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Theorem muldisc 25481
Description: Multiplication by a difference of scalars. (Contributed by FL, 12-Sep-2010.)
Hypotheses
Ref Expression
muldisc.1  |-  X  =  ran  ( 1st `  ( 1st `  R ) )
muldisc.2  |-  + t  =  ( 1st `  ( 1st `  R ) )
muldisc.4  |-  + w  =  ( 1st `  ( 2nd `  R ) )
muldisc.5  |-  . w  =  ( 2nd `  ( 2nd `  R ) )
muldisc.6  |-  W  =  ran  ( 1st `  ( 2nd `  R ) )
muldisc.7  |-  - t  =  (  /g  `  + t )
muldisc.8  |-  - w  =  (  /g  `  + w )
muldisc.9  |-  . t  =  ( 2nd `  ( 1st `  R ) )
Assertion
Ref Expression
muldisc  |-  ( ( R  e.  Vec  /\  <. + t ,  . t >.  e.  RingOps )  ->  A. u  e.  W  A. x  e.  X  A. y  e.  X  ( (
x - t y
) . w u
)  =  ( ( x . w u
) - w (
y . w u
) ) )
Distinct variable groups:    u, + t, x, y    u, . t, x, y    u, R, x, y    x, W, y    y, X
Allowed substitution hints:    W( u)    X( x, u)    - t( x, y, u)    + w( x, y, u)    - w( x, y, u)    . w( x, y, u)

Proof of Theorem muldisc
StepHypRef Expression
1 muldisc.2 . . . . . . . . . . . . 13  |-  + t  =  ( 1st `  ( 1st `  R ) )
2 fvex 5539 . . . . . . . . . . . . 13  |-  ( 1st `  ( 1st `  R
) )  e.  _V
31, 2eqeltri 2353 . . . . . . . . . . . 12  |-  + t  e.  _V
4 muldisc.9 . . . . . . . . . . . . 13  |-  . t  =  ( 2nd `  ( 1st `  R ) )
5 fvex 5539 . . . . . . . . . . . . 13  |-  ( 2nd `  ( 1st `  R
) )  e.  _V
64, 5eqeltri 2353 . . . . . . . . . . . 12  |-  . t  e.  _V
73, 6op1st 6128 . . . . . . . . . . 11  |-  ( 1st `  <. + t ,  . t >. )  =  + t
87eqcomi 2287 . . . . . . . . . 10  |-  + t  =  ( 1st `  <. + t ,  . t >. )
98rngogrpo 21057 . . . . . . . . 9  |-  ( <. + t ,  . t >.  e.  RingOps  ->  + t  e.  GrpOp
)
109adantl 452 . . . . . . . 8  |-  ( ( R  e.  Vec  /\  <. + t ,  . t >.  e.  RingOps )  ->  + t  e.  GrpOp )
1110ad2antrr 706 . . . . . . 7  |-  ( ( ( ( R  e. 
Vec  /\  <. + t ,  . t >.  e.  RingOps )  /\  ( u  e.  W  /\  x  e.  X ) )  /\  y  e.  X )  ->  + t  e.  GrpOp )
12 muldisc.1 . . . . . . . . . . 11  |-  X  =  ran  ( 1st `  ( 1st `  R ) )
1312eleq2i 2347 . . . . . . . . . 10  |-  ( x  e.  X  <->  x  e.  ran  ( 1st `  ( 1st `  R ) ) )
1413biimpi 186 . . . . . . . . 9  |-  ( x  e.  X  ->  x  e.  ran  ( 1st `  ( 1st `  R ) ) )
1514adantl 452 . . . . . . . 8  |-  ( ( u  e.  W  /\  x  e.  X )  ->  x  e.  ran  ( 1st `  ( 1st `  R
) ) )
1615ad2antlr 707 . . . . . . 7  |-  ( ( ( ( R  e. 
Vec  /\  <. + t ,  . t >.  e.  RingOps )  /\  ( u  e.  W  /\  x  e.  X ) )  /\  y  e.  X )  ->  x  e.  ran  ( 1st `  ( 1st `  R
) ) )
1712eleq2i 2347 . . . . . . . . 9  |-  ( y  e.  X  <->  y  e.  ran  ( 1st `  ( 1st `  R ) ) )
1817biimpi 186 . . . . . . . 8  |-  ( y  e.  X  ->  y  e.  ran  ( 1st `  ( 1st `  R ) ) )
1918adantl 452 . . . . . . 7  |-  ( ( ( ( R  e. 
Vec  /\  <. + t ,  . t >.  e.  RingOps )  /\  ( u  e.  W  /\  x  e.  X ) )  /\  y  e.  X )  ->  y  e.  ran  ( 1st `  ( 1st `  R
) ) )
201rneqi 4905 . . . . . . . . 9  |-  ran  + t  =  ran  ( 1st `  ( 1st `  R
) )
2120eqcomi 2287 . . . . . . . 8  |-  ran  ( 1st `  ( 1st `  R
) )  =  ran  + t
22 eqid 2283 . . . . . . . 8  |-  ( inv `  + t )  =  ( inv `  + t )
23 muldisc.7 . . . . . . . 8  |-  - t  =  (  /g  `  + t )
2421, 22, 23grpodivval 20910 . . . . . . 7  |-  ( ( + t  e.  GrpOp  /\  x  e.  ran  ( 1st `  ( 1st `  R
) )  /\  y  e.  ran  ( 1st `  ( 1st `  R ) ) )  ->  ( x - t y )  =  ( x + t
( ( inv `  + t ) `  y
) ) )
2511, 16, 19, 24syl3anc 1182 . . . . . 6  |-  ( ( ( ( R  e. 
Vec  /\  <. + t ,  . t >.  e.  RingOps )  /\  ( u  e.  W  /\  x  e.  X ) )  /\  y  e.  X )  ->  ( x - t
y )  =  ( x + t (
( inv `  + t ) `  y
) ) )
2625oveq1d 5873 . . . . 5  |-  ( ( ( ( R  e. 
Vec  /\  <. + t ,  . t >.  e.  RingOps )  /\  ( u  e.  W  /\  x  e.  X ) )  /\  y  e.  X )  ->  ( ( x - t y ) . w u )  =  ( ( x + t ( ( inv `  + t ) `  y ) ) . w u ) )
27 simplll 734 . . . . . 6  |-  ( ( ( ( R  e. 
Vec  /\  <. + t ,  . t >.  e.  RingOps )  /\  ( u  e.  W  /\  x  e.  X ) )  /\  y  e.  X )  ->  R  e.  Vec  )
28 simplrl 736 . . . . . 6  |-  ( ( ( ( R  e. 
Vec  /\  <. + t ,  . t >.  e.  RingOps )  /\  ( u  e.  W  /\  x  e.  X ) )  /\  y  e.  X )  ->  u  e.  W )
29 simplrr 737 . . . . . 6  |-  ( ( ( ( R  e. 
Vec  /\  <. + t ,  . t >.  e.  RingOps )  /\  ( u  e.  W  /\  x  e.  X ) )  /\  y  e.  X )  ->  x  e.  X )
301, 9syl5eqelr 2368 . . . . . . . 8  |-  ( <. + t ,  . t >.  e.  RingOps  ->  ( 1st `  ( 1st `  R ) )  e.  GrpOp )
3130ad2antlr 707 . . . . . . 7  |-  ( ( ( R  e.  Vec  /\ 
<. + t ,  . t >.  e.  RingOps )  /\  ( u  e.  W  /\  x  e.  X
) )  ->  ( 1st `  ( 1st `  R
) )  e.  GrpOp )
321fveq2i 5528 . . . . . . . 8  |-  ( inv `  + t )  =  ( inv `  ( 1st `  ( 1st `  R
) ) )
3312, 32grpoinvcl 20893 . . . . . . 7  |-  ( ( ( 1st `  ( 1st `  R ) )  e.  GrpOp  /\  y  e.  X )  ->  (
( inv `  + t ) `  y
)  e.  X )
3431, 33sylan 457 . . . . . 6  |-  ( ( ( ( R  e. 
Vec  /\  <. + t ,  . t >.  e.  RingOps )  /\  ( u  e.  W  /\  x  e.  X ) )  /\  y  e.  X )  ->  ( ( inv `  + t ) `  y
)  e.  X )
35 muldisc.4 . . . . . . 7  |-  + w  =  ( 1st `  ( 2nd `  R ) )
36 muldisc.5 . . . . . . 7  |-  . w  =  ( 2nd `  ( 2nd `  R ) )
37 muldisc.6 . . . . . . 7  |-  W  =  ran  ( 1st `  ( 2nd `  R ) )
3812, 1, 35, 36, 37vecax5b 25459 . . . . . 6  |-  ( ( R  e.  Vec  /\  ( u  e.  W  /\  x  e.  X  /\  ( ( inv `  + t ) `  y
)  e.  X ) )  ->  ( (
x + t (
( inv `  + t ) `  y
) ) . w
u )  =  ( ( x . w
u ) + w
( ( ( inv `  + t ) `  y ) . w
u ) ) )
3927, 28, 29, 34, 38syl13anc 1184 . . . . 5  |-  ( ( ( ( R  e. 
Vec  /\  <. + t ,  . t >.  e.  RingOps )  /\  ( u  e.  W  /\  x  e.  X ) )  /\  y  e.  X )  ->  ( ( x + t ( ( inv `  + t ) `  y ) ) . w u )  =  ( ( x . w u ) + w ( ( ( inv `  + t
) `  y ) . w u ) ) )
40 simpllr 735 . . . . . . 7  |-  ( ( ( ( R  e. 
Vec  /\  <. + t ,  . t >.  e.  RingOps )  /\  ( u  e.  W  /\  x  e.  X ) )  /\  y  e.  X )  -> 
<. + t ,  . t >.  e.  RingOps )
411eqcomi 2287 . . . . . . . . 9  |-  ( 1st `  ( 1st `  R
) )  =  + t
4241rneqi 4905 . . . . . . . 8  |-  ran  ( 1st `  ( 1st `  R
) )  =  ran  + t
4335fveq2i 5528 . . . . . . . 8  |-  ( inv `  + w )  =  ( inv `  ( 1st `  ( 2nd `  R
) ) )
4442, 37, 36, 22, 43, 1, 4mulinvsca 25480 . . . . . . 7  |-  ( ( R  e.  Vec  /\  <. + t ,  . t >.  e.  RingOps  /\  ( y  e.  ran  ( 1st `  ( 1st `  R ) )  /\  u  e.  W
) )  ->  (
( ( inv `  + t ) `  y
) . w u
)  =  ( ( inv `  + w
) `  ( y . w u ) ) )
4527, 40, 19, 28, 44syl112anc 1186 . . . . . 6  |-  ( ( ( ( R  e. 
Vec  /\  <. + t ,  . t >.  e.  RingOps )  /\  ( u  e.  W  /\  x  e.  X ) )  /\  y  e.  X )  ->  ( ( ( inv `  + t ) `  y ) . w
u )  =  ( ( inv `  + w ) `  (
y . w u
) ) )
4645oveq2d 5874 . . . . 5  |-  ( ( ( ( R  e. 
Vec  /\  <. + t ,  . t >.  e.  RingOps )  /\  ( u  e.  W  /\  x  e.  X ) )  /\  y  e.  X )  ->  ( ( x . w u ) + w ( ( ( inv `  + t
) `  y ) . w u ) )  =  ( ( x . w u ) + w ( ( inv `  + w
) `  ( y . w u ) ) ) )
4726, 39, 463eqtrd 2319 . . . 4  |-  ( ( ( ( R  e. 
Vec  /\  <. + t ,  . t >.  e.  RingOps )  /\  ( u  e.  W  /\  x  e.  X ) )  /\  y  e.  X )  ->  ( ( x - t y ) . w u )  =  ( ( x . w u ) + w ( ( inv `  + w ) `  ( y . w
u ) ) ) )
4835vecax1 25453 . . . . . . . 8  |-  ( R  e.  Vec  ->  + w  e.  AbelOp )
49 ablogrpo 20951 . . . . . . . 8  |-  ( + w  e.  AbelOp  ->  + w  e.  GrpOp )
5048, 49syl 15 . . . . . . 7  |-  ( R  e.  Vec  ->  + w  e.  GrpOp )
5150adantr 451 . . . . . 6  |-  ( ( R  e.  Vec  /\  <. + t ,  . t >.  e.  RingOps )  ->  + w  e.  GrpOp )
5251ad2antrr 706 . . . . 5  |-  ( ( ( ( R  e. 
Vec  /\  <. + t ,  . t >.  e.  RingOps )  /\  ( u  e.  W  /\  x  e.  X ) )  /\  y  e.  X )  ->  + w  e.  GrpOp )
5312, 37, 36prvs 25478 . . . . . . . . . . . 12  |-  ( ( R  e.  Vec  /\  x  e.  X  /\  u  e.  W )  ->  ( x . w
u )  e.  W
)
54533exp 1150 . . . . . . . . . . 11  |-  ( R  e.  Vec  ->  (
x  e.  X  -> 
( u  e.  W  ->  ( x . w
u )  e.  W
) ) )
5554com3l 75 . . . . . . . . . 10  |-  ( x  e.  X  ->  (
u  e.  W  -> 
( R  e.  Vec  ->  ( x . w
u )  e.  W
) ) )
5655impcom 419 . . . . . . . . 9  |-  ( ( u  e.  W  /\  x  e.  X )  ->  ( R  e.  Vec  ->  ( x . w
u )  e.  W
) )
5756com12 27 . . . . . . . 8  |-  ( R  e.  Vec  ->  (
( u  e.  W  /\  x  e.  X
)  ->  ( x . w u )  e.  W ) )
5857adantr 451 . . . . . . 7  |-  ( ( R  e.  Vec  /\  <. + t ,  . t >.  e.  RingOps )  ->  (
( u  e.  W  /\  x  e.  X
)  ->  ( x . w u )  e.  W ) )
5958imp 418 . . . . . 6  |-  ( ( ( R  e.  Vec  /\ 
<. + t ,  . t >.  e.  RingOps )  /\  ( u  e.  W  /\  x  e.  X
) )  ->  (
x . w u
)  e.  W )
6059adantr 451 . . . . 5  |-  ( ( ( ( R  e. 
Vec  /\  <. + t ,  . t >.  e.  RingOps )  /\  ( u  e.  W  /\  x  e.  X ) )  /\  y  e.  X )  ->  ( x . w
u )  e.  W
)
6112, 37, 36prvs 25478 . . . . . . . . . . 11  |-  ( ( R  e.  Vec  /\  y  e.  X  /\  u  e.  W )  ->  ( y . w
u )  e.  W
)
62613exp 1150 . . . . . . . . . 10  |-  ( R  e.  Vec  ->  (
y  e.  X  -> 
( u  e.  W  ->  ( y . w
u )  e.  W
) ) )
6362adantr 451 . . . . . . . . 9  |-  ( ( R  e.  Vec  /\  <. + t ,  . t >.  e.  RingOps )  ->  (
y  e.  X  -> 
( u  e.  W  ->  ( y . w
u )  e.  W
) ) )
6463com3r 73 . . . . . . . 8  |-  ( u  e.  W  ->  (
( R  e.  Vec  /\ 
<. + t ,  . t >.  e.  RingOps )  -> 
( y  e.  X  ->  ( y . w
u )  e.  W
) ) )
6564adantr 451 . . . . . . 7  |-  ( ( u  e.  W  /\  x  e.  X )  ->  ( ( R  e. 
Vec  /\  <. + t ,  . t >.  e.  RingOps )  ->  ( y  e.  X  ->  ( y . w u )  e.  W ) ) )
6665impcom 419 . . . . . 6  |-  ( ( ( R  e.  Vec  /\ 
<. + t ,  . t >.  e.  RingOps )  /\  ( u  e.  W  /\  x  e.  X
) )  ->  (
y  e.  X  -> 
( y . w
u )  e.  W
) )
6766imp 418 . . . . 5  |-  ( ( ( ( R  e. 
Vec  /\  <. + t ,  . t >.  e.  RingOps )  /\  ( u  e.  W  /\  x  e.  X ) )  /\  y  e.  X )  ->  ( y . w
u )  e.  W
)
6835eqcomi 2287 . . . . . . . 8  |-  ( 1st `  ( 2nd `  R
) )  =  + w
6968rneqi 4905 . . . . . . 7  |-  ran  ( 1st `  ( 2nd `  R
) )  =  ran  + w
7037, 69eqtri 2303 . . . . . 6  |-  W  =  ran  + w
71 eqid 2283 . . . . . 6  |-  ( inv `  + w )  =  ( inv `  + w )
72 muldisc.8 . . . . . 6  |-  - w  =  (  /g  `  + w )
7370, 71, 72grpodivval 20910 . . . . 5  |-  ( ( + w  e.  GrpOp  /\  ( x . w
u )  e.  W  /\  ( y . w
u )  e.  W
)  ->  ( (
x . w u
) - w (
y . w u
) )  =  ( ( x . w
u ) + w
( ( inv `  + w ) `  (
y . w u
) ) ) )
7452, 60, 67, 73syl3anc 1182 . . . 4  |-  ( ( ( ( R  e. 
Vec  /\  <. + t ,  . t >.  e.  RingOps )  /\  ( u  e.  W  /\  x  e.  X ) )  /\  y  e.  X )  ->  ( ( x . w u ) - w ( y . w u ) )  =  ( ( x . w u ) + w ( ( inv `  + w
) `  ( y . w u ) ) ) )
7547, 74eqtr4d 2318 . . 3  |-  ( ( ( ( R  e. 
Vec  /\  <. + t ,  . t >.  e.  RingOps )  /\  ( u  e.  W  /\  x  e.  X ) )  /\  y  e.  X )  ->  ( ( x - t y ) . w u )  =  ( ( x . w u ) - w ( y . w u ) ) )
7675ralrimiva 2626 . 2  |-  ( ( ( R  e.  Vec  /\ 
<. + t ,  . t >.  e.  RingOps )  /\  ( u  e.  W  /\  x  e.  X
) )  ->  A. y  e.  X  ( (
x - t y
) . w u
)  =  ( ( x . w u
) - w (
y . w u
) ) )
7776ralrimivva 2635 1  |-  ( ( R  e.  Vec  /\  <. + t ,  . t >.  e.  RingOps )  ->  A. u  e.  W  A. x  e.  X  A. y  e.  X  ( (
x - t y
) . w u
)  =  ( ( x . w u
) - w (
y . w u
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   _Vcvv 2788   <.cop 3643   ran crn 4690   ` cfv 5255  (class class class)co 5858   1stc1st 6120   2ndc2nd 6121   GrpOpcgr 20853   invcgn 20855    /g cgs 20856   AbelOpcablo 20948   RingOpscrngo 21042    Vec cvec 25449
This theorem is referenced by:  vecax5c  25483
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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