Theorem crngcom 14301

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 2238	. . . . 5 |- (mulGrp ` R ) = (mulGrp ` R )
2	1	crngmgp 14291	. . . 4 |- ( R e. CRing -> (mulGrp ` R ) e. CMnd )
3	2	3ad2ant1 1049	. . 3 |- ( ( R e. CRing /\ X e. B /\ Y e. B ) -> (mulGrp ` R ) e. CMnd )
4		simp2 1029	. . . 4 |- ( ( R e. CRing /\ X e. B /\ Y e. B ) -> X e. B )
5		ringcl.b	. . . . . 6 |- B = ( Base ` R )
6	1, 5	mgpbasg 14207	. . . . 5 |- ( R e. CRing -> B = ( Base ` (mulGrp ` R ) ) )
7	6	3ad2ant1 1049	. . . 4 |- ( ( R e. CRing /\ X e. B /\ Y e. B ) -> B = ( Base ` (mulGrp ` R ) ) )
8	4, 7	eleqtrd 2317	. . 3 |- ( ( R e. CRing /\ X e. B /\ Y e. B ) -> X e. ( Base ` (mulGrp ` R ) ) )
9		simp3 1030	. . . 4 |- ( ( R e. CRing /\ X e. B /\ Y e. B ) -> Y e. B )
10	9, 7	eleqtrd 2317	. . 3 |- ( ( R e. CRing /\ X e. B /\ Y e. B ) -> Y e. ( Base ` (mulGrp ` R ) ) )
11		eqid 2238	. . . 4 |- ( Base ` (mulGrp ` R ) ) = ( Base ` (mulGrp ` R ) )
12		eqid 2238	. . . 4 |- ( +g ` (mulGrp ` R ) ) = ( +g ` (mulGrp ` R ) )
13	11, 12	cmncom 14088	. . 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 1278	. 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 14204	. . . 4 |- ( R e. CRing -> .x. = ( +g ` (mulGrp ` R ) ) )
17	16	3ad2ant1 1049	. . 3 |- ( ( R e. CRing /\ X e. B /\ Y e. B ) -> .x. = ( +g ` (mulGrp ` R ) ) )
18	17	oveqd 6096	. 2 |- ( ( R e. CRing /\ X e. B /\ Y e. B ) -> ( X .x. Y ) = ( X ( +g ` (mulGrp ` R ) ) Y ) )
19	17	oveqd 6096	. 2 |- ( ( R e. CRing /\ X e. B /\ Y e. B ) -> ( Y .x. X ) = ( Y ( +g ` (mulGrp ` R ) ) X ) )
20	14, 18, 19	3eqtr4d 2281	1 |- ( ( R e. CRing /\ X e. B /\ Y e. B ) -> ( X .x. Y ) = ( Y .x. X ) )

This proof depends on syntax axioms:   -> wi 4   /\ w3a 1009   = wceq 1402   e. wcel 2209   ` cfv 5375  (class class class)co 6079   Basecbs 13335   +g cplusg 13414   .rcmulr 13415  CMndccmn 14070  mulGrpcmgp 14200   CRingccrg 14284

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-sep 4247  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-cnex 8264  ax-resscn 8265  ax-1cn 8266  ax-1re 8267  ax-icn 8268  ax-addcl 8269  ax-addrcl 8270  ax-mulcl 8271  ax-addcom 8273  ax-addass 8275  ax-i2m1 8278  ax-0lt1 8279  ax-0id 8281  ax-rnegex 8282  ax-pre-ltirr 8285  ax-pre-ltadd 8289

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-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 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-br 4129  df-opab 4191  df-mpt 4192  df-id 4436  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-iota 5335  df-fun 5377  df-fn 5378  df-fv 5383  df-ov 6082  df-oprab 6083  df-mpo 6084  df-pnf 8356  df-mnf 8357  df-ltxr 8359  df-inn 9288  df-2 9346  df-3 9347  df-ndx 13338  df-slot 13339  df-base 13341  df-sets 13342  df-plusg 13427  df-mulr 13428  df-cmn 14072  df-mgp 14201  df-cring 14286

This theorem is used by:  crngoppr  14360  unitmulclb  14404  rdivmuldivd  14434  rmodislmodlem  14670  quscrng  14853