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Theorem lmod0vs 15657
Description: Zero times a vector is the zero vector. Equation 1a of [Kreyszig] p. 51. (ax-hvmul0 21582 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 445 . . . . 5  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  W  e.  LMod )
2 lmod0vs.f . . . . . . . 8  |-  F  =  (Scalar `  W )
32lmodrng 15629 . . . . . . 7  |-  ( W  e.  LMod  ->  F  e. 
Ring )
43adantr 453 . . . . . 6  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  F  e.  Ring )
5 eqid 2284 . . . . . . 7  |-  ( Base `  F )  =  (
Base `  F )
6 lmod0vs.o . . . . . . 7  |-  O  =  ( 0g `  F
)
75, 6rng0cl 15356 . . . . . 6  |-  ( F  e.  Ring  ->  O  e.  ( Base `  F
) )
84, 7syl 17 . . . . 5  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  O  e.  ( Base `  F
) )
9 simpr 449 . . . . 5  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  X  e.  V )
10 lmod0vs.v . . . . . 6  |-  V  =  ( Base `  W
)
11 eqid 2284 . . . . . 6  |-  ( +g  `  W )  =  ( +g  `  W )
12 lmod0vs.s . . . . . 6  |-  .x.  =  ( .s `  W )
13 eqid 2284 . . . . . 6  |-  ( +g  `  F )  =  ( +g  `  F )
1410, 11, 2, 12, 5, 13lmodvsdir 15646 . . . . 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 15340 . . . . . . 7  |-  ( F  e.  Ring  ->  F  e. 
Grp )
174, 16syl 17 . . . . . 6  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  F  e.  Grp )
185, 13, 6grplid 14506 . . . . . 6  |-  ( ( F  e.  Grp  /\  O  e.  ( Base `  F ) )  -> 
( O ( +g  `  F ) O )  =  O )
1917, 8, 18syl2anc 644 . . . . 5  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  ( O ( +g  `  F
) O )  =  O )
2019oveq1d 5834 . . . 4  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  (
( O ( +g  `  F ) O ) 
.x.  X )  =  ( O  .x.  X
) )
2115, 20eqtr3d 2318 . . 3  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  (
( O  .x.  X
) ( +g  `  W
) ( O  .x.  X ) )  =  ( O  .x.  X
) )
2210, 2, 12, 5lmodvscl 15638 . . . . 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 15656 . . . 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 458 . . 3  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  (
( ( O  .x.  X ) ( +g  `  W ) ( O 
.x.  X ) )  =  ( O  .x.  X )  <->  .0.  =  ( O  .x.  X ) ) )
2721, 26mpbid 203 . 2  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  .0.  =  ( O  .x.  X ) )
2827eqcomd 2289 1  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  ( O  .x.  X )  =  .0.  )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360    = wceq 1624    e. wcel 1685   ` cfv 5221  (class class class)co 5819   Basecbs 13142   +g cplusg 13202  Scalarcsca 13205   .scvsca 13206   0gc0g 13394   Grpcgrp 14356   Ringcrg 15331   LModclmod 15621
This theorem is referenced by:  lmodvs0  15658  lmodvneg1  15661  lvecvs0or  15855  lssvs0or  15857  lspsneleq  15862  lspdisj  15872  lspfixed  15875  lspexch  15876  lspsolvlem  15889  lspsolv  15890  mplcoe1  16203  mplbas2  16206  ply1scl0  16359  ply1coe  16362  clm0vs  18582  plypf1  19588  lcomfsup  26167  uvcresum  26641  frlmsslsp  26647  frlmup1  26649  frlmup2  26650  lshpkrlem1  28567  ldual0vs  28617  lclkrlem1  30963  lcd0vs  31072  baerlem3lem1  31164  baerlem5blem1  31166  hdmap14lem2a  31327  hdmap14lem4a  31331  hdmap14lem6  31333  hgmapval0  31352
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-reu 2551  df-rmo 2552  df-rab 2553  df-v 2791  df-sbc 2993  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-br 4025  df-opab 4079  df-mpt 4080  df-id 4308  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fv 5229  df-ov 5822  df-iota 6252  df-riota 6299  df-0g 13398  df-mnd 14361  df-grp 14483  df-rng 15334  df-lmod 15623
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