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Theorem slmdvs0 27928
Description: Anything times the zero vector is the zero vector. Equation 1b of [Kreyszig] p. 51. (hvmul0 26068 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 27910 . . . 4  |-  ( W  e. SLMod  ->  F  e. SRing )
3 slmdvs0.k . . . . 5  |-  K  =  ( Base `  F
)
4 eqid 2457 . . . . 5  |-  ( .r
`  F )  =  ( .r `  F
)
5 eqid 2457 . . . . 5  |-  ( 0g
`  F )  =  ( 0g `  F
)
63, 4, 5srgrz 17304 . . . 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 6311 . 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 17297 . . . . 5  |-  ( F  e. SRing  ->  ( 0g `  F )  e.  K
)
1311, 12syl 16 . . . 4  |-  ( ( W  e. SLMod  /\  X  e.  K )  ->  ( 0g `  F )  e.  K )
14 eqid 2457 . . . . . 6  |-  ( Base `  W )  =  (
Base `  W )
15 slmdvs0.z . . . . . 6  |-  .0.  =  ( 0g `  W )
1614, 15slmd0vcl 27924 . . . . 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 27920 . . . 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 1230 . . 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 27927 . . . . 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 6312 . . 3  |-  ( ( W  e. SLMod  /\  X  e.  K )  ->  ( X  .x.  ( ( 0g
`  F )  .x.  .0.  ) )  =  ( X  .x.  .0.  )
)
2420, 23eqtrd 2498 . 2  |-  ( ( W  e. SLMod  /\  X  e.  K )  ->  (
( X ( .r
`  F ) ( 0g `  F ) )  .x.  .0.  )  =  ( X  .x.  .0.  ) )
258, 24, 223eqtr3d 2506 1  |-  ( ( W  e. SLMod  /\  X  e.  K )  ->  ( X  .x.  .0.  )  =  .0.  )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1395    e. wcel 1819   ` cfv 5594  (class class class)co 6296   Basecbs 14644   .rcmulr 14713  Scalarcsca 14715   .scvsca 14716   0gc0g 14857  SRingcsrg 17284  SLModcslmd 27903
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-iota 5557  df-fun 5596  df-fv 5602  df-riota 6258  df-ov 6299  df-0g 14859  df-mgm 15999  df-sgrp 16038  df-mnd 16048  df-cmn 16927  df-srg 17285  df-slmd 27904
This theorem is referenced by:  gsumvsca1  27934
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