Theorem soinxp 4714

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 4713	. . 3 |- ( R Po A <-> ( R i^i ( A X. A ) ) Po A )
2		brinxp 4712	. . . . . . . 8 |- ( ( x e. A /\ y e. A ) -> ( x R y <-> x ( R i^i ( A X. A ) ) y ) )
3	2	3adant3 1019	. . . . . . 7 |- ( ( x e. A /\ y e. A /\ z e. A ) -> ( x R y <-> x ( R i^i ( A X. A ) ) y ) )
4		brinxp 4712	. . . . . . . . 9 |- ( ( x e. A /\ z e. A ) -> ( x R z <-> x ( R i^i ( A X. A ) ) z ) )
5	4	3adant2 1018	. . . . . . . 8 |- ( ( x e. A /\ y e. A /\ z e. A ) -> ( x R z <-> x ( R i^i ( A X. A ) ) z ) )
6		brinxp 4712	. . . . . . . . . 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 1017	. . . . . . . 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 794	. . . . . . 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 1206	. . . . 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 2512	. . . 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 2504	. . 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 4315	. 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 4315	. 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 709   /\ w3a 980   e. wcel 2160   A.wral 2468   i^i cin 3143   class class class wbr 4018   Po wpo 4312   Or wor 4313   X. cxp 4642

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-br 4019  df-opab 4080  df-po 4314  df-iso 4315  df-xp 4650

This theorem is referenced by: (None)