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Theorem slmdvs0 26241
Description: Anything times the zero vector is the zero vector. Equation 1b of [Kreyszig] p. 51. (hvmul0 24426 analog.) (Contributed by NM, 12-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.)
Hypotheses
Ref Expression
slmdvs0.f  |-  F  =  (Scalar `  W )
slmdvs0.s  |-  .x.  =  ( .s `  W )
slmdvs0.k  |-  K  =  ( Base `  F
)
slmdvs0.z  |-  .0.  =  ( 0g `  W )
Assertion
Ref Expression
slmdvs0  |-  ( ( W  e. SLMod  /\  X  e.  K )  ->  ( X  .x.  .0.  )  =  .0.  )

Proof of Theorem slmdvs0
StepHypRef Expression
1 slmdvs0.f . . . . 5  |-  F  =  (Scalar `  W )
21slmdsrg 26223 . . . 4  |-  ( W  e. SLMod  ->  F  e. SRing )
3 slmdvs0.k . . . . 5  |-  K  =  ( Base `  F
)
4 eqid 2443 . . . . 5  |-  ( .r
`  F )  =  ( .r `  F
)
5 eqid 2443 . . . . 5  |-  ( 0g
`  F )  =  ( 0g `  F
)
63, 4, 5srgrz 16627 . . . 4  |-  ( ( F  e. SRing  /\  X  e.  K )  ->  ( X ( .r `  F ) ( 0g
`  F ) )  =  ( 0g `  F ) )
72, 6sylan 471 . . 3  |-  ( ( W  e. SLMod  /\  X  e.  K )  ->  ( X ( .r `  F ) ( 0g
`  F ) )  =  ( 0g `  F ) )
87oveq1d 6106 . 2  |-  ( ( W  e. SLMod  /\  X  e.  K )  ->  (
( X ( .r
`  F ) ( 0g `  F ) )  .x.  .0.  )  =  ( ( 0g
`  F )  .x.  .0.  ) )
9 simpl 457 . . . 4  |-  ( ( W  e. SLMod  /\  X  e.  K )  ->  W  e. SLMod )
10 simpr 461 . . . 4  |-  ( ( W  e. SLMod  /\  X  e.  K )  ->  X  e.  K )
112adantr 465 . . . . 5  |-  ( ( W  e. SLMod  /\  X  e.  K )  ->  F  e. SRing )
123, 5srg0cl 16620 . . . . 5  |-  ( F  e. SRing  ->  ( 0g `  F )  e.  K
)
1311, 12syl 16 . . . 4  |-  ( ( W  e. SLMod  /\  X  e.  K )  ->  ( 0g `  F )  e.  K )
14 eqid 2443 . . . . . 6  |-  ( Base `  W )  =  (
Base `  W )
15 slmdvs0.z . . . . . 6  |-  .0.  =  ( 0g `  W )
1614, 15slmd0vcl 26237 . . . . 5  |-  ( W  e. SLMod  ->  .0.  e.  ( Base `  W ) )
1716adantr 465 . . . 4  |-  ( ( W  e. SLMod  /\  X  e.  K )  ->  .0.  e.  ( Base `  W
) )
18 slmdvs0.s . . . . 5  |-  .x.  =  ( .s `  W )
1914, 1, 18, 3, 4slmdvsass 26233 . . . 4  |-  ( ( W  e. SLMod  /\  ( X  e.  K  /\  ( 0g `  F )  e.  K  /\  .0.  e.  ( Base `  W
) ) )  -> 
( ( X ( .r `  F ) ( 0g `  F
) )  .x.  .0.  )  =  ( X  .x.  ( ( 0g `  F )  .x.  .0.  ) ) )
209, 10, 13, 17, 19syl13anc 1220 . . 3  |-  ( ( W  e. SLMod  /\  X  e.  K )  ->  (
( X ( .r
`  F ) ( 0g `  F ) )  .x.  .0.  )  =  ( X  .x.  ( ( 0g `  F )  .x.  .0.  ) ) )
2114, 1, 18, 5, 15slmd0vs 26240 . . . . 5  |-  ( ( W  e. SLMod  /\  .0.  e.  ( Base `  W )
)  ->  ( ( 0g `  F )  .x.  .0.  )  =  .0.  )
2217, 21syldan 470 . . . 4  |-  ( ( W  e. SLMod  /\  X  e.  K )  ->  (
( 0g `  F
)  .x.  .0.  )  =  .0.  )
2322oveq2d 6107 . . 3  |-  ( ( W  e. SLMod  /\  X  e.  K )  ->  ( X  .x.  ( ( 0g
`  F )  .x.  .0.  ) )  =  ( X  .x.  .0.  )
)
2420, 23eqtrd 2475 . 2  |-  ( ( W  e. SLMod  /\  X  e.  K )  ->  (
( X ( .r
`  F ) ( 0g `  F ) )  .x.  .0.  )  =  ( X  .x.  .0.  ) )
258, 24, 223eqtr3d 2483 1  |-  ( ( W  e. SLMod  /\  X  e.  K )  ->  ( X  .x.  .0.  )  =  .0.  )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   ` cfv 5418  (class class class)co 6091   Basecbs 14174   .rcmulr 14239  Scalarcsca 14241   .scvsca 14242   0gc0g 14378  SRingcsrg 16607  SLModcslmd 26216
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-iota 5381  df-fun 5420  df-fv 5426  df-riota 6052  df-ov 6094  df-0g 14380  df-mnd 15415  df-cmn 16279  df-srg 16608  df-slmd 26217
This theorem is referenced by:  gsumvsca1  26251
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