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Theorem mulveczer 25479
Description: Multiplication of a vector by zero. (Contributed by FL, 12-Sep-2010.)
Hypotheses
Ref Expression
mulveczer.1  |-  W  =  ran  ( 1st `  ( 2nd `  R ) )
mulveczer.2  |-  0 t  =  (GId `  + t )
mulveczer.3  |-  + t  =  ( 1st `  ( 1st `  R ) )
mulveczer.4  |-  . t  =  ( 2nd `  ( 1st `  R ) )
mulveczer.5  |-  . w  =  ( 2nd `  ( 2nd `  R ) )
mulveczer.6  |-  0 w  =  (GId `  ( 1st `  ( 2nd `  R
) ) )
Assertion
Ref Expression
mulveczer  |-  ( ( R  e.  Vec  /\  <. + t ,  . t >.  e.  RingOps  /\  U  e.  W )  ->  (
0 t . w U )  =  0 w )

Proof of Theorem mulveczer
StepHypRef Expression
1 mulveczer.3 . . . . . . . . . . 11  |-  + t  =  ( 1st `  ( 1st `  R ) )
2 fvex 5539 . . . . . . . . . . 11  |-  ( 1st `  ( 1st `  R
) )  e.  _V
31, 2eqeltri 2353 . . . . . . . . . 10  |-  + t  e.  _V
4 mulveczer.4 . . . . . . . . . . 11  |-  . t  =  ( 2nd `  ( 1st `  R ) )
5 fvex 5539 . . . . . . . . . . 11  |-  ( 2nd `  ( 1st `  R
) )  e.  _V
64, 5eqeltri 2353 . . . . . . . . . 10  |-  . t  e.  _V
73, 6op1st 6128 . . . . . . . . 9  |-  ( 1st `  <. + t ,  . t >. )  =  + t
87eqcomi 2287 . . . . . . . 8  |-  + t  =  ( 1st `  <. + t ,  . t >. )
9 eqid 2283 . . . . . . . 8  |-  ran  + t  =  ran  + t
10 mulveczer.2 . . . . . . . 8  |-  0 t  =  (GId `  + t )
118, 9, 10rngo0cl 21065 . . . . . . 7  |-  ( <. + t ,  . t >.  e.  RingOps  ->  0 t  e.  ran  + t )
128, 9, 10rngo0lid 21067 . . . . . . . 8  |-  ( (
<. + t ,  . t >.  e.  RingOps  /\  0 t  e.  ran  + t
)  ->  ( 0 t + t 0 t
)  =  0 t
)
1312eqcomd 2288 . . . . . . 7  |-  ( (
<. + t ,  . t >.  e.  RingOps  /\  0 t  e.  ran  + t
)  ->  0 t  =  ( 0 t + t 0 t )
)
1411, 13mpdan 649 . . . . . 6  |-  ( <. + t ,  . t >.  e.  RingOps  ->  0 t  =  ( 0 t + t 0 t )
)
15143ad2ant2 977 . . . . 5  |-  ( ( R  e.  Vec  /\  <. + t ,  . t >.  e.  RingOps  /\  U  e.  W )  ->  0 t  =  ( 0 t + t 0 t
) )
1615oveq1d 5873 . . . 4  |-  ( ( R  e.  Vec  /\  <. + t ,  . t >.  e.  RingOps  /\  U  e.  W )  ->  (
0 t . w U )  =  ( ( 0 t + t 0 t ) . w U ) )
17 simp1 955 . . . . 5  |-  ( ( R  e.  Vec  /\  <. + t ,  . t >.  e.  RingOps  /\  U  e.  W )  ->  R  e.  Vec  )
18 simp3 957 . . . . 5  |-  ( ( R  e.  Vec  /\  <. + t ,  . t >.  e.  RingOps  /\  U  e.  W )  ->  U  e.  W )
191eqcomi 2287 . . . . . . . 8  |-  ( 1st `  ( 1st `  R
) )  =  + t
2019rneqi 4905 . . . . . . 7  |-  ran  ( 1st `  ( 1st `  R
) )  =  ran  + t
218, 20, 10rngo0cl 21065 . . . . . 6  |-  ( <. + t ,  . t >.  e.  RingOps  ->  0 t  e.  ran  ( 1st `  ( 1st `  R ) ) )
22213ad2ant2 977 . . . . 5  |-  ( ( R  e.  Vec  /\  <. + t ,  . t >.  e.  RingOps  /\  U  e.  W )  ->  0 t  e.  ran  ( 1st `  ( 1st `  R
) ) )
23 eqid 2283 . . . . . 6  |-  ran  ( 1st `  ( 1st `  R
) )  =  ran  ( 1st `  ( 1st `  R ) )
24 eqid 2283 . . . . . 6  |-  ( 1st `  ( 2nd `  R
) )  =  ( 1st `  ( 2nd `  R ) )
25 mulveczer.5 . . . . . 6  |-  . w  =  ( 2nd `  ( 2nd `  R ) )
26 mulveczer.1 . . . . . 6  |-  W  =  ran  ( 1st `  ( 2nd `  R ) )
2723, 1, 24, 25, 26vecax5b 25459 . . . . 5  |-  ( ( R  e.  Vec  /\  ( U  e.  W  /\  0 t  e.  ran  ( 1st `  ( 1st `  R ) )  /\  0 t  e.  ran  ( 1st `  ( 1st `  R ) ) ) )  ->  ( (
0 t + t
0 t ) . w U )  =  ( ( 0 t . w U ) ( 1st `  ( 2nd `  R ) ) ( 0 t . w U ) ) )
2817, 18, 22, 22, 27syl13anc 1184 . . . 4  |-  ( ( R  e.  Vec  /\  <. + t ,  . t >.  e.  RingOps  /\  U  e.  W )  ->  (
( 0 t + t 0 t ) . w U )  =  ( ( 0 t . w U ) ( 1st `  ( 2nd `  R ) ) ( 0 t . w U ) ) )
2916, 28eqtrd 2315 . . 3  |-  ( ( R  e.  Vec  /\  <. + t ,  . t >.  e.  RingOps  /\  U  e.  W )  ->  (
0 t . w U )  =  ( ( 0 t . w U ) ( 1st `  ( 2nd `  R
) ) ( 0 t . w U
) ) )
3024, 25, 26, 9, 1prodvs 25468 . . . . 5  |-  ( ( R  e.  Vec  /\  0 t  e.  ran  + t  /\  U  e.  W )  ->  (
0 t . w U )  e.  W
)
3111, 30syl3an2 1216 . . . 4  |-  ( ( R  e.  Vec  /\  <. + t ,  . t >.  e.  RingOps  /\  U  e.  W )  ->  (
0 t . w U )  e.  W
)
32 mulveczer.6 . . . . 5  |-  0 w  =  (GId `  ( 1st `  ( 2nd `  R
) ) )
33 eqid 2283 . . . . 5  |-  (  /g  `  ( 1st `  ( 2nd `  R ) ) )  =  (  /g  `  ( 1st `  ( 2nd `  R ) ) )
3432, 24, 33, 26mvecrtol2 25477 . . . 4  |-  ( ( R  e.  Vec  /\  ( ( 0 t . w U )  e.  W  /\  ( 0 t . w U
)  e.  W  /\  ( 0 t . w U )  e.  W
) )  ->  (
( 0 t . w U )  =  ( ( 0 t . w U ) ( 1st `  ( 2nd `  R
) ) ( 0 t . w U
) )  <->  ( (
0 t . w U ) (  /g  `  ( 1st `  ( 2nd `  R ) ) ) ( 0 t . w U ) )  =  ( 0 t . w U ) ) )
3517, 31, 31, 31, 34syl13anc 1184 . . 3  |-  ( ( R  e.  Vec  /\  <. + t ,  . t >.  e.  RingOps  /\  U  e.  W )  ->  (
( 0 t . w U )  =  ( ( 0 t . w U ) ( 1st `  ( 2nd `  R
) ) ( 0 t . w U
) )  <->  ( (
0 t . w U ) (  /g  `  ( 1st `  ( 2nd `  R ) ) ) ( 0 t . w U ) )  =  ( 0 t . w U ) ) )
3629, 35mpbid 201 . 2  |-  ( ( R  e.  Vec  /\  <. + t ,  . t >.  e.  RingOps  /\  U  e.  W )  ->  (
( 0 t . w U ) (  /g  `  ( 1st `  ( 2nd `  R ) ) ) ( 0 t . w U ) )  =  ( 0 t . w U ) )
3732, 24, 33, 26vwit 25471 . . 3  |-  ( ( R  e.  Vec  /\  ( 0 t . w U )  e.  W
)  ->  ( (
0 t . w U ) (  /g  `  ( 1st `  ( 2nd `  R ) ) ) ( 0 t . w U ) )  =  0 w )
3817, 31, 37syl2anc 642 . 2  |-  ( ( R  e.  Vec  /\  <. + t ,  . t >.  e.  RingOps  /\  U  e.  W )  ->  (
( 0 t . w U ) (  /g  `  ( 1st `  ( 2nd `  R ) ) ) ( 0 t . w U ) )  =  0 w )
3936, 38eqtr3d 2317 1  |-  ( ( R  e.  Vec  /\  <. + t ,  . t >.  e.  RingOps  /\  U  e.  W )  ->  (
0 t . w U )  =  0 w )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   _Vcvv 2788   <.cop 3643   ran crn 4690   ` cfv 5255  (class class class)co 5858   1stc1st 6120   2ndc2nd 6121  GIdcgi 20854    /g cgs 20856   RingOpscrngo 21042    Vec cvec 25449
This theorem is referenced by:  mulinvsca  25480  svli2  25484
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-grpo 20858  df-gid 20859  df-ginv 20860  df-gdiv 20861  df-ablo 20949  df-rngo 21043  df-vec 25450
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