Step |
Hyp |
Ref |
Expression |
1 |
|
fourierdlem6.a |
|- ( ph -> A e. RR ) |
2 |
|
fourierdlem6.b |
|- ( ph -> B e. RR ) |
3 |
|
fourierdlem6.altb |
|- ( ph -> A < B ) |
4 |
|
fourierdlem6.t |
|- T = ( B - A ) |
5 |
|
fourierdlem6.5 |
|- ( ph -> X e. RR ) |
6 |
|
fourierdlem6.i |
|- ( ph -> I e. ZZ ) |
7 |
|
fourierdlem6.j |
|- ( ph -> J e. ZZ ) |
8 |
|
fourierdlem6.iltj |
|- ( ph -> I < J ) |
9 |
|
fourierdlem6.iel |
|- ( ph -> ( X + ( I x. T ) ) e. ( A [,] B ) ) |
10 |
|
fourierdlem6.jel |
|- ( ph -> ( X + ( J x. T ) ) e. ( A [,] B ) ) |
11 |
7
|
zred |
|- ( ph -> J e. RR ) |
12 |
6
|
zred |
|- ( ph -> I e. RR ) |
13 |
11 12
|
resubcld |
|- ( ph -> ( J - I ) e. RR ) |
14 |
2 1
|
resubcld |
|- ( ph -> ( B - A ) e. RR ) |
15 |
4 14
|
eqeltrid |
|- ( ph -> T e. RR ) |
16 |
13 15
|
remulcld |
|- ( ph -> ( ( J - I ) x. T ) e. RR ) |
17 |
1 2
|
posdifd |
|- ( ph -> ( A < B <-> 0 < ( B - A ) ) ) |
18 |
3 17
|
mpbid |
|- ( ph -> 0 < ( B - A ) ) |
19 |
18 4
|
breqtrrdi |
|- ( ph -> 0 < T ) |
20 |
15 19
|
elrpd |
|- ( ph -> T e. RR+ ) |
21 |
1 2 10 9
|
iccsuble |
|- ( ph -> ( ( X + ( J x. T ) ) - ( X + ( I x. T ) ) ) <_ ( B - A ) ) |
22 |
11
|
recnd |
|- ( ph -> J e. CC ) |
23 |
12
|
recnd |
|- ( ph -> I e. CC ) |
24 |
15
|
recnd |
|- ( ph -> T e. CC ) |
25 |
22 23 24
|
subdird |
|- ( ph -> ( ( J - I ) x. T ) = ( ( J x. T ) - ( I x. T ) ) ) |
26 |
5
|
recnd |
|- ( ph -> X e. CC ) |
27 |
11 15
|
remulcld |
|- ( ph -> ( J x. T ) e. RR ) |
28 |
27
|
recnd |
|- ( ph -> ( J x. T ) e. CC ) |
29 |
12 15
|
remulcld |
|- ( ph -> ( I x. T ) e. RR ) |
30 |
29
|
recnd |
|- ( ph -> ( I x. T ) e. CC ) |
31 |
26 28 30
|
pnpcand |
|- ( ph -> ( ( X + ( J x. T ) ) - ( X + ( I x. T ) ) ) = ( ( J x. T ) - ( I x. T ) ) ) |
32 |
25 31
|
eqtr4d |
|- ( ph -> ( ( J - I ) x. T ) = ( ( X + ( J x. T ) ) - ( X + ( I x. T ) ) ) ) |
33 |
4
|
a1i |
|- ( ph -> T = ( B - A ) ) |
34 |
21 32 33
|
3brtr4d |
|- ( ph -> ( ( J - I ) x. T ) <_ T ) |
35 |
16 15 20 34
|
lediv1dd |
|- ( ph -> ( ( ( J - I ) x. T ) / T ) <_ ( T / T ) ) |
36 |
13
|
recnd |
|- ( ph -> ( J - I ) e. CC ) |
37 |
19
|
gt0ne0d |
|- ( ph -> T =/= 0 ) |
38 |
36 24 37
|
divcan4d |
|- ( ph -> ( ( ( J - I ) x. T ) / T ) = ( J - I ) ) |
39 |
24 37
|
dividd |
|- ( ph -> ( T / T ) = 1 ) |
40 |
35 38 39
|
3brtr3d |
|- ( ph -> ( J - I ) <_ 1 ) |
41 |
|
1red |
|- ( ph -> 1 e. RR ) |
42 |
11 12 41
|
lesubadd2d |
|- ( ph -> ( ( J - I ) <_ 1 <-> J <_ ( I + 1 ) ) ) |
43 |
40 42
|
mpbid |
|- ( ph -> J <_ ( I + 1 ) ) |
44 |
|
zltp1le |
|- ( ( I e. ZZ /\ J e. ZZ ) -> ( I < J <-> ( I + 1 ) <_ J ) ) |
45 |
6 7 44
|
syl2anc |
|- ( ph -> ( I < J <-> ( I + 1 ) <_ J ) ) |
46 |
8 45
|
mpbid |
|- ( ph -> ( I + 1 ) <_ J ) |
47 |
12 41
|
readdcld |
|- ( ph -> ( I + 1 ) e. RR ) |
48 |
11 47
|
letri3d |
|- ( ph -> ( J = ( I + 1 ) <-> ( J <_ ( I + 1 ) /\ ( I + 1 ) <_ J ) ) ) |
49 |
43 46 48
|
mpbir2and |
|- ( ph -> J = ( I + 1 ) ) |