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Metamath Proof Explorer

Theorem xlemul2a

Description: Extended real version of lemul2a . (Contributed by Mario Carneiro, 8-Sep-2015)

Ref Expression
Assertion xlemul2a 
|- ( ( ( A e. RR* /\ B e. RR* /\ ( C e. RR* /\ 0 <_ C ) ) /\ A <_ B ) -> ( C *e A ) <_ ( C *e B ) )

Proof

Step Hyp Ref Expression
1 xlemul1a 
 |-  ( ( ( A e. RR* /\ B e. RR* /\ ( C e. RR* /\ 0 <_ C ) ) /\ A <_ B ) -> ( A *e C ) <_ ( B *e C ) )
2 simpl1 
 |-  ( ( ( A e. RR* /\ B e. RR* /\ ( C e. RR* /\ 0 <_ C ) ) /\ A <_ B ) -> A e. RR* )
3 simpl3l 
 |-  ( ( ( A e. RR* /\ B e. RR* /\ ( C e. RR* /\ 0 <_ C ) ) /\ A <_ B ) -> C e. RR* )
4 xmulcom 
 |-  ( ( A e. RR* /\ C e. RR* ) -> ( A *e C ) = ( C *e A ) )
5 2 3 4 syl2anc 
 |-  ( ( ( A e. RR* /\ B e. RR* /\ ( C e. RR* /\ 0 <_ C ) ) /\ A <_ B ) -> ( A *e C ) = ( C *e A ) )
6 simpl2 
 |-  ( ( ( A e. RR* /\ B e. RR* /\ ( C e. RR* /\ 0 <_ C ) ) /\ A <_ B ) -> B e. RR* )
7 xmulcom 
 |-  ( ( B e. RR* /\ C e. RR* ) -> ( B *e C ) = ( C *e B ) )
8 6 3 7 syl2anc 
 |-  ( ( ( A e. RR* /\ B e. RR* /\ ( C e. RR* /\ 0 <_ C ) ) /\ A <_ B ) -> ( B *e C ) = ( C *e B ) )
9 1 5 8 3brtr3d 
 |-  ( ( ( A e. RR* /\ B e. RR* /\ ( C e. RR* /\ 0 <_ C ) ) /\ A <_ B ) -> ( C *e A ) <_ ( C *e B ) )