Metamath Proof Explorer

Theorem divlt1lt

Description: A real number divided by a positive real number is less than 1 iff the real number is less than the positive real number. (Contributed by AV, 25-May-2020)

Ref Expression
Assertion divlt1lt 
|- ( ( A e. RR /\ B e. RR+ ) -> ( ( A / B ) < 1 <-> A < B ) )

Proof

Step Hyp Ref Expression
1 simpl 
 |-  ( ( A e. RR /\ B e. RR+ ) -> A e. RR )
2 rpregt0 
 |-  ( B e. RR+ -> ( B e. RR /\ 0 < B ) )
3 2 adantl 
 |-  ( ( A e. RR /\ B e. RR+ ) -> ( B e. RR /\ 0 < B ) )
4 1re 
 |-  1 e. RR
5 0lt1 
 |-  0 < 1
6 4 5 pm3.2i 
 |-  ( 1 e. RR /\ 0 < 1 )
7 6 a1i 
 |-  ( ( A e. RR /\ B e. RR+ ) -> ( 1 e. RR /\ 0 < 1 ) )
8 ltdiv23 
 |-  ( ( A e. RR /\ ( B e. RR /\ 0 < B ) /\ ( 1 e. RR /\ 0 < 1 ) ) -> ( ( A / B ) < 1 <-> ( A / 1 ) < B ) )
9 1 3 7 8 syl3anc 
 |-  ( ( A e. RR /\ B e. RR+ ) -> ( ( A / B ) < 1 <-> ( A / 1 ) < B ) )
10 recn 
 |-  ( A e. RR -> A e. CC )
11 10 div1d 
 |-  ( A e. RR -> ( A / 1 ) = A )
12 11 adantr 
 |-  ( ( A e. RR /\ B e. RR+ ) -> ( A / 1 ) = A )
13 12 breq1d 
 |-  ( ( A e. RR /\ B e. RR+ ) -> ( ( A / 1 ) < B <-> A < B ) )
14 9 13 bitrd 
 |-  ( ( A e. RR /\ B e. RR+ ) -> ( ( A / B ) < 1 <-> A < B ) )