Theorem crngcom 13993

Description: A commutative ring's multiplication operation is commutative. (Contributed by Mario Carneiro, 7-Jan-2015.)

Hypotheses

Ref	Expression
ringcl.b	|- B = ( Base ` R )
ringcl.t	|- .x. = ( .r ` R )

Assertion

Ref	Expression
crngcom	|- ( ( R e. CRing /\ X e. B /\ Y e. B ) -> ( X .x. Y ) = ( Y .x. X ) )

Proof of Theorem crngcom

Step	Hyp	Ref	Expression
1		eqid 2229	. . . . 5 |- (mulGrp ` R ) = (mulGrp ` R )
2	1	crngmgp 13983	. . . 4 |- ( R e. CRing -> (mulGrp ` R ) e. CMnd )
3	2	3ad2ant1 1042	. . 3 |- ( ( R e. CRing /\ X e. B /\ Y e. B ) -> (mulGrp ` R ) e. CMnd )
4		simp2 1022	. . . 4 |- ( ( R e. CRing /\ X e. B /\ Y e. B ) -> X e. B )
5		ringcl.b	. . . . . 6 |- B = ( Base ` R )
6	1, 5	mgpbasg 13905	. . . . 5 |- ( R e. CRing -> B = ( Base ` (mulGrp ` R ) ) )
7	6	3ad2ant1 1042	. . . 4 |- ( ( R e. CRing /\ X e. B /\ Y e. B ) -> B = ( Base ` (mulGrp ` R ) ) )
8	4, 7	eleqtrd 2308	. . 3 |- ( ( R e. CRing /\ X e. B /\ Y e. B ) -> X e. ( Base ` (mulGrp ` R ) ) )
9		simp3 1023	. . . 4 |- ( ( R e. CRing /\ X e. B /\ Y e. B ) -> Y e. B )
10	9, 7	eleqtrd 2308	. . 3 |- ( ( R e. CRing /\ X e. B /\ Y e. B ) -> Y e. ( Base ` (mulGrp ` R ) ) )
11		eqid 2229	. . . 4 |- ( Base ` (mulGrp ` R ) ) = ( Base ` (mulGrp ` R ) )
12		eqid 2229	. . . 4 |- ( +g ` (mulGrp ` R ) ) = ( +g ` (mulGrp ` R ) )
13	11, 12	cmncom 13855	. . 3 |- ( ( (mulGrp ` R ) e. CMnd /\ X e. ( Base ` (mulGrp ` R ) ) /\ Y e. ( Base ` (mulGrp ` R ) ) ) -> ( X ( +g ` (mulGrp ` R ) ) Y ) = ( Y ( +g ` (mulGrp ` R ) ) X ) )
14	3, 8, 10, 13	syl3anc 1271	. 2 |- ( ( R e. CRing /\ X e. B /\ Y e. B ) -> ( X ( +g ` (mulGrp ` R ) ) Y ) = ( Y ( +g ` (mulGrp ` R ) ) X ) )
15		ringcl.t	. . . . 5 |- .x. = ( .r ` R )
16	1, 15	mgpplusgg 13903	. . . 4 |- ( R e. CRing -> .x. = ( +g ` (mulGrp ` R ) ) )
17	16	3ad2ant1 1042	. . 3 |- ( ( R e. CRing /\ X e. B /\ Y e. B ) -> .x. = ( +g ` (mulGrp ` R ) ) )
18	17	oveqd 6024	. 2 |- ( ( R e. CRing /\ X e. B /\ Y e. B ) -> ( X .x. Y ) = ( X ( +g ` (mulGrp ` R ) ) Y ) )
19	17	oveqd 6024	. 2 |- ( ( R e. CRing /\ X e. B /\ Y e. B ) -> ( Y .x. X ) = ( Y ( +g ` (mulGrp ` R ) ) X ) )
20	14, 18, 19	3eqtr4d 2272	1 |- ( ( R e. CRing /\ X e. B /\ Y e. B ) -> ( X .x. Y ) = ( Y .x. X ) )

Syntax hints:   -> wi 4   /\ w3a 1002   = wceq 1395   e. wcel 2200   ` cfv 5318  (class class class)co 6007   Basecbs 13048   +g cplusg 13126   .rcmulr 13127  CMndccmn 13837  mulGrpcmgp 13899   CRingccrg 13976

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-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8101  ax-resscn 8102  ax-1cn 8103  ax-1re 8104  ax-icn 8105  ax-addcl 8106  ax-addrcl 8107  ax-mulcl 8108  ax-addcom 8110  ax-addass 8112  ax-i2m1 8115  ax-0lt1 8116  ax-0id 8118  ax-rnegex 8119  ax-pre-ltirr 8122  ax-pre-ltadd 8126

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-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-iota 5278  df-fun 5320  df-fn 5321  df-fv 5326  df-ov 6010  df-oprab 6011  df-mpo 6012  df-pnf 8194  df-mnf 8195  df-ltxr 8197  df-inn 9122  df-2 9180  df-3 9181  df-ndx 13051  df-slot 13052  df-base 13054  df-sets 13055  df-plusg 13139  df-mulr 13140  df-cmn 13839  df-mgp 13900  df-cring 13978

This theorem is referenced by:  crngoppr  14051  unitmulclb  14094  rdivmuldivd  14124  rmodislmodlem  14330  quscrng  14513