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Theorem rngrghm 15704
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 2435 . 2  |-  ( +g  `  R )  =  ( +g  `  R )
3 rnggrp 15661 . . 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 15669 . . . . 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 2435 . . 3  |-  ( x  e.  B  |->  ( x 
.x.  X ) )  =  ( x  e.  B  |->  ( x  .x.  X ) )
108, 9fmptd 5885 . 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 15675 . . . . 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 15683 . . . . . 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 6080 . . . . 5  |-  ( x  =  ( y ( +g  `  R ) z )  ->  (
x  .x.  X )  =  ( ( y ( +g  `  R
) z )  .x.  X ) )
19 ovex 6098 . . . . 5  |-  ( ( y ( +g  `  R
) z )  .x.  X )  e.  _V
2018, 9, 19fvmpt 5798 . . . 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 6080 . . . . . 6  |-  ( x  =  y  ->  (
x  .x.  X )  =  ( y  .x.  X ) )
23 ovex 6098 . . . . . 6  |-  ( y 
.x.  X )  e. 
_V
2422, 9, 23fvmpt 5798 . . . . 5  |-  ( y  e.  B  ->  (
( x  e.  B  |->  ( x  .x.  X
) ) `  y
)  =  ( y 
.x.  X ) )
25 oveq1 6080 . . . . . 6  |-  ( x  =  z  ->  (
x  .x.  X )  =  ( z  .x.  X ) )
26 ovex 6098 . . . . . 6  |-  ( z 
.x.  X )  e. 
_V
2725, 9, 26fvmpt 5798 . . . . 5  |-  ( z  e.  B  ->  (
( x  e.  B  |->  ( x  .x.  X
) ) `  z
)  =  ( z 
.x.  X ) )
2824, 27oveqan12d 6092 . . . 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 2477 . 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 15007 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 1652    e. wcel 1725    e. cmpt 4258   ` cfv 5446  (class class class)co 6073   Basecbs 13461   +g cplusg 13521   .rcmulr 13522   Grpcgrp 14677    GrpHom cghm 14995   Ringcrg 15652
This theorem is referenced by:  gsummulc1  15705  fidomndrnglem  16358
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-riota 6541  df-recs 6625  df-rdg 6660  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-2 10050  df-ndx 13464  df-slot 13465  df-base 13466  df-sets 13467  df-plusg 13534  df-mnd 14682  df-grp 14804  df-ghm 14996  df-mgp 15641  df-rng 15655
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