Theorem soinxp 4681

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 4680	. . 3 |- ( R Po A <-> ( R i^i ( A X. A ) ) Po A )
2		brinxp 4679	. . . . . . . 8 |- ( ( x e. A /\ y e. A ) -> ( x R y <-> x ( R i^i ( A X. A ) ) y ) )
3	2	3adant3 1012	. . . . . . 7 |- ( ( x e. A /\ y e. A /\ z e. A ) -> ( x R y <-> x ( R i^i ( A X. A ) ) y ) )
4		brinxp 4679	. . . . . . . . 9 |- ( ( x e. A /\ z e. A ) -> ( x R z <-> x ( R i^i ( A X. A ) ) z ) )
5	4	3adant2 1011	. . . . . . . 8 |- ( ( x e. A /\ y e. A /\ z e. A ) -> ( x R z <-> x ( R i^i ( A X. A ) ) z ) )
6		brinxp 4679	. . . . . . . . . 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 1010	. . . . . . . 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 788	. . . . . . 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 1199	. . . . 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 2492	. . . 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 2484	. . 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 457	. 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 4282	. 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 4282	. 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 703   /\ w3a 973   e. wcel 2141   A.wral 2448   i^i cin 3120   class class class wbr 3989   Po wpo 4279   Or wor 4280   X. cxp 4609

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-po 4281  df-iso 4282  df-xp 4617

This theorem is referenced by: (None)