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Theorem mulgass3 17154
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 2441 . . . . . 6  |-  (oppr `  R
)  =  (oppr `  R
)
21opprring 17148 . . . . 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 1001 . . . 4  |-  ( ( R  e.  Ring  /\  ( N  e.  ZZ  /\  X  e.  B  /\  Y  e.  B ) )  ->  N  e.  ZZ )
5 simpr3 1003 . . . 4  |-  ( ( R  e.  Ring  /\  ( N  e.  ZZ  /\  X  e.  B  /\  Y  e.  B ) )  ->  Y  e.  B )
6 simpr2 1002 . . . 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 17146 . . . . 5  |-  B  =  ( Base `  (oppr `  R
) )
9 eqid 2441 . . . . 5  |-  (.g `  (oppr `  R
) )  =  (.g `  (oppr
`  R ) )
10 eqid 2441 . . . . 5  |-  ( .r
`  (oppr
`  R ) )  =  ( .r `  (oppr `  R ) )
118, 9, 10mulgass2 17115 . . . 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 1229 . . 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 17143 . . 3  |-  ( ( N (.g `  (oppr
`  R ) ) Y ) ( .r
`  (oppr
`  R ) ) X )  =  ( X  .X.  ( N
(.g `  (oppr
`  R ) ) Y ) )
157, 13, 1, 10opprmul 17143 . . . 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 2505 . 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 3506 . . . . . 6  |-  B  C_  _V
2221a1i 11 . . . . 5  |-  ( ( R  e.  Ring  /\  ( N  e.  ZZ  /\  X  e.  B  /\  Y  e.  B ) )  ->  B  C_  _V )
23 ovex 6305 . . . . . 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 2441 . . . . . . . 8  |-  ( +g  `  R )  =  ( +g  `  R )
261, 25oppradd 17147 . . . . . . 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 16044 . . . 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 2492 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 972    = wceq 1381    e. wcel 1802   _Vcvv 3093    C_ wss 3458   ` cfv 5574  (class class class)co 6277   ZZcz 10865   Basecbs 14504   +g cplusg 14569   .rcmulr 14570  .gcmg 15925   Ringcrg 17066  opprcoppr 17139
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4544  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573  ax-inf2 8056  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-pss 3474  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-tp 4015  df-op 4017  df-uni 4231  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-tr 4527  df-eprel 4777  df-id 4781  df-po 4786  df-so 4787  df-fr 4824  df-we 4826  df-ord 4867  df-on 4868  df-lim 4869  df-suc 4870  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6682  df-1st 6781  df-2nd 6782  df-tpos 6953  df-recs 7040  df-rdg 7074  df-er 7309  df-en 7515  df-dom 7516  df-sdom 7517  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9807  df-neg 9808  df-nn 10538  df-2 10595  df-3 10596  df-n0 10797  df-z 10866  df-uz 11086  df-fz 11677  df-seq 12082  df-ndx 14507  df-slot 14508  df-base 14509  df-sets 14510  df-plusg 14582  df-mulr 14583  df-0g 14711  df-mgm 15741  df-sgrp 15780  df-mnd 15790  df-grp 15926  df-minusg 15927  df-mulg 15929  df-mgp 17010  df-ur 17022  df-ring 17068  df-oppr 17140
This theorem is referenced by:  zlmassa  18428
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