Metamath Proof Explorer


Theorem xpdom1g

Description: Dominance law for Cartesian product. Theorem 6L(c) of Enderton p. 149. (Contributed by NM, 25-Mar-2006) (Revised by Mario Carneiro, 26-Apr-2015)

Ref Expression
Assertion xpdom1g
|- ( ( C e. V /\ A ~<_ B ) -> ( A X. C ) ~<_ ( B X. C ) )

Proof

Step Hyp Ref Expression
1 reldom
 |-  Rel ~<_
2 1 brrelex1i
 |-  ( A ~<_ B -> A e. _V )
3 xpcomeng
 |-  ( ( A e. _V /\ C e. V ) -> ( A X. C ) ~~ ( C X. A ) )
4 3 ancoms
 |-  ( ( C e. V /\ A e. _V ) -> ( A X. C ) ~~ ( C X. A ) )
5 2 4 sylan2
 |-  ( ( C e. V /\ A ~<_ B ) -> ( A X. C ) ~~ ( C X. A ) )
6 xpdom2g
 |-  ( ( C e. V /\ A ~<_ B ) -> ( C X. A ) ~<_ ( C X. B ) )
7 1 brrelex2i
 |-  ( A ~<_ B -> B e. _V )
8 xpcomeng
 |-  ( ( C e. V /\ B e. _V ) -> ( C X. B ) ~~ ( B X. C ) )
9 7 8 sylan2
 |-  ( ( C e. V /\ A ~<_ B ) -> ( C X. B ) ~~ ( B X. C ) )
10 domentr
 |-  ( ( ( C X. A ) ~<_ ( C X. B ) /\ ( C X. B ) ~~ ( B X. C ) ) -> ( C X. A ) ~<_ ( B X. C ) )
11 6 9 10 syl2anc
 |-  ( ( C e. V /\ A ~<_ B ) -> ( C X. A ) ~<_ ( B X. C ) )
12 endomtr
 |-  ( ( ( A X. C ) ~~ ( C X. A ) /\ ( C X. A ) ~<_ ( B X. C ) ) -> ( A X. C ) ~<_ ( B X. C ) )
13 5 11 12 syl2anc
 |-  ( ( C e. V /\ A ~<_ B ) -> ( A X. C ) ~<_ ( B X. C ) )