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Theorem rngrghm 17028
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 2460 . 2  |-  ( +g  `  R )  =  ( +g  `  R )
3 rnggrp 16984 . . 3  |-  ( R  e.  Ring  ->  R  e. 
Grp )
43adantr 465 . 2  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  R  e.  Grp )
5 rnglghm.t . . . . . 6  |-  .x.  =  ( .r `  R )
61, 5rngcl 16992 . . . . 5  |-  ( ( R  e.  Ring  /\  x  e.  B  /\  X  e.  B )  ->  (
x  .x.  X )  e.  B )
763expa 1191 . . . 4  |-  ( ( ( R  e.  Ring  /\  x  e.  B )  /\  X  e.  B
)  ->  ( x  .x.  X )  e.  B
)
87an32s 802 . . 3  |-  ( ( ( R  e.  Ring  /\  X  e.  B )  /\  x  e.  B
)  ->  ( x  .x.  X )  e.  B
)
9 eqid 2460 . . 3  |-  ( x  e.  B  |->  ( x 
.x.  X ) )  =  ( x  e.  B  |->  ( x  .x.  X ) )
108, 9fmptd 6036 . 2  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  (
x  e.  B  |->  ( x  .x.  X ) ) : B --> B )
11 df-3an 970 . . . . 5  |-  ( ( y  e.  B  /\  z  e.  B  /\  X  e.  B )  <->  ( ( y  e.  B  /\  z  e.  B
)  /\  X  e.  B ) )
121, 2, 5rngdir 16998 . . . . 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 476 . . . 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 805 . . 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 17006 . . . . . 6  |-  ( ( R  e.  Ring  /\  y  e.  B  /\  z  e.  B )  ->  (
y ( +g  `  R
) z )  e.  B )
16153expb 1192 . . . . 5  |-  ( ( R  e.  Ring  /\  (
y  e.  B  /\  z  e.  B )
)  ->  ( y
( +g  `  R ) z )  e.  B
)
1716adantlr 714 . . . 4  |-  ( ( ( R  e.  Ring  /\  X  e.  B )  /\  ( y  e.  B  /\  z  e.  B ) )  -> 
( y ( +g  `  R ) z )  e.  B )
18 oveq1 6282 . . . . 5  |-  ( x  =  ( y ( +g  `  R ) z )  ->  (
x  .x.  X )  =  ( ( y ( +g  `  R
) z )  .x.  X ) )
19 ovex 6300 . . . . 5  |-  ( ( y ( +g  `  R
) z )  .x.  X )  e.  _V
2018, 9, 19fvmpt 5941 . . . 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 6282 . . . . . 6  |-  ( x  =  y  ->  (
x  .x.  X )  =  ( y  .x.  X ) )
23 ovex 6300 . . . . . 6  |-  ( y 
.x.  X )  e. 
_V
2422, 9, 23fvmpt 5941 . . . . 5  |-  ( y  e.  B  ->  (
( x  e.  B  |->  ( x  .x.  X
) ) `  y
)  =  ( y 
.x.  X ) )
25 oveq1 6282 . . . . . 6  |-  ( x  =  z  ->  (
x  .x.  X )  =  ( z  .x.  X ) )
26 ovex 6300 . . . . . 6  |-  ( z 
.x.  X )  e. 
_V
2725, 9, 26fvmpt 5941 . . . . 5  |-  ( z  e.  B  ->  (
( x  e.  B  |->  ( x  .x.  X
) ) `  z
)  =  ( z 
.x.  X ) )
2824, 27oveqan12d 6294 . . . 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 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 ) )  =  ( ( y  .x.  X ) ( +g  `  R ) ( z 
.x.  X ) ) )
3014, 21, 293eqtr4d 2511 . 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 16064 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 968    = wceq 1374    e. wcel 1762    |-> cmpt 4498   ` cfv 5579  (class class class)co 6275   Basecbs 14479   +g cplusg 14544   .rcmulr 14545   Grpcgrp 15716    GrpHom cghm 16052   Ringcrg 16979
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-recs 7032  df-rdg 7066  df-er 7301  df-en 7507  df-dom 7508  df-sdom 7509  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-nn 10526  df-2 10583  df-ndx 14482  df-slot 14483  df-base 14484  df-sets 14485  df-plusg 14557  df-mnd 15721  df-grp 15851  df-ghm 16053  df-mgp 16925  df-rng 16981
This theorem is referenced by:  gsummulc1  17029  gsummulc1OLD  17031  fidomndrnglem  17719
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