Theorem ringrghm 14068

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 2229	. 2 |- ( +g ` R ) = ( +g ` R )
3		ringgrp 14007	. . 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 14019	. . . . 5 |- ( ( R e. Ring /\ x e. B /\ X e. B ) -> ( x .x. X ) e. B )
7	6	3expa 1227	. . . 4 |- ( ( ( R e. Ring /\ x e. B ) /\ X e. B ) -> ( x .x. X ) e. B )
8	7	an32s 568	. . 3 |- ( ( ( R e. Ring /\ X e. B ) /\ x e. B ) -> ( x .x. X ) e. B )
9	8	fmpttd 5798	. 2 |- ( ( R e. Ring /\ X e. B ) -> ( x e. B |-> ( x .x. X ) ) : B --> B )
10		df-3an 1004	. . . . 5 |- ( ( y e. B /\ z e. B /\ X e. B ) <-> ( ( y e. B /\ z e. B ) /\ X e. B ) )
11	1, 2, 5	ringdir 14025	. . . . 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 571	. . 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 2229	. . . 4 |- ( x e. B |-> ( x .x. X ) ) = ( x e. B |-> ( x .x. X ) )
15		oveq1 6020	. . . 4 |- ( x = ( y ( +g ` R ) z ) -> ( x .x. X ) = ( ( y ( +g ` R ) z ) .x. X ) )
16	1, 2	ringacl 14036	. . . . . 6 |- ( ( R e. Ring /\ y e. B /\ z e. B ) -> ( y ( +g ` R ) z ) e. B )
17	16	3expb 1228	. . . . 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 528	. . . . 5 |- ( ( ( R e. Ring /\ X e. B ) /\ ( y e. B /\ z e. B ) ) -> X e. B )
21	1, 5	ringcl 14019	. . . . 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 1271	. . . 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 5736	. . 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 6020	. . . . 5 |- ( x = y -> ( x .x. X ) = ( y .x. X ) )
25		simprl 529	. . . . 5 |- ( ( ( R e. Ring /\ X e. B ) /\ ( y e. B /\ z e. B ) ) -> y e. B )
26	1, 5	ringcl 14019	. . . . . 6 |- ( ( R e. Ring /\ y e. B /\ X e. B ) -> ( y .x. X ) e. B )
27	19, 25, 20, 26	syl3anc 1271	. . . . 5 |- ( ( ( R e. Ring /\ X e. B ) /\ ( y e. B /\ z e. B ) ) -> ( y .x. X ) e. B )
28	14, 24, 25, 27	fvmptd3 5736	. . . 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 6020	. . . . 5 |- ( x = z -> ( x .x. X ) = ( z .x. X ) )
30		simprr 531	. . . . 5 |- ( ( ( R e. Ring /\ X e. B ) /\ ( y e. B /\ z e. B ) ) -> z e. B )
31	1, 5	ringcl 14019	. . . . . 6 |- ( ( R e. Ring /\ z e. B /\ X e. B ) -> ( z .x. X ) e. B )
32	19, 30, 20, 31	syl3anc 1271	. . . . 5 |- ( ( ( R e. Ring /\ X e. B ) /\ ( y e. B /\ z e. B ) ) -> ( z .x. X ) e. B )
33	14, 29, 30, 32	fvmptd3 5736	. . . 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 6031	. . 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 2272	. 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 13832	1 |- ( ( R e. Ring /\ X e. B ) -> ( x e. B |-> ( x .x. X ) ) e. ( R GrpHom R ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1002   = wceq 1395   e. wcel 2200   |-> cmpt 4148   ` cfv 5324  (class class class)co 6013   Basecbs 13075   +g cplusg 13153   .rcmulr 13154   Grpcgrp 13576   GrpHom cghm 13820   Ringcrg 14002

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-cnex 8116  ax-resscn 8117  ax-1cn 8118  ax-1re 8119  ax-icn 8120  ax-addcl 8121  ax-addrcl 8122  ax-mulcl 8123  ax-addcom 8125  ax-addass 8127  ax-i2m1 8130  ax-0lt1 8131  ax-0id 8133  ax-rnegex 8134  ax-pre-ltirr 8137  ax-pre-ltadd 8141

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-ov 6016  df-oprab 6017  df-mpo 6018  df-pnf 8209  df-mnf 8210  df-ltxr 8212  df-inn 9137  df-2 9195  df-3 9196  df-ndx 13078  df-slot 13079  df-base 13081  df-sets 13082  df-plusg 13166  df-mulr 13167  df-mgm 13432  df-sgrp 13478  df-mnd 13493  df-grp 13579  df-ghm 13821  df-mgp 13927  df-ring 14004

This theorem is referenced by: (None)