Theorem ringlghm 14136

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 14076	. . 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 14088	. . . 4 |- ( ( R e. Ring /\ X e. B /\ x e. B ) -> ( X .x. x ) e. B )
7	6	3expa 1230	. . 3 |- ( ( ( R e. Ring /\ X e. B ) /\ x e. B ) -> ( X .x. x ) e. B )
8	7	fmpttd 5810	. 2 |- ( ( R e. Ring /\ X e. B ) -> ( x e. B |-> ( X .x. x ) ) : B --> B )
9		3anass 1009	. . . . 5 |- ( ( X e. B /\ y e. B /\ z e. B ) <-> ( X e. B /\ ( y e. B /\ z e. B ) ) )
10	1, 2, 5	ringdi 14093	. . . . 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 6036	. . . 4 |- ( x = ( y ( +g ` R ) z ) -> ( X .x. x ) = ( X .x. ( y ( +g ` R ) z ) ) )
15	1, 2	ringacl 14105	. . . . . 6 |- ( ( R e. Ring /\ y e. B /\ z e. B ) -> ( y ( +g ` R ) z ) e. B )
16	15	3expb 1231	. . . . 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 14088	. . . . 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 1274	. . . 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 5749	. . 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 6036	. . . . 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 14088	. . . . . 6 |- ( ( R e. Ring /\ X e. B /\ y e. B ) -> ( X .x. y ) e. B )
26	18, 19, 24, 25	syl3anc 1274	. . . . 5 |- ( ( ( R e. Ring /\ X e. B ) /\ ( y e. B /\ z e. B ) ) -> ( X .x. y ) e. B )
27	13, 23, 24, 26	fvmptd3 5749	. . . 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 6036	. . . . 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 14088	. . . . . 6 |- ( ( R e. Ring /\ X e. B /\ z e. B ) -> ( X .x. z ) e. B )
31	18, 19, 29, 30	syl3anc 1274	. . . . 5 |- ( ( ( R e. Ring /\ X e. B ) /\ ( y e. B /\ z e. B ) ) -> ( X .x. z ) e. B )
32	13, 28, 29, 31	fvmptd3 5749	. . . 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 6046	. . 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 13900	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 2202   |-> cmpt 4155   ` cfv 5333  (class class class)co 6028   Basecbs 13143   +g cplusg 13221   .rcmulr 13222   Grpcgrp 13644   GrpHom cghm 13888   Ringcrg 14071

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 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8166  ax-resscn 8167  ax-1cn 8168  ax-1re 8169  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-addcom 8175  ax-addass 8177  ax-i2m1 8180  ax-0lt1 8181  ax-0id 8183  ax-rnegex 8184  ax-pre-ltirr 8187  ax-pre-ltadd 8191

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-ov 6031  df-oprab 6032  df-mpo 6033  df-pnf 8259  df-mnf 8260  df-ltxr 8262  df-inn 9187  df-2 9245  df-3 9246  df-ndx 13146  df-slot 13147  df-base 13149  df-sets 13150  df-plusg 13234  df-mulr 13235  df-mgm 13500  df-sgrp 13546  df-mnd 13561  df-grp 13647  df-ghm 13889  df-mgp 13996  df-ring 14073

This theorem is referenced by: (None)