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Theorem lmod0vs 15946
Description: Zero times a vector is the zero vector. Equation 1a of [Kreyszig] p. 51. (ax-hvmul0 22474 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 444 . . . . 5  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  W  e.  LMod )
2 lmod0vs.f . . . . . . . 8  |-  F  =  (Scalar `  W )
32lmodrng 15921 . . . . . . 7  |-  ( W  e.  LMod  ->  F  e. 
Ring )
43adantr 452 . . . . . 6  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  F  e.  Ring )
5 eqid 2412 . . . . . . 7  |-  ( Base `  F )  =  (
Base `  F )
6 lmod0vs.o . . . . . . 7  |-  O  =  ( 0g `  F
)
75, 6rng0cl 15648 . . . . . 6  |-  ( F  e.  Ring  ->  O  e.  ( Base `  F
) )
84, 7syl 16 . . . . 5  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  O  e.  ( Base `  F
) )
9 simpr 448 . . . . 5  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  X  e.  V )
10 lmod0vs.v . . . . . 6  |-  V  =  ( Base `  W
)
11 eqid 2412 . . . . . 6  |-  ( +g  `  W )  =  ( +g  `  W )
12 lmod0vs.s . . . . . 6  |-  .x.  =  ( .s `  W )
13 eqid 2412 . . . . . 6  |-  ( +g  `  F )  =  ( +g  `  F )
1410, 11, 2, 12, 5, 13lmodvsdir 15937 . . . . 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 1186 . . . 4  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  (
( O ( +g  `  F ) O ) 
.x.  X )  =  ( ( O  .x.  X ) ( +g  `  W ) ( O 
.x.  X ) ) )
16 rnggrp 15632 . . . . . . 7  |-  ( F  e.  Ring  ->  F  e. 
Grp )
174, 16syl 16 . . . . . 6  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  F  e.  Grp )
185, 13, 6grplid 14798 . . . . . 6  |-  ( ( F  e.  Grp  /\  O  e.  ( Base `  F ) )  -> 
( O ( +g  `  F ) O )  =  O )
1917, 8, 18syl2anc 643 . . . . 5  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  ( O ( +g  `  F
) O )  =  O )
2019oveq1d 6063 . . . 4  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  (
( O ( +g  `  F ) O ) 
.x.  X )  =  ( O  .x.  X
) )
2115, 20eqtr3d 2446 . . 3  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  (
( O  .x.  X
) ( +g  `  W
) ( O  .x.  X ) )  =  ( O  .x.  X
) )
2210, 2, 12, 5lmodvscl 15930 . . . . 5  |-  ( ( W  e.  LMod  /\  O  e.  ( Base `  F
)  /\  X  e.  V )  ->  ( O  .x.  X )  e.  V )
231, 8, 9, 22syl3anc 1184 . . . 4  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  ( O  .x.  X )  e.  V )
24 lmod0vs.z . . . . 5  |-  .0.  =  ( 0g `  W )
2510, 11, 24lmod0vid 15945 . . . 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 457 . . 3  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  (
( ( O  .x.  X ) ( +g  `  W ) ( O 
.x.  X ) )  =  ( O  .x.  X )  <->  .0.  =  ( O  .x.  X ) ) )
2721, 26mpbid 202 . 2  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  .0.  =  ( O  .x.  X ) )
2827eqcomd 2417 1  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  ( O  .x.  X )  =  .0.  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721   ` cfv 5421  (class class class)co 6048   Basecbs 13432   +g cplusg 13492  Scalarcsca 13495   .scvsca 13496   0gc0g 13686   Grpcgrp 14648   Ringcrg 15623   LModclmod 15913
This theorem is referenced by:  lmodvs0  15947  lmodvneg1  15950  lvecvs0or  16143  lssvs0or  16145  lspsneleq  16150  lspdisj  16160  lspfixed  16163  lspexch  16164  lspsolvlem  16177  lspsolv  16178  mplcoe1  16491  mplbas2  16494  ply1scl0  16644  ply1coe  16647  clm0vs  19076  plypf1  20092  lcomfsup  26645  uvcresum  27118  frlmsslsp  27124  frlmup1  27126  frlmup2  27127  lshpkrlem1  29605  ldual0vs  29655  lclkrlem1  32001  lcd0vs  32110  baerlem3lem1  32202  baerlem5blem1  32204  hdmap14lem2a  32365  hdmap14lem4a  32369  hdmap14lem6  32371  hgmapval0  32390
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-iota 5385  df-fun 5423  df-fv 5429  df-ov 6051  df-riota 6516  df-0g 13690  df-mnd 14653  df-grp 14775  df-rng 15626  df-lmod 15915
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