Theorem infxpdom

Description: Dominance law for multiplication with an infinite cardinal. (Contributed by NM, 26-Mar-2006) (Revised by Mario Carneiro, 29-Apr-2015)

	Ref	Expression
Assertion	infxpdom	|- ( ( A e. dom card /\ _om ~<_ A /\ B ~<_ A ) -> ( A X. B ) ~<_ A )

Proof

Step	Hyp	Ref	Expression
1		xpdom2g	|- ( ( A e. dom card /\ B ~<_ A ) -> ( A X. B ) ~<_ ( A X. A ) )
2	1	3adant2	|- ( ( A e. dom card /\ _om ~<_ A /\ B ~<_ A ) -> ( A X. B ) ~<_ ( A X. A ) )
3		infxpidm2	|- ( ( A e. dom card /\ _om ~<_ A ) -> ( A X. A ) ~~ A )
4	3	3adant3	|- ( ( A e. dom card /\ _om ~<_ A /\ B ~<_ A ) -> ( A X. A ) ~~ A )
5		domentr	|- ( ( ( A X. B ) ~<_ ( A X. A ) /\ ( A X. A ) ~~ A ) -> ( A X. B ) ~<_ A )
6	2 4 5	syl2anc	|- ( ( A e. dom card /\ _om ~<_ A /\ B ~<_ A ) -> ( A X. B ) ~<_ A )