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Theorem lmod0vs 14083
Description: Zero times a vector is the zero vector. Equation 1a of [Kreyszig] p. 51. (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 109 . . . . 5  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  W  e.  LMod )
2 lmod0vs.f . . . . . . . 8  |-  F  =  (Scalar `  W )
32lmodring 14057 . . . . . . 7  |-  ( W  e.  LMod  ->  F  e. 
Ring )
43adantr 276 . . . . . 6  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  F  e.  Ring )
5 eqid 2205 . . . . . . 7  |-  ( Base `  F )  =  (
Base `  F )
6 lmod0vs.o . . . . . . 7  |-  O  =  ( 0g `  F
)
75, 6ring0cl 13783 . . . . . 6  |-  ( F  e.  Ring  ->  O  e.  ( Base `  F
) )
84, 7syl 14 . . . . 5  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  O  e.  ( Base `  F
) )
9 simpr 110 . . . . 5  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  X  e.  V )
10 lmod0vs.v . . . . . 6  |-  V  =  ( Base `  W
)
11 eqid 2205 . . . . . 6  |-  ( +g  `  W )  =  ( +g  `  W )
12 lmod0vs.s . . . . . 6  |-  .x.  =  ( .s `  W )
13 eqid 2205 . . . . . 6  |-  ( +g  `  F )  =  ( +g  `  F )
1410, 11, 2, 12, 5, 13lmodvsdir 14074 . . . . 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 1252 . . . 4  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  (
( O ( +g  `  F ) O ) 
.x.  X )  =  ( ( O  .x.  X ) ( +g  `  W ) ( O 
.x.  X ) ) )
16 ringgrp 13763 . . . . . . 7  |-  ( F  e.  Ring  ->  F  e. 
Grp )
174, 16syl 14 . . . . . 6  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  F  e.  Grp )
185, 13, 6grplid 13363 . . . . . 6  |-  ( ( F  e.  Grp  /\  O  e.  ( Base `  F ) )  -> 
( O ( +g  `  F ) O )  =  O )
1917, 8, 18syl2anc 411 . . . . 5  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  ( O ( +g  `  F
) O )  =  O )
2019oveq1d 5959 . . . 4  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  (
( O ( +g  `  F ) O ) 
.x.  X )  =  ( O  .x.  X
) )
2115, 20eqtr3d 2240 . . 3  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  (
( O  .x.  X
) ( +g  `  W
) ( O  .x.  X ) )  =  ( O  .x.  X
) )
2210, 2, 12, 5lmodvscl 14067 . . . . 5  |-  ( ( W  e.  LMod  /\  O  e.  ( Base `  F
)  /\  X  e.  V )  ->  ( O  .x.  X )  e.  V )
231, 8, 9, 22syl3anc 1250 . . . 4  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  ( O  .x.  X )  e.  V )
24 lmod0vs.z . . . . 5  |-  .0.  =  ( 0g `  W )
2510, 11, 24lmod0vid 14082 . . . 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 282 . . 3  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  (
( ( O  .x.  X ) ( +g  `  W ) ( O 
.x.  X ) )  =  ( O  .x.  X )  <->  .0.  =  ( O  .x.  X ) ) )
2721, 26mpbid 147 . 2  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  .0.  =  ( O  .x.  X ) )
2827eqcomd 2211 1  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  ( O  .x.  X )  =  .0.  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    = wceq 1373    e. wcel 2176   ` cfv 5271  (class class class)co 5944   Basecbs 12832   +g cplusg 12909  Scalarcsca 12912   .scvsca 12913   0gc0g 13088   Grpcgrp 13332   Ringcrg 13758   LModclmod 14049
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-cnex 8016  ax-resscn 8017  ax-1re 8019  ax-addrcl 8022
This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-reu 2491  df-rmo 2492  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-br 4045  df-opab 4106  df-mpt 4107  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-iota 5232  df-fun 5273  df-fn 5274  df-fv 5279  df-riota 5899  df-ov 5947  df-inn 9037  df-2 9095  df-3 9096  df-4 9097  df-5 9098  df-6 9099  df-ndx 12835  df-slot 12836  df-base 12838  df-plusg 12922  df-mulr 12923  df-sca 12925  df-vsca 12926  df-0g 13090  df-mgm 13188  df-sgrp 13234  df-mnd 13249  df-grp 13335  df-ring 13760  df-lmod 14051
This theorem is referenced by:  lmodvs0  14084  lmodvsmmulgdi  14085  lmodvneg1  14092
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