Step |
Hyp |
Ref |
Expression |
1 |
|
fsumfldivdiag.1 |
|- ( ph -> A e. RR ) |
2 |
|
simprr |
|- ( ( ph /\ ( n e. ( 1 ... ( |_ ` A ) ) /\ m e. ( 1 ... ( |_ ` ( A / n ) ) ) ) ) -> m e. ( 1 ... ( |_ ` ( A / n ) ) ) ) |
3 |
1
|
adantr |
|- ( ( ph /\ ( n e. ( 1 ... ( |_ ` A ) ) /\ m e. ( 1 ... ( |_ ` ( A / n ) ) ) ) ) -> A e. RR ) |
4 |
|
simprl |
|- ( ( ph /\ ( n e. ( 1 ... ( |_ ` A ) ) /\ m e. ( 1 ... ( |_ ` ( A / n ) ) ) ) ) -> n e. ( 1 ... ( |_ ` A ) ) ) |
5 |
|
fznnfl |
|- ( A e. RR -> ( n e. ( 1 ... ( |_ ` A ) ) <-> ( n e. NN /\ n <_ A ) ) ) |
6 |
3 5
|
syl |
|- ( ( ph /\ ( n e. ( 1 ... ( |_ ` A ) ) /\ m e. ( 1 ... ( |_ ` ( A / n ) ) ) ) ) -> ( n e. ( 1 ... ( |_ ` A ) ) <-> ( n e. NN /\ n <_ A ) ) ) |
7 |
4 6
|
mpbid |
|- ( ( ph /\ ( n e. ( 1 ... ( |_ ` A ) ) /\ m e. ( 1 ... ( |_ ` ( A / n ) ) ) ) ) -> ( n e. NN /\ n <_ A ) ) |
8 |
7
|
simpld |
|- ( ( ph /\ ( n e. ( 1 ... ( |_ ` A ) ) /\ m e. ( 1 ... ( |_ ` ( A / n ) ) ) ) ) -> n e. NN ) |
9 |
3 8
|
nndivred |
|- ( ( ph /\ ( n e. ( 1 ... ( |_ ` A ) ) /\ m e. ( 1 ... ( |_ ` ( A / n ) ) ) ) ) -> ( A / n ) e. RR ) |
10 |
|
fznnfl |
|- ( ( A / n ) e. RR -> ( m e. ( 1 ... ( |_ ` ( A / n ) ) ) <-> ( m e. NN /\ m <_ ( A / n ) ) ) ) |
11 |
9 10
|
syl |
|- ( ( ph /\ ( n e. ( 1 ... ( |_ ` A ) ) /\ m e. ( 1 ... ( |_ ` ( A / n ) ) ) ) ) -> ( m e. ( 1 ... ( |_ ` ( A / n ) ) ) <-> ( m e. NN /\ m <_ ( A / n ) ) ) ) |
12 |
2 11
|
mpbid |
|- ( ( ph /\ ( n e. ( 1 ... ( |_ ` A ) ) /\ m e. ( 1 ... ( |_ ` ( A / n ) ) ) ) ) -> ( m e. NN /\ m <_ ( A / n ) ) ) |
13 |
12
|
simpld |
|- ( ( ph /\ ( n e. ( 1 ... ( |_ ` A ) ) /\ m e. ( 1 ... ( |_ ` ( A / n ) ) ) ) ) -> m e. NN ) |
14 |
13
|
nnred |
|- ( ( ph /\ ( n e. ( 1 ... ( |_ ` A ) ) /\ m e. ( 1 ... ( |_ ` ( A / n ) ) ) ) ) -> m e. RR ) |
15 |
12
|
simprd |
|- ( ( ph /\ ( n e. ( 1 ... ( |_ ` A ) ) /\ m e. ( 1 ... ( |_ ` ( A / n ) ) ) ) ) -> m <_ ( A / n ) ) |
16 |
3
|
recnd |
|- ( ( ph /\ ( n e. ( 1 ... ( |_ ` A ) ) /\ m e. ( 1 ... ( |_ ` ( A / n ) ) ) ) ) -> A e. CC ) |
17 |
16
|
mulid2d |
|- ( ( ph /\ ( n e. ( 1 ... ( |_ ` A ) ) /\ m e. ( 1 ... ( |_ ` ( A / n ) ) ) ) ) -> ( 1 x. A ) = A ) |
18 |
8
|
nnge1d |
|- ( ( ph /\ ( n e. ( 1 ... ( |_ ` A ) ) /\ m e. ( 1 ... ( |_ ` ( A / n ) ) ) ) ) -> 1 <_ n ) |
19 |
|
1red |
|- ( ( ph /\ ( n e. ( 1 ... ( |_ ` A ) ) /\ m e. ( 1 ... ( |_ ` ( A / n ) ) ) ) ) -> 1 e. RR ) |
20 |
8
|
nnred |
|- ( ( ph /\ ( n e. ( 1 ... ( |_ ` A ) ) /\ m e. ( 1 ... ( |_ ` ( A / n ) ) ) ) ) -> n e. RR ) |
21 |
|
0red |
|- ( ( ph /\ ( n e. ( 1 ... ( |_ ` A ) ) /\ m e. ( 1 ... ( |_ ` ( A / n ) ) ) ) ) -> 0 e. RR ) |
22 |
8 13
|
nnmulcld |
|- ( ( ph /\ ( n e. ( 1 ... ( |_ ` A ) ) /\ m e. ( 1 ... ( |_ ` ( A / n ) ) ) ) ) -> ( n x. m ) e. NN ) |
23 |
22
|
nnred |
|- ( ( ph /\ ( n e. ( 1 ... ( |_ ` A ) ) /\ m e. ( 1 ... ( |_ ` ( A / n ) ) ) ) ) -> ( n x. m ) e. RR ) |
24 |
22
|
nngt0d |
|- ( ( ph /\ ( n e. ( 1 ... ( |_ ` A ) ) /\ m e. ( 1 ... ( |_ ` ( A / n ) ) ) ) ) -> 0 < ( n x. m ) ) |
25 |
8
|
nngt0d |
|- ( ( ph /\ ( n e. ( 1 ... ( |_ ` A ) ) /\ m e. ( 1 ... ( |_ ` ( A / n ) ) ) ) ) -> 0 < n ) |
26 |
|
lemuldiv2 |
|- ( ( m e. RR /\ A e. RR /\ ( n e. RR /\ 0 < n ) ) -> ( ( n x. m ) <_ A <-> m <_ ( A / n ) ) ) |
27 |
14 3 20 25 26
|
syl112anc |
|- ( ( ph /\ ( n e. ( 1 ... ( |_ ` A ) ) /\ m e. ( 1 ... ( |_ ` ( A / n ) ) ) ) ) -> ( ( n x. m ) <_ A <-> m <_ ( A / n ) ) ) |
28 |
15 27
|
mpbird |
|- ( ( ph /\ ( n e. ( 1 ... ( |_ ` A ) ) /\ m e. ( 1 ... ( |_ ` ( A / n ) ) ) ) ) -> ( n x. m ) <_ A ) |
29 |
21 23 3 24 28
|
ltletrd |
|- ( ( ph /\ ( n e. ( 1 ... ( |_ ` A ) ) /\ m e. ( 1 ... ( |_ ` ( A / n ) ) ) ) ) -> 0 < A ) |
30 |
|
lemul1 |
|- ( ( 1 e. RR /\ n e. RR /\ ( A e. RR /\ 0 < A ) ) -> ( 1 <_ n <-> ( 1 x. A ) <_ ( n x. A ) ) ) |
31 |
19 20 3 29 30
|
syl112anc |
|- ( ( ph /\ ( n e. ( 1 ... ( |_ ` A ) ) /\ m e. ( 1 ... ( |_ ` ( A / n ) ) ) ) ) -> ( 1 <_ n <-> ( 1 x. A ) <_ ( n x. A ) ) ) |
32 |
18 31
|
mpbid |
|- ( ( ph /\ ( n e. ( 1 ... ( |_ ` A ) ) /\ m e. ( 1 ... ( |_ ` ( A / n ) ) ) ) ) -> ( 1 x. A ) <_ ( n x. A ) ) |
33 |
17 32
|
eqbrtrrd |
|- ( ( ph /\ ( n e. ( 1 ... ( |_ ` A ) ) /\ m e. ( 1 ... ( |_ ` ( A / n ) ) ) ) ) -> A <_ ( n x. A ) ) |
34 |
|
ledivmul |
|- ( ( A e. RR /\ A e. RR /\ ( n e. RR /\ 0 < n ) ) -> ( ( A / n ) <_ A <-> A <_ ( n x. A ) ) ) |
35 |
3 3 20 25 34
|
syl112anc |
|- ( ( ph /\ ( n e. ( 1 ... ( |_ ` A ) ) /\ m e. ( 1 ... ( |_ ` ( A / n ) ) ) ) ) -> ( ( A / n ) <_ A <-> A <_ ( n x. A ) ) ) |
36 |
33 35
|
mpbird |
|- ( ( ph /\ ( n e. ( 1 ... ( |_ ` A ) ) /\ m e. ( 1 ... ( |_ ` ( A / n ) ) ) ) ) -> ( A / n ) <_ A ) |
37 |
14 9 3 15 36
|
letrd |
|- ( ( ph /\ ( n e. ( 1 ... ( |_ ` A ) ) /\ m e. ( 1 ... ( |_ ` ( A / n ) ) ) ) ) -> m <_ A ) |
38 |
|
fznnfl |
|- ( A e. RR -> ( m e. ( 1 ... ( |_ ` A ) ) <-> ( m e. NN /\ m <_ A ) ) ) |
39 |
3 38
|
syl |
|- ( ( ph /\ ( n e. ( 1 ... ( |_ ` A ) ) /\ m e. ( 1 ... ( |_ ` ( A / n ) ) ) ) ) -> ( m e. ( 1 ... ( |_ ` A ) ) <-> ( m e. NN /\ m <_ A ) ) ) |
40 |
13 37 39
|
mpbir2and |
|- ( ( ph /\ ( n e. ( 1 ... ( |_ ` A ) ) /\ m e. ( 1 ... ( |_ ` ( A / n ) ) ) ) ) -> m e. ( 1 ... ( |_ ` A ) ) ) |
41 |
13
|
nngt0d |
|- ( ( ph /\ ( n e. ( 1 ... ( |_ ` A ) ) /\ m e. ( 1 ... ( |_ ` ( A / n ) ) ) ) ) -> 0 < m ) |
42 |
|
lemuldiv |
|- ( ( n e. RR /\ A e. RR /\ ( m e. RR /\ 0 < m ) ) -> ( ( n x. m ) <_ A <-> n <_ ( A / m ) ) ) |
43 |
20 3 14 41 42
|
syl112anc |
|- ( ( ph /\ ( n e. ( 1 ... ( |_ ` A ) ) /\ m e. ( 1 ... ( |_ ` ( A / n ) ) ) ) ) -> ( ( n x. m ) <_ A <-> n <_ ( A / m ) ) ) |
44 |
28 43
|
mpbid |
|- ( ( ph /\ ( n e. ( 1 ... ( |_ ` A ) ) /\ m e. ( 1 ... ( |_ ` ( A / n ) ) ) ) ) -> n <_ ( A / m ) ) |
45 |
3 13
|
nndivred |
|- ( ( ph /\ ( n e. ( 1 ... ( |_ ` A ) ) /\ m e. ( 1 ... ( |_ ` ( A / n ) ) ) ) ) -> ( A / m ) e. RR ) |
46 |
|
fznnfl |
|- ( ( A / m ) e. RR -> ( n e. ( 1 ... ( |_ ` ( A / m ) ) ) <-> ( n e. NN /\ n <_ ( A / m ) ) ) ) |
47 |
45 46
|
syl |
|- ( ( ph /\ ( n e. ( 1 ... ( |_ ` A ) ) /\ m e. ( 1 ... ( |_ ` ( A / n ) ) ) ) ) -> ( n e. ( 1 ... ( |_ ` ( A / m ) ) ) <-> ( n e. NN /\ n <_ ( A / m ) ) ) ) |
48 |
8 44 47
|
mpbir2and |
|- ( ( ph /\ ( n e. ( 1 ... ( |_ ` A ) ) /\ m e. ( 1 ... ( |_ ` ( A / n ) ) ) ) ) -> n e. ( 1 ... ( |_ ` ( A / m ) ) ) ) |
49 |
40 48
|
jca |
|- ( ( ph /\ ( n e. ( 1 ... ( |_ ` A ) ) /\ m e. ( 1 ... ( |_ ` ( A / n ) ) ) ) ) -> ( m e. ( 1 ... ( |_ ` A ) ) /\ n e. ( 1 ... ( |_ ` ( A / m ) ) ) ) ) |
50 |
49
|
ex |
|- ( ph -> ( ( n e. ( 1 ... ( |_ ` A ) ) /\ m e. ( 1 ... ( |_ ` ( A / n ) ) ) ) -> ( m e. ( 1 ... ( |_ ` A ) ) /\ n e. ( 1 ... ( |_ ` ( A / m ) ) ) ) ) ) |