Theorem soinxp 4840

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 4839	. . 3 |- ( R Po A <-> ( R i^i ( A X. A ) ) Po A )
2		brinxp 4838	. . . . . . . 8 |- ( ( x e. A /\ y e. A ) -> ( x R y <-> x ( R i^i ( A X. A ) ) y ) )
3	2	3adant3 1048	. . . . . . 7 |- ( ( x e. A /\ y e. A /\ z e. A ) -> ( x R y <-> x ( R i^i ( A X. A ) ) y ) )
4		brinxp 4838	. . . . . . . . 9 |- ( ( x e. A /\ z e. A ) -> ( x R z <-> x ( R i^i ( A X. A ) ) z ) )
5	4	3adant2 1047	. . . . . . . 8 |- ( ( x e. A /\ y e. A /\ z e. A ) -> ( x R z <-> x ( R i^i ( A X. A ) ) z ) )
6		brinxp 4838	. . . . . . . . . 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 1046	. . . . . . . 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 805	. . . . . . 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 1235	. . . . 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 2572	. . . 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 2564	. . 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 464	. 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 4437	. 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 4437	. 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 720   /\ w3a 1009   e. wcel 2209   A.wral 2528   i^i cin 3219   class class class wbr 4125   Po wpo 4434   Or wor 4435   X. cxp 4767

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-br 4126  df-opab 4188  df-po 4436  df-iso 4437  df-xp 4775

This theorem is referenced by: (None)