Theorem soinxp 4802

Description: Intersection of linear order with cross product of its field. (Contributed by Mario Carneiro, 10-Jul-2014.)

Assertion

Ref	Expression
soinxp	|- ( R Or A <-> ( R i^i ( A X. A ) ) Or A )

Proof of Theorem soinxp

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

Step	Hyp	Ref	Expression
1		poinxp 4801	. . 3 |- ( R Po A <-> ( R i^i ( A X. A ) ) Po A )
2		brinxp 4800	. . . . . . . 8 |- ( ( x e. A /\ y e. A ) -> ( x R y <-> x ( R i^i ( A X. A ) ) y ) )
3	2	3adant3 1044	. . . . . . 7 |- ( ( x e. A /\ y e. A /\ z e. A ) -> ( x R y <-> x ( R i^i ( A X. A ) ) y ) )
4		brinxp 4800	. . . . . . . . 9 |- ( ( x e. A /\ z e. A ) -> ( x R z <-> x ( R i^i ( A X. A ) ) z ) )
5	4	3adant2 1043	. . . . . . . 8 |- ( ( x e. A /\ y e. A /\ z e. A ) -> ( x R z <-> x ( R i^i ( A X. A ) ) z ) )
6		brinxp 4800	. . . . . . . . . 10 |- ( ( z e. A /\ y e. A ) -> ( z R y <-> z ( R i^i ( A X. A ) ) y ) )
7	6	ancoms 268	. . . . . . . . 9 |- ( ( y e. A /\ z e. A ) -> ( z R y <-> z ( R i^i ( A X. A ) ) y ) )
8	7	3adant1 1042	. . . . . . . 8 |- ( ( x e. A /\ y e. A /\ z e. A ) -> ( z R y <-> z ( R i^i ( A X. A ) ) y ) )
9	5, 8	orbi12d 801	. . . . . . 7 |- ( ( x e. A /\ y e. A /\ z e. A ) -> ( ( x R z \/ z R y ) <-> ( x ( R i^i ( A X. A ) ) z \/ z ( R i^i ( A X. A ) ) y ) ) )
10	3, 9	imbi12d 234	. . . . . 6 |- ( ( x e. A /\ y e. A /\ z e. A ) -> ( ( x R y -> ( x R z \/ z R y ) ) <-> ( x ( R i^i ( A X. A ) ) y -> ( x ( R i^i ( A X. A ) ) z \/ z ( R i^i ( A X. A ) ) y ) ) ) )
11	10	3expb 1231	. . . . 5 |- ( ( x e. A /\ ( y e. A /\ z e. A ) ) -> ( ( x R y -> ( x R z \/ z R y ) ) <-> ( x ( R i^i ( A X. A ) ) y -> ( x ( R i^i ( A X. A ) ) z \/ z ( R i^i ( A X. A ) ) y ) ) ) )
12	11	2ralbidva 2555	. . . 4 |- ( x e. A -> ( A. y e. A A. z e. A ( x R y -> ( x R z \/ z R y ) ) <-> A. y e. A A. z e. A ( x ( R i^i ( A X. A ) ) y -> ( x ( R i^i ( A X. A ) ) z \/ z ( R i^i ( A X. A ) ) y ) ) ) )
13	12	ralbiia 2547	. . 3 |- ( A. x e. A A. y e. A A. z e. A ( x R y -> ( x R z \/ z R y ) ) <-> A. x e. A A. y e. A A. z e. A ( x ( R i^i ( A X. A ) ) y -> ( x ( R i^i ( A X. A ) ) z \/ z ( R i^i ( A X. A ) ) y ) ) )
14	1, 13	anbi12i 460	. 2 |- ( ( R Po A /\ A. x e. A A. y e. A A. z e. A ( x R y -> ( x R z \/ z R y ) ) ) <-> ( ( R i^i ( A X. A ) ) Po A /\ A. x e. A A. y e. A A. z e. A ( x ( R i^i ( A X. A ) ) y -> ( x ( R i^i ( A X. A ) ) z \/ z ( R i^i ( A X. A ) ) y ) ) ) )
15		df-iso 4400	. 2 |- ( R Or A <-> ( R Po A /\ A. x e. A A. y e. A A. z e. A ( x R y -> ( x R z \/ z R y ) ) ) )
16		df-iso 4400	. 2 |- ( ( R i^i ( A X. A ) ) Or A <-> ( ( R i^i ( A X. A ) ) Po A /\ A. x e. A A. y e. A A. z e. A ( x ( R i^i ( A X. A ) ) y -> ( x ( R i^i ( A X. A ) ) z \/ z ( R i^i ( A X. A ) ) y ) ) ) )
17	14, 15, 16	3bitr4i 212	1 |- ( R Or A <-> ( R i^i ( A X. A ) ) Or A )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 716   /\ w3a 1005   e. wcel 2202   A.wral 2511   i^i cin 3200   class class class wbr 4093   Po wpo 4397   Or wor 4398   X. cxp 4729

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-br 4094  df-opab 4156  df-po 4399  df-iso 4400  df-xp 4737

This theorem is referenced by: (None)