Theorem rngoi 33698

Description: The properties of a unital ring. (Contributed by Steve Rodriguez, 8-Sep-2007.) (Proof shortened by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)

Hypotheses

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

Assertion

Ref	Expression
rngoi	|- ( R e. RingOps -> ( ( G e. AbelOp /\ H : ( X X. X ) --> X ) /\ ( A. x e. X A. y e. X A. z e. X ( ( ( x H y ) H z ) = ( x H ( y H z ) ) /\ ( x H ( y G z ) ) = ( ( x H y ) G ( x H z ) ) /\ ( ( x G y ) H z ) = ( ( x H z ) G ( y H z ) ) ) /\ E. x e. X A. y e. X ( ( x H y ) = y /\ ( y H x ) = y ) ) ) )

Distinct variable groups:   x, y, z, G   x, H, y, z   x, X, y, z   x, R

Allowed substitution hints:   R( y, z)

Proof of Theorem rngoi

Step	Hyp	Ref	Expression
1		relrngo 33695	. . . . 5 |- Rel RingOps
2		1st2nd 7214	. . . . 5 |- ( ( Rel RingOps /\ R e. RingOps ) -> R = <. ( 1st ` R ) , ( 2nd ` R ) >. )
3	1, 2	mpan 706	. . . 4 |- ( R e. RingOps -> R = <. ( 1st ` R ) , ( 2nd ` R ) >. )
4		ringi.1	. . . . 5 |- G = ( 1st ` R )
5		ringi.2	. . . . 5 |- H = ( 2nd ` R )
6	4, 5	opeq12i 4407	. . . 4 |- <. G , H >. = <. ( 1st ` R ) , ( 2nd ` R ) >.
7	3, 6	syl6reqr 2675	. . 3 |- ( R e. RingOps -> <. G , H >. = R )
8		id 22	. . 3 |- ( R e. RingOps -> R e. RingOps )
9	7, 8	eqeltrd 2701	. 2 |- ( R e. RingOps -> <. G , H >. e. RingOps )
10		fvex 6201	. . . 4 |- ( 2nd ` R ) e. _V
11	5, 10	eqeltri 2697	. . 3 |- H e. _V
12		ringi.3	. . . 4 |- X = ran G
13	12	isrngo 33696	. . 3 |- ( H e. _V -> ( <. G , H >. e. RingOps <-> ( ( G e. AbelOp /\ H : ( X X. X ) --> X ) /\ ( A. x e. X A. y e. X A. z e. X ( ( ( x H y ) H z ) = ( x H ( y H z ) ) /\ ( x H ( y G z ) ) = ( ( x H y ) G ( x H z ) ) /\ ( ( x G y ) H z ) = ( ( x H z ) G ( y H z ) ) ) /\ E. x e. X A. y e. X ( ( x H y ) = y /\ ( y H x ) = y ) ) ) ) )
14	11, 13	ax-mp 5	. 2 |- ( <. G , H >. e. RingOps <-> ( ( G e. AbelOp /\ H : ( X X. X ) --> X ) /\ ( A. x e. X A. y e. X A. z e. X ( ( ( x H y ) H z ) = ( x H ( y H z ) ) /\ ( x H ( y G z ) ) = ( ( x H y ) G ( x H z ) ) /\ ( ( x G y ) H z ) = ( ( x H z ) G ( y H z ) ) ) /\ E. x e. X A. y e. X ( ( x H y ) = y /\ ( y H x ) = y ) ) ) )
15	9, 14	sylib 208	1 |- ( R e. RingOps -> ( ( G e. AbelOp /\ H : ( X X. X ) --> X ) /\ ( A. x e. X A. y e. X A. z e. X ( ( ( x H y ) H z ) = ( x H ( y H z ) ) /\ ( x H ( y G z ) ) = ( ( x H y ) G ( x H z ) ) /\ ( ( x G y ) H z ) = ( ( x H z ) G ( y H z ) ) ) /\ E. x e. X A. y e. X ( ( x H y ) = y /\ ( y H x ) = y ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   _Vcvv 3200   <.cop 4183   X. cxp 5112   ran crn 5115   Rel wrel 5119   -->wf 5884   ` cfv 5888  (class class class)co 6650   1stc1st 7166   2ndc2nd 7167   AbelOpcablo 27398   RingOpscrngo 33693

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-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-1st 7168  df-2nd 7169  df-rngo 33694

This theorem is referenced by:  rngosm  33699  rngoid  33701  rngoideu  33702  rngodi  33703  rngodir  33704  rngoass  33705  rngoablo  33707  rngorn1eq  33733  rngomndo  33734