Theorem iscrngo2 33796

Description: The predicate "is a commutative ring". (Contributed by Jeff Madsen, 8-Jun-2010.)

Hypotheses

Ref	Expression
iscring2.1	|- G = ( 1st ` R )
iscring2.2	|- H = ( 2nd ` R )
iscring2.3	|- X = ran G

Assertion

Ref	Expression
iscrngo2	|- ( R e. CRingOps <-> ( R e. RingOps /\ A. x e. X A. y e. X ( x H y ) = ( y H x ) ) )

Distinct variable groups:   x, R, y   x, X, y

Allowed substitution hints:   G( x, y)   H( x, y)

Proof of Theorem iscrngo2

Step	Hyp	Ref	Expression
1		iscrngo 33795	. 2 |- ( R e. CRingOps <-> ( R e. RingOps /\ R e. Com2 ) )
2		relrngo 33695	. . . . 5 |- Rel RingOps
3		1st2nd 7214	. . . . 5 |- ( ( Rel RingOps /\ R e. RingOps ) -> R = <. ( 1st ` R ) , ( 2nd ` R ) >. )
4	2, 3	mpan 706	. . . 4 |- ( R e. RingOps -> R = <. ( 1st ` R ) , ( 2nd ` R ) >. )
5		eleq1 2689	. . . . 5 |- ( R = <. ( 1st ` R ) , ( 2nd ` R ) >. -> ( R e. Com2 <-> <. ( 1st ` R ) , ( 2nd ` R ) >. e. Com2 ) )
6		iscring2.3	. . . . . . . 8 |- X = ran G
7		iscring2.1	. . . . . . . . 9 |- G = ( 1st ` R )
8	7	rneqi 5352	. . . . . . . 8 |- ran G = ran ( 1st ` R )
9	6, 8	eqtri 2644	. . . . . . 7 |- X = ran ( 1st ` R )
10	9	raleqi 3142	. . . . . 6 |- ( A. x e. X A. y e. ran ( 1st ` R ) ( x ( 2nd ` R ) y ) = ( y ( 2nd ` R ) x ) <-> A. x e. ran ( 1st ` R ) A. y e. ran ( 1st ` R ) ( x ( 2nd ` R ) y ) = ( y ( 2nd ` R ) x ) )
11		iscring2.2	. . . . . . . . . 10 |- H = ( 2nd ` R )
12	11	oveqi 6663	. . . . . . . . 9 |- ( x H y ) = ( x ( 2nd ` R ) y )
13	11	oveqi 6663	. . . . . . . . 9 |- ( y H x ) = ( y ( 2nd ` R ) x )
14	12, 13	eqeq12i 2636	. . . . . . . 8 |- ( ( x H y ) = ( y H x ) <-> ( x ( 2nd ` R ) y ) = ( y ( 2nd ` R ) x ) )
15	9, 14	raleqbii 2990	. . . . . . 7 |- ( A. y e. X ( x H y ) = ( y H x ) <-> A. y e. ran ( 1st ` R ) ( x ( 2nd ` R ) y ) = ( y ( 2nd ` R ) x ) )
16	15	ralbii 2980	. . . . . 6 |- ( A. x e. X A. y e. X ( x H y ) = ( y H x ) <-> A. x e. X A. y e. ran ( 1st ` R ) ( x ( 2nd ` R ) y ) = ( y ( 2nd ` R ) x ) )
17		fvex 6201	. . . . . . 7 |- ( 1st ` R ) e. _V
18		fvex 6201	. . . . . . 7 |- ( 2nd ` R ) e. _V
19		iscom2 33794	. . . . . . 7 |- ( ( ( 1st ` R ) e. _V /\ ( 2nd ` R ) e. _V ) -> ( <. ( 1st ` R ) , ( 2nd ` R ) >. e. Com2 <-> A. x e. ran ( 1st ` R ) A. y e. ran ( 1st ` R ) ( x ( 2nd ` R ) y ) = ( y ( 2nd ` R ) x ) ) )
20	17, 18, 19	mp2an 708	. . . . . 6 |- ( <. ( 1st ` R ) , ( 2nd ` R ) >. e. Com2 <-> A. x e. ran ( 1st ` R ) A. y e. ran ( 1st ` R ) ( x ( 2nd ` R ) y ) = ( y ( 2nd ` R ) x ) )
21	10, 16, 20	3bitr4ri 293	. . . . 5 |- ( <. ( 1st ` R ) , ( 2nd ` R ) >. e. Com2 <-> A. x e. X A. y e. X ( x H y ) = ( y H x ) )
22	5, 21	syl6bb 276	. . . 4 |- ( R = <. ( 1st ` R ) , ( 2nd ` R ) >. -> ( R e. Com2 <-> A. x e. X A. y e. X ( x H y ) = ( y H x ) ) )
23	4, 22	syl 17	. . 3 |- ( R e. RingOps -> ( R e. Com2 <-> A. x e. X A. y e. X ( x H y ) = ( y H x ) ) )
24	23	pm5.32i 669	. 2 |- ( ( R e. RingOps /\ R e. Com2 ) <-> ( R e. RingOps /\ A. x e. X A. y e. X ( x H y ) = ( y H x ) ) )
25	1, 24	bitri 264	1 |- ( R e. CRingOps <-> ( R e. RingOps /\ A. x e. X A. y e. X ( x H y ) = ( y H x ) ) )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   <.cop 4183   ran crn 5115   Rel wrel 5119   ` cfv 5888  (class class class)co 6650   1stc1st 7166   2ndc2nd 7167   RingOpscrngo 33693   Com2ccm2 33788  CRingOpsccring 33792

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-1st 7168  df-2nd 7169  df-rngo 33694  df-com2 33789  df-crngo 33793

This theorem is referenced by:  crngocom  33800  crngohomfo  33805