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Theorem slmd0vs 28205
Description: Zero times a vector is the zero vector. Equation 1a of [Kreyszig] p. 51. (ax-hvmul0 26327 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 455 . . . 4  |-  ( ( W  e. SLMod  /\  X  e.  V )  ->  W  e. SLMod )
2 slmd0vs.f . . . . . 6  |-  F  =  (Scalar `  W )
3 eqid 2402 . . . . . 6  |-  ( Base `  F )  =  (
Base `  F )
4 slmd0vs.o . . . . . 6  |-  O  =  ( 0g `  F
)
52, 3, 4slmd0cl 28199 . . . . 5  |-  ( W  e. SLMod  ->  O  e.  (
Base `  F )
)
65adantr 463 . . . 4  |-  ( ( W  e. SLMod  /\  X  e.  V )  ->  O  e.  ( Base `  F
) )
7 simpr 459 . . . 4  |-  ( ( W  e. SLMod  /\  X  e.  V )  ->  X  e.  V )
8 slmd0vs.v . . . . 5  |-  V  =  ( Base `  W
)
9 eqid 2402 . . . . 5  |-  ( +g  `  W )  =  ( +g  `  W )
10 slmd0vs.s . . . . 5  |-  .x.  =  ( .s `  W )
11 slmd0vs.z . . . . 5  |-  .0.  =  ( 0g `  W )
12 eqid 2402 . . . . 5  |-  ( +g  `  F )  =  ( +g  `  F )
13 eqid 2402 . . . . 5  |-  ( .r
`  F )  =  ( .r `  F
)
14 eqid 2402 . . . . 5  |-  ( 1r
`  F )  =  ( 1r `  F
)
158, 9, 10, 11, 2, 3, 12, 13, 14, 4slmdlema 28184 . . . 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 1239 . . 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 461 . 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 1011 1  |-  ( ( W  e. SLMod  /\  X  e.  V )  ->  ( O  .x.  X )  =  .0.  )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842   ` cfv 5568  (class class class)co 6277   Basecbs 14839   +g cplusg 14907   .rcmulr 14908  Scalarcsca 14910   .scvsca 14911   0gc0g 15052   1rcur 17471  SLModcslmd 28181
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-iota 5532  df-fun 5570  df-fv 5576  df-riota 6239  df-ov 6280  df-0g 15054  df-mgm 16194  df-sgrp 16233  df-mnd 16243  df-cmn 17122  df-srg 17476  df-slmd 28182
This theorem is referenced by:  slmdvs0  28206  gsumvsca2  28212
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