Theorem itgioocnicc 37951

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 3878	. . . . . . . . 9 |- ( x = A -> if ( x = A , R , if ( x = B , L , ( F ` x ) ) ) = R )
5		iftrue 3878	. . . . . . . . 9 |- ( x = A -> if ( x = A , R , if ( x = B , L , ( ( F |` ( A (,) B ) ) ` x ) ) ) = R )
6	4, 5	eqtr4d 2508	. . . . . . . 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 473	. . . . . . 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 3878	. . . . . . . . . . . 12 |- ( x = B -> if ( x = B , L , ( F ` x ) ) = L )
9		iftrue 3878	. . . . . . . . . . . 12 |- ( x = B -> if ( x = B , L , ( ( F |` ( A (,) B ) ) ` x ) ) = L )
10	8, 9	eqtr4d 2508	. . . . . . . . . . 11 |- ( x = B -> if ( x = B , L , ( F ` x ) ) = if ( x = B , L , ( ( F |` ( A (,) B ) ) ` x ) ) )
11	10	adantl 473	. . . . . . . . . 10 |- ( ( -. x = A /\ x = B ) -> if ( x = B , L , ( F ` x ) ) = if ( x = B , L , ( ( F |` ( A (,) B ) ) ` x ) ) )
12	11	ifeq2d 3891	. . . . . . . . 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 728	. . . . . . . 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 3881	. . . . . . . . . 10 |- ( -. x = A -> if ( x = A , R , if ( x = B , L , ( F ` x ) ) ) = if ( x = B , L , ( F ` x ) ) )
15	14	ad2antlr 741	. . . . . . . . 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 3881	. . . . . . . . . 10 |- ( -. x = B -> if ( x = B , L , ( F ` x ) ) = ( F ` x ) )
17	16	adantl 473	. . . . . . . . 9 |- ( ( ( ( ph /\ x e. ( A [,] B ) ) /\ -. x = A ) /\ -. x = B ) -> if ( x = B , L , ( F ` x ) ) = ( F ` x ) )
18		iffalse 3881	. . . . . . . . . . 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 741	. . . . . . . . . 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 3881	. . . . . . . . . . 11 |- ( -. x = B -> if ( x = B , L , ( ( F |` ( A (,) B ) ) ` x ) ) = ( ( F |` ( A (,) B ) ) ` x ) )
21	20	adantl 473	. . . . . . . . . 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 472	. . . . . . . . . . . . . 14 |- ( ( ph /\ x e. ( A [,] B ) ) -> A e. RR )
23	22	rexrd 9708	. . . . . . . . . . . . 13 |- ( ( ph /\ x e. ( A [,] B ) ) -> A e. RR* )
24	23	ad2antrr 740	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ x e. ( A [,] B ) ) /\ -. x = A ) /\ -. x = B ) -> A e. RR* )
25	2	rexrd 9708	. . . . . . . . . . . . 13 |- ( ph -> B e. RR* )
26	25	ad3antrrr 744	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ x e. ( A [,] B ) ) /\ -. x = A ) /\ -. x = B ) -> B e. RR* )
27	2	adantr 472	. . . . . . . . . . . . . 14 |- ( ( ph /\ x e. ( A [,] B ) ) -> B e. RR )
28		simpr 468	. . . . . . . . . . . . . 14 |- ( ( ph /\ x e. ( A [,] B ) ) -> x e. ( A [,] B ) )
29		eliccre 37699	. . . . . . . . . . . . . 14 |- ( ( A e. RR /\ B e. RR /\ x e. ( A [,] B ) ) -> x e. RR )
30	22, 27, 28, 29	syl3anc 1292	. . . . . . . . . . . . 13 |- ( ( ph /\ x e. ( A [,] B ) ) -> x e. RR )
31	30	ad2antrr 740	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ x e. ( A [,] B ) ) /\ -. x = A ) /\ -. x = B ) -> x e. RR )
32	1	ad2antrr 740	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ -. x = A ) -> A e. RR )
33	30	adantr 472	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ -. x = A ) -> x e. RR )
34	25	adantr 472	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ x e. ( A [,] B ) ) -> B e. RR* )
35		iccgelb 11716	. . . . . . . . . . . . . . . 16 |- ( ( A e. RR* /\ B e. RR* /\ x e. ( A [,] B ) ) -> A <_ x )
36	23, 34, 28, 35	syl3anc 1292	. . . . . . . . . . . . . . 15 |- ( ( ph /\ x e. ( A [,] B ) ) -> A <_ x )
37	36	adantr 472	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ -. x = A ) -> A <_ x )
38		neqne 2651	. . . . . . . . . . . . . . 15 |- ( -. x = A -> x =/= A )
39	38	adantl 473	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ -. x = A ) -> x =/= A )
40	32, 33, 37, 39	leneltd 9806	. . . . . . . . . . . . 13 |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ -. x = A ) -> A < x )
41	40	adantr 472	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ x e. ( A [,] B ) ) /\ -. x = A ) /\ -. x = B ) -> A < x )
42	30	adantr 472	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ -. x = B ) -> x e. RR )
43	2	ad2antrr 740	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ -. x = B ) -> B e. RR )
44		iccleub 11715	. . . . . . . . . . . . . . . 16 |- ( ( A e. RR* /\ B e. RR* /\ x e. ( A [,] B ) ) -> x <_ B )
45	23, 34, 28, 44	syl3anc 1292	. . . . . . . . . . . . . . 15 |- ( ( ph /\ x e. ( A [,] B ) ) -> x <_ B )
46	45	adantr 472	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ -. x = B ) -> x <_ B )
47		eqcom 2478	. . . . . . . . . . . . . . . . . 18 |- ( x = B <-> B = x )
48	47	notbii 303	. . . . . . . . . . . . . . . . 17 |- ( -. x = B <-> -. B = x )
49	48	biimpi 199	. . . . . . . . . . . . . . . 16 |- ( -. x = B -> -. B = x )
50	49	neqned 2650	. . . . . . . . . . . . . . 15 |- ( -. x = B -> B =/= x )
51	50	adantl 473	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ -. x = B ) -> B =/= x )
52	42, 43, 46, 51	leneltd 9806	. . . . . . . . . . . . 13 |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ -. x = B ) -> x < B )
53	52	adantlr 729	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ x e. ( A [,] B ) ) /\ -. x = A ) /\ -. x = B ) -> x < B )
54	24, 26, 31, 41, 53	eliood 37691	. . . . . . . . . . 11 |- ( ( ( ( ph /\ x e. ( A [,] B ) ) /\ -. x = A ) /\ -. x = B ) -> x e. ( A (,) B ) )
55		fvres 5893	. . . . . . . . . . 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 2510	. . . . . . . . 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 2509	. . . . . . . 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 808	. . . . . . 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 808	. . . . . 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 4482	. . . . 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 2517	. . . 4 |- ( ph -> G = ( x e. ( A [,] B ) |-> if ( x = A , R , if ( x = B , L , ( ( F |` ( A (,) B ) ) ` x ) ) ) ) )
63		nfv 1769	. . . . 5 |- F/ x ph
64		eqid 2471	. . . . 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 37869	. . . 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 2549	. . 3 |- ( ph -> G e. ( ( A [,] B ) -cn-> CC ) )
70		cniccibl 22877	. . 3 |- ( ( A e. RR /\ B e. RR /\ G e. ( ( A [,] B ) -cn-> CC ) ) -> G e. L^1 )
71	1, 2, 69, 70	syl3anc 1292	. 2 |- ( ph -> G e. L^1 )
72	4	adantl 473	. . . . . . . . 9 |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ x = A ) -> if ( x = A , R , if ( x = B , L , ( F ` x ) ) ) = R )
73		limccl 22909	. . . . . . . . . . 11 |- ( ( F |` ( A (,) B ) ) lim CC A ) C_ CC
74	73, 67	sseldi 3416	. . . . . . . . . 10 |- ( ph -> R e. CC )
75	74	ad2antrr 740	. . . . . . . . 9 |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ x = A ) -> R e. CC )
76	72, 75	eqeltrd 2549	. . . . . . . 8 |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ x = A ) -> if ( x = A , R , if ( x = B , L , ( F ` x ) ) ) e. CC )
77	14, 8	sylan9eq 2525	. . . . . . . . . . 11 |- ( ( -. x = A /\ x = B ) -> if ( x = A , R , if ( x = B , L , ( F ` x ) ) ) = L )
78	77	adantll 728	. . . . . . . . . 10 |- ( ( ( ( ph /\ x e. ( A [,] B ) ) /\ -. x = A ) /\ x = B ) -> if ( x = A , R , if ( x = B , L , ( F ` x ) ) ) = L )
79		limccl 22909	. . . . . . . . . . . 12 |- ( ( F |` ( A (,) B ) ) lim CC B ) C_ CC
80	79, 66	sseldi 3416	. . . . . . . . . . 11 |- ( ph -> L e. CC )
81	80	ad3antrrr 744	. . . . . . . . . 10 |- ( ( ( ( ph /\ x e. ( A [,] B ) ) /\ -. x = A ) /\ x = B ) -> L e. CC )
82	78, 81	eqeltrd 2549	. . . . . . . . 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 2525	. . . . . . . . . . 11 |- ( ( -. x = A /\ -. x = B ) -> if ( x = A , R , if ( x = B , L , ( F ` x ) ) ) = ( F ` x ) )
84	83	adantll 728	. . . . . . . . . 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 2477	. . . . . . . . . . 11 |- ( ( ( ( ph /\ x e. ( A [,] B ) ) /\ -. x = A ) /\ -. x = B ) -> ( F ` x ) = ( ( F |` ( A (,) B ) ) ` x ) )
86		cncff 22003	. . . . . . . . . . . . . 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 744	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ x e. ( A [,] B ) ) /\ -. x = A ) /\ -. x = B ) -> ( F |` ( A (,) B ) ) : ( A (,) B ) --> CC )
89	88, 54	ffvelrnd 6038	. . . . . . . . . . 11 |- ( ( ( ( ph /\ x e. ( A [,] B ) ) /\ -. x = A ) /\ -. x = B ) -> ( ( F |` ( A (,) B ) ) ` x ) e. CC )
90	85, 89	eqeltrd 2549	. . . . . . . . . 10 |- ( ( ( ( ph /\ x e. ( A [,] B ) ) /\ -. x = A ) /\ -. x = B ) -> ( F ` x ) e. CC )
91	84, 90	eqeltrd 2549	. . . . . . . . 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 808	. . . . . . . 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 808	. . . . . . 7 |- ( ( ph /\ x e. ( A [,] B ) ) -> if ( x = A , R , if ( x = B , L , ( F ` x ) ) ) e. CC )
94	3	fvmpt2 5972	. . . . . . 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 673	. . . . . 6 |- ( ( ph /\ x e. ( A [,] B ) ) -> ( G ` x ) = if ( x = A , R , if ( x = B , L , ( F ` x ) ) ) )
96	95, 93	eqeltrd 2549	. . . . 5 |- ( ( ph /\ x e. ( A [,] B ) ) -> ( G ` x ) e. CC )
97	1, 2, 96	itgioo 22852	. . . 4 |- ( ph -> S. ( A (,) B ) ( G ` x ) _d x = S. ( A [,] B ) ( G ` x ) _d x )
98	97	eqcomd 2477	. . 3 |- ( ph -> S. ( A [,] B ) ( G ` x ) _d x = S. ( A (,) B ) ( G ` x ) _d x )
99		ioossicc 11745	. . . . . . 7 |- ( A (,) B ) C_ ( A [,] B )
100	99	sseli 3414	. . . . . 6 |- ( x e. ( A (,) B ) -> x e. ( A [,] B ) )
101	100, 95	sylan2 482	. . . . 5 |- ( ( ph /\ x e. ( A (,) B ) ) -> ( G ` x ) = if ( x = A , R , if ( x = B , L , ( F ` x ) ) ) )
102	1	adantr 472	. . . . . . . 8 |- ( ( ph /\ x e. ( A (,) B ) ) -> A e. RR )
103		eliooord 11719	. . . . . . . . . 10 |- ( x e. ( A (,) B ) -> ( A < x /\ x < B ) )
104	103	simpld 466	. . . . . . . . 9 |- ( x e. ( A (,) B ) -> A < x )
105	104	adantl 473	. . . . . . . 8 |- ( ( ph /\ x e. ( A (,) B ) ) -> A < x )
106	102, 105	gtned 9787	. . . . . . 7 |- ( ( ph /\ x e. ( A (,) B ) ) -> x =/= A )
107	106	neneqd 2648	. . . . . 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 482	. . . . . . . 8 |- ( ( ph /\ x e. ( A (,) B ) ) -> x e. RR )
110	103	simprd 470	. . . . . . . . 9 |- ( x e. ( A (,) B ) -> x < B )
111	110	adantl 473	. . . . . . . 8 |- ( ( ph /\ x e. ( A (,) B ) ) -> x < B )
112	109, 111	ltned 9788	. . . . . . 7 |- ( ( ph /\ x e. ( A (,) B ) ) -> x =/= B )
113	112	neneqd 2648	. . . . . 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 2509	. . . 4 |- ( ( ph /\ x e. ( A (,) B ) ) -> ( G ` x ) = ( F ` x ) )
116	115	itgeq2dv 22818	. . 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 472	. . . . 5 |- ( ( ph /\ x e. ( A [,] B ) ) -> F : dom F --> CC )
119		itgioocnicc.fdom	. . . . . 6 |- ( ph -> ( A [,] B ) C_ dom F )
120	119	sselda 3418	. . . . 5 |- ( ( ph /\ x e. ( A [,] B ) ) -> x e. dom F )
121	118, 120	ffvelrnd 6038	. . . 4 |- ( ( ph /\ x e. ( A [,] B ) ) -> ( F ` x ) e. CC )
122	1, 2, 121	itgioo 22852	. . 3 |- ( ph -> S. ( A (,) B ) ( F ` x ) _d x = S. ( A [,] B ) ( F ` x ) _d x )
123	98, 116, 122	3eqtrd 2509	. 2 |- ( ph -> S. ( A [,] B ) ( G ` x ) _d x = S. ( A [,] B ) ( F ` x ) _d x )
124	71, 123	jca 541	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 376   = wceq 1452   e. wcel 1904   =/= wne 2641   C_ wss 3390   ifcif 3872   class class class wbr 4395   |-> cmpt 4454   dom cdm 4839   |` cres 4841   -->wf 5585   ` cfv 5589  (class class class)co 6308   CCcc 9555   RRcr 9556   RR*cxr 9692   < clt 9693   <_ cle 9694   (,)cioo 11660   [,]cicc 11663   -cn->ccncf 21986   L^1cibl 22654   S.citg 22655   lim CC climc 22896

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164  ax-cc 8883  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635  ax-addf 9636  ax-mulf 9637

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-iin 4272  df-disj 4367  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-of 6550  df-ofr 6551  df-om 6712  df-1st 6812  df-2nd 6813  df-supp 6934  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-2o 7201  df-oadd 7204  df-omul 7205  df-er 7381  df-map 7492  df-pm 7493  df-ixp 7541  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-fsupp 7902  df-fi 7943  df-sup 7974  df-inf 7975  df-oi 8043  df-card 8391  df-acn 8394  df-cda 8616  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-4 10692  df-5 10693  df-6 10694  df-7 10695  df-8 10696  df-9 10697  df-10 10698  df-n0 10894  df-z 10962  df-dec 11075  df-uz 11183  df-q 11288  df-rp 11326  df-xneg 11432  df-xadd 11433  df-xmul 11434  df-ioo 11664  df-ioc 11665  df-ico 11666  df-icc 11667  df-fz 11811  df-fzo 11943  df-fl 12061  df-mod 12130  df-seq 12252  df-exp 12311  df-hash 12554  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-limsup 13603  df-clim 13629  df-rlim 13630  df-sum 13830  df-struct 15201  df-ndx 15202  df-slot 15203  df-base 15204  df-sets 15205  df-ress 15206  df-plusg 15281  df-mulr 15282  df-starv 15283  df-sca 15284  df-vsca 15285  df-ip 15286  df-tset 15287  df-ple 15288  df-ds 15290  df-unif 15291  df-hom 15292  df-cco 15293  df-rest 15399  df-topn 15400  df-0g 15418  df-gsum 15419  df-topgen 15420  df-pt 15421  df-prds 15424  df-xrs 15478  df-qtop 15484  df-imas 15485  df-xps 15488  df-mre 15570  df-mrc 15571  df-acs 15573  df-mgm 16566  df-sgrp 16605  df-mnd 16615  df-submnd 16661  df-mulg 16754  df-cntz 17049  df-cmn 17510  df-psmet 19039  df-xmet 19040  df-met 19041  df-bl 19042  df-mopn 19043  df-cnfld 19048  df-top 19998  df-bases 19999  df-topon 20000  df-topsp 20001  df-cld 20111  df-ntr 20112  df-cls 20113  df-cn 20320  df-cnp 20321  df-cmp 20479  df-tx 20654  df-hmeo 20847  df-xms 21413  df-ms 21414  df-tms 21415  df-cncf 21988  df-ovol 22494  df-vol 22496  df-mbf 22656  df-itg1 22657  df-itg2 22658  df-ibl 22659  df-itg 22660  df-0p 22707  df-limc 22900

This theorem is referenced by:  fourierdlem81  38163