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Theorem rnglghm 15713
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 2438 . 2  |-  ( +g  `  R )  =  ( +g  `  R )
3 rnggrp 15671 . . 3  |-  ( R  e.  Ring  ->  R  e. 
Grp )
43adantr 453 . 2  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  R  e.  Grp )
5 rnglghm.t . . . . 5  |-  .x.  =  ( .r `  R )
61, 5rngcl 15679 . . . 4  |-  ( ( R  e.  Ring  /\  X  e.  B  /\  x  e.  B )  ->  ( X  .x.  x )  e.  B )
763expa 1154 . . 3  |-  ( ( ( R  e.  Ring  /\  X  e.  B )  /\  x  e.  B
)  ->  ( X  .x.  x )  e.  B
)
8 eqid 2438 . . 3  |-  ( x  e.  B  |->  ( X 
.x.  x ) )  =  ( x  e.  B  |->  ( X  .x.  x ) )
97, 8fmptd 5895 . 2  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  (
x  e.  B  |->  ( X  .x.  x ) ) : B --> B )
10 3anass 941 . . . . 5  |-  ( ( X  e.  B  /\  y  e.  B  /\  z  e.  B )  <->  ( X  e.  B  /\  ( y  e.  B  /\  z  e.  B
) ) )
111, 2, 5rngdi 15684 . . . . 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 464 . . . 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 631 . . 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 15693 . . . . . 6  |-  ( ( R  e.  Ring  /\  y  e.  B  /\  z  e.  B )  ->  (
y ( +g  `  R
) z )  e.  B )
15143expb 1155 . . . . 5  |-  ( ( R  e.  Ring  /\  (
y  e.  B  /\  z  e.  B )
)  ->  ( y
( +g  `  R ) z )  e.  B
)
1615adantlr 697 . . . 4  |-  ( ( ( R  e.  Ring  /\  X  e.  B )  /\  ( y  e.  B  /\  z  e.  B ) )  -> 
( y ( +g  `  R ) z )  e.  B )
17 oveq2 6091 . . . . 5  |-  ( x  =  ( y ( +g  `  R ) z )  ->  ( X  .x.  x )  =  ( X  .x.  (
y ( +g  `  R
) z ) ) )
18 ovex 6108 . . . . 5  |-  ( X 
.x.  ( y ( +g  `  R ) z ) )  e. 
_V
1917, 8, 18fvmpt 5808 . . . 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 16 . . 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 6091 . . . . . 6  |-  ( x  =  y  ->  ( X  .x.  x )  =  ( X  .x.  y
) )
22 ovex 6108 . . . . . 6  |-  ( X 
.x.  y )  e. 
_V
2321, 8, 22fvmpt 5808 . . . . 5  |-  ( y  e.  B  ->  (
( x  e.  B  |->  ( X  .x.  x
) ) `  y
)  =  ( X 
.x.  y ) )
24 oveq2 6091 . . . . . 6  |-  ( x  =  z  ->  ( X  .x.  x )  =  ( X  .x.  z
) )
25 ovex 6108 . . . . . 6  |-  ( X 
.x.  z )  e. 
_V
2624, 8, 25fvmpt 5808 . . . . 5  |-  ( z  e.  B  ->  (
( x  e.  B  |->  ( X  .x.  x
) ) `  z
)  =  ( X 
.x.  z ) )
2723, 26oveqan12d 6102 . . . 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 454 . . 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 2480 . 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 15017 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 360    /\ w3a 937    = wceq 1653    e. wcel 1726    e. cmpt 4268   ` cfv 5456  (class class class)co 6083   Basecbs 13471   +g cplusg 13531   .rcmulr 13532   Grpcgrp 14687    GrpHom cghm 15005   Ringcrg 15662
This theorem is referenced by:  gsummulc2  15716
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-riota 6551  df-recs 6635  df-rdg 6670  df-er 6907  df-en 7112  df-dom 7113  df-sdom 7114  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-nn 10003  df-2 10060  df-ndx 13474  df-slot 13475  df-base 13476  df-sets 13477  df-plusg 13544  df-mnd 14692  df-grp 14814  df-ghm 15006  df-mgp 15651  df-rng 15665
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