Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  itgeq12dv Unicode version

Theorem itgeq12dv 23905
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 5957 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  A )  ->  ( C  /  ( _i ^
k ) )  =  ( D  /  (
_i ^ k ) ) )
32fveq2d 5609 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  A )  ->  (
Re `  ( C  /  ( _i ^
k ) ) )  =  ( Re `  ( D  /  (
_i ^ k ) ) ) )
43breq2d 4114 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  A )  ->  (
0  <_  ( Re `  ( C  /  (
_i ^ k ) ) )  <->  0  <_  ( Re `  ( D  /  ( _i ^
k ) ) ) ) )
54pm5.32da 622 . . . . . . . 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 2425 . . . . . . . . 9  |-  ( ph  ->  ( x  e.  A  <->  x  e.  B ) )
87anbi1d 685 . . . . . . . 8  |-  ( ph  ->  ( ( x  e.  A  /\  0  <_ 
( Re `  ( D  /  ( _i ^
k ) ) ) )  <->  ( x  e.  B  /\  0  <_ 
( Re `  ( D  /  ( _i ^
k ) ) ) ) ) )
95, 8bitrd 244 . . . . . . 7  |-  ( ph  ->  ( ( x  e.  A  /\  0  <_ 
( Re `  ( C  /  ( _i ^
k ) ) ) )  <->  ( x  e.  B  /\  0  <_ 
( Re `  ( D  /  ( _i ^
k ) ) ) ) ) )
103adantrr 697 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  A  /\  0  <_  ( Re `  ( C  /  ( _i ^
k ) ) ) ) )  ->  (
Re `  ( C  /  ( _i ^
k ) ) )  =  ( Re `  ( D  /  (
_i ^ k ) ) ) )
11 eqidd 2359 . . . . . . . 8  |-  ( ph  ->  0  =  0 )
1211adantr 451 . . . . . . 7  |-  ( (
ph  /\  -.  (
x  e.  A  /\  0  <_  ( Re `  ( C  /  (
_i ^ k ) ) ) ) )  ->  0  =  0 )
139, 10, 12ifbieq12d2 23198 . . . . . 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 ) )
1413mpteq2dv 4186 . . . . 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 ) ) )
1514fveq2d 5609 . . . 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 ) ) ) )
1615oveq2d 5958 . . 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 ) ) ) ) )
1716sumeq2sdv 12268 . 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 ) ) ) ) )
18 eqid 2358 . . 3  |-  ( Re
`  ( C  / 
( _i ^ k
) ) )  =  ( Re `  ( C  /  ( _i ^
k ) ) )
1918dfitg 19222 . 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 ) ) ) )
20 eqid 2358 . . 3  |-  ( Re
`  ( D  / 
( _i ^ k
) ) )  =  ( Re `  ( D  /  ( _i ^
k ) ) )
2120dfitg 19222 . 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 ) ) ) )
2217, 19, 213eqtr4g 2415 1  |-  ( ph  ->  S. A C  _d x  =  S. B D  _d x )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    = wceq 1642    e. wcel 1710   ifcif 3641   class class class wbr 4102    e. cmpt 4156   ` cfv 5334  (class class class)co 5942   RRcr 8823   0cc0 8824   _ici 8826    x. cmul 8829    <_ cle 8955    / cdiv 9510   3c3 9883   ...cfz 10871   ^cexp 11194   Recre 11672   sum_csu 12249   S.2citg2 19069   S.citg 19071
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591  ax-cnex 8880  ax-resscn 8881  ax-1cn 8882  ax-icn 8883  ax-addcl 8884  ax-addrcl 8885  ax-mulcl 8886  ax-mulrcl 8887  ax-mulcom 8888  ax-addass 8889  ax-mulass 8890  ax-distr 8891  ax-i2m1 8892  ax-1ne0 8893  ax-1rid 8894  ax-rnegex 8895  ax-rrecex 8896  ax-cnre 8897  ax-pre-lttri 8898  ax-pre-lttrn 8899  ax-pre-ltadd 8900  ax-pre-mulgt0 8901
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3907  df-iun 3986  df-br 4103  df-opab 4157  df-mpt 4158  df-tr 4193  df-eprel 4384  df-id 4388  df-po 4393  df-so 4394  df-fr 4431  df-we 4433  df-ord 4474  df-on 4475  df-lim 4476  df-suc 4477  df-om 4736  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-ov 5945  df-oprab 5946  df-mpt2 5947  df-1st 6206  df-2nd 6207  df-riota 6388  df-recs 6472  df-rdg 6507  df-er 6744  df-en 6949  df-dom 6950  df-sdom 6951  df-pnf 8956  df-mnf 8957  df-xr 8958  df-ltxr 8959  df-le 8960  df-sub 9126  df-neg 9127  df-nn 9834  df-n0 10055  df-z 10114  df-uz 10320  df-fz 10872  df-seq 11136  df-sum 12250  df-itg 19077
  Copyright terms: Public domain W3C validator