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Theorem slmdvs0 28535
Description: Anything times the zero vector is the zero vector. Equation 1b of [Kreyszig] p. 51. (hvmul0 26662 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 28517 . . . 4  |-  ( W  e. SLMod  ->  F  e. SRing )
3 slmdvs0.k . . . . 5  |-  K  =  ( Base `  F
)
4 eqid 2422 . . . . 5  |-  ( .r
`  F )  =  ( .r `  F
)
5 eqid 2422 . . . . 5  |-  ( 0g
`  F )  =  ( 0g `  F
)
63, 4, 5srgrz 17746 . . . 4  |-  ( ( F  e. SRing  /\  X  e.  K )  ->  ( X ( .r `  F ) ( 0g
`  F ) )  =  ( 0g `  F ) )
72, 6sylan 473 . . 3  |-  ( ( W  e. SLMod  /\  X  e.  K )  ->  ( X ( .r `  F ) ( 0g
`  F ) )  =  ( 0g `  F ) )
87oveq1d 6316 . 2  |-  ( ( W  e. SLMod  /\  X  e.  K )  ->  (
( X ( .r
`  F ) ( 0g `  F ) )  .x.  .0.  )  =  ( ( 0g
`  F )  .x.  .0.  ) )
9 simpl 458 . . . 4  |-  ( ( W  e. SLMod  /\  X  e.  K )  ->  W  e. SLMod )
10 simpr 462 . . . 4  |-  ( ( W  e. SLMod  /\  X  e.  K )  ->  X  e.  K )
112adantr 466 . . . . 5  |-  ( ( W  e. SLMod  /\  X  e.  K )  ->  F  e. SRing )
123, 5srg0cl 17739 . . . . 5  |-  ( F  e. SRing  ->  ( 0g `  F )  e.  K
)
1311, 12syl 17 . . . 4  |-  ( ( W  e. SLMod  /\  X  e.  K )  ->  ( 0g `  F )  e.  K )
14 eqid 2422 . . . . . 6  |-  ( Base `  W )  =  (
Base `  W )
15 slmdvs0.z . . . . . 6  |-  .0.  =  ( 0g `  W )
1614, 15slmd0vcl 28531 . . . . 5  |-  ( W  e. SLMod  ->  .0.  e.  ( Base `  W ) )
1716adantr 466 . . . 4  |-  ( ( W  e. SLMod  /\  X  e.  K )  ->  .0.  e.  ( Base `  W
) )
18 slmdvs0.s . . . . 5  |-  .x.  =  ( .s `  W )
1914, 1, 18, 3, 4slmdvsass 28527 . . . 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 1266 . . 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 28534 . . . . 5  |-  ( ( W  e. SLMod  /\  .0.  e.  ( Base `  W )
)  ->  ( ( 0g `  F )  .x.  .0.  )  =  .0.  )
2217, 21syldan 472 . . . 4  |-  ( ( W  e. SLMod  /\  X  e.  K )  ->  (
( 0g `  F
)  .x.  .0.  )  =  .0.  )
2322oveq2d 6317 . . 3  |-  ( ( W  e. SLMod  /\  X  e.  K )  ->  ( X  .x.  ( ( 0g
`  F )  .x.  .0.  ) )  =  ( X  .x.  .0.  )
)
2420, 23eqtrd 2463 . 2  |-  ( ( W  e. SLMod  /\  X  e.  K )  ->  (
( X ( .r
`  F ) ( 0g `  F ) )  .x.  .0.  )  =  ( X  .x.  .0.  ) )
258, 24, 223eqtr3d 2471 1  |-  ( ( W  e. SLMod  /\  X  e.  K )  ->  ( X  .x.  .0.  )  =  .0.  )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1868   ` cfv 5597  (class class class)co 6301   Basecbs 15108   .rcmulr 15178  Scalarcsca 15180   .scvsca 15181   0gc0g 15325  SRingcsrg 17726  SLModcslmd 28510
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-sep 4543  ax-nul 4551  ax-pow 4598  ax-pr 4656
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-ral 2780  df-rex 2781  df-reu 2782  df-rmo 2783  df-rab 2784  df-v 3083  df-sbc 3300  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3910  df-sn 3997  df-pr 3999  df-op 4003  df-uni 4217  df-br 4421  df-opab 4480  df-mpt 4481  df-id 4764  df-xp 4855  df-rel 4856  df-cnv 4857  df-co 4858  df-dm 4859  df-iota 5561  df-fun 5599  df-fv 5605  df-riota 6263  df-ov 6304  df-0g 15327  df-mgm 16475  df-sgrp 16514  df-mnd 16524  df-cmn 17419  df-srg 17727  df-slmd 28511
This theorem is referenced by:  gsumvsca1  28540
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