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Mirrors > Home > MPE Home > Th. List > lspid | Structured version Visualization version GIF version |
Description: The span of a subspace is itself. (spanid 30331 analog.) (Contributed by NM, 15-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.) |
Ref | Expression |
---|---|
lspid.s | β’ π = (LSubSpβπ) |
lspid.n | β’ π = (LSpanβπ) |
Ref | Expression |
---|---|
lspid | β’ ((π β LMod β§ π β π) β (πβπ) = π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2733 | . . . 4 β’ (Baseβπ) = (Baseβπ) | |
2 | lspid.s | . . . 4 β’ π = (LSubSpβπ) | |
3 | 1, 2 | lssss 20412 | . . 3 β’ (π β π β π β (Baseβπ)) |
4 | lspid.n | . . . 4 β’ π = (LSpanβπ) | |
5 | 1, 2, 4 | lspval 20451 | . . 3 β’ ((π β LMod β§ π β (Baseβπ)) β (πβπ) = β© {π‘ β π β£ π β π‘}) |
6 | 3, 5 | sylan2 594 | . 2 β’ ((π β LMod β§ π β π) β (πβπ) = β© {π‘ β π β£ π β π‘}) |
7 | intmin 4930 | . . 3 β’ (π β π β β© {π‘ β π β£ π β π‘} = π) | |
8 | 7 | adantl 483 | . 2 β’ ((π β LMod β§ π β π) β β© {π‘ β π β£ π β π‘} = π) |
9 | 6, 8 | eqtrd 2773 | 1 β’ ((π β LMod β§ π β π) β (πβπ) = π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 {crab 3406 β wss 3911 β© cint 4908 βcfv 6497 Basecbs 17088 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: lspidm 20462 lspssp 20464 lspsn0 20484 lspsolvlem 20619 lbsextlem3 20637 islshpsm 37488 lshpnel2N 37493 lssats 37520 lkrlsp3 37612 dochspocN 39889 dochsatshp 39960 filnm 41460 |
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