Metamath Proof Explorer


Theorem divge1b

Description: The ratio of a real number to a positive real number is greater than or equal to 1 iff the divisor (the positive real number) is less than or equal to the dividend (the real number). (Contributed by AV, 26-May-2020)

Ref Expression
Assertion divge1b
|- ( ( A e. RR+ /\ B e. RR ) -> ( A <_ B <-> 1 <_ ( B / A ) ) )

Proof

Step Hyp Ref Expression
1 rpcn
 |-  ( A e. RR+ -> A e. CC )
2 1 mulid2d
 |-  ( A e. RR+ -> ( 1 x. A ) = A )
3 2 eqcomd
 |-  ( A e. RR+ -> A = ( 1 x. A ) )
4 3 adantr
 |-  ( ( A e. RR+ /\ B e. RR ) -> A = ( 1 x. A ) )
5 4 breq1d
 |-  ( ( A e. RR+ /\ B e. RR ) -> ( A <_ B <-> ( 1 x. A ) <_ B ) )
6 1red
 |-  ( ( A e. RR+ /\ B e. RR ) -> 1 e. RR )
7 simpr
 |-  ( ( A e. RR+ /\ B e. RR ) -> B e. RR )
8 rpregt0
 |-  ( A e. RR+ -> ( A e. RR /\ 0 < A ) )
9 8 adantr
 |-  ( ( A e. RR+ /\ B e. RR ) -> ( A e. RR /\ 0 < A ) )
10 lemuldiv
 |-  ( ( 1 e. RR /\ B e. RR /\ ( A e. RR /\ 0 < A ) ) -> ( ( 1 x. A ) <_ B <-> 1 <_ ( B / A ) ) )
11 6 7 9 10 syl3anc
 |-  ( ( A e. RR+ /\ B e. RR ) -> ( ( 1 x. A ) <_ B <-> 1 <_ ( B / A ) ) )
12 5 11 bitrd
 |-  ( ( A e. RR+ /\ B e. RR ) -> ( A <_ B <-> 1 <_ ( B / A ) ) )