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Theorem rngrghm 15639
Description: Right-multiplication in a ring by a fixed element of the ring is a group homomorphism. (It is not usually a ring homomorphism.) (Contributed by Mario Carneiro, 4-May-2015.)
Hypotheses
Ref Expression
rnglghm.b  |-  B  =  ( Base `  R
)
rnglghm.t  |-  .x.  =  ( .r `  R )
Assertion
Ref Expression
rngrghm  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  (
x  e.  B  |->  ( x  .x.  X ) )  e.  ( R 
GrpHom  R ) )
Distinct variable groups:    x, B    x, R    x,  .x.    x, X

Proof of Theorem rngrghm
Dummy variables  y 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rnglghm.b . 2  |-  B  =  ( Base `  R
)
2 eqid 2387 . 2  |-  ( +g  `  R )  =  ( +g  `  R )
3 rnggrp 15596 . . 3  |-  ( R  e.  Ring  ->  R  e. 
Grp )
43adantr 452 . 2  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  R  e.  Grp )
5 rnglghm.t . . . . . 6  |-  .x.  =  ( .r `  R )
61, 5rngcl 15604 . . . . 5  |-  ( ( R  e.  Ring  /\  x  e.  B  /\  X  e.  B )  ->  (
x  .x.  X )  e.  B )
763expa 1153 . . . 4  |-  ( ( ( R  e.  Ring  /\  x  e.  B )  /\  X  e.  B
)  ->  ( x  .x.  X )  e.  B
)
87an32s 780 . . 3  |-  ( ( ( R  e.  Ring  /\  X  e.  B )  /\  x  e.  B
)  ->  ( x  .x.  X )  e.  B
)
9 eqid 2387 . . 3  |-  ( x  e.  B  |->  ( x 
.x.  X ) )  =  ( x  e.  B  |->  ( x  .x.  X ) )
108, 9fmptd 5832 . 2  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  (
x  e.  B  |->  ( x  .x.  X ) ) : B --> B )
11 df-3an 938 . . . . 5  |-  ( ( y  e.  B  /\  z  e.  B  /\  X  e.  B )  <->  ( ( y  e.  B  /\  z  e.  B
)  /\  X  e.  B ) )
121, 2, 5rngdir 15610 . . . . 5  |-  ( ( R  e.  Ring  /\  (
y  e.  B  /\  z  e.  B  /\  X  e.  B )
)  ->  ( (
y ( +g  `  R
) z )  .x.  X )  =  ( ( y  .x.  X
) ( +g  `  R
) ( z  .x.  X ) ) )
1311, 12sylan2br 463 . . . 4  |-  ( ( R  e.  Ring  /\  (
( y  e.  B  /\  z  e.  B
)  /\  X  e.  B ) )  -> 
( ( y ( +g  `  R ) z )  .x.  X
)  =  ( ( y  .x.  X ) ( +g  `  R
) ( z  .x.  X ) ) )
1413anass1rs 783 . . 3  |-  ( ( ( R  e.  Ring  /\  X  e.  B )  /\  ( y  e.  B  /\  z  e.  B ) )  -> 
( ( y ( +g  `  R ) z )  .x.  X
)  =  ( ( y  .x.  X ) ( +g  `  R
) ( z  .x.  X ) ) )
151, 2rngacl 15618 . . . . . 6  |-  ( ( R  e.  Ring  /\  y  e.  B  /\  z  e.  B )  ->  (
y ( +g  `  R
) z )  e.  B )
16153expb 1154 . . . . 5  |-  ( ( R  e.  Ring  /\  (
y  e.  B  /\  z  e.  B )
)  ->  ( y
( +g  `  R ) z )  e.  B
)
1716adantlr 696 . . . 4  |-  ( ( ( R  e.  Ring  /\  X  e.  B )  /\  ( y  e.  B  /\  z  e.  B ) )  -> 
( y ( +g  `  R ) z )  e.  B )
18 oveq1 6027 . . . . 5  |-  ( x  =  ( y ( +g  `  R ) z )  ->  (
x  .x.  X )  =  ( ( y ( +g  `  R
) z )  .x.  X ) )
19 ovex 6045 . . . . 5  |-  ( ( y ( +g  `  R
) z )  .x.  X )  e.  _V
2018, 9, 19fvmpt 5745 . . . 4  |-  ( ( y ( +g  `  R
) z )  e.  B  ->  ( (
x  e.  B  |->  ( x  .x.  X ) ) `  ( y ( +g  `  R
) z ) )  =  ( ( y ( +g  `  R
) z )  .x.  X ) )
2117, 20syl 16 . . 3  |-  ( ( ( R  e.  Ring  /\  X  e.  B )  /\  ( y  e.  B  /\  z  e.  B ) )  -> 
( ( x  e.  B  |->  ( x  .x.  X ) ) `  ( y ( +g  `  R ) z ) )  =  ( ( y ( +g  `  R
) z )  .x.  X ) )
22 oveq1 6027 . . . . . 6  |-  ( x  =  y  ->  (
x  .x.  X )  =  ( y  .x.  X ) )
23 ovex 6045 . . . . . 6  |-  ( y 
.x.  X )  e. 
_V
2422, 9, 23fvmpt 5745 . . . . 5  |-  ( y  e.  B  ->  (
( x  e.  B  |->  ( x  .x.  X
) ) `  y
)  =  ( y 
.x.  X ) )
25 oveq1 6027 . . . . . 6  |-  ( x  =  z  ->  (
x  .x.  X )  =  ( z  .x.  X ) )
26 ovex 6045 . . . . . 6  |-  ( z 
.x.  X )  e. 
_V
2725, 9, 26fvmpt 5745 . . . . 5  |-  ( z  e.  B  ->  (
( x  e.  B  |->  ( x  .x.  X
) ) `  z
)  =  ( z 
.x.  X ) )
2824, 27oveqan12d 6039 . . . 4  |-  ( ( y  e.  B  /\  z  e.  B )  ->  ( ( ( x  e.  B  |->  ( x 
.x.  X ) ) `
 y ) ( +g  `  R ) ( ( x  e.  B  |->  ( x  .x.  X ) ) `  z ) )  =  ( ( y  .x.  X ) ( +g  `  R ) ( z 
.x.  X ) ) )
2928adantl 453 . . 3  |-  ( ( ( R  e.  Ring  /\  X  e.  B )  /\  ( y  e.  B  /\  z  e.  B ) )  -> 
( ( ( x  e.  B  |->  ( x 
.x.  X ) ) `
 y ) ( +g  `  R ) ( ( x  e.  B  |->  ( x  .x.  X ) ) `  z ) )  =  ( ( y  .x.  X ) ( +g  `  R ) ( z 
.x.  X ) ) )
3014, 21, 293eqtr4d 2429 . 2  |-  ( ( ( R  e.  Ring  /\  X  e.  B )  /\  ( y  e.  B  /\  z  e.  B ) )  -> 
( ( x  e.  B  |->  ( x  .x.  X ) ) `  ( y ( +g  `  R ) z ) )  =  ( ( ( x  e.  B  |->  ( x  .x.  X
) ) `  y
) ( +g  `  R
) ( ( x  e.  B  |->  ( x 
.x.  X ) ) `
 z ) ) )
311, 1, 2, 2, 4, 4, 10, 30isghmd 14942 1  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  (
x  e.  B  |->  ( x  .x.  X ) )  e.  ( R 
GrpHom  R ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717    e. cmpt 4207   ` cfv 5394  (class class class)co 6020   Basecbs 13396   +g cplusg 13456   .rcmulr 13457   Grpcgrp 14612    GrpHom cghm 14930   Ringcrg 15587
This theorem is referenced by:  gsummulc1  15640  fidomndrnglem  16293
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 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-cnex 8979  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-pre-mulgt0 9000
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 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-riota 6485  df-recs 6569  df-rdg 6604  df-er 6841  df-en 7046  df-dom 7047  df-sdom 7048  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059  df-sub 9225  df-neg 9226  df-nn 9933  df-2 9990  df-ndx 13399  df-slot 13400  df-base 13401  df-sets 13402  df-plusg 13469  df-mnd 14617  df-grp 14739  df-ghm 14931  df-mgp 15576  df-rng 15590
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