Theorem ringlghm 14368

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 2238	. 2 |- ( +g ` R ) = ( +g ` R )
3		ringgrp 14307	. . 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 14319	. . . 4 |- ( ( R e. Ring /\ X e. B /\ x e. B ) -> ( X .x. x ) e. B )
7	6	3expa 1234	. . 3 |- ( ( ( R e. Ring /\ X e. B ) /\ x e. B ) -> ( X .x. x ) e. B )
8	7	fmpttd 5863	. 2 |- ( ( R e. Ring /\ X e. B ) -> ( x e. B |-> ( X .x. x ) ) : B --> B )
9		3anass 1013	. . . . 5 |- ( ( X e. B /\ y e. B /\ z e. B ) <-> ( X e. B /\ ( y e. B /\ z e. B ) ) )
10	1, 2, 5	ringdi 14325	. . . . 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 404	. . 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 2238	. . . 4 |- ( x e. B |-> ( X .x. x ) ) = ( x e. B |-> ( X .x. x ) )
14		oveq2 6093	. . . 4 |- ( x = ( y ( +g ` R ) z ) -> ( X .x. x ) = ( X .x. ( y ( +g ` R ) z ) ) )
15	1, 2	ringacl 14337	. . . . . 6 |- ( ( R e. Ring /\ y e. B /\ z e. B ) -> ( y ( +g ` R ) z ) e. B )
16	15	3expb 1235	. . . . 5 |- ( ( R e. Ring /\ ( y e. B /\ z e. B ) ) -> ( y ( +g ` R ) z ) e. B )
17	16	adantlr 481	. . . 4 |- ( ( ( R e. Ring /\ X e. B ) /\ ( y e. B /\ z e. B ) ) -> ( y ( +g ` R ) z ) e. B )
18		simpll 531	. . . . 5 |- ( ( ( R e. Ring /\ X e. B ) /\ ( y e. B /\ z e. B ) ) -> R e. Ring )
19		simplr 533	. . . . 5 |- ( ( ( R e. Ring /\ X e. B ) /\ ( y e. B /\ z e. B ) ) -> X e. B )
20	1, 5	ringcl 14319	. . . . 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 1278	. . . 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 5799	. . 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 6093	. . . . 5 |- ( x = y -> ( X .x. x ) = ( X .x. y ) )
24		simprl 535	. . . . 5 |- ( ( ( R e. Ring /\ X e. B ) /\ ( y e. B /\ z e. B ) ) -> y e. B )
25	1, 5	ringcl 14319	. . . . . 6 |- ( ( R e. Ring /\ X e. B /\ y e. B ) -> ( X .x. y ) e. B )
26	18, 19, 24, 25	syl3anc 1278	. . . . 5 |- ( ( ( R e. Ring /\ X e. B ) /\ ( y e. B /\ z e. B ) ) -> ( X .x. y ) e. B )
27	13, 23, 24, 26	fvmptd3 5799	. . . 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 6093	. . . . 5 |- ( x = z -> ( X .x. x ) = ( X .x. z ) )
29		simprr 537	. . . . 5 |- ( ( ( R e. Ring /\ X e. B ) /\ ( y e. B /\ z e. B ) ) -> z e. B )
30	1, 5	ringcl 14319	. . . . . 6 |- ( ( R e. Ring /\ X e. B /\ z e. B ) -> ( X .x. z ) e. B )
31	18, 19, 29, 30	syl3anc 1278	. . . . 5 |- ( ( ( R e. Ring /\ X e. B ) /\ ( y e. B /\ z e. B ) ) -> ( X .x. z ) e. B )
32	13, 28, 29, 31	fvmptd3 5799	. . . 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 6103	. . 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 2281	. 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 14057	1 |- ( ( R e. Ring /\ X e. B ) -> ( x e. B |-> ( X .x. x ) ) e. ( R GrpHom R ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   /\ w3a 1009   = wceq 1402   e. wcel 2209   |-> cmpt 4192   ` cfv 5377  (class class class)co 6085   Basecbs 13354   +g cplusg 13433   .rcmulr 13434   Grpcgrp 13807   GrpHom cghm 14045   Ringcrg 14302

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-addcom 8279  ax-addass 8281  ax-i2m1 8284  ax-0lt1 8285  ax-0id 8287  ax-rnegex 8288  ax-pre-ltirr 8291  ax-pre-ltadd 8295

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-ov 6088  df-oprab 6089  df-mpo 6090  df-pnf 8362  df-mnf 8363  df-ltxr 8365  df-inn 9306  df-2 9364  df-3 9365  df-ndx 13357  df-slot 13358  df-base 13360  df-sets 13361  df-plusg 13446  df-mulr 13447  df-mgm 13678  df-sgrp 13719  df-mnd 13732  df-grp 13810  df-ghm 14046  df-mgp 14220  df-ring 14304

This theorem is used by: (None)