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Theorem lmod0vs 15663
Description: Zero times a vector is the zero vector. Equation 1a of [Kreyszig] p. 51. (ax-hvmul0 21590 analog.) (Contributed by NM, 12-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
Hypotheses
Ref Expression
lmod0vs.v  |-  V  =  ( Base `  W
)
lmod0vs.f  |-  F  =  (Scalar `  W )
lmod0vs.s  |-  .x.  =  ( .s `  W )
lmod0vs.o  |-  O  =  ( 0g `  F
)
lmod0vs.z  |-  .0.  =  ( 0g `  W )
Assertion
Ref Expression
lmod0vs  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  ( O  .x.  X )  =  .0.  )

Proof of Theorem lmod0vs
StepHypRef Expression
1 simpl 443 . . . . 5  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  W  e.  LMod )
2 lmod0vs.f . . . . . . . 8  |-  F  =  (Scalar `  W )
32lmodrng 15635 . . . . . . 7  |-  ( W  e.  LMod  ->  F  e. 
Ring )
43adantr 451 . . . . . 6  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  F  e.  Ring )
5 eqid 2283 . . . . . . 7  |-  ( Base `  F )  =  (
Base `  F )
6 lmod0vs.o . . . . . . 7  |-  O  =  ( 0g `  F
)
75, 6rng0cl 15362 . . . . . 6  |-  ( F  e.  Ring  ->  O  e.  ( Base `  F
) )
84, 7syl 15 . . . . 5  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  O  e.  ( Base `  F
) )
9 simpr 447 . . . . 5  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  X  e.  V )
10 lmod0vs.v . . . . . 6  |-  V  =  ( Base `  W
)
11 eqid 2283 . . . . . 6  |-  ( +g  `  W )  =  ( +g  `  W )
12 lmod0vs.s . . . . . 6  |-  .x.  =  ( .s `  W )
13 eqid 2283 . . . . . 6  |-  ( +g  `  F )  =  ( +g  `  F )
1410, 11, 2, 12, 5, 13lmodvsdir 15652 . . . . 5  |-  ( ( W  e.  LMod  /\  ( O  e.  ( Base `  F )  /\  O  e.  ( Base `  F
)  /\  X  e.  V ) )  -> 
( ( O ( +g  `  F ) O )  .x.  X
)  =  ( ( O  .x.  X ) ( +g  `  W
) ( O  .x.  X ) ) )
151, 8, 8, 9, 14syl13anc 1184 . . . 4  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  (
( O ( +g  `  F ) O ) 
.x.  X )  =  ( ( O  .x.  X ) ( +g  `  W ) ( O 
.x.  X ) ) )
16 rnggrp 15346 . . . . . . 7  |-  ( F  e.  Ring  ->  F  e. 
Grp )
174, 16syl 15 . . . . . 6  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  F  e.  Grp )
185, 13, 6grplid 14512 . . . . . 6  |-  ( ( F  e.  Grp  /\  O  e.  ( Base `  F ) )  -> 
( O ( +g  `  F ) O )  =  O )
1917, 8, 18syl2anc 642 . . . . 5  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  ( O ( +g  `  F
) O )  =  O )
2019oveq1d 5873 . . . 4  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  (
( O ( +g  `  F ) O ) 
.x.  X )  =  ( O  .x.  X
) )
2115, 20eqtr3d 2317 . . 3  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  (
( O  .x.  X
) ( +g  `  W
) ( O  .x.  X ) )  =  ( O  .x.  X
) )
2210, 2, 12, 5lmodvscl 15644 . . . . 5  |-  ( ( W  e.  LMod  /\  O  e.  ( Base `  F
)  /\  X  e.  V )  ->  ( O  .x.  X )  e.  V )
231, 8, 9, 22syl3anc 1182 . . . 4  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  ( O  .x.  X )  e.  V )
24 lmod0vs.z . . . . 5  |-  .0.  =  ( 0g `  W )
2510, 11, 24lmod0vid 15662 . . . 4  |-  ( ( W  e.  LMod  /\  ( O  .x.  X )  e.  V )  ->  (
( ( O  .x.  X ) ( +g  `  W ) ( O 
.x.  X ) )  =  ( O  .x.  X )  <->  .0.  =  ( O  .x.  X ) ) )
2623, 25syldan 456 . . 3  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  (
( ( O  .x.  X ) ( +g  `  W ) ( O 
.x.  X ) )  =  ( O  .x.  X )  <->  .0.  =  ( O  .x.  X ) ) )
2721, 26mpbid 201 . 2  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  .0.  =  ( O  .x.  X ) )
2827eqcomd 2288 1  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  ( O  .x.  X )  =  .0.  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   ` cfv 5255  (class class class)co 5858   Basecbs 13148   +g cplusg 13208  Scalarcsca 13211   .scvsca 13212   0gc0g 13400   Grpcgrp 14362   Ringcrg 15337   LModclmod 15627
This theorem is referenced by:  lmodvs0  15664  lmodvneg1  15667  lvecvs0or  15861  lssvs0or  15863  lspsneleq  15868  lspdisj  15878  lspfixed  15881  lspexch  15882  lspsolvlem  15895  lspsolv  15896  mplcoe1  16209  mplbas2  16212  ply1scl0  16365  ply1coe  16368  clm0vs  18588  plypf1  19594  lcomfsup  26768  uvcresum  27242  frlmsslsp  27248  frlmup1  27250  frlmup2  27251  lshpkrlem1  29300  ldual0vs  29350  lclkrlem1  31696  lcd0vs  31805  baerlem3lem1  31897  baerlem5blem1  31899  hdmap14lem2a  32060  hdmap14lem4a  32064  hdmap14lem6  32066  hgmapval0  32085
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-riota 6304  df-0g 13404  df-mnd 14367  df-grp 14489  df-rng 15340  df-lmod 15629
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