Theorem ringrghm 14198

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
ringlghm.b	|- B = ( Base ` R )
ringlghm.t	|- .x. = ( .r ` R )

Assertion

Ref	Expression
ringrghm	|- ( ( 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 ringrghm

Dummy variables y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ringlghm.b	. 2 |- B = ( Base ` R )
2		eqid 2232	. 2 |- ( +g ` R ) = ( +g ` R )
3		ringgrp 14137	. . 3 |- ( R e. Ring -> R e. Grp )
4	3	adantr 276	. 2 |- ( ( R e. Ring /\ X e. B ) -> R e. Grp )
5		ringlghm.t	. . . . . 6 |- .x. = ( .r ` R )
6	1, 5	ringcl 14149	. . . . 5 |- ( ( R e. Ring /\ x e. B /\ X e. B ) -> ( x .x. X ) e. B )
7	6	3expa 1230	. . . 4 |- ( ( ( R e. Ring /\ x e. B ) /\ X e. B ) -> ( x .x. X ) e. B )
8	7	an32s 570	. . 3 |- ( ( ( R e. Ring /\ X e. B ) /\ x e. B ) -> ( x .x. X ) e. B )
9	8	fmpttd 5831	. 2 |- ( ( R e. Ring /\ X e. B ) -> ( x e. B |-> ( x .x. X ) ) : B --> B )
10		df-3an 1007	. . . . 5 |- ( ( y e. B /\ z e. B /\ X e. B ) <-> ( ( y e. B /\ z e. B ) /\ X e. B ) )
11	1, 2, 5	ringdir 14155	. . . . 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 ) ) )
12	10, 11	sylan2br 288	. . . 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 ) ) )
13	12	anass1rs 573	. . 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 ) ) )
14		eqid 2232	. . . 4 |- ( x e. B |-> ( x .x. X ) ) = ( x e. B |-> ( x .x. X ) )
15		oveq1 6056	. . . 4 |- ( x = ( y ( +g ` R ) z ) -> ( x .x. X ) = ( ( y ( +g ` R ) z ) .x. X ) )
16	1, 2	ringacl 14166	. . . . . 6 |- ( ( R e. Ring /\ y e. B /\ z e. B ) -> ( y ( +g ` R ) z ) e. B )
17	16	3expb 1231	. . . . 5 |- ( ( R e. Ring /\ ( y e. B /\ z e. B ) ) -> ( y ( +g ` R ) z ) e. B )
18	17	adantlr 477	. . . 4 |- ( ( ( R e. Ring /\ X e. B ) /\ ( y e. B /\ z e. B ) ) -> ( y ( +g ` R ) z ) e. B )
19		simpll 527	. . . . 5 |- ( ( ( R e. Ring /\ X e. B ) /\ ( y e. B /\ z e. B ) ) -> R e. Ring )
20		simplr 529	. . . . 5 |- ( ( ( R e. Ring /\ X e. B ) /\ ( y e. B /\ z e. B ) ) -> X e. B )
21	1, 5	ringcl 14149	. . . . 5 |- ( ( R e. Ring /\ ( y ( +g ` R ) z ) e. B /\ X e. B ) -> ( ( y ( +g ` R ) z ) .x. X ) e. B )
22	19, 18, 20, 21	syl3anc 1274	. . . 4 |- ( ( ( R e. Ring /\ X e. B ) /\ ( y e. B /\ z e. B ) ) -> ( ( y ( +g ` R ) z ) .x. X ) e. B )
23	14, 15, 18, 22	fvmptd3 5770	. . 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 ) )
24		oveq1 6056	. . . . 5 |- ( x = y -> ( x .x. X ) = ( y .x. X ) )
25		simprl 531	. . . . 5 |- ( ( ( R e. Ring /\ X e. B ) /\ ( y e. B /\ z e. B ) ) -> y e. B )
26	1, 5	ringcl 14149	. . . . . 6 |- ( ( R e. Ring /\ y e. B /\ X e. B ) -> ( y .x. X ) e. B )
27	19, 25, 20, 26	syl3anc 1274	. . . . 5 |- ( ( ( R e. Ring /\ X e. B ) /\ ( y e. B /\ z e. B ) ) -> ( y .x. X ) e. B )
28	14, 24, 25, 27	fvmptd3 5770	. . . 4 |- ( ( ( R e. Ring /\ X e. B ) /\ ( y e. B /\ z e. B ) ) -> ( ( x e. B |-> ( x .x. X ) ) ` y ) = ( y .x. X ) )
29		oveq1 6056	. . . . 5 |- ( x = z -> ( x .x. X ) = ( z .x. X ) )
30		simprr 533	. . . . 5 |- ( ( ( R e. Ring /\ X e. B ) /\ ( y e. B /\ z e. B ) ) -> z e. B )
31	1, 5	ringcl 14149	. . . . . 6 |- ( ( R e. Ring /\ z e. B /\ X e. B ) -> ( z .x. X ) e. B )
32	19, 30, 20, 31	syl3anc 1274	. . . . 5 |- ( ( ( R e. Ring /\ X e. B ) /\ ( y e. B /\ z e. B ) ) -> ( z .x. X ) e. B )
33	14, 29, 30, 32	fvmptd3 5770	. . . 4 |- ( ( ( R e. Ring /\ X e. B ) /\ ( y e. B /\ z e. B ) ) -> ( ( x e. B |-> ( x .x. X ) ) ` z ) = ( z .x. X ) )
34	28, 33	oveq12d 6067	. . 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 ) ) )
35	13, 23, 34	3eqtr4d 2275	. 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 ) ) )
36	1, 1, 2, 2, 4, 4, 9, 35	isghmd 13961	1 |- ( ( R e. Ring /\ X e. B ) -> ( x e. B |-> ( x .x. X ) ) e. ( R GrpHom R ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1005   = wceq 1398   e. wcel 2203   |-> cmpt 4170   ` cfv 5351  (class class class)co 6049   Basecbs 13204   +g cplusg 13282   .rcmulr 13283   Grpcgrp 13705   GrpHom cghm 13949   Ringcrg 14132

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4224  ax-sep 4227  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-cnex 8217  ax-resscn 8218  ax-1cn 8219  ax-1re 8220  ax-icn 8221  ax-addcl 8222  ax-addrcl 8223  ax-mulcl 8224  ax-addcom 8226  ax-addass 8228  ax-i2m1 8231  ax-0lt1 8232  ax-0id 8234  ax-rnegex 8235  ax-pre-ltirr 8238  ax-pre-ltadd 8242

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3508  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-iun 3992  df-br 4109  df-opab 4171  df-mpt 4172  df-id 4413  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-f1 5356  df-fo 5357  df-f1o 5358  df-fv 5359  df-ov 6052  df-oprab 6053  df-mpo 6054  df-pnf 8309  df-mnf 8310  df-ltxr 8312  df-inn 9237  df-2 9295  df-3 9296  df-ndx 13207  df-slot 13208  df-base 13210  df-sets 13211  df-plusg 13295  df-mulr 13296  df-mgm 13561  df-sgrp 13607  df-mnd 13622  df-grp 13708  df-ghm 13950  df-mgp 14057  df-ring 14134

This theorem is referenced by: (None)