Step |
Hyp |
Ref |
Expression |
1 |
|
fourierdlem52.tf |
|- ( ph -> T e. Fin ) |
2 |
|
fourierdlem52.n |
|- N = ( ( # ` T ) - 1 ) |
3 |
|
fourierdlem52.s |
|- S = ( iota f f Isom < , < ( ( 0 ... N ) , T ) ) |
4 |
|
fourierdlem52.a |
|- ( ph -> A e. RR ) |
5 |
|
fourierdlem52.b |
|- ( ph -> B e. RR ) |
6 |
|
fourierdlem52.t |
|- ( ph -> T C_ ( A [,] B ) ) |
7 |
|
fourierdlem52.at |
|- ( ph -> A e. T ) |
8 |
|
fourierdlem52.bt |
|- ( ph -> B e. T ) |
9 |
4 5
|
iccssred |
|- ( ph -> ( A [,] B ) C_ RR ) |
10 |
6 9
|
sstrd |
|- ( ph -> T C_ RR ) |
11 |
1 10 3 2
|
fourierdlem36 |
|- ( ph -> S Isom < , < ( ( 0 ... N ) , T ) ) |
12 |
|
isof1o |
|- ( S Isom < , < ( ( 0 ... N ) , T ) -> S : ( 0 ... N ) -1-1-onto-> T ) |
13 |
|
f1of |
|- ( S : ( 0 ... N ) -1-1-onto-> T -> S : ( 0 ... N ) --> T ) |
14 |
11 12 13
|
3syl |
|- ( ph -> S : ( 0 ... N ) --> T ) |
15 |
14 6
|
fssd |
|- ( ph -> S : ( 0 ... N ) --> ( A [,] B ) ) |
16 |
|
f1ofo |
|- ( S : ( 0 ... N ) -1-1-onto-> T -> S : ( 0 ... N ) -onto-> T ) |
17 |
11 12 16
|
3syl |
|- ( ph -> S : ( 0 ... N ) -onto-> T ) |
18 |
|
foelrn |
|- ( ( S : ( 0 ... N ) -onto-> T /\ A e. T ) -> E. j e. ( 0 ... N ) A = ( S ` j ) ) |
19 |
17 7 18
|
syl2anc |
|- ( ph -> E. j e. ( 0 ... N ) A = ( S ` j ) ) |
20 |
|
elfzle1 |
|- ( j e. ( 0 ... N ) -> 0 <_ j ) |
21 |
20
|
adantl |
|- ( ( ph /\ j e. ( 0 ... N ) ) -> 0 <_ j ) |
22 |
11
|
adantr |
|- ( ( ph /\ j e. ( 0 ... N ) ) -> S Isom < , < ( ( 0 ... N ) , T ) ) |
23 |
|
ressxr |
|- RR C_ RR* |
24 |
10 23
|
sstrdi |
|- ( ph -> T C_ RR* ) |
25 |
24
|
adantr |
|- ( ( ph /\ j e. ( 0 ... N ) ) -> T C_ RR* ) |
26 |
|
fzssz |
|- ( 0 ... N ) C_ ZZ |
27 |
|
zssre |
|- ZZ C_ RR |
28 |
27 23
|
sstri |
|- ZZ C_ RR* |
29 |
26 28
|
sstri |
|- ( 0 ... N ) C_ RR* |
30 |
25 29
|
jctil |
|- ( ( ph /\ j e. ( 0 ... N ) ) -> ( ( 0 ... N ) C_ RR* /\ T C_ RR* ) ) |
31 |
|
hashcl |
|- ( T e. Fin -> ( # ` T ) e. NN0 ) |
32 |
1 31
|
syl |
|- ( ph -> ( # ` T ) e. NN0 ) |
33 |
7
|
ne0d |
|- ( ph -> T =/= (/) ) |
34 |
|
hashge1 |
|- ( ( T e. Fin /\ T =/= (/) ) -> 1 <_ ( # ` T ) ) |
35 |
1 33 34
|
syl2anc |
|- ( ph -> 1 <_ ( # ` T ) ) |
36 |
|
elnnnn0c |
|- ( ( # ` T ) e. NN <-> ( ( # ` T ) e. NN0 /\ 1 <_ ( # ` T ) ) ) |
37 |
32 35 36
|
sylanbrc |
|- ( ph -> ( # ` T ) e. NN ) |
38 |
|
nnm1nn0 |
|- ( ( # ` T ) e. NN -> ( ( # ` T ) - 1 ) e. NN0 ) |
39 |
37 38
|
syl |
|- ( ph -> ( ( # ` T ) - 1 ) e. NN0 ) |
40 |
2 39
|
eqeltrid |
|- ( ph -> N e. NN0 ) |
41 |
|
nn0uz |
|- NN0 = ( ZZ>= ` 0 ) |
42 |
40 41
|
eleqtrdi |
|- ( ph -> N e. ( ZZ>= ` 0 ) ) |
43 |
|
eluzfz1 |
|- ( N e. ( ZZ>= ` 0 ) -> 0 e. ( 0 ... N ) ) |
44 |
42 43
|
syl |
|- ( ph -> 0 e. ( 0 ... N ) ) |
45 |
44
|
anim1i |
|- ( ( ph /\ j e. ( 0 ... N ) ) -> ( 0 e. ( 0 ... N ) /\ j e. ( 0 ... N ) ) ) |
46 |
|
leisorel |
|- ( ( S Isom < , < ( ( 0 ... N ) , T ) /\ ( ( 0 ... N ) C_ RR* /\ T C_ RR* ) /\ ( 0 e. ( 0 ... N ) /\ j e. ( 0 ... N ) ) ) -> ( 0 <_ j <-> ( S ` 0 ) <_ ( S ` j ) ) ) |
47 |
22 30 45 46
|
syl3anc |
|- ( ( ph /\ j e. ( 0 ... N ) ) -> ( 0 <_ j <-> ( S ` 0 ) <_ ( S ` j ) ) ) |
48 |
21 47
|
mpbid |
|- ( ( ph /\ j e. ( 0 ... N ) ) -> ( S ` 0 ) <_ ( S ` j ) ) |
49 |
48
|
3adant3 |
|- ( ( ph /\ j e. ( 0 ... N ) /\ A = ( S ` j ) ) -> ( S ` 0 ) <_ ( S ` j ) ) |
50 |
|
eqcom |
|- ( A = ( S ` j ) <-> ( S ` j ) = A ) |
51 |
50
|
biimpi |
|- ( A = ( S ` j ) -> ( S ` j ) = A ) |
52 |
51
|
3ad2ant3 |
|- ( ( ph /\ j e. ( 0 ... N ) /\ A = ( S ` j ) ) -> ( S ` j ) = A ) |
53 |
49 52
|
breqtrd |
|- ( ( ph /\ j e. ( 0 ... N ) /\ A = ( S ` j ) ) -> ( S ` 0 ) <_ A ) |
54 |
53
|
rexlimdv3a |
|- ( ph -> ( E. j e. ( 0 ... N ) A = ( S ` j ) -> ( S ` 0 ) <_ A ) ) |
55 |
19 54
|
mpd |
|- ( ph -> ( S ` 0 ) <_ A ) |
56 |
4
|
rexrd |
|- ( ph -> A e. RR* ) |
57 |
5
|
rexrd |
|- ( ph -> B e. RR* ) |
58 |
15 44
|
ffvelrnd |
|- ( ph -> ( S ` 0 ) e. ( A [,] B ) ) |
59 |
|
iccgelb |
|- ( ( A e. RR* /\ B e. RR* /\ ( S ` 0 ) e. ( A [,] B ) ) -> A <_ ( S ` 0 ) ) |
60 |
56 57 58 59
|
syl3anc |
|- ( ph -> A <_ ( S ` 0 ) ) |
61 |
9 58
|
sseldd |
|- ( ph -> ( S ` 0 ) e. RR ) |
62 |
61 4
|
letri3d |
|- ( ph -> ( ( S ` 0 ) = A <-> ( ( S ` 0 ) <_ A /\ A <_ ( S ` 0 ) ) ) ) |
63 |
55 60 62
|
mpbir2and |
|- ( ph -> ( S ` 0 ) = A ) |
64 |
|
eluzfz2 |
|- ( N e. ( ZZ>= ` 0 ) -> N e. ( 0 ... N ) ) |
65 |
42 64
|
syl |
|- ( ph -> N e. ( 0 ... N ) ) |
66 |
15 65
|
ffvelrnd |
|- ( ph -> ( S ` N ) e. ( A [,] B ) ) |
67 |
|
iccleub |
|- ( ( A e. RR* /\ B e. RR* /\ ( S ` N ) e. ( A [,] B ) ) -> ( S ` N ) <_ B ) |
68 |
56 57 66 67
|
syl3anc |
|- ( ph -> ( S ` N ) <_ B ) |
69 |
|
foelrn |
|- ( ( S : ( 0 ... N ) -onto-> T /\ B e. T ) -> E. j e. ( 0 ... N ) B = ( S ` j ) ) |
70 |
17 8 69
|
syl2anc |
|- ( ph -> E. j e. ( 0 ... N ) B = ( S ` j ) ) |
71 |
|
simp3 |
|- ( ( ph /\ j e. ( 0 ... N ) /\ B = ( S ` j ) ) -> B = ( S ` j ) ) |
72 |
|
elfzle2 |
|- ( j e. ( 0 ... N ) -> j <_ N ) |
73 |
72
|
3ad2ant2 |
|- ( ( ph /\ j e. ( 0 ... N ) /\ B = ( S ` j ) ) -> j <_ N ) |
74 |
11
|
3ad2ant1 |
|- ( ( ph /\ j e. ( 0 ... N ) /\ B = ( S ` j ) ) -> S Isom < , < ( ( 0 ... N ) , T ) ) |
75 |
30
|
3adant3 |
|- ( ( ph /\ j e. ( 0 ... N ) /\ B = ( S ` j ) ) -> ( ( 0 ... N ) C_ RR* /\ T C_ RR* ) ) |
76 |
|
simp2 |
|- ( ( ph /\ j e. ( 0 ... N ) /\ B = ( S ` j ) ) -> j e. ( 0 ... N ) ) |
77 |
65
|
3ad2ant1 |
|- ( ( ph /\ j e. ( 0 ... N ) /\ B = ( S ` j ) ) -> N e. ( 0 ... N ) ) |
78 |
|
leisorel |
|- ( ( S Isom < , < ( ( 0 ... N ) , T ) /\ ( ( 0 ... N ) C_ RR* /\ T C_ RR* ) /\ ( j e. ( 0 ... N ) /\ N e. ( 0 ... N ) ) ) -> ( j <_ N <-> ( S ` j ) <_ ( S ` N ) ) ) |
79 |
74 75 76 77 78
|
syl112anc |
|- ( ( ph /\ j e. ( 0 ... N ) /\ B = ( S ` j ) ) -> ( j <_ N <-> ( S ` j ) <_ ( S ` N ) ) ) |
80 |
73 79
|
mpbid |
|- ( ( ph /\ j e. ( 0 ... N ) /\ B = ( S ` j ) ) -> ( S ` j ) <_ ( S ` N ) ) |
81 |
71 80
|
eqbrtrd |
|- ( ( ph /\ j e. ( 0 ... N ) /\ B = ( S ` j ) ) -> B <_ ( S ` N ) ) |
82 |
81
|
rexlimdv3a |
|- ( ph -> ( E. j e. ( 0 ... N ) B = ( S ` j ) -> B <_ ( S ` N ) ) ) |
83 |
70 82
|
mpd |
|- ( ph -> B <_ ( S ` N ) ) |
84 |
9 66
|
sseldd |
|- ( ph -> ( S ` N ) e. RR ) |
85 |
84 5
|
letri3d |
|- ( ph -> ( ( S ` N ) = B <-> ( ( S ` N ) <_ B /\ B <_ ( S ` N ) ) ) ) |
86 |
68 83 85
|
mpbir2and |
|- ( ph -> ( S ` N ) = B ) |
87 |
15 63 86
|
jca31 |
|- ( ph -> ( ( S : ( 0 ... N ) --> ( A [,] B ) /\ ( S ` 0 ) = A ) /\ ( S ` N ) = B ) ) |