Step |
Hyp |
Ref |
Expression |
1 |
|
fourierdlem14.1 |
|- ( ph -> A e. RR ) |
2 |
|
fourierdlem14.2 |
|- ( ph -> B e. RR ) |
3 |
|
fourierdlem14.x |
|- ( ph -> X e. RR ) |
4 |
|
fourierdlem14.p |
|- P = ( m e. NN |-> { p e. ( RR ^m ( 0 ... m ) ) | ( ( ( p ` 0 ) = ( A + X ) /\ ( p ` m ) = ( B + X ) ) /\ A. i e. ( 0 ..^ m ) ( p ` i ) < ( p ` ( i + 1 ) ) ) } ) |
5 |
|
fourierdlem14.o |
|- O = ( m e. NN |-> { p e. ( RR ^m ( 0 ... m ) ) | ( ( ( p ` 0 ) = A /\ ( p ` m ) = B ) /\ A. i e. ( 0 ..^ m ) ( p ` i ) < ( p ` ( i + 1 ) ) ) } ) |
6 |
|
fourierdlem14.m |
|- ( ph -> M e. NN ) |
7 |
|
fourierdlem14.v |
|- ( ph -> V e. ( P ` M ) ) |
8 |
|
fourierdlem14.q |
|- Q = ( i e. ( 0 ... M ) |-> ( ( V ` i ) - X ) ) |
9 |
4
|
fourierdlem2 |
|- ( M e. NN -> ( V e. ( P ` M ) <-> ( V e. ( RR ^m ( 0 ... M ) ) /\ ( ( ( V ` 0 ) = ( A + X ) /\ ( V ` M ) = ( B + X ) ) /\ A. i e. ( 0 ..^ M ) ( V ` i ) < ( V ` ( i + 1 ) ) ) ) ) ) |
10 |
6 9
|
syl |
|- ( ph -> ( V e. ( P ` M ) <-> ( V e. ( RR ^m ( 0 ... M ) ) /\ ( ( ( V ` 0 ) = ( A + X ) /\ ( V ` M ) = ( B + X ) ) /\ A. i e. ( 0 ..^ M ) ( V ` i ) < ( V ` ( i + 1 ) ) ) ) ) ) |
11 |
7 10
|
mpbid |
|- ( ph -> ( V e. ( RR ^m ( 0 ... M ) ) /\ ( ( ( V ` 0 ) = ( A + X ) /\ ( V ` M ) = ( B + X ) ) /\ A. i e. ( 0 ..^ M ) ( V ` i ) < ( V ` ( i + 1 ) ) ) ) ) |
12 |
11
|
simpld |
|- ( ph -> V e. ( RR ^m ( 0 ... M ) ) ) |
13 |
|
elmapi |
|- ( V e. ( RR ^m ( 0 ... M ) ) -> V : ( 0 ... M ) --> RR ) |
14 |
12 13
|
syl |
|- ( ph -> V : ( 0 ... M ) --> RR ) |
15 |
14
|
ffvelrnda |
|- ( ( ph /\ i e. ( 0 ... M ) ) -> ( V ` i ) e. RR ) |
16 |
3
|
adantr |
|- ( ( ph /\ i e. ( 0 ... M ) ) -> X e. RR ) |
17 |
15 16
|
resubcld |
|- ( ( ph /\ i e. ( 0 ... M ) ) -> ( ( V ` i ) - X ) e. RR ) |
18 |
17 8
|
fmptd |
|- ( ph -> Q : ( 0 ... M ) --> RR ) |
19 |
|
reex |
|- RR e. _V |
20 |
19
|
a1i |
|- ( ph -> RR e. _V ) |
21 |
|
ovex |
|- ( 0 ... M ) e. _V |
22 |
21
|
a1i |
|- ( ph -> ( 0 ... M ) e. _V ) |
23 |
20 22
|
elmapd |
|- ( ph -> ( Q e. ( RR ^m ( 0 ... M ) ) <-> Q : ( 0 ... M ) --> RR ) ) |
24 |
18 23
|
mpbird |
|- ( ph -> Q e. ( RR ^m ( 0 ... M ) ) ) |
25 |
8
|
a1i |
|- ( ph -> Q = ( i e. ( 0 ... M ) |-> ( ( V ` i ) - X ) ) ) |
26 |
|
fveq2 |
|- ( i = 0 -> ( V ` i ) = ( V ` 0 ) ) |
27 |
26
|
oveq1d |
|- ( i = 0 -> ( ( V ` i ) - X ) = ( ( V ` 0 ) - X ) ) |
28 |
27
|
adantl |
|- ( ( ph /\ i = 0 ) -> ( ( V ` i ) - X ) = ( ( V ` 0 ) - X ) ) |
29 |
|
0zd |
|- ( ph -> 0 e. ZZ ) |
30 |
6
|
nnzd |
|- ( ph -> M e. ZZ ) |
31 |
29 30 29
|
3jca |
|- ( ph -> ( 0 e. ZZ /\ M e. ZZ /\ 0 e. ZZ ) ) |
32 |
|
0le0 |
|- 0 <_ 0 |
33 |
32
|
a1i |
|- ( ph -> 0 <_ 0 ) |
34 |
|
0red |
|- ( ph -> 0 e. RR ) |
35 |
6
|
nnred |
|- ( ph -> M e. RR ) |
36 |
6
|
nngt0d |
|- ( ph -> 0 < M ) |
37 |
34 35 36
|
ltled |
|- ( ph -> 0 <_ M ) |
38 |
31 33 37
|
jca32 |
|- ( ph -> ( ( 0 e. ZZ /\ M e. ZZ /\ 0 e. ZZ ) /\ ( 0 <_ 0 /\ 0 <_ M ) ) ) |
39 |
|
elfz2 |
|- ( 0 e. ( 0 ... M ) <-> ( ( 0 e. ZZ /\ M e. ZZ /\ 0 e. ZZ ) /\ ( 0 <_ 0 /\ 0 <_ M ) ) ) |
40 |
38 39
|
sylibr |
|- ( ph -> 0 e. ( 0 ... M ) ) |
41 |
14 40
|
ffvelrnd |
|- ( ph -> ( V ` 0 ) e. RR ) |
42 |
41 3
|
resubcld |
|- ( ph -> ( ( V ` 0 ) - X ) e. RR ) |
43 |
25 28 40 42
|
fvmptd |
|- ( ph -> ( Q ` 0 ) = ( ( V ` 0 ) - X ) ) |
44 |
11
|
simprd |
|- ( ph -> ( ( ( V ` 0 ) = ( A + X ) /\ ( V ` M ) = ( B + X ) ) /\ A. i e. ( 0 ..^ M ) ( V ` i ) < ( V ` ( i + 1 ) ) ) ) |
45 |
44
|
simpld |
|- ( ph -> ( ( V ` 0 ) = ( A + X ) /\ ( V ` M ) = ( B + X ) ) ) |
46 |
45
|
simpld |
|- ( ph -> ( V ` 0 ) = ( A + X ) ) |
47 |
46
|
oveq1d |
|- ( ph -> ( ( V ` 0 ) - X ) = ( ( A + X ) - X ) ) |
48 |
1
|
recnd |
|- ( ph -> A e. CC ) |
49 |
3
|
recnd |
|- ( ph -> X e. CC ) |
50 |
48 49
|
pncand |
|- ( ph -> ( ( A + X ) - X ) = A ) |
51 |
43 47 50
|
3eqtrd |
|- ( ph -> ( Q ` 0 ) = A ) |
52 |
|
fveq2 |
|- ( i = M -> ( V ` i ) = ( V ` M ) ) |
53 |
52
|
oveq1d |
|- ( i = M -> ( ( V ` i ) - X ) = ( ( V ` M ) - X ) ) |
54 |
53
|
adantl |
|- ( ( ph /\ i = M ) -> ( ( V ` i ) - X ) = ( ( V ` M ) - X ) ) |
55 |
29 30 30
|
3jca |
|- ( ph -> ( 0 e. ZZ /\ M e. ZZ /\ M e. ZZ ) ) |
56 |
35
|
leidd |
|- ( ph -> M <_ M ) |
57 |
55 37 56
|
jca32 |
|- ( ph -> ( ( 0 e. ZZ /\ M e. ZZ /\ M e. ZZ ) /\ ( 0 <_ M /\ M <_ M ) ) ) |
58 |
|
elfz2 |
|- ( M e. ( 0 ... M ) <-> ( ( 0 e. ZZ /\ M e. ZZ /\ M e. ZZ ) /\ ( 0 <_ M /\ M <_ M ) ) ) |
59 |
57 58
|
sylibr |
|- ( ph -> M e. ( 0 ... M ) ) |
60 |
14 59
|
ffvelrnd |
|- ( ph -> ( V ` M ) e. RR ) |
61 |
60 3
|
resubcld |
|- ( ph -> ( ( V ` M ) - X ) e. RR ) |
62 |
25 54 59 61
|
fvmptd |
|- ( ph -> ( Q ` M ) = ( ( V ` M ) - X ) ) |
63 |
45
|
simprd |
|- ( ph -> ( V ` M ) = ( B + X ) ) |
64 |
63
|
oveq1d |
|- ( ph -> ( ( V ` M ) - X ) = ( ( B + X ) - X ) ) |
65 |
2
|
recnd |
|- ( ph -> B e. CC ) |
66 |
65 49
|
pncand |
|- ( ph -> ( ( B + X ) - X ) = B ) |
67 |
62 64 66
|
3eqtrd |
|- ( ph -> ( Q ` M ) = B ) |
68 |
51 67
|
jca |
|- ( ph -> ( ( Q ` 0 ) = A /\ ( Q ` M ) = B ) ) |
69 |
|
elfzofz |
|- ( i e. ( 0 ..^ M ) -> i e. ( 0 ... M ) ) |
70 |
69 15
|
sylan2 |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( V ` i ) e. RR ) |
71 |
14
|
adantr |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> V : ( 0 ... M ) --> RR ) |
72 |
|
fzofzp1 |
|- ( i e. ( 0 ..^ M ) -> ( i + 1 ) e. ( 0 ... M ) ) |
73 |
72
|
adantl |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( i + 1 ) e. ( 0 ... M ) ) |
74 |
71 73
|
ffvelrnd |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( V ` ( i + 1 ) ) e. RR ) |
75 |
3
|
adantr |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> X e. RR ) |
76 |
44
|
simprd |
|- ( ph -> A. i e. ( 0 ..^ M ) ( V ` i ) < ( V ` ( i + 1 ) ) ) |
77 |
76
|
r19.21bi |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( V ` i ) < ( V ` ( i + 1 ) ) ) |
78 |
70 74 75 77
|
ltsub1dd |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( ( V ` i ) - X ) < ( ( V ` ( i + 1 ) ) - X ) ) |
79 |
69
|
adantl |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> i e. ( 0 ... M ) ) |
80 |
69 17
|
sylan2 |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( ( V ` i ) - X ) e. RR ) |
81 |
8
|
fvmpt2 |
|- ( ( i e. ( 0 ... M ) /\ ( ( V ` i ) - X ) e. RR ) -> ( Q ` i ) = ( ( V ` i ) - X ) ) |
82 |
79 80 81
|
syl2anc |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( Q ` i ) = ( ( V ` i ) - X ) ) |
83 |
|
fveq2 |
|- ( i = j -> ( V ` i ) = ( V ` j ) ) |
84 |
83
|
oveq1d |
|- ( i = j -> ( ( V ` i ) - X ) = ( ( V ` j ) - X ) ) |
85 |
84
|
cbvmptv |
|- ( i e. ( 0 ... M ) |-> ( ( V ` i ) - X ) ) = ( j e. ( 0 ... M ) |-> ( ( V ` j ) - X ) ) |
86 |
8 85
|
eqtri |
|- Q = ( j e. ( 0 ... M ) |-> ( ( V ` j ) - X ) ) |
87 |
86
|
a1i |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> Q = ( j e. ( 0 ... M ) |-> ( ( V ` j ) - X ) ) ) |
88 |
|
fveq2 |
|- ( j = ( i + 1 ) -> ( V ` j ) = ( V ` ( i + 1 ) ) ) |
89 |
88
|
oveq1d |
|- ( j = ( i + 1 ) -> ( ( V ` j ) - X ) = ( ( V ` ( i + 1 ) ) - X ) ) |
90 |
89
|
adantl |
|- ( ( ( ph /\ i e. ( 0 ..^ M ) ) /\ j = ( i + 1 ) ) -> ( ( V ` j ) - X ) = ( ( V ` ( i + 1 ) ) - X ) ) |
91 |
74 75
|
resubcld |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( ( V ` ( i + 1 ) ) - X ) e. RR ) |
92 |
87 90 73 91
|
fvmptd |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( Q ` ( i + 1 ) ) = ( ( V ` ( i + 1 ) ) - X ) ) |
93 |
78 82 92
|
3brtr4d |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( Q ` i ) < ( Q ` ( i + 1 ) ) ) |
94 |
93
|
ralrimiva |
|- ( ph -> A. i e. ( 0 ..^ M ) ( Q ` i ) < ( Q ` ( i + 1 ) ) ) |
95 |
24 68 94
|
jca32 |
|- ( ph -> ( Q e. ( RR ^m ( 0 ... M ) ) /\ ( ( ( Q ` 0 ) = A /\ ( Q ` M ) = B ) /\ A. i e. ( 0 ..^ M ) ( Q ` i ) < ( Q ` ( i + 1 ) ) ) ) ) |
96 |
5
|
fourierdlem2 |
|- ( M e. NN -> ( Q e. ( O ` M ) <-> ( Q e. ( RR ^m ( 0 ... M ) ) /\ ( ( ( Q ` 0 ) = A /\ ( Q ` M ) = B ) /\ A. i e. ( 0 ..^ M ) ( Q ` i ) < ( Q ` ( i + 1 ) ) ) ) ) ) |
97 |
6 96
|
syl |
|- ( ph -> ( Q e. ( O ` M ) <-> ( Q e. ( RR ^m ( 0 ... M ) ) /\ ( ( ( Q ` 0 ) = A /\ ( Q ` M ) = B ) /\ A. i e. ( 0 ..^ M ) ( Q ` i ) < ( Q ` ( i + 1 ) ) ) ) ) ) |
98 |
95 97
|
mpbird |
|- ( ph -> Q e. ( O ` M ) ) |