Step |
Hyp |
Ref |
Expression |
1 |
|
fourierdlem34.p |
|- P = ( 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 ) ) ) } ) |
2 |
|
fourierdlem34.m |
|- ( ph -> M e. NN ) |
3 |
|
fourierdlem34.q |
|- ( ph -> Q e. ( P ` M ) ) |
4 |
1
|
fourierdlem2 |
|- ( M e. NN -> ( Q e. ( P ` M ) <-> ( Q e. ( RR ^m ( 0 ... M ) ) /\ ( ( ( Q ` 0 ) = A /\ ( Q ` M ) = B ) /\ A. i e. ( 0 ..^ M ) ( Q ` i ) < ( Q ` ( i + 1 ) ) ) ) ) ) |
5 |
2 4
|
syl |
|- ( ph -> ( Q e. ( P ` M ) <-> ( Q e. ( RR ^m ( 0 ... M ) ) /\ ( ( ( Q ` 0 ) = A /\ ( Q ` M ) = B ) /\ A. i e. ( 0 ..^ M ) ( Q ` i ) < ( Q ` ( i + 1 ) ) ) ) ) ) |
6 |
3 5
|
mpbid |
|- ( ph -> ( Q e. ( RR ^m ( 0 ... M ) ) /\ ( ( ( Q ` 0 ) = A /\ ( Q ` M ) = B ) /\ A. i e. ( 0 ..^ M ) ( Q ` i ) < ( Q ` ( i + 1 ) ) ) ) ) |
7 |
6
|
simpld |
|- ( ph -> Q e. ( RR ^m ( 0 ... M ) ) ) |
8 |
|
elmapi |
|- ( Q e. ( RR ^m ( 0 ... M ) ) -> Q : ( 0 ... M ) --> RR ) |
9 |
7 8
|
syl |
|- ( ph -> Q : ( 0 ... M ) --> RR ) |
10 |
|
simplr |
|- ( ( ( ( ( ph /\ i e. ( 0 ... M ) ) /\ j e. ( 0 ... M ) ) /\ ( Q ` i ) = ( Q ` j ) ) /\ -. i = j ) -> ( Q ` i ) = ( Q ` j ) ) |
11 |
9
|
ffvelrnda |
|- ( ( ph /\ i e. ( 0 ... M ) ) -> ( Q ` i ) e. RR ) |
12 |
11
|
ad2antrr |
|- ( ( ( ( ph /\ i e. ( 0 ... M ) ) /\ j e. ( 0 ... M ) ) /\ i < j ) -> ( Q ` i ) e. RR ) |
13 |
9
|
ffvelrnda |
|- ( ( ph /\ k e. ( 0 ... M ) ) -> ( Q ` k ) e. RR ) |
14 |
13
|
ad4ant14 |
|- ( ( ( ( ph /\ i e. ( 0 ... M ) ) /\ i < j ) /\ k e. ( 0 ... M ) ) -> ( Q ` k ) e. RR ) |
15 |
14
|
adantllr |
|- ( ( ( ( ( ph /\ i e. ( 0 ... M ) ) /\ j e. ( 0 ... M ) ) /\ i < j ) /\ k e. ( 0 ... M ) ) -> ( Q ` k ) e. RR ) |
16 |
|
eleq1w |
|- ( i = k -> ( i e. ( 0 ..^ M ) <-> k e. ( 0 ..^ M ) ) ) |
17 |
16
|
anbi2d |
|- ( i = k -> ( ( ph /\ i e. ( 0 ..^ M ) ) <-> ( ph /\ k e. ( 0 ..^ M ) ) ) ) |
18 |
|
fveq2 |
|- ( i = k -> ( Q ` i ) = ( Q ` k ) ) |
19 |
|
oveq1 |
|- ( i = k -> ( i + 1 ) = ( k + 1 ) ) |
20 |
19
|
fveq2d |
|- ( i = k -> ( Q ` ( i + 1 ) ) = ( Q ` ( k + 1 ) ) ) |
21 |
18 20
|
breq12d |
|- ( i = k -> ( ( Q ` i ) < ( Q ` ( i + 1 ) ) <-> ( Q ` k ) < ( Q ` ( k + 1 ) ) ) ) |
22 |
17 21
|
imbi12d |
|- ( i = k -> ( ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( Q ` i ) < ( Q ` ( i + 1 ) ) ) <-> ( ( ph /\ k e. ( 0 ..^ M ) ) -> ( Q ` k ) < ( Q ` ( k + 1 ) ) ) ) ) |
23 |
6
|
simprrd |
|- ( ph -> A. i e. ( 0 ..^ M ) ( Q ` i ) < ( Q ` ( i + 1 ) ) ) |
24 |
23
|
r19.21bi |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( Q ` i ) < ( Q ` ( i + 1 ) ) ) |
25 |
22 24
|
chvarvv |
|- ( ( ph /\ k e. ( 0 ..^ M ) ) -> ( Q ` k ) < ( Q ` ( k + 1 ) ) ) |
26 |
25
|
ad4ant14 |
|- ( ( ( ( ph /\ i e. ( 0 ... M ) ) /\ i < j ) /\ k e. ( 0 ..^ M ) ) -> ( Q ` k ) < ( Q ` ( k + 1 ) ) ) |
27 |
26
|
adantllr |
|- ( ( ( ( ( ph /\ i e. ( 0 ... M ) ) /\ j e. ( 0 ... M ) ) /\ i < j ) /\ k e. ( 0 ..^ M ) ) -> ( Q ` k ) < ( Q ` ( k + 1 ) ) ) |
28 |
|
simpllr |
|- ( ( ( ( ph /\ i e. ( 0 ... M ) ) /\ j e. ( 0 ... M ) ) /\ i < j ) -> i e. ( 0 ... M ) ) |
29 |
|
simplr |
|- ( ( ( ( ph /\ i e. ( 0 ... M ) ) /\ j e. ( 0 ... M ) ) /\ i < j ) -> j e. ( 0 ... M ) ) |
30 |
|
simpr |
|- ( ( ( ( ph /\ i e. ( 0 ... M ) ) /\ j e. ( 0 ... M ) ) /\ i < j ) -> i < j ) |
31 |
15 27 28 29 30
|
monoords |
|- ( ( ( ( ph /\ i e. ( 0 ... M ) ) /\ j e. ( 0 ... M ) ) /\ i < j ) -> ( Q ` i ) < ( Q ` j ) ) |
32 |
12 31
|
ltned |
|- ( ( ( ( ph /\ i e. ( 0 ... M ) ) /\ j e. ( 0 ... M ) ) /\ i < j ) -> ( Q ` i ) =/= ( Q ` j ) ) |
33 |
32
|
neneqd |
|- ( ( ( ( ph /\ i e. ( 0 ... M ) ) /\ j e. ( 0 ... M ) ) /\ i < j ) -> -. ( Q ` i ) = ( Q ` j ) ) |
34 |
33
|
adantlr |
|- ( ( ( ( ( ph /\ i e. ( 0 ... M ) ) /\ j e. ( 0 ... M ) ) /\ -. i = j ) /\ i < j ) -> -. ( Q ` i ) = ( Q ` j ) ) |
35 |
|
simpll |
|- ( ( ( ( ( ph /\ i e. ( 0 ... M ) ) /\ j e. ( 0 ... M ) ) /\ -. i = j ) /\ -. i < j ) -> ( ( ph /\ i e. ( 0 ... M ) ) /\ j e. ( 0 ... M ) ) ) |
36 |
|
elfzelz |
|- ( j e. ( 0 ... M ) -> j e. ZZ ) |
37 |
36
|
zred |
|- ( j e. ( 0 ... M ) -> j e. RR ) |
38 |
37
|
ad3antlr |
|- ( ( ( ( ( ph /\ i e. ( 0 ... M ) ) /\ j e. ( 0 ... M ) ) /\ -. i = j ) /\ -. i < j ) -> j e. RR ) |
39 |
|
elfzelz |
|- ( i e. ( 0 ... M ) -> i e. ZZ ) |
40 |
39
|
zred |
|- ( i e. ( 0 ... M ) -> i e. RR ) |
41 |
40
|
ad4antlr |
|- ( ( ( ( ( ph /\ i e. ( 0 ... M ) ) /\ j e. ( 0 ... M ) ) /\ -. i = j ) /\ -. i < j ) -> i e. RR ) |
42 |
|
neqne |
|- ( -. i = j -> i =/= j ) |
43 |
42
|
necomd |
|- ( -. i = j -> j =/= i ) |
44 |
43
|
ad2antlr |
|- ( ( ( ( ( ph /\ i e. ( 0 ... M ) ) /\ j e. ( 0 ... M ) ) /\ -. i = j ) /\ -. i < j ) -> j =/= i ) |
45 |
|
simpr |
|- ( ( ( ( ( ph /\ i e. ( 0 ... M ) ) /\ j e. ( 0 ... M ) ) /\ -. i = j ) /\ -. i < j ) -> -. i < j ) |
46 |
38 41 44 45
|
lttri5d |
|- ( ( ( ( ( ph /\ i e. ( 0 ... M ) ) /\ j e. ( 0 ... M ) ) /\ -. i = j ) /\ -. i < j ) -> j < i ) |
47 |
9
|
ffvelrnda |
|- ( ( ph /\ j e. ( 0 ... M ) ) -> ( Q ` j ) e. RR ) |
48 |
47
|
adantr |
|- ( ( ( ph /\ j e. ( 0 ... M ) ) /\ j < i ) -> ( Q ` j ) e. RR ) |
49 |
48
|
adantllr |
|- ( ( ( ( ph /\ i e. ( 0 ... M ) ) /\ j e. ( 0 ... M ) ) /\ j < i ) -> ( Q ` j ) e. RR ) |
50 |
|
simp-4l |
|- ( ( ( ( ( ph /\ i e. ( 0 ... M ) ) /\ j e. ( 0 ... M ) ) /\ j < i ) /\ k e. ( 0 ... M ) ) -> ph ) |
51 |
50 13
|
sylancom |
|- ( ( ( ( ( ph /\ i e. ( 0 ... M ) ) /\ j e. ( 0 ... M ) ) /\ j < i ) /\ k e. ( 0 ... M ) ) -> ( Q ` k ) e. RR ) |
52 |
|
simp-4l |
|- ( ( ( ( ( ph /\ i e. ( 0 ... M ) ) /\ j e. ( 0 ... M ) ) /\ j < i ) /\ k e. ( 0 ..^ M ) ) -> ph ) |
53 |
52 25
|
sylancom |
|- ( ( ( ( ( ph /\ i e. ( 0 ... M ) ) /\ j e. ( 0 ... M ) ) /\ j < i ) /\ k e. ( 0 ..^ M ) ) -> ( Q ` k ) < ( Q ` ( k + 1 ) ) ) |
54 |
|
simplr |
|- ( ( ( ( ph /\ i e. ( 0 ... M ) ) /\ j e. ( 0 ... M ) ) /\ j < i ) -> j e. ( 0 ... M ) ) |
55 |
|
simpllr |
|- ( ( ( ( ph /\ i e. ( 0 ... M ) ) /\ j e. ( 0 ... M ) ) /\ j < i ) -> i e. ( 0 ... M ) ) |
56 |
|
simpr |
|- ( ( ( ( ph /\ i e. ( 0 ... M ) ) /\ j e. ( 0 ... M ) ) /\ j < i ) -> j < i ) |
57 |
51 53 54 55 56
|
monoords |
|- ( ( ( ( ph /\ i e. ( 0 ... M ) ) /\ j e. ( 0 ... M ) ) /\ j < i ) -> ( Q ` j ) < ( Q ` i ) ) |
58 |
49 57
|
gtned |
|- ( ( ( ( ph /\ i e. ( 0 ... M ) ) /\ j e. ( 0 ... M ) ) /\ j < i ) -> ( Q ` i ) =/= ( Q ` j ) ) |
59 |
58
|
neneqd |
|- ( ( ( ( ph /\ i e. ( 0 ... M ) ) /\ j e. ( 0 ... M ) ) /\ j < i ) -> -. ( Q ` i ) = ( Q ` j ) ) |
60 |
35 46 59
|
syl2anc |
|- ( ( ( ( ( ph /\ i e. ( 0 ... M ) ) /\ j e. ( 0 ... M ) ) /\ -. i = j ) /\ -. i < j ) -> -. ( Q ` i ) = ( Q ` j ) ) |
61 |
34 60
|
pm2.61dan |
|- ( ( ( ( ph /\ i e. ( 0 ... M ) ) /\ j e. ( 0 ... M ) ) /\ -. i = j ) -> -. ( Q ` i ) = ( Q ` j ) ) |
62 |
61
|
adantlr |
|- ( ( ( ( ( ph /\ i e. ( 0 ... M ) ) /\ j e. ( 0 ... M ) ) /\ ( Q ` i ) = ( Q ` j ) ) /\ -. i = j ) -> -. ( Q ` i ) = ( Q ` j ) ) |
63 |
10 62
|
condan |
|- ( ( ( ( ph /\ i e. ( 0 ... M ) ) /\ j e. ( 0 ... M ) ) /\ ( Q ` i ) = ( Q ` j ) ) -> i = j ) |
64 |
63
|
ex |
|- ( ( ( ph /\ i e. ( 0 ... M ) ) /\ j e. ( 0 ... M ) ) -> ( ( Q ` i ) = ( Q ` j ) -> i = j ) ) |
65 |
64
|
ralrimiva |
|- ( ( ph /\ i e. ( 0 ... M ) ) -> A. j e. ( 0 ... M ) ( ( Q ` i ) = ( Q ` j ) -> i = j ) ) |
66 |
65
|
ralrimiva |
|- ( ph -> A. i e. ( 0 ... M ) A. j e. ( 0 ... M ) ( ( Q ` i ) = ( Q ` j ) -> i = j ) ) |
67 |
|
dff13 |
|- ( Q : ( 0 ... M ) -1-1-> RR <-> ( Q : ( 0 ... M ) --> RR /\ A. i e. ( 0 ... M ) A. j e. ( 0 ... M ) ( ( Q ` i ) = ( Q ` j ) -> i = j ) ) ) |
68 |
9 66 67
|
sylanbrc |
|- ( ph -> Q : ( 0 ... M ) -1-1-> RR ) |