Step |
Hyp |
Ref |
Expression |
1 |
|
ltp1 |
|- ( A e. RR -> A < ( A + 1 ) ) |
2 |
|
recn |
|- ( A e. RR -> A e. CC ) |
3 |
|
ax-1cn |
|- 1 e. CC |
4 |
|
addcom |
|- ( ( A e. CC /\ 1 e. CC ) -> ( A + 1 ) = ( 1 + A ) ) |
5 |
2 3 4
|
sylancl |
|- ( A e. RR -> ( A + 1 ) = ( 1 + A ) ) |
6 |
1 5
|
breqtrd |
|- ( A e. RR -> A < ( 1 + A ) ) |
7 |
6
|
adantr |
|- ( ( A e. RR /\ 0 <_ A ) -> A < ( 1 + A ) ) |
8 |
2
|
adantr |
|- ( ( A e. RR /\ 0 <_ A ) -> A e. CC ) |
9 |
|
1re |
|- 1 e. RR |
10 |
|
readdcl |
|- ( ( 1 e. RR /\ A e. RR ) -> ( 1 + A ) e. RR ) |
11 |
9 10
|
mpan |
|- ( A e. RR -> ( 1 + A ) e. RR ) |
12 |
11
|
adantr |
|- ( ( A e. RR /\ 0 <_ A ) -> ( 1 + A ) e. RR ) |
13 |
12
|
recnd |
|- ( ( A e. RR /\ 0 <_ A ) -> ( 1 + A ) e. CC ) |
14 |
|
0lt1 |
|- 0 < 1 |
15 |
|
addgtge0 |
|- ( ( ( 1 e. RR /\ A e. RR ) /\ ( 0 < 1 /\ 0 <_ A ) ) -> 0 < ( 1 + A ) ) |
16 |
14 15
|
mpanr1 |
|- ( ( ( 1 e. RR /\ A e. RR ) /\ 0 <_ A ) -> 0 < ( 1 + A ) ) |
17 |
9 16
|
mpanl1 |
|- ( ( A e. RR /\ 0 <_ A ) -> 0 < ( 1 + A ) ) |
18 |
17
|
gt0ne0d |
|- ( ( A e. RR /\ 0 <_ A ) -> ( 1 + A ) =/= 0 ) |
19 |
8 13 18
|
divcan1d |
|- ( ( A e. RR /\ 0 <_ A ) -> ( ( A / ( 1 + A ) ) x. ( 1 + A ) ) = A ) |
20 |
11
|
recnd |
|- ( A e. RR -> ( 1 + A ) e. CC ) |
21 |
20
|
mulid2d |
|- ( A e. RR -> ( 1 x. ( 1 + A ) ) = ( 1 + A ) ) |
22 |
21
|
adantr |
|- ( ( A e. RR /\ 0 <_ A ) -> ( 1 x. ( 1 + A ) ) = ( 1 + A ) ) |
23 |
7 19 22
|
3brtr4d |
|- ( ( A e. RR /\ 0 <_ A ) -> ( ( A / ( 1 + A ) ) x. ( 1 + A ) ) < ( 1 x. ( 1 + A ) ) ) |
24 |
|
simpl |
|- ( ( A e. RR /\ 0 <_ A ) -> A e. RR ) |
25 |
24 12 18
|
redivcld |
|- ( ( A e. RR /\ 0 <_ A ) -> ( A / ( 1 + A ) ) e. RR ) |
26 |
|
ltmul1 |
|- ( ( ( A / ( 1 + A ) ) e. RR /\ 1 e. RR /\ ( ( 1 + A ) e. RR /\ 0 < ( 1 + A ) ) ) -> ( ( A / ( 1 + A ) ) < 1 <-> ( ( A / ( 1 + A ) ) x. ( 1 + A ) ) < ( 1 x. ( 1 + A ) ) ) ) |
27 |
9 26
|
mp3an2 |
|- ( ( ( A / ( 1 + A ) ) e. RR /\ ( ( 1 + A ) e. RR /\ 0 < ( 1 + A ) ) ) -> ( ( A / ( 1 + A ) ) < 1 <-> ( ( A / ( 1 + A ) ) x. ( 1 + A ) ) < ( 1 x. ( 1 + A ) ) ) ) |
28 |
25 12 17 27
|
syl12anc |
|- ( ( A e. RR /\ 0 <_ A ) -> ( ( A / ( 1 + A ) ) < 1 <-> ( ( A / ( 1 + A ) ) x. ( 1 + A ) ) < ( 1 x. ( 1 + A ) ) ) ) |
29 |
23 28
|
mpbird |
|- ( ( A e. RR /\ 0 <_ A ) -> ( A / ( 1 + A ) ) < 1 ) |