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Theorem mulveczer 25582
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 5555 . . . . . . . . . . 11  |-  ( 1st `  ( 1st `  R
) )  e.  _V
31, 2eqeltri 2366 . . . . . . . . . 10  |-  + t  e.  _V
4 mulveczer.4 . . . . . . . . . . 11  |-  . t  =  ( 2nd `  ( 1st `  R ) )
5 fvex 5555 . . . . . . . . . . 11  |-  ( 2nd `  ( 1st `  R
) )  e.  _V
64, 5eqeltri 2366 . . . . . . . . . 10  |-  . t  e.  _V
73, 6op1st 6144 . . . . . . . . 9  |-  ( 1st `  <. + t ,  . t >. )  =  + t
87eqcomi 2300 . . . . . . . 8  |-  + t  =  ( 1st `  <. + t ,  . t >. )
9 eqid 2296 . . . . . . . 8  |-  ran  + t  =  ran  + t
10 mulveczer.2 . . . . . . . 8  |-  0 t  =  (GId `  + t )
118, 9, 10rngo0cl 21081 . . . . . . 7  |-  ( <. + t ,  . t >.  e.  RingOps  ->  0 t  e.  ran  + t )
128, 9, 10rngo0lid 21083 . . . . . . . 8  |-  ( (
<. + t ,  . t >.  e.  RingOps  /\  0 t  e.  ran  + t
)  ->  ( 0 t + t 0 t
)  =  0 t
)
1312eqcomd 2301 . . . . . . 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 5889 . . . 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 2300 . . . . . . . 8  |-  ( 1st `  ( 1st `  R
) )  =  + t
2019rneqi 4921 . . . . . . 7  |-  ran  ( 1st `  ( 1st `  R
) )  =  ran  + t
218, 20, 10rngo0cl 21081 . . . . . 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 2296 . . . . . 6  |-  ran  ( 1st `  ( 1st `  R
) )  =  ran  ( 1st `  ( 1st `  R ) )
24 eqid 2296 . . . . . 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 25562 . . . . 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 2328 . . 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 25571 . . . . 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 2296 . . . . 5  |-  (  /g  `  ( 1st `  ( 2nd `  R ) ) )  =  (  /g  `  ( 1st `  ( 2nd `  R ) ) )
3432, 24, 33, 26mvecrtol2 25580 . . . 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 25574 . . 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 2330 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 1632    e. wcel 1696   _Vcvv 2801   <.cop 3656   ran crn 4706   ` cfv 5271  (class class class)co 5874   1stc1st 6136   2ndc2nd 6137  GIdcgi 20870    /g cgs 20872   RingOpscrngo 21058    Vec cvec 25552
This theorem is referenced by:  mulinvsca  25583  svli2  25587
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-grpo 20874  df-gid 20875  df-ginv 20876  df-gdiv 20877  df-ablo 20965  df-rngo 21059  df-vec 25553
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