Nearby theorems
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| Mirrors > Home > ILE Home > Th. List > lspsnvsi | GIF version | ||
| Description: Span of a scalar product of a singleton. (Contributed by NM, 23-Apr-2014.) (Proof shortened by Mario Carneiro, 4-Sep-2014.) |
| Ref | Expression |
|---|---|
| lspsn.f | β’ πΉ = (Scalarβπ) |
| lspsn.k | β’ πΎ = (BaseβπΉ) |
| lspsn.v | β’ π = (Baseβπ) |
| lspsn.t | β’ Β· = ( Β·π βπ) |
| lspsn.n | β’ π = (LSpanβπ) |
| Ref | Expression |
|---|---|
| lspsnvsi | β’ ((π β LMod β§ π β πΎ β§ π β π) β (πβ{(π Β· π)}) β (πβ{π})) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | eqid 2187 | . 2 β’ (LSubSpβπ) = (LSubSpβπ) | |
| 2 | lspsn.n | . 2 β’ π = (LSpanβπ) | |
| 3 | simp1 998 | . 2 β’ ((π β LMod β§ π β πΎ β§ π β π) β π β LMod) | |
| 4 | simp3 1000 | . . . 4 β’ ((π β LMod β§ π β πΎ β§ π β π) β π β π) | |
| 5 | 4 | snssd 3749 | . . 3 β’ ((π β LMod β§ π β πΎ β§ π β π) β {π} β π) |
| 6 | lspsn.v | . . . 4 β’ π = (Baseβπ) | |
| 7 | 6, 1, 2 | lspcl 13543 | . . 3 β’ ((π β LMod β§ {π} β π) β (πβ{π}) β (LSubSpβπ)) |
| 8 | 3, 5, 7 | syl2anc 411 | . 2 β’ ((π β LMod β§ π β πΎ β§ π β π) β (πβ{π}) β (LSubSpβπ)) |
| 9 | lspsn.t | . . 3 β’ Β· = ( Β·π βπ) | |
| 10 | lspsn.f | . . 3 β’ πΉ = (Scalarβπ) | |
| 11 | lspsn.k | . . 3 β’ πΎ = (BaseβπΉ) | |
| 12 | simp2 999 | . . 3 β’ ((π β LMod β§ π β πΎ β§ π β π) β π β πΎ) | |
| 13 | 6, 9, 10, 11, 2, 3, 12, 4 | lspsneli 13567 | . 2 β’ ((π β LMod β§ π β πΎ β§ π β π) β (π Β· π) β (πβ{π})) |
| 14 | 1, 2, 3, 8, 13 | lspsnel5a 13562 | 1 β’ ((π β LMod β§ π β πΎ β§ π β π) β (πβ{(π Β· π)}) β (πβ{π})) |
| Colors of variables: wff set class |
| Syntax hints: β wi 4 β§ w3a 979 = wceq 1363 β wcel 2158 β wss 3141 {csn 3604 βcfv 5228 (class class class)co 5888 Basecbs 12475 Scalarcsca 12553 Β·π cvsca 12554 LModclmod 13439 LSubSpclss 13504 LSpanclspn 13538 |
| 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 2160 ax-14 2161 ax-ext 2169 ax-coll 4130 ax-sep 4133 ax-pow 4186 ax-pr 4221 ax-un 4445 ax-setind 4548 ax-cnex 7915 ax-resscn 7916 ax-1cn 7917 ax-1re 7918 ax-icn 7919 ax-addcl 7920 ax-addrcl 7921 ax-mulcl 7922 ax-addcom 7924 ax-addass 7926 ax-i2m1 7929 ax-0lt1 7930 ax-0id 7932 ax-rnegex 7933 ax-pre-ltirr 7936 ax-pre-ltadd 7940 |
| 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 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ne 2358 df-nel 2453 df-ral 2470 df-rex 2471 df-reu 2472 df-rmo 2473 df-rab 2474 df-v 2751 df-sbc 2975 df-csb 3070 df-dif 3143 df-un 3145 df-in 3147 df-ss 3154 df-nul 3435 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-int 3857 df-iun 3900 df-br 4016 df-opab 4077 df-mpt 4078 df-id 4305 df-xp 4644 df-rel 4645 df-cnv 4646 df-co 4647 df-dm 4648 df-rn 4649 df-res 4650 df-ima 4651 df-iota 5190 df-fun 5230 df-fn 5231 df-f 5232 df-f1 5233 df-fo 5234 df-f1o 5235 df-fv 5236 df-riota 5844 df-ov 5891 df-oprab 5892 df-mpo 5893 df-1st 6154 df-2nd 6155 df-pnf 8007 df-mnf 8008 df-ltxr 8010 df-inn 8933 df-2 8991 df-3 8992 df-4 8993 df-5 8994 df-6 8995 df-ndx 12478 df-slot 12479 df-base 12481 df-sets 12482 df-plusg 12563 df-mulr 12564 df-sca 12566 df-vsca 12567 df-0g 12724 df-mgm 12793 df-sgrp 12826 df-mnd 12837 df-grp 12899 df-minusg 12900 df-sbg 12901 df-mgp 13163 df-ur 13197 df-ring 13235 df-lmod 13441 df-lssm 13505 df-lsp 13539 |
| This theorem is referenced by: lspsnneg 13572 |
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