This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

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 ) ) )