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Theorem mulgass3 17065
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 2462 . . . . . 6  |-  (oppr `  R
)  =  (oppr `  R
)
21opprrng 17059 . . . . 5  |-  ( R  e.  Ring  ->  (oppr `  R
)  e.  Ring )
32adantr 465 . . . 4  |-  ( ( R  e.  Ring  /\  ( N  e.  ZZ  /\  X  e.  B  /\  Y  e.  B ) )  -> 
(oppr `  R )  e.  Ring )
4 simpr1 997 . . . 4  |-  ( ( R  e.  Ring  /\  ( N  e.  ZZ  /\  X  e.  B  /\  Y  e.  B ) )  ->  N  e.  ZZ )
5 simpr3 999 . . . 4  |-  ( ( R  e.  Ring  /\  ( N  e.  ZZ  /\  X  e.  B  /\  Y  e.  B ) )  ->  Y  e.  B )
6 simpr2 998 . . . 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 17057 . . . . 5  |-  B  =  ( Base `  (oppr `  R
) )
9 eqid 2462 . . . . 5  |-  (.g `  (oppr `  R
) )  =  (.g `  (oppr
`  R ) )
10 eqid 2462 . . . . 5  |-  ( .r
`  (oppr
`  R ) )  =  ( .r `  (oppr `  R ) )
118, 9, 10mulgass2 17028 . . . 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 1225 . . 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 17054 . . 3  |-  ( ( N (.g `  (oppr
`  R ) ) Y ) ( .r
`  (oppr
`  R ) ) X )  =  ( X  .X.  ( N
(.g `  (oppr
`  R ) ) Y ) )
157, 13, 1, 10opprmul 17054 . . . 4  |-  ( Y ( .r `  (oppr `  R
) ) X )  =  ( X  .X.  Y )
1615oveq2i 6288 . . 3  |-  ( N (.g `  (oppr
`  R ) ) ( Y ( .r
`  (oppr
`  R ) ) X ) )  =  ( N (.g `  (oppr `  R
) ) ( X 
.X.  Y ) )
1712, 14, 163eqtr3g 2526 . 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 3519 . . . . . 6  |-  B  C_  _V
2221a1i 11 . . . . 5  |-  ( ( R  e.  Ring  /\  ( N  e.  ZZ  /\  X  e.  B  /\  Y  e.  B ) )  ->  B  C_  _V )
23 ovex 6302 . . . . . 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 2462 . . . . . . . 8  |-  ( +g  `  R )  =  ( +g  `  R )
261, 25oppradd 17058 . . . . . . 7  |-  ( +g  `  R )  =  ( +g  `  (oppr `  R
) )
2726oveqi 6290 . . . . . 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 15970 . . . 4  |-  ( ( R  e.  Ring  /\  ( N  e.  ZZ  /\  X  e.  B  /\  Y  e.  B ) )  ->  .x.  =  (.g `  (oppr
`  R ) ) )
3029oveqd 6294 . . 3  |-  ( ( R  e.  Ring  /\  ( N  e.  ZZ  /\  X  e.  B  /\  Y  e.  B ) )  -> 
( N  .x.  Y
)  =  ( N (.g `  (oppr
`  R ) ) Y ) )
3130oveq2d 6293 . 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 6294 . 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 2513 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762   _Vcvv 3108    C_ wss 3471   ` cfv 5581  (class class class)co 6277   ZZcz 10855   Basecbs 14481   +g cplusg 14546   .rcmulr 14547  .gcmg 15722   Ringcrg 16981  opprcoppr 17050
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-inf2 8049  ax-cnex 9539  ax-resscn 9540  ax-1cn 9541  ax-icn 9542  ax-addcl 9543  ax-addrcl 9544  ax-mulcl 9545  ax-mulrcl 9546  ax-mulcom 9547  ax-addass 9548  ax-mulass 9549  ax-distr 9550  ax-i2m1 9551  ax-1ne0 9552  ax-1rid 9553  ax-rnegex 9554  ax-rrecex 9555  ax-cnre 9556  ax-pre-lttri 9557  ax-pre-lttrn 9558  ax-pre-ltadd 9559  ax-pre-mulgt0 9560
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-nel 2660  df-ral 2814  df-rex 2815  df-reu 2816  df-rmo 2817  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-tp 4027  df-op 4029  df-uni 4241  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-tr 4536  df-eprel 4786  df-id 4790  df-po 4795  df-so 4796  df-fr 4833  df-we 4835  df-ord 4876  df-on 4877  df-lim 4878  df-suc 4879  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6674  df-1st 6776  df-2nd 6777  df-tpos 6947  df-recs 7034  df-rdg 7068  df-er 7303  df-en 7509  df-dom 7510  df-sdom 7511  df-pnf 9621  df-mnf 9622  df-xr 9623  df-ltxr 9624  df-le 9625  df-sub 9798  df-neg 9799  df-nn 10528  df-2 10585  df-3 10586  df-n0 10787  df-z 10856  df-uz 11074  df-fz 11664  df-seq 12066  df-ndx 14484  df-slot 14485  df-base 14486  df-sets 14487  df-plusg 14559  df-mulr 14560  df-0g 14688  df-mnd 15723  df-grp 15853  df-minusg 15854  df-mulg 15856  df-mgp 16927  df-ur 16939  df-rng 16983  df-oppr 17051
This theorem is referenced by:  zlmassa  18323
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