![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > lmod0vs | Structured version Visualization version GIF version |
Description: Zero times a vector is the zero vector. Equation 1a of [Kreyszig] p. 51. (ax-hvmul0 29994 analog.) (Contributed by NM, 12-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) |
Ref | Expression |
---|---|
lmod0vs.v | β’ π = (Baseβπ) |
lmod0vs.f | β’ πΉ = (Scalarβπ) |
lmod0vs.s | β’ Β· = ( Β·π βπ) |
lmod0vs.o | β’ π = (0gβπΉ) |
lmod0vs.z | β’ 0 = (0gβπ) |
Ref | Expression |
---|---|
lmod0vs | β’ ((π β LMod β§ π β π) β (π Β· π) = 0 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 484 | . . . . 5 β’ ((π β LMod β§ π β π) β π β LMod) | |
2 | lmod0vs.f | . . . . . . . 8 β’ πΉ = (Scalarβπ) | |
3 | 2 | lmodring 20344 | . . . . . . 7 β’ (π β LMod β πΉ β Ring) |
4 | 3 | adantr 482 | . . . . . 6 β’ ((π β LMod β§ π β π) β πΉ β Ring) |
5 | eqid 2733 | . . . . . . 7 β’ (BaseβπΉ) = (BaseβπΉ) | |
6 | lmod0vs.o | . . . . . . 7 β’ π = (0gβπΉ) | |
7 | 5, 6 | ring0cl 19995 | . . . . . 6 β’ (πΉ β Ring β π β (BaseβπΉ)) |
8 | 4, 7 | syl 17 | . . . . 5 β’ ((π β LMod β§ π β π) β π β (BaseβπΉ)) |
9 | simpr 486 | . . . . 5 β’ ((π β LMod β§ π β π) β π β π) | |
10 | lmod0vs.v | . . . . . 6 β’ π = (Baseβπ) | |
11 | eqid 2733 | . . . . . 6 β’ (+gβπ) = (+gβπ) | |
12 | lmod0vs.s | . . . . . 6 β’ Β· = ( Β·π βπ) | |
13 | eqid 2733 | . . . . . 6 β’ (+gβπΉ) = (+gβπΉ) | |
14 | 10, 11, 2, 12, 5, 13 | lmodvsdir 20361 | . . . . 5 β’ ((π β LMod β§ (π β (BaseβπΉ) β§ π β (BaseβπΉ) β§ π β π)) β ((π(+gβπΉ)π) Β· π) = ((π Β· π)(+gβπ)(π Β· π))) |
15 | 1, 8, 8, 9, 14 | syl13anc 1373 | . . . 4 β’ ((π β LMod β§ π β π) β ((π(+gβπΉ)π) Β· π) = ((π Β· π)(+gβπ)(π Β· π))) |
16 | ringgrp 19974 | . . . . . . 7 β’ (πΉ β Ring β πΉ β Grp) | |
17 | 4, 16 | syl 17 | . . . . . 6 β’ ((π β LMod β§ π β π) β πΉ β Grp) |
18 | 5, 13, 6 | grplid 18785 | . . . . . 6 β’ ((πΉ β Grp β§ π β (BaseβπΉ)) β (π(+gβπΉ)π) = π) |
19 | 17, 8, 18 | syl2anc 585 | . . . . 5 β’ ((π β LMod β§ π β π) β (π(+gβπΉ)π) = π) |
20 | 19 | oveq1d 7373 | . . . 4 β’ ((π β LMod β§ π β π) β ((π(+gβπΉ)π) Β· π) = (π Β· π)) |
21 | 15, 20 | eqtr3d 2775 | . . 3 β’ ((π β LMod β§ π β π) β ((π Β· π)(+gβπ)(π Β· π)) = (π Β· π)) |
22 | 10, 2, 12, 5 | lmodvscl 20354 | . . . . 5 β’ ((π β LMod β§ π β (BaseβπΉ) β§ π β π) β (π Β· π) β π) |
23 | 1, 8, 9, 22 | syl3anc 1372 | . . . 4 β’ ((π β LMod β§ π β π) β (π Β· π) β π) |
24 | lmod0vs.z | . . . . 5 β’ 0 = (0gβπ) | |
25 | 10, 11, 24 | lmod0vid 20369 | . . . 4 β’ ((π β LMod β§ (π Β· π) β π) β (((π Β· π)(+gβπ)(π Β· π)) = (π Β· π) β 0 = (π Β· π))) |
26 | 23, 25 | syldan 592 | . . 3 β’ ((π β LMod β§ π β π) β (((π Β· π)(+gβπ)(π Β· π)) = (π Β· π) β 0 = (π Β· π))) |
27 | 21, 26 | mpbid 231 | . 2 β’ ((π β LMod β§ π β π) β 0 = (π Β· π)) |
28 | 27 | eqcomd 2739 | 1 β’ ((π β LMod β§ π β π) β (π Β· π) = 0 ) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 397 = wceq 1542 β wcel 2107 βcfv 6497 (class class class)co 7358 Basecbs 17088 +gcplusg 17138 Scalarcsca 17141 Β·π cvsca 17142 0gc0g 17326 Grpcgrp 18753 Ringcrg 19969 LModclmod 20336 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pr 5385 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-iota 6449 df-fun 6499 df-fv 6505 df-riota 7314 df-ov 7361 df-0g 17328 df-mgm 18502 df-sgrp 18551 df-mnd 18562 df-grp 18756 df-ring 19971 df-lmod 20338 |
This theorem is referenced by: lmodvs0 20371 lmodvsmmulgdi 20372 lcomfsupp 20377 lmodvneg1 20380 mptscmfsupp0 20402 lvecvs0or 20585 lssvs0or 20587 lspsneleq 20592 lspdisj 20602 lspfixed 20605 lspexch 20606 lspsolvlem 20619 lspsolv 20620 uvcresum 21215 frlmsslsp 21218 frlmup1 21220 frlmup2 21221 ascl0 21303 mplcoe1 21454 mplbas2 21459 ply10s0 21643 ply1scl0 21677 gsummoncoe1 21691 pmatcollpwscmatlem1 22154 idpm2idmp 22166 mp2pm2mplem4 22174 pm2mpmhmlem1 22183 monmat2matmon 22189 cpmidpmatlem3 22237 clm0vs 24474 plypf1 25589 lmodslmd 32088 evls1fpws 32320 ply1degltdimlem 32374 lbsdiflsp0 32378 fedgmullem2 32382 lshpkrlem1 37618 ldual0vs 37668 lclkrlem1 40015 lcd0vs 40124 baerlem3lem1 40216 baerlem5blem1 40218 hdmap14lem2a 40376 hdmap14lem4a 40380 hdmap14lem6 40382 hgmapval0 40401 prjspersym 40988 prjspreln0 40990 prjspner1 41007 lmod0rng 46252 scmsuppss 46534 lmodvsmdi 46544 ply1mulgsumlem4 46556 lincval1 46586 lincvalsc0 46588 linc0scn0 46590 linc1 46592 ldepsprlem 46639 |
Copyright terms: Public domain | W3C validator |