Theorem itgioocnicc 40193

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 4092	. . . . . . . . 9 |- ( x = A -> if ( x = A , R , if ( x = B , L , ( F ` x ) ) ) = R )
5		iftrue 4092	. . . . . . . . 9 |- ( x = A -> if ( x = A , R , if ( x = B , L , ( ( F |` ( A (,) B ) ) ` x ) ) ) = R )
6	4, 5	eqtr4d 2659	. . . . . . . 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 482	. . . . . . 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 4092	. . . . . . . . . . . 12 |- ( x = B -> if ( x = B , L , ( F ` x ) ) = L )
9		iftrue 4092	. . . . . . . . . . . 12 |- ( x = B -> if ( x = B , L , ( ( F |` ( A (,) B ) ) ` x ) ) = L )
10	8, 9	eqtr4d 2659	. . . . . . . . . . 11 |- ( x = B -> if ( x = B , L , ( F ` x ) ) = if ( x = B , L , ( ( F |` ( A (,) B ) ) ` x ) ) )
11	10	adantl 482	. . . . . . . . . 10 |- ( ( -. x = A /\ x = B ) -> if ( x = B , L , ( F ` x ) ) = if ( x = B , L , ( ( F |` ( A (,) B ) ) ` x ) ) )
12	11	ifeq2d 4105	. . . . . . . . 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 750	. . . . . . . 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 4095	. . . . . . . . . 10 |- ( -. x = A -> if ( x = A , R , if ( x = B , L , ( F ` x ) ) ) = if ( x = B , L , ( F ` x ) ) )
15	14	ad2antlr 763	. . . . . . . . 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 4095	. . . . . . . . . 10 |- ( -. x = B -> if ( x = B , L , ( F ` x ) ) = ( F ` x ) )
17	16	adantl 482	. . . . . . . . 9 |- ( ( ( ( ph /\ x e. ( A [,] B ) ) /\ -. x = A ) /\ -. x = B ) -> if ( x = B , L , ( F ` x ) ) = ( F ` x ) )
18		iffalse 4095	. . . . . . . . . . 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 763	. . . . . . . . . 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 4095	. . . . . . . . . . 11 |- ( -. x = B -> if ( x = B , L , ( ( F |` ( A (,) B ) ) ` x ) ) = ( ( F |` ( A (,) B ) ) ` x ) )
21	20	adantl 482	. . . . . . . . . 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 481	. . . . . . . . . . . . . 14 |- ( ( ph /\ x e. ( A [,] B ) ) -> A e. RR )
23	22	rexrd 10089	. . . . . . . . . . . . 13 |- ( ( ph /\ x e. ( A [,] B ) ) -> A e. RR* )
24	23	ad2antrr 762	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ x e. ( A [,] B ) ) /\ -. x = A ) /\ -. x = B ) -> A e. RR* )
25	2	rexrd 10089	. . . . . . . . . . . . 13 |- ( ph -> B e. RR* )
26	25	ad3antrrr 766	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ x e. ( A [,] B ) ) /\ -. x = A ) /\ -. x = B ) -> B e. RR* )
27	2	adantr 481	. . . . . . . . . . . . . 14 |- ( ( ph /\ x e. ( A [,] B ) ) -> B e. RR )
28		simpr 477	. . . . . . . . . . . . . 14 |- ( ( ph /\ x e. ( A [,] B ) ) -> x e. ( A [,] B ) )
29		eliccre 39728	. . . . . . . . . . . . . 14 |- ( ( A e. RR /\ B e. RR /\ x e. ( A [,] B ) ) -> x e. RR )
30	22, 27, 28, 29	syl3anc 1326	. . . . . . . . . . . . 13 |- ( ( ph /\ x e. ( A [,] B ) ) -> x e. RR )
31	30	ad2antrr 762	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ x e. ( A [,] B ) ) /\ -. x = A ) /\ -. x = B ) -> x e. RR )
32	1	ad2antrr 762	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ -. x = A ) -> A e. RR )
33	30	adantr 481	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ -. x = A ) -> x e. RR )
34	25	adantr 481	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ x e. ( A [,] B ) ) -> B e. RR* )
35		iccgelb 12230	. . . . . . . . . . . . . . . 16 |- ( ( A e. RR* /\ B e. RR* /\ x e. ( A [,] B ) ) -> A <_ x )
36	23, 34, 28, 35	syl3anc 1326	. . . . . . . . . . . . . . 15 |- ( ( ph /\ x e. ( A [,] B ) ) -> A <_ x )
37	36	adantr 481	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ -. x = A ) -> A <_ x )
38		neqne 2802	. . . . . . . . . . . . . . 15 |- ( -. x = A -> x =/= A )
39	38	adantl 482	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ -. x = A ) -> x =/= A )
40	32, 33, 37, 39	leneltd 10191	. . . . . . . . . . . . 13 |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ -. x = A ) -> A < x )
41	40	adantr 481	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ x e. ( A [,] B ) ) /\ -. x = A ) /\ -. x = B ) -> A < x )
42	30	adantr 481	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ -. x = B ) -> x e. RR )
43	2	ad2antrr 762	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ -. x = B ) -> B e. RR )
44		iccleub 12229	. . . . . . . . . . . . . . . 16 |- ( ( A e. RR* /\ B e. RR* /\ x e. ( A [,] B ) ) -> x <_ B )
45	23, 34, 28, 44	syl3anc 1326	. . . . . . . . . . . . . . 15 |- ( ( ph /\ x e. ( A [,] B ) ) -> x <_ B )
46	45	adantr 481	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ -. x = B ) -> x <_ B )
47		eqcom 2629	. . . . . . . . . . . . . . . . . 18 |- ( x = B <-> B = x )
48	47	notbii 310	. . . . . . . . . . . . . . . . 17 |- ( -. x = B <-> -. B = x )
49	48	biimpi 206	. . . . . . . . . . . . . . . 16 |- ( -. x = B -> -. B = x )
50	49	neqned 2801	. . . . . . . . . . . . . . 15 |- ( -. x = B -> B =/= x )
51	50	adantl 482	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ -. x = B ) -> B =/= x )
52	42, 43, 46, 51	leneltd 10191	. . . . . . . . . . . . 13 |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ -. x = B ) -> x < B )
53	52	adantlr 751	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ x e. ( A [,] B ) ) /\ -. x = A ) /\ -. x = B ) -> x < B )
54	24, 26, 31, 41, 53	eliood 39720	. . . . . . . . . . 11 |- ( ( ( ( ph /\ x e. ( A [,] B ) ) /\ -. x = A ) /\ -. x = B ) -> x e. ( A (,) B ) )
55		fvres 6207	. . . . . . . . . . 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 2661	. . . . . . . . 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 2660	. . . . . . . 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 832	. . . . . . 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 832	. . . . . 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 4744	. . . . 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 2668	. . . 4 |- ( ph -> G = ( x e. ( A [,] B ) |-> if ( x = A , R , if ( x = B , L , ( ( F |` ( A (,) B ) ) ` x ) ) ) ) )
63		nfv 1843	. . . . 5 |- F/ x ph
64		eqid 2622	. . . . 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 40107	. . . 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 2701	. . 3 |- ( ph -> G e. ( ( A [,] B ) -cn-> CC ) )
70		cniccibl 23607	. . 3 |- ( ( A e. RR /\ B e. RR /\ G e. ( ( A [,] B ) -cn-> CC ) ) -> G e. L^1 )
71	1, 2, 69, 70	syl3anc 1326	. 2 |- ( ph -> G e. L^1 )
72	4	adantl 482	. . . . . . . . 9 |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ x = A ) -> if ( x = A , R , if ( x = B , L , ( F ` x ) ) ) = R )
73		limccl 23639	. . . . . . . . . . 11 |- ( ( F |` ( A (,) B ) ) lim CC A ) C_ CC
74	73, 67	sseldi 3601	. . . . . . . . . 10 |- ( ph -> R e. CC )
75	74	ad2antrr 762	. . . . . . . . 9 |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ x = A ) -> R e. CC )
76	72, 75	eqeltrd 2701	. . . . . . . 8 |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ x = A ) -> if ( x = A , R , if ( x = B , L , ( F ` x ) ) ) e. CC )
77	14, 8	sylan9eq 2676	. . . . . . . . . . 11 |- ( ( -. x = A /\ x = B ) -> if ( x = A , R , if ( x = B , L , ( F ` x ) ) ) = L )
78	77	adantll 750	. . . . . . . . . 10 |- ( ( ( ( ph /\ x e. ( A [,] B ) ) /\ -. x = A ) /\ x = B ) -> if ( x = A , R , if ( x = B , L , ( F ` x ) ) ) = L )
79		limccl 23639	. . . . . . . . . . . 12 |- ( ( F |` ( A (,) B ) ) lim CC B ) C_ CC
80	79, 66	sseldi 3601	. . . . . . . . . . 11 |- ( ph -> L e. CC )
81	80	ad3antrrr 766	. . . . . . . . . 10 |- ( ( ( ( ph /\ x e. ( A [,] B ) ) /\ -. x = A ) /\ x = B ) -> L e. CC )
82	78, 81	eqeltrd 2701	. . . . . . . . 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 2676	. . . . . . . . . . 11 |- ( ( -. x = A /\ -. x = B ) -> if ( x = A , R , if ( x = B , L , ( F ` x ) ) ) = ( F ` x ) )
84	83	adantll 750	. . . . . . . . . 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 2628	. . . . . . . . . . 11 |- ( ( ( ( ph /\ x e. ( A [,] B ) ) /\ -. x = A ) /\ -. x = B ) -> ( F ` x ) = ( ( F |` ( A (,) B ) ) ` x ) )
86		cncff 22696	. . . . . . . . . . . . . 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 766	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ x e. ( A [,] B ) ) /\ -. x = A ) /\ -. x = B ) -> ( F |` ( A (,) B ) ) : ( A (,) B ) --> CC )
89	88, 54	ffvelrnd 6360	. . . . . . . . . . 11 |- ( ( ( ( ph /\ x e. ( A [,] B ) ) /\ -. x = A ) /\ -. x = B ) -> ( ( F |` ( A (,) B ) ) ` x ) e. CC )
90	85, 89	eqeltrd 2701	. . . . . . . . . 10 |- ( ( ( ( ph /\ x e. ( A [,] B ) ) /\ -. x = A ) /\ -. x = B ) -> ( F ` x ) e. CC )
91	84, 90	eqeltrd 2701	. . . . . . . . 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 832	. . . . . . . 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 832	. . . . . . 7 |- ( ( ph /\ x e. ( A [,] B ) ) -> if ( x = A , R , if ( x = B , L , ( F ` x ) ) ) e. CC )
94	3	fvmpt2 6291	. . . . . . 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 693	. . . . . 6 |- ( ( ph /\ x e. ( A [,] B ) ) -> ( G ` x ) = if ( x = A , R , if ( x = B , L , ( F ` x ) ) ) )
96	95, 93	eqeltrd 2701	. . . . 5 |- ( ( ph /\ x e. ( A [,] B ) ) -> ( G ` x ) e. CC )
97	1, 2, 96	itgioo 23582	. . . 4 |- ( ph -> S. ( A (,) B ) ( G ` x ) _d x = S. ( A [,] B ) ( G ` x ) _d x )
98	97	eqcomd 2628	. . 3 |- ( ph -> S. ( A [,] B ) ( G ` x ) _d x = S. ( A (,) B ) ( G ` x ) _d x )
99		ioossicc 12259	. . . . . . 7 |- ( A (,) B ) C_ ( A [,] B )
100	99	sseli 3599	. . . . . 6 |- ( x e. ( A (,) B ) -> x e. ( A [,] B ) )
101	100, 95	sylan2 491	. . . . 5 |- ( ( ph /\ x e. ( A (,) B ) ) -> ( G ` x ) = if ( x = A , R , if ( x = B , L , ( F ` x ) ) ) )
102	1	adantr 481	. . . . . . . 8 |- ( ( ph /\ x e. ( A (,) B ) ) -> A e. RR )
103		eliooord 12233	. . . . . . . . . 10 |- ( x e. ( A (,) B ) -> ( A < x /\ x < B ) )
104	103	simpld 475	. . . . . . . . 9 |- ( x e. ( A (,) B ) -> A < x )
105	104	adantl 482	. . . . . . . 8 |- ( ( ph /\ x e. ( A (,) B ) ) -> A < x )
106	102, 105	gtned 10172	. . . . . . 7 |- ( ( ph /\ x e. ( A (,) B ) ) -> x =/= A )
107	106	neneqd 2799	. . . . . 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 491	. . . . . . . 8 |- ( ( ph /\ x e. ( A (,) B ) ) -> x e. RR )
110	103	simprd 479	. . . . . . . . 9 |- ( x e. ( A (,) B ) -> x < B )
111	110	adantl 482	. . . . . . . 8 |- ( ( ph /\ x e. ( A (,) B ) ) -> x < B )
112	109, 111	ltned 10173	. . . . . . 7 |- ( ( ph /\ x e. ( A (,) B ) ) -> x =/= B )
113	112	neneqd 2799	. . . . . 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 2660	. . . 4 |- ( ( ph /\ x e. ( A (,) B ) ) -> ( G ` x ) = ( F ` x ) )
116	115	itgeq2dv 23548	. . 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 481	. . . . 5 |- ( ( ph /\ x e. ( A [,] B ) ) -> F : dom F --> CC )
119		itgioocnicc.fdom	. . . . . 6 |- ( ph -> ( A [,] B ) C_ dom F )
120	119	sselda 3603	. . . . 5 |- ( ( ph /\ x e. ( A [,] B ) ) -> x e. dom F )
121	118, 120	ffvelrnd 6360	. . . 4 |- ( ( ph /\ x e. ( A [,] B ) ) -> ( F ` x ) e. CC )
122	1, 2, 121	itgioo 23582	. . 3 |- ( ph -> S. ( A (,) B ) ( F ` x ) _d x = S. ( A [,] B ) ( F ` x ) _d x )
123	98, 116, 122	3eqtrd 2660	. 2 |- ( ph -> S. ( A [,] B ) ( G ` x ) _d x = S. ( A [,] B ) ( F ` x ) _d x )
124	71, 123	jca 554	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 384   = wceq 1483   e. wcel 1990   =/= wne 2794   C_ wss 3574   ifcif 4086   class class class wbr 4653   |-> cmpt 4729   dom cdm 5114   |` cres 5116   -->wf 5884   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   RR*cxr 10073   < clt 10074   <_ cle 10075   (,)cioo 12175   [,]cicc 12178   -cn->ccncf 22679   L^1cibl 23386   S.citg 23387   lim CC climc 23626

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-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cc 9257  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  ax-pre-sup 10014  ax-addf 10015  ax-mulf 10016

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-rmo 2920  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-int 4476  df-iun 4522  df-iin 4523  df-disj 4621  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-se 5074  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-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-ofr 6898  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-omul 7565  df-er 7742  df-map 7859  df-pm 7860  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-fi 8317  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-acn 8768  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ioo 12179  df-ioc 12180  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-limsup 14202  df-clim 14219  df-rlim 14220  df-sum 14417  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-hom 15966  df-cco 15967  df-rest 16083  df-topn 16084  df-0g 16102  df-gsum 16103  df-topgen 16104  df-pt 16105  df-prds 16108  df-xrs 16162  df-qtop 16167  df-imas 16168  df-xps 16170  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-mulg 17541  df-cntz 17750  df-cmn 18195  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-cnfld 19747  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cld 20823  df-ntr 20824  df-cls 20825  df-cn 21031  df-cnp 21032  df-cmp 21190  df-tx 21365  df-hmeo 21558  df-xms 22125  df-ms 22126  df-tms 22127  df-cncf 22681  df-ovol 23233  df-vol 23234  df-mbf 23388  df-itg1 23389  df-itg2 23390  df-ibl 23391  df-itg 23392  df-0p 23437  df-limc 23630

This theorem is referenced by:  fourierdlem81  40404