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Theorem fourierdlem25 38106
Description: If  C is not in the range of the partition, then it is in an open interval induced by the partition. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
Hypotheses
Ref Expression
fourierdlem25.m  |-  ( ph  ->  M  e.  NN )
fourierdlem25.qf  |-  ( ph  ->  Q : ( 0 ... M ) --> RR )
fourierdlem25.cel  |-  ( ph  ->  C  e.  ( ( Q `  0 ) [,] ( Q `  M ) ) )
fourierdlem25.cnel  |-  ( ph  ->  -.  C  e.  ran  Q )
fourierdlem25.i  |-  I  =  sup ( { k  e.  ( 0..^ M )  |  ( Q `
 k )  < 
C } ,  RR ,  <  )
Assertion
Ref Expression
fourierdlem25  |-  ( ph  ->  E. j  e.  ( 0..^ M ) C  e.  ( ( Q `
 j ) (,) ( Q `  (
j  +  1 ) ) ) )
Distinct variable groups:    C, k    C, j    j, I    k, I    k, M    j, M    Q, k    Q, j
Allowed substitution hints:    ph( j, k)

Proof of Theorem fourierdlem25
Dummy variables  h  m are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fourierdlem25.i . . 3  |-  I  =  sup ( { k  e.  ( 0..^ M )  |  ( Q `
 k )  < 
C } ,  RR ,  <  )
2 ssrab2 3500 . . . 4  |-  { k  e.  ( 0..^ M )  |  ( Q `
 k )  < 
C }  C_  (
0..^ M )
3 ltso 9732 . . . . . 6  |-  <  Or  RR
43a1i 11 . . . . 5  |-  ( ph  ->  <  Or  RR )
5 fzofi 12225 . . . . . . 7  |-  ( 0..^ M )  e.  Fin
6 ssfi 7810 . . . . . . 7  |-  ( ( ( 0..^ M )  e.  Fin  /\  {
k  e.  ( 0..^ M )  |  ( Q `  k )  <  C }  C_  ( 0..^ M ) )  ->  { k  e.  ( 0..^ M )  |  ( Q `  k )  <  C }  e.  Fin )
75, 2, 6mp2an 686 . . . . . 6  |-  { k  e.  ( 0..^ M )  |  ( Q `
 k )  < 
C }  e.  Fin
87a1i 11 . . . . 5  |-  ( ph  ->  { k  e.  ( 0..^ M )  |  ( Q `  k
)  <  C }  e.  Fin )
9 0zd 10973 . . . . . . . 8  |-  ( ph  ->  0  e.  ZZ )
10 fourierdlem25.m . . . . . . . . 9  |-  ( ph  ->  M  e.  NN )
1110nnzd 11062 . . . . . . . 8  |-  ( ph  ->  M  e.  ZZ )
1210nngt0d 10675 . . . . . . . 8  |-  ( ph  ->  0  <  M )
13 fzolb 11953 . . . . . . . 8  |-  ( 0  e.  ( 0..^ M )  <->  ( 0  e.  ZZ  /\  M  e.  ZZ  /\  0  < 
M ) )
149, 11, 12, 13syl3anbrc 1214 . . . . . . 7  |-  ( ph  ->  0  e.  ( 0..^ M ) )
15 fourierdlem25.qf . . . . . . . . 9  |-  ( ph  ->  Q : ( 0 ... M ) --> RR )
16 elfzofz 11962 . . . . . . . . . 10  |-  ( 0  e.  ( 0..^ M )  ->  0  e.  ( 0 ... M
) )
1714, 16syl 17 . . . . . . . . 9  |-  ( ph  ->  0  e.  ( 0 ... M ) )
1815, 17ffvelrnd 6038 . . . . . . . 8  |-  ( ph  ->  ( Q `  0
)  e.  RR )
1910nnnn0d 10949 . . . . . . . . . . . . 13  |-  ( ph  ->  M  e.  NN0 )
20 nn0uz 11217 . . . . . . . . . . . . 13  |-  NN0  =  ( ZZ>= `  0 )
2119, 20syl6eleq 2559 . . . . . . . . . . . 12  |-  ( ph  ->  M  e.  ( ZZ>= ` 
0 ) )
22 eluzfz2 11833 . . . . . . . . . . . 12  |-  ( M  e.  ( ZZ>= `  0
)  ->  M  e.  ( 0 ... M
) )
2321, 22syl 17 . . . . . . . . . . 11  |-  ( ph  ->  M  e.  ( 0 ... M ) )
2415, 23ffvelrnd 6038 . . . . . . . . . 10  |-  ( ph  ->  ( Q `  M
)  e.  RR )
2518, 24iccssred 37698 . . . . . . . . 9  |-  ( ph  ->  ( ( Q ` 
0 ) [,] ( Q `  M )
)  C_  RR )
26 fourierdlem25.cel . . . . . . . . 9  |-  ( ph  ->  C  e.  ( ( Q `  0 ) [,] ( Q `  M ) ) )
2725, 26sseldd 3419 . . . . . . . 8  |-  ( ph  ->  C  e.  RR )
2818rexrd 9708 . . . . . . . . 9  |-  ( ph  ->  ( Q `  0
)  e.  RR* )
2924rexrd 9708 . . . . . . . . 9  |-  ( ph  ->  ( Q `  M
)  e.  RR* )
30 iccgelb 11716 . . . . . . . . 9  |-  ( ( ( Q `  0
)  e.  RR*  /\  ( Q `  M )  e.  RR*  /\  C  e.  ( ( Q ` 
0 ) [,] ( Q `  M )
) )  ->  ( Q `  0 )  <_  C )
3128, 29, 26, 30syl3anc 1292 . . . . . . . 8  |-  ( ph  ->  ( Q `  0
)  <_  C )
32 fourierdlem25.cnel . . . . . . . . . 10  |-  ( ph  ->  -.  C  e.  ran  Q )
33 simpr 468 . . . . . . . . . . 11  |-  ( (
ph  /\  C  =  ( Q `  0 ) )  ->  C  =  ( Q `  0 ) )
34 ffn 5739 . . . . . . . . . . . . . 14  |-  ( Q : ( 0 ... M ) --> RR  ->  Q  Fn  ( 0 ... M ) )
3515, 34syl 17 . . . . . . . . . . . . 13  |-  ( ph  ->  Q  Fn  ( 0 ... M ) )
3635adantr 472 . . . . . . . . . . . 12  |-  ( (
ph  /\  C  =  ( Q `  0 ) )  ->  Q  Fn  ( 0 ... M
) )
3717adantr 472 . . . . . . . . . . . 12  |-  ( (
ph  /\  C  =  ( Q `  0 ) )  ->  0  e.  ( 0 ... M
) )
38 fnfvelrn 6034 . . . . . . . . . . . 12  |-  ( ( Q  Fn  ( 0 ... M )  /\  0  e.  ( 0 ... M ) )  ->  ( Q ` 
0 )  e.  ran  Q )
3936, 37, 38syl2anc 673 . . . . . . . . . . 11  |-  ( (
ph  /\  C  =  ( Q `  0 ) )  ->  ( Q `  0 )  e. 
ran  Q )
4033, 39eqeltrd 2549 . . . . . . . . . 10  |-  ( (
ph  /\  C  =  ( Q `  0 ) )  ->  C  e.  ran  Q )
4132, 40mtand 671 . . . . . . . . 9  |-  ( ph  ->  -.  C  =  ( Q `  0 ) )
4241neqned 2650 . . . . . . . 8  |-  ( ph  ->  C  =/=  ( Q `
 0 ) )
4318, 27, 31, 42leneltd 9806 . . . . . . 7  |-  ( ph  ->  ( Q `  0
)  <  C )
44 fveq2 5879 . . . . . . . . 9  |-  ( k  =  0  ->  ( Q `  k )  =  ( Q ` 
0 ) )
4544breq1d 4405 . . . . . . . 8  |-  ( k  =  0  ->  (
( Q `  k
)  <  C  <->  ( Q `  0 )  < 
C ) )
4645elrab 3184 . . . . . . 7  |-  ( 0  e.  { k  e.  ( 0..^ M )  |  ( Q `  k )  <  C } 
<->  ( 0  e.  ( 0..^ M )  /\  ( Q `  0 )  <  C ) )
4714, 43, 46sylanbrc 677 . . . . . 6  |-  ( ph  ->  0  e.  { k  e.  ( 0..^ M )  |  ( Q `
 k )  < 
C } )
48 ne0i 3728 . . . . . 6  |-  ( 0  e.  { k  e.  ( 0..^ M )  |  ( Q `  k )  <  C }  ->  { k  e.  ( 0..^ M )  |  ( Q `  k )  <  C }  =/=  (/) )
4947, 48syl 17 . . . . 5  |-  ( ph  ->  { k  e.  ( 0..^ M )  |  ( Q `  k
)  <  C }  =/=  (/) )
50 fzossfz 11965 . . . . . . . 8  |-  ( 0..^ M )  C_  (
0 ... M )
51 fzssz 11827 . . . . . . . . 9  |-  ( 0 ... M )  C_  ZZ
52 zssre 10968 . . . . . . . . 9  |-  ZZ  C_  RR
5351, 52sstri 3427 . . . . . . . 8  |-  ( 0 ... M )  C_  RR
5450, 53sstri 3427 . . . . . . 7  |-  ( 0..^ M )  C_  RR
552, 54sstri 3427 . . . . . 6  |-  { k  e.  ( 0..^ M )  |  ( Q `
 k )  < 
C }  C_  RR
5655a1i 11 . . . . 5  |-  ( ph  ->  { k  e.  ( 0..^ M )  |  ( Q `  k
)  <  C }  C_  RR )
57 fisupcl 8003 . . . . 5  |-  ( (  <  Or  RR  /\  ( { k  e.  ( 0..^ M )  |  ( Q `  k
)  <  C }  e.  Fin  /\  { k  e.  ( 0..^ M )  |  ( Q `
 k )  < 
C }  =/=  (/)  /\  {
k  e.  ( 0..^ M )  |  ( Q `  k )  <  C }  C_  RR ) )  ->  sup ( { k  e.  ( 0..^ M )  |  ( Q `  k
)  <  C } ,  RR ,  <  )  e.  { k  e.  ( 0..^ M )  |  ( Q `  k
)  <  C }
)
584, 8, 49, 56, 57syl13anc 1294 . . . 4  |-  ( ph  ->  sup ( { k  e.  ( 0..^ M )  |  ( Q `
 k )  < 
C } ,  RR ,  <  )  e.  {
k  e.  ( 0..^ M )  |  ( Q `  k )  <  C } )
592, 58sseldi 3416 . . 3  |-  ( ph  ->  sup ( { k  e.  ( 0..^ M )  |  ( Q `
 k )  < 
C } ,  RR ,  <  )  e.  ( 0..^ M ) )
601, 59syl5eqel 2553 . 2  |-  ( ph  ->  I  e.  ( 0..^ M ) )
6150, 60sseldi 3416 . . . . 5  |-  ( ph  ->  I  e.  ( 0 ... M ) )
6215, 61ffvelrnd 6038 . . . 4  |-  ( ph  ->  ( Q `  I
)  e.  RR )
6362rexrd 9708 . . 3  |-  ( ph  ->  ( Q `  I
)  e.  RR* )
64 fzofzp1 12037 . . . . . 6  |-  ( I  e.  ( 0..^ M )  ->  ( I  +  1 )  e.  ( 0 ... M
) )
6560, 64syl 17 . . . . 5  |-  ( ph  ->  ( I  +  1 )  e.  ( 0 ... M ) )
6615, 65ffvelrnd 6038 . . . 4  |-  ( ph  ->  ( Q `  (
I  +  1 ) )  e.  RR )
6766rexrd 9708 . . 3  |-  ( ph  ->  ( Q `  (
I  +  1 ) )  e.  RR* )
681, 58syl5eqel 2553 . . . . 5  |-  ( ph  ->  I  e.  { k  e.  ( 0..^ M )  |  ( Q `
 k )  < 
C } )
69 fveq2 5879 . . . . . . 7  |-  ( k  =  I  ->  ( Q `  k )  =  ( Q `  I ) )
7069breq1d 4405 . . . . . 6  |-  ( k  =  I  ->  (
( Q `  k
)  <  C  <->  ( Q `  I )  <  C
) )
7170elrab 3184 . . . . 5  |-  ( I  e.  { k  e.  ( 0..^ M )  |  ( Q `  k )  <  C } 
<->  ( I  e.  ( 0..^ M )  /\  ( Q `  I )  <  C ) )
7268, 71sylib 201 . . . 4  |-  ( ph  ->  ( I  e.  ( 0..^ M )  /\  ( Q `  I )  <  C ) )
7372simprd 470 . . 3  |-  ( ph  ->  ( Q `  I
)  <  C )
7454, 60sseldi 3416 . . . . . . . . 9  |-  ( ph  ->  I  e.  RR )
75 ltp1 10465 . . . . . . . . . 10  |-  ( I  e.  RR  ->  I  <  ( I  +  1 ) )
76 id 22 . . . . . . . . . . 11  |-  ( I  e.  RR  ->  I  e.  RR )
77 peano2re 9824 . . . . . . . . . . 11  |-  ( I  e.  RR  ->  (
I  +  1 )  e.  RR )
7876, 77ltnled 9799 . . . . . . . . . 10  |-  ( I  e.  RR  ->  (
I  <  ( I  +  1 )  <->  -.  (
I  +  1 )  <_  I ) )
7975, 78mpbid 215 . . . . . . . . 9  |-  ( I  e.  RR  ->  -.  ( I  +  1
)  <_  I )
8074, 79syl 17 . . . . . . . 8  |-  ( ph  ->  -.  ( I  + 
1 )  <_  I
)
8150, 51sstri 3427 . . . . . . . . . . . 12  |-  ( 0..^ M )  C_  ZZ
822, 81sstri 3427 . . . . . . . . . . 11  |-  { k  e.  ( 0..^ M )  |  ( Q `
 k )  < 
C }  C_  ZZ
8382a1i 11 . . . . . . . . . 10  |-  ( (
ph  /\  ( Q `  ( I  +  1 ) )  <  C
)  ->  { k  e.  ( 0..^ M )  |  ( Q `  k )  <  C }  C_  ZZ )
84 elrabi 3181 . . . . . . . . . . . . . . 15  |-  ( h  e.  { k  e.  ( 0..^ M )  |  ( Q `  k )  <  C }  ->  h  e.  ( 0..^ M ) )
85 elfzo0le 11987 . . . . . . . . . . . . . . 15  |-  ( h  e.  ( 0..^ M )  ->  h  <_  M )
8684, 85syl 17 . . . . . . . . . . . . . 14  |-  ( h  e.  { k  e.  ( 0..^ M )  |  ( Q `  k )  <  C }  ->  h  <_  M
)
8786adantl 473 . . . . . . . . . . . . 13  |-  ( (
ph  /\  h  e.  { k  e.  ( 0..^ M )  |  ( Q `  k )  <  C } )  ->  h  <_  M
)
8887ralrimiva 2809 . . . . . . . . . . . 12  |-  ( ph  ->  A. h  e.  {
k  e.  ( 0..^ M )  |  ( Q `  k )  <  C } h  <_  M )
89 breq2 4399 . . . . . . . . . . . . . 14  |-  ( m  =  M  ->  (
h  <_  m  <->  h  <_  M ) )
9089ralbidv 2829 . . . . . . . . . . . . 13  |-  ( m  =  M  ->  ( A. h  e.  { k  e.  ( 0..^ M )  |  ( Q `
 k )  < 
C } h  <_  m 
<-> 
A. h  e.  {
k  e.  ( 0..^ M )  |  ( Q `  k )  <  C } h  <_  M ) )
9190rspcev 3136 . . . . . . . . . . . 12  |-  ( ( M  e.  ZZ  /\  A. h  e.  { k  e.  ( 0..^ M )  |  ( Q `
 k )  < 
C } h  <_  M )  ->  E. m  e.  ZZ  A. h  e. 
{ k  e.  ( 0..^ M )  |  ( Q `  k
)  <  C }
h  <_  m )
9211, 88, 91syl2anc 673 . . . . . . . . . . 11  |-  ( ph  ->  E. m  e.  ZZ  A. h  e.  { k  e.  ( 0..^ M )  |  ( Q `
 k )  < 
C } h  <_  m )
9392adantr 472 . . . . . . . . . 10  |-  ( (
ph  /\  ( Q `  ( I  +  1 ) )  <  C
)  ->  E. m  e.  ZZ  A. h  e. 
{ k  e.  ( 0..^ M )  |  ( Q `  k
)  <  C }
h  <_  m )
94 elfzuz 11822 . . . . . . . . . . . . . 14  |-  ( ( I  +  1 )  e.  ( 0 ... M )  ->  (
I  +  1 )  e.  ( ZZ>= `  0
) )
9565, 94syl 17 . . . . . . . . . . . . 13  |-  ( ph  ->  ( I  +  1 )  e.  ( ZZ>= ` 
0 ) )
9695adantr 472 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( Q `  ( I  +  1 ) )  <  C
)  ->  ( I  +  1 )  e.  ( ZZ>= `  0 )
)
9711adantr 472 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( Q `  ( I  +  1 ) )  <  C
)  ->  M  e.  ZZ )
9853, 65sseldi 3416 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( I  +  1 )  e.  RR )
9998adantr 472 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( Q `  ( I  +  1 ) )  <  C
)  ->  ( I  +  1 )  e.  RR )
10097zred 11063 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( Q `  ( I  +  1 ) )  <  C
)  ->  M  e.  RR )
101 elfzle2 11829 . . . . . . . . . . . . . . 15  |-  ( ( I  +  1 )  e.  ( 0 ... M )  ->  (
I  +  1 )  <_  M )
10265, 101syl 17 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( I  +  1 )  <_  M )
103102adantr 472 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( Q `  ( I  +  1 ) )  <  C
)  ->  ( I  +  1 )  <_  M )
104 simpr 468 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  ( Q `  ( I  +  1 ) )  <  C
)  ->  ( Q `  ( I  +  1 ) )  <  C
)
10566adantr 472 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  ( Q `  ( I  +  1 ) )  <  C
)  ->  ( Q `  ( I  +  1 ) )  e.  RR )
10627adantr 472 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  ( Q `  ( I  +  1 ) )  <  C
)  ->  C  e.  RR )
107105, 106ltnled 9799 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  ( Q `  ( I  +  1 ) )  <  C
)  ->  ( ( Q `  ( I  +  1 ) )  <  C  <->  -.  C  <_  ( Q `  (
I  +  1 ) ) ) )
108104, 107mpbid 215 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( Q `  ( I  +  1 ) )  <  C
)  ->  -.  C  <_  ( Q `  (
I  +  1 ) ) )
109 iccleub 11715 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( Q `  0
)  e.  RR*  /\  ( Q `  M )  e.  RR*  /\  C  e.  ( ( Q ` 
0 ) [,] ( Q `  M )
) )  ->  C  <_  ( Q `  M
) )
11028, 29, 26, 109syl3anc 1292 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  C  <_  ( Q `  M ) )
111110adantr 472 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  M  =  ( I  +  1
) )  ->  C  <_  ( Q `  M
) )
112 fveq2 5879 . . . . . . . . . . . . . . . . . 18  |-  ( M  =  ( I  + 
1 )  ->  ( Q `  M )  =  ( Q `  ( I  +  1
) ) )
113112adantl 473 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  M  =  ( I  +  1
) )  ->  ( Q `  M )  =  ( Q `  ( I  +  1
) ) )
114111, 113breqtrd 4420 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  M  =  ( I  +  1
) )  ->  C  <_  ( Q `  (
I  +  1 ) ) )
115114adantlr 729 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  ( Q `  ( I  +  1 ) )  <  C )  /\  M  =  ( I  +  1 ) )  ->  C  <_  ( Q `  ( I  +  1 ) ) )
116108, 115mtand 671 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( Q `  ( I  +  1 ) )  <  C
)  ->  -.  M  =  ( I  + 
1 ) )
117116neqned 2650 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( Q `  ( I  +  1 ) )  <  C
)  ->  M  =/=  ( I  +  1
) )
11899, 100, 103, 117leneltd 9806 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( Q `  ( I  +  1 ) )  <  C
)  ->  ( I  +  1 )  < 
M )
119 elfzo2 11950 . . . . . . . . . . . 12  |-  ( ( I  +  1 )  e.  ( 0..^ M )  <->  ( ( I  +  1 )  e.  ( ZZ>= `  0 )  /\  M  e.  ZZ  /\  ( I  +  1 )  <  M ) )
12096, 97, 118, 119syl3anbrc 1214 . . . . . . . . . . 11  |-  ( (
ph  /\  ( Q `  ( I  +  1 ) )  <  C
)  ->  ( I  +  1 )  e.  ( 0..^ M ) )
121 fveq2 5879 . . . . . . . . . . . . 13  |-  ( k  =  ( I  + 
1 )  ->  ( Q `  k )  =  ( Q `  ( I  +  1
) ) )
122121breq1d 4405 . . . . . . . . . . . 12  |-  ( k  =  ( I  + 
1 )  ->  (
( Q `  k
)  <  C  <->  ( Q `  ( I  +  1 ) )  <  C
) )
123122elrab 3184 . . . . . . . . . . 11  |-  ( ( I  +  1 )  e.  { k  e.  ( 0..^ M )  |  ( Q `  k )  <  C } 
<->  ( ( I  + 
1 )  e.  ( 0..^ M )  /\  ( Q `  ( I  +  1 ) )  <  C ) )
124120, 104, 123sylanbrc 677 . . . . . . . . . 10  |-  ( (
ph  /\  ( Q `  ( I  +  1 ) )  <  C
)  ->  ( I  +  1 )  e. 
{ k  e.  ( 0..^ M )  |  ( Q `  k
)  <  C }
)
125 suprzub 11278 . . . . . . . . . 10  |-  ( ( { k  e.  ( 0..^ M )  |  ( Q `  k
)  <  C }  C_  ZZ  /\  E. m  e.  ZZ  A. h  e. 
{ k  e.  ( 0..^ M )  |  ( Q `  k
)  <  C }
h  <_  m  /\  ( I  +  1
)  e.  { k  e.  ( 0..^ M )  |  ( Q `
 k )  < 
C } )  -> 
( I  +  1 )  <_  sup ( { k  e.  ( 0..^ M )  |  ( Q `  k
)  <  C } ,  RR ,  <  )
)
12683, 93, 124, 125syl3anc 1292 . . . . . . . . 9  |-  ( (
ph  /\  ( Q `  ( I  +  1 ) )  <  C
)  ->  ( I  +  1 )  <_  sup ( { k  e.  ( 0..^ M )  |  ( Q `  k )  <  C } ,  RR ,  <  ) )
127126, 1syl6breqr 4436 . . . . . . . 8  |-  ( (
ph  /\  ( Q `  ( I  +  1 ) )  <  C
)  ->  ( I  +  1 )  <_  I )
12880, 127mtand 671 . . . . . . 7  |-  ( ph  ->  -.  ( Q `  ( I  +  1
) )  <  C
)
129 eqcom 2478 . . . . . . . . . . 11  |-  ( ( Q `  ( I  +  1 ) )  =  C  <->  C  =  ( Q `  ( I  +  1 ) ) )
130129biimpi 199 . . . . . . . . . 10  |-  ( ( Q `  ( I  +  1 ) )  =  C  ->  C  =  ( Q `  ( I  +  1
) ) )
131130adantl 473 . . . . . . . . 9  |-  ( (
ph  /\  ( Q `  ( I  +  1 ) )  =  C )  ->  C  =  ( Q `  ( I  +  1 ) ) )
13235adantr 472 . . . . . . . . . 10  |-  ( (
ph  /\  ( Q `  ( I  +  1 ) )  =  C )  ->  Q  Fn  ( 0 ... M
) )
13365adantr 472 . . . . . . . . . 10  |-  ( (
ph  /\  ( Q `  ( I  +  1 ) )  =  C )  ->  ( I  +  1 )  e.  ( 0 ... M
) )
134 fnfvelrn 6034 . . . . . . . . . 10  |-  ( ( Q  Fn  ( 0 ... M )  /\  ( I  +  1
)  e.  ( 0 ... M ) )  ->  ( Q `  ( I  +  1
) )  e.  ran  Q )
135132, 133, 134syl2anc 673 . . . . . . . . 9  |-  ( (
ph  /\  ( Q `  ( I  +  1 ) )  =  C )  ->  ( Q `  ( I  +  1 ) )  e.  ran  Q )
136131, 135eqeltrd 2549 . . . . . . . 8  |-  ( (
ph  /\  ( Q `  ( I  +  1 ) )  =  C )  ->  C  e.  ran  Q )
13732, 136mtand 671 . . . . . . 7  |-  ( ph  ->  -.  ( Q `  ( I  +  1
) )  =  C )
138128, 137jca 541 . . . . . 6  |-  ( ph  ->  ( -.  ( Q `
 ( I  + 
1 ) )  < 
C  /\  -.  ( Q `  ( I  +  1 ) )  =  C ) )
139 pm4.56 503 . . . . . 6  |-  ( ( -.  ( Q `  ( I  +  1
) )  <  C  /\  -.  ( Q `  ( I  +  1
) )  =  C )  <->  -.  ( ( Q `  ( I  +  1 ) )  <  C  \/  ( Q `  ( I  +  1 ) )  =  C ) )
140138, 139sylib 201 . . . . 5  |-  ( ph  ->  -.  ( ( Q `
 ( I  + 
1 ) )  < 
C  \/  ( Q `
 ( I  + 
1 ) )  =  C ) )
14166, 27leloed 9795 . . . . 5  |-  ( ph  ->  ( ( Q `  ( I  +  1
) )  <_  C  <->  ( ( Q `  (
I  +  1 ) )  <  C  \/  ( Q `  ( I  +  1 ) )  =  C ) ) )
142140, 141mtbird 308 . . . 4  |-  ( ph  ->  -.  ( Q `  ( I  +  1
) )  <_  C
)
14327, 66ltnled 9799 . . . 4  |-  ( ph  ->  ( C  <  ( Q `  ( I  +  1 ) )  <->  -.  ( Q `  (
I  +  1 ) )  <_  C )
)
144142, 143mpbird 240 . . 3  |-  ( ph  ->  C  <  ( Q `
 ( I  + 
1 ) ) )
14563, 67, 27, 73, 144eliood 37691 . 2  |-  ( ph  ->  C  e.  ( ( Q `  I ) (,) ( Q `  ( I  +  1
) ) ) )
146 fveq2 5879 . . . . 5  |-  ( j  =  I  ->  ( Q `  j )  =  ( Q `  I ) )
147 oveq1 6315 . . . . . 6  |-  ( j  =  I  ->  (
j  +  1 )  =  ( I  + 
1 ) )
148147fveq2d 5883 . . . . 5  |-  ( j  =  I  ->  ( Q `  ( j  +  1 ) )  =  ( Q `  ( I  +  1
) ) )
149146, 148oveq12d 6326 . . . 4  |-  ( j  =  I  ->  (
( Q `  j
) (,) ( Q `
 ( j  +  1 ) ) )  =  ( ( Q `
 I ) (,) ( Q `  (
I  +  1 ) ) ) )
150149eleq2d 2534 . . 3  |-  ( j  =  I  ->  ( C  e.  ( ( Q `  j ) (,) ( Q `  (
j  +  1 ) ) )  <->  C  e.  ( ( Q `  I ) (,) ( Q `  ( I  +  1 ) ) ) ) )
151150rspcev 3136 . 2  |-  ( ( I  e.  ( 0..^ M )  /\  C  e.  ( ( Q `  I ) (,) ( Q `  ( I  +  1 ) ) ) )  ->  E. j  e.  ( 0..^ M ) C  e.  ( ( Q `  j ) (,) ( Q `  ( j  +  1 ) ) ) )
15260, 145, 151syl2anc 673 1  |-  ( ph  ->  E. j  e.  ( 0..^ M ) C  e.  ( ( Q `
 j ) (,) ( Q `  (
j  +  1 ) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 375    /\ wa 376    = wceq 1452    e. wcel 1904    =/= wne 2641   A.wral 2756   E.wrex 2757   {crab 2760    C_ wss 3390   (/)c0 3722   class class class wbr 4395    Or wor 4759   ran crn 4840    Fn wfn 5584   -->wf 5585   ` cfv 5589  (class class class)co 6308   Fincfn 7587   supcsup 7972   RRcr 9556   0cc0 9557   1c1 9558    + caddc 9560   RR*cxr 9692    < clt 9693    <_ cle 9694   NNcn 10631   NN0cn0 10893   ZZcz 10961   ZZ>=cuz 11182   (,)cioo 11660   [,]cicc 11663   ...cfz 11810  ..^cfzo 11942
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-sup 7974  df-inf 7975  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-n0 10894  df-z 10962  df-uz 11183  df-ioo 11664  df-icc 11667  df-fz 11811  df-fzo 11943
This theorem is referenced by:  fourierdlem41  38123  fourierdlem48  38130  fourierdlem49  38131  fourierdlem70  38152  fourierdlem71  38153  fourierdlem103  38185  fourierdlem104  38186
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