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

Step	Hyp	Ref	Expression
1		ioossicc 11720	. . . . . . 7 |- ( A (,) B ) C_ ( A [,] B )
2	1	sseli 3428	. . . . . 6 |- ( x e. ( A (,) B ) -> x e. ( A [,] B ) )
3	2	adantl 468	. . . . 5 |- ( ( ph /\ x e. ( A (,) B ) ) -> x e. ( A [,] B ) )
4		ditgeqiooicc.2	. . . . . . . . . . 11 |- ( ph -> A e. RR )
5	4	adantr 467	. . . . . . . . . 10 |- ( ( ph /\ x e. ( A (,) B ) ) -> A e. RR )
6		simpr 463	. . . . . . . . . . . 12 |- ( ( ph /\ x e. ( A (,) B ) ) -> x e. ( A (,) B ) )
7	5	rexrd 9690	. . . . . . . . . . . . 13 |- ( ( ph /\ x e. ( A (,) B ) ) -> A e. RR* )
8		ditgeqiooicc.3	. . . . . . . . . . . . . . 15 |- ( ph -> B e. RR )
9	8	adantr 467	. . . . . . . . . . . . . 14 |- ( ( ph /\ x e. ( A (,) B ) ) -> B e. RR )
10	9	rexrd 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 ) ) )
12	7, 10, 11	syl2anc 667	. . . . . . . . . . . 12 |- ( ( ph /\ x e. ( A (,) B ) ) -> ( x e. ( A (,) B ) <-> ( x e. RR /\ A < x /\ x < B ) ) )
13	6, 12	mpbid 214	. . . . . . . . . . 11 |- ( ( ph /\ x e. ( A (,) B ) ) -> ( x e. RR /\ A < x /\ x < B ) )
14	13	simp2d 1021	. . . . . . . . . 10 |- ( ( ph /\ x e. ( A (,) B ) ) -> A < x )
15	5, 14	gtned 9770	. . . . . . . . 9 |- ( ( ph /\ x e. ( A (,) B ) ) -> x =/= A )
16	15	neneqd 2629	. . . . . . . 8 |- ( ( ph /\ x e. ( A (,) B ) ) -> -. x = A )
17	16	iffalsed 3892	. . . . . . 7 |- ( ( ph /\ x e. ( A (,) B ) ) -> if ( x = A , R , if ( x = B , L , ( F ` x ) ) ) = if ( x = B , L , ( F ` x ) ) )
18	13	simp1d 1020	. . . . . . . . . 10 |- ( ( ph /\ x e. ( A (,) B ) ) -> x e. RR )
19	13	simp3d 1022	. . . . . . . . . 10 |- ( ( ph /\ x e. ( A (,) B ) ) -> x < B )
20	18, 19	ltned 9771	. . . . . . . . 9 |- ( ( ph /\ x e. ( A (,) B ) ) -> x =/= B )
21	20	neneqd 2629	. . . . . . . 8 |- ( ( ph /\ x e. ( A (,) B ) ) -> -. x = B )
22	21	iffalsed 3892	. . . . . . 7 |- ( ( ph /\ x e. ( A (,) B ) ) -> if ( x = B , L , ( F ` x ) ) = ( F ` x ) )
23	17, 22	eqtrd 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 )
25	24	fnvinran 37335	. . . . . 6 |- ( ( ph /\ x e. ( A (,) B ) ) -> ( F ` x ) e. RR )
26	23, 25	eqeltrd 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 ) ) ) )
28	27	fvmpt2 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 ) ) ) )
29	3, 26, 28	syl2anc 667	. . . 4 |- ( ( ph /\ x e. ( A (,) B ) ) -> ( G ` x ) = if ( x = A , R , if ( x = B , L , ( F ` x ) ) ) )
30	29, 17, 22	3eqtrrd 2490	. . 3 |- ( ( ph /\ x e. ( A (,) B ) ) -> ( F ` x ) = ( G ` x ) )
31	30	itgeq2dv 22739	. 2 |- ( ph -> S. ( A (,) B ) ( F ` x ) _d x = S. ( A (,) B ) ( G ` x ) _d x )
32		ditgeqiooicc.4	. . 3 |- ( ph -> A <_ B )
33	32	ditgpos 22811	. 2 |- ( ph -> S__ [ A -> B ] ( F ` x ) _d x = S. ( A (,) B ) ( F ` x ) _d x )
34	32	ditgpos 22811	. 2 |- ( ph -> S__ [ A -> B ] ( G ` x ) _d x = S. ( A (,) B ) ( G ` x ) _d x )
35	31, 33, 34	3eqtr4d 2495	1 |- ( ph -> S__ [ A -> B ] ( F ` x ) _d x = S__ [ A -> B ] ( G ` x ) _d x )

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)