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Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  itgsubsticclem Structured version   Unicode version

Theorem itgsubsticclem 31616
Description: lemma for itgsubsticc 31617 (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 5872 . . . 4  |-  ( u  =  w  ->  ( G `  u )  =  ( G `  w ) )
2 nfcv 2629 . . . 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 4542 . . . . . 6  |-  F/_ u
( u  e.  RR  |->  if ( u  e.  ( K [,] L ) ,  ( F `  u ) ,  if ( u  <  K , 
( F `  K
) ,  ( F `
 L ) ) ) )
53, 4nfcxfr 2627 . . . . 5  |-  F/_ u G
6 nfcv 2629 . . . . 5  |-  F/_ u w
75, 6nffv 5879 . . . 4  |-  F/_ u
( G `  w
)
81, 2, 7cbvditg 22126 . . 3  |-  S__ [ K  ->  L ] ( G `  u )  _d u  =  S__
[ K  ->  L ] ( G `  w )  _d w
9 itgsubsticclem.11 . . . 4  |-  ( ph  ->  K  <_  L )
10 ioossicc 11622 . . . . . . . . . 10  |-  ( K (,) L )  C_  ( K [,] L )
1110sseli 3505 . . . . . . . . 9  |-  ( u  e.  ( K (,) L )  ->  u  e.  ( K [,] L
) )
1211adantl 466 . . . . . . . 8  |-  ( (
ph  /\  u  e.  ( K (,) L ) )  ->  u  e.  ( K [,] L ) )
13 itgsubsticclem.9 . . . . . . . . . . . 12  |-  ( ph  ->  K  e.  RR )
14 itgsubsticclem.10 . . . . . . . . . . . 12  |-  ( ph  ->  L  e.  RR )
1513, 14jca 532 . . . . . . . . . . 11  |-  ( ph  ->  ( K  e.  RR  /\  L  e.  RR ) )
16 iccssre 11618 . . . . . . . . . . 11  |-  ( ( K  e.  RR  /\  L  e.  RR )  ->  ( K [,] L
)  C_  RR )
1715, 16syl 16 . . . . . . . . . 10  |-  ( ph  ->  ( K [,] L
)  C_  RR )
1817adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  u  e.  ( K (,) L ) )  ->  ( K [,] L )  C_  RR )
1918sseld 3508 . . . . . . . 8  |-  ( (
ph  /\  u  e.  ( K (,) L ) )  ->  ( u  e.  ( K [,] L
)  ->  u  e.  RR ) )
2012, 19mpd 15 . . . . . . 7  |-  ( (
ph  /\  u  e.  ( K (,) L ) )  ->  u  e.  RR )
21 iftrue 3951 . . . . . . . . 9  |-  ( u  e.  ( K [,] L )  ->  if ( u  e.  ( K [,] L ) ,  ( F `  u
) ,  if ( u  <  K , 
( F `  K
) ,  ( F `
 L ) ) )  =  ( F `
 u ) )
2212, 21syl 16 . . . . . . . 8  |-  ( (
ph  /\  u  e.  ( K (,) L ) )  ->  if (
u  e.  ( K [,] L ) ,  ( F `  u
) ,  if ( u  <  K , 
( F `  K
) ,  ( F `
 L ) ) )  =  ( F `
 u ) )
23 itgsubsticclem.8 . . . . . . . . . . . . . . 15  |-  ( ph  ->  F  e.  ( ( K [,] L )
-cn-> CC ) )
24 cncff 21265 . . . . . . . . . . . . . . 15  |-  ( F  e.  ( ( K [,] L ) -cn-> CC )  ->  F :
( K [,] L
) --> CC )
2523, 24syl 16 . . . . . . . . . . . . . 14  |-  ( ph  ->  F : ( K [,] L ) --> CC )
26 itgsubsticclem.1 . . . . . . . . . . . . . . . 16  |-  F  =  ( u  e.  ( K [,] L ) 
|->  C )
2726a1i 11 . . . . . . . . . . . . . . 15  |-  ( ph  ->  F  =  ( u  e.  ( K [,] L )  |->  C ) )
2827feq1d 5723 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( F : ( K [,] L ) --> CC  <->  ( u  e.  ( K [,] L
)  |->  C ) : ( K [,] L
) --> CC ) )
2925, 28mpbid 210 . . . . . . . . . . . . 13  |-  ( ph  ->  ( u  e.  ( K [,] L ) 
|->  C ) : ( K [,] L ) --> CC )
3029mptex2 31346 . . . . . . . . . . . 12  |-  ( (
ph  /\  u  e.  ( K [,] L ) )  ->  C  e.  CC )
3112, 30syldan 470 . . . . . . . . . . 11  |-  ( (
ph  /\  u  e.  ( K (,) L ) )  ->  C  e.  CC )
3212, 31jca 532 . . . . . . . . . 10  |-  ( (
ph  /\  u  e.  ( K (,) L ) )  ->  ( u  e.  ( K [,] L
)  /\  C  e.  CC ) )
3326fvmpt2 5964 . . . . . . . . . 10  |-  ( ( u  e.  ( K [,] L )  /\  C  e.  CC )  ->  ( F `  u
)  =  C )
3432, 33syl 16 . . . . . . . . 9  |-  ( (
ph  /\  u  e.  ( K (,) L ) )  ->  ( F `  u )  =  C )
3534, 31eqeltrd 2555 . . . . . . . 8  |-  ( (
ph  /\  u  e.  ( K (,) L ) )  ->  ( F `  u )  e.  CC )
3622, 35eqeltrd 2555 . . . . . . 7  |-  ( (
ph  /\  u  e.  ( K (,) L ) )  ->  if (
u  e.  ( K [,] L ) ,  ( F `  u
) ,  if ( u  <  K , 
( F `  K
) ,  ( F `
 L ) ) )  e.  CC )
3720, 36jca 532 . . . . . 6  |-  ( (
ph  /\  u  e.  ( K (,) L ) )  ->  ( u  e.  RR  /\  if ( u  e.  ( K [,] L ) ,  ( F `  u
) ,  if ( u  <  K , 
( F `  K
) ,  ( F `
 L ) ) )  e.  CC ) )
383fvmpt2 5964 . . . . . 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 ) ) ) )
3937, 38syl 16 . . . . 5  |-  ( (
ph  /\  u  e.  ( K (,) L ) )  ->  ( G `  u )  =  if ( u  e.  ( K [,] L ) ,  ( F `  u ) ,  if ( u  <  K , 
( F `  K
) ,  ( F `
 L ) ) ) )
4039, 22, 343eqtrd 2512 . . . 4  |-  ( (
ph  /\  u  e.  ( K (,) L ) )  ->  ( G `  u )  =  C )
419, 40ditgeq3d 31605 . . 3  |-  ( ph  ->  S__ [ K  ->  L ] ( G `  u )  _d u  =  S__ [ K  ->  L ] C  _d u )
42 itgsubsticclem.3 . . . 4  |-  ( ph  ->  X  e.  RR )
43 itgsubsticclem.4 . . . 4  |-  ( ph  ->  Y  e.  RR )
44 itgsubsticclem.5 . . . 4  |-  ( ph  ->  X  <_  Y )
45 mnfxr 11335 . . . . 5  |- -oo  e.  RR*
4645a1i 11 . . . 4  |-  ( ph  -> -oo  e.  RR* )
47 pnfxr 11333 . . . . 5  |- +oo  e.  RR*
4847a1i 11 . . . 4  |-  ( ph  -> +oo  e.  RR* )
49 itgsubsticclem.6 . . . . 5  |-  ( ph  ->  ( x  e.  ( X [,] Y ) 
|->  A )  e.  ( ( X [,] Y
) -cn-> ( K [,] L ) ) )
50 ioomax 11611 . . . . . . . . . . 11  |-  ( -oo (,) +oo )  =  RR
5150eqcomi 2480 . . . . . . . . . 10  |-  RR  =  ( -oo (,) +oo )
5251a1i 11 . . . . . . . . 9  |-  ( ph  ->  RR  =  ( -oo (,) +oo ) )
5317, 52sseqtrd 3545 . . . . . . . 8  |-  ( ph  ->  ( K [,] L
)  C_  ( -oo (,) +oo ) )
54 ax-resscn 9561 . . . . . . . . . 10  |-  RR  C_  CC
5554a1i 11 . . . . . . . . 9  |-  ( ph  ->  RR  C_  CC )
5652, 55eqsstr3d 3544 . . . . . . . 8  |-  ( ph  ->  ( -oo (,) +oo )  C_  CC )
5753, 56jca 532 . . . . . . 7  |-  ( ph  ->  ( ( K [,] L )  C_  ( -oo (,) +oo )  /\  ( -oo (,) +oo )  C_  CC ) )
58 cncfss 21271 . . . . . . 7  |-  ( ( ( K [,] L
)  C_  ( -oo (,) +oo )  /\  ( -oo (,) +oo )  C_  CC )  ->  ( ( X [,] Y )
-cn-> ( K [,] L
) )  C_  (
( X [,] Y
) -cn-> ( -oo (,) +oo ) ) )
5957, 58syl 16 . . . . . 6  |-  ( ph  ->  ( ( X [,] Y ) -cn-> ( K [,] L ) ) 
C_  ( ( X [,] Y ) -cn-> ( -oo (,) +oo )
) )
6059sseld 3508 . . . . 5  |-  ( ph  ->  ( ( x  e.  ( X [,] Y
)  |->  A )  e.  ( ( X [,] Y ) -cn-> ( K [,] L ) )  ->  ( x  e.  ( X [,] Y
)  |->  A )  e.  ( ( X [,] Y ) -cn-> ( -oo (,) +oo ) ) ) )
6149, 60mpd 15 . . . 4  |-  ( ph  ->  ( x  e.  ( X [,] Y ) 
|->  A )  e.  ( ( X [,] Y
) -cn-> ( -oo (,) +oo ) ) )
62 itgsubsticclem.7 . . . 4  |-  ( ph  ->  ( x  e.  ( X (,) Y ) 
|->  B )  e.  ( ( ( X (,) Y ) -cn-> CC )  i^i  L^1 ) )
63 nfmpt1 4542 . . . . . . . . . . . . 13  |-  F/_ u
( u  e.  ( K [,] L ) 
|->  C )
6426, 63nfcxfr 2627 . . . . . . . . . . . 12  |-  F/_ u F
65 eqid 2467 . . . . . . . . . . . 12  |-  ( topGen ` 
ran  (,) )  =  (
topGen `  ran  (,) )
66 eqid 2467 . . . . . . . . . . . 12  |-  U. ( TopOpen
` fld
)  =  U. ( TopOpen
` fld
)
67 eqid 2467 . . . . . . . . . . . . . . 15  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
6867cnfldtopon 21158 . . . . . . . . . . . . . 14  |-  ( TopOpen ` fld )  e.  (TopOn `  CC )
6968topontopi 19301 . . . . . . . . . . . . 13  |-  ( TopOpen ` fld )  e.  Top
7069a1i 11 . . . . . . . . . . . 12  |-  ( ph  ->  ( TopOpen ` fld )  e.  Top )
7117, 55sstrd 3519 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( K [,] L
)  C_  CC )
72 ssid 3528 . . . . . . . . . . . . . . . . 17  |-  CC  C_  CC
7372a1i 11 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  CC  C_  CC )
7471, 73jca 532 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( ( K [,] L )  C_  CC  /\  CC  C_  CC )
)
75 eqid 2467 . . . . . . . . . . . . . . . 16  |-  ( (
TopOpen ` fld )t  ( K [,] L
) )  =  ( ( TopOpen ` fld )t  ( K [,] L ) )
76 toponuni 19297 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
TopOpen ` fld )  e.  (TopOn `  CC )  ->  CC  =  U. ( TopOpen ` fld ) )
7768, 76ax-mp 5 . . . . . . . . . . . . . . . . . . 19  |-  CC  =  U. ( TopOpen ` fld )
7877restid 14706 . . . . . . . . . . . . . . . . . 18  |-  ( (
TopOpen ` fld )  e.  Top  ->  ( ( TopOpen ` fld )t  CC )  =  (
TopOpen ` fld ) )
7969, 78ax-mp 5 . . . . . . . . . . . . . . . . 17  |-  ( (
TopOpen ` fld )t  CC )  =  (
TopOpen ` fld )
8079eqcomi 2480 . . . . . . . . . . . . . . . 16  |-  ( TopOpen ` fld )  =  ( ( TopOpen ` fld )t  CC )
8167, 75, 80cncfcn 21281 . . . . . . . . . . . . . . 15  |-  ( ( ( K [,] L
)  C_  CC  /\  CC  C_  CC )  ->  (
( K [,] L
) -cn-> CC )  =  ( ( ( TopOpen ` fld )t  ( K [,] L ) )  Cn  ( TopOpen ` fld ) ) )
8274, 81syl 16 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( ( K [,] L ) -cn-> CC )  =  ( ( (
TopOpen ` fld )t  ( K [,] L
) )  Cn  ( TopOpen
` fld
) ) )
83 reex 9595 . . . . . . . . . . . . . . . . . . . 20  |-  RR  e.  _V
8483a1i 11 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  RR  e.  _V )
8570, 17, 843jca 1176 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  ( ( TopOpen ` fld )  e.  Top  /\  ( K [,] L
)  C_  RR  /\  RR  e.  _V ) )
86 restabs 19534 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( TopOpen ` fld )  e.  Top  /\  ( K [,] L
)  C_  RR  /\  RR  e.  _V )  ->  (
( ( TopOpen ` fld )t  RR )t  ( K [,] L ) )  =  ( ( TopOpen ` fld )t  ( K [,] L ) ) )
8785, 86syl 16 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( ( ( TopOpen ` fld )t  RR )t  ( K [,] L ) )  =  ( (
TopOpen ` fld )t  ( K [,] L
) ) )
8887eqcomd 2475 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( ( TopOpen ` fld )t  ( K [,] L ) )  =  ( ( ( TopOpen ` fld )t  RR )t  ( K [,] L ) ) )
8967tgioo2 21176 . . . . . . . . . . . . . . . . . . 19  |-  ( topGen ` 
ran  (,) )  =  ( ( TopOpen ` fld )t  RR )
9089eqcomi 2480 . . . . . . . . . . . . . . . . . 18  |-  ( (
TopOpen ` fld )t  RR )  =  (
topGen `  ran  (,) )
9190a1i 11 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( ( TopOpen ` fld )t  RR )  =  (
topGen `  ran  (,) )
)
9291oveq1d 6310 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( ( ( TopOpen ` fld )t  RR )t  ( K [,] L ) )  =  ( (
topGen `  ran  (,) )t  ( K [,] L ) ) )
9388, 92eqtrd 2508 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( ( TopOpen ` fld )t  ( K [,] L ) )  =  ( ( topGen `  ran  (,) )t  ( K [,] L
) ) )
9493oveq1d 6310 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( ( ( TopOpen ` fld )t  ( K [,] L ) )  Cn  ( TopOpen ` fld ) )  =  ( ( ( topGen `  ran  (,) )t  ( K [,] L
) )  Cn  ( TopOpen
` fld
) ) )
9582, 94eqtrd 2508 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( K [,] L ) -cn-> CC )  =  ( ( (
topGen `  ran  (,) )t  ( K [,] L ) )  Cn  ( TopOpen ` fld ) ) )
9623, 95eleqtrd 2557 . . . . . . . . . . . 12  |-  ( ph  ->  F  e.  ( ( ( topGen `  ran  (,) )t  ( K [,] L ) )  Cn  ( TopOpen ` fld ) ) )
9764, 65, 66, 3, 13, 14, 9, 70, 96icccncfext 31549 . . . . . . . . . . 11  |-  ( ph  ->  ( G  e.  ( ( topGen `  ran  (,) )  Cn  ( ( TopOpen ` fld )t  ran  F ) )  /\  ( G  |`  ( K [,] L ) )  =  F ) )
9897simpld 459 . . . . . . . . . 10  |-  ( ph  ->  G  e.  ( (
topGen `  ran  (,) )  Cn  ( ( TopOpen ` fld )t  ran  F ) ) )
99 uniretop 21137 . . . . . . . . . . 11  |-  RR  =  U. ( topGen `  ran  (,) )
100 eqid 2467 . . . . . . . . . . 11  |-  U. (
( TopOpen ` fld )t  ran  F )  = 
U. ( ( TopOpen ` fld )t  ran  F )
10199, 100cnf 19615 . . . . . . . . . 10  |-  ( G  e.  ( ( topGen ` 
ran  (,) )  Cn  (
( TopOpen ` fld )t  ran  F ) )  ->  G : RR --> U. ( ( TopOpen ` fld )t  ran  F ) )
10298, 101syl 16 . . . . . . . . 9  |-  ( ph  ->  G : RR --> U. (
( TopOpen ` fld )t  ran  F ) )
10352feq2d 5724 . . . . . . . . 9  |-  ( ph  ->  ( G : RR --> U. ( ( TopOpen ` fld )t  ran  F )  <->  G :
( -oo (,) +oo ) --> U. ( ( TopOpen ` fld )t  ran  F ) ) )
104102, 103mpbid 210 . . . . . . . 8  |-  ( ph  ->  G : ( -oo (,) +oo ) --> U. (
( TopOpen ` fld )t  ran  F ) )
105 ffn 5737 . . . . . . . 8  |-  ( G : ( -oo (,) +oo ) --> U. ( ( TopOpen ` fld )t  ran  F )  ->  G  Fn  ( -oo (,) +oo )
)
106104, 105syl 16 . . . . . . 7  |-  ( ph  ->  G  Fn  ( -oo (,) +oo ) )
107 dffn5 5919 . . . . . . 7  |-  ( G  Fn  ( -oo (,) +oo )  <->  G  =  (
w  e.  ( -oo (,) +oo )  |->  ( G `
 w ) ) )
108106, 107sylib 196 . . . . . 6  |-  ( ph  ->  G  =  ( w  e.  ( -oo (,) +oo )  |->  ( G `  w ) ) )
109108eqcomd 2475 . . . . 5  |-  ( ph  ->  ( w  e.  ( -oo (,) +oo )  |->  ( G `  w
) )  =  G )
110 frn 5743 . . . . . . . . . 10  |-  ( F : ( K [,] L ) --> CC  ->  ran 
F  C_  CC )
11125, 110syl 16 . . . . . . . . 9  |-  ( ph  ->  ran  F  C_  CC )
11251oveq2i 6306 . . . . . . . . . . 11  |-  ( (
TopOpen ` fld )t  RR )  =  ( ( TopOpen ` fld )t  ( -oo (,) +oo ) )
11389, 112eqtri 2496 . . . . . . . . . 10  |-  ( topGen ` 
ran  (,) )  =  ( ( TopOpen ` fld )t  ( -oo (,) +oo ) )
114 eqid 2467 . . . . . . . . . 10  |-  ( (
TopOpen ` fld )t 
ran  F )  =  ( ( TopOpen ` fld )t  ran  F )
11567, 113, 114cncfcn 21281 . . . . . . . . 9  |-  ( ( ( -oo (,) +oo )  C_  CC  /\  ran  F 
C_  CC )  -> 
( ( -oo (,) +oo ) -cn-> ran  F )  =  ( ( topGen ` 
ran  (,) )  Cn  (
( TopOpen ` fld )t  ran  F ) ) )
11656, 111, 115syl2anc 661 . . . . . . . 8  |-  ( ph  ->  ( ( -oo (,) +oo ) -cn-> ran  F )  =  ( ( topGen ` 
ran  (,) )  Cn  (
( TopOpen ` fld )t  ran  F ) ) )
117116eqcomd 2475 . . . . . . 7  |-  ( ph  ->  ( ( topGen `  ran  (,) )  Cn  ( (
TopOpen ` fld )t 
ran  F ) )  =  ( ( -oo (,) +oo ) -cn-> ran  F
) )
11898, 117eleqtrd 2557 . . . . . 6  |-  ( ph  ->  G  e.  ( ( -oo (,) +oo ) -cn->
ran  F ) )
119111, 73jca 532 . . . . . . . 8  |-  ( ph  ->  ( ran  F  C_  CC  /\  CC  C_  CC ) )
120 cncfss 21271 . . . . . . . 8  |-  ( ( ran  F  C_  CC  /\  CC  C_  CC )  ->  ( ( -oo (,) +oo ) -cn-> ran  F )  C_  ( ( -oo (,) +oo ) -cn-> CC ) )
121119, 120syl 16 . . . . . . 7  |-  ( ph  ->  ( ( -oo (,) +oo ) -cn-> ran  F )  C_  ( ( -oo (,) +oo ) -cn-> CC ) )
122121sseld 3508 . . . . . 6  |-  ( ph  ->  ( G  e.  ( ( -oo (,) +oo ) -cn-> ran  F )  ->  G  e.  ( ( -oo (,) +oo ) -cn->
CC ) ) )
123118, 122mpd 15 . . . . 5  |-  ( ph  ->  G  e.  ( ( -oo (,) +oo ) -cn->
CC ) )
124109, 123eqeltrd 2555 . . . 4  |-  ( ph  ->  ( w  e.  ( -oo (,) +oo )  |->  ( G `  w
) )  e.  ( ( -oo (,) +oo ) -cn-> CC ) )
125 itgsubsticclem.12 . . . 4  |-  ( ph  ->  ( RR  _D  (
x  e.  ( X [,] Y )  |->  A ) )  =  ( x  e.  ( X (,) Y )  |->  B ) )
126 fveq2 5872 . . . 4  |-  ( w  =  A  ->  ( G `  w )  =  ( G `  A ) )
127 itgsubsticclem.14 . . . 4  |-  ( x  =  X  ->  A  =  K )
128 itgsubsticclem.15 . . . 4  |-  ( x  =  Y  ->  A  =  L )
12942, 43, 44, 46, 48, 61, 62, 124, 125, 126, 127, 128itgsubst 22318 . . 3  |-  ( ph  ->  S__ [ K  ->  L ] ( G `  w )  _d w  =  S__ [ X  ->  Y ] ( ( G `  A )  x.  B )  _d x )
1308, 41, 1293eqtr3a 2532 . 2  |-  ( ph  ->  S__ [ K  ->  L ] C  _d u  =  S__ [ X  ->  Y ] ( ( G `  A )  x.  B )  _d x )
1313a1i 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 ) ) ) ) )
132 simpr 461 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  ( X (,) Y
) )  /\  u  =  A )  ->  u  =  A )
13368a1i 11 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( TopOpen ` fld )  e.  (TopOn `  CC ) )
13442, 43jca 532 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  ( X  e.  RR  /\  Y  e.  RR ) )
135 iccssre 11618 . . . . . . . . . . . . . . . . . 18  |-  ( ( X  e.  RR  /\  Y  e.  RR )  ->  ( X [,] Y
)  C_  RR )
136134, 135syl 16 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( X [,] Y
)  C_  RR )
137136, 55sstrd 3519 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( X [,] Y
)  C_  CC )
138133, 137jca 532 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( ( TopOpen ` fld )  e.  (TopOn `  CC )  /\  ( X [,] Y )  C_  CC ) )
139 resttopon 19530 . . . . . . . . . . . . . . 15  |-  ( ( ( TopOpen ` fld )  e.  (TopOn `  CC )  /\  ( X [,] Y )  C_  CC )  ->  ( (
TopOpen ` fld )t  ( X [,] Y
) )  e.  (TopOn `  ( X [,] Y
) ) )
140138, 139syl 16 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( ( TopOpen ` fld )t  ( X [,] Y ) )  e.  (TopOn `  ( X [,] Y ) ) )
141133, 71jca 532 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( ( TopOpen ` fld )  e.  (TopOn `  CC )  /\  ( K [,] L )  C_  CC ) )
142 resttopon 19530 . . . . . . . . . . . . . . 15  |-  ( ( ( TopOpen ` fld )  e.  (TopOn `  CC )  /\  ( K [,] L )  C_  CC )  ->  ( (
TopOpen ` fld )t  ( K [,] L
) )  e.  (TopOn `  ( K [,] L
) ) )
143141, 142syl 16 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( ( TopOpen ` fld )t  ( K [,] L ) )  e.  (TopOn `  ( K [,] L ) ) )
144137, 71jca 532 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( ( X [,] Y )  C_  CC  /\  ( K [,] L
)  C_  CC )
)
145 eqid 2467 . . . . . . . . . . . . . . . . 17  |-  ( (
TopOpen ` fld )t  ( X [,] Y
) )  =  ( ( TopOpen ` fld )t  ( X [,] Y ) )
14667, 145, 75cncfcn 21281 . . . . . . . . . . . . . . . 16  |-  ( ( ( X [,] Y
)  C_  CC  /\  ( K [,] L )  C_  CC )  ->  ( ( X [,] Y )
-cn-> ( K [,] L
) )  =  ( ( ( TopOpen ` fld )t  ( X [,] Y ) )  Cn  ( ( TopOpen ` fld )t  ( K [,] L ) ) ) )
147144, 146syl 16 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( ( X [,] Y ) -cn-> ( K [,] L ) )  =  ( ( (
TopOpen ` fld )t  ( X [,] Y
) )  Cn  (
( TopOpen ` fld )t  ( K [,] L ) ) ) )
14849, 147eleqtrd 2557 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( x  e.  ( X [,] Y ) 
|->  A )  e.  ( ( ( TopOpen ` fld )t  ( X [,] Y ) )  Cn  ( ( TopOpen ` fld )t  ( K [,] L ) ) ) )
149140, 143, 1483jca 1176 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( ( 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 ) ) ) ) )
150 cnf2 19618 . . . . . . . . . . . . 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 ) )
151149, 150syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  ( x  e.  ( X [,] Y ) 
|->  A ) : ( X [,] Y ) --> ( K [,] L
) )
152151adantr 465 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( X (,) Y ) )  ->  ( x  e.  ( X [,] Y
)  |->  A ) : ( X [,] Y
) --> ( K [,] L ) )
153 eqid 2467 . . . . . . . . . . . 12  |-  ( x  e.  ( X [,] Y )  |->  A )  =  ( x  e.  ( X [,] Y
)  |->  A )
154153fmpt 6053 . . . . . . . . . . 11  |-  ( A. x  e.  ( X [,] Y ) A  e.  ( K [,] L
)  <->  ( x  e.  ( X [,] Y
)  |->  A ) : ( X [,] Y
) --> ( K [,] L ) )
155152, 154sylibr 212 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( X (,) Y ) )  ->  A. x  e.  ( X [,] Y
) A  e.  ( K [,] L ) )
156 ioossicc 11622 . . . . . . . . . . . 12  |-  ( X (,) Y )  C_  ( X [,] Y )
157156sseli 3505 . . . . . . . . . . 11  |-  ( x  e.  ( X (,) Y )  ->  x  e.  ( X [,] Y
) )
158157adantl 466 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( X (,) Y ) )  ->  x  e.  ( X [,] Y ) )
159 rsp 2833 . . . . . . . . . 10  |-  ( A. x  e.  ( X [,] Y ) A  e.  ( K [,] L
)  ->  ( x  e.  ( X [,] Y
)  ->  A  e.  ( K [,] L ) ) )
160155, 158, 159sylc 60 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( X (,) Y ) )  ->  A  e.  ( K [,] L ) )
161160adantr 465 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  ( X (,) Y
) )  /\  u  =  A )  ->  A  e.  ( K [,] L
) )
162132, 161eqeltrd 2555 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  ( X (,) Y
) )  /\  u  =  A )  ->  u  e.  ( K [,] L
) )
163162, 21syl 16 . . . . . 6  |-  ( ( ( ph  /\  x  e.  ( X (,) Y
) )  /\  u  =  A )  ->  if ( u  e.  ( K [,] L ) ,  ( F `  u
) ,  if ( u  <  K , 
( F `  K
) ,  ( F `
 L ) ) )  =  ( F `
 u ) )
164 simpll 753 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  ( X (,) Y
) )  /\  u  =  A )  ->  ph )
165164, 162jca 532 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  ( X (,) Y
) )  /\  u  =  A )  ->  ( ph  /\  u  e.  ( K [,] L ) ) )
166165, 30syl 16 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  ( X (,) Y
) )  /\  u  =  A )  ->  C  e.  CC )
167162, 166jca 532 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  ( X (,) Y
) )  /\  u  =  A )  ->  (
u  e.  ( K [,] L )  /\  C  e.  CC )
)
168167, 33syl 16 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  ( X (,) Y
) )  /\  u  =  A )  ->  ( F `  u )  =  C )
169 itgsubsticclem.13 . . . . . . . 8  |-  ( u  =  A  ->  C  =  E )
170132, 169syl 16 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  ( X (,) Y
) )  /\  u  =  A )  ->  C  =  E )
171168, 170eqtrd 2508 . . . . . 6  |-  ( ( ( ph  /\  x  e.  ( X (,) Y
) )  /\  u  =  A )  ->  ( F `  u )  =  E )
172163, 171eqtrd 2508 . . . . 5  |-  ( ( ( ph  /\  x  e.  ( X (,) Y
) )  /\  u  =  A )  ->  if ( u  e.  ( K [,] L ) ,  ( F `  u
) ,  if ( u  <  K , 
( F `  K
) ,  ( F `
 L ) ) )  =  E )
17317adantr 465 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( X (,) Y ) )  ->  ( K [,] L )  C_  RR )
174173sseld 3508 . . . . . 6  |-  ( (
ph  /\  x  e.  ( X (,) Y ) )  ->  ( A  e.  ( K [,] L
)  ->  A  e.  RR ) )
175160, 174mpd 15 . . . . 5  |-  ( (
ph  /\  x  e.  ( X (,) Y ) )  ->  A  e.  RR )
176 elex 3127 . . . . . . . 8  |-  ( A  e.  ( K [,] L )  ->  A  e.  _V )
177160, 176syl 16 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( X (,) Y ) )  ->  A  e.  _V )
178 isset 3122 . . . . . . 7  |-  ( A  e.  _V  <->  E. u  u  =  A )
179177, 178sylib 196 . . . . . 6  |-  ( (
ph  /\  x  e.  ( X (,) Y ) )  ->  E. u  u  =  A )
180171eqcomd 2475 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  ( X (,) Y
) )  /\  u  =  A )  ->  E  =  ( F `  u ) )
181168, 166eqeltrd 2555 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  ( X (,) Y
) )  /\  u  =  A )  ->  ( F `  u )  e.  CC )
182180, 181eqeltrd 2555 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  ( X (,) Y
) )  /\  u  =  A )  ->  E  e.  CC )
183182ex 434 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( X (,) Y ) )  ->  ( u  =  A  ->  E  e.  CC ) )
184183exlimdv 1700 . . . . . 6  |-  ( (
ph  /\  x  e.  ( X (,) Y ) )  ->  ( E. u  u  =  A  ->  E  e.  CC ) )
185179, 184mpd 15 . . . . 5  |-  ( (
ph  /\  x  e.  ( X (,) Y ) )  ->  E  e.  CC )
186131, 172, 175, 185fvmptd 5962 . . . 4  |-  ( (
ph  /\  x  e.  ( X (,) Y ) )  ->  ( G `  A )  =  E )
187186oveq1d 6310 . . 3  |-  ( (
ph  /\  x  e.  ( X (,) Y ) )  ->  ( ( G `  A )  x.  B )  =  ( E  x.  B ) )
18844, 187ditgeq3d 31605 . 2  |-  ( ph  ->  S__ [ X  ->  Y ] ( ( G `
 A )  x.  B )  _d x  =  S__ [ X  ->  Y ] ( E  x.  B )  _d x )
189130, 188eqtrd 2508 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 369    /\ w3a 973    = wceq 1379   E.wex 1596    e. wcel 1767   A.wral 2817   _Vcvv 3118    i^i cin 3480    C_ wss 3481   ifcif 3945   U.cuni 4251   class class class wbr 4453    |-> cmpt 4511   ran crn 5006    |` cres 5007    Fn wfn 5589   -->wf 5590   ` cfv 5594  (class class class)co 6295   CCcc 9502   RRcr 9503    x. cmul 9509   +oocpnf 9637   -oocmnf 9638   RR*cxr 9639    < clt 9640    <_ cle 9641   (,)cioo 11541   [,]cicc 11544   ↾t crest 14693   TopOpenctopn 14694   topGenctg 14710  ℂfldccnfld 18290   Topctop 19263  TopOnctopon 19264    Cn ccn 19593   -cn->ccncf 21248   L^1cibl 21894   S__cdit 22118    _D cdv 22135
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-inf2 8070  ax-cc 8827  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-pre-sup 9582  ax-addf 9583  ax-mulf 9584
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-iin 4334  df-disj 4424  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-se 4845  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-of 6535  df-ofr 6536  df-om 6696  df-1st 6795  df-2nd 6796  df-supp 6914  df-recs 7054  df-rdg 7088  df-1o 7142  df-2o 7143  df-oadd 7146  df-omul 7147  df-er 7323  df-map 7434  df-pm 7435  df-ixp 7482  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-fsupp 7842  df-fi 7883  df-sup 7913  df-oi 7947  df-card 8332  df-acn 8335  df-cda 8560  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-div 10219  df-nn 10549  df-2 10606  df-3 10607  df-4 10608  df-5 10609  df-6 10610  df-7 10611  df-8 10612  df-9 10613  df-10 10614  df-n0 10808  df-z 10877  df-dec 10989  df-uz 11095  df-q 11195  df-rp 11233  df-xneg 11330  df-xadd 11331  df-xmul 11332  df-ioo 11545  df-ioc 11546  df-ico 11547  df-icc 11548  df-fz 11685  df-fzo 11805  df-fl 11909  df-mod 11977  df-seq 12088  df-exp 12147  df-hash 12386  df-cj 12912  df-re 12913  df-im 12914  df-sqrt 13048  df-abs 13049  df-limsup 13274  df-clim 13291  df-rlim 13292  df-sum 13489  df-struct 14509  df-ndx 14510  df-slot 14511  df-base 14512  df-sets 14513  df-ress 14514  df-plusg 14585  df-mulr 14586  df-starv 14587  df-sca 14588  df-vsca 14589  df-ip 14590  df-tset 14591  df-ple 14592  df-ds 14594  df-unif 14595  df-hom 14596  df-cco 14597  df-rest 14695  df-topn 14696  df-0g 14714  df-gsum 14715  df-topgen 14716  df-pt 14717  df-prds 14720  df-xrs 14774  df-qtop 14779  df-imas 14780  df-xps 14782  df-mre 14858  df-mrc 14859  df-acs 14861  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-submnd 15840  df-mulg 15932  df-cntz 16227  df-cmn 16673  df-psmet 18281  df-xmet 18282  df-met 18283  df-bl 18284  df-mopn 18285  df-fbas 18286  df-fg 18287  df-cnfld 18291  df-top 19268  df-bases 19270  df-topon 19271  df-topsp 19272  df-cld 19388  df-ntr 19389  df-cls 19390  df-nei 19467  df-lp 19505  df-perf 19506  df-cn 19596  df-cnp 19597  df-haus 19684  df-cmp 19755  df-tx 19931  df-hmeo 20124  df-fil 20215  df-fm 20307  df-flim 20308  df-flf 20309  df-xms 20691  df-ms 20692  df-tms 20693  df-cncf 21250  df-ovol 21744  df-vol 21745  df-mbf 21896  df-itg1 21897  df-itg2 21898  df-ibl 21899  df-itg 21900  df-0p 21945  df-ditg 22119  df-limc 22138  df-dv 22139
This theorem is referenced by:  itgsubsticc  31617
  Copyright terms: Public domain W3C validator