Theorem xpsless

Description: Closure of the ordering in a binary structure product. (Contributed by Mario Carneiro, 15-Aug-2015)

	Ref	Expression
Hypotheses	xpsle.t	|- T = ( R Xs. S )
	xpsle.x	|- X = ( Base ` R )
	xpsle.y	|- Y = ( Base ` S )
	xpsle.1	|- ( ph -> R e. V )
	xpsle.2	|- ( ph -> S e. W )
	xpsle.p	|- .<_ = ( le ` T )
Assertion	xpsless	|- ( ph -> .<_ C_ ( ( X X. Y ) X. ( X X. Y ) ) )

Proof

Step	Hyp	Ref	Expression
1		xpsle.t	|- T = ( R Xs. S )
2		xpsle.x	|- X = ( Base ` R )
3		xpsle.y	|- Y = ( Base ` S )
4		xpsle.1	|- ( ph -> R e. V )
5		xpsle.2	|- ( ph -> S e. W )
6		xpsle.p	|- .<_ = ( le ` T )
7		eqid	|- ( x e. X , y e. Y |-> { <. (/) , x >. , <. 1o , y >. } ) = ( x e. X , y e. Y |-> { <. (/) , x >. , <. 1o , y >. } )
8		eqid	|- ( Scalar ` R ) = ( Scalar ` R )
9		eqid	|- ( ( Scalar ` R ) Xs_ { <. (/) , R >. , <. 1o , S >. } ) = ( ( Scalar ` R ) Xs_ { <. (/) , R >. , <. 1o , S >. } )
10	1 2 3 4 5 7 8 9	xpsval	|- ( ph -> T = ( `' ( x e. X , y e. Y |-> { <. (/) , x >. , <. 1o , y >. } ) "s ( ( Scalar ` R ) Xs_ { <. (/) , R >. , <. 1o , S >. } ) ) )
11	1 2 3 4 5 7 8 9	xpsrnbas	|- ( ph -> ran ( x e. X , y e. Y |-> { <. (/) , x >. , <. 1o , y >. } ) = ( Base ` ( ( Scalar ` R ) Xs_ { <. (/) , R >. , <. 1o , S >. } ) ) )
12	7	xpsff1o2	|- ( x e. X , y e. Y |-> { <. (/) , x >. , <. 1o , y >. } ) : ( X X. Y ) -1-1-onto-> ran ( x e. X , y e. Y |-> { <. (/) , x >. , <. 1o , y >. } )
13		f1ocnv	|- ( ( x e. X , y e. Y |-> { <. (/) , x >. , <. 1o , y >. } ) : ( X X. Y ) -1-1-onto-> ran ( x e. X , y e. Y |-> { <. (/) , x >. , <. 1o , y >. } ) -> `' ( x e. X , y e. Y |-> { <. (/) , x >. , <. 1o , y >. } ) : ran ( x e. X , y e. Y |-> { <. (/) , x >. , <. 1o , y >. } ) -1-1-onto-> ( X X. Y ) )
14	12 13	mp1i	|- ( ph -> `' ( x e. X , y e. Y |-> { <. (/) , x >. , <. 1o , y >. } ) : ran ( x e. X , y e. Y |-> { <. (/) , x >. , <. 1o , y >. } ) -1-1-onto-> ( X X. Y ) )
15		f1ofo	|- ( `' ( x e. X , y e. Y |-> { <. (/) , x >. , <. 1o , y >. } ) : ran ( x e. X , y e. Y |-> { <. (/) , x >. , <. 1o , y >. } ) -1-1-onto-> ( X X. Y ) -> `' ( x e. X , y e. Y |-> { <. (/) , x >. , <. 1o , y >. } ) : ran ( x e. X , y e. Y |-> { <. (/) , x >. , <. 1o , y >. } ) -onto-> ( X X. Y ) )
16	14 15	syl	|- ( ph -> `' ( x e. X , y e. Y |-> { <. (/) , x >. , <. 1o , y >. } ) : ran ( x e. X , y e. Y |-> { <. (/) , x >. , <. 1o , y >. } ) -onto-> ( X X. Y ) )
17		ovexd	|- ( ph -> ( ( Scalar ` R ) Xs_ { <. (/) , R >. , <. 1o , S >. } ) e. _V )
18	10 11 16 17 6	imasless	|- ( ph -> .<_ C_ ( ( X X. Y ) X. ( X X. Y ) ) )