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 2189 | . 2 β’ (LSubSpβπ) = (LSubSpβπ) | |
| 2 | lspsn.n | . 2 β’ π = (LSpanβπ) | |
| 3 | simp1 999 | . 2 β’ ((π β LMod β§ π β πΎ β§ π β π) β π β LMod) | |
| 4 | simp3 1001 | . . . 4 β’ ((π β LMod β§ π β πΎ β§ π β π) β π β π) | |
| 5 | 4 | snssd 3752 | . . 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 13668 | . . 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 1000 | . . 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 13692 | . 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 13687 | 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 980 = wceq 1364 β wcel 2160 β wss 3144 {csn 3607 βcfv 5231 (class class class)co 5891 Basecbs 12480 Scalarcsca 12558 Β·π cvsca 12559 LModclmod 13564 LSubSpclss 13629 LSpanclspn 13663 |
| 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 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-coll 4133 ax-sep 4136 ax-pow 4189 ax-pr 4224 ax-un 4448 ax-setind 4551 ax-cnex 7920 ax-resscn 7921 ax-1cn 7922 ax-1re 7923 ax-icn 7924 ax-addcl 7925 ax-addrcl 7926 ax-mulcl 7927 ax-addcom 7929 ax-addass 7931 ax-i2m1 7934 ax-0lt1 7935 ax-0id 7937 ax-rnegex 7938 ax-pre-ltirr 7941 ax-pre-ltadd 7945 |
| This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rmo 2476 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-iun 3903 df-br 4019 df-opab 4080 df-mpt 4081 df-id 4308 df-xp 4647 df-rel 4648 df-cnv 4649 df-co 4650 df-dm 4651 df-rn 4652 df-res 4653 df-ima 4654 df-iota 5193 df-fun 5233 df-fn 5234 df-f 5235 df-f1 5236 df-fo 5237 df-f1o 5238 df-fv 5239 df-riota 5847 df-ov 5894 df-oprab 5895 df-mpo 5896 df-1st 6159 df-2nd 6160 df-pnf 8012 df-mnf 8013 df-ltxr 8015 df-inn 8938 df-2 8996 df-3 8997 df-4 8998 df-5 8999 df-6 9000 df-ndx 12483 df-slot 12484 df-base 12486 df-sets 12487 df-plusg 12568 df-mulr 12569 df-sca 12571 df-vsca 12572 df-0g 12729 df-mgm 12798 df-sgrp 12831 df-mnd 12844 df-grp 12914 df-minusg 12915 df-sbg 12916 df-mgp 13236 df-ur 13275 df-ring 13313 df-lmod 13566 df-lssm 13630 df-lsp 13664 |
| This theorem is referenced by: lspsnneg 13697 |
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