Theorem iscom2 33794

Description: A device to add commutativity to various sorts of rings. (Contributed by FL, 6-Sep-2009.) (New usage is discouraged.)

Assertion

Ref	Expression
iscom2	|- ( ( G e. A /\ H e. B ) -> ( <. G , H >. e. Com2 <-> A. a e. ran G A. b e. ran G ( a H b ) = ( b H a ) ) )

Distinct variable groups:   G, a, b   H, a, b

Allowed substitution hints:   A( a, b)   B( a, b)

Proof of Theorem iscom2

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-com2 33789	. . . 4 |- Com2 = { <. x , y >. | A. a e. ran x A. b e. ran x ( a y b ) = ( b y a ) }
2	1	a1i 11	. . 3 |- ( ( G e. A /\ H e. B ) -> Com2 = { <. x , y >. | A. a e. ran x A. b e. ran x ( a y b ) = ( b y a ) } )
3	2	eleq2d 2687	. 2 |- ( ( G e. A /\ H e. B ) -> ( <. G , H >. e. Com2 <-> <. G , H >. e. { <. x , y >. | A. a e. ran x A. b e. ran x ( a y b ) = ( b y a ) } ) )
4		rneq 5351	. . . 4 |- ( x = G -> ran x = ran G )
5	4	raleqdv 3144	. . . 4 |- ( x = G -> ( A. b e. ran x ( a y b ) = ( b y a ) <-> A. b e. ran G ( a y b ) = ( b y a ) ) )
6	4, 5	raleqbidv 3152	. . 3 |- ( x = G -> ( A. a e. ran x A. b e. ran x ( a y b ) = ( b y a ) <-> A. a e. ran G A. b e. ran G ( a y b ) = ( b y a ) ) )
7		oveq 6656	. . . . 5 |- ( y = H -> ( a y b ) = ( a H b ) )
8		oveq 6656	. . . . 5 |- ( y = H -> ( b y a ) = ( b H a ) )
9	7, 8	eqeq12d 2637	. . . 4 |- ( y = H -> ( ( a y b ) = ( b y a ) <-> ( a H b ) = ( b H a ) ) )
10	9	2ralbidv 2989	. . 3 |- ( y = H -> ( A. a e. ran G A. b e. ran G ( a y b ) = ( b y a ) <-> A. a e. ran G A. b e. ran G ( a H b ) = ( b H a ) ) )
11	6, 10	opelopabg 4993	. 2 |- ( ( G e. A /\ H e. B ) -> ( <. G , H >. e. { <. x , y >. | A. a e. ran x A. b e. ran x ( a y b ) = ( b y a ) } <-> A. a e. ran G A. b e. ran G ( a H b ) = ( b H a ) ) )
12	3, 11	bitrd 268	1 |- ( ( G e. A /\ H e. B ) -> ( <. G , H >. e. Com2 <-> A. a e. ran G A. b e. ran G ( a H b ) = ( b H a ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   <.cop 4183   {copab 4712   ran crn 5115  (class class class)co 6650   Com2ccm2 33788

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

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-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-cnv 5122  df-dm 5124  df-rn 5125  df-iota 5851  df-fv 5896  df-ov 6653  df-com2 33789

This theorem is referenced by:  iscrngo2  33796  iscringd  33797