Theorem mdbr 29153

Description: Binary relation expressing <. A , B >. is a modular pair. Definition 1.1 of [MaedaMaeda] p. 1. (Contributed by NM, 14-Jun-2004.) (New usage is discouraged.)

Assertion

Ref	Expression
mdbr	|- ( ( A e. CH /\ B e. CH ) -> ( A MH B <-> A. x e. CH ( x C_ B -> ( ( x vH A ) i^i B ) = ( x vH ( A i^i B ) ) ) ) )

Distinct variable groups:   x, A   x, B

Proof of Theorem mdbr

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

Step	Hyp	Ref	Expression
1		eleq1 2689	. . . . 5 |- ( y = A -> ( y e. CH <-> A e. CH ) )
2	1	anbi1d 741	. . . 4 |- ( y = A -> ( ( y e. CH /\ z e. CH ) <-> ( A e. CH /\ z e. CH ) ) )
3		oveq2 6658	. . . . . . . 8 |- ( y = A -> ( x vH y ) = ( x vH A ) )
4	3	ineq1d 3813	. . . . . . 7 |- ( y = A -> ( ( x vH y ) i^i z ) = ( ( x vH A ) i^i z ) )
5		ineq1 3807	. . . . . . . 8 |- ( y = A -> ( y i^i z ) = ( A i^i z ) )
6	5	oveq2d 6666	. . . . . . 7 |- ( y = A -> ( x vH ( y i^i z ) ) = ( x vH ( A i^i z ) ) )
7	4, 6	eqeq12d 2637	. . . . . 6 |- ( y = A -> ( ( ( x vH y ) i^i z ) = ( x vH ( y i^i z ) ) <-> ( ( x vH A ) i^i z ) = ( x vH ( A i^i z ) ) ) )
8	7	imbi2d 330	. . . . 5 |- ( y = A -> ( ( x C_ z -> ( ( x vH y ) i^i z ) = ( x vH ( y i^i z ) ) ) <-> ( x C_ z -> ( ( x vH A ) i^i z ) = ( x vH ( A i^i z ) ) ) ) )
9	8	ralbidv 2986	. . . 4 |- ( y = A -> ( A. x e. CH ( x C_ z -> ( ( x vH y ) i^i z ) = ( x vH ( y i^i z ) ) ) <-> A. x e. CH ( x C_ z -> ( ( x vH A ) i^i z ) = ( x vH ( A i^i z ) ) ) ) )
10	2, 9	anbi12d 747	. . 3 |- ( y = A -> ( ( ( y e. CH /\ z e. CH ) /\ A. x e. CH ( x C_ z -> ( ( x vH y ) i^i z ) = ( x vH ( y i^i z ) ) ) ) <-> ( ( A e. CH /\ z e. CH ) /\ A. x e. CH ( x C_ z -> ( ( x vH A ) i^i z ) = ( x vH ( A i^i z ) ) ) ) ) )
11		eleq1 2689	. . . . 5 |- ( z = B -> ( z e. CH <-> B e. CH ) )
12	11	anbi2d 740	. . . 4 |- ( z = B -> ( ( A e. CH /\ z e. CH ) <-> ( A e. CH /\ B e. CH ) ) )
13		sseq2 3627	. . . . . 6 |- ( z = B -> ( x C_ z <-> x C_ B ) )
14		ineq2 3808	. . . . . . 7 |- ( z = B -> ( ( x vH A ) i^i z ) = ( ( x vH A ) i^i B ) )
15		ineq2 3808	. . . . . . . 8 |- ( z = B -> ( A i^i z ) = ( A i^i B ) )
16	15	oveq2d 6666	. . . . . . 7 |- ( z = B -> ( x vH ( A i^i z ) ) = ( x vH ( A i^i B ) ) )
17	14, 16	eqeq12d 2637	. . . . . 6 |- ( z = B -> ( ( ( x vH A ) i^i z ) = ( x vH ( A i^i z ) ) <-> ( ( x vH A ) i^i B ) = ( x vH ( A i^i B ) ) ) )
18	13, 17	imbi12d 334	. . . . 5 |- ( z = B -> ( ( x C_ z -> ( ( x vH A ) i^i z ) = ( x vH ( A i^i z ) ) ) <-> ( x C_ B -> ( ( x vH A ) i^i B ) = ( x vH ( A i^i B ) ) ) ) )
19	18	ralbidv 2986	. . . 4 |- ( z = B -> ( A. x e. CH ( x C_ z -> ( ( x vH A ) i^i z ) = ( x vH ( A i^i z ) ) ) <-> A. x e. CH ( x C_ B -> ( ( x vH A ) i^i B ) = ( x vH ( A i^i B ) ) ) ) )
20	12, 19	anbi12d 747	. . 3 |- ( z = B -> ( ( ( A e. CH /\ z e. CH ) /\ A. x e. CH ( x C_ z -> ( ( x vH A ) i^i z ) = ( x vH ( A i^i z ) ) ) ) <-> ( ( A e. CH /\ B e. CH ) /\ A. x e. CH ( x C_ B -> ( ( x vH A ) i^i B ) = ( x vH ( A i^i B ) ) ) ) ) )
21		df-md 29139	. . 3 |- MH = { <. y , z >. | ( ( y e. CH /\ z e. CH ) /\ A. x e. CH ( x C_ z -> ( ( x vH y ) i^i z ) = ( x vH ( y i^i z ) ) ) ) }
22	10, 20, 21	brabg 4994	. 2 |- ( ( A e. CH /\ B e. CH ) -> ( A MH B <-> ( ( A e. CH /\ B e. CH ) /\ A. x e. CH ( x C_ B -> ( ( x vH A ) i^i B ) = ( x vH ( A i^i B ) ) ) ) ) )
23	22	bianabs 924	1 |- ( ( A e. CH /\ B e. CH ) -> ( A MH B <-> A. x e. CH ( x C_ B -> ( ( x vH A ) i^i B ) = ( x vH ( A i^i B ) ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   i^i cin 3573   C_ wss 3574   class class class wbr 4653  (class class class)co 6650   CHcch 27786   vH chj 27790   MH cmd 27823

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-iota 5851  df-fv 5896  df-ov 6653  df-md 29139

This theorem is referenced by:  mdi  29154  mdbr2  29155  mdbr3  29156  dmdmd  29159  mddmd2  29168  mdsl1i  29180