Theorem itgioocnicc 37854

Description: The integral of a piecewise continuous function F on an open interval is equal to the integral of the continuous function G, in the corresponding closed interval. G is equal to F on the open interval, but it is continuous at the two boundaries, also. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Hypotheses

Ref	Expression
itgioocnicc.a	|- ( ph -> A e. RR )
itgioocnicc.b	|- ( ph -> B e. RR )
itgioocnicc.f	|- ( ph -> F : dom F --> CC )
itgioocnicc.fcn	|- ( ph -> ( F |` ( A (,) B ) ) e. ( ( A (,) B ) -cn-> CC ) )
itgioocnicc.fdom	|- ( ph -> ( A [,] B ) C_ dom F )
itgioocnicc.r	|- ( ph -> R e. ( ( F |` ( A (,) B ) ) lim CC A ) )
itgioocnicc.l	|- ( ph -> L e. ( ( F |` ( A (,) B ) ) lim CC B ) )
itgioocnicc.g	|- G = ( x e. ( A [,] B ) |-> if ( x = A , R , if ( x = B , L , ( F ` x ) ) ) )

Assertion

Ref	Expression
itgioocnicc	|- ( ph -> ( G e. L^1 /\ S. ( A [,] B ) ( G ` x ) _d x = S. ( A [,] B ) ( F ` x ) _d x ) )

Distinct variable groups:   x, A   x, B   x, F   x, L   x, R   ph, x

Allowed substitution hint:   G( x)

Proof of Theorem itgioocnicc

Step	Hyp	Ref	Expression
1		itgioocnicc.a	. . 3 |- ( ph -> A e. RR )
2		itgioocnicc.b	. . 3 |- ( ph -> B e. RR )
3		itgioocnicc.g	. . . . 5 |- G = ( x e. ( A [,] B ) |-> if ( x = A , R , if ( x = B , L , ( F ` x ) ) ) )
4		iftrue 3887	. . . . . . . . 9 |- ( x = A -> if ( x = A , R , if ( x = B , L , ( F ` x ) ) ) = R )
5		iftrue 3887	. . . . . . . . 9 |- ( x = A -> if ( x = A , R , if ( x = B , L , ( ( F |` ( A (,) B ) ) ` x ) ) ) = R )
6	4, 5	eqtr4d 2488	. . . . . . . 8 |- ( x = A -> if ( x = A , R , if ( x = B , L , ( F ` x ) ) ) = if ( x = A , R , if ( x = B , L , ( ( F |` ( A (,) B ) ) ` x ) ) ) )
7	6	adantl 468	. . . . . . 7 |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ x = A ) -> if ( x = A , R , if ( x = B , L , ( F ` x ) ) ) = if ( x = A , R , if ( x = B , L , ( ( F |` ( A (,) B ) ) ` x ) ) ) )
8		iftrue 3887	. . . . . . . . . . . 12 |- ( x = B -> if ( x = B , L , ( F ` x ) ) = L )
9		iftrue 3887	. . . . . . . . . . . 12 |- ( x = B -> if ( x = B , L , ( ( F |` ( A (,) B ) ) ` x ) ) = L )
10	8, 9	eqtr4d 2488	. . . . . . . . . . 11 |- ( x = B -> if ( x = B , L , ( F ` x ) ) = if ( x = B , L , ( ( F |` ( A (,) B ) ) ` x ) ) )
11	10	adantl 468	. . . . . . . . . 10 |- ( ( -. x = A /\ x = B ) -> if ( x = B , L , ( F ` x ) ) = if ( x = B , L , ( ( F |` ( A (,) B ) ) ` x ) ) )
12	11	ifeq2d 3900	. . . . . . . . 9 |- ( ( -. x = A /\ x = B ) -> if ( x = A , R , if ( x = B , L , ( F ` x ) ) ) = if ( x = A , R , if ( x = B , L , ( ( F |` ( A (,) B ) ) ` x ) ) ) )
13	12	adantll 720	. . . . . . . 8 |- ( ( ( ( ph /\ x e. ( A [,] B ) ) /\ -. x = A ) /\ x = B ) -> if ( x = A , R , if ( x = B , L , ( F ` x ) ) ) = if ( x = A , R , if ( x = B , L , ( ( F |` ( A (,) B ) ) ` x ) ) ) )
14		iffalse 3890	. . . . . . . . . 10 |- ( -. x = A -> if ( x = A , R , if ( x = B , L , ( F ` x ) ) ) = if ( x = B , L , ( F ` x ) ) )
15	14	ad2antlr 733	. . . . . . . . 9 |- ( ( ( ( ph /\ x e. ( A [,] B ) ) /\ -. x = A ) /\ -. x = B ) -> if ( x = A , R , if ( x = B , L , ( F ` x ) ) ) = if ( x = B , L , ( F ` x ) ) )
16		iffalse 3890	. . . . . . . . . 10 |- ( -. x = B -> if ( x = B , L , ( F ` x ) ) = ( F ` x ) )
17	16	adantl 468	. . . . . . . . 9 |- ( ( ( ( ph /\ x e. ( A [,] B ) ) /\ -. x = A ) /\ -. x = B ) -> if ( x = B , L , ( F ` x ) ) = ( F ` x ) )
18		iffalse 3890	. . . . . . . . . . 11 |- ( -. x = A -> if ( x = A , R , if ( x = B , L , ( ( F |` ( A (,) B ) ) ` x ) ) ) = if ( x = B , L , ( ( F |` ( A (,) B ) ) ` x ) ) )
19	18	ad2antlr 733	. . . . . . . . . 10 |- ( ( ( ( ph /\ x e. ( A [,] B ) ) /\ -. x = A ) /\ -. x = B ) -> if ( x = A , R , if ( x = B , L , ( ( F |` ( A (,) B ) ) ` x ) ) ) = if ( x = B , L , ( ( F |` ( A (,) B ) ) ` x ) ) )
20		iffalse 3890	. . . . . . . . . . 11 |- ( -. x = B -> if ( x = B , L , ( ( F |` ( A (,) B ) ) ` x ) ) = ( ( F |` ( A (,) B ) ) ` x ) )
21	20	adantl 468	. . . . . . . . . 10 |- ( ( ( ( ph /\ x e. ( A [,] B ) ) /\ -. x = A ) /\ -. x = B ) -> if ( x = B , L , ( ( F |` ( A (,) B ) ) ` x ) ) = ( ( F |` ( A (,) B ) ) ` x ) )
22	1	adantr 467	. . . . . . . . . . . . . 14 |- ( ( ph /\ x e. ( A [,] B ) ) -> A e. RR )
23	22	rexrd 9690	. . . . . . . . . . . . 13 |- ( ( ph /\ x e. ( A [,] B ) ) -> A e. RR* )
24	23	ad2antrr 732	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ x e. ( A [,] B ) ) /\ -. x = A ) /\ -. x = B ) -> A e. RR* )
25	2	rexrd 9690	. . . . . . . . . . . . 13 |- ( ph -> B e. RR* )
26	25	ad3antrrr 736	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ x e. ( A [,] B ) ) /\ -. x = A ) /\ -. x = B ) -> B e. RR* )
27	2	adantr 467	. . . . . . . . . . . . . 14 |- ( ( ph /\ x e. ( A [,] B ) ) -> B e. RR )
28		simpr 463	. . . . . . . . . . . . . 14 |- ( ( ph /\ x e. ( A [,] B ) ) -> x e. ( A [,] B ) )
29		eliccre 37603	. . . . . . . . . . . . . 14 |- ( ( A e. RR /\ B e. RR /\ x e. ( A [,] B ) ) -> x e. RR )
30	22, 27, 28, 29	syl3anc 1268	. . . . . . . . . . . . 13 |- ( ( ph /\ x e. ( A [,] B ) ) -> x e. RR )
31	30	ad2antrr 732	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ x e. ( A [,] B ) ) /\ -. x = A ) /\ -. x = B ) -> x e. RR )
32	1	ad2antrr 732	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ -. x = A ) -> A e. RR )
33	30	adantr 467	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ -. x = A ) -> x e. RR )
34	25	adantr 467	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ x e. ( A [,] B ) ) -> B e. RR* )
35		iccgelb 11691	. . . . . . . . . . . . . . . 16 |- ( ( A e. RR* /\ B e. RR* /\ x e. ( A [,] B ) ) -> A <_ x )
36	23, 34, 28, 35	syl3anc 1268	. . . . . . . . . . . . . . 15 |- ( ( ph /\ x e. ( A [,] B ) ) -> A <_ x )
37	36	adantr 467	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ -. x = A ) -> A <_ x )
38		neqne 37374	. . . . . . . . . . . . . . 15 |- ( -. x = A -> x =/= A )
39	38	adantl 468	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ -. x = A ) -> x =/= A )
40	32, 33, 37, 39	leneltd 9789	. . . . . . . . . . . . 13 |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ -. x = A ) -> A < x )
41	40	adantr 467	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ x e. ( A [,] B ) ) /\ -. x = A ) /\ -. x = B ) -> A < x )
42	30	adantr 467	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ -. x = B ) -> x e. RR )
43	2	ad2antrr 732	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ -. x = B ) -> B e. RR )
44		iccleub 11690	. . . . . . . . . . . . . . . 16 |- ( ( A e. RR* /\ B e. RR* /\ x e. ( A [,] B ) ) -> x <_ B )
45	23, 34, 28, 44	syl3anc 1268	. . . . . . . . . . . . . . 15 |- ( ( ph /\ x e. ( A [,] B ) ) -> x <_ B )
46	45	adantr 467	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ -. x = B ) -> x <_ B )
47		eqcom 2458	. . . . . . . . . . . . . . . . . 18 |- ( x = B <-> B = x )
48	47	notbii 298	. . . . . . . . . . . . . . . . 17 |- ( -. x = B <-> -. B = x )
49	48	biimpi 198	. . . . . . . . . . . . . . . 16 |- ( -. x = B -> -. B = x )
50	49	neqned 2631	. . . . . . . . . . . . . . 15 |- ( -. x = B -> B =/= x )
51	50	adantl 468	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ -. x = B ) -> B =/= x )
52	42, 43, 46, 51	leneltd 9789	. . . . . . . . . . . . 13 |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ -. x = B ) -> x < B )
53	52	adantlr 721	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ x e. ( A [,] B ) ) /\ -. x = A ) /\ -. x = B ) -> x < B )
54	24, 26, 31, 41, 53	eliood 37595	. . . . . . . . . . 11 |- ( ( ( ( ph /\ x e. ( A [,] B ) ) /\ -. x = A ) /\ -. x = B ) -> x e. ( A (,) B ) )
55		fvres 5879	. . . . . . . . . . 11 |- ( x e. ( A (,) B ) -> ( ( F |` ( A (,) B ) ) ` x ) = ( F ` x ) )
56	54, 55	syl 17	. . . . . . . . . 10 |- ( ( ( ( ph /\ x e. ( A [,] B ) ) /\ -. x = A ) /\ -. x = B ) -> ( ( F |` ( A (,) B ) ) ` x ) = ( F ` x ) )
57	19, 21, 56	3eqtrrd 2490	. . . . . . . . 9 |- ( ( ( ( ph /\ x e. ( A [,] B ) ) /\ -. x = A ) /\ -. x = B ) -> ( F ` x ) = if ( x = A , R , if ( x = B , L , ( ( F |` ( A (,) B ) ) ` x ) ) ) )
58	15, 17, 57	3eqtrd 2489	. . . . . . . 8 |- ( ( ( ( ph /\ x e. ( A [,] B ) ) /\ -. x = A ) /\ -. x = B ) -> if ( x = A , R , if ( x = B , L , ( F ` x ) ) ) = if ( x = A , R , if ( x = B , L , ( ( F |` ( A (,) B ) ) ` x ) ) ) )
59	13, 58	pm2.61dan 800	. . . . . . 7 |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ -. x = A ) -> if ( x = A , R , if ( x = B , L , ( F ` x ) ) ) = if ( x = A , R , if ( x = B , L , ( ( F |` ( A (,) B ) ) ` x ) ) ) )
60	7, 59	pm2.61dan 800	. . . . . 6 |- ( ( ph /\ x e. ( A [,] B ) ) -> if ( x = A , R , if ( x = B , L , ( F ` x ) ) ) = if ( x = A , R , if ( x = B , L , ( ( F |` ( A (,) B ) ) ` x ) ) ) )
61	60	mpteq2dva 4489	. . . . 5 |- ( ph -> ( x e. ( A [,] B ) |-> if ( x = A , R , if ( x = B , L , ( F ` x ) ) ) ) = ( x e. ( A [,] B ) |-> if ( x = A , R , if ( x = B , L , ( ( F |` ( A (,) B ) ) ` x ) ) ) ) )
62	3, 61	syl5eq 2497	. . . 4 |- ( ph -> G = ( x e. ( A [,] B ) |-> if ( x = A , R , if ( x = B , L , ( ( F |` ( A (,) B ) ) ` x ) ) ) ) )
63		nfv 1761	. . . . 5 |- F/ x ph
64		eqid 2451	. . . . 5 |- ( x e. ( A [,] B ) |-> if ( x = A , R , if ( x = B , L , ( ( F |` ( A (,) B ) ) ` x ) ) ) ) = ( x e. ( A [,] B ) |-> if ( x = A , R , if ( x = B , L , ( ( F |` ( A (,) B ) ) ` x ) ) ) )
65		itgioocnicc.fcn	. . . . 5 |- ( ph -> ( F |` ( A (,) B ) ) e. ( ( A (,) B ) -cn-> CC ) )
66		itgioocnicc.l	. . . . 5 |- ( ph -> L e. ( ( F |` ( A (,) B ) ) lim CC B ) )
67		itgioocnicc.r	. . . . 5 |- ( ph -> R e. ( ( F |` ( A (,) B ) ) lim CC A ) )
68	63, 64, 1, 2, 65, 66, 67	cncfiooicc 37772	. . . 4 |- ( ph -> ( x e. ( A [,] B ) |-> if ( x = A , R , if ( x = B , L , ( ( F |` ( A (,) B ) ) ` x ) ) ) ) e. ( ( A [,] B ) -cn-> CC ) )
69	62, 68	eqeltrd 2529	. . 3 |- ( ph -> G e. ( ( A [,] B ) -cn-> CC ) )
70		cniccibl 22798	. . 3 |- ( ( A e. RR /\ B e. RR /\ G e. ( ( A [,] B ) -cn-> CC ) ) -> G e. L^1 )
71	1, 2, 69, 70	syl3anc 1268	. 2 |- ( ph -> G e. L^1 )
72	4	adantl 468	. . . . . . . . 9 |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ x = A ) -> if ( x = A , R , if ( x = B , L , ( F ` x ) ) ) = R )
73		limccl 22830	. . . . . . . . . . 11 |- ( ( F |` ( A (,) B ) ) lim CC A ) C_ CC
74	73, 67	sseldi 3430	. . . . . . . . . 10 |- ( ph -> R e. CC )
75	74	ad2antrr 732	. . . . . . . . 9 |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ x = A ) -> R e. CC )
76	72, 75	eqeltrd 2529	. . . . . . . 8 |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ x = A ) -> if ( x = A , R , if ( x = B , L , ( F ` x ) ) ) e. CC )
77	14, 8	sylan9eq 2505	. . . . . . . . . . 11 |- ( ( -. x = A /\ x = B ) -> if ( x = A , R , if ( x = B , L , ( F ` x ) ) ) = L )
78	77	adantll 720	. . . . . . . . . 10 |- ( ( ( ( ph /\ x e. ( A [,] B ) ) /\ -. x = A ) /\ x = B ) -> if ( x = A , R , if ( x = B , L , ( F ` x ) ) ) = L )
79		limccl 22830	. . . . . . . . . . . 12 |- ( ( F |` ( A (,) B ) ) lim CC B ) C_ CC
80	79, 66	sseldi 3430	. . . . . . . . . . 11 |- ( ph -> L e. CC )
81	80	ad3antrrr 736	. . . . . . . . . 10 |- ( ( ( ( ph /\ x e. ( A [,] B ) ) /\ -. x = A ) /\ x = B ) -> L e. CC )
82	78, 81	eqeltrd 2529	. . . . . . . . 9 |- ( ( ( ( ph /\ x e. ( A [,] B ) ) /\ -. x = A ) /\ x = B ) -> if ( x = A , R , if ( x = B , L , ( F ` x ) ) ) e. CC )
83	14, 16	sylan9eq 2505	. . . . . . . . . . 11 |- ( ( -. x = A /\ -. x = B ) -> if ( x = A , R , if ( x = B , L , ( F ` x ) ) ) = ( F ` x ) )
84	83	adantll 720	. . . . . . . . . 10 |- ( ( ( ( ph /\ x e. ( A [,] B ) ) /\ -. x = A ) /\ -. x = B ) -> if ( x = A , R , if ( x = B , L , ( F ` x ) ) ) = ( F ` x ) )
85	56	eqcomd 2457	. . . . . . . . . . 11 |- ( ( ( ( ph /\ x e. ( A [,] B ) ) /\ -. x = A ) /\ -. x = B ) -> ( F ` x ) = ( ( F |` ( A (,) B ) ) ` x ) )
86		cncff 21925	. . . . . . . . . . . . . 14 |- ( ( F |` ( A (,) B ) ) e. ( ( A (,) B ) -cn-> CC ) -> ( F |` ( A (,) B ) ) : ( A (,) B ) --> CC )
87	65, 86	syl 17	. . . . . . . . . . . . 13 |- ( ph -> ( F |` ( A (,) B ) ) : ( A (,) B ) --> CC )
88	87	ad3antrrr 736	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ x e. ( A [,] B ) ) /\ -. x = A ) /\ -. x = B ) -> ( F |` ( A (,) B ) ) : ( A (,) B ) --> CC )
89	88, 54	ffvelrnd 6023	. . . . . . . . . . 11 |- ( ( ( ( ph /\ x e. ( A [,] B ) ) /\ -. x = A ) /\ -. x = B ) -> ( ( F |` ( A (,) B ) ) ` x ) e. CC )
90	85, 89	eqeltrd 2529	. . . . . . . . . 10 |- ( ( ( ( ph /\ x e. ( A [,] B ) ) /\ -. x = A ) /\ -. x = B ) -> ( F ` x ) e. CC )
91	84, 90	eqeltrd 2529	. . . . . . . . 9 |- ( ( ( ( ph /\ x e. ( A [,] B ) ) /\ -. x = A ) /\ -. x = B ) -> if ( x = A , R , if ( x = B , L , ( F ` x ) ) ) e. CC )
92	82, 91	pm2.61dan 800	. . . . . . . 8 |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ -. x = A ) -> if ( x = A , R , if ( x = B , L , ( F ` x ) ) ) e. CC )
93	76, 92	pm2.61dan 800	. . . . . . 7 |- ( ( ph /\ x e. ( A [,] B ) ) -> if ( x = A , R , if ( x = B , L , ( F ` x ) ) ) e. CC )
94	3	fvmpt2 5957	. . . . . . 7 |- ( ( x e. ( A [,] B ) /\ if ( x = A , R , if ( x = B , L , ( F ` x ) ) ) e. CC ) -> ( G ` x ) = if ( x = A , R , if ( x = B , L , ( F ` x ) ) ) )
95	28, 93, 94	syl2anc 667	. . . . . 6 |- ( ( ph /\ x e. ( A [,] B ) ) -> ( G ` x ) = if ( x = A , R , if ( x = B , L , ( F ` x ) ) ) )
96	95, 93	eqeltrd 2529	. . . . 5 |- ( ( ph /\ x e. ( A [,] B ) ) -> ( G ` x ) e. CC )
97	1, 2, 96	itgioo 22773	. . . 4 |- ( ph -> S. ( A (,) B ) ( G ` x ) _d x = S. ( A [,] B ) ( G ` x ) _d x )
98	97	eqcomd 2457	. . 3 |- ( ph -> S. ( A [,] B ) ( G ` x ) _d x = S. ( A (,) B ) ( G ` x ) _d x )
99		ioossicc 11720	. . . . . . 7 |- ( A (,) B ) C_ ( A [,] B )
100	99	sseli 3428	. . . . . 6 |- ( x e. ( A (,) B ) -> x e. ( A [,] B ) )
101	100, 95	sylan2 477	. . . . 5 |- ( ( ph /\ x e. ( A (,) B ) ) -> ( G ` x ) = if ( x = A , R , if ( x = B , L , ( F ` x ) ) ) )
102	1	adantr 467	. . . . . . . 8 |- ( ( ph /\ x e. ( A (,) B ) ) -> A e. RR )
103		eliooord 11694	. . . . . . . . . 10 |- ( x e. ( A (,) B ) -> ( A < x /\ x < B ) )
104	103	simpld 461	. . . . . . . . 9 |- ( x e. ( A (,) B ) -> A < x )
105	104	adantl 468	. . . . . . . 8 |- ( ( ph /\ x e. ( A (,) B ) ) -> A < x )
106	102, 105	gtned 9770	. . . . . . 7 |- ( ( ph /\ x e. ( A (,) B ) ) -> x =/= A )
107	106	neneqd 2629	. . . . . 6 |- ( ( ph /\ x e. ( A (,) B ) ) -> -. x = A )
108	107, 14	syl 17	. . . . 5 |- ( ( ph /\ x e. ( A (,) B ) ) -> if ( x = A , R , if ( x = B , L , ( F ` x ) ) ) = if ( x = B , L , ( F ` x ) ) )
109	100, 30	sylan2 477	. . . . . . . 8 |- ( ( ph /\ x e. ( A (,) B ) ) -> x e. RR )
110	103	simprd 465	. . . . . . . . 9 |- ( x e. ( A (,) B ) -> x < B )
111	110	adantl 468	. . . . . . . 8 |- ( ( ph /\ x e. ( A (,) B ) ) -> x < B )
112	109, 111	ltned 9771	. . . . . . 7 |- ( ( ph /\ x e. ( A (,) B ) ) -> x =/= B )
113	112	neneqd 2629	. . . . . 6 |- ( ( ph /\ x e. ( A (,) B ) ) -> -. x = B )
114	113, 16	syl 17	. . . . 5 |- ( ( ph /\ x e. ( A (,) B ) ) -> if ( x = B , L , ( F ` x ) ) = ( F ` x ) )
115	101, 108, 114	3eqtrd 2489	. . . 4 |- ( ( ph /\ x e. ( A (,) B ) ) -> ( G ` x ) = ( F ` x ) )
116	115	itgeq2dv 22739	. . 3 |- ( ph -> S. ( A (,) B ) ( G ` x ) _d x = S. ( A (,) B ) ( F ` x ) _d x )
117		itgioocnicc.f	. . . . . 6 |- ( ph -> F : dom F --> CC )
118	117	adantr 467	. . . . 5 |- ( ( ph /\ x e. ( A [,] B ) ) -> F : dom F --> CC )
119		itgioocnicc.fdom	. . . . . 6 |- ( ph -> ( A [,] B ) C_ dom F )
120	119	sselda 3432	. . . . 5 |- ( ( ph /\ x e. ( A [,] B ) ) -> x e. dom F )
121	118, 120	ffvelrnd 6023	. . . 4 |- ( ( ph /\ x e. ( A [,] B ) ) -> ( F ` x ) e. CC )
122	1, 2, 121	itgioo 22773	. . 3 |- ( ph -> S. ( A (,) B ) ( F ` x ) _d x = S. ( A [,] B ) ( F ` x ) _d x )
123	98, 116, 122	3eqtrd 2489	. 2 |- ( ph -> S. ( A [,] B ) ( G ` x ) _d x = S. ( A [,] B ) ( F ` x ) _d x )
124	71, 123	jca 535	1 |- ( ph -> ( G e. L^1 /\ S. ( A [,] B ) ( G ` x ) _d x = S. ( A [,] B ) ( F ` x ) _d x ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 371   = wceq 1444   e. wcel 1887   =/= wne 2622   C_ wss 3404   ifcif 3881   class class class wbr 4402   |-> cmpt 4461   dom cdm 4834   |` cres 4836   -->wf 5578   ` cfv 5582  (class class class)co 6290   CCcc 9537   RRcr 9538   RR*cxr 9674   < clt 9675   <_ cle 9676   (,)cioo 11635   [,]cicc 11638   -cn->ccncf 21908   L^1cibl 22575   S.citg 22576   lim CC climc 22817

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-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-inf2 8146  ax-cc 8865  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  ax-pre-sup 9617  ax-addf 9618  ax-mulf 9619

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-rmo 2745  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-int 4235  df-iun 4280  df-iin 4281  df-disj 4374  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-se 4794  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-isom 5591  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-of 6531  df-ofr 6532  df-om 6693  df-1st 6793  df-2nd 6794  df-supp 6915  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-2o 7183  df-oadd 7186  df-omul 7187  df-er 7363  df-map 7474  df-pm 7475  df-ixp 7523  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-fsupp 7884  df-fi 7925  df-sup 7956  df-inf 7957  df-oi 8025  df-card 8373  df-acn 8376  df-cda 8598  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-9 10675  df-10 10676  df-n0 10870  df-z 10938  df-dec 11052  df-uz 11160  df-q 11265  df-rp 11303  df-xneg 11409  df-xadd 11410  df-xmul 11411  df-ioo 11639  df-ioc 11640  df-ico 11641  df-icc 11642  df-fz 11785  df-fzo 11916  df-fl 12028  df-mod 12097  df-seq 12214  df-exp 12273  df-hash 12516  df-cj 13162  df-re 13163  df-im 13164  df-sqrt 13298  df-abs 13299  df-limsup 13526  df-clim 13552  df-rlim 13553  df-sum 13753  df-struct 15123  df-ndx 15124  df-slot 15125  df-base 15126  df-sets 15127  df-ress 15128  df-plusg 15203  df-mulr 15204  df-starv 15205  df-sca 15206  df-vsca 15207  df-ip 15208  df-tset 15209  df-ple 15210  df-ds 15212  df-unif 15213  df-hom 15214  df-cco 15215  df-rest 15321  df-topn 15322  df-0g 15340  df-gsum 15341  df-topgen 15342  df-pt 15343  df-prds 15346  df-xrs 15400  df-qtop 15406  df-imas 15407  df-xps 15410  df-mre 15492  df-mrc 15493  df-acs 15495  df-mgm 16488  df-sgrp 16527  df-mnd 16537  df-submnd 16583  df-mulg 16676  df-cntz 16971  df-cmn 17432  df-psmet 18962  df-xmet 18963  df-met 18964  df-bl 18965  df-mopn 18966  df-cnfld 18971  df-top 19921  df-bases 19922  df-topon 19923  df-topsp 19924  df-cld 20034  df-ntr 20035  df-cls 20036  df-cn 20243  df-cnp 20244  df-cmp 20402  df-tx 20577  df-hmeo 20770  df-xms 21335  df-ms 21336  df-tms 21337  df-cncf 21910  df-ovol 22416  df-vol 22418  df-mbf 22577  df-itg1 22578  df-itg2 22579  df-ibl 22580  df-itg 22581  df-0p 22628  df-limc 22821

This theorem is referenced by:  fourierdlem81  38051