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Theorem mulgass3 13317
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 2187 . . . . . 6  |-  (oppr `  R
)  =  (oppr `  R
)
21opprring 13311 . . . . 5  |-  ( R  e.  Ring  ->  (oppr `  R
)  e.  Ring )
32adantr 276 . . . 4  |-  ( ( R  e.  Ring  /\  ( N  e.  ZZ  /\  X  e.  B  /\  Y  e.  B ) )  -> 
(oppr `  R )  e.  Ring )
4 simpr1 1004 . . . 4  |-  ( ( R  e.  Ring  /\  ( N  e.  ZZ  /\  X  e.  B  /\  Y  e.  B ) )  ->  N  e.  ZZ )
5 simpr3 1006 . . . . 5  |-  ( ( R  e.  Ring  /\  ( N  e.  ZZ  /\  X  e.  B  /\  Y  e.  B ) )  ->  Y  e.  B )
6 mulgass3.b . . . . . . 7  |-  B  =  ( Base `  R
)
71, 6opprbasg 13308 . . . . . 6  |-  ( R  e.  Ring  ->  B  =  ( Base `  (oppr `  R
) ) )
87adantr 276 . . . . 5  |-  ( ( R  e.  Ring  /\  ( N  e.  ZZ  /\  X  e.  B  /\  Y  e.  B ) )  ->  B  =  ( Base `  (oppr
`  R ) ) )
95, 8eleqtrd 2266 . . . 4  |-  ( ( R  e.  Ring  /\  ( N  e.  ZZ  /\  X  e.  B  /\  Y  e.  B ) )  ->  Y  e.  ( Base `  (oppr
`  R ) ) )
10 simpr2 1005 . . . . 5  |-  ( ( R  e.  Ring  /\  ( N  e.  ZZ  /\  X  e.  B  /\  Y  e.  B ) )  ->  X  e.  B )
1110, 8eleqtrd 2266 . . . 4  |-  ( ( R  e.  Ring  /\  ( N  e.  ZZ  /\  X  e.  B  /\  Y  e.  B ) )  ->  X  e.  ( Base `  (oppr
`  R ) ) )
12 eqid 2187 . . . . 5  |-  ( Base `  (oppr
`  R ) )  =  ( Base `  (oppr `  R
) )
13 eqid 2187 . . . . 5  |-  (.g `  (oppr `  R
) )  =  (.g `  (oppr
`  R ) )
14 eqid 2187 . . . . 5  |-  ( .r
`  (oppr
`  R ) )  =  ( .r `  (oppr `  R ) )
1512, 13, 14mulgass2 13293 . . . 4  |-  ( ( (oppr
`  R )  e. 
Ring  /\  ( N  e.  ZZ  /\  Y  e.  ( Base `  (oppr `  R
) )  /\  X  e.  ( Base `  (oppr `  R
) ) ) )  ->  ( ( N (.g `  (oppr
`  R ) ) Y ) ( .r
`  (oppr
`  R ) ) X )  =  ( N (.g `  (oppr
`  R ) ) ( Y ( .r
`  (oppr
`  R ) ) X ) ) )
163, 4, 9, 11, 15syl13anc 1250 . . 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 ) ) )
17 simpl 109 . . . 4  |-  ( ( R  e.  Ring  /\  ( N  e.  ZZ  /\  X  e.  B  /\  Y  e.  B ) )  ->  R  e.  Ring )
183ringgrpd 13242 . . . . . 6  |-  ( ( R  e.  Ring  /\  ( N  e.  ZZ  /\  X  e.  B  /\  Y  e.  B ) )  -> 
(oppr `  R )  e.  Grp )
1912, 13, 18, 4, 9mulgcld 13034 . . . . 5  |-  ( ( R  e.  Ring  /\  ( N  e.  ZZ  /\  X  e.  B  /\  Y  e.  B ) )  -> 
( N (.g `  (oppr `  R
) ) Y )  e.  ( Base `  (oppr `  R
) ) )
2019, 8eleqtrrd 2267 . . . 4  |-  ( ( R  e.  Ring  /\  ( N  e.  ZZ  /\  X  e.  B  /\  Y  e.  B ) )  -> 
( N (.g `  (oppr `  R
) ) Y )  e.  B )
21 mulgass3.t . . . . 5  |-  .X.  =  ( .r `  R )
226, 21, 1, 14opprmulg 13304 . . . 4  |-  ( ( R  e.  Ring  /\  ( N (.g `  (oppr
`  R ) ) Y )  e.  B  /\  X  e.  B
)  ->  ( ( N (.g `  (oppr
`  R ) ) Y ) ( .r
`  (oppr
`  R ) ) X )  =  ( X  .X.  ( N
(.g `  (oppr
`  R ) ) Y ) ) )
2317, 20, 10, 22syl3anc 1248 . . 3  |-  ( ( R  e.  Ring  /\  ( N  e.  ZZ  /\  X  e.  B  /\  Y  e.  B ) )  -> 
( ( N (.g `  (oppr
`  R ) ) Y ) ( .r
`  (oppr
`  R ) ) X )  =  ( X  .X.  ( N
(.g `  (oppr
`  R ) ) Y ) ) )
246, 21, 1, 14opprmulg 13304 . . . . 5  |-  ( ( R  e.  Ring  /\  Y  e.  B  /\  X  e.  B )  ->  ( Y ( .r `  (oppr `  R ) ) X )  =  ( X 
.X.  Y ) )
2517, 5, 10, 24syl3anc 1248 . . . 4  |-  ( ( R  e.  Ring  /\  ( N  e.  ZZ  /\  X  e.  B  /\  Y  e.  B ) )  -> 
( Y ( .r
`  (oppr
`  R ) ) X )  =  ( X  .X.  Y )
)
2625oveq2d 5904 . . 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 ) ) ( X  .X.  Y
) ) )
2716, 23, 263eqtr3d 2228 . 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 ) ) )
28 mulgass3.m . . . . . 6  |-  .x.  =  (.g
`  R )
2928a1i 9 . . . . 5  |-  ( ( R  e.  Ring  /\  ( N  e.  ZZ  /\  X  e.  B  /\  Y  e.  B ) )  ->  .x.  =  (.g `  R ) )
30 eqidd 2188 . . . . 5  |-  ( ( R  e.  Ring  /\  ( N  e.  ZZ  /\  X  e.  B  /\  Y  e.  B ) )  -> 
(.g `  (oppr
`  R ) )  =  (.g `  (oppr
`  R ) ) )
316a1i 9 . . . . 5  |-  ( ( R  e.  Ring  /\  ( N  e.  ZZ  /\  X  e.  B  /\  Y  e.  B ) )  ->  B  =  ( Base `  R ) )
32 ssidd 3188 . . . . 5  |-  ( ( R  e.  Ring  /\  ( N  e.  ZZ  /\  X  e.  B  /\  Y  e.  B ) )  ->  B  C_  B )
33 eqid 2187 . . . . . . . 8  |-  ( +g  `  R )  =  ( +g  `  R )
346, 33ringacl 13267 . . . . . . 7  |-  ( ( R  e.  Ring  /\  x  e.  B  /\  y  e.  B )  ->  (
x ( +g  `  R
) y )  e.  B )
35343expb 1205 . . . . . 6  |-  ( ( R  e.  Ring  /\  (
x  e.  B  /\  y  e.  B )
)  ->  ( x
( +g  `  R ) y )  e.  B
)
3635adantlr 477 . . . . 5  |-  ( ( ( R  e.  Ring  /\  ( N  e.  ZZ  /\  X  e.  B  /\  Y  e.  B )
)  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( x ( +g  `  R ) y )  e.  B )
371, 33oppraddg 13309 . . . . . . 7  |-  ( R  e.  Ring  ->  ( +g  `  R )  =  ( +g  `  (oppr `  R
) ) )
3837oveqdr 5916 . . . . . 6  |-  ( ( R  e.  Ring  /\  ( N  e.  ZZ  /\  X  e.  B  /\  Y  e.  B ) )  -> 
( x ( +g  `  R ) y )  =  ( x ( +g  `  (oppr `  R
) ) y ) )
3938adantr 276 . . . . 5  |-  ( ( ( R  e.  Ring  /\  ( N  e.  ZZ  /\  X  e.  B  /\  Y  e.  B )
)  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( x ( +g  `  R ) y )  =  ( x ( +g  `  (oppr `  R
) ) y ) )
4029, 30, 17, 3, 31, 8, 32, 36, 39mulgpropdg 13054 . . . 4  |-  ( ( R  e.  Ring  /\  ( N  e.  ZZ  /\  X  e.  B  /\  Y  e.  B ) )  ->  .x.  =  (.g `  (oppr
`  R ) ) )
4140oveqd 5905 . . 3  |-  ( ( R  e.  Ring  /\  ( N  e.  ZZ  /\  X  e.  B  /\  Y  e.  B ) )  -> 
( N  .x.  Y
)  =  ( N (.g `  (oppr
`  R ) ) Y ) )
4241oveq2d 5904 . 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 ) ) )
4340oveqd 5905 . 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
) ) )
4427, 42, 433eqtr4d 2230 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 104    /\ w3a 979    = wceq 1363    e. wcel 2158   ` cfv 5228  (class class class)co 5888   ZZcz 9266   Basecbs 12475   +g cplusg 12550   .rcmulr 12551  .gcmg 13011   Ringcrg 13233  opprcoppr 13300
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-coll 4130  ax-sep 4133  ax-nul 4141  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-iinf 4599  ax-cnex 7915  ax-resscn 7916  ax-1cn 7917  ax-1re 7918  ax-icn 7919  ax-addcl 7920  ax-addrcl 7921  ax-mulcl 7922  ax-addcom 7924  ax-addass 7926  ax-distr 7928  ax-i2m1 7929  ax-0lt1 7930  ax-0id 7932  ax-rnegex 7933  ax-cnre 7935  ax-pre-ltirr 7936  ax-pre-ltwlin 7937  ax-pre-lttrn 7938  ax-pre-ltadd 7940
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 980  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-nel 2453  df-ral 2470  df-rex 2471  df-reu 2472  df-rmo 2473  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-if 3547  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-tr 4114  df-id 4305  df-iord 4378  df-on 4380  df-ilim 4381  df-suc 4383  df-iom 4602  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-riota 5844  df-ov 5891  df-oprab 5892  df-mpo 5893  df-1st 6154  df-2nd 6155  df-tpos 6259  df-recs 6319  df-frec 6405  df-pnf 8007  df-mnf 8008  df-xr 8009  df-ltxr 8010  df-le 8011  df-sub 8143  df-neg 8144  df-inn 8933  df-2 8991  df-3 8992  df-n0 9190  df-z 9267  df-uz 9542  df-fz 10022  df-seqfrec 10459  df-ndx 12478  df-slot 12479  df-base 12481  df-sets 12482  df-plusg 12563  df-mulr 12564  df-0g 12724  df-mgm 12793  df-sgrp 12826  df-mnd 12837  df-grp 12899  df-minusg 12900  df-mulg 13012  df-mgp 13163  df-ur 13197  df-ring 13235  df-oppr 13301
This theorem is referenced by: (None)
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