Step |
Hyp |
Ref |
Expression |
1 |
|
sqrlearg.1 |
|- ( ph -> A e. RR ) |
2 |
|
0re |
|- 0 e. RR |
3 |
2
|
a1i |
|- ( ( ph /\ ( A ^ 2 ) <_ A ) -> 0 e. RR ) |
4 |
|
simpr |
|- ( ( ph /\ -. A <_ 1 ) -> -. A <_ 1 ) |
5 |
|
1red |
|- ( ( ph /\ -. A <_ 1 ) -> 1 e. RR ) |
6 |
1
|
adantr |
|- ( ( ph /\ -. A <_ 1 ) -> A e. RR ) |
7 |
5 6
|
ltnled |
|- ( ( ph /\ -. A <_ 1 ) -> ( 1 < A <-> -. A <_ 1 ) ) |
8 |
4 7
|
mpbird |
|- ( ( ph /\ -. A <_ 1 ) -> 1 < A ) |
9 |
|
1red |
|- ( ( ph /\ 1 < A ) -> 1 e. RR ) |
10 |
1
|
adantr |
|- ( ( ph /\ 1 < A ) -> A e. RR ) |
11 |
2
|
a1i |
|- ( ( ph /\ 1 < A ) -> 0 e. RR ) |
12 |
|
0lt1 |
|- 0 < 1 |
13 |
12
|
a1i |
|- ( ( ph /\ 1 < A ) -> 0 < 1 ) |
14 |
|
simpr |
|- ( ( ph /\ 1 < A ) -> 1 < A ) |
15 |
11 9 10 13 14
|
lttrd |
|- ( ( ph /\ 1 < A ) -> 0 < A ) |
16 |
10 15
|
elrpd |
|- ( ( ph /\ 1 < A ) -> A e. RR+ ) |
17 |
9 10 16 14
|
ltmul2dd |
|- ( ( ph /\ 1 < A ) -> ( A x. 1 ) < ( A x. A ) ) |
18 |
1
|
recnd |
|- ( ph -> A e. CC ) |
19 |
18
|
mulid1d |
|- ( ph -> ( A x. 1 ) = A ) |
20 |
19
|
adantr |
|- ( ( ph /\ 1 < A ) -> ( A x. 1 ) = A ) |
21 |
18
|
sqvald |
|- ( ph -> ( A ^ 2 ) = ( A x. A ) ) |
22 |
21
|
eqcomd |
|- ( ph -> ( A x. A ) = ( A ^ 2 ) ) |
23 |
22
|
adantr |
|- ( ( ph /\ 1 < A ) -> ( A x. A ) = ( A ^ 2 ) ) |
24 |
20 23
|
breq12d |
|- ( ( ph /\ 1 < A ) -> ( ( A x. 1 ) < ( A x. A ) <-> A < ( A ^ 2 ) ) ) |
25 |
17 24
|
mpbid |
|- ( ( ph /\ 1 < A ) -> A < ( A ^ 2 ) ) |
26 |
8 25
|
syldan |
|- ( ( ph /\ -. A <_ 1 ) -> A < ( A ^ 2 ) ) |
27 |
26
|
adantlr |
|- ( ( ( ph /\ ( A ^ 2 ) <_ A ) /\ -. A <_ 1 ) -> A < ( A ^ 2 ) ) |
28 |
|
simpr |
|- ( ( ph /\ ( A ^ 2 ) <_ A ) -> ( A ^ 2 ) <_ A ) |
29 |
1
|
resqcld |
|- ( ph -> ( A ^ 2 ) e. RR ) |
30 |
29
|
adantr |
|- ( ( ph /\ ( A ^ 2 ) <_ A ) -> ( A ^ 2 ) e. RR ) |
31 |
1
|
adantr |
|- ( ( ph /\ ( A ^ 2 ) <_ A ) -> A e. RR ) |
32 |
30 31
|
lenltd |
|- ( ( ph /\ ( A ^ 2 ) <_ A ) -> ( ( A ^ 2 ) <_ A <-> -. A < ( A ^ 2 ) ) ) |
33 |
28 32
|
mpbid |
|- ( ( ph /\ ( A ^ 2 ) <_ A ) -> -. A < ( A ^ 2 ) ) |
34 |
33
|
adantr |
|- ( ( ( ph /\ ( A ^ 2 ) <_ A ) /\ -. A <_ 1 ) -> -. A < ( A ^ 2 ) ) |
35 |
27 34
|
condan |
|- ( ( ph /\ ( A ^ 2 ) <_ A ) -> A <_ 1 ) |
36 |
|
1red |
|- ( ( ph /\ A <_ 1 ) -> 1 e. RR ) |
37 |
35 36
|
syldan |
|- ( ( ph /\ ( A ^ 2 ) <_ A ) -> 1 e. RR ) |
38 |
31
|
sqge0d |
|- ( ( ph /\ ( A ^ 2 ) <_ A ) -> 0 <_ ( A ^ 2 ) ) |
39 |
3 30 31 38 28
|
letrd |
|- ( ( ph /\ ( A ^ 2 ) <_ A ) -> 0 <_ A ) |
40 |
3 37 31 39 35
|
eliccd |
|- ( ( ph /\ ( A ^ 2 ) <_ A ) -> A e. ( 0 [,] 1 ) ) |
41 |
40
|
ex |
|- ( ph -> ( ( A ^ 2 ) <_ A -> A e. ( 0 [,] 1 ) ) ) |
42 |
|
unitssre |
|- ( 0 [,] 1 ) C_ RR |
43 |
42
|
sseli |
|- ( A e. ( 0 [,] 1 ) -> A e. RR ) |
44 |
|
1red |
|- ( A e. ( 0 [,] 1 ) -> 1 e. RR ) |
45 |
|
0xr |
|- 0 e. RR* |
46 |
45
|
a1i |
|- ( A e. ( 0 [,] 1 ) -> 0 e. RR* ) |
47 |
44
|
rexrd |
|- ( A e. ( 0 [,] 1 ) -> 1 e. RR* ) |
48 |
|
id |
|- ( A e. ( 0 [,] 1 ) -> A e. ( 0 [,] 1 ) ) |
49 |
46 47 48
|
iccgelbd |
|- ( A e. ( 0 [,] 1 ) -> 0 <_ A ) |
50 |
46 47 48
|
iccleubd |
|- ( A e. ( 0 [,] 1 ) -> A <_ 1 ) |
51 |
43 44 43 49 50
|
lemul2ad |
|- ( A e. ( 0 [,] 1 ) -> ( A x. A ) <_ ( A x. 1 ) ) |
52 |
51
|
adantl |
|- ( ( ph /\ A e. ( 0 [,] 1 ) ) -> ( A x. A ) <_ ( A x. 1 ) ) |
53 |
22
|
adantr |
|- ( ( ph /\ A e. ( 0 [,] 1 ) ) -> ( A x. A ) = ( A ^ 2 ) ) |
54 |
19
|
adantr |
|- ( ( ph /\ A e. ( 0 [,] 1 ) ) -> ( A x. 1 ) = A ) |
55 |
53 54
|
breq12d |
|- ( ( ph /\ A e. ( 0 [,] 1 ) ) -> ( ( A x. A ) <_ ( A x. 1 ) <-> ( A ^ 2 ) <_ A ) ) |
56 |
52 55
|
mpbid |
|- ( ( ph /\ A e. ( 0 [,] 1 ) ) -> ( A ^ 2 ) <_ A ) |
57 |
56
|
ex |
|- ( ph -> ( A e. ( 0 [,] 1 ) -> ( A ^ 2 ) <_ A ) ) |
58 |
41 57
|
impbid |
|- ( ph -> ( ( A ^ 2 ) <_ A <-> A e. ( 0 [,] 1 ) ) ) |