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Theorem itgsubsticclem 37852
Description: lemma for itgsubsticc 37853. (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 5865 . . . 4  |-  ( u  =  w  ->  ( G `  u )  =  ( G `  w ) )
2 nfcv 2592 . . . 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 4492 . . . . . 6  |-  F/_ u
( u  e.  RR  |->  if ( u  e.  ( K [,] L ) ,  ( F `  u ) ,  if ( u  <  K , 
( F `  K
) ,  ( F `
 L ) ) ) )
53, 4nfcxfr 2590 . . . . 5  |-  F/_ u G
6 nfcv 2592 . . . . 5  |-  F/_ u w
75, 6nffv 5872 . . . 4  |-  F/_ u
( G `  w
)
81, 2, 7cbvditg 22809 . . 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 37602 . . . . . . . 8  |-  ( ph  ->  ( K [,] L
)  C_  RR )
1312adantr 467 . . . . . . 7  |-  ( (
ph  /\  u  e.  ( K (,) L ) )  ->  ( K [,] L )  C_  RR )
14 ioossicc 11720 . . . . . . . . 9  |-  ( K (,) L )  C_  ( K [,] L )
1514sseli 3428 . . . . . . . 8  |-  ( u  e.  ( K (,) L )  ->  u  e.  ( K [,] L
) )
1615adantl 468 . . . . . . 7  |-  ( (
ph  /\  u  e.  ( K (,) L ) )  ->  u  e.  ( K [,] L ) )
1713, 16sseldd 3433 . . . . . 6  |-  ( (
ph  /\  u  e.  ( K (,) L ) )  ->  u  e.  RR )
1816iftrued 3889 . . . . . . 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 21925 . . . . . . . . . . . . 13  |-  ( F  e.  ( ( K [,] L ) -cn-> CC )  ->  F :
( K [,] L
) --> CC )
2321, 22syl 17 . . . . . . . . . . . 12  |-  ( ph  ->  F : ( K [,] L ) --> CC )
2420, 23feq1dd 37430 . . . . . . . . . . 11  |-  ( ph  ->  ( u  e.  ( K [,] L ) 
|->  C ) : ( K [,] L ) --> CC )
2524mptex2 37433 . . . . . . . . . 10  |-  ( (
ph  /\  u  e.  ( K [,] L ) )  ->  C  e.  CC )
2616, 25syldan 473 . . . . . . . . 9  |-  ( (
ph  /\  u  e.  ( K (,) L ) )  ->  C  e.  CC )
2719fvmpt2 5957 . . . . . . . . 9  |-  ( ( u  e.  ( K [,] L )  /\  C  e.  CC )  ->  ( F `  u
)  =  C )
2816, 26, 27syl2anc 667 . . . . . . . 8  |-  ( (
ph  /\  u  e.  ( K (,) L ) )  ->  ( F `  u )  =  C )
2928, 26eqeltrd 2529 . . . . . . 7  |-  ( (
ph  /\  u  e.  ( K (,) L ) )  ->  ( F `  u )  e.  CC )
3018, 29eqeltrd 2529 . . . . . 6  |-  ( (
ph  /\  u  e.  ( K (,) L ) )  ->  if (
u  e.  ( K [,] L ) ,  ( F `  u
) ,  if ( u  <  K , 
( F `  K
) ,  ( F `
 L ) ) )  e.  CC )
313fvmpt2 5957 . . . . . 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 667 . . . . 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 2489 . . . 4  |-  ( (
ph  /\  u  e.  ( K (,) L ) )  ->  ( G `  u )  =  C )
349, 33ditgeq3d 37841 . . 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 11414 . . . . 5  |- -oo  e.  RR*
3938a1i 11 . . . 4  |-  ( ph  -> -oo  e.  RR* )
40 pnfxr 11412 . . . . 5  |- +oo  e.  RR*
4140a1i 11 . . . 4  |-  ( ph  -> +oo  e.  RR* )
42 ioomax 11709 . . . . . . . . 9  |-  ( -oo (,) +oo )  =  RR
4342eqcomi 2460 . . . . . . . 8  |-  RR  =  ( -oo (,) +oo )
4443a1i 11 . . . . . . 7  |-  ( ph  ->  RR  =  ( -oo (,) +oo ) )
4512, 44sseqtrd 3468 . . . . . 6  |-  ( ph  ->  ( K [,] L
)  C_  ( -oo (,) +oo ) )
46 ax-resscn 9596 . . . . . . 7  |-  RR  C_  CC
4744, 46syl6eqssr 3483 . . . . . 6  |-  ( ph  ->  ( -oo (,) +oo )  C_  CC )
48 cncfss 21931 . . . . . 6  |-  ( ( ( K [,] L
)  C_  ( -oo (,) +oo )  /\  ( -oo (,) +oo )  C_  CC )  ->  ( ( X [,] Y )
-cn-> ( K [,] L
) )  C_  (
( X [,] Y
) -cn-> ( -oo (,) +oo ) ) )
4945, 47, 48syl2anc 667 . . . . 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 3433 . . . 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 4492 . . . . . . . . . . 11  |-  F/_ u
( u  e.  ( K [,] L ) 
|->  C )
5419, 53nfcxfr 2590 . . . . . . . . . 10  |-  F/_ u F
55 eqid 2451 . . . . . . . . . 10  |-  ( topGen ` 
ran  (,) )  =  (
topGen `  ran  (,) )
56 eqid 2451 . . . . . . . . . 10  |-  U. ( TopOpen
` fld
)  =  U. ( TopOpen
` fld
)
57 eqid 2451 . . . . . . . . . . . 12  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
5857cnfldtop 21804 . . . . . . . . . . 11  |-  ( TopOpen ` fld )  e.  Top
5958a1i 11 . . . . . . . . . 10  |-  ( ph  ->  ( TopOpen ` fld )  e.  Top )
6012, 46syl6ss 3444 . . . . . . . . . . . . 13  |-  ( ph  ->  ( K [,] L
)  C_  CC )
61 ssid 3451 . . . . . . . . . . . . 13  |-  CC  C_  CC
62 eqid 2451 . . . . . . . . . . . . . 14  |-  ( (
TopOpen ` fld )t  ( K [,] L
) )  =  ( ( TopOpen ` fld )t  ( K [,] L ) )
63 unicntop 37371 . . . . . . . . . . . . . . . . 17  |-  CC  =  U. ( TopOpen ` fld )
6463restid 15332 . . . . . . . . . . . . . . . 16  |-  ( (
TopOpen ` fld )  e.  Top  ->  ( ( TopOpen ` fld )t  CC )  =  (
TopOpen ` fld ) )
6558, 64ax-mp 5 . . . . . . . . . . . . . . 15  |-  ( (
TopOpen ` fld )t  CC )  =  (
TopOpen ` fld )
6665eqcomi 2460 . . . . . . . . . . . . . 14  |-  ( TopOpen ` fld )  =  ( ( TopOpen ` fld )t  CC )
6757, 62, 66cncfcn 21941 . . . . . . . . . . . . 13  |-  ( ( ( K [,] L
)  C_  CC  /\  CC  C_  CC )  ->  (
( K [,] L
) -cn-> CC )  =  ( ( ( TopOpen ` fld )t  ( K [,] L ) )  Cn  ( TopOpen ` fld ) ) )
6860, 61, 67sylancl 668 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( K [,] L ) -cn-> CC )  =  ( ( (
TopOpen ` fld )t  ( K [,] L
) )  Cn  ( TopOpen
` fld
) ) )
69 reex 9630 . . . . . . . . . . . . . . . 16  |-  RR  e.  _V
7069a1i 11 . . . . . . . . . . . . . . 15  |-  ( ph  ->  RR  e.  _V )
71 restabs 20181 . . . . . . . . . . . . . . 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 1268 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( ( ( TopOpen ` fld )t  RR )t  ( K [,] L ) )  =  ( (
TopOpen ` fld )t  ( K [,] L
) ) )
7357tgioo2 21821 . . . . . . . . . . . . . . . . 17  |-  ( topGen ` 
ran  (,) )  =  ( ( TopOpen ` fld )t  RR )
7473eqcomi 2460 . . . . . . . . . . . . . . . 16  |-  ( (
TopOpen ` fld )t  RR )  =  (
topGen `  ran  (,) )
7574a1i 11 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( ( TopOpen ` fld )t  RR )  =  (
topGen `  ran  (,) )
)
7675oveq1d 6305 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( ( ( TopOpen ` fld )t  RR )t  ( K [,] L ) )  =  ( (
topGen `  ran  (,) )t  ( K [,] L ) ) )
7772, 76eqtr3d 2487 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( TopOpen ` fld )t  ( K [,] L ) )  =  ( ( topGen `  ran  (,) )t  ( K [,] L
) ) )
7877oveq1d 6305 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( ( TopOpen ` fld )t  ( K [,] L ) )  Cn  ( TopOpen ` fld ) )  =  ( ( ( topGen `  ran  (,) )t  ( K [,] L
) )  Cn  ( TopOpen
` fld
) ) )
7968, 78eqtrd 2485 . . . . . . . . . . 11  |-  ( ph  ->  ( ( K [,] L ) -cn-> CC )  =  ( ( (
topGen `  ran  (,) )t  ( K [,] L ) )  Cn  ( TopOpen ` fld ) ) )
8021, 79eleqtrd 2531 . . . . . . . . . 10  |-  ( ph  ->  F  e.  ( ( ( topGen `  ran  (,) )t  ( K [,] L ) )  Cn  ( TopOpen ` fld ) ) )
8154, 55, 56, 3, 10, 11, 9, 59, 80icccncfext 37765 . . . . . . . . 9  |-  ( ph  ->  ( G  e.  ( ( topGen `  ran  (,) )  Cn  ( ( TopOpen ` fld )t  ran  F ) )  /\  ( G  |`  ( K [,] L ) )  =  F ) )
8281simpld 461 . . . . . . . 8  |-  ( ph  ->  G  e.  ( (
topGen `  ran  (,) )  Cn  ( ( TopOpen ` fld )t  ran  F ) ) )
83 uniretop 21783 . . . . . . . . 9  |-  RR  =  U. ( topGen `  ran  (,) )
84 eqid 2451 . . . . . . . . 9  |-  U. (
( TopOpen ` fld )t  ran  F )  = 
U. ( ( TopOpen ` fld )t  ran  F )
8583, 84cnf 20262 . . . . . . . 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 5715 . . . . . . 7  |-  ( ph  ->  ( G : RR --> U. ( ( TopOpen ` fld )t  ran  F )  <->  G :
( -oo (,) +oo ) --> U. ( ( TopOpen ` fld )t  ran  F ) ) )
8886, 87mpbid 214 . . . . . 6  |-  ( ph  ->  G : ( -oo (,) +oo ) --> U. (
( TopOpen ` fld )t  ran  F ) )
8988feqmptd 5918 . . . . 5  |-  ( ph  ->  G  =  ( w  e.  ( -oo (,) +oo )  |->  ( G `  w ) ) )
90 frn 5735 . . . . . . . 8  |-  ( F : ( K [,] L ) --> CC  ->  ran 
F  C_  CC )
9123, 90syl 17 . . . . . . 7  |-  ( ph  ->  ran  F  C_  CC )
92 cncfss 21931 . . . . . . 7  |-  ( ( ran  F  C_  CC  /\  CC  C_  CC )  ->  ( ( -oo (,) +oo ) -cn-> ran  F )  C_  ( ( -oo (,) +oo ) -cn-> CC ) )
9391, 61, 92sylancl 668 . . . . . 6  |-  ( ph  ->  ( ( -oo (,) +oo ) -cn-> ran  F )  C_  ( ( -oo (,) +oo ) -cn-> CC ) )
9443oveq2i 6301 . . . . . . . . . . 11  |-  ( (
TopOpen ` fld )t  RR )  =  ( ( TopOpen ` fld )t  ( -oo (,) +oo ) )
9573, 94eqtri 2473 . . . . . . . . . 10  |-  ( topGen ` 
ran  (,) )  =  ( ( TopOpen ` fld )t  ( -oo (,) +oo ) )
96 eqid 2451 . . . . . . . . . 10  |-  ( (
TopOpen ` fld )t 
ran  F )  =  ( ( TopOpen ` fld )t  ran  F )
9757, 95, 96cncfcn 21941 . . . . . . . . 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 667 . . . . . . . 8  |-  ( ph  ->  ( ( -oo (,) +oo ) -cn-> ran  F )  =  ( ( topGen ` 
ran  (,) )  Cn  (
( TopOpen ` fld )t  ran  F ) ) )
9998eqcomd 2457 . . . . . . 7  |-  ( ph  ->  ( ( topGen `  ran  (,) )  Cn  ( (
TopOpen ` fld )t 
ran  F ) )  =  ( ( -oo (,) +oo ) -cn-> ran  F
) )
10082, 99eleqtrd 2531 . . . . . 6  |-  ( ph  ->  G  e.  ( ( -oo (,) +oo ) -cn->
ran  F ) )
10193, 100sseldd 3433 . . . . 5  |-  ( ph  ->  G  e.  ( ( -oo (,) +oo ) -cn->
CC ) )
10289, 101eqeltrrd 2530 . . . 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 5865 . . . 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 23001 . . 3  |-  ( ph  ->  S__ [ K  ->  L ] ( G `  w )  _d w  =  S__ [ X  ->  Y ] ( ( G `  A )  x.  B )  _d x )
1088, 34, 1073eqtr3a 2509 . 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 463 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  ( X (,) Y
) )  /\  u  =  A )  ->  u  =  A )
11157cnfldtopon 21803 . . . . . . . . . . . . . 14  |-  ( TopOpen ` fld )  e.  (TopOn `  CC )
11235, 36iccssred 37602 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( X [,] Y
)  C_  RR )
113112, 46syl6ss 3444 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( X [,] Y
)  C_  CC )
114 resttopon 20177 . . . . . . . . . . . . . 14  |-  ( ( ( TopOpen ` fld )  e.  (TopOn `  CC )  /\  ( X [,] Y )  C_  CC )  ->  ( (
TopOpen ` fld )t  ( X [,] Y
) )  e.  (TopOn `  ( X [,] Y
) ) )
115111, 113, 114sylancr 669 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( TopOpen ` fld )t  ( X [,] Y ) )  e.  (TopOn `  ( X [,] Y ) ) )
116 resttopon 20177 . . . . . . . . . . . . . 14  |-  ( ( ( TopOpen ` fld )  e.  (TopOn `  CC )  /\  ( K [,] L )  C_  CC )  ->  ( (
TopOpen ` fld )t  ( K [,] L
) )  e.  (TopOn `  ( K [,] L
) ) )
117111, 60, 116sylancr 669 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( TopOpen ` fld )t  ( K [,] L ) )  e.  (TopOn `  ( K [,] L ) ) )
118 eqid 2451 . . . . . . . . . . . . . . . 16  |-  ( (
TopOpen ` fld )t  ( X [,] Y
) )  =  ( ( TopOpen ` fld )t  ( X [,] Y ) )
11957, 118, 62cncfcn 21941 . . . . . . . . . . . . . . 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 667 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( ( X [,] Y ) -cn-> ( K [,] L ) )  =  ( ( (
TopOpen ` fld )t  ( X [,] Y
) )  Cn  (
( TopOpen ` fld )t  ( K [,] L ) ) ) )
12150, 120eleqtrd 2531 . . . . . . . . . . . . 13  |-  ( ph  ->  ( x  e.  ( X [,] Y ) 
|->  A )  e.  ( ( ( TopOpen ` fld )t  ( X [,] Y ) )  Cn  ( ( TopOpen ` fld )t  ( K [,] L ) ) ) )
122 cnf2 20265 . . . . . . . . . . . . 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 1268 . . . . . . . . . . . 12  |-  ( ph  ->  ( x  e.  ( X [,] Y ) 
|->  A ) : ( X [,] Y ) --> ( K [,] L
) )
124123adantr 467 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( X (,) Y ) )  ->  ( x  e.  ( X [,] Y
)  |->  A ) : ( X [,] Y
) --> ( K [,] L ) )
125 eqid 2451 . . . . . . . . . . . 12  |-  ( x  e.  ( X [,] Y )  |->  A )  =  ( x  e.  ( X [,] Y
)  |->  A )
126125fmpt 6043 . . . . . . . . . . 11  |-  ( A. x  e.  ( X [,] Y ) A  e.  ( K [,] L
)  <->  ( x  e.  ( X [,] Y
)  |->  A ) : ( X [,] Y
) --> ( K [,] L ) )
127124, 126sylibr 216 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( X (,) Y ) )  ->  A. x  e.  ( X [,] Y
) A  e.  ( K [,] L ) )
128 ioossicc 11720 . . . . . . . . . . . 12  |-  ( X (,) Y )  C_  ( X [,] Y )
129128sseli 3428 . . . . . . . . . . 11  |-  ( x  e.  ( X (,) Y )  ->  x  e.  ( X [,] Y
) )
130129adantl 468 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( X (,) Y ) )  ->  x  e.  ( X [,] Y ) )
131 rsp 2754 . . . . . . . . . 10  |-  ( A. x  e.  ( X [,] Y ) A  e.  ( K [,] L
)  ->  ( x  e.  ( X [,] Y
)  ->  A  e.  ( K [,] L ) ) )
132127, 130, 131sylc 62 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( X (,) Y ) )  ->  A  e.  ( K [,] L ) )
133132adantr 467 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  ( X (,) Y
) )  /\  u  =  A )  ->  A  e.  ( K [,] L
) )
134110, 133eqeltrd 2529 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  ( X (,) Y
) )  /\  u  =  A )  ->  u  e.  ( K [,] L
) )
135134iftrued 3889 . . . . . 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 760 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  ( X (,) Y
) )  /\  u  =  A )  ->  ph )
137136, 134, 25syl2anc 667 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  ( X (,) Y
) )  /\  u  =  A )  ->  C  e.  CC )
138134, 137, 27syl2anc 667 . . . . . 6  |-  ( ( ( ph  /\  x  e.  ( X (,) Y
) )  /\  u  =  A )  ->  ( F `  u )  =  C )
139 itgsubsticclem.13 . . . . . . 7  |-  ( u  =  A  ->  C  =  E )
140139adantl 468 . . . . . 6  |-  ( ( ( ph  /\  x  e.  ( X (,) Y
) )  /\  u  =  A )  ->  C  =  E )
141135, 138, 1403eqtrd 2489 . . . . 5  |-  ( ( ( ph  /\  x  e.  ( X (,) Y
) )  /\  u  =  A )  ->  if ( u  e.  ( K [,] L ) ,  ( F `  u
) ,  if ( u  <  K , 
( F `  K
) ,  ( F `
 L ) ) )  =  E )
14212adantr 467 . . . . . 6  |-  ( (
ph  /\  x  e.  ( X (,) Y ) )  ->  ( K [,] L )  C_  RR )
143142, 132sseldd 3433 . . . . 5  |-  ( (
ph  /\  x  e.  ( X (,) Y ) )  ->  A  e.  RR )
144 elex 3054 . . . . . . . 8  |-  ( A  e.  ( K [,] L )  ->  A  e.  _V )
145132, 144syl 17 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( X (,) Y ) )  ->  A  e.  _V )
146 isset 3049 . . . . . . 7  |-  ( A  e.  _V  <->  E. u  u  =  A )
147145, 146sylib 200 . . . . . 6  |-  ( (
ph  /\  x  e.  ( X (,) Y ) )  ->  E. u  u  =  A )
148140, 137eqeltrrd 2530 . . . . . 6  |-  ( ( ( ph  /\  x  e.  ( X (,) Y
) )  /\  u  =  A )  ->  E  e.  CC )
149147, 148exlimddv 1781 . . . . 5  |-  ( (
ph  /\  x  e.  ( X (,) Y ) )  ->  E  e.  CC )
150109, 141, 143, 149fvmptd 5954 . . . 4  |-  ( (
ph  /\  x  e.  ( X (,) Y ) )  ->  ( G `  A )  =  E )
151150oveq1d 6305 . . 3  |-  ( (
ph  /\  x  e.  ( X (,) Y ) )  ->  ( ( G `  A )  x.  B )  =  ( E  x.  B ) )
15237, 151ditgeq3d 37841 . 2  |-  ( ph  ->  S__ [ X  ->  Y ] ( ( G `
 A )  x.  B )  _d x  =  S__ [ X  ->  Y ] ( E  x.  B )  _d x )
153108, 152eqtrd 2485 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 371    = wceq 1444   E.wex 1663    e. wcel 1887   A.wral 2737   _Vcvv 3045    i^i cin 3403    C_ wss 3404   ifcif 3881   U.cuni 4198   class class class wbr 4402    |-> cmpt 4461   ran crn 4835    |` cres 4836   -->wf 5578   ` cfv 5582  (class class class)co 6290   CCcc 9537   RRcr 9538    x. cmul 9544   +oocpnf 9672   -oocmnf 9673   RR*cxr 9674    < clt 9675    <_ cle 9676   (,)cioo 11635   [,]cicc 11638   ↾t crest 15319   TopOpenctopn 15320   topGenctg 15336  ℂfldccnfld 18970   Topctop 19917  TopOnctopon 19918    Cn ccn 20240   -cn->ccncf 21908   L^1cibl 22575   S__cdit 22801    _D cdv 22818
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-inf2 8146  ax-cc 8865  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-pre-sup 9617  ax-addf 9618  ax-mulf 9619
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-fal 1450  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-iin 4281  df-disj 4374  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-se 4794  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-isom 5591  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-of 6531  df-ofr 6532  df-om 6693  df-1st 6793  df-2nd 6794  df-supp 6915  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-2o 7183  df-oadd 7186  df-omul 7187  df-er 7363  df-map 7474  df-pm 7475  df-ixp 7523  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-fsupp 7884  df-fi 7925  df-sup 7956  df-inf 7957  df-oi 8025  df-card 8373  df-acn 8376  df-cda 8598  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-9 10675  df-10 10676  df-n0 10870  df-z 10938  df-dec 11052  df-uz 11160  df-q 11265  df-rp 11303  df-xneg 11409  df-xadd 11410  df-xmul 11411  df-ioo 11639  df-ioc 11640  df-ico 11641  df-icc 11642  df-fz 11785  df-fzo 11916  df-fl 12028  df-mod 12097  df-seq 12214  df-exp 12273  df-hash 12516  df-cj 13162  df-re 13163  df-im 13164  df-sqrt 13298  df-abs 13299  df-limsup 13526  df-clim 13552  df-rlim 13553  df-sum 13753  df-struct 15123  df-ndx 15124  df-slot 15125  df-base 15126  df-sets 15127  df-ress 15128  df-plusg 15203  df-mulr 15204  df-starv 15205  df-sca 15206  df-vsca 15207  df-ip 15208  df-tset 15209  df-ple 15210  df-ds 15212  df-unif 15213  df-hom 15214  df-cco 15215  df-rest 15321  df-topn 15322  df-0g 15340  df-gsum 15341  df-topgen 15342  df-pt 15343  df-prds 15346  df-xrs 15400  df-qtop 15406  df-imas 15407  df-xps 15410  df-mre 15492  df-mrc 15493  df-acs 15495  df-mgm 16488  df-sgrp 16527  df-mnd 16537  df-submnd 16583  df-mulg 16676  df-cntz 16971  df-cmn 17432  df-psmet 18962  df-xmet 18963  df-met 18964  df-bl 18965  df-mopn 18966  df-fbas 18967  df-fg 18968  df-cnfld 18971  df-top 19921  df-bases 19922  df-topon 19923  df-topsp 19924  df-cld 20034  df-ntr 20035  df-cls 20036  df-nei 20114  df-lp 20152  df-perf 20153  df-cn 20243  df-cnp 20244  df-haus 20331  df-cmp 20402  df-tx 20577  df-hmeo 20770  df-fil 20861  df-fm 20953  df-flim 20954  df-flf 20955  df-xms 21335  df-ms 21336  df-tms 21337  df-cncf 21910  df-ovol 22416  df-vol 22418  df-mbf 22577  df-itg1 22578  df-itg2 22579  df-ibl 22580  df-itg 22581  df-0p 22628  df-ditg 22802  df-limc 22821  df-dv 22822
This theorem is referenced by:  itgsubsticc  37853
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