Step |
Hyp |
Ref |
Expression |
1 |
|
recgt0 |
|- ( ( A e. RR /\ 0 < A ) -> 0 < ( 1 / A ) ) |
2 |
|
gt0ne0 |
|- ( ( A e. RR /\ 0 < A ) -> A =/= 0 ) |
3 |
|
rereccl |
|- ( ( A e. RR /\ A =/= 0 ) -> ( 1 / A ) e. RR ) |
4 |
2 3
|
syldan |
|- ( ( A e. RR /\ 0 < A ) -> ( 1 / A ) e. RR ) |
5 |
|
1re |
|- 1 e. RR |
6 |
|
ltaddpos |
|- ( ( ( 1 / A ) e. RR /\ 1 e. RR ) -> ( 0 < ( 1 / A ) <-> 1 < ( 1 + ( 1 / A ) ) ) ) |
7 |
4 5 6
|
sylancl |
|- ( ( A e. RR /\ 0 < A ) -> ( 0 < ( 1 / A ) <-> 1 < ( 1 + ( 1 / A ) ) ) ) |
8 |
1 7
|
mpbid |
|- ( ( A e. RR /\ 0 < A ) -> 1 < ( 1 + ( 1 / A ) ) ) |
9 |
|
readdcl |
|- ( ( 1 e. RR /\ ( 1 / A ) e. RR ) -> ( 1 + ( 1 / A ) ) e. RR ) |
10 |
5 4 9
|
sylancr |
|- ( ( A e. RR /\ 0 < A ) -> ( 1 + ( 1 / A ) ) e. RR ) |
11 |
|
0lt1 |
|- 0 < 1 |
12 |
|
0re |
|- 0 e. RR |
13 |
|
lttr |
|- ( ( 0 e. RR /\ 1 e. RR /\ ( 1 + ( 1 / A ) ) e. RR ) -> ( ( 0 < 1 /\ 1 < ( 1 + ( 1 / A ) ) ) -> 0 < ( 1 + ( 1 / A ) ) ) ) |
14 |
12 5 10 13
|
mp3an12i |
|- ( ( A e. RR /\ 0 < A ) -> ( ( 0 < 1 /\ 1 < ( 1 + ( 1 / A ) ) ) -> 0 < ( 1 + ( 1 / A ) ) ) ) |
15 |
11 14
|
mpani |
|- ( ( A e. RR /\ 0 < A ) -> ( 1 < ( 1 + ( 1 / A ) ) -> 0 < ( 1 + ( 1 / A ) ) ) ) |
16 |
8 15
|
mpd |
|- ( ( A e. RR /\ 0 < A ) -> 0 < ( 1 + ( 1 / A ) ) ) |
17 |
|
recgt1 |
|- ( ( ( 1 + ( 1 / A ) ) e. RR /\ 0 < ( 1 + ( 1 / A ) ) ) -> ( 1 < ( 1 + ( 1 / A ) ) <-> ( 1 / ( 1 + ( 1 / A ) ) ) < 1 ) ) |
18 |
10 16 17
|
syl2anc |
|- ( ( A e. RR /\ 0 < A ) -> ( 1 < ( 1 + ( 1 / A ) ) <-> ( 1 / ( 1 + ( 1 / A ) ) ) < 1 ) ) |
19 |
8 18
|
mpbid |
|- ( ( A e. RR /\ 0 < A ) -> ( 1 / ( 1 + ( 1 / A ) ) ) < 1 ) |
20 |
|
ltaddpos |
|- ( ( 1 e. RR /\ ( 1 / A ) e. RR ) -> ( 0 < 1 <-> ( 1 / A ) < ( ( 1 / A ) + 1 ) ) ) |
21 |
5 4 20
|
sylancr |
|- ( ( A e. RR /\ 0 < A ) -> ( 0 < 1 <-> ( 1 / A ) < ( ( 1 / A ) + 1 ) ) ) |
22 |
11 21
|
mpbii |
|- ( ( A e. RR /\ 0 < A ) -> ( 1 / A ) < ( ( 1 / A ) + 1 ) ) |
23 |
4
|
recnd |
|- ( ( A e. RR /\ 0 < A ) -> ( 1 / A ) e. CC ) |
24 |
|
ax-1cn |
|- 1 e. CC |
25 |
|
addcom |
|- ( ( ( 1 / A ) e. CC /\ 1 e. CC ) -> ( ( 1 / A ) + 1 ) = ( 1 + ( 1 / A ) ) ) |
26 |
23 24 25
|
sylancl |
|- ( ( A e. RR /\ 0 < A ) -> ( ( 1 / A ) + 1 ) = ( 1 + ( 1 / A ) ) ) |
27 |
22 26
|
breqtrd |
|- ( ( A e. RR /\ 0 < A ) -> ( 1 / A ) < ( 1 + ( 1 / A ) ) ) |
28 |
|
simpl |
|- ( ( A e. RR /\ 0 < A ) -> A e. RR ) |
29 |
|
simpr |
|- ( ( A e. RR /\ 0 < A ) -> 0 < A ) |
30 |
|
ltrec1 |
|- ( ( ( A e. RR /\ 0 < A ) /\ ( ( 1 + ( 1 / A ) ) e. RR /\ 0 < ( 1 + ( 1 / A ) ) ) ) -> ( ( 1 / A ) < ( 1 + ( 1 / A ) ) <-> ( 1 / ( 1 + ( 1 / A ) ) ) < A ) ) |
31 |
28 29 10 16 30
|
syl22anc |
|- ( ( A e. RR /\ 0 < A ) -> ( ( 1 / A ) < ( 1 + ( 1 / A ) ) <-> ( 1 / ( 1 + ( 1 / A ) ) ) < A ) ) |
32 |
27 31
|
mpbid |
|- ( ( A e. RR /\ 0 < A ) -> ( 1 / ( 1 + ( 1 / A ) ) ) < A ) |
33 |
19 32
|
jca |
|- ( ( A e. RR /\ 0 < A ) -> ( ( 1 / ( 1 + ( 1 / A ) ) ) < 1 /\ ( 1 / ( 1 + ( 1 / A ) ) ) < A ) ) |