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Mirrors > Home > ILE Home > Th. List > lspidm | GIF version |
Description: The span of a set of vectors is idempotent. (Contributed by NM, 22-Feb-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) |
Ref | Expression |
---|---|
lspss.v | β’ π = (Baseβπ) |
lspss.n | β’ π = (LSpanβπ) |
Ref | Expression |
---|---|
lspidm | β’ ((π β LMod β§ π β π) β (πβ(πβπ)) = (πβπ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lspss.v | . . 3 β’ π = (Baseβπ) | |
2 | eqid 2188 | . . 3 β’ (LSubSpβπ) = (LSubSpβπ) | |
3 | lspss.n | . . 3 β’ π = (LSpanβπ) | |
4 | 1, 2, 3 | lspcl 13667 | . 2 β’ ((π β LMod β§ π β π) β (πβπ) β (LSubSpβπ)) |
5 | 2, 3 | lspid 13673 | . 2 β’ ((π β LMod β§ (πβπ) β (LSubSpβπ)) β (πβ(πβπ)) = (πβπ)) |
6 | 4, 5 | syldan 282 | 1 β’ ((π β LMod β§ π β π) β (πβ(πβπ)) = (πβπ)) |
Colors of variables: wff set class |
Syntax hints: β wi 4 β§ wa 104 = wceq 1363 β wcel 2159 β wss 3143 βcfv 5230 Basecbs 12479 LModclmod 13563 LSubSpclss 13628 LSpanclspn 13662 |
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-in1 615 ax-in2 616 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2161 ax-14 2162 ax-ext 2170 ax-coll 4132 ax-sep 4135 ax-pow 4188 ax-pr 4223 ax-un 4447 ax-setind 4550 ax-cnex 7919 ax-resscn 7920 ax-1cn 7921 ax-1re 7922 ax-icn 7923 ax-addcl 7924 ax-addrcl 7925 ax-mulcl 7926 ax-addcom 7928 ax-addass 7930 ax-i2m1 7933 ax-0lt1 7934 ax-0id 7936 ax-rnegex 7937 ax-pre-ltirr 7940 ax-pre-ltadd 7944 |
This theorem depends on definitions: df-bi 117 df-3an 981 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-eu 2040 df-mo 2041 df-clab 2175 df-cleq 2181 df-clel 2184 df-nfc 2320 df-ne 2360 df-nel 2455 df-ral 2472 df-rex 2473 df-reu 2474 df-rmo 2475 df-rab 2476 df-v 2753 df-sbc 2977 df-csb 3072 df-dif 3145 df-un 3147 df-in 3149 df-ss 3156 df-nul 3437 df-pw 3591 df-sn 3612 df-pr 3613 df-op 3615 df-uni 3824 df-int 3859 df-iun 3902 df-br 4018 df-opab 4079 df-mpt 4080 df-id 4307 df-xp 4646 df-rel 4647 df-cnv 4648 df-co 4649 df-dm 4650 df-rn 4651 df-res 4652 df-ima 4653 df-iota 5192 df-fun 5232 df-fn 5233 df-f 5234 df-f1 5235 df-fo 5236 df-f1o 5237 df-fv 5238 df-riota 5846 df-ov 5893 df-oprab 5894 df-mpo 5895 df-1st 6158 df-2nd 6159 df-pnf 8011 df-mnf 8012 df-ltxr 8014 df-inn 8937 df-2 8995 df-3 8996 df-4 8997 df-5 8998 df-6 8999 df-ndx 12482 df-slot 12483 df-base 12485 df-sets 12486 df-plusg 12567 df-mulr 12568 df-sca 12570 df-vsca 12571 df-0g 12728 df-mgm 12797 df-sgrp 12830 df-mnd 12843 df-grp 12913 df-minusg 12914 df-sbg 12915 df-mgp 13235 df-ur 13274 df-ring 13312 df-lmod 13565 df-lssm 13629 df-lsp 13663 |
This theorem is referenced by: lspun 13678 lspun0 13701 |
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