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Metamath Proof Explorer

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 )