Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  itgsubsticclem Structured version   Visualization version   Unicode version

Theorem itgsubsticclem 37949
Description: lemma for itgsubsticc 37950. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
Hypotheses
Ref Expression
itgsubsticclem.1  |-  F  =  ( u  e.  ( K [,] L ) 
|->  C )
itgsubsticclem.2  |-  G  =  ( u  e.  RR  |->  if ( u  e.  ( K [,] L ) ,  ( F `  u ) ,  if ( u  <  K , 
( F `  K
) ,  ( F `
 L ) ) ) )
itgsubsticclem.3  |-  ( ph  ->  X  e.  RR )
itgsubsticclem.4  |-  ( ph  ->  Y  e.  RR )
itgsubsticclem.5  |-  ( ph  ->  X  <_  Y )
itgsubsticclem.6  |-  ( ph  ->  ( x  e.  ( X [,] Y ) 
|->  A )  e.  ( ( X [,] Y
) -cn-> ( K [,] L ) ) )
itgsubsticclem.7  |-  ( ph  ->  ( x  e.  ( X (,) Y ) 
|->  B )  e.  ( ( ( X (,) Y ) -cn-> CC )  i^i  L^1 ) )
itgsubsticclem.8  |-  ( ph  ->  F  e.  ( ( K [,] L )
-cn-> CC ) )
itgsubsticclem.9  |-  ( ph  ->  K  e.  RR )
itgsubsticclem.10  |-  ( ph  ->  L  e.  RR )
itgsubsticclem.11  |-  ( ph  ->  K  <_  L )
itgsubsticclem.12  |-  ( ph  ->  ( RR  _D  (
x  e.  ( X [,] Y )  |->  A ) )  =  ( x  e.  ( X (,) Y )  |->  B ) )
itgsubsticclem.13  |-  ( u  =  A  ->  C  =  E )
itgsubsticclem.14  |-  ( x  =  X  ->  A  =  K )
itgsubsticclem.15  |-  ( x  =  Y  ->  A  =  L )
Assertion
Ref Expression
itgsubsticclem  |-  ( ph  ->  S__ [ K  ->  L ] C  _d u  =  S__ [ X  ->  Y ] ( E  x.  B )  _d x )
Distinct variable groups:    u, A    u, E    x, G    u, K, x    u, L, x   
u, X, x    u, Y, x    ph, u, x
Allowed substitution hints:    A( x)    B( x, u)    C( x, u)    E( x)    F( x, u)    G( u)

Proof of Theorem itgsubsticclem
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 fveq2 5879 . . . 4  |-  ( u  =  w  ->  ( G `  u )  =  ( G `  w ) )
2 nfcv 2612 . . . 4  |-  F/_ w
( G `  u
)
3 itgsubsticclem.2 . . . . . 6  |-  G  =  ( u  e.  RR  |->  if ( u  e.  ( K [,] L ) ,  ( F `  u ) ,  if ( u  <  K , 
( F `  K
) ,  ( F `
 L ) ) ) )
4 nfmpt1 4485 . . . . . 6  |-  F/_ u
( u  e.  RR  |->  if ( u  e.  ( K [,] L ) ,  ( F `  u ) ,  if ( u  <  K , 
( F `  K
) ,  ( F `
 L ) ) ) )
53, 4nfcxfr 2610 . . . . 5  |-  F/_ u G
6 nfcv 2612 . . . . 5  |-  F/_ u w
75, 6nffv 5886 . . . 4  |-  F/_ u
( G `  w
)
81, 2, 7cbvditg 22888 . . 3  |-  S__ [ K  ->  L ] ( G `  u )  _d u  =  S__
[ K  ->  L ] ( G `  w )  _d w
9 itgsubsticclem.11 . . . 4  |-  ( ph  ->  K  <_  L )
10 itgsubsticclem.9 . . . . . . . . 9  |-  ( ph  ->  K  e.  RR )
11 itgsubsticclem.10 . . . . . . . . 9  |-  ( ph  ->  L  e.  RR )
1210, 11iccssred 37698 . . . . . . . 8  |-  ( ph  ->  ( K [,] L
)  C_  RR )
1312adantr 472 . . . . . . 7  |-  ( (
ph  /\  u  e.  ( K (,) L ) )  ->  ( K [,] L )  C_  RR )
14 ioossicc 11745 . . . . . . . . 9  |-  ( K (,) L )  C_  ( K [,] L )
1514sseli 3414 . . . . . . . 8  |-  ( u  e.  ( K (,) L )  ->  u  e.  ( K [,] L
) )
1615adantl 473 . . . . . . 7  |-  ( (
ph  /\  u  e.  ( K (,) L ) )  ->  u  e.  ( K [,] L ) )
1713, 16sseldd 3419 . . . . . 6  |-  ( (
ph  /\  u  e.  ( K (,) L ) )  ->  u  e.  RR )
1816iftrued 3880 . . . . . . 7  |-  ( (
ph  /\  u  e.  ( K (,) L ) )  ->  if (
u  e.  ( K [,] L ) ,  ( F `  u
) ,  if ( u  <  K , 
( F `  K
) ,  ( F `
 L ) ) )  =  ( F `
 u ) )
19 itgsubsticclem.1 . . . . . . . . . . . . 13  |-  F  =  ( u  e.  ( K [,] L ) 
|->  C )
2019a1i 11 . . . . . . . . . . . 12  |-  ( ph  ->  F  =  ( u  e.  ( K [,] L )  |->  C ) )
21 itgsubsticclem.8 . . . . . . . . . . . . 13  |-  ( ph  ->  F  e.  ( ( K [,] L )
-cn-> CC ) )
22 cncff 22003 . . . . . . . . . . . . 13  |-  ( F  e.  ( ( K [,] L ) -cn-> CC )  ->  F :
( K [,] L
) --> CC )
2321, 22syl 17 . . . . . . . . . . . 12  |-  ( ph  ->  F : ( K [,] L ) --> CC )
2420, 23feq1dd 37503 . . . . . . . . . . 11  |-  ( ph  ->  ( u  e.  ( K [,] L ) 
|->  C ) : ( K [,] L ) --> CC )
2524mptex2 37506 . . . . . . . . . 10  |-  ( (
ph  /\  u  e.  ( K [,] L ) )  ->  C  e.  CC )
2616, 25syldan 478 . . . . . . . . 9  |-  ( (
ph  /\  u  e.  ( K (,) L ) )  ->  C  e.  CC )
2719fvmpt2 5972 . . . . . . . . 9  |-  ( ( u  e.  ( K [,] L )  /\  C  e.  CC )  ->  ( F `  u
)  =  C )
2816, 26, 27syl2anc 673 . . . . . . . 8  |-  ( (
ph  /\  u  e.  ( K (,) L ) )  ->  ( F `  u )  =  C )
2928, 26eqeltrd 2549 . . . . . . 7  |-  ( (
ph  /\  u  e.  ( K (,) L ) )  ->  ( F `  u )  e.  CC )
3018, 29eqeltrd 2549 . . . . . 6  |-  ( (
ph  /\  u  e.  ( K (,) L ) )  ->  if (
u  e.  ( K [,] L ) ,  ( F `  u
) ,  if ( u  <  K , 
( F `  K
) ,  ( F `
 L ) ) )  e.  CC )
313fvmpt2 5972 . . . . . 6  |-  ( ( u  e.  RR  /\  if ( u  e.  ( K [,] L ) ,  ( F `  u ) ,  if ( u  <  K , 
( F `  K
) ,  ( F `
 L ) ) )  e.  CC )  ->  ( G `  u )  =  if ( u  e.  ( K [,] L ) ,  ( F `  u ) ,  if ( u  <  K , 
( F `  K
) ,  ( F `
 L ) ) ) )
3217, 30, 31syl2anc 673 . . . . 5  |-  ( (
ph  /\  u  e.  ( K (,) L ) )  ->  ( G `  u )  =  if ( u  e.  ( K [,] L ) ,  ( F `  u ) ,  if ( u  <  K , 
( F `  K
) ,  ( F `
 L ) ) ) )
3332, 18, 283eqtrd 2509 . . . 4  |-  ( (
ph  /\  u  e.  ( K (,) L ) )  ->  ( G `  u )  =  C )
349, 33ditgeq3d 37938 . . 3  |-  ( ph  ->  S__ [ K  ->  L ] ( G `  u )  _d u  =  S__ [ K  ->  L ] C  _d u )
35 itgsubsticclem.3 . . . 4  |-  ( ph  ->  X  e.  RR )
36 itgsubsticclem.4 . . . 4  |-  ( ph  ->  Y  e.  RR )
37 itgsubsticclem.5 . . . 4  |-  ( ph  ->  X  <_  Y )
38 mnfxr 11437 . . . . 5  |- -oo  e.  RR*
3938a1i 11 . . . 4  |-  ( ph  -> -oo  e.  RR* )
40 pnfxr 11435 . . . . 5  |- +oo  e.  RR*
4140a1i 11 . . . 4  |-  ( ph  -> +oo  e.  RR* )
42 ioomax 11734 . . . . . . . . 9  |-  ( -oo (,) +oo )  =  RR
4342eqcomi 2480 . . . . . . . 8  |-  RR  =  ( -oo (,) +oo )
4443a1i 11 . . . . . . 7  |-  ( ph  ->  RR  =  ( -oo (,) +oo ) )
4512, 44sseqtrd 3454 . . . . . 6  |-  ( ph  ->  ( K [,] L
)  C_  ( -oo (,) +oo ) )
46 ax-resscn 9614 . . . . . . 7  |-  RR  C_  CC
4744, 46syl6eqssr 3469 . . . . . 6  |-  ( ph  ->  ( -oo (,) +oo )  C_  CC )
48 cncfss 22009 . . . . . 6  |-  ( ( ( K [,] L
)  C_  ( -oo (,) +oo )  /\  ( -oo (,) +oo )  C_  CC )  ->  ( ( X [,] Y )
-cn-> ( K [,] L
) )  C_  (
( X [,] Y
) -cn-> ( -oo (,) +oo ) ) )
4945, 47, 48syl2anc 673 . . . . 5  |-  ( ph  ->  ( ( X [,] Y ) -cn-> ( K [,] L ) ) 
C_  ( ( X [,] Y ) -cn-> ( -oo (,) +oo )
) )
50 itgsubsticclem.6 . . . . 5  |-  ( ph  ->  ( x  e.  ( X [,] Y ) 
|->  A )  e.  ( ( X [,] Y
) -cn-> ( K [,] L ) ) )
5149, 50sseldd 3419 . . . 4  |-  ( ph  ->  ( x  e.  ( X [,] Y ) 
|->  A )  e.  ( ( X [,] Y
) -cn-> ( -oo (,) +oo ) ) )
52 itgsubsticclem.7 . . . 4  |-  ( ph  ->  ( x  e.  ( X (,) Y ) 
|->  B )  e.  ( ( ( X (,) Y ) -cn-> CC )  i^i  L^1 ) )
53 nfmpt1 4485 . . . . . . . . . . 11  |-  F/_ u
( u  e.  ( K [,] L ) 
|->  C )
5419, 53nfcxfr 2610 . . . . . . . . . 10  |-  F/_ u F
55 eqid 2471 . . . . . . . . . 10  |-  ( topGen ` 
ran  (,) )  =  (
topGen `  ran  (,) )
56 eqid 2471 . . . . . . . . . 10  |-  U. ( TopOpen
` fld
)  =  U. ( TopOpen
` fld
)
57 eqid 2471 . . . . . . . . . . . 12  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
5857cnfldtop 21882 . . . . . . . . . . 11  |-  ( TopOpen ` fld )  e.  Top
5958a1i 11 . . . . . . . . . 10  |-  ( ph  ->  ( TopOpen ` fld )  e.  Top )
6012, 46syl6ss 3430 . . . . . . . . . . . . 13  |-  ( ph  ->  ( K [,] L
)  C_  CC )
61 ssid 3437 . . . . . . . . . . . . 13  |-  CC  C_  CC
62 eqid 2471 . . . . . . . . . . . . . 14  |-  ( (
TopOpen ` fld )t  ( K [,] L
) )  =  ( ( TopOpen ` fld )t  ( K [,] L ) )
63 unicntop 37433 . . . . . . . . . . . . . . . . 17  |-  CC  =  U. ( TopOpen ` fld )
6463restid 15410 . . . . . . . . . . . . . . . 16  |-  ( (
TopOpen ` fld )  e.  Top  ->  ( ( TopOpen ` fld )t  CC )  =  (
TopOpen ` fld ) )
6558, 64ax-mp 5 . . . . . . . . . . . . . . 15  |-  ( (
TopOpen ` fld )t  CC )  =  (
TopOpen ` fld )
6665eqcomi 2480 . . . . . . . . . . . . . 14  |-  ( TopOpen ` fld )  =  ( ( TopOpen ` fld )t  CC )
6757, 62, 66cncfcn 22019 . . . . . . . . . . . . 13  |-  ( ( ( K [,] L
)  C_  CC  /\  CC  C_  CC )  ->  (
( K [,] L
) -cn-> CC )  =  ( ( ( TopOpen ` fld )t  ( K [,] L ) )  Cn  ( TopOpen ` fld ) ) )
6860, 61, 67sylancl 675 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( K [,] L ) -cn-> CC )  =  ( ( (
TopOpen ` fld )t  ( K [,] L
) )  Cn  ( TopOpen
` fld
) ) )
69 reex 9648 . . . . . . . . . . . . . . . 16  |-  RR  e.  _V
7069a1i 11 . . . . . . . . . . . . . . 15  |-  ( ph  ->  RR  e.  _V )
71 restabs 20258 . . . . . . . . . . . . . . 15  |-  ( ( ( TopOpen ` fld )  e.  Top  /\  ( K [,] L
)  C_  RR  /\  RR  e.  _V )  ->  (
( ( TopOpen ` fld )t  RR )t  ( K [,] L ) )  =  ( ( TopOpen ` fld )t  ( K [,] L ) ) )
7259, 12, 70, 71syl3anc 1292 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( ( ( TopOpen ` fld )t  RR )t  ( K [,] L ) )  =  ( (
TopOpen ` fld )t  ( K [,] L
) ) )
7357tgioo2 21899 . . . . . . . . . . . . . . . . 17  |-  ( topGen ` 
ran  (,) )  =  ( ( TopOpen ` fld )t  RR )
7473eqcomi 2480 . . . . . . . . . . . . . . . 16  |-  ( (
TopOpen ` fld )t  RR )  =  (
topGen `  ran  (,) )
7574a1i 11 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( ( TopOpen ` fld )t  RR )  =  (
topGen `  ran  (,) )
)
7675oveq1d 6323 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( ( ( TopOpen ` fld )t  RR )t  ( K [,] L ) )  =  ( (
topGen `  ran  (,) )t  ( K [,] L ) ) )
7772, 76eqtr3d 2507 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( TopOpen ` fld )t  ( K [,] L ) )  =  ( ( topGen `  ran  (,) )t  ( K [,] L
) ) )
7877oveq1d 6323 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( ( TopOpen ` fld )t  ( K [,] L ) )  Cn  ( TopOpen ` fld ) )  =  ( ( ( topGen `  ran  (,) )t  ( K [,] L
) )  Cn  ( TopOpen
` fld
) ) )
7968, 78eqtrd 2505 . . . . . . . . . . 11  |-  ( ph  ->  ( ( K [,] L ) -cn-> CC )  =  ( ( (
topGen `  ran  (,) )t  ( K [,] L ) )  Cn  ( TopOpen ` fld ) ) )
8021, 79eleqtrd 2551 . . . . . . . . . 10  |-  ( ph  ->  F  e.  ( ( ( topGen `  ran  (,) )t  ( K [,] L ) )  Cn  ( TopOpen ` fld ) ) )
8154, 55, 56, 3, 10, 11, 9, 59, 80icccncfext 37862 . . . . . . . . 9  |-  ( ph  ->  ( G  e.  ( ( topGen `  ran  (,) )  Cn  ( ( TopOpen ` fld )t  ran  F ) )  /\  ( G  |`  ( K [,] L ) )  =  F ) )
8281simpld 466 . . . . . . . 8  |-  ( ph  ->  G  e.  ( (
topGen `  ran  (,) )  Cn  ( ( TopOpen ` fld )t  ran  F ) ) )
83 uniretop 21861 . . . . . . . . 9  |-  RR  =  U. ( topGen `  ran  (,) )
84 eqid 2471 . . . . . . . . 9  |-  U. (
( TopOpen ` fld )t  ran  F )  = 
U. ( ( TopOpen ` fld )t  ran  F )
8583, 84cnf 20339 . . . . . . . 8  |-  ( G  e.  ( ( topGen ` 
ran  (,) )  Cn  (
( TopOpen ` fld )t  ran  F ) )  ->  G : RR --> U. ( ( TopOpen ` fld )t  ran  F ) )
8682, 85syl 17 . . . . . . 7  |-  ( ph  ->  G : RR --> U. (
( TopOpen ` fld )t  ran  F ) )
8744feq2d 5725 . . . . . . 7  |-  ( ph  ->  ( G : RR --> U. ( ( TopOpen ` fld )t  ran  F )  <->  G :
( -oo (,) +oo ) --> U. ( ( TopOpen ` fld )t  ran  F ) ) )
8886, 87mpbid 215 . . . . . 6  |-  ( ph  ->  G : ( -oo (,) +oo ) --> U. (
( TopOpen ` fld )t  ran  F ) )
8988feqmptd 5932 . . . . 5  |-  ( ph  ->  G  =  ( w  e.  ( -oo (,) +oo )  |->  ( G `  w ) ) )
90 frn 5747 . . . . . . . 8  |-  ( F : ( K [,] L ) --> CC  ->  ran 
F  C_  CC )
9123, 90syl 17 . . . . . . 7  |-  ( ph  ->  ran  F  C_  CC )
92 cncfss 22009 . . . . . . 7  |-  ( ( ran  F  C_  CC  /\  CC  C_  CC )  ->  ( ( -oo (,) +oo ) -cn-> ran  F )  C_  ( ( -oo (,) +oo ) -cn-> CC ) )
9391, 61, 92sylancl 675 . . . . . 6  |-  ( ph  ->  ( ( -oo (,) +oo ) -cn-> ran  F )  C_  ( ( -oo (,) +oo ) -cn-> CC ) )
9443oveq2i 6319 . . . . . . . . . . 11  |-  ( (
TopOpen ` fld )t  RR )  =  ( ( TopOpen ` fld )t  ( -oo (,) +oo ) )
9573, 94eqtri 2493 . . . . . . . . . 10  |-  ( topGen ` 
ran  (,) )  =  ( ( TopOpen ` fld )t  ( -oo (,) +oo ) )
96 eqid 2471 . . . . . . . . . 10  |-  ( (
TopOpen ` fld )t 
ran  F )  =  ( ( TopOpen ` fld )t  ran  F )
9757, 95, 96cncfcn 22019 . . . . . . . . 9  |-  ( ( ( -oo (,) +oo )  C_  CC  /\  ran  F 
C_  CC )  -> 
( ( -oo (,) +oo ) -cn-> ran  F )  =  ( ( topGen ` 
ran  (,) )  Cn  (
( TopOpen ` fld )t  ran  F ) ) )
9847, 91, 97syl2anc 673 . . . . . . . 8  |-  ( ph  ->  ( ( -oo (,) +oo ) -cn-> ran  F )  =  ( ( topGen ` 
ran  (,) )  Cn  (
( TopOpen ` fld )t  ran  F ) ) )
9998eqcomd 2477 . . . . . . 7  |-  ( ph  ->  ( ( topGen `  ran  (,) )  Cn  ( (
TopOpen ` fld )t 
ran  F ) )  =  ( ( -oo (,) +oo ) -cn-> ran  F
) )
10082, 99eleqtrd 2551 . . . . . 6  |-  ( ph  ->  G  e.  ( ( -oo (,) +oo ) -cn->
ran  F ) )
10193, 100sseldd 3419 . . . . 5  |-  ( ph  ->  G  e.  ( ( -oo (,) +oo ) -cn->
CC ) )
10289, 101eqeltrrd 2550 . . . 4  |-  ( ph  ->  ( w  e.  ( -oo (,) +oo )  |->  ( G `  w
) )  e.  ( ( -oo (,) +oo ) -cn-> CC ) )
103 itgsubsticclem.12 . . . 4  |-  ( ph  ->  ( RR  _D  (
x  e.  ( X [,] Y )  |->  A ) )  =  ( x  e.  ( X (,) Y )  |->  B ) )
104 fveq2 5879 . . . 4  |-  ( w  =  A  ->  ( G `  w )  =  ( G `  A ) )
105 itgsubsticclem.14 . . . 4  |-  ( x  =  X  ->  A  =  K )
106 itgsubsticclem.15 . . . 4  |-  ( x  =  Y  ->  A  =  L )
10735, 36, 37, 39, 41, 51, 52, 102, 103, 104, 105, 106itgsubst 23080 . . 3  |-  ( ph  ->  S__ [ K  ->  L ] ( G `  w )  _d w  =  S__ [ X  ->  Y ] ( ( G `  A )  x.  B )  _d x )
1088, 34, 1073eqtr3a 2529 . 2  |-  ( ph  ->  S__ [ K  ->  L ] C  _d u  =  S__ [ X  ->  Y ] ( ( G `  A )  x.  B )  _d x )
1093a1i 11 . . . . 5  |-  ( (
ph  /\  x  e.  ( X (,) Y ) )  ->  G  =  ( u  e.  RR  |->  if ( u  e.  ( K [,] L ) ,  ( F `  u ) ,  if ( u  <  K , 
( F `  K
) ,  ( F `
 L ) ) ) ) )
110 simpr 468 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  ( X (,) Y
) )  /\  u  =  A )  ->  u  =  A )
11157cnfldtopon 21881 . . . . . . . . . . . . . 14  |-  ( TopOpen ` fld )  e.  (TopOn `  CC )
11235, 36iccssred 37698 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( X [,] Y
)  C_  RR )
113112, 46syl6ss 3430 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( X [,] Y
)  C_  CC )
114 resttopon 20254 . . . . . . . . . . . . . 14  |-  ( ( ( TopOpen ` fld )  e.  (TopOn `  CC )  /\  ( X [,] Y )  C_  CC )  ->  ( (
TopOpen ` fld )t  ( X [,] Y
) )  e.  (TopOn `  ( X [,] Y
) ) )
115111, 113, 114sylancr 676 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( TopOpen ` fld )t  ( X [,] Y ) )  e.  (TopOn `  ( X [,] Y ) ) )
116 resttopon 20254 . . . . . . . . . . . . . 14  |-  ( ( ( TopOpen ` fld )  e.  (TopOn `  CC )  /\  ( K [,] L )  C_  CC )  ->  ( (
TopOpen ` fld )t  ( K [,] L
) )  e.  (TopOn `  ( K [,] L
) ) )
117111, 60, 116sylancr 676 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( TopOpen ` fld )t  ( K [,] L ) )  e.  (TopOn `  ( K [,] L ) ) )
118 eqid 2471 . . . . . . . . . . . . . . . 16  |-  ( (
TopOpen ` fld )t  ( X [,] Y
) )  =  ( ( TopOpen ` fld )t  ( X [,] Y ) )
11957, 118, 62cncfcn 22019 . . . . . . . . . . . . . . 15  |-  ( ( ( X [,] Y
)  C_  CC  /\  ( K [,] L )  C_  CC )  ->  ( ( X [,] Y )
-cn-> ( K [,] L
) )  =  ( ( ( TopOpen ` fld )t  ( X [,] Y ) )  Cn  ( ( TopOpen ` fld )t  ( K [,] L ) ) ) )
120113, 60, 119syl2anc 673 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( ( X [,] Y ) -cn-> ( K [,] L ) )  =  ( ( (
TopOpen ` fld )t  ( X [,] Y
) )  Cn  (
( TopOpen ` fld )t  ( K [,] L ) ) ) )
12150, 120eleqtrd 2551 . . . . . . . . . . . . 13  |-  ( ph  ->  ( x  e.  ( X [,] Y ) 
|->  A )  e.  ( ( ( TopOpen ` fld )t  ( X [,] Y ) )  Cn  ( ( TopOpen ` fld )t  ( K [,] L ) ) ) )
122 cnf2 20342 . . . . . . . . . . . . 13  |-  ( ( ( ( TopOpen ` fld )t  ( X [,] Y ) )  e.  (TopOn `  ( X [,] Y ) )  /\  ( ( TopOpen ` fld )t  ( K [,] L ) )  e.  (TopOn `  ( K [,] L ) )  /\  ( x  e.  ( X [,] Y )  |->  A )  e.  ( ( ( TopOpen ` fld )t  ( X [,] Y ) )  Cn  ( ( TopOpen ` fld )t  ( K [,] L ) ) ) )  ->  ( x  e.  ( X [,] Y
)  |->  A ) : ( X [,] Y
) --> ( K [,] L ) )
123115, 117, 121, 122syl3anc 1292 . . . . . . . . . . . 12  |-  ( ph  ->  ( x  e.  ( X [,] Y ) 
|->  A ) : ( X [,] Y ) --> ( K [,] L
) )
124123adantr 472 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( X (,) Y ) )  ->  ( x  e.  ( X [,] Y
)  |->  A ) : ( X [,] Y
) --> ( K [,] L ) )
125 eqid 2471 . . . . . . . . . . . 12  |-  ( x  e.  ( X [,] Y )  |->  A )  =  ( x  e.  ( X [,] Y
)  |->  A )
126125fmpt 6058 . . . . . . . . . . 11  |-  ( A. x  e.  ( X [,] Y ) A  e.  ( K [,] L
)  <->  ( x  e.  ( X [,] Y
)  |->  A ) : ( X [,] Y
) --> ( K [,] L ) )
127124, 126sylibr 217 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( X (,) Y ) )  ->  A. x  e.  ( X [,] Y
) A  e.  ( K [,] L ) )
128 ioossicc 11745 . . . . . . . . . . . 12  |-  ( X (,) Y )  C_  ( X [,] Y )
129128sseli 3414 . . . . . . . . . . 11  |-  ( x  e.  ( X (,) Y )  ->  x  e.  ( X [,] Y
) )
130129adantl 473 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( X (,) Y ) )  ->  x  e.  ( X [,] Y ) )
131 rsp 2773 . . . . . . . . . 10  |-  ( A. x  e.  ( X [,] Y ) A  e.  ( K [,] L
)  ->  ( x  e.  ( X [,] Y
)  ->  A  e.  ( K [,] L ) ) )
132127, 130, 131sylc 61 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( X (,) Y ) )  ->  A  e.  ( K [,] L ) )
133132adantr 472 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  ( X (,) Y
) )  /\  u  =  A )  ->  A  e.  ( K [,] L
) )
134110, 133eqeltrd 2549 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  ( X (,) Y
) )  /\  u  =  A )  ->  u  e.  ( K [,] L
) )
135134iftrued 3880 . . . . . 6  |-  ( ( ( ph  /\  x  e.  ( X (,) Y
) )  /\  u  =  A )  ->  if ( u  e.  ( K [,] L ) ,  ( F `  u
) ,  if ( u  <  K , 
( F `  K
) ,  ( F `
 L ) ) )  =  ( F `
 u ) )
136 simpll 768 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  ( X (,) Y
) )  /\  u  =  A )  ->  ph )
137136, 134, 25syl2anc 673 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  ( X (,) Y
) )  /\  u  =  A )  ->  C  e.  CC )
138134, 137, 27syl2anc 673 . . . . . 6  |-  ( ( ( ph  /\  x  e.  ( X (,) Y
) )  /\  u  =  A )  ->  ( F `  u )  =  C )
139 itgsubsticclem.13 . . . . . . 7  |-  ( u  =  A  ->  C  =  E )
140139adantl 473 . . . . . 6  |-  ( ( ( ph  /\  x  e.  ( X (,) Y
) )  /\  u  =  A )  ->  C  =  E )
141135, 138, 1403eqtrd 2509 . . . . 5  |-  ( ( ( ph  /\  x  e.  ( X (,) Y
) )  /\  u  =  A )  ->  if ( u  e.  ( K [,] L ) ,  ( F `  u
) ,  if ( u  <  K , 
( F `  K
) ,  ( F `
 L ) ) )  =  E )
14212adantr 472 . . . . . 6  |-  ( (
ph  /\  x  e.  ( X (,) Y ) )  ->  ( K [,] L )  C_  RR )
143142, 132sseldd 3419 . . . . 5  |-  ( (
ph  /\  x  e.  ( X (,) Y ) )  ->  A  e.  RR )
144 elex 3040 . . . . . . . 8  |-  ( A  e.  ( K [,] L )  ->  A  e.  _V )
145132, 144syl 17 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( X (,) Y ) )  ->  A  e.  _V )
146 isset 3035 . . . . . . 7  |-  ( A  e.  _V  <->  E. u  u  =  A )
147145, 146sylib 201 . . . . . 6  |-  ( (
ph  /\  x  e.  ( X (,) Y ) )  ->  E. u  u  =  A )
148140, 137eqeltrrd 2550 . . . . . 6  |-  ( ( ( ph  /\  x  e.  ( X (,) Y
) )  /\  u  =  A )  ->  E  e.  CC )
149147, 148exlimddv 1789 . . . . 5  |-  ( (
ph  /\  x  e.  ( X (,) Y ) )  ->  E  e.  CC )
150109, 141, 143, 149fvmptd 5969 . . . 4  |-  ( (
ph  /\  x  e.  ( X (,) Y ) )  ->  ( G `  A )  =  E )
151150oveq1d 6323 . . 3  |-  ( (
ph  /\  x  e.  ( X (,) Y ) )  ->  ( ( G `  A )  x.  B )  =  ( E  x.  B ) )
15237, 151ditgeq3d 37938 . 2  |-  ( ph  ->  S__ [ X  ->  Y ] ( ( G `
 A )  x.  B )  _d x  =  S__ [ X  ->  Y ] ( E  x.  B )  _d x )
153108, 152eqtrd 2505 1  |-  ( ph  ->  S__ [ K  ->  L ] C  _d u  =  S__ [ X  ->  Y ] ( E  x.  B )  _d x )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 376    = wceq 1452   E.wex 1671    e. wcel 1904   A.wral 2756   _Vcvv 3031    i^i cin 3389    C_ wss 3390   ifcif 3872   U.cuni 4190   class class class wbr 4395    |-> cmpt 4454   ran crn 4840    |` cres 4841   -->wf 5585   ` cfv 5589  (class class class)co 6308   CCcc 9555   RRcr 9556    x. cmul 9562   +oocpnf 9690   -oocmnf 9691   RR*cxr 9692    < clt 9693    <_ cle 9694   (,)cioo 11660   [,]cicc 11663   ↾t crest 15397   TopOpenctopn 15398   topGenctg 15414  ℂfldccnfld 19047   Topctop 19994  TopOnctopon 19995    Cn ccn 20317   -cn->ccncf 21986   L^1cibl 22654   S__cdit 22880    _D cdv 22897
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-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164  ax-cc 8883  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  ax-pre-sup 9635  ax-addf 9636  ax-mulf 9637
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  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-int 4227  df-iun 4271  df-iin 4272  df-disj 4367  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-se 4799  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-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-of 6550  df-ofr 6551  df-om 6712  df-1st 6812  df-2nd 6813  df-supp 6934  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-2o 7201  df-oadd 7204  df-omul 7205  df-er 7381  df-map 7492  df-pm 7493  df-ixp 7541  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-fsupp 7902  df-fi 7943  df-sup 7974  df-inf 7975  df-oi 8043  df-card 8391  df-acn 8394  df-cda 8616  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-4 10692  df-5 10693  df-6 10694  df-7 10695  df-8 10696  df-9 10697  df-10 10698  df-n0 10894  df-z 10962  df-dec 11075  df-uz 11183  df-q 11288  df-rp 11326  df-xneg 11432  df-xadd 11433  df-xmul 11434  df-ioo 11664  df-ioc 11665  df-ico 11666  df-icc 11667  df-fz 11811  df-fzo 11943  df-fl 12061  df-mod 12130  df-seq 12252  df-exp 12311  df-hash 12554  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-limsup 13603  df-clim 13629  df-rlim 13630  df-sum 13830  df-struct 15201  df-ndx 15202  df-slot 15203  df-base 15204  df-sets 15205  df-ress 15206  df-plusg 15281  df-mulr 15282  df-starv 15283  df-sca 15284  df-vsca 15285  df-ip 15286  df-tset 15287  df-ple 15288  df-ds 15290  df-unif 15291  df-hom 15292  df-cco 15293  df-rest 15399  df-topn 15400  df-0g 15418  df-gsum 15419  df-topgen 15420  df-pt 15421  df-prds 15424  df-xrs 15478  df-qtop 15484  df-imas 15485  df-xps 15488  df-mre 15570  df-mrc 15571  df-acs 15573  df-mgm 16566  df-sgrp 16605  df-mnd 16615  df-submnd 16661  df-mulg 16754  df-cntz 17049  df-cmn 17510  df-psmet 19039  df-xmet 19040  df-met 19041  df-bl 19042  df-mopn 19043  df-fbas 19044  df-fg 19045  df-cnfld 19048  df-top 19998  df-bases 19999  df-topon 20000  df-topsp 20001  df-cld 20111  df-ntr 20112  df-cls 20113  df-nei 20191  df-lp 20229  df-perf 20230  df-cn 20320  df-cnp 20321  df-haus 20408  df-cmp 20479  df-tx 20654  df-hmeo 20847  df-fil 20939  df-fm 21031  df-flim 21032  df-flf 21033  df-xms 21413  df-ms 21414  df-tms 21415  df-cncf 21988  df-ovol 22494  df-vol 22496  df-mbf 22656  df-itg1 22657  df-itg2 22658  df-ibl 22659  df-itg 22660  df-0p 22707  df-ditg 22881  df-limc 22900  df-dv 22901
This theorem is referenced by:  itgsubsticc  37950
  Copyright terms: Public domain W3C validator