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Theorem mulgass3 15670
Description: An associative property between group multiple and ring multiplication. (Contributed by Mario Carneiro, 14-Jun-2015.)
Hypotheses
Ref Expression
mulgass3.b  |-  B  =  ( Base `  R
)
mulgass3.m  |-  .x.  =  (.g
`  R )
mulgass3.t  |-  .X.  =  ( .r `  R )
Assertion
Ref Expression
mulgass3  |-  ( ( R  e.  Ring  /\  ( N  e.  ZZ  /\  X  e.  B  /\  Y  e.  B ) )  -> 
( X  .X.  ( N  .x.  Y ) )  =  ( N  .x.  ( X  .X.  Y ) ) )

Proof of Theorem mulgass3
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2388 . . . . . 6  |-  (oppr `  R
)  =  (oppr `  R
)
21opprrng 15664 . . . . 5  |-  ( R  e.  Ring  ->  (oppr `  R
)  e.  Ring )
32adantr 452 . . . 4  |-  ( ( R  e.  Ring  /\  ( N  e.  ZZ  /\  X  e.  B  /\  Y  e.  B ) )  -> 
(oppr `  R )  e.  Ring )
4 simpr1 963 . . . 4  |-  ( ( R  e.  Ring  /\  ( N  e.  ZZ  /\  X  e.  B  /\  Y  e.  B ) )  ->  N  e.  ZZ )
5 simpr3 965 . . . 4  |-  ( ( R  e.  Ring  /\  ( N  e.  ZZ  /\  X  e.  B  /\  Y  e.  B ) )  ->  Y  e.  B )
6 simpr2 964 . . . 4  |-  ( ( R  e.  Ring  /\  ( N  e.  ZZ  /\  X  e.  B  /\  Y  e.  B ) )  ->  X  e.  B )
7 mulgass3.b . . . . . 6  |-  B  =  ( Base `  R
)
81, 7opprbas 15662 . . . . 5  |-  B  =  ( Base `  (oppr `  R
) )
9 eqid 2388 . . . . 5  |-  (.g `  (oppr `  R
) )  =  (.g `  (oppr
`  R ) )
10 eqid 2388 . . . . 5  |-  ( .r
`  (oppr
`  R ) )  =  ( .r `  (oppr `  R ) )
118, 9, 10mulgass2 15638 . . . 4  |-  ( ( (oppr
`  R )  e. 
Ring  /\  ( N  e.  ZZ  /\  Y  e.  B  /\  X  e.  B ) )  -> 
( ( N (.g `  (oppr
`  R ) ) Y ) ( .r
`  (oppr
`  R ) ) X )  =  ( N (.g `  (oppr
`  R ) ) ( Y ( .r
`  (oppr
`  R ) ) X ) ) )
123, 4, 5, 6, 11syl13anc 1186 . . 3  |-  ( ( R  e.  Ring  /\  ( N  e.  ZZ  /\  X  e.  B  /\  Y  e.  B ) )  -> 
( ( N (.g `  (oppr
`  R ) ) Y ) ( .r
`  (oppr
`  R ) ) X )  =  ( N (.g `  (oppr
`  R ) ) ( Y ( .r
`  (oppr
`  R ) ) X ) ) )
13 mulgass3.t . . . 4  |-  .X.  =  ( .r `  R )
147, 13, 1, 10opprmul 15659 . . 3  |-  ( ( N (.g `  (oppr
`  R ) ) Y ) ( .r
`  (oppr
`  R ) ) X )  =  ( X  .X.  ( N
(.g `  (oppr
`  R ) ) Y ) )
157, 13, 1, 10opprmul 15659 . . . 4  |-  ( Y ( .r `  (oppr `  R
) ) X )  =  ( X  .X.  Y )
1615oveq2i 6032 . . 3  |-  ( N (.g `  (oppr
`  R ) ) ( Y ( .r
`  (oppr
`  R ) ) X ) )  =  ( N (.g `  (oppr `  R
) ) ( X 
.X.  Y ) )
1712, 14, 163eqtr3g 2443 . 2  |-  ( ( R  e.  Ring  /\  ( N  e.  ZZ  /\  X  e.  B  /\  Y  e.  B ) )  -> 
( X  .X.  ( N (.g `  (oppr
`  R ) ) Y ) )  =  ( N (.g `  (oppr `  R
) ) ( X 
.X.  Y ) ) )
18 mulgass3.m . . . . 5  |-  .x.  =  (.g
`  R )
197a1i 11 . . . . 5  |-  ( ( R  e.  Ring  /\  ( N  e.  ZZ  /\  X  e.  B  /\  Y  e.  B ) )  ->  B  =  ( Base `  R ) )
208a1i 11 . . . . 5  |-  ( ( R  e.  Ring  /\  ( N  e.  ZZ  /\  X  e.  B  /\  Y  e.  B ) )  ->  B  =  ( Base `  (oppr
`  R ) ) )
21 ssv 3312 . . . . . 6  |-  B  C_  _V
2221a1i 11 . . . . 5  |-  ( ( R  e.  Ring  /\  ( N  e.  ZZ  /\  X  e.  B  /\  Y  e.  B ) )  ->  B  C_  _V )
23 ovex 6046 . . . . . 6  |-  ( x ( +g  `  R
) y )  e. 
_V
2423a1i 11 . . . . 5  |-  ( ( ( R  e.  Ring  /\  ( N  e.  ZZ  /\  X  e.  B  /\  Y  e.  B )
)  /\  ( x  e.  _V  /\  y  e. 
_V ) )  -> 
( x ( +g  `  R ) y )  e.  _V )
25 eqid 2388 . . . . . . . 8  |-  ( +g  `  R )  =  ( +g  `  R )
261, 25oppradd 15663 . . . . . . 7  |-  ( +g  `  R )  =  ( +g  `  (oppr `  R
) )
2726oveqi 6034 . . . . . 6  |-  ( x ( +g  `  R
) y )  =  ( x ( +g  `  (oppr
`  R ) ) y )
2827a1i 11 . . . . 5  |-  ( ( ( R  e.  Ring  /\  ( N  e.  ZZ  /\  X  e.  B  /\  Y  e.  B )
)  /\  ( x  e.  _V  /\  y  e. 
_V ) )  -> 
( x ( +g  `  R ) y )  =  ( x ( +g  `  (oppr `  R
) ) y ) )
2918, 9, 19, 20, 22, 24, 28mulgpropd 14851 . . . 4  |-  ( ( R  e.  Ring  /\  ( N  e.  ZZ  /\  X  e.  B  /\  Y  e.  B ) )  ->  .x.  =  (.g `  (oppr
`  R ) ) )
3029oveqd 6038 . . 3  |-  ( ( R  e.  Ring  /\  ( N  e.  ZZ  /\  X  e.  B  /\  Y  e.  B ) )  -> 
( N  .x.  Y
)  =  ( N (.g `  (oppr
`  R ) ) Y ) )
3130oveq2d 6037 . 2  |-  ( ( R  e.  Ring  /\  ( N  e.  ZZ  /\  X  e.  B  /\  Y  e.  B ) )  -> 
( X  .X.  ( N  .x.  Y ) )  =  ( X  .X.  ( N (.g `  (oppr
`  R ) ) Y ) ) )
3229oveqd 6038 . 2  |-  ( ( R  e.  Ring  /\  ( N  e.  ZZ  /\  X  e.  B  /\  Y  e.  B ) )  -> 
( N  .x.  ( X  .X.  Y ) )  =  ( N (.g `  (oppr
`  R ) ) ( X  .X.  Y
) ) )
3317, 31, 323eqtr4d 2430 1  |-  ( ( R  e.  Ring  /\  ( N  e.  ZZ  /\  X  e.  B  /\  Y  e.  B ) )  -> 
( X  .X.  ( N  .x.  Y ) )  =  ( N  .x.  ( X  .X.  Y ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   _Vcvv 2900    C_ wss 3264   ` cfv 5395  (class class class)co 6021   ZZcz 10215   Basecbs 13397   +g cplusg 13457   .rcmulr 13458  .gcmg 14617   Ringcrg 15588  opprcoppr 15655
This theorem is referenced by:  zlmassa  16729
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-inf2 7530  ax-cnex 8980  ax-resscn 8981  ax-1cn 8982  ax-icn 8983  ax-addcl 8984  ax-addrcl 8985  ax-mulcl 8986  ax-mulrcl 8987  ax-mulcom 8988  ax-addass 8989  ax-mulass 8990  ax-distr 8991  ax-i2m1 8992  ax-1ne0 8993  ax-1rid 8994  ax-rnegex 8995  ax-rrecex 8996  ax-cnre 8997  ax-pre-lttri 8998  ax-pre-lttrn 8999  ax-pre-ltadd 9000  ax-pre-mulgt0 9001
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rmo 2658  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-1st 6289  df-2nd 6290  df-tpos 6416  df-riota 6486  df-recs 6570  df-rdg 6605  df-er 6842  df-en 7047  df-dom 7048  df-sdom 7049  df-pnf 9056  df-mnf 9057  df-xr 9058  df-ltxr 9059  df-le 9060  df-sub 9226  df-neg 9227  df-nn 9934  df-2 9991  df-3 9992  df-n0 10155  df-z 10216  df-uz 10422  df-fz 10977  df-seq 11252  df-ndx 13400  df-slot 13401  df-base 13402  df-sets 13403  df-plusg 13470  df-mulr 13471  df-0g 13655  df-mnd 14618  df-grp 14740  df-minusg 14741  df-mulg 14743  df-mgp 15577  df-rng 15591  df-ur 15593  df-oppr 15656
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