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Theorem slmd0vs 27417
Description: Zero times a vector is the zero vector. Equation 1a of [Kreyszig] p. 51. (ax-hvmul0 25591 analog.) (Contributed by NM, 12-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.)
Hypotheses
Ref Expression
slmd0vs.v  |-  V  =  ( Base `  W
)
slmd0vs.f  |-  F  =  (Scalar `  W )
slmd0vs.s  |-  .x.  =  ( .s `  W )
slmd0vs.o  |-  O  =  ( 0g `  F
)
slmd0vs.z  |-  .0.  =  ( 0g `  W )
Assertion
Ref Expression
slmd0vs  |-  ( ( W  e. SLMod  /\  X  e.  V )  ->  ( O  .x.  X )  =  .0.  )

Proof of Theorem slmd0vs
StepHypRef Expression
1 simpl 457 . . . 4  |-  ( ( W  e. SLMod  /\  X  e.  V )  ->  W  e. SLMod )
2 slmd0vs.f . . . . . 6  |-  F  =  (Scalar `  W )
3 eqid 2462 . . . . . 6  |-  ( Base `  F )  =  (
Base `  F )
4 slmd0vs.o . . . . . 6  |-  O  =  ( 0g `  F
)
52, 3, 4slmd0cl 27411 . . . . 5  |-  ( W  e. SLMod  ->  O  e.  (
Base `  F )
)
61, 5syl 16 . . . 4  |-  ( ( W  e. SLMod  /\  X  e.  V )  ->  O  e.  ( Base `  F
) )
7 simpr 461 . . . 4  |-  ( ( W  e. SLMod  /\  X  e.  V )  ->  X  e.  V )
8 slmd0vs.v . . . . 5  |-  V  =  ( Base `  W
)
9 eqid 2462 . . . . 5  |-  ( +g  `  W )  =  ( +g  `  W )
10 slmd0vs.s . . . . 5  |-  .x.  =  ( .s `  W )
11 slmd0vs.z . . . . 5  |-  .0.  =  ( 0g `  W )
12 eqid 2462 . . . . 5  |-  ( +g  `  F )  =  ( +g  `  F )
13 eqid 2462 . . . . 5  |-  ( .r
`  F )  =  ( .r `  F
)
14 eqid 2462 . . . . 5  |-  ( 1r
`  F )  =  ( 1r `  F
)
158, 9, 10, 11, 2, 3, 12, 13, 14, 4slmdlema 27396 . . . 4  |-  ( ( W  e. SLMod  /\  ( O  e.  ( Base `  F )  /\  O  e.  ( Base `  F
) )  /\  ( X  e.  V  /\  X  e.  V )
)  ->  ( (
( O  .x.  X
)  e.  V  /\  ( O  .x.  ( X ( +g  `  W
) X ) )  =  ( ( O 
.x.  X ) ( +g  `  W ) ( O  .x.  X
) )  /\  (
( O ( +g  `  F ) O ) 
.x.  X )  =  ( ( O  .x.  X ) ( +g  `  W ) ( O 
.x.  X ) ) )  /\  ( ( ( O ( .r
`  F ) O )  .x.  X )  =  ( O  .x.  ( O  .x.  X ) )  /\  ( ( 1r `  F ) 
.x.  X )  =  X  /\  ( O 
.x.  X )  =  .0.  ) ) )
161, 6, 6, 7, 7, 15syl122anc 1232 . . 3  |-  ( ( W  e. SLMod  /\  X  e.  V )  ->  (
( ( O  .x.  X )  e.  V  /\  ( O  .x.  ( X ( +g  `  W
) X ) )  =  ( ( O 
.x.  X ) ( +g  `  W ) ( O  .x.  X
) )  /\  (
( O ( +g  `  F ) O ) 
.x.  X )  =  ( ( O  .x.  X ) ( +g  `  W ) ( O 
.x.  X ) ) )  /\  ( ( ( O ( .r
`  F ) O )  .x.  X )  =  ( O  .x.  ( O  .x.  X ) )  /\  ( ( 1r `  F ) 
.x.  X )  =  X  /\  ( O 
.x.  X )  =  .0.  ) ) )
1716simprd 463 . 2  |-  ( ( W  e. SLMod  /\  X  e.  V )  ->  (
( ( O ( .r `  F ) O )  .x.  X
)  =  ( O 
.x.  ( O  .x.  X ) )  /\  ( ( 1r `  F )  .x.  X
)  =  X  /\  ( O  .x.  X )  =  .0.  ) )
1817simp3d 1005 1  |-  ( ( W  e. SLMod  /\  X  e.  V )  ->  ( O  .x.  X )  =  .0.  )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762   ` cfv 5581  (class class class)co 6277   Basecbs 14481   +g cplusg 14546   .rcmulr 14547  Scalarcsca 14549   .scvsca 14550   0gc0g 14686   1rcur 16938  SLModcslmd 27393
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-reu 2816  df-rmo 2817  df-rab 2818  df-v 3110  df-sbc 3327  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-br 4443  df-opab 4501  df-mpt 4502  df-id 4790  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-iota 5544  df-fun 5583  df-fv 5589  df-riota 6238  df-ov 6280  df-0g 14688  df-mnd 15723  df-cmn 16591  df-srg 16943  df-slmd 27394
This theorem is referenced by:  slmdvs0  27418  gsumvsca2  27425
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