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Mirrors > Home > ILE Home > Th. List > lspid | GIF version |
Description: The span of a subspace is itself. (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 2177 | . . . 4 β’ (Baseβπ) = (Baseβπ) | |
2 | lspid.s | . . . 4 β’ π = (LSubSpβπ) | |
3 | 1, 2 | lssssg 13485 | . . 3 β’ ((π β LMod β§ π β π) β π β (Baseβπ)) |
4 | lspid.n | . . . 4 β’ π = (LSpanβπ) | |
5 | 1, 2, 4 | lspval 13515 | . . 3 β’ ((π β LMod β§ π β (Baseβπ)) β (πβπ) = β© {π‘ β π β£ π β π‘}) |
6 | 3, 5 | syldan 282 | . 2 β’ ((π β LMod β§ π β π) β (πβπ) = β© {π‘ β π β£ π β π‘}) |
7 | intmin 3866 | . . 3 β’ (π β π β β© {π‘ β π β£ π β π‘} = π) | |
8 | 7 | adantl 277 | . 2 β’ ((π β LMod β§ π β π) β β© {π‘ β π β£ π β π‘} = π) |
9 | 6, 8 | eqtrd 2210 | 1 β’ ((π β LMod β§ π β π) β (πβπ) = π) |
Colors of variables: wff set class |
Syntax hints: β wi 4 β§ wa 104 = wceq 1353 β wcel 2148 {crab 2459 β wss 3131 β© cint 3846 βcfv 5218 Basecbs 12465 LModclmod 13415 LSubSpclss 13480 LSpanclspn 13511 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4120 ax-sep 4123 ax-pow 4176 ax-pr 4211 ax-un 4435 ax-cnex 7905 ax-resscn 7906 ax-1re 7908 ax-addrcl 7911 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2741 df-sbc 2965 df-csb 3060 df-un 3135 df-in 3137 df-ss 3144 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-int 3847 df-iun 3890 df-br 4006 df-opab 4067 df-mpt 4068 df-id 4295 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-rn 4639 df-res 4640 df-ima 4641 df-iota 5180 df-fun 5220 df-fn 5221 df-f 5222 df-f1 5223 df-fo 5224 df-f1o 5225 df-fv 5226 df-riota 5834 df-ov 5881 df-inn 8923 df-2 8981 df-3 8982 df-4 8983 df-5 8984 df-6 8985 df-ndx 12468 df-slot 12469 df-base 12471 df-plusg 12552 df-mulr 12553 df-sca 12555 df-vsca 12556 df-0g 12713 df-mgm 12782 df-sgrp 12815 df-mnd 12826 df-grp 12888 df-lmod 13417 df-lssm 13481 df-lsp 13512 |
This theorem is referenced by: lspidm 13526 lspssp 13528 lspsn0 13547 |
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