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Theorem itgperiod 31962
Description: The integral of a periodic function, with period  T stays the same if the domain of integration is shifted. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
Hypotheses
Ref Expression
itgperiod.a  |-  ( ph  ->  A  e.  RR )
itgperiod.b  |-  ( ph  ->  B  e.  RR )
itgperiod.aleb  |-  ( ph  ->  A  <_  B )
itgperiod.t  |-  ( ph  ->  T  e.  RR+ )
itgperiod.f  |-  ( ph  ->  F : RR --> CC )
itgperiod.fper  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( F `  ( x  +  T
) )  =  ( F `  x ) )
itgperiod.fcn  |-  ( ph  ->  ( F  |`  ( A [,] B ) )  e.  ( ( A [,] B ) -cn-> CC ) )
Assertion
Ref Expression
itgperiod  |-  ( ph  ->  S. ( ( A  +  T ) [,] ( B  +  T
) ) ( F `
 x )  _d x  =  S. ( A [,] B ) ( F `  x
)  _d x )
Distinct variable groups:    x, A    x, B    x, F    x, T    ph, x

Proof of Theorem itgperiod
Dummy variables  w  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 itgperiod.a . . . . 5  |-  ( ph  ->  A  e.  RR )
2 itgperiod.b . . . . 5  |-  ( ph  ->  B  e.  RR )
3 itgperiod.t . . . . . 6  |-  ( ph  ->  T  e.  RR+ )
43rpred 11281 . . . . 5  |-  ( ph  ->  T  e.  RR )
5 itgperiod.aleb . . . . 5  |-  ( ph  ->  A  <_  B )
61, 2, 4, 5leadd1dd 10187 . . . 4  |-  ( ph  ->  ( A  +  T
)  <_  ( B  +  T ) )
76ditgpos 22386 . . 3  |-  ( ph  ->  S__ [ ( A  +  T )  -> 
( B  +  T
) ] ( F `
 x )  _d x  =  S. ( ( A  +  T
) (,) ( B  +  T ) ) ( F `  x
)  _d x )
81, 4readdcld 9640 . . . 4  |-  ( ph  ->  ( A  +  T
)  e.  RR )
92, 4readdcld 9640 . . . 4  |-  ( ph  ->  ( B  +  T
)  e.  RR )
10 itgperiod.f . . . . . 6  |-  ( ph  ->  F : RR --> CC )
1110adantr 465 . . . . 5  |-  ( (
ph  /\  x  e.  ( ( A  +  T ) [,] ( B  +  T )
) )  ->  F : RR --> CC )
128adantr 465 . . . . . 6  |-  ( (
ph  /\  x  e.  ( ( A  +  T ) [,] ( B  +  T )
) )  ->  ( A  +  T )  e.  RR )
139adantr 465 . . . . . 6  |-  ( (
ph  /\  x  e.  ( ( A  +  T ) [,] ( B  +  T )
) )  ->  ( B  +  T )  e.  RR )
14 simpr 461 . . . . . 6  |-  ( (
ph  /\  x  e.  ( ( A  +  T ) [,] ( B  +  T )
) )  ->  x  e.  ( ( A  +  T ) [,] ( B  +  T )
) )
15 eliccre 31722 . . . . . 6  |-  ( ( ( A  +  T
)  e.  RR  /\  ( B  +  T
)  e.  RR  /\  x  e.  ( ( A  +  T ) [,] ( B  +  T
) ) )  ->  x  e.  RR )
1612, 13, 14, 15syl3anc 1228 . . . . 5  |-  ( (
ph  /\  x  e.  ( ( A  +  T ) [,] ( B  +  T )
) )  ->  x  e.  RR )
1711, 16ffvelrnd 6033 . . . 4  |-  ( (
ph  /\  x  e.  ( ( A  +  T ) [,] ( B  +  T )
) )  ->  ( F `  x )  e.  CC )
188, 9, 17itgioo 22348 . . 3  |-  ( ph  ->  S. ( ( A  +  T ) (,) ( B  +  T
) ) ( F `
 x )  _d x  =  S. ( ( A  +  T
) [,] ( B  +  T ) ) ( F `  x
)  _d x )
197, 18eqtr2d 2499 . 2  |-  ( ph  ->  S. ( ( A  +  T ) [,] ( B  +  T
) ) ( F `
 x )  _d x  =  S__ [
( A  +  T
)  ->  ( B  +  T ) ] ( F `  x )  _d x )
20 eqid 2457 . . . 4  |-  ( y  e.  CC  |->  ( y  +  T ) )  =  ( y  e.  CC  |->  ( y  +  T ) )
214recnd 9639 . . . . 5  |-  ( ph  ->  T  e.  CC )
2220addccncf 21546 . . . . 5  |-  ( T  e.  CC  ->  (
y  e.  CC  |->  ( y  +  T ) )  e.  ( CC
-cn-> CC ) )
2321, 22syl 16 . . . 4  |-  ( ph  ->  ( y  e.  CC  |->  ( y  +  T
) )  e.  ( CC -cn-> CC ) )
241, 2iccssred 31721 . . . . 5  |-  ( ph  ->  ( A [,] B
)  C_  RR )
25 ax-resscn 9566 . . . . 5  |-  RR  C_  CC
2624, 25syl6ss 3511 . . . 4  |-  ( ph  ->  ( A [,] B
)  C_  CC )
278, 9iccssred 31721 . . . . 5  |-  ( ph  ->  ( ( A  +  T ) [,] ( B  +  T )
)  C_  RR )
2827, 25syl6ss 3511 . . . 4  |-  ( ph  ->  ( ( A  +  T ) [,] ( B  +  T )
)  C_  CC )
298adantr 465 . . . . 5  |-  ( (
ph  /\  y  e.  ( A [,] B ) )  ->  ( A  +  T )  e.  RR )
309adantr 465 . . . . 5  |-  ( (
ph  /\  y  e.  ( A [,] B ) )  ->  ( B  +  T )  e.  RR )
3124sselda 3499 . . . . . 6  |-  ( (
ph  /\  y  e.  ( A [,] B ) )  ->  y  e.  RR )
324adantr 465 . . . . . 6  |-  ( (
ph  /\  y  e.  ( A [,] B ) )  ->  T  e.  RR )
3331, 32readdcld 9640 . . . . 5  |-  ( (
ph  /\  y  e.  ( A [,] B ) )  ->  ( y  +  T )  e.  RR )
341adantr 465 . . . . . 6  |-  ( (
ph  /\  y  e.  ( A [,] B ) )  ->  A  e.  RR )
35 simpr 461 . . . . . . . 8  |-  ( (
ph  /\  y  e.  ( A [,] B ) )  ->  y  e.  ( A [,] B ) )
362adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  y  e.  ( A [,] B ) )  ->  B  e.  RR )
37 elicc2 11614 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( y  e.  ( A [,] B )  <-> 
( y  e.  RR  /\  A  <_  y  /\  y  <_  B ) ) )
3834, 36, 37syl2anc 661 . . . . . . . 8  |-  ( (
ph  /\  y  e.  ( A [,] B ) )  ->  ( y  e.  ( A [,] B
)  <->  ( y  e.  RR  /\  A  <_ 
y  /\  y  <_  B ) ) )
3935, 38mpbid 210 . . . . . . 7  |-  ( (
ph  /\  y  e.  ( A [,] B ) )  ->  ( y  e.  RR  /\  A  <_ 
y  /\  y  <_  B ) )
4039simp2d 1009 . . . . . 6  |-  ( (
ph  /\  y  e.  ( A [,] B ) )  ->  A  <_  y )
4134, 31, 32, 40leadd1dd 10187 . . . . 5  |-  ( (
ph  /\  y  e.  ( A [,] B ) )  ->  ( A  +  T )  <_  (
y  +  T ) )
4239simp3d 1010 . . . . . 6  |-  ( (
ph  /\  y  e.  ( A [,] B ) )  ->  y  <_  B )
4331, 36, 32, 42leadd1dd 10187 . . . . 5  |-  ( (
ph  /\  y  e.  ( A [,] B ) )  ->  ( y  +  T )  <_  ( B  +  T )
)
4429, 30, 33, 41, 43eliccd 31720 . . . 4  |-  ( (
ph  /\  y  e.  ( A [,] B ) )  ->  ( y  +  T )  e.  ( ( A  +  T
) [,] ( B  +  T ) ) )
4520, 23, 26, 28, 44cncfmptssg 31854 . . 3  |-  ( ph  ->  ( y  e.  ( A [,] B ) 
|->  ( y  +  T
) )  e.  ( ( A [,] B
) -cn-> ( ( A  +  T ) [,] ( B  +  T
) ) ) )
46 eqeq1 2461 . . . . . . . 8  |-  ( w  =  x  ->  (
w  =  ( z  +  T )  <->  x  =  ( z  +  T
) ) )
4746rexbidv 2968 . . . . . . 7  |-  ( w  =  x  ->  ( E. z  e.  ( A [,] B ) w  =  ( z  +  T )  <->  E. z  e.  ( A [,] B
) x  =  ( z  +  T ) ) )
48 oveq1 6303 . . . . . . . . 9  |-  ( z  =  y  ->  (
z  +  T )  =  ( y  +  T ) )
4948eqeq2d 2471 . . . . . . . 8  |-  ( z  =  y  ->  (
x  =  ( z  +  T )  <->  x  =  ( y  +  T
) ) )
5049cbvrexv 3085 . . . . . . 7  |-  ( E. z  e.  ( A [,] B ) x  =  ( z  +  T )  <->  E. y  e.  ( A [,] B
) x  =  ( y  +  T ) )
5147, 50syl6bb 261 . . . . . 6  |-  ( w  =  x  ->  ( E. z  e.  ( A [,] B ) w  =  ( z  +  T )  <->  E. y  e.  ( A [,] B
) x  =  ( y  +  T ) ) )
5251cbvrabv 3108 . . . . 5  |-  { w  e.  CC  |  E. z  e.  ( A [,] B
) w  =  ( z  +  T ) }  =  { x  e.  CC  |  E. y  e.  ( A [,] B
) x  =  ( y  +  T ) }
53 ffdm 5751 . . . . . . 7  |-  ( F : RR --> CC  ->  ( F : dom  F --> CC  /\  dom  F  C_  RR ) )
5410, 53syl 16 . . . . . 6  |-  ( ph  ->  ( F : dom  F --> CC  /\  dom  F  C_  RR ) )
5554simpld 459 . . . . 5  |-  ( ph  ->  F : dom  F --> CC )
56 simp3 998 . . . . . . . . . 10  |-  ( (
ph  /\  z  e.  ( A [,] B )  /\  w  =  ( z  +  T ) )  ->  w  =  ( z  +  T
) )
5724sselda 3499 . . . . . . . . . . . 12  |-  ( (
ph  /\  z  e.  ( A [,] B ) )  ->  z  e.  RR )
584adantr 465 . . . . . . . . . . . 12  |-  ( (
ph  /\  z  e.  ( A [,] B ) )  ->  T  e.  RR )
5957, 58readdcld 9640 . . . . . . . . . . 11  |-  ( (
ph  /\  z  e.  ( A [,] B ) )  ->  ( z  +  T )  e.  RR )
60593adant3 1016 . . . . . . . . . 10  |-  ( (
ph  /\  z  e.  ( A [,] B )  /\  w  =  ( z  +  T ) )  ->  ( z  +  T )  e.  RR )
6156, 60eqeltrd 2545 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  ( A [,] B )  /\  w  =  ( z  +  T ) )  ->  w  e.  RR )
6261rexlimdv3a 2951 . . . . . . . 8  |-  ( ph  ->  ( E. z  e.  ( A [,] B
) w  =  ( z  +  T )  ->  w  e.  RR ) )
6362ralrimivw 2872 . . . . . . 7  |-  ( ph  ->  A. w  e.  CC  ( E. z  e.  ( A [,] B ) w  =  ( z  +  T )  ->  w  e.  RR )
)
64 rabss 3573 . . . . . . 7  |-  ( { w  e.  CC  |  E. z  e.  ( A [,] B ) w  =  ( z  +  T ) }  C_  RR 
<-> 
A. w  e.  CC  ( E. z  e.  ( A [,] B ) w  =  ( z  +  T )  ->  w  e.  RR )
)
6563, 64sylibr 212 . . . . . 6  |-  ( ph  ->  { w  e.  CC  |  E. z  e.  ( A [,] B ) w  =  ( z  +  T ) } 
C_  RR )
66 fdm 5741 . . . . . . 7  |-  ( F : RR --> CC  ->  dom 
F  =  RR )
6710, 66syl 16 . . . . . 6  |-  ( ph  ->  dom  F  =  RR )
6865, 67sseqtr4d 3536 . . . . 5  |-  ( ph  ->  { w  e.  CC  |  E. z  e.  ( A [,] B ) w  =  ( z  +  T ) } 
C_  dom  F )
69 itgperiod.fper . . . . 5  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( F `  ( x  +  T
) )  =  ( F `  x ) )
70 itgperiod.fcn . . . . 5  |-  ( ph  ->  ( F  |`  ( A [,] B ) )  e.  ( ( A [,] B ) -cn-> CC ) )
7126, 4, 52, 55, 68, 69, 70cncfperiod 31863 . . . 4  |-  ( ph  ->  ( F  |`  { w  e.  CC  |  E. z  e.  ( A [,] B
) w  =  ( z  +  T ) } )  e.  ( { w  e.  CC  |  E. z  e.  ( A [,] B ) w  =  ( z  +  T ) }
-cn-> CC ) )
7247elrab 3257 . . . . . . . . 9  |-  ( x  e.  { w  e.  CC  |  E. z  e.  ( A [,] B
) w  =  ( z  +  T ) }  <->  ( x  e.  CC  /\  E. z  e.  ( A [,] B
) x  =  ( z  +  T ) ) )
73 simprr 757 . . . . . . . . . 10  |-  ( (
ph  /\  ( x  e.  CC  /\  E. z  e.  ( A [,] B
) x  =  ( z  +  T ) ) )  ->  E. z  e.  ( A [,] B
) x  =  ( z  +  T ) )
74 nfv 1708 . . . . . . . . . . . 12  |-  F/ z
ph
75 nfv 1708 . . . . . . . . . . . . 13  |-  F/ z  x  e.  CC
76 nfre1 2918 . . . . . . . . . . . . 13  |-  F/ z E. z  e.  ( A [,] B ) x  =  ( z  +  T )
7775, 76nfan 1929 . . . . . . . . . . . 12  |-  F/ z ( x  e.  CC  /\ 
E. z  e.  ( A [,] B ) x  =  ( z  +  T ) )
7874, 77nfan 1929 . . . . . . . . . . 11  |-  F/ z ( ph  /\  (
x  e.  CC  /\  E. z  e.  ( A [,] B ) x  =  ( z  +  T ) ) )
79 nfv 1708 . . . . . . . . . . 11  |-  F/ z  x  e.  ( ( A  +  T ) [,] ( B  +  T ) )
80 simp3 998 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  z  e.  ( A [,] B )  /\  x  =  ( z  +  T ) )  ->  x  =  ( z  +  T
) )
811adantr 465 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  z  e.  ( A [,] B ) )  ->  A  e.  RR )
82 simpr 461 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  z  e.  ( A [,] B ) )  ->  z  e.  ( A [,] B ) )
832adantr 465 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
ph  /\  z  e.  ( A [,] B ) )  ->  B  e.  RR )
84 elicc2 11614 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( z  e.  ( A [,] B )  <-> 
( z  e.  RR  /\  A  <_  z  /\  z  <_  B ) ) )
8581, 83, 84syl2anc 661 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  z  e.  ( A [,] B ) )  ->  ( z  e.  ( A [,] B
)  <->  ( z  e.  RR  /\  A  <_ 
z  /\  z  <_  B ) ) )
8682, 85mpbid 210 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  z  e.  ( A [,] B ) )  ->  ( z  e.  RR  /\  A  <_ 
z  /\  z  <_  B ) )
8786simp2d 1009 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  z  e.  ( A [,] B ) )  ->  A  <_  z )
8881, 57, 58, 87leadd1dd 10187 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  z  e.  ( A [,] B ) )  ->  ( A  +  T )  <_  (
z  +  T ) )
8986simp3d 1010 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  z  e.  ( A [,] B ) )  ->  z  <_  B )
9057, 83, 58, 89leadd1dd 10187 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  z  e.  ( A [,] B ) )  ->  ( z  +  T )  <_  ( B  +  T )
)
9159, 88, 903jca 1176 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  z  e.  ( A [,] B ) )  ->  ( (
z  +  T )  e.  RR  /\  ( A  +  T )  <_  ( z  +  T
)  /\  ( z  +  T )  <_  ( B  +  T )
) )
92913adant3 1016 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  z  e.  ( A [,] B )  /\  x  =  ( z  +  T ) )  ->  ( (
z  +  T )  e.  RR  /\  ( A  +  T )  <_  ( z  +  T
)  /\  ( z  +  T )  <_  ( B  +  T )
) )
9383ad2ant1 1017 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  z  e.  ( A [,] B )  /\  x  =  ( z  +  T ) )  ->  ( A  +  T )  e.  RR )
9493ad2ant1 1017 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  z  e.  ( A [,] B )  /\  x  =  ( z  +  T ) )  ->  ( B  +  T )  e.  RR )
95 elicc2 11614 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  +  T
)  e.  RR  /\  ( B  +  T
)  e.  RR )  ->  ( ( z  +  T )  e.  ( ( A  +  T ) [,] ( B  +  T )
)  <->  ( ( z  +  T )  e.  RR  /\  ( A  +  T )  <_ 
( z  +  T
)  /\  ( z  +  T )  <_  ( B  +  T )
) ) )
9693, 94, 95syl2anc 661 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  z  e.  ( A [,] B )  /\  x  =  ( z  +  T ) )  ->  ( (
z  +  T )  e.  ( ( A  +  T ) [,] ( B  +  T
) )  <->  ( (
z  +  T )  e.  RR  /\  ( A  +  T )  <_  ( z  +  T
)  /\  ( z  +  T )  <_  ( B  +  T )
) ) )
9792, 96mpbird 232 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  z  e.  ( A [,] B )  /\  x  =  ( z  +  T ) )  ->  ( z  +  T )  e.  ( ( A  +  T
) [,] ( B  +  T ) ) )
9880, 97eqeltrd 2545 . . . . . . . . . . . . 13  |-  ( (
ph  /\  z  e.  ( A [,] B )  /\  x  =  ( z  +  T ) )  ->  x  e.  ( ( A  +  T ) [,] ( B  +  T )
) )
99983exp 1195 . . . . . . . . . . . 12  |-  ( ph  ->  ( z  e.  ( A [,] B )  ->  ( x  =  ( z  +  T
)  ->  x  e.  ( ( A  +  T ) [,] ( B  +  T )
) ) ) )
10099adantr 465 . . . . . . . . . . 11  |-  ( (
ph  /\  ( x  e.  CC  /\  E. z  e.  ( A [,] B
) x  =  ( z  +  T ) ) )  ->  (
z  e.  ( A [,] B )  -> 
( x  =  ( z  +  T )  ->  x  e.  ( ( A  +  T
) [,] ( B  +  T ) ) ) ) )
10178, 79, 100rexlimd 2941 . . . . . . . . . 10  |-  ( (
ph  /\  ( x  e.  CC  /\  E. z  e.  ( A [,] B
) x  =  ( z  +  T ) ) )  ->  ( E. z  e.  ( A [,] B ) x  =  ( z  +  T )  ->  x  e.  ( ( A  +  T ) [,] ( B  +  T )
) ) )
10273, 101mpd 15 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  CC  /\  E. z  e.  ( A [,] B
) x  =  ( z  +  T ) ) )  ->  x  e.  ( ( A  +  T ) [,] ( B  +  T )
) )
10372, 102sylan2b 475 . . . . . . . 8  |-  ( (
ph  /\  x  e.  { w  e.  CC  |  E. z  e.  ( A [,] B ) w  =  ( z  +  T ) } )  ->  x  e.  ( ( A  +  T
) [,] ( B  +  T ) ) )
10416recnd 9639 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( ( A  +  T ) [,] ( B  +  T )
) )  ->  x  e.  CC )
1051adantr 465 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( ( A  +  T ) [,] ( B  +  T )
) )  ->  A  e.  RR )
1062adantr 465 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( ( A  +  T ) [,] ( B  +  T )
) )  ->  B  e.  RR )
1074adantr 465 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  ( ( A  +  T ) [,] ( B  +  T )
) )  ->  T  e.  RR )
10816, 107resubcld 10008 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( ( A  +  T ) [,] ( B  +  T )
) )  ->  (
x  -  T )  e.  RR )
1091recnd 9639 . . . . . . . . . . . . . . 15  |-  ( ph  ->  A  e.  CC )
110109, 21pncand 9951 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( ( A  +  T )  -  T
)  =  A )
111110eqcomd 2465 . . . . . . . . . . . . 13  |-  ( ph  ->  A  =  ( ( A  +  T )  -  T ) )
112111adantr 465 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  ( ( A  +  T ) [,] ( B  +  T )
) )  ->  A  =  ( ( A  +  T )  -  T ) )
113 elicc2 11614 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  +  T
)  e.  RR  /\  ( B  +  T
)  e.  RR )  ->  ( x  e.  ( ( A  +  T ) [,] ( B  +  T )
)  <->  ( x  e.  RR  /\  ( A  +  T )  <_  x  /\  x  <_  ( B  +  T )
) ) )
11412, 13, 113syl2anc 661 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  x  e.  ( ( A  +  T ) [,] ( B  +  T )
) )  ->  (
x  e.  ( ( A  +  T ) [,] ( B  +  T ) )  <->  ( x  e.  RR  /\  ( A  +  T )  <_  x  /\  x  <_  ( B  +  T )
) ) )
11514, 114mpbid 210 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  x  e.  ( ( A  +  T ) [,] ( B  +  T )
) )  ->  (
x  e.  RR  /\  ( A  +  T
)  <_  x  /\  x  <_  ( B  +  T ) ) )
116115simp2d 1009 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  ( ( A  +  T ) [,] ( B  +  T )
) )  ->  ( A  +  T )  <_  x )
11712, 16, 107, 116lesub1dd 10189 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  ( ( A  +  T ) [,] ( B  +  T )
) )  ->  (
( A  +  T
)  -  T )  <_  ( x  -  T ) )
118112, 117eqbrtrd 4476 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( ( A  +  T ) [,] ( B  +  T )
) )  ->  A  <_  ( x  -  T
) )
119115simp3d 1010 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  ( ( A  +  T ) [,] ( B  +  T )
) )  ->  x  <_  ( B  +  T
) )
12016, 13, 107, 119lesub1dd 10189 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  ( ( A  +  T ) [,] ( B  +  T )
) )  ->  (
x  -  T )  <_  ( ( B  +  T )  -  T ) )
1212recnd 9639 . . . . . . . . . . . . . 14  |-  ( ph  ->  B  e.  CC )
122121, 21pncand 9951 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( B  +  T )  -  T
)  =  B )
123122adantr 465 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  ( ( A  +  T ) [,] ( B  +  T )
) )  ->  (
( B  +  T
)  -  T )  =  B )
124120, 123breqtrd 4480 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( ( A  +  T ) [,] ( B  +  T )
) )  ->  (
x  -  T )  <_  B )
125105, 106, 108, 118, 124eliccd 31720 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( ( A  +  T ) [,] ( B  +  T )
) )  ->  (
x  -  T )  e.  ( A [,] B ) )
12621adantr 465 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  ( ( A  +  T ) [,] ( B  +  T )
) )  ->  T  e.  CC )
127104, 126npcand 9954 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( ( A  +  T ) [,] ( B  +  T )
) )  ->  (
( x  -  T
)  +  T )  =  x )
128127eqcomd 2465 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( ( A  +  T ) [,] ( B  +  T )
) )  ->  x  =  ( ( x  -  T )  +  T ) )
129 oveq1 6303 . . . . . . . . . . . 12  |-  ( z  =  ( x  -  T )  ->  (
z  +  T )  =  ( ( x  -  T )  +  T ) )
130129eqeq2d 2471 . . . . . . . . . . 11  |-  ( z  =  ( x  -  T )  ->  (
x  =  ( z  +  T )  <->  x  =  ( ( x  -  T )  +  T
) ) )
131130rspcev 3210 . . . . . . . . . 10  |-  ( ( ( x  -  T
)  e.  ( A [,] B )  /\  x  =  ( (
x  -  T )  +  T ) )  ->  E. z  e.  ( A [,] B ) x  =  ( z  +  T ) )
132125, 128, 131syl2anc 661 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( ( A  +  T ) [,] ( B  +  T )
) )  ->  E. z  e.  ( A [,] B
) x  =  ( z  +  T ) )
133104, 132, 72sylanbrc 664 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( ( A  +  T ) [,] ( B  +  T )
) )  ->  x  e.  { w  e.  CC  |  E. z  e.  ( A [,] B ) w  =  ( z  +  T ) } )
134103, 133impbida 832 . . . . . . 7  |-  ( ph  ->  ( x  e.  {
w  e.  CC  |  E. z  e.  ( A [,] B ) w  =  ( z  +  T ) }  <->  x  e.  ( ( A  +  T ) [,] ( B  +  T )
) ) )
135134eqrdv 2454 . . . . . 6  |-  ( ph  ->  { w  e.  CC  |  E. z  e.  ( A [,] B ) w  =  ( z  +  T ) }  =  ( ( A  +  T ) [,] ( B  +  T
) ) )
136135reseq2d 5283 . . . . 5  |-  ( ph  ->  ( F  |`  { w  e.  CC  |  E. z  e.  ( A [,] B
) w  =  ( z  +  T ) } )  =  ( F  |`  ( ( A  +  T ) [,] ( B  +  T
) ) ) )
137135, 68eqsstr3d 3534 . . . . . 6  |-  ( ph  ->  ( ( A  +  T ) [,] ( B  +  T )
)  C_  dom  F )
13855, 137feqresmpt 5927 . . . . 5  |-  ( ph  ->  ( F  |`  (
( A  +  T
) [,] ( B  +  T ) ) )  =  ( x  e.  ( ( A  +  T ) [,] ( B  +  T
) )  |->  ( F `
 x ) ) )
139136, 138eqtr2d 2499 . . . 4  |-  ( ph  ->  ( x  e.  ( ( A  +  T
) [,] ( B  +  T ) ) 
|->  ( F `  x
) )  =  ( F  |`  { w  e.  CC  |  E. z  e.  ( A [,] B
) w  =  ( z  +  T ) } ) )
1401, 2, 4iccshift 31740 . . . . 5  |-  ( ph  ->  ( ( A  +  T ) [,] ( B  +  T )
)  =  { w  e.  CC  |  E. z  e.  ( A [,] B
) w  =  ( z  +  T ) } )
141140oveq1d 6311 . . . 4  |-  ( ph  ->  ( ( ( A  +  T ) [,] ( B  +  T
) ) -cn-> CC )  =  ( { w  e.  CC  |  E. z  e.  ( A [,] B
) w  =  ( z  +  T ) } -cn-> CC ) )
14271, 139, 1413eltr4d 2560 . . 3  |-  ( ph  ->  ( x  e.  ( ( A  +  T
) [,] ( B  +  T ) ) 
|->  ( F `  x
) )  e.  ( ( ( A  +  T ) [,] ( B  +  T )
) -cn-> CC ) )
143 ioosscn 31709 . . . . . 6  |-  ( A (,) B )  C_  CC
144143a1i 11 . . . . 5  |-  ( ph  ->  ( A (,) B
)  C_  CC )
145 1cnd 9629 . . . . 5  |-  ( ph  ->  1  e.  CC )
146 ssid 3518 . . . . . 6  |-  CC  C_  CC
147146a1i 11 . . . . 5  |-  ( ph  ->  CC  C_  CC )
148144, 145, 147constcncfg 31855 . . . 4  |-  ( ph  ->  ( y  e.  ( A (,) B ) 
|->  1 )  e.  ( ( A (,) B
) -cn-> CC ) )
149 fconstmpt 5052 . . . . 5  |-  ( ( A (,) B )  X.  { 1 } )  =  ( y  e.  ( A (,) B )  |->  1 )
150 ioombl 22101 . . . . . . 7  |-  ( A (,) B )  e. 
dom  vol
151150a1i 11 . . . . . 6  |-  ( ph  ->  ( A (,) B
)  e.  dom  vol )
152 ioovolcl 22105 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( vol `  ( A (,) B ) )  e.  RR )
1531, 2, 152syl2anc 661 . . . . . 6  |-  ( ph  ->  ( vol `  ( A (,) B ) )  e.  RR )
154 iblconst 22350 . . . . . 6  |-  ( ( ( A (,) B
)  e.  dom  vol  /\  ( vol `  ( A (,) B ) )  e.  RR  /\  1  e.  CC )  ->  (
( A (,) B
)  X.  { 1 } )  e.  L^1 )
155151, 153, 145, 154syl3anc 1228 . . . . 5  |-  ( ph  ->  ( ( A (,) B )  X.  {
1 } )  e.  L^1 )
156149, 155syl5eqelr 2550 . . . 4  |-  ( ph  ->  ( y  e.  ( A (,) B ) 
|->  1 )  e.  L^1 )
157148, 156elind 3684 . . 3  |-  ( ph  ->  ( y  e.  ( A (,) B ) 
|->  1 )  e.  ( ( ( A (,) B ) -cn-> CC )  i^i  L^1 ) )
15824resmptd 5335 . . . . . . 7  |-  ( ph  ->  ( ( y  e.  RR  |->  ( y  +  T ) )  |`  ( A [,] B ) )  =  ( y  e.  ( A [,] B )  |->  ( y  +  T ) ) )
159158eqcomd 2465 . . . . . 6  |-  ( ph  ->  ( y  e.  ( A [,] B ) 
|->  ( y  +  T
) )  =  ( ( y  e.  RR  |->  ( y  +  T
) )  |`  ( A [,] B ) ) )
160159oveq2d 6312 . . . . 5  |-  ( ph  ->  ( RR  _D  (
y  e.  ( A [,] B )  |->  ( y  +  T ) ) )  =  ( RR  _D  ( ( y  e.  RR  |->  ( y  +  T ) )  |`  ( A [,] B ) ) ) )
16125a1i 11 . . . . . 6  |-  ( ph  ->  RR  C_  CC )
162161sselda 3499 . . . . . . . 8  |-  ( (
ph  /\  y  e.  RR )  ->  y  e.  CC )
16321adantr 465 . . . . . . . 8  |-  ( (
ph  /\  y  e.  RR )  ->  T  e.  CC )
164162, 163addcld 9632 . . . . . . 7  |-  ( (
ph  /\  y  e.  RR )  ->  ( y  +  T )  e.  CC )
165 eqid 2457 . . . . . . 7  |-  ( y  e.  RR  |->  ( y  +  T ) )  =  ( y  e.  RR  |->  ( y  +  T ) )
166164, 165fmptd 6056 . . . . . 6  |-  ( ph  ->  ( y  e.  RR  |->  ( y  +  T
) ) : RR --> CC )
167 ssid 3518 . . . . . . 7  |-  RR  C_  RR
168167a1i 11 . . . . . 6  |-  ( ph  ->  RR  C_  RR )
169 eqid 2457 . . . . . . 7  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
170169tgioo2 21434 . . . . . . 7  |-  ( topGen ` 
ran  (,) )  =  ( ( TopOpen ` fld )t  RR )
171169, 170dvres 22441 . . . . . 6  |-  ( ( ( RR  C_  CC  /\  ( y  e.  RR  |->  ( y  +  T
) ) : RR --> CC )  /\  ( RR  C_  RR  /\  ( A [,] B )  C_  RR ) )  ->  ( RR  _D  ( ( y  e.  RR  |->  ( y  +  T ) )  |`  ( A [,] B
) ) )  =  ( ( RR  _D  ( y  e.  RR  |->  ( y  +  T
) ) )  |`  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) ) )
172161, 166, 168, 24, 171syl22anc 1229 . . . . 5  |-  ( ph  ->  ( RR  _D  (
( y  e.  RR  |->  ( y  +  T
) )  |`  ( A [,] B ) ) )  =  ( ( RR  _D  ( y  e.  RR  |->  ( y  +  T ) ) )  |`  ( ( int `  ( topGen `  ran  (,) ) ) `  ( A [,] B ) ) ) )
173160, 172eqtrd 2498 . . . 4  |-  ( ph  ->  ( RR  _D  (
y  e.  ( A [,] B )  |->  ( y  +  T ) ) )  =  ( ( RR  _D  (
y  e.  RR  |->  ( y  +  T ) ) )  |`  (
( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) ) )
174 iccntr 21452 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) )  =  ( A (,) B
) )
1751, 2, 174syl2anc 661 . . . . 5  |-  ( ph  ->  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) )  =  ( A (,) B
) )
176175reseq2d 5283 . . . 4  |-  ( ph  ->  ( ( RR  _D  ( y  e.  RR  |->  ( y  +  T
) ) )  |`  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) )  =  ( ( RR 
_D  ( y  e.  RR  |->  ( y  +  T ) ) )  |`  ( A (,) B
) ) )
177 reelprrecn 9601 . . . . . . . 8  |-  RR  e.  { RR ,  CC }
178177a1i 11 . . . . . . 7  |-  ( ph  ->  RR  e.  { RR ,  CC } )
179 1cnd 9629 . . . . . . 7  |-  ( (
ph  /\  y  e.  RR )  ->  1  e.  CC )
180178dvmptid 22486 . . . . . . 7  |-  ( ph  ->  ( RR  _D  (
y  e.  RR  |->  y ) )  =  ( y  e.  RR  |->  1 ) )
181 0cnd 9606 . . . . . . 7  |-  ( (
ph  /\  y  e.  RR )  ->  0  e.  CC )
182178, 21dvmptc 22487 . . . . . . 7  |-  ( ph  ->  ( RR  _D  (
y  e.  RR  |->  T ) )  =  ( y  e.  RR  |->  0 ) )
183178, 162, 179, 180, 163, 181, 182dvmptadd 22489 . . . . . 6  |-  ( ph  ->  ( RR  _D  (
y  e.  RR  |->  ( y  +  T ) ) )  =  ( y  e.  RR  |->  ( 1  +  0 ) ) )
184183reseq1d 5282 . . . . 5  |-  ( ph  ->  ( ( RR  _D  ( y  e.  RR  |->  ( y  +  T
) ) )  |`  ( A (,) B ) )  =  ( ( y  e.  RR  |->  ( 1  +  0 ) )  |`  ( A (,) B ) ) )
185 ioossre 11611 . . . . . . 7  |-  ( A (,) B )  C_  RR
186185a1i 11 . . . . . 6  |-  ( ph  ->  ( A (,) B
)  C_  RR )
187186resmptd 5335 . . . . 5  |-  ( ph  ->  ( ( y  e.  RR  |->  ( 1  +  0 ) )  |`  ( A (,) B ) )  =  ( y  e.  ( A (,) B )  |->  ( 1  +  0 ) ) )
188 1p0e1 10669 . . . . . . 7  |-  ( 1  +  0 )  =  1
189188mpteq2i 4540 . . . . . 6  |-  ( y  e.  ( A (,) B )  |->  ( 1  +  0 ) )  =  ( y  e.  ( A (,) B
)  |->  1 )
190189a1i 11 . . . . 5  |-  ( ph  ->  ( y  e.  ( A (,) B ) 
|->  ( 1  +  0 ) )  =  ( y  e.  ( A (,) B )  |->  1 ) )
191184, 187, 1903eqtrd 2502 . . . 4  |-  ( ph  ->  ( ( RR  _D  ( y  e.  RR  |->  ( y  +  T
) ) )  |`  ( A (,) B ) )  =  ( y  e.  ( A (,) B )  |->  1 ) )
192173, 176, 1913eqtrd 2502 . . 3  |-  ( ph  ->  ( RR  _D  (
y  e.  ( A [,] B )  |->  ( y  +  T ) ) )  =  ( y  e.  ( A (,) B )  |->  1 ) )
193 fveq2 5872 . . 3  |-  ( x  =  ( y  +  T )  ->  ( F `  x )  =  ( F `  ( y  +  T
) ) )
194 oveq1 6303 . . 3  |-  ( y  =  A  ->  (
y  +  T )  =  ( A  +  T ) )
195 oveq1 6303 . . 3  |-  ( y  =  B  ->  (
y  +  T )  =  ( B  +  T ) )
1961, 2, 5, 45, 142, 157, 192, 193, 194, 195, 8, 9itgsubsticc 31957 . 2  |-  ( ph  ->  S__ [ ( A  +  T )  -> 
( B  +  T
) ] ( F `
 x )  _d x  =  S__ [ A  ->  B ] ( ( F `  (
y  +  T ) )  x.  1 )  _d y )
1975ditgpos 22386 . . 3  |-  ( ph  ->  S__ [ A  ->  B ] ( ( F `
 ( y  +  T ) )  x.  1 )  _d y  =  S. ( A (,) B ) ( ( F `  (
y  +  T ) )  x.  1 )  _d y )
19810adantr 465 . . . . . 6  |-  ( (
ph  /\  y  e.  ( A [,] B ) )  ->  F : RR
--> CC )
199198, 33ffvelrnd 6033 . . . . 5  |-  ( (
ph  /\  y  e.  ( A [,] B ) )  ->  ( F `  ( y  +  T
) )  e.  CC )
200 1cnd 9629 . . . . 5  |-  ( (
ph  /\  y  e.  ( A [,] B ) )  ->  1  e.  CC )
201199, 200mulcld 9633 . . . 4  |-  ( (
ph  /\  y  e.  ( A [,] B ) )  ->  ( ( F `  ( y  +  T ) )  x.  1 )  e.  CC )
2021, 2, 201itgioo 22348 . . 3  |-  ( ph  ->  S. ( A (,) B ) ( ( F `  ( y  +  T ) )  x.  1 )  _d y  =  S. ( A [,] B ) ( ( F `  ( y  +  T
) )  x.  1 )  _d y )
203 oveq1 6303 . . . . . . 7  |-  ( y  =  x  ->  (
y  +  T )  =  ( x  +  T ) )
204203fveq2d 5876 . . . . . 6  |-  ( y  =  x  ->  ( F `  ( y  +  T ) )  =  ( F `  (
x  +  T ) ) )
205204oveq1d 6311 . . . . 5  |-  ( y  =  x  ->  (
( F `  (
y  +  T ) )  x.  1 )  =  ( ( F `
 ( x  +  T ) )  x.  1 ) )
206205cbvitgv 22309 . . . 4  |-  S. ( A [,] B ) ( ( F `  ( y  +  T
) )  x.  1 )  _d y  =  S. ( A [,] B ) ( ( F `  ( x  +  T ) )  x.  1 )  _d x
20710adantr 465 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  F : RR
--> CC )
20824sselda 3499 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  x  e.  RR )
2094adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  T  e.  RR )
210208, 209readdcld 9640 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( x  +  T )  e.  RR )
211207, 210ffvelrnd 6033 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( F `  ( x  +  T
) )  e.  CC )
212211mulid1d 9630 . . . . . 6  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( ( F `  ( x  +  T ) )  x.  1 )  =  ( F `  ( x  +  T ) ) )
213212, 69eqtrd 2498 . . . . 5  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( ( F `  ( x  +  T ) )  x.  1 )  =  ( F `  x ) )
214213itgeq2dv 22314 . . . 4  |-  ( ph  ->  S. ( A [,] B ) ( ( F `  ( x  +  T ) )  x.  1 )  _d x  =  S. ( A [,] B ) ( F `  x
)  _d x )
215206, 214syl5eq 2510 . . 3  |-  ( ph  ->  S. ( A [,] B ) ( ( F `  ( y  +  T ) )  x.  1 )  _d y  =  S. ( A [,] B ) ( F `  x
)  _d x )
216197, 202, 2153eqtrd 2502 . 2  |-  ( ph  ->  S__ [ A  ->  B ] ( ( F `
 ( y  +  T ) )  x.  1 )  _d y  =  S. ( A [,] B ) ( F `  x )  _d x )
21719, 196, 2163eqtrd 2502 1  |-  ( ph  ->  S. ( ( A  +  T ) [,] ( B  +  T
) ) ( F `
 x )  _d x  =  S. ( A [,] B ) ( F `  x
)  _d x )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819   A.wral 2807   E.wrex 2808   {crab 2811    C_ wss 3471   {csn 4032   {cpr 4034   class class class wbr 4456    |-> cmpt 4515    X. cxp 5006   dom cdm 5008   ran crn 5009    |` cres 5010   -->wf 5590   ` cfv 5594  (class class class)co 6296   CCcc 9507   RRcr 9508   0cc0 9509   1c1 9510    + caddc 9512    x. cmul 9514    <_ cle 9646    - cmin 9824   RR+crp 11245   (,)cioo 11554   [,]cicc 11557   TopOpenctopn 14839   topGenctg 14855  ℂfldccnfld 18547   intcnt 19645   -cn->ccncf 21506   volcvol 22001   L^1cibl 22152   S.citg 22153   S__cdit 22376    _D cdv 22393
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-inf2 8075  ax-cc 8832  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587  ax-addf 9588  ax-mulf 9589
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-fal 1401  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-iin 4335  df-disj 4428  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-se 4848  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  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 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-of 6539  df-ofr 6540  df-om 6700  df-1st 6799  df-2nd 6800  df-supp 6918  df-recs 7060  df-rdg 7094  df-1o 7148  df-2o 7149  df-oadd 7152  df-omul 7153  df-er 7329  df-map 7440  df-pm 7441  df-ixp 7489  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-fsupp 7848  df-fi 7889  df-sup 7919  df-oi 7953  df-card 8337  df-acn 8340  df-cda 8565  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-3 10616  df-4 10617  df-5 10618  df-6 10619  df-7 10620  df-8 10621  df-9 10622  df-10 10623  df-n0 10817  df-z 10886  df-dec 11001  df-uz 11107  df-q 11208  df-rp 11246  df-xneg 11343  df-xadd 11344  df-xmul 11345  df-ioo 11558  df-ioc 11559  df-ico 11560  df-icc 11561  df-fz 11698  df-fzo 11822  df-fl 11932  df-mod 12000  df-seq 12111  df-exp 12170  df-hash 12409  df-cj 12944  df-re 12945  df-im 12946  df-sqrt 13080  df-abs 13081  df-limsup 13306  df-clim 13323  df-rlim 13324  df-sum 13521  df-struct 14646  df-ndx 14647  df-slot 14648  df-base 14649  df-sets 14650  df-ress 14651  df-plusg 14725  df-mulr 14726  df-starv 14727  df-sca 14728  df-vsca 14729  df-ip 14730  df-tset 14731  df-ple 14732  df-ds 14734  df-unif 14735  df-hom 14736  df-cco 14737  df-rest 14840  df-topn 14841  df-0g 14859  df-gsum 14860  df-topgen 14861  df-pt 14862  df-prds 14865  df-xrs 14919  df-qtop 14924  df-imas 14925  df-xps 14927  df-mre 15003  df-mrc 15004  df-acs 15006  df-mgm 15999  df-sgrp 16038  df-mnd 16048  df-submnd 16094  df-mulg 16187  df-cntz 16482  df-cmn 16927  df-psmet 18538  df-xmet 18539  df-met 18540  df-bl 18541  df-mopn 18542  df-fbas 18543  df-fg 18544  df-cnfld 18548  df-top 19526  df-bases 19528  df-topon 19529  df-topsp 19530  df-cld 19647  df-ntr 19648  df-cls 19649  df-nei 19726  df-lp 19764  df-perf 19765  df-cn 19855  df-cnp 19856  df-haus 19943  df-cmp 20014  df-tx 20189  df-hmeo 20382  df-fil 20473  df-fm 20565  df-flim 20566  df-flf 20567  df-xms 20949  df-ms 20950  df-tms 20951  df-cncf 21508  df-ovol 22002  df-vol 22003  df-mbf 22154  df-itg1 22155  df-itg2 22156  df-ibl 22157  df-itg 22158  df-0p 22203  df-ditg 22377  df-limc 22396  df-dv 22397
This theorem is referenced by: (None)
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