Theorem ringlghm 14080

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

Assertion

Ref	Expression
ringlghm	|- ( ( 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 ringlghm

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 2231	. 2 |- ( +g ` R ) = ( +g ` R )
3		ringgrp 14020	. . 3 |- ( R e. Ring -> R e. Grp )
4	3	adantr 276	. 2 |- ( ( R e. Ring /\ X e. B ) -> R e. Grp )
5		ringlghm.t	. . . . 5 |- .x. = ( .r ` R )
6	1, 5	ringcl 14032	. . . 4 |- ( ( R e. Ring /\ X e. B /\ x e. B ) -> ( X .x. x ) e. B )
7	6	3expa 1229	. . 3 |- ( ( ( R e. Ring /\ X e. B ) /\ x e. B ) -> ( X .x. x ) e. B )
8	7	fmpttd 5802	. 2 |- ( ( R e. Ring /\ X e. B ) -> ( x e. B |-> ( X .x. x ) ) : B --> B )
9		3anass 1008	. . . . 5 |- ( ( X e. B /\ y e. B /\ z e. B ) <-> ( X e. B /\ ( y e. B /\ z e. B ) ) )
10	1, 2, 5	ringdi 14037	. . . . 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 ) ) )
11	9, 10	sylan2br 288	. . . 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 ) ) )
12	11	anassrs 400	. . 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 ) ) )
13		eqid 2231	. . . 4 |- ( x e. B |-> ( X .x. x ) ) = ( x e. B |-> ( X .x. x ) )
14		oveq2 6026	. . . 4 |- ( x = ( y ( +g ` R ) z ) -> ( X .x. x ) = ( X .x. ( y ( +g ` R ) z ) ) )
15	1, 2	ringacl 14049	. . . . . 6 |- ( ( R e. Ring /\ y e. B /\ z e. B ) -> ( y ( +g ` R ) z ) e. B )
16	15	3expb 1230	. . . . 5 |- ( ( R e. Ring /\ ( y e. B /\ z e. B ) ) -> ( y ( +g ` R ) z ) e. B )
17	16	adantlr 477	. . . 4 |- ( ( ( R e. Ring /\ X e. B ) /\ ( y e. B /\ z e. B ) ) -> ( y ( +g ` R ) z ) e. B )
18		simpll 527	. . . . 5 |- ( ( ( R e. Ring /\ X e. B ) /\ ( y e. B /\ z e. B ) ) -> R e. Ring )
19		simplr 529	. . . . 5 |- ( ( ( R e. Ring /\ X e. B ) /\ ( y e. B /\ z e. B ) ) -> X e. B )
20	1, 5	ringcl 14032	. . . . 5 |- ( ( R e. Ring /\ X e. B /\ ( y ( +g ` R ) z ) e. B ) -> ( X .x. ( y ( +g ` R ) z ) ) e. B )
21	18, 19, 17, 20	syl3anc 1273	. . . 4 |- ( ( ( R e. Ring /\ X e. B ) /\ ( y e. B /\ z e. B ) ) -> ( X .x. ( y ( +g ` R ) z ) ) e. B )
22	13, 14, 17, 21	fvmptd3 5740	. . 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 ) ) )
23		oveq2 6026	. . . . 5 |- ( x = y -> ( X .x. x ) = ( X .x. y ) )
24		simprl 531	. . . . 5 |- ( ( ( R e. Ring /\ X e. B ) /\ ( y e. B /\ z e. B ) ) -> y e. B )
25	1, 5	ringcl 14032	. . . . . 6 |- ( ( R e. Ring /\ X e. B /\ y e. B ) -> ( X .x. y ) e. B )
26	18, 19, 24, 25	syl3anc 1273	. . . . 5 |- ( ( ( R e. Ring /\ X e. B ) /\ ( y e. B /\ z e. B ) ) -> ( X .x. y ) e. B )
27	13, 23, 24, 26	fvmptd3 5740	. . . 4 |- ( ( ( R e. Ring /\ X e. B ) /\ ( y e. B /\ z e. B ) ) -> ( ( x e. B |-> ( X .x. x ) ) ` y ) = ( X .x. y ) )
28		oveq2 6026	. . . . 5 |- ( x = z -> ( X .x. x ) = ( X .x. z ) )
29		simprr 533	. . . . 5 |- ( ( ( R e. Ring /\ X e. B ) /\ ( y e. B /\ z e. B ) ) -> z e. B )
30	1, 5	ringcl 14032	. . . . . 6 |- ( ( R e. Ring /\ X e. B /\ z e. B ) -> ( X .x. z ) e. B )
31	18, 19, 29, 30	syl3anc 1273	. . . . 5 |- ( ( ( R e. Ring /\ X e. B ) /\ ( y e. B /\ z e. B ) ) -> ( X .x. z ) e. B )
32	13, 28, 29, 31	fvmptd3 5740	. . . 4 |- ( ( ( R e. Ring /\ X e. B ) /\ ( y e. B /\ z e. B ) ) -> ( ( x e. B |-> ( X .x. x ) ) ` z ) = ( X .x. z ) )
33	27, 32	oveq12d 6036	. . 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 ) ) )
34	12, 22, 33	3eqtr4d 2274	. 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 ) ) )
35	1, 1, 2, 2, 4, 4, 8, 34	isghmd 13844	1 |- ( ( R e. Ring /\ X e. B ) -> ( x e. B |-> ( X .x. x ) ) e. ( R GrpHom R ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1004   = wceq 1397   e. wcel 2202   |-> cmpt 4150   ` cfv 5326  (class class class)co 6018   Basecbs 13087   +g cplusg 13165   .rcmulr 13166   Grpcgrp 13588   GrpHom cghm 13832   Ringcrg 14015

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-addcom 8132  ax-addass 8134  ax-i2m1 8137  ax-0lt1 8138  ax-0id 8140  ax-rnegex 8141  ax-pre-ltirr 8144  ax-pre-ltadd 8148

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-ov 6021  df-oprab 6022  df-mpo 6023  df-pnf 8216  df-mnf 8217  df-ltxr 8219  df-inn 9144  df-2 9202  df-3 9203  df-ndx 13090  df-slot 13091  df-base 13093  df-sets 13094  df-plusg 13178  df-mulr 13179  df-mgm 13444  df-sgrp 13490  df-mnd 13505  df-grp 13591  df-ghm 13833  df-mgp 13940  df-ring 14017

This theorem is referenced by: (None)