Metamath Proof Explorer

Theorem mulgt1

Description: The product of two numbers greater than 1 is greater than 1. (Contributed by NM, 13-Feb-2005)

Ref Expression
Assertion mulgt1 
|- ( ( ( A e. RR /\ B e. RR ) /\ ( 1 < A /\ 1 < B ) ) -> 1 < ( A x. B ) )

Proof

Step Hyp Ref Expression
1 simpl 
 |-  ( ( 1 < A /\ 1 < B ) -> 1 < A )
2 1 a1i 
 |-  ( ( A e. RR /\ B e. RR ) -> ( ( 1 < A /\ 1 < B ) -> 1 < A ) )
3 0lt1 
 |-  0 < 1
4 0re 
 |-  0 e. RR
5 1re 
 |-  1 e. RR
6 lttr 
 |-  ( ( 0 e. RR /\ 1 e. RR /\ A e. RR ) -> ( ( 0 < 1 /\ 1 < A ) -> 0 < A ) )
7 4 5 6 mp3an12 
 |-  ( A e. RR -> ( ( 0 < 1 /\ 1 < A ) -> 0 < A ) )
8 3 7 mpani 
 |-  ( A e. RR -> ( 1 < A -> 0 < A ) )
9 8 adantr 
 |-  ( ( A e. RR /\ B e. RR ) -> ( 1 < A -> 0 < A ) )
10 ltmul2 
 |-  ( ( 1 e. RR /\ B e. RR /\ ( A e. RR /\ 0 < A ) ) -> ( 1 < B <-> ( A x. 1 ) < ( A x. B ) ) )
11 10 biimpd 
 |-  ( ( 1 e. RR /\ B e. RR /\ ( A e. RR /\ 0 < A ) ) -> ( 1 < B -> ( A x. 1 ) < ( A x. B ) ) )
12 5 11 mp3an1 
 |-  ( ( B e. RR /\ ( A e. RR /\ 0 < A ) ) -> ( 1 < B -> ( A x. 1 ) < ( A x. B ) ) )
13 12 exp32 
 |-  ( B e. RR -> ( A e. RR -> ( 0 < A -> ( 1 < B -> ( A x. 1 ) < ( A x. B ) ) ) ) )
14 13 impcom 
 |-  ( ( A e. RR /\ B e. RR ) -> ( 0 < A -> ( 1 < B -> ( A x. 1 ) < ( A x. B ) ) ) )
15 9 14 syld 
 |-  ( ( A e. RR /\ B e. RR ) -> ( 1 < A -> ( 1 < B -> ( A x. 1 ) < ( A x. B ) ) ) )
16 15 impd 
 |-  ( ( A e. RR /\ B e. RR ) -> ( ( 1 < A /\ 1 < B ) -> ( A x. 1 ) < ( A x. B ) ) )
17 ax-1rid 
 |-  ( A e. RR -> ( A x. 1 ) = A )
18 17 adantr 
 |-  ( ( A e. RR /\ B e. RR ) -> ( A x. 1 ) = A )
19 18 breq1d 
 |-  ( ( A e. RR /\ B e. RR ) -> ( ( A x. 1 ) < ( A x. B ) <-> A < ( A x. B ) ) )
20 16 19 sylibd 
 |-  ( ( A e. RR /\ B e. RR ) -> ( ( 1 < A /\ 1 < B ) -> A < ( A x. B ) ) )
21 2 20 jcad 
 |-  ( ( A e. RR /\ B e. RR ) -> ( ( 1 < A /\ 1 < B ) -> ( 1 < A /\ A < ( A x. B ) ) ) )
22 remulcl 
 |-  ( ( A e. RR /\ B e. RR ) -> ( A x. B ) e. RR )
23 lttr 
 |-  ( ( 1 e. RR /\ A e. RR /\ ( A x. B ) e. RR ) -> ( ( 1 < A /\ A < ( A x. B ) ) -> 1 < ( A x. B ) ) )
24 5 23 mp3an1 
 |-  ( ( A e. RR /\ ( A x. B ) e. RR ) -> ( ( 1 < A /\ A < ( A x. B ) ) -> 1 < ( A x. B ) ) )
25 22 24 syldan 
 |-  ( ( A e. RR /\ B e. RR ) -> ( ( 1 < A /\ A < ( A x. B ) ) -> 1 < ( A x. B ) ) )
26 21 25 syld 
 |-  ( ( A e. RR /\ B e. RR ) -> ( ( 1 < A /\ 1 < B ) -> 1 < ( A x. B ) ) )
27 26 imp 
 |-  ( ( ( A e. RR /\ B e. RR ) /\ ( 1 < A /\ 1 < B ) ) -> 1 < ( A x. B ) )