Theorem ditgeqiooicc 40176

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 12259	. . . . . . 7 |- ( A (,) B ) C_ ( A [,] B )
2	1	sseli 3599	. . . . . 6 |- ( x e. ( A (,) B ) -> x e. ( A [,] B ) )
3	2	adantl 482	. . . . 5 |- ( ( ph /\ x e. ( A (,) B ) ) -> x e. ( A [,] B ) )
4		ditgeqiooicc.2	. . . . . . . . . . 11 |- ( ph -> A e. RR )
5	4	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ x e. ( A (,) B ) ) -> A e. RR )
6		simpr 477	. . . . . . . . . . . 12 |- ( ( ph /\ x e. ( A (,) B ) ) -> x e. ( A (,) B ) )
7	5	rexrd 10089	. . . . . . . . . . . . 13 |- ( ( ph /\ x e. ( A (,) B ) ) -> A e. RR* )
8		ditgeqiooicc.3	. . . . . . . . . . . . . . 15 |- ( ph -> B e. RR )
9	8	adantr 481	. . . . . . . . . . . . . 14 |- ( ( ph /\ x e. ( A (,) B ) ) -> B e. RR )
10	9	rexrd 10089	. . . . . . . . . . . . 13 |- ( ( ph /\ x e. ( A (,) B ) ) -> B e. RR* )
11		elioo2 12216	. . . . . . . . . . . . 13 |- ( ( A e. RR* /\ B e. RR* ) -> ( x e. ( A (,) B ) <-> ( x e. RR /\ A < x /\ x < B ) ) )
12	7, 10, 11	syl2anc 693	. . . . . . . . . . . 12 |- ( ( ph /\ x e. ( A (,) B ) ) -> ( x e. ( A (,) B ) <-> ( x e. RR /\ A < x /\ x < B ) ) )
13	6, 12	mpbid 222	. . . . . . . . . . 11 |- ( ( ph /\ x e. ( A (,) B ) ) -> ( x e. RR /\ A < x /\ x < B ) )
14	13	simp2d 1074	. . . . . . . . . 10 |- ( ( ph /\ x e. ( A (,) B ) ) -> A < x )
15	5, 14	gtned 10172	. . . . . . . . 9 |- ( ( ph /\ x e. ( A (,) B ) ) -> x =/= A )
16	15	neneqd 2799	. . . . . . . 8 |- ( ( ph /\ x e. ( A (,) B ) ) -> -. x = A )
17	16	iffalsed 4097	. . . . . . 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 1073	. . . . . . . . . 10 |- ( ( ph /\ x e. ( A (,) B ) ) -> x e. RR )
19	13	simp3d 1075	. . . . . . . . . 10 |- ( ( ph /\ x e. ( A (,) B ) ) -> x < B )
20	18, 19	ltned 10173	. . . . . . . . 9 |- ( ( ph /\ x e. ( A (,) B ) ) -> x =/= B )
21	20	neneqd 2799	. . . . . . . 8 |- ( ( ph /\ x e. ( A (,) B ) ) -> -. x = B )
22	21	iffalsed 4097	. . . . . . 7 |- ( ( ph /\ x e. ( A (,) B ) ) -> if ( x = B , L , ( F ` x ) ) = ( F ` x ) )
23	17, 22	eqtrd 2656	. . . . . 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	ffvelrnda 6359	. . . . . 6 |- ( ( ph /\ x e. ( A (,) B ) ) -> ( F ` x ) e. RR )
26	23, 25	eqeltrd 2701	. . . . 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 6291	. . . . 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 693	. . . 4 |- ( ( ph /\ x e. ( A (,) B ) ) -> ( G ` x ) = if ( x = A , R , if ( x = B , L , ( F ` x ) ) ) )
30	29, 17, 22	3eqtrrd 2661	. . 3 |- ( ( ph /\ x e. ( A (,) B ) ) -> ( F ` x ) = ( G ` x ) )
31	30	itgeq2dv 23548	. 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 23620	. 2 |- ( ph -> S__ [ A -> B ] ( F ` x ) _d x = S. ( A (,) B ) ( F ` x ) _d x )
34	32	ditgpos 23620	. 2 |- ( ph -> S__ [ A -> B ] ( G ` x ) _d x = S. ( A (,) B ) ( G ` x ) _d x )
35	31, 33, 34	3eqtr4d 2666	1 |- ( ph -> S__ [ A -> B ] ( F ` x ) _d x = S__ [ A -> B ] ( G ` x ) _d x )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   ifcif 4086   class class class wbr 4653   |-> cmpt 4729   -->wf 5884   ` cfv 5888  (class class class)co 6650   RRcr 9935   RR*cxr 10073   < clt 10074   <_ cle 10075   (,)cioo 12175   [,]cicc 12178   S.citg 23387   S__cdit 23610

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-ioo 12179  df-icc 12182  df-fz 12327  df-seq 12802  df-sum 14417  df-itg 23392  df-ditg 23611

This theorem is referenced by: (None)