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Theorem itgsbtaddcnst 31667
Description: Integral substitution, adding a constant to the function's argument. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
Hypotheses
Ref Expression
itgsbtaddcnst.a  |-  ( ph  ->  A  e.  RR )
itgsbtaddcnst.b  |-  ( ph  ->  B  e.  RR )
itgsbtaddcnst.aleb  |-  ( ph  ->  A  <_  B )
itgsbtaddcnst.x  |-  ( ph  ->  X  e.  RR )
itgsbtaddcnst.f  |-  ( ph  ->  F  e.  ( ( A [,] B )
-cn-> CC ) )
Assertion
Ref Expression
itgsbtaddcnst  |-  ( ph  ->  S__ [ ( A  -  X )  -> 
( B  -  X
) ] ( F `
 ( X  +  s ) )  _d s  =  S__ [ A  ->  B ] ( F `  t )  _d t )
Distinct variable groups:    A, s,
t    B, s, t    F, s, t    X, s, t    ph, s, t

Proof of Theorem itgsbtaddcnst
StepHypRef Expression
1 itgsbtaddcnst.a . . 3  |-  ( ph  ->  A  e.  RR )
2 itgsbtaddcnst.b . . 3  |-  ( ph  ->  B  e.  RR )
3 itgsbtaddcnst.aleb . . 3  |-  ( ph  ->  A  <_  B )
41, 2iccssred 31471 . . . . . . . . 9  |-  ( ph  ->  ( A [,] B
)  C_  RR )
54sselda 3486 . . . . . . . 8  |-  ( (
ph  /\  t  e.  ( A [,] B ) )  ->  t  e.  RR )
65recnd 9620 . . . . . . 7  |-  ( (
ph  /\  t  e.  ( A [,] B ) )  ->  t  e.  CC )
7 itgsbtaddcnst.x . . . . . . . . 9  |-  ( ph  ->  X  e.  RR )
87recnd 9620 . . . . . . . 8  |-  ( ph  ->  X  e.  CC )
98adantr 465 . . . . . . 7  |-  ( (
ph  /\  t  e.  ( A [,] B ) )  ->  X  e.  CC )
106, 9negsubd 9937 . . . . . 6  |-  ( (
ph  /\  t  e.  ( A [,] B ) )  ->  ( t  +  -u X )  =  ( t  -  X
) )
1110eqcomd 2449 . . . . 5  |-  ( (
ph  /\  t  e.  ( A [,] B ) )  ->  ( t  -  X )  =  ( t  +  -u X
) )
1211mpteq2dva 4519 . . . 4  |-  ( ph  ->  ( t  e.  ( A [,] B ) 
|->  ( t  -  X
) )  =  ( t  e.  ( A [,] B )  |->  ( t  +  -u X
) ) )
131adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  t  e.  ( A [,] B ) )  ->  A  e.  RR )
147adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  t  e.  ( A [,] B ) )  ->  X  e.  RR )
1513, 14resubcld 9988 . . . . . . . 8  |-  ( (
ph  /\  t  e.  ( A [,] B ) )  ->  ( A  -  X )  e.  RR )
162adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  t  e.  ( A [,] B ) )  ->  B  e.  RR )
1716, 14resubcld 9988 . . . . . . . 8  |-  ( (
ph  /\  t  e.  ( A [,] B ) )  ->  ( B  -  X )  e.  RR )
185, 14resubcld 9988 . . . . . . . 8  |-  ( (
ph  /\  t  e.  ( A [,] B ) )  ->  ( t  -  X )  e.  RR )
19 simpr 461 . . . . . . . . . . 11  |-  ( (
ph  /\  t  e.  ( A [,] B ) )  ->  t  e.  ( A [,] B ) )
201, 2jca 532 . . . . . . . . . . . . 13  |-  ( ph  ->  ( A  e.  RR  /\  B  e.  RR ) )
2120adantr 465 . . . . . . . . . . . 12  |-  ( (
ph  /\  t  e.  ( A [,] B ) )  ->  ( A  e.  RR  /\  B  e.  RR ) )
22 elicc2 11593 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( t  e.  ( A [,] B )  <-> 
( t  e.  RR  /\  A  <_  t  /\  t  <_  B ) ) )
2321, 22syl 16 . . . . . . . . . . 11  |-  ( (
ph  /\  t  e.  ( A [,] B ) )  ->  ( t  e.  ( A [,] B
)  <->  ( t  e.  RR  /\  A  <_ 
t  /\  t  <_  B ) ) )
2419, 23mpbid 210 . . . . . . . . . 10  |-  ( (
ph  /\  t  e.  ( A [,] B ) )  ->  ( t  e.  RR  /\  A  <_ 
t  /\  t  <_  B ) )
2524simp2d 1008 . . . . . . . . 9  |-  ( (
ph  /\  t  e.  ( A [,] B ) )  ->  A  <_  t )
2613, 5, 14, 25lesub1dd 10169 . . . . . . . 8  |-  ( (
ph  /\  t  e.  ( A [,] B ) )  ->  ( A  -  X )  <_  (
t  -  X ) )
2724simp3d 1009 . . . . . . . . 9  |-  ( (
ph  /\  t  e.  ( A [,] B ) )  ->  t  <_  B )
285, 16, 14, 27lesub1dd 10169 . . . . . . . 8  |-  ( (
ph  /\  t  e.  ( A [,] B ) )  ->  ( t  -  X )  <_  ( B  -  X )
)
2915, 17, 18, 26, 28eliccd 31470 . . . . . . 7  |-  ( (
ph  /\  t  e.  ( A [,] B ) )  ->  ( t  -  X )  e.  ( ( A  -  X
) [,] ( B  -  X ) ) )
30 eqid 2441 . . . . . . 7  |-  ( t  e.  ( A [,] B )  |->  ( t  -  X ) )  =  ( t  e.  ( A [,] B
)  |->  ( t  -  X ) )
3129, 30fmptd 6036 . . . . . 6  |-  ( ph  ->  ( t  e.  ( A [,] B ) 
|->  ( t  -  X
) ) : ( A [,] B ) --> ( ( A  -  X ) [,] ( B  -  X )
) )
3212, 31feq1dd 31388 . . . . 5  |-  ( ph  ->  ( t  e.  ( A [,] B ) 
|->  ( t  +  -u X ) ) : ( A [,] B
) --> ( ( A  -  X ) [,] ( B  -  X
) ) )
331, 7resubcld 9988 . . . . . . . 8  |-  ( ph  ->  ( A  -  X
)  e.  RR )
342, 7resubcld 9988 . . . . . . . 8  |-  ( ph  ->  ( B  -  X
)  e.  RR )
3533, 34iccssred 31471 . . . . . . 7  |-  ( ph  ->  ( ( A  -  X ) [,] ( B  -  X )
)  C_  RR )
36 ax-resscn 9547 . . . . . . 7  |-  RR  C_  CC
3735, 36syl6ss 3498 . . . . . 6  |-  ( ph  ->  ( ( A  -  X ) [,] ( B  -  X )
)  C_  CC )
384, 36syl6ss 3498 . . . . . . . . 9  |-  ( ph  ->  ( A [,] B
)  C_  CC )
3938resmptd 5311 . . . . . . . 8  |-  ( ph  ->  ( ( t  e.  CC  |->  ( t  -  X ) )  |`  ( A [,] B ) )  =  ( t  e.  ( A [,] B )  |->  ( t  -  X ) ) )
40 ssid 3505 . . . . . . . . . . . . 13  |-  CC  C_  CC
41 cncfmptid 21282 . . . . . . . . . . . . 13  |-  ( ( CC  C_  CC  /\  CC  C_  CC )  ->  (
t  e.  CC  |->  t )  e.  ( CC
-cn-> CC ) )
4240, 40, 41mp2an 672 . . . . . . . . . . . 12  |-  ( t  e.  CC  |->  t )  e.  ( CC -cn-> CC )
4342a1i 11 . . . . . . . . . . 11  |-  ( X  e.  CC  ->  (
t  e.  CC  |->  t )  e.  ( CC
-cn-> CC ) )
4440a1i 11 . . . . . . . . . . . 12  |-  ( X  e.  CC  ->  CC  C_  CC )
45 id 22 . . . . . . . . . . . 12  |-  ( X  e.  CC  ->  X  e.  CC )
4644, 45, 44constcncfg 31576 . . . . . . . . . . 11  |-  ( X  e.  CC  ->  (
t  e.  CC  |->  X )  e.  ( CC
-cn-> CC ) )
4743, 46subcncf 31574 . . . . . . . . . 10  |-  ( X  e.  CC  ->  (
t  e.  CC  |->  ( t  -  X ) )  e.  ( CC
-cn-> CC ) )
488, 47syl 16 . . . . . . . . 9  |-  ( ph  ->  ( t  e.  CC  |->  ( t  -  X
) )  e.  ( CC -cn-> CC ) )
49 rescncf 21267 . . . . . . . . 9  |-  ( ( A [,] B ) 
C_  CC  ->  ( ( t  e.  CC  |->  ( t  -  X ) )  e.  ( CC
-cn-> CC )  ->  (
( t  e.  CC  |->  ( t  -  X
) )  |`  ( A [,] B ) )  e.  ( ( A [,] B ) -cn-> CC ) ) )
5038, 48, 49sylc 60 . . . . . . . 8  |-  ( ph  ->  ( ( t  e.  CC  |->  ( t  -  X ) )  |`  ( A [,] B ) )  e.  ( ( A [,] B )
-cn-> CC ) )
5139, 50eqeltrrd 2530 . . . . . . 7  |-  ( ph  ->  ( t  e.  ( A [,] B ) 
|->  ( t  -  X
) )  e.  ( ( A [,] B
) -cn-> CC ) )
5212, 51eqeltrrd 2530 . . . . . 6  |-  ( ph  ->  ( t  e.  ( A [,] B ) 
|->  ( t  +  -u X ) )  e.  ( ( A [,] B ) -cn-> CC ) )
53 cncffvrn 21268 . . . . . 6  |-  ( ( ( ( A  -  X ) [,] ( B  -  X )
)  C_  CC  /\  (
t  e.  ( A [,] B )  |->  ( t  +  -u X
) )  e.  ( ( A [,] B
) -cn-> CC ) )  -> 
( ( t  e.  ( A [,] B
)  |->  ( t  + 
-u X ) )  e.  ( ( A [,] B ) -cn-> ( ( A  -  X
) [,] ( B  -  X ) ) )  <->  ( t  e.  ( A [,] B
)  |->  ( t  + 
-u X ) ) : ( A [,] B ) --> ( ( A  -  X ) [,] ( B  -  X ) ) ) )
5437, 52, 53syl2anc 661 . . . . 5  |-  ( ph  ->  ( ( t  e.  ( A [,] B
)  |->  ( t  + 
-u X ) )  e.  ( ( A [,] B ) -cn-> ( ( A  -  X
) [,] ( B  -  X ) ) )  <->  ( t  e.  ( A [,] B
)  |->  ( t  + 
-u X ) ) : ( A [,] B ) --> ( ( A  -  X ) [,] ( B  -  X ) ) ) )
5532, 54mpbird 232 . . . 4  |-  ( ph  ->  ( t  e.  ( A [,] B ) 
|->  ( t  +  -u X ) )  e.  ( ( A [,] B ) -cn-> ( ( A  -  X ) [,] ( B  -  X ) ) ) )
5612, 55eqeltrd 2529 . . 3  |-  ( ph  ->  ( t  e.  ( A [,] B ) 
|->  ( t  -  X
) )  e.  ( ( A [,] B
) -cn-> ( ( A  -  X ) [,] ( B  -  X
) ) ) )
57 eqid 2441 . . . . 5  |-  ( s  e.  CC  |->  ( X  +  s ) )  =  ( s  e.  CC  |->  ( X  +  s ) )
588adantr 465 . . . . . . . 8  |-  ( (
ph  /\  s  e.  CC )  ->  X  e.  CC )
59 simpr 461 . . . . . . . 8  |-  ( (
ph  /\  s  e.  CC )  ->  s  e.  CC )
6058, 59addcomd 9780 . . . . . . 7  |-  ( (
ph  /\  s  e.  CC )  ->  ( X  +  s )  =  ( s  +  X
) )
6160mpteq2dva 4519 . . . . . 6  |-  ( ph  ->  ( s  e.  CC  |->  ( X  +  s
) )  =  ( s  e.  CC  |->  ( s  +  X ) ) )
62 eqid 2441 . . . . . . . 8  |-  ( s  e.  CC  |->  ( s  +  X ) )  =  ( s  e.  CC  |->  ( s  +  X ) )
6362addccncf 21286 . . . . . . 7  |-  ( X  e.  CC  ->  (
s  e.  CC  |->  ( s  +  X ) )  e.  ( CC
-cn-> CC ) )
648, 63syl 16 . . . . . 6  |-  ( ph  ->  ( s  e.  CC  |->  ( s  +  X
) )  e.  ( CC -cn-> CC ) )
6561, 64eqeltrd 2529 . . . . 5  |-  ( ph  ->  ( s  e.  CC  |->  ( X  +  s
) )  e.  ( CC -cn-> CC ) )
661adantr 465 . . . . . 6  |-  ( (
ph  /\  s  e.  ( ( A  -  X ) [,] ( B  -  X )
) )  ->  A  e.  RR )
672adantr 465 . . . . . 6  |-  ( (
ph  /\  s  e.  ( ( A  -  X ) [,] ( B  -  X )
) )  ->  B  e.  RR )
687adantr 465 . . . . . . 7  |-  ( (
ph  /\  s  e.  ( ( A  -  X ) [,] ( B  -  X )
) )  ->  X  e.  RR )
6935sselda 3486 . . . . . . 7  |-  ( (
ph  /\  s  e.  ( ( A  -  X ) [,] ( B  -  X )
) )  ->  s  e.  RR )
7068, 69readdcld 9621 . . . . . 6  |-  ( (
ph  /\  s  e.  ( ( A  -  X ) [,] ( B  -  X )
) )  ->  ( X  +  s )  e.  RR )
71 simpr 461 . . . . . . . . 9  |-  ( (
ph  /\  s  e.  ( ( A  -  X ) [,] ( B  -  X )
) )  ->  s  e.  ( ( A  -  X ) [,] ( B  -  X )
) )
7233adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  s  e.  ( ( A  -  X ) [,] ( B  -  X )
) )  ->  ( A  -  X )  e.  RR )
7334adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  s  e.  ( ( A  -  X ) [,] ( B  -  X )
) )  ->  ( B  -  X )  e.  RR )
74 elicc2 11593 . . . . . . . . . 10  |-  ( ( ( A  -  X
)  e.  RR  /\  ( B  -  X
)  e.  RR )  ->  ( s  e.  ( ( A  -  X ) [,] ( B  -  X )
)  <->  ( s  e.  RR  /\  ( A  -  X )  <_ 
s  /\  s  <_  ( B  -  X ) ) ) )
7572, 73, 74syl2anc 661 . . . . . . . . 9  |-  ( (
ph  /\  s  e.  ( ( A  -  X ) [,] ( B  -  X )
) )  ->  (
s  e.  ( ( A  -  X ) [,] ( B  -  X ) )  <->  ( s  e.  RR  /\  ( A  -  X )  <_ 
s  /\  s  <_  ( B  -  X ) ) ) )
7671, 75mpbid 210 . . . . . . . 8  |-  ( (
ph  /\  s  e.  ( ( A  -  X ) [,] ( B  -  X )
) )  ->  (
s  e.  RR  /\  ( A  -  X
)  <_  s  /\  s  <_  ( B  -  X ) ) )
7776simp2d 1008 . . . . . . 7  |-  ( (
ph  /\  s  e.  ( ( A  -  X ) [,] ( B  -  X )
) )  ->  ( A  -  X )  <_  s )
7866, 68, 69lesubadd2d 10152 . . . . . . 7  |-  ( (
ph  /\  s  e.  ( ( A  -  X ) [,] ( B  -  X )
) )  ->  (
( A  -  X
)  <_  s  <->  A  <_  ( X  +  s ) ) )
7977, 78mpbid 210 . . . . . 6  |-  ( (
ph  /\  s  e.  ( ( A  -  X ) [,] ( B  -  X )
) )  ->  A  <_  ( X  +  s ) )
8076simp3d 1009 . . . . . . 7  |-  ( (
ph  /\  s  e.  ( ( A  -  X ) [,] ( B  -  X )
) )  ->  s  <_  ( B  -  X
) )
8168, 69, 67leaddsub2d 10155 . . . . . . 7  |-  ( (
ph  /\  s  e.  ( ( A  -  X ) [,] ( B  -  X )
) )  ->  (
( X  +  s )  <_  B  <->  s  <_  ( B  -  X ) ) )
8280, 81mpbird 232 . . . . . 6  |-  ( (
ph  /\  s  e.  ( ( A  -  X ) [,] ( B  -  X )
) )  ->  ( X  +  s )  <_  B )
8366, 67, 70, 79, 82eliccd 31470 . . . . 5  |-  ( (
ph  /\  s  e.  ( ( A  -  X ) [,] ( B  -  X )
) )  ->  ( X  +  s )  e.  ( A [,] B
) )
8457, 65, 37, 38, 83cncfmptssg 31575 . . . 4  |-  ( ph  ->  ( s  e.  ( ( A  -  X
) [,] ( B  -  X ) ) 
|->  ( X  +  s ) )  e.  ( ( ( A  -  X ) [,] ( B  -  X )
) -cn-> ( A [,] B ) ) )
85 itgsbtaddcnst.f . . . 4  |-  ( ph  ->  F  e.  ( ( A [,] B )
-cn-> CC ) )
8684, 85cncfcompt 31588 . . 3  |-  ( ph  ->  ( s  e.  ( ( A  -  X
) [,] ( B  -  X ) ) 
|->  ( F `  ( X  +  s )
) )  e.  ( ( ( A  -  X ) [,] ( B  -  X )
) -cn-> CC ) )
87 ax-1cn 9548 . . . . . 6  |-  1  e.  CC
88 ioosscn 31459 . . . . . 6  |-  ( A (,) B )  C_  CC
89 cncfmptc 21281 . . . . . 6  |-  ( ( 1  e.  CC  /\  ( A (,) B ) 
C_  CC  /\  CC  C_  CC )  ->  ( t  e.  ( A (,) B )  |->  1 )  e.  ( ( A (,) B ) -cn-> CC ) )
9087, 88, 40, 89mp3an 1323 . . . . 5  |-  ( t  e.  ( A (,) B )  |->  1 )  e.  ( ( A (,) B ) -cn-> CC )
9190a1i 11 . . . 4  |-  ( ph  ->  ( t  e.  ( A (,) B ) 
|->  1 )  e.  ( ( A (,) B
) -cn-> CC ) )
92 fconstmpt 5029 . . . . 5  |-  ( ( A (,) B )  X.  { 1 } )  =  ( t  e.  ( A (,) B )  |->  1 )
93 ioombl 21841 . . . . . . 7  |-  ( A (,) B )  e. 
dom  vol
9493a1i 11 . . . . . 6  |-  ( ph  ->  ( A (,) B
)  e.  dom  vol )
95 volioo 31633 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  ( vol `  ( A (,) B ) )  =  ( B  -  A
) )
961, 2, 3, 95syl3anc 1227 . . . . . . 7  |-  ( ph  ->  ( vol `  ( A (,) B ) )  =  ( B  -  A ) )
972, 1resubcld 9988 . . . . . . 7  |-  ( ph  ->  ( B  -  A
)  e.  RR )
9896, 97eqeltrd 2529 . . . . . 6  |-  ( ph  ->  ( vol `  ( A (,) B ) )  e.  RR )
99 1cnd 9610 . . . . . 6  |-  ( ph  ->  1  e.  CC )
100 iblconst 22090 . . . . . 6  |-  ( ( ( A (,) B
)  e.  dom  vol  /\  ( vol `  ( A (,) B ) )  e.  RR  /\  1  e.  CC )  ->  (
( A (,) B
)  X.  { 1 } )  e.  L^1 )
10194, 98, 99, 100syl3anc 1227 . . . . 5  |-  ( ph  ->  ( ( A (,) B )  X.  {
1 } )  e.  L^1 )
10292, 101syl5eqelr 2534 . . . 4  |-  ( ph  ->  ( t  e.  ( A (,) B ) 
|->  1 )  e.  L^1 )
10391, 102elind 3670 . . 3  |-  ( ph  ->  ( t  e.  ( A (,) B ) 
|->  1 )  e.  ( ( ( A (,) B ) -cn-> CC )  i^i  L^1 ) )
10436a1i 11 . . . . 5  |-  ( ph  ->  RR  C_  CC )
10518recnd 9620 . . . . 5  |-  ( (
ph  /\  t  e.  ( A [,] B ) )  ->  ( t  -  X )  e.  CC )
106 eqid 2441 . . . . . 6  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
107106tgioo2 21174 . . . . 5  |-  ( topGen ` 
ran  (,) )  =  ( ( TopOpen ` fld )t  RR )
108 iccntr 21192 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) )  =  ( A (,) B
) )
10920, 108syl 16 . . . . 5  |-  ( ph  ->  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) )  =  ( A (,) B
) )
110104, 4, 105, 107, 106, 109dvmptntr 22240 . . . 4  |-  ( ph  ->  ( RR  _D  (
t  e.  ( A [,] B )  |->  ( t  -  X ) ) )  =  ( RR  _D  ( t  e.  ( A (,) B )  |->  ( t  -  X ) ) ) )
111 reelprrecn 9582 . . . . . 6  |-  RR  e.  { RR ,  CC }
112111a1i 11 . . . . 5  |-  ( ph  ->  RR  e.  { RR ,  CC } )
113 ioossre 11590 . . . . . . . 8  |-  ( A (,) B )  C_  RR
114113sseli 3482 . . . . . . 7  |-  ( t  e.  ( A (,) B )  ->  t  e.  RR )
115114adantl 466 . . . . . 6  |-  ( (
ph  /\  t  e.  ( A (,) B ) )  ->  t  e.  RR )
116115recnd 9620 . . . . 5  |-  ( (
ph  /\  t  e.  ( A (,) B ) )  ->  t  e.  CC )
117 1cnd 9610 . . . . 5  |-  ( (
ph  /\  t  e.  ( A (,) B ) )  ->  1  e.  CC )
118104sselda 3486 . . . . . 6  |-  ( (
ph  /\  t  e.  RR )  ->  t  e.  CC )
119 1cnd 9610 . . . . . 6  |-  ( (
ph  /\  t  e.  RR )  ->  1  e.  CC )
120112dvmptid 22226 . . . . . 6  |-  ( ph  ->  ( RR  _D  (
t  e.  RR  |->  t ) )  =  ( t  e.  RR  |->  1 ) )
121113a1i 11 . . . . . 6  |-  ( ph  ->  ( A (,) B
)  C_  RR )
122 iooretop 21139 . . . . . . 7  |-  ( A (,) B )  e.  ( topGen `  ran  (,) )
123122a1i 11 . . . . . 6  |-  ( ph  ->  ( A (,) B
)  e.  ( topGen ` 
ran  (,) ) )
124112, 118, 119, 120, 121, 107, 106, 123dvmptres 22232 . . . . 5  |-  ( ph  ->  ( RR  _D  (
t  e.  ( A (,) B )  |->  t ) )  =  ( t  e.  ( A (,) B )  |->  1 ) )
1258adantr 465 . . . . 5  |-  ( (
ph  /\  t  e.  ( A (,) B ) )  ->  X  e.  CC )
126 0cnd 9587 . . . . 5  |-  ( (
ph  /\  t  e.  ( A (,) B ) )  ->  0  e.  CC )
1278adantr 465 . . . . . 6  |-  ( (
ph  /\  t  e.  RR )  ->  X  e.  CC )
128 0cnd 9587 . . . . . 6  |-  ( (
ph  /\  t  e.  RR )  ->  0  e.  CC )
129112, 8dvmptc 22227 . . . . . 6  |-  ( ph  ->  ( RR  _D  (
t  e.  RR  |->  X ) )  =  ( t  e.  RR  |->  0 ) )
130112, 127, 128, 129, 121, 107, 106, 123dvmptres 22232 . . . . 5  |-  ( ph  ->  ( RR  _D  (
t  e.  ( A (,) B )  |->  X ) )  =  ( t  e.  ( A (,) B )  |->  0 ) )
131112, 116, 117, 124, 125, 126, 130dvmptsub 22236 . . . 4  |-  ( ph  ->  ( RR  _D  (
t  e.  ( A (,) B )  |->  ( t  -  X ) ) )  =  ( t  e.  ( A (,) B )  |->  ( 1  -  0 ) ) )
132117subid1d 9920 . . . . 5  |-  ( (
ph  /\  t  e.  ( A (,) B ) )  ->  ( 1  -  0 )  =  1 )
133132mpteq2dva 4519 . . . 4  |-  ( ph  ->  ( t  e.  ( A (,) B ) 
|->  ( 1  -  0 ) )  =  ( t  e.  ( A (,) B )  |->  1 ) )
134110, 131, 1333eqtrd 2486 . . 3  |-  ( ph  ->  ( RR  _D  (
t  e.  ( A [,] B )  |->  ( t  -  X ) ) )  =  ( t  e.  ( A (,) B )  |->  1 ) )
135 oveq2 6285 . . . 4  |-  ( s  =  ( t  -  X )  ->  ( X  +  s )  =  ( X  +  ( t  -  X
) ) )
136135fveq2d 5856 . . 3  |-  ( s  =  ( t  -  X )  ->  ( F `  ( X  +  s ) )  =  ( F `  ( X  +  (
t  -  X ) ) ) )
137 oveq1 6284 . . 3  |-  ( t  =  A  ->  (
t  -  X )  =  ( A  -  X ) )
138 oveq1 6284 . . 3  |-  ( t  =  B  ->  (
t  -  X )  =  ( B  -  X ) )
1391, 2, 3, 56, 86, 103, 134, 136, 137, 138, 33, 34itgsubsticc 31661 . 2  |-  ( ph  ->  S__ [ ( A  -  X )  -> 
( B  -  X
) ] ( F `
 ( X  +  s ) )  _d s  =  S__ [ A  ->  B ] ( ( F `  ( X  +  ( t  -  X ) ) )  x.  1 )  _d t )
140125, 116pncan3d 9934 . . . . . 6  |-  ( (
ph  /\  t  e.  ( A (,) B ) )  ->  ( X  +  ( t  -  X ) )  =  t )
141140fveq2d 5856 . . . . 5  |-  ( (
ph  /\  t  e.  ( A (,) B ) )  ->  ( F `  ( X  +  ( t  -  X ) ) )  =  ( F `  t ) )
142141oveq1d 6292 . . . 4  |-  ( (
ph  /\  t  e.  ( A (,) B ) )  ->  ( ( F `  ( X  +  ( t  -  X ) ) )  x.  1 )  =  ( ( F `  t )  x.  1 ) )
143 cncff 21263 . . . . . . . 8  |-  ( F  e.  ( ( A [,] B ) -cn-> CC )  ->  F :
( A [,] B
) --> CC )
14485, 143syl 16 . . . . . . 7  |-  ( ph  ->  F : ( A [,] B ) --> CC )
145144adantr 465 . . . . . 6  |-  ( (
ph  /\  t  e.  ( A (,) B ) )  ->  F :
( A [,] B
) --> CC )
146 ioossicc 11614 . . . . . . . 8  |-  ( A (,) B )  C_  ( A [,] B )
147146sseli 3482 . . . . . . 7  |-  ( t  e.  ( A (,) B )  ->  t  e.  ( A [,] B
) )
148147adantl 466 . . . . . 6  |-  ( (
ph  /\  t  e.  ( A (,) B ) )  ->  t  e.  ( A [,] B ) )
149145, 148ffvelrnd 6013 . . . . 5  |-  ( (
ph  /\  t  e.  ( A (,) B ) )  ->  ( F `  t )  e.  CC )
150149mulid1d 9611 . . . 4  |-  ( (
ph  /\  t  e.  ( A (,) B ) )  ->  ( ( F `  t )  x.  1 )  =  ( F `  t ) )
151142, 150eqtrd 2482 . . 3  |-  ( (
ph  /\  t  e.  ( A (,) B ) )  ->  ( ( F `  ( X  +  ( t  -  X ) ) )  x.  1 )  =  ( F `  t
) )
1523, 151ditgeq3d 31649 . 2  |-  ( ph  ->  S__ [ A  ->  B ] ( ( F `
 ( X  +  ( t  -  X
) ) )  x.  1 )  _d t  =  S__ [ A  ->  B ] ( F `
 t )  _d t )
153139, 152eqtrd 2482 1  |-  ( ph  ->  S__ [ ( A  -  X )  -> 
( B  -  X
) ] ( F `
 ( X  +  s ) )  _d s  =  S__ [ A  ->  B ] ( F `  t )  _d t )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 972    = wceq 1381    e. wcel 1802    C_ wss 3458   {csn 4010   {cpr 4012   class class class wbr 4433    |-> cmpt 4491    X. cxp 4983   dom cdm 4985   ran crn 4986    |` cres 4987   -->wf 5570   ` cfv 5574  (class class class)co 6277   CCcc 9488   RRcr 9489   0cc0 9490   1c1 9491    + caddc 9493    x. cmul 9495    <_ cle 9627    - cmin 9805   -ucneg 9806   (,)cioo 11533   [,]cicc 11536   TopOpenctopn 14691   topGenctg 14707  ℂfldccnfld 18288   intcnt 19384   -cn->ccncf 21246   volcvol 21741   L^1cibl 21892   S__cdit 22116    _D cdv 22133
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4544  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573  ax-inf2 8056  ax-cc 8813  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567  ax-pre-sup 9568  ax-addf 9569  ax-mulf 9570
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-fal 1387  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-pss 3474  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-tp 4015  df-op 4017  df-uni 4231  df-int 4268  df-iun 4313  df-iin 4314  df-disj 4404  df-br 4434  df-opab 4492  df-mpt 4493  df-tr 4527  df-eprel 4777  df-id 4781  df-po 4786  df-so 4787  df-fr 4824  df-se 4825  df-we 4826  df-ord 4867  df-on 4868  df-lim 4869  df-suc 4870  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-isom 5583  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-of 6521  df-ofr 6522  df-om 6682  df-1st 6781  df-2nd 6782  df-supp 6900  df-recs 7040  df-rdg 7074  df-1o 7128  df-2o 7129  df-oadd 7132  df-omul 7133  df-er 7309  df-map 7420  df-pm 7421  df-ixp 7468  df-en 7515  df-dom 7516  df-sdom 7517  df-fin 7518  df-fsupp 7828  df-fi 7869  df-sup 7899  df-oi 7933  df-card 8318  df-acn 8321  df-cda 8546  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9807  df-neg 9808  df-div 10208  df-nn 10538  df-2 10595  df-3 10596  df-4 10597  df-5 10598  df-6 10599  df-7 10600  df-8 10601  df-9 10602  df-10 10603  df-n0 10797  df-z 10866  df-dec 10980  df-uz 11086  df-q 11187  df-rp 11225  df-xneg 11322  df-xadd 11323  df-xmul 11324  df-ioo 11537  df-ioc 11538  df-ico 11539  df-icc 11540  df-fz 11677  df-fzo 11799  df-fl 11903  df-mod 11971  df-seq 12082  df-exp 12141  df-hash 12380  df-cj 12906  df-re 12907  df-im 12908  df-sqrt 13042  df-abs 13043  df-limsup 13268  df-clim 13285  df-rlim 13286  df-sum 13483  df-struct 14506  df-ndx 14507  df-slot 14508  df-base 14509  df-sets 14510  df-ress 14511  df-plusg 14582  df-mulr 14583  df-starv 14584  df-sca 14585  df-vsca 14586  df-ip 14587  df-tset 14588  df-ple 14589  df-ds 14591  df-unif 14592  df-hom 14593  df-cco 14594  df-rest 14692  df-topn 14693  df-0g 14711  df-gsum 14712  df-topgen 14713  df-pt 14714  df-prds 14717  df-xrs 14771  df-qtop 14776  df-imas 14777  df-xps 14779  df-mre 14855  df-mrc 14856  df-acs 14858  df-mgm 15741  df-sgrp 15780  df-mnd 15790  df-submnd 15836  df-mulg 15929  df-cntz 16224  df-cmn 16669  df-psmet 18279  df-xmet 18280  df-met 18281  df-bl 18282  df-mopn 18283  df-fbas 18284  df-fg 18285  df-cnfld 18289  df-top 19266  df-bases 19268  df-topon 19269  df-topsp 19270  df-cld 19386  df-ntr 19387  df-cls 19388  df-nei 19465  df-lp 19503  df-perf 19504  df-cn 19594  df-cnp 19595  df-haus 19682  df-cmp 19753  df-tx 19929  df-hmeo 20122  df-fil 20213  df-fm 20305  df-flim 20306  df-flf 20307  df-xms 20689  df-ms 20690  df-tms 20691  df-cncf 21248  df-ovol 21742  df-vol 21743  df-mbf 21894  df-itg1 21895  df-itg2 21896  df-ibl 21897  df-itg 21898  df-0p 21943  df-ditg 22117  df-limc 22136  df-dv 22137
This theorem is referenced by:  fourierdlem82  31856
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