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Theorem rnglghm 15404
Description: Left-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
rnglghm  |-  ( ( 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 rnglghm
Dummy variables  y 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rnglghm.b . 2  |-  B  =  ( Base `  R
)
2 eqid 2296 . 2  |-  ( +g  `  R )  =  ( +g  `  R )
3 rnggrp 15362 . . 3  |-  ( R  e.  Ring  ->  R  e. 
Grp )
43adantr 451 . 2  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  R  e.  Grp )
5 rnglghm.t . . . . 5  |-  .x.  =  ( .r `  R )
61, 5rngcl 15370 . . . 4  |-  ( ( R  e.  Ring  /\  X  e.  B  /\  x  e.  B )  ->  ( X  .x.  x )  e.  B )
763expa 1151 . . 3  |-  ( ( ( R  e.  Ring  /\  X  e.  B )  /\  x  e.  B
)  ->  ( X  .x.  x )  e.  B
)
8 eqid 2296 . . 3  |-  ( x  e.  B  |->  ( X 
.x.  x ) )  =  ( x  e.  B  |->  ( X  .x.  x ) )
97, 8fmptd 5700 . 2  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  (
x  e.  B  |->  ( X  .x.  x ) ) : B --> B )
10 3anass 938 . . . . 5  |-  ( ( X  e.  B  /\  y  e.  B  /\  z  e.  B )  <->  ( X  e.  B  /\  ( y  e.  B  /\  z  e.  B
) ) )
111, 2, 5rngdi 15375 . . . . 5  |-  ( ( R  e.  Ring  /\  ( X  e.  B  /\  y  e.  B  /\  z  e.  B )
)  ->  ( X  .x.  ( y ( +g  `  R ) z ) )  =  ( ( X  .x.  y ) ( +g  `  R
) ( X  .x.  z ) ) )
1210, 11sylan2br 462 . . . 4  |-  ( ( R  e.  Ring  /\  ( X  e.  B  /\  ( y  e.  B  /\  z  e.  B
) ) )  -> 
( X  .x.  (
y ( +g  `  R
) z ) )  =  ( ( X 
.x.  y ) ( +g  `  R ) ( X  .x.  z
) ) )
1312anassrs 629 . . 3  |-  ( ( ( R  e.  Ring  /\  X  e.  B )  /\  ( y  e.  B  /\  z  e.  B ) )  -> 
( X  .x.  (
y ( +g  `  R
) z ) )  =  ( ( X 
.x.  y ) ( +g  `  R ) ( X  .x.  z
) ) )
141, 2rngacl 15384 . . . . . 6  |-  ( ( R  e.  Ring  /\  y  e.  B  /\  z  e.  B )  ->  (
y ( +g  `  R
) z )  e.  B )
15143expb 1152 . . . . 5  |-  ( ( R  e.  Ring  /\  (
y  e.  B  /\  z  e.  B )
)  ->  ( y
( +g  `  R ) z )  e.  B
)
1615adantlr 695 . . . 4  |-  ( ( ( R  e.  Ring  /\  X  e.  B )  /\  ( y  e.  B  /\  z  e.  B ) )  -> 
( y ( +g  `  R ) z )  e.  B )
17 oveq2 5882 . . . . 5  |-  ( x  =  ( y ( +g  `  R ) z )  ->  ( X  .x.  x )  =  ( X  .x.  (
y ( +g  `  R
) z ) ) )
18 ovex 5899 . . . . 5  |-  ( X 
.x.  ( y ( +g  `  R ) z ) )  e. 
_V
1917, 8, 18fvmpt 5618 . . . 4  |-  ( ( y ( +g  `  R
) z )  e.  B  ->  ( (
x  e.  B  |->  ( X  .x.  x ) ) `  ( y ( +g  `  R
) z ) )  =  ( X  .x.  ( y ( +g  `  R ) z ) ) )
2016, 19syl 15 . . 3  |-  ( ( ( R  e.  Ring  /\  X  e.  B )  /\  ( y  e.  B  /\  z  e.  B ) )  -> 
( ( x  e.  B  |->  ( X  .x.  x ) ) `  ( y ( +g  `  R ) z ) )  =  ( X 
.x.  ( y ( +g  `  R ) z ) ) )
21 oveq2 5882 . . . . . 6  |-  ( x  =  y  ->  ( X  .x.  x )  =  ( X  .x.  y
) )
22 ovex 5899 . . . . . 6  |-  ( X 
.x.  y )  e. 
_V
2321, 8, 22fvmpt 5618 . . . . 5  |-  ( y  e.  B  ->  (
( x  e.  B  |->  ( X  .x.  x
) ) `  y
)  =  ( X 
.x.  y ) )
24 oveq2 5882 . . . . . 6  |-  ( x  =  z  ->  ( X  .x.  x )  =  ( X  .x.  z
) )
25 ovex 5899 . . . . . 6  |-  ( X 
.x.  z )  e. 
_V
2624, 8, 25fvmpt 5618 . . . . 5  |-  ( z  e.  B  ->  (
( x  e.  B  |->  ( X  .x.  x
) ) `  z
)  =  ( X 
.x.  z ) )
2723, 26oveqan12d 5893 . . . 4  |-  ( ( y  e.  B  /\  z  e.  B )  ->  ( ( ( x  e.  B  |->  ( X 
.x.  x ) ) `
 y ) ( +g  `  R ) ( ( x  e.  B  |->  ( X  .x.  x ) ) `  z ) )  =  ( ( X  .x.  y ) ( +g  `  R ) ( X 
.x.  z ) ) )
2827adantl 452 . . 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 ) )  =  ( ( X  .x.  y ) ( +g  `  R ) ( X 
.x.  z ) ) )
2913, 20, 283eqtr4d 2338 . 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 ) ) )
301, 1, 2, 2, 4, 4, 9, 29isghmd 14708 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 358    /\ w3a 934    = wceq 1632    e. wcel 1696    e. cmpt 4093   ` cfv 5271  (class class class)co 5874   Basecbs 13164   +g cplusg 13224   .rcmulr 13225   Grpcgrp 14378    GrpHom cghm 14696   Ringcrg 15353
This theorem is referenced by:  gsummulc2  15407
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-plusg 13237  df-mnd 14383  df-grp 14505  df-ghm 14697  df-mgp 15342  df-rng 15356
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