Theorem soinxp 4674

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 4673	. . 3 |- ( R Po A <-> ( R i^i ( A X. A ) ) Po A )
2		brinxp 4672	. . . . . . . 8 |- ( ( x e. A /\ y e. A ) -> ( x R y <-> x ( R i^i ( A X. A ) ) y ) )
3	2	3adant3 1007	. . . . . . 7 |- ( ( x e. A /\ y e. A /\ z e. A ) -> ( x R y <-> x ( R i^i ( A X. A ) ) y ) )
4		brinxp 4672	. . . . . . . . 9 |- ( ( x e. A /\ z e. A ) -> ( x R z <-> x ( R i^i ( A X. A ) ) z ) )
5	4	3adant2 1006	. . . . . . . 8 |- ( ( x e. A /\ y e. A /\ z e. A ) -> ( x R z <-> x ( R i^i ( A X. A ) ) z ) )
6		brinxp 4672	. . . . . . . . . 10 |- ( ( z e. A /\ y e. A ) -> ( z R y <-> z ( R i^i ( A X. A ) ) y ) )
7	6	ancoms 266	. . . . . . . . 9 |- ( ( y e. A /\ z e. A ) -> ( z R y <-> z ( R i^i ( A X. A ) ) y ) )
8	7	3adant1 1005	. . . . . . . 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 783	. . . . . . 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 233	. . . . . 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 1194	. . . . 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 2488	. . . 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 2480	. . 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 456	. 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 4275	. 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 4275	. 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 211	1 |- ( R Or A <-> ( R i^i ( A X. A ) ) Or A )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 698   /\ w3a 968   e. wcel 2136   A.wral 2444   i^i cin 3115   class class class wbr 3982   Po wpo 4272   Or wor 4273   X. cxp 4602

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-opab 4044  df-po 4274  df-iso 4275  df-xp 4610

This theorem is referenced by: (None)