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Theorem ditgeqiooicc 37837
Description: A function  F on an open interval, has the same directed integral as its extension  G on the closed interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
Hypotheses
Ref Expression
ditgeqiooicc.1  |-  G  =  ( x  e.  ( A [,] B ) 
|->  if ( x  =  A ,  R ,  if ( x  =  B ,  L ,  ( F `  x ) ) ) )
ditgeqiooicc.2  |-  ( ph  ->  A  e.  RR )
ditgeqiooicc.3  |-  ( ph  ->  B  e.  RR )
ditgeqiooicc.4  |-  ( ph  ->  A  <_  B )
ditgeqiooicc.5  |-  ( ph  ->  F : ( A (,) B ) --> RR )
Assertion
Ref Expression
ditgeqiooicc  |-  ( ph  ->  S__ [ A  ->  B ] ( F `  x )  _d x  =  S__ [ A  ->  B ] ( G `
 x )  _d x )
Distinct variable groups:    x, A    x, B    ph, x
Allowed substitution hints:    R( x)    F( x)    G( x)    L( x)

Proof of Theorem ditgeqiooicc
StepHypRef Expression
1 ioossicc 11720 . . . . . . 7  |-  ( A (,) B )  C_  ( A [,] B )
21sseli 3428 . . . . . 6  |-  ( x  e.  ( A (,) B )  ->  x  e.  ( A [,] B
) )
32adantl 468 . . . . 5  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  x  e.  ( A [,] B ) )
4 ditgeqiooicc.2 . . . . . . . . . . 11  |-  ( ph  ->  A  e.  RR )
54adantr 467 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  A  e.  RR )
6 simpr 463 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  x  e.  ( A (,) B ) )
75rexrd 9690 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  A  e.  RR* )
8 ditgeqiooicc.3 . . . . . . . . . . . . . . 15  |-  ( ph  ->  B  e.  RR )
98adantr 467 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  B  e.  RR )
109rexrd 9690 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  B  e.  RR* )
11 elioo2 11677 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
x  e.  ( A (,) B )  <->  ( x  e.  RR  /\  A  < 
x  /\  x  <  B ) ) )
127, 10, 11syl2anc 667 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  ( x  e.  ( A (,) B
)  <->  ( x  e.  RR  /\  A  < 
x  /\  x  <  B ) ) )
136, 12mpbid 214 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  ( x  e.  RR  /\  A  < 
x  /\  x  <  B ) )
1413simp2d 1021 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  A  <  x )
155, 14gtned 9770 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  x  =/=  A )
1615neneqd 2629 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  -.  x  =  A )
1716iffalsed 3892 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  if (
x  =  A ,  R ,  if (
x  =  B ,  L ,  ( F `  x ) ) )  =  if ( x  =  B ,  L ,  ( F `  x ) ) )
1813simp1d 1020 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  x  e.  RR )
1913simp3d 1022 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  x  <  B )
2018, 19ltned 9771 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  x  =/=  B )
2120neneqd 2629 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  -.  x  =  B )
2221iffalsed 3892 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  if (
x  =  B ,  L ,  ( F `  x ) )  =  ( F `  x
) )
2317, 22eqtrd 2485 . . . . . 6  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  if (
x  =  A ,  R ,  if (
x  =  B ,  L ,  ( F `  x ) ) )  =  ( F `  x ) )
24 ditgeqiooicc.5 . . . . . . 7  |-  ( ph  ->  F : ( A (,) B ) --> RR )
2524fnvinran 37335 . . . . . 6  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  ( F `  x )  e.  RR )
2623, 25eqeltrd 2529 . . . . 5  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  if (
x  =  A ,  R ,  if (
x  =  B ,  L ,  ( F `  x ) ) )  e.  RR )
27 ditgeqiooicc.1 . . . . . 6  |-  G  =  ( x  e.  ( A [,] B ) 
|->  if ( x  =  A ,  R ,  if ( x  =  B ,  L ,  ( F `  x ) ) ) )
2827fvmpt2 5957 . . . . 5  |-  ( ( x  e.  ( A [,] B )  /\  if ( x  =  A ,  R ,  if ( x  =  B ,  L ,  ( F `
 x ) ) )  e.  RR )  ->  ( G `  x )  =  if ( x  =  A ,  R ,  if ( x  =  B ,  L ,  ( F `
 x ) ) ) )
293, 26, 28syl2anc 667 . . . 4  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  ( G `  x )  =  if ( x  =  A ,  R ,  if ( x  =  B ,  L ,  ( F `
 x ) ) ) )
3029, 17, 223eqtrrd 2490 . . 3  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  ( F `  x )  =  ( G `  x ) )
3130itgeq2dv 22739 . 2  |-  ( ph  ->  S. ( A (,) B ) ( F `
 x )  _d x  =  S. ( A (,) B ) ( G `  x
)  _d x )
32 ditgeqiooicc.4 . . 3  |-  ( ph  ->  A  <_  B )
3332ditgpos 22811 . 2  |-  ( ph  ->  S__ [ A  ->  B ] ( F `  x )  _d x  =  S. ( A (,) B ) ( F `  x )  _d x )
3432ditgpos 22811 . 2  |-  ( ph  ->  S__ [ A  ->  B ] ( G `  x )  _d x  =  S. ( A (,) B ) ( G `  x )  _d x )
3531, 33, 343eqtr4d 2495 1  |-  ( ph  ->  S__ [ A  ->  B ] ( F `  x )  _d x  =  S__ [ A  ->  B ] ( G `
 x )  _d x )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    /\ w3a 985    = wceq 1444    e. wcel 1887   ifcif 3881   class class class wbr 4402    |-> cmpt 4461   -->wf 5578   ` cfv 5582  (class class class)co 6290   RRcr 9538   RR*cxr 9674    < clt 9675    <_ cle 9676   (,)cioo 11635   [,]cicc 11638   S.citg 22576   S__cdit 22801
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-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  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
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-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-iun 4280  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-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-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-1st 6793  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-er 7363  df-en 7570  df-dom 7571  df-sdom 7572  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-nn 10610  df-n0 10870  df-z 10938  df-uz 11160  df-ioo 11639  df-icc 11642  df-fz 11785  df-seq 12214  df-sum 13753  df-itg 22581  df-ditg 22802
This theorem is referenced by: (None)
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