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Theorem mulinvsca 25480
Description: Multiplication by the inverse of a scalar. (Contributed by FL, 12-Sep-2010.)
Hypotheses
Ref Expression
mulinvsca.1  |-  X  =  ran  + t
mulinvsca.2  |-  W  =  ran  ( 1st `  ( 2nd `  R ) )
mulinvsca.3  |-  . w  =  ( 2nd `  ( 2nd `  R ) )
mulinvsca.4  |-  ~ t  =  ( inv `  + t )
mulinvsca.5  |-  ~ w  =  ( inv `  ( 1st `  ( 2nd `  R
) ) )
mulinvsca.6  |-  + t  =  ( 1st `  ( 1st `  R ) )
mulinvsca.7  |-  . t  =  ( 2nd `  ( 1st `  R ) )
Assertion
Ref Expression
mulinvsca  |-  ( ( R  e.  Vec  /\  <. + t ,  . t >.  e.  RingOps  /\  ( A  e.  X  /\  U  e.  W ) )  -> 
( ( ~ t `  A ) . w U )  =  ( ~ w `  ( A . w U ) ) )

Proof of Theorem mulinvsca
StepHypRef Expression
1 simp1 955 . . . 4  |-  ( ( R  e.  Vec  /\  <. + t ,  . t >.  e.  RingOps  /\  ( A  e.  X  /\  U  e.  W ) )  ->  R  e.  Vec  )
2 simp3r 984 . . . 4  |-  ( ( R  e.  Vec  /\  <. + t ,  . t >.  e.  RingOps  /\  ( A  e.  X  /\  U  e.  W ) )  ->  U  e.  W )
3 simp3l 983 . . . 4  |-  ( ( R  e.  Vec  /\  <. + t ,  . t >.  e.  RingOps  /\  ( A  e.  X  /\  U  e.  W ) )  ->  A  e.  X )
4 mulinvsca.1 . . . . . . 7  |-  X  =  ran  + t
5 mulinvsca.4 . . . . . . 7  |-  ~ t  =  ( inv `  + t )
64, 5rngoinvcl 25421 . . . . . 6  |-  ( (
<. + t ,  . t >.  e.  RingOps  /\  A  e.  X )  ->  ( ~ t `  A )  e.  X )
76adantrr 697 . . . . 5  |-  ( (
<. + t ,  . t >.  e.  RingOps  /\  ( A  e.  X  /\  U  e.  W )
)  ->  ( ~ t `  A )  e.  X )
873adant1 973 . . . 4  |-  ( ( R  e.  Vec  /\  <. + t ,  . t >.  e.  RingOps  /\  ( A  e.  X  /\  U  e.  W ) )  -> 
( ~ t `  A )  e.  X
)
9 mulinvsca.6 . . . . . . 7  |-  + t  =  ( 1st `  ( 1st `  R ) )
109rneqi 4905 . . . . . 6  |-  ran  + t  =  ran  ( 1st `  ( 1st `  R
) )
114, 10eqtri 2303 . . . . 5  |-  X  =  ran  ( 1st `  ( 1st `  R ) )
12 eqid 2283 . . . . 5  |-  ( 1st `  ( 2nd `  R
) )  =  ( 1st `  ( 2nd `  R ) )
13 mulinvsca.3 . . . . 5  |-  . w  =  ( 2nd `  ( 2nd `  R ) )
14 mulinvsca.2 . . . . 5  |-  W  =  ran  ( 1st `  ( 2nd `  R ) )
1511, 9, 12, 13, 14vecax5b 25459 . . . 4  |-  ( ( R  e.  Vec  /\  ( U  e.  W  /\  A  e.  X  /\  ( ~ t `  A )  e.  X
) )  ->  (
( A + t
( ~ t `  A ) ) . w U )  =  ( ( A . w U ) ( 1st `  ( 2nd `  R
) ) ( ( ~ t `  A
) . w U
) ) )
161, 2, 3, 8, 15syl13anc 1184 . . 3  |-  ( ( R  e.  Vec  /\  <. + t ,  . t >.  e.  RingOps  /\  ( A  e.  X  /\  U  e.  W ) )  -> 
( ( A + t ( ~ t `  A ) ) . w U )  =  ( ( A . w U ) ( 1st `  ( 2nd `  R
) ) ( ( ~ t `  A
) . w U
) ) )
1712, 13, 14, 4, 9prodvs 25468 . . . . . . . 8  |-  ( ( R  e.  Vec  /\  A  e.  X  /\  U  e.  W )  ->  ( A . w U )  e.  W
)
18173expb 1152 . . . . . . 7  |-  ( ( R  e.  Vec  /\  ( A  e.  X  /\  U  e.  W
) )  ->  ( A . w U )  e.  W )
19183adant2 974 . . . . . 6  |-  ( ( R  e.  Vec  /\  <. + t ,  . t >.  e.  RingOps  /\  ( A  e.  X  /\  U  e.  W ) )  -> 
( A . w U )  e.  W
)
201, 19jca 518 . . . . 5  |-  ( ( R  e.  Vec  /\  <. + t ,  . t >.  e.  RingOps  /\  ( A  e.  X  /\  U  e.  W ) )  -> 
( R  e.  Vec  /\  ( A . w U )  e.  W
) )
21 eqid 2283 . . . . . . 7  |-  (GId `  ( 1st `  ( 2nd `  R ) ) )  =  (GId `  ( 1st `  ( 2nd `  R
) ) )
22 eqid 2283 . . . . . . 7  |-  (  /g  `  ( 1st `  ( 2nd `  R ) ) )  =  (  /g  `  ( 1st `  ( 2nd `  R ) ) )
2321, 12, 22, 14vwit 25471 . . . . . 6  |-  ( ( R  e.  Vec  /\  ( A . w U
)  e.  W )  ->  ( ( A . w U ) (  /g  `  ( 1st `  ( 2nd `  R
) ) ) ( A . w U
) )  =  (GId
`  ( 1st `  ( 2nd `  R ) ) ) )
24 simpl 443 . . . . . . . 8  |-  ( ( ( ( A . w U ) (  /g  `  ( 1st `  ( 2nd `  R ) ) ) ( A . w U ) )  =  (GId `  ( 1st `  ( 2nd `  R
) ) )  /\  ( R  e.  Vec  /\ 
<. + t ,  . t >.  e.  RingOps  /\  ( A  e.  X  /\  U  e.  W )
) )  ->  (
( A . w U ) (  /g  `  ( 1st `  ( 2nd `  R ) ) ) ( A . w U ) )  =  (GId `  ( 1st `  ( 2nd `  R
) ) ) )
25 eqid 2283 . . . . . . . . . . . 12  |-  (GId `  + t )  =  (GId
`  + t )
26 mulinvsca.7 . . . . . . . . . . . 12  |-  . t  =  ( 2nd `  ( 1st `  R ) )
2714, 25, 9, 26, 13, 21mulveczer 25479 . . . . . . . . . . 11  |-  ( ( R  e.  Vec  /\  <. + t ,  . t >.  e.  RingOps  /\  U  e.  W )  ->  (
(GId `  + t ) . w U )  =  (GId `  ( 1st `  ( 2nd `  R
) ) ) )
28273adant3l 1178 . . . . . . . . . 10  |-  ( ( R  e.  Vec  /\  <. + t ,  . t >.  e.  RingOps  /\  ( A  e.  X  /\  U  e.  W ) )  -> 
( (GId `  + t ) . w U )  =  (GId
`  ( 1st `  ( 2nd `  R ) ) ) )
29 fvex 5539 . . . . . . . . . . . . . . . . . 18  |-  ( 1st `  ( 1st `  R
) )  e.  _V
309, 29eqeltri 2353 . . . . . . . . . . . . . . . . 17  |-  + t  e.  _V
31 fvex 5539 . . . . . . . . . . . . . . . . . 18  |-  ( 2nd `  ( 1st `  R
) )  e.  _V
3226, 31eqeltri 2353 . . . . . . . . . . . . . . . . 17  |-  . t  e.  _V
3330, 32op1st 6128 . . . . . . . . . . . . . . . 16  |-  ( 1st `  <. + t ,  . t >. )  =  + t
3433eqcomi 2287 . . . . . . . . . . . . . . 15  |-  + t  =  ( 1st `  <. + t ,  . t >. )
3534rngogrpo 21057 . . . . . . . . . . . . . 14  |-  ( <. + t ,  . t >.  e.  RingOps  ->  + t  e.  GrpOp
)
36353ad2ant2 977 . . . . . . . . . . . . 13  |-  ( ( R  e.  Vec  /\  <. + t ,  . t >.  e.  RingOps  /\  ( A  e.  X  /\  U  e.  W ) )  ->  + t  e.  GrpOp )
374, 25, 5grporinv 20896 . . . . . . . . . . . . 13  |-  ( ( + t  e.  GrpOp  /\  A  e.  X )  ->  ( A + t ( ~ t `  A ) )  =  (GId `  + t )
)
3836, 3, 37syl2anc 642 . . . . . . . . . . . 12  |-  ( ( R  e.  Vec  /\  <. + t ,  . t >.  e.  RingOps  /\  ( A  e.  X  /\  U  e.  W ) )  -> 
( A + t
( ~ t `  A ) )  =  (GId `  + t )
)
3938eqcomd 2288 . . . . . . . . . . 11  |-  ( ( R  e.  Vec  /\  <. + t ,  . t >.  e.  RingOps  /\  ( A  e.  X  /\  U  e.  W ) )  -> 
(GId `  + t )  =  ( A + t ( ~ t `  A ) ) )
4039oveq1d 5873 . . . . . . . . . 10  |-  ( ( R  e.  Vec  /\  <. + t ,  . t >.  e.  RingOps  /\  ( A  e.  X  /\  U  e.  W ) )  -> 
( (GId `  + t ) . w U )  =  ( ( A + t
( ~ t `  A ) ) . w U ) )
4128, 40eqtr3d 2317 . . . . . . . . 9  |-  ( ( R  e.  Vec  /\  <. + t ,  . t >.  e.  RingOps  /\  ( A  e.  X  /\  U  e.  W ) )  -> 
(GId `  ( 1st `  ( 2nd `  R
) ) )  =  ( ( A + t ( ~ t `  A ) ) . w U ) )
4241adantl 452 . . . . . . . 8  |-  ( ( ( ( A . w U ) (  /g  `  ( 1st `  ( 2nd `  R ) ) ) ( A . w U ) )  =  (GId `  ( 1st `  ( 2nd `  R
) ) )  /\  ( R  e.  Vec  /\ 
<. + t ,  . t >.  e.  RingOps  /\  ( A  e.  X  /\  U  e.  W )
) )  ->  (GId `  ( 1st `  ( 2nd `  R ) ) )  =  ( ( A + t ( ~ t `  A ) ) . w U
) )
4324, 42eqtrd 2315 . . . . . . 7  |-  ( ( ( ( A . w U ) (  /g  `  ( 1st `  ( 2nd `  R ) ) ) ( A . w U ) )  =  (GId `  ( 1st `  ( 2nd `  R
) ) )  /\  ( R  e.  Vec  /\ 
<. + t ,  . t >.  e.  RingOps  /\  ( A  e.  X  /\  U  e.  W )
) )  ->  (
( A . w U ) (  /g  `  ( 1st `  ( 2nd `  R ) ) ) ( A . w U ) )  =  ( ( A + t ( ~ t `  A ) ) . w U ) )
4443ex 423 . . . . . 6  |-  ( ( ( A . w U ) (  /g  `  ( 1st `  ( 2nd `  R ) ) ) ( A . w U ) )  =  (GId `  ( 1st `  ( 2nd `  R
) ) )  -> 
( ( R  e. 
Vec  /\  <. + t ,  . t >.  e.  RingOps  /\  ( A  e.  X  /\  U  e.  W
) )  ->  (
( A . w U ) (  /g  `  ( 1st `  ( 2nd `  R ) ) ) ( A . w U ) )  =  ( ( A + t ( ~ t `  A ) ) . w U ) ) )
4523, 44syl 15 . . . . 5  |-  ( ( R  e.  Vec  /\  ( A . w U
)  e.  W )  ->  ( ( R  e.  Vec  /\  <. + t ,  . t >.  e.  RingOps  /\  ( A  e.  X  /\  U  e.  W ) )  -> 
( ( A . w U ) (  /g  `  ( 1st `  ( 2nd `  R ) ) ) ( A . w U ) )  =  ( ( A + t ( ~ t `  A ) ) . w U ) ) )
4620, 45mpcom 32 . . . 4  |-  ( ( R  e.  Vec  /\  <. + t ,  . t >.  e.  RingOps  /\  ( A  e.  X  /\  U  e.  W ) )  -> 
( ( A . w U ) (  /g  `  ( 1st `  ( 2nd `  R ) ) ) ( A . w U ) )  =  ( ( A + t ( ~ t `  A ) ) . w U ) )
4711, 14, 13prvs 25478 . . . . . 6  |-  ( ( R  e.  Vec  /\  A  e.  X  /\  U  e.  W )  ->  ( A . w U )  e.  W
)
481, 3, 2, 47syl3anc 1182 . . . . 5  |-  ( ( R  e.  Vec  /\  <. + t ,  . t >.  e.  RingOps  /\  ( A  e.  X  /\  U  e.  W ) )  -> 
( A . w U )  e.  W
)
49 mulinvsca.5 . . . . . 6  |-  ~ w  =  ( inv `  ( 1st `  ( 2nd `  R
) ) )
5021, 12, 22, 14, 49sub2vec 25472 . . . . 5  |-  ( ( R  e.  Vec  /\  ( ( A . w U )  e.  W  /\  ( A . w U )  e.  W
) )  ->  (
( A . w U ) (  /g  `  ( 1st `  ( 2nd `  R ) ) ) ( A . w U ) )  =  ( ( A . w U ) ( 1st `  ( 2nd `  R
) ) ( ~ w `  ( A . w U ) ) ) )
511, 48, 48, 50syl12anc 1180 . . . 4  |-  ( ( R  e.  Vec  /\  <. + t ,  . t >.  e.  RingOps  /\  ( A  e.  X  /\  U  e.  W ) )  -> 
( ( A . w U ) (  /g  `  ( 1st `  ( 2nd `  R ) ) ) ( A . w U ) )  =  ( ( A . w U ) ( 1st `  ( 2nd `  R
) ) ( ~ w `  ( A . w U ) ) ) )
5246, 51eqtr3d 2317 . . 3  |-  ( ( R  e.  Vec  /\  <. + t ,  . t >.  e.  RingOps  /\  ( A  e.  X  /\  U  e.  W ) )  -> 
( ( A + t ( ~ t `  A ) ) . w U )  =  ( ( A . w U ) ( 1st `  ( 2nd `  R
) ) ( ~ w `  ( A . w U ) ) ) )
5316, 52eqtr3d 2317 . 2  |-  ( ( R  e.  Vec  /\  <. + t ,  . t >.  e.  RingOps  /\  ( A  e.  X  /\  U  e.  W ) )  -> 
( ( A . w U ) ( 1st `  ( 2nd `  R
) ) ( ( ~ t `  A
) . w U
) )  =  ( ( A . w U ) ( 1st `  ( 2nd `  R
) ) ( ~ w `  ( A . w U ) ) ) )
5412vecax1 25453 . . . . 5  |-  ( R  e.  Vec  ->  ( 1st `  ( 2nd `  R
) )  e.  AbelOp )
55 ablogrpo 20951 . . . . 5  |-  ( ( 1st `  ( 2nd `  R ) )  e. 
AbelOp  ->  ( 1st `  ( 2nd `  R ) )  e.  GrpOp )
5654, 55syl 15 . . . 4  |-  ( R  e.  Vec  ->  ( 1st `  ( 2nd `  R
) )  e.  GrpOp )
57563ad2ant1 976 . . 3  |-  ( ( R  e.  Vec  /\  <. + t ,  . t >.  e.  RingOps  /\  ( A  e.  X  /\  U  e.  W ) )  -> 
( 1st `  ( 2nd `  R ) )  e.  GrpOp )
5811, 14, 13prvs 25478 . . . 4  |-  ( ( R  e.  Vec  /\  ( ~ t `  A
)  e.  X  /\  U  e.  W )  ->  ( ( ~ t `  A ) . w U )  e.  W
)
591, 8, 2, 58syl3anc 1182 . . 3  |-  ( ( R  e.  Vec  /\  <. + t ,  . t >.  e.  RingOps  /\  ( A  e.  X  /\  U  e.  W ) )  -> 
( ( ~ t `  A ) . w U )  e.  W
)
6014, 49claddinvvec 25460 . . . 4  |-  ( ( R  e.  Vec  /\  ( A . w U
)  e.  W )  ->  ( ~ w `  ( A . w U ) )  e.  W )
611, 19, 60syl2anc 642 . . 3  |-  ( ( R  e.  Vec  /\  <. + t ,  . t >.  e.  RingOps  /\  ( A  e.  X  /\  U  e.  W ) )  -> 
( ~ w `  ( A . w U
) )  e.  W
)
6214grpolcan 20900 . . 3  |-  ( ( ( 1st `  ( 2nd `  R ) )  e.  GrpOp  /\  ( (
( ~ t `  A ) . w U )  e.  W  /\  ( ~ w `  ( A . w U
) )  e.  W  /\  ( A . w U )  e.  W
) )  ->  (
( ( A . w U ) ( 1st `  ( 2nd `  R
) ) ( ( ~ t `  A
) . w U
) )  =  ( ( A . w U ) ( 1st `  ( 2nd `  R
) ) ( ~ w `  ( A . w U ) ) )  <->  ( ( ~ t `  A ) . w U )  =  ( ~ w `  ( A . w U ) ) ) )
6357, 59, 61, 19, 62syl13anc 1184 . 2  |-  ( ( R  e.  Vec  /\  <. + t ,  . t >.  e.  RingOps  /\  ( A  e.  X  /\  U  e.  W ) )  -> 
( ( ( A . w U ) ( 1st `  ( 2nd `  R ) ) ( ( ~ t `  A ) . w U ) )  =  ( ( A . w U ) ( 1st `  ( 2nd `  R
) ) ( ~ w `  ( A . w U ) ) )  <->  ( ( ~ t `  A ) . w U )  =  ( ~ w `  ( A . w U ) ) ) )
6453, 63mpbid 201 1  |-  ( ( R  e.  Vec  /\  <. + t ,  . t >.  e.  RingOps  /\  ( A  e.  X  /\  U  e.  W ) )  -> 
( ( ~ t `  A ) . w U )  =  ( ~ w `  ( A . w U ) ) )
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   GrpOpcgr 20853  GIdcgi 20854   invcgn 20855    /g cgs 20856   AbelOpcablo 20948   RingOpscrngo 21042    Vec cvec 25449
This theorem is referenced by:  muldisc  25481
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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