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Mirrors > Home > MPE Home > Th. List > lmodvs0 | Structured version Visualization version GIF version |
Description: Anything times the zero vector is the zero vector. Equation 1b of [Kreyszig] p. 51. (hvmul0 30015 analog.) (Contributed by NM, 12-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) |
Ref | Expression |
---|---|
lmodvs0.f | β’ πΉ = (Scalarβπ) |
lmodvs0.s | β’ Β· = ( Β·π βπ) |
lmodvs0.k | β’ πΎ = (BaseβπΉ) |
lmodvs0.z | β’ 0 = (0gβπ) |
Ref | Expression |
---|---|
lmodvs0 | β’ ((π β LMod β§ π β πΎ) β (π Β· 0 ) = 0 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lmodvs0.f | . . . . 5 β’ πΉ = (Scalarβπ) | |
2 | 1 | lmodring 20373 | . . . 4 β’ (π β LMod β πΉ β Ring) |
3 | lmodvs0.k | . . . . 5 β’ πΎ = (BaseβπΉ) | |
4 | eqid 2733 | . . . . 5 β’ (.rβπΉ) = (.rβπΉ) | |
5 | eqid 2733 | . . . . 5 β’ (0gβπΉ) = (0gβπΉ) | |
6 | 3, 4, 5 | ringrz 20020 | . . . 4 β’ ((πΉ β Ring β§ π β πΎ) β (π(.rβπΉ)(0gβπΉ)) = (0gβπΉ)) |
7 | 2, 6 | sylan 581 | . . 3 β’ ((π β LMod β§ π β πΎ) β (π(.rβπΉ)(0gβπΉ)) = (0gβπΉ)) |
8 | 7 | oveq1d 7376 | . 2 β’ ((π β LMod β§ π β πΎ) β ((π(.rβπΉ)(0gβπΉ)) Β· 0 ) = ((0gβπΉ) Β· 0 )) |
9 | simpl 484 | . . . 4 β’ ((π β LMod β§ π β πΎ) β π β LMod) | |
10 | simpr 486 | . . . 4 β’ ((π β LMod β§ π β πΎ) β π β πΎ) | |
11 | 2 | adantr 482 | . . . . 5 β’ ((π β LMod β§ π β πΎ) β πΉ β Ring) |
12 | 3, 5 | ring0cl 19998 | . . . . 5 β’ (πΉ β Ring β (0gβπΉ) β πΎ) |
13 | 11, 12 | syl 17 | . . . 4 β’ ((π β LMod β§ π β πΎ) β (0gβπΉ) β πΎ) |
14 | eqid 2733 | . . . . . 6 β’ (Baseβπ) = (Baseβπ) | |
15 | lmodvs0.z | . . . . . 6 β’ 0 = (0gβπ) | |
16 | 14, 15 | lmod0vcl 20395 | . . . . 5 β’ (π β LMod β 0 β (Baseβπ)) |
17 | 16 | adantr 482 | . . . 4 β’ ((π β LMod β§ π β πΎ) β 0 β (Baseβπ)) |
18 | lmodvs0.s | . . . . 5 β’ Β· = ( Β·π βπ) | |
19 | 14, 1, 18, 3, 4 | lmodvsass 20391 | . . . 4 β’ ((π β LMod β§ (π β πΎ β§ (0gβπΉ) β πΎ β§ 0 β (Baseβπ))) β ((π(.rβπΉ)(0gβπΉ)) Β· 0 ) = (π Β· ((0gβπΉ) Β· 0 ))) |
20 | 9, 10, 13, 17, 19 | syl13anc 1373 | . . 3 β’ ((π β LMod β§ π β πΎ) β ((π(.rβπΉ)(0gβπΉ)) Β· 0 ) = (π Β· ((0gβπΉ) Β· 0 ))) |
21 | 14, 1, 18, 5, 15 | lmod0vs 20399 | . . . . 5 β’ ((π β LMod β§ 0 β (Baseβπ)) β ((0gβπΉ) Β· 0 ) = 0 ) |
22 | 17, 21 | syldan 592 | . . . 4 β’ ((π β LMod β§ π β πΎ) β ((0gβπΉ) Β· 0 ) = 0 ) |
23 | 22 | oveq2d 7377 | . . 3 β’ ((π β LMod β§ π β πΎ) β (π Β· ((0gβπΉ) Β· 0 )) = (π Β· 0 )) |
24 | 20, 23 | eqtrd 2773 | . 2 β’ ((π β LMod β§ π β πΎ) β ((π(.rβπΉ)(0gβπΉ)) Β· 0 ) = (π Β· 0 )) |
25 | 8, 24, 22 | 3eqtr3d 2781 | 1 β’ ((π β LMod β§ π β πΎ) β (π Β· 0 ) = 0 ) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 βcfv 6500 (class class class)co 7361 Basecbs 17091 .rcmulr 17142 Scalarcsca 17144 Β·π cvsca 17145 0gc0g 17329 Ringcrg 19972 LModclmod 20365 |
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 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 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-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7807 df-2nd 7926 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-er 8654 df-en 8890 df-dom 8891 df-sdom 8892 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-le 11203 df-sub 11395 df-neg 11396 df-nn 12162 df-2 12224 df-sets 17044 df-slot 17062 df-ndx 17074 df-base 17092 df-plusg 17154 df-0g 17331 df-mgm 18505 df-sgrp 18554 df-mnd 18565 df-grp 18759 df-mgp 19905 df-ring 19974 df-lmod 20367 |
This theorem is referenced by: lmodfopne 20404 lsssn0 20452 lmodvsinv2 20542 0lmhm 20545 lvecvs0or 20614 dsmmlss 21173 pmatcollpwfi 22154 pmatcollpw3fi1lem1 22158 pm2mp 22197 chfacfscmul0 22230 ttgbtwnid 27881 0nellinds 32213 lcdvs0N 40129 hdmap14lem13 40393 lmodvsmdi 46548 linc0scn0 46594 |
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