6 | lsscl.s | . . . 4 β’ π = (LSubSpβπ) | |
7 | 1, 2, 3, 4, 5, 6 | islss 20410 | . . 3 β’ (π β π β (π β (Baseβπ) β§ π β β β§ βπ₯ β π΅ βπ β π βπ β π ((π₯ Β· π) + π) β π)) |
8 | 7 | simp3bi 1148 | . 2 β’ (π β π β βπ₯ β π΅ βπ β π βπ β π ((π₯ Β· π) + π) β π) |
9 | oveq1 7365 | . . . . 5 β’ (π₯ = π β (π₯ Β· π) = (π Β· π)) | |
10 | 9 | oveq1d 7373 | . . . 4 β’ (π₯ = π β ((π₯ Β· π) + π) = ((π Β· π) + π)) |
11 | 10 | eleq1d 2819 | . . 3 β’ (π₯ = π β (((π₯ Β· π) + π) β π β ((π Β· π) + π) β π)) |
12 | oveq2 7366 | . . . . 5 β’ (π = π β (π Β· π) = (π Β· π)) | |
13 | 12 | oveq1d 7373 | . . . 4 β’ (π = π β ((π Β· π) + π) = ((π Β· π) + π)) |
14 | 13 | eleq1d 2819 | . . 3 β’ (π = π β (((π Β· π) + π) β π β ((π Β· π) + π) β π)) |
15 | oveq2 7366 | . . . 4 β’ (π = π β ((π Β· π) + π) = ((π Β· π) + π)) | |
16 | 15 | eleq1d 2819 | . . 3 β’ (π = π β (((π Β· π) + π) β π β ((π Β· π) + π) β π)) |
17 | 11, 14, 16 | rspc3v 3592 | . 2 β’ ((π β π΅ β§ π β π β§ π β π) β (βπ₯ β π΅ βπ β π βπ β π ((π₯ Β· π) + π) β π β ((π Β· π) + π) β π)) |
18 | 8, 17 | mpan9 508 | 1 β’ ((π β π β§ (π β π΅ β§ π β π β§ π β π)) β ((π Β· π) + π) β π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 β§ w3a 1088 = wceq 1542 β wcel 2107 β wne 2940 βwral 3061 β wss 3911 β c0 4283 βcfv 6497 (class class class)co 7358 Basecbs 17088 +gcplusg 17138 Scalarcsca 17141 Β·π cvsca 17142 LSubSpclss 20407 |
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-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-rab 3407 df-v 3446 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-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-iota 6449 df-fun 6499 df-fv 6505 df-ov 7361 df-lss 20408 |
This theorem is referenced by: lssvsubcl 20419 lssvacl 20430 lssvscl 20431 islss3 20435 lssintcl 20440 lspsolvlem 20619 lbsextlem2 20636 isphld 21074 |
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