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Theorem rnglghm 16801
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 2451 . 2  |-  ( +g  `  R )  =  ( +g  `  R )
3 rnggrp 16758 . . 3  |-  ( R  e.  Ring  ->  R  e. 
Grp )
43adantr 465 . 2  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  R  e.  Grp )
5 rnglghm.t . . . . 5  |-  .x.  =  ( .r `  R )
61, 5rngcl 16766 . . . 4  |-  ( ( R  e.  Ring  /\  X  e.  B  /\  x  e.  B )  ->  ( X  .x.  x )  e.  B )
763expa 1188 . . 3  |-  ( ( ( R  e.  Ring  /\  X  e.  B )  /\  x  e.  B
)  ->  ( X  .x.  x )  e.  B
)
8 eqid 2451 . . 3  |-  ( x  e.  B  |->  ( X 
.x.  x ) )  =  ( x  e.  B  |->  ( X  .x.  x ) )
97, 8fmptd 5968 . 2  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  (
x  e.  B  |->  ( X  .x.  x ) ) : B --> B )
10 3anass 969 . . . . 5  |-  ( ( X  e.  B  /\  y  e.  B  /\  z  e.  B )  <->  ( X  e.  B  /\  ( y  e.  B  /\  z  e.  B
) ) )
111, 2, 5rngdi 16771 . . . . 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 476 . . . 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 648 . . 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 16780 . . . . . 6  |-  ( ( R  e.  Ring  /\  y  e.  B  /\  z  e.  B )  ->  (
y ( +g  `  R
) z )  e.  B )
15143expb 1189 . . . . 5  |-  ( ( R  e.  Ring  /\  (
y  e.  B  /\  z  e.  B )
)  ->  ( y
( +g  `  R ) z )  e.  B
)
1615adantlr 714 . . . 4  |-  ( ( ( R  e.  Ring  /\  X  e.  B )  /\  ( y  e.  B  /\  z  e.  B ) )  -> 
( y ( +g  `  R ) z )  e.  B )
17 oveq2 6200 . . . . 5  |-  ( x  =  ( y ( +g  `  R ) z )  ->  ( X  .x.  x )  =  ( X  .x.  (
y ( +g  `  R
) z ) ) )
18 ovex 6217 . . . . 5  |-  ( X 
.x.  ( y ( +g  `  R ) z ) )  e. 
_V
1917, 8, 18fvmpt 5875 . . . 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 6200 . . . . . 6  |-  ( x  =  y  ->  ( X  .x.  x )  =  ( X  .x.  y
) )
22 ovex 6217 . . . . . 6  |-  ( X 
.x.  y )  e. 
_V
2321, 8, 22fvmpt 5875 . . . . 5  |-  ( y  e.  B  ->  (
( x  e.  B  |->  ( X  .x.  x
) ) `  y
)  =  ( X 
.x.  y ) )
24 oveq2 6200 . . . . . 6  |-  ( x  =  z  ->  ( X  .x.  x )  =  ( X  .x.  z
) )
25 ovex 6217 . . . . . 6  |-  ( X 
.x.  z )  e. 
_V
2624, 8, 25fvmpt 5875 . . . . 5  |-  ( z  e.  B  ->  (
( x  e.  B  |->  ( X  .x.  x
) ) `  z
)  =  ( X 
.x.  z ) )
2723, 26oveqan12d 6211 . . . 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 466 . . 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 2502 . 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 15860 1  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  (
x  e.  B  |->  ( X  .x.  x ) )  e.  ( R 
GrpHom  R ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758    |-> cmpt 4450   ` cfv 5518  (class class class)co 6192   Basecbs 14278   +g cplusg 14342   .rcmulr 14343   Grpcgrp 15514    GrpHom cghm 15848   Ringcrg 16753
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4503  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474  ax-cnex 9441  ax-resscn 9442  ax-1cn 9443  ax-icn 9444  ax-addcl 9445  ax-addrcl 9446  ax-mulcl 9447  ax-mulrcl 9448  ax-mulcom 9449  ax-addass 9450  ax-mulass 9451  ax-distr 9452  ax-i2m1 9453  ax-1ne0 9454  ax-1rid 9455  ax-rnegex 9456  ax-rrecex 9457  ax-cnre 9458  ax-pre-lttri 9459  ax-pre-lttrn 9460  ax-pre-ltadd 9461  ax-pre-mulgt0 9462
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-pss 3444  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-tp 3982  df-op 3984  df-uni 4192  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4486  df-eprel 4732  df-id 4736  df-po 4741  df-so 4742  df-fr 4779  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-riota 6153  df-ov 6195  df-oprab 6196  df-mpt2 6197  df-om 6579  df-recs 6934  df-rdg 6968  df-er 7203  df-en 7413  df-dom 7414  df-sdom 7415  df-pnf 9523  df-mnf 9524  df-xr 9525  df-ltxr 9526  df-le 9527  df-sub 9700  df-neg 9701  df-nn 10426  df-2 10483  df-ndx 14281  df-slot 14282  df-base 14283  df-sets 14284  df-plusg 14355  df-mnd 15519  df-grp 15649  df-ghm 15849  df-mgp 16699  df-rng 16755
This theorem is referenced by:  gsummulc2  16804  gsummulc2OLD  16806
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