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Theorem rngrghm 16628
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 2433 . 2  |-  ( +g  `  R )  =  ( +g  `  R )
3 rnggrp 16585 . . 3  |-  ( R  e.  Ring  ->  R  e. 
Grp )
43adantr 462 . 2  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  R  e.  Grp )
5 rnglghm.t . . . . . 6  |-  .x.  =  ( .r `  R )
61, 5rngcl 16593 . . . . 5  |-  ( ( R  e.  Ring  /\  x  e.  B  /\  X  e.  B )  ->  (
x  .x.  X )  e.  B )
763expa 1180 . . . 4  |-  ( ( ( R  e.  Ring  /\  x  e.  B )  /\  X  e.  B
)  ->  ( x  .x.  X )  e.  B
)
87an32s 795 . . 3  |-  ( ( ( R  e.  Ring  /\  X  e.  B )  /\  x  e.  B
)  ->  ( x  .x.  X )  e.  B
)
9 eqid 2433 . . 3  |-  ( x  e.  B  |->  ( x 
.x.  X ) )  =  ( x  e.  B  |->  ( x  .x.  X ) )
108, 9fmptd 5855 . 2  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  (
x  e.  B  |->  ( x  .x.  X ) ) : B --> B )
11 df-3an 960 . . . . 5  |-  ( ( y  e.  B  /\  z  e.  B  /\  X  e.  B )  <->  ( ( y  e.  B  /\  z  e.  B
)  /\  X  e.  B ) )
121, 2, 5rngdir 16599 . . . . 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 473 . . . 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 798 . . 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 16607 . . . . . 6  |-  ( ( R  e.  Ring  /\  y  e.  B  /\  z  e.  B )  ->  (
y ( +g  `  R
) z )  e.  B )
16153expb 1181 . . . . 5  |-  ( ( R  e.  Ring  /\  (
y  e.  B  /\  z  e.  B )
)  ->  ( y
( +g  `  R ) z )  e.  B
)
1716adantlr 707 . . . 4  |-  ( ( ( R  e.  Ring  /\  X  e.  B )  /\  ( y  e.  B  /\  z  e.  B ) )  -> 
( y ( +g  `  R ) z )  e.  B )
18 oveq1 6087 . . . . 5  |-  ( x  =  ( y ( +g  `  R ) z )  ->  (
x  .x.  X )  =  ( ( y ( +g  `  R
) z )  .x.  X ) )
19 ovex 6105 . . . . 5  |-  ( ( y ( +g  `  R
) z )  .x.  X )  e.  _V
2018, 9, 19fvmpt 5762 . . . 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 6087 . . . . . 6  |-  ( x  =  y  ->  (
x  .x.  X )  =  ( y  .x.  X ) )
23 ovex 6105 . . . . . 6  |-  ( y 
.x.  X )  e. 
_V
2422, 9, 23fvmpt 5762 . . . . 5  |-  ( y  e.  B  ->  (
( x  e.  B  |->  ( x  .x.  X
) ) `  y
)  =  ( y 
.x.  X ) )
25 oveq1 6087 . . . . . 6  |-  ( x  =  z  ->  (
x  .x.  X )  =  ( z  .x.  X ) )
26 ovex 6105 . . . . . 6  |-  ( z 
.x.  X )  e. 
_V
2725, 9, 26fvmpt 5762 . . . . 5  |-  ( z  e.  B  ->  (
( x  e.  B  |->  ( x  .x.  X
) ) `  z
)  =  ( z 
.x.  X ) )
2824, 27oveqan12d 6099 . . . 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 463 . . 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 2475 . 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 15735 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 958    = wceq 1362    e. wcel 1755    e. cmpt 4338   ` cfv 5406  (class class class)co 6080   Basecbs 14156   +g cplusg 14220   .rcmulr 14221   Grpcgrp 15392    GrpHom cghm 15723   Ringcrg 16576
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-cnex 9325  ax-resscn 9326  ax-1cn 9327  ax-icn 9328  ax-addcl 9329  ax-addrcl 9330  ax-mulcl 9331  ax-mulrcl 9332  ax-mulcom 9333  ax-addass 9334  ax-mulass 9335  ax-distr 9336  ax-i2m1 9337  ax-1ne0 9338  ax-1rid 9339  ax-rnegex 9340  ax-rrecex 9341  ax-cnre 9342  ax-pre-lttri 9343  ax-pre-lttrn 9344  ax-pre-ltadd 9345  ax-pre-mulgt0 9346
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-uni 4080  df-iun 4161  df-br 4281  df-opab 4339  df-mpt 4340  df-tr 4374  df-eprel 4619  df-id 4623  df-po 4628  df-so 4629  df-fr 4666  df-we 4668  df-ord 4709  df-on 4710  df-lim 4711  df-suc 4712  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-om 6466  df-recs 6818  df-rdg 6852  df-er 7089  df-en 7299  df-dom 7300  df-sdom 7301  df-pnf 9407  df-mnf 9408  df-xr 9409  df-ltxr 9410  df-le 9411  df-sub 9584  df-neg 9585  df-nn 10310  df-2 10367  df-ndx 14159  df-slot 14160  df-base 14161  df-sets 14162  df-plusg 14233  df-mnd 15397  df-grp 15524  df-ghm 15724  df-mgp 16565  df-rng 16579
This theorem is referenced by:  gsummulc1  16629  gsummulc1OLD  16631  fidomndrnglem  17299
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