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Theorem itgeq12dv 27936
Description: Equality theorem for an integral. (Contributed by Thierry Arnoux, 14-Feb-2017.)
Hypotheses
Ref Expression
itgeq12dv.2  |-  ( ph  ->  A  =  B )
itgeq12dv.1  |-  ( (
ph  /\  x  e.  A )  ->  C  =  D )
Assertion
Ref Expression
itgeq12dv  |-  ( ph  ->  S. A C  _d x  =  S. B D  _d x )
Distinct variable group:    ph, x
Allowed substitution hints:    A( x)    B( x)    C( x)    D( x)

Proof of Theorem itgeq12dv
Dummy variable  k is distinct from all other variables.
StepHypRef Expression
1 itgeq12dv.1 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  A )  ->  C  =  D )
21oveq1d 6299 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  A )  ->  ( C  /  ( _i ^
k ) )  =  ( D  /  (
_i ^ k ) ) )
32fveq2d 5870 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  A )  ->  (
Re `  ( C  /  ( _i ^
k ) ) )  =  ( Re `  ( D  /  (
_i ^ k ) ) ) )
43breq2d 4459 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  A )  ->  (
0  <_  ( Re `  ( C  /  (
_i ^ k ) ) )  <->  0  <_  ( Re `  ( D  /  ( _i ^
k ) ) ) ) )
54pm5.32da 641 . . . . . . . 8  |-  ( ph  ->  ( ( x  e.  A  /\  0  <_ 
( Re `  ( C  /  ( _i ^
k ) ) ) )  <->  ( x  e.  A  /\  0  <_ 
( Re `  ( D  /  ( _i ^
k ) ) ) ) ) )
6 itgeq12dv.2 . . . . . . . . . 10  |-  ( ph  ->  A  =  B )
76eleq2d 2537 . . . . . . . . 9  |-  ( ph  ->  ( x  e.  A  <->  x  e.  B ) )
87anbi1d 704 . . . . . . . 8  |-  ( ph  ->  ( ( x  e.  A  /\  0  <_ 
( Re `  ( D  /  ( _i ^
k ) ) ) )  <->  ( x  e.  B  /\  0  <_ 
( Re `  ( D  /  ( _i ^
k ) ) ) ) ) )
95, 8bitrd 253 . . . . . . 7  |-  ( ph  ->  ( ( x  e.  A  /\  0  <_ 
( Re `  ( C  /  ( _i ^
k ) ) ) )  <->  ( x  e.  B  /\  0  <_ 
( Re `  ( D  /  ( _i ^
k ) ) ) ) ) )
103adantrr 716 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  A  /\  0  <_  ( Re `  ( C  /  ( _i ^
k ) ) ) ) )  ->  (
Re `  ( C  /  ( _i ^
k ) ) )  =  ( Re `  ( D  /  (
_i ^ k ) ) ) )
11 eqidd 2468 . . . . . . 7  |-  ( (
ph  /\  -.  (
x  e.  A  /\  0  <_  ( Re `  ( C  /  (
_i ^ k ) ) ) ) )  ->  0  =  0 )
129, 10, 11ifbieq12d2 27122 . . . . . 6  |-  ( ph  ->  if ( ( x  e.  A  /\  0  <_  ( Re `  ( C  /  ( _i ^
k ) ) ) ) ,  ( Re
`  ( C  / 
( _i ^ k
) ) ) ,  0 )  =  if ( ( x  e.  B  /\  0  <_ 
( Re `  ( D  /  ( _i ^
k ) ) ) ) ,  ( Re
`  ( D  / 
( _i ^ k
) ) ) ,  0 ) )
1312mpteq2dv 4534 . . . . 5  |-  ( ph  ->  ( x  e.  RR  |->  if ( ( x  e.  A  /\  0  <_ 
( Re `  ( C  /  ( _i ^
k ) ) ) ) ,  ( Re
`  ( C  / 
( _i ^ k
) ) ) ,  0 ) )  =  ( x  e.  RR  |->  if ( ( x  e.  B  /\  0  <_ 
( Re `  ( D  /  ( _i ^
k ) ) ) ) ,  ( Re
`  ( D  / 
( _i ^ k
) ) ) ,  0 ) ) )
1413fveq2d 5870 . . . 4  |-  ( ph  ->  ( S.2 `  (
x  e.  RR  |->  if ( ( x  e.  A  /\  0  <_ 
( Re `  ( C  /  ( _i ^
k ) ) ) ) ,  ( Re
`  ( C  / 
( _i ^ k
) ) ) ,  0 ) ) )  =  ( S.2 `  (
x  e.  RR  |->  if ( ( x  e.  B  /\  0  <_ 
( Re `  ( D  /  ( _i ^
k ) ) ) ) ,  ( Re
`  ( D  / 
( _i ^ k
) ) ) ,  0 ) ) ) )
1514oveq2d 6300 . . 3  |-  ( ph  ->  ( ( _i ^
k )  x.  ( S.2 `  ( x  e.  RR  |->  if ( ( x  e.  A  /\  0  <_  ( Re `  ( C  /  (
_i ^ k ) ) ) ) ,  ( Re `  ( C  /  ( _i ^
k ) ) ) ,  0 ) ) ) )  =  ( ( _i ^ k
)  x.  ( S.2 `  ( x  e.  RR  |->  if ( ( x  e.  B  /\  0  <_ 
( Re `  ( D  /  ( _i ^
k ) ) ) ) ,  ( Re
`  ( D  / 
( _i ^ k
) ) ) ,  0 ) ) ) ) )
1615sumeq2sdv 13489 . 2  |-  ( ph  -> 
sum_ k  e.  ( 0 ... 3 ) ( ( _i ^
k )  x.  ( S.2 `  ( x  e.  RR  |->  if ( ( x  e.  A  /\  0  <_  ( Re `  ( C  /  (
_i ^ k ) ) ) ) ,  ( Re `  ( C  /  ( _i ^
k ) ) ) ,  0 ) ) ) )  =  sum_ k  e.  ( 0 ... 3 ) ( ( _i ^ k
)  x.  ( S.2 `  ( x  e.  RR  |->  if ( ( x  e.  B  /\  0  <_ 
( Re `  ( D  /  ( _i ^
k ) ) ) ) ,  ( Re
`  ( D  / 
( _i ^ k
) ) ) ,  0 ) ) ) ) )
17 eqid 2467 . . 3  |-  ( Re
`  ( C  / 
( _i ^ k
) ) )  =  ( Re `  ( C  /  ( _i ^
k ) ) )
1817dfitg 21939 . 2  |-  S. A C  _d x  =  sum_ k  e.  ( 0 ... 3 ) ( ( _i ^ k
)  x.  ( S.2 `  ( x  e.  RR  |->  if ( ( x  e.  A  /\  0  <_ 
( Re `  ( C  /  ( _i ^
k ) ) ) ) ,  ( Re
`  ( C  / 
( _i ^ k
) ) ) ,  0 ) ) ) )
19 eqid 2467 . . 3  |-  ( Re
`  ( D  / 
( _i ^ k
) ) )  =  ( Re `  ( D  /  ( _i ^
k ) ) )
2019dfitg 21939 . 2  |-  S. B D  _d x  =  sum_ k  e.  ( 0 ... 3 ) ( ( _i ^ k
)  x.  ( S.2 `  ( x  e.  RR  |->  if ( ( x  e.  B  /\  0  <_ 
( Re `  ( D  /  ( _i ^
k ) ) ) ) ,  ( Re
`  ( D  / 
( _i ^ k
) ) ) ,  0 ) ) ) )
2116, 18, 203eqtr4g 2533 1  |-  ( ph  ->  S. A C  _d x  =  S. B D  _d x )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   ifcif 3939   class class class wbr 4447    |-> cmpt 4505   ` cfv 5588  (class class class)co 6284   RRcr 9491   0cc0 9492   _ici 9494    x. cmul 9497    <_ cle 9629    / cdiv 10206   3c3 10586   ...cfz 11672   ^cexp 12134   Recre 12893   sum_csu 13471   S.2citg2 21788   S.citg 21790
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-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569
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 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-om 6685  df-1st 6784  df-2nd 6785  df-recs 7042  df-rdg 7076  df-er 7311  df-en 7517  df-dom 7518  df-sdom 7519  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-nn 10537  df-n0 10796  df-z 10865  df-uz 11083  df-fz 11673  df-seq 12076  df-sum 13472  df-itg 21795
This theorem is referenced by: (None)
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