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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 31150 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 2728 | . . . 4 β’ (Baseβπ) = (Baseβπ) | |
2 | lspid.s | . . . 4 β’ π = (LSubSpβπ) | |
3 | 1, 2 | lssss 20813 | . . 3 β’ (π β π β π β (Baseβπ)) |
4 | lspid.n | . . . 4 β’ π = (LSpanβπ) | |
5 | 1, 2, 4 | lspval 20852 | . . 3 β’ ((π β LMod β§ π β (Baseβπ)) β (πβπ) = β© {π‘ β π β£ π β π‘}) |
6 | 3, 5 | sylan2 592 | . 2 β’ ((π β LMod β§ π β π) β (πβπ) = β© {π‘ β π β£ π β π‘}) |
7 | intmin 4966 | . . 3 β’ (π β π β β© {π‘ β π β£ π β π‘} = π) | |
8 | 7 | adantl 481 | . 2 β’ ((π β LMod β§ π β π) β β© {π‘ β π β£ π β π‘} = π) |
9 | 6, 8 | eqtrd 2768 | 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 395 = wceq 1534 β wcel 2099 {crab 3428 β wss 3945 β© cint 4944 βcfv 6542 Basecbs 17173 LModclmod 20736 LSubSpclss 20808 LSpanclspn 20848 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-0g 17416 df-mgm 18593 df-sgrp 18672 df-mnd 18688 df-grp 18886 df-lmod 20738 df-lss 20809 df-lsp 20849 |
This theorem is referenced by: lspidm 20863 lspssp 20865 lspsn0 20885 lspsolvlem 21023 lbsextlem3 21041 islshpsm 38446 lshpnel2N 38451 lssats 38478 lkrlsp3 38570 dochspocN 40847 dochsatshp 40918 filnm 42508 |
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