Theorem ftc2 23807

Description: The Fundamental Theorem of Calculus, part two. If F is a function continuous on [ A , B ] and continuously differentiable on ( A , B ), then the integral of the derivative of F is equal to F ( B ) - F ( A ). This is part of Metamath 100 proof #15. (Contributed by Mario Carneiro, 2-Sep-2014.)

Hypotheses

Ref	Expression
ftc2.a	|- ( ph -> A e. RR )
ftc2.b	|- ( ph -> B e. RR )
ftc2.le	|- ( ph -> A <_ B )
ftc2.c	|- ( ph -> ( RR _D F ) e. ( ( A (,) B ) -cn-> CC ) )
ftc2.i	|- ( ph -> ( RR _D F ) e. L^1 )
ftc2.f	|- ( ph -> F e. ( ( A [,] B ) -cn-> CC ) )

Assertion

Ref	Expression
ftc2	|- ( ph -> S. ( A (,) B ) ( ( RR _D F ) ` t ) _d t = ( ( F ` B ) - ( F ` A ) ) )

Distinct variable groups:   t, A   t, B   t, F   ph, t

Proof of Theorem ftc2

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ftc2.a	. . . . . . 7 |- ( ph -> A e. RR )
2	1	rexrd 10089	. . . . . 6 |- ( ph -> A e. RR* )
3		ftc2.b	. . . . . . 7 |- ( ph -> B e. RR )
4	3	rexrd 10089	. . . . . 6 |- ( ph -> B e. RR* )
5		ftc2.le	. . . . . 6 |- ( ph -> A <_ B )
6		ubicc2 12289	. . . . . 6 |- ( ( A e. RR* /\ B e. RR* /\ A <_ B ) -> B e. ( A [,] B ) )
7	2, 4, 5, 6	syl3anc 1326	. . . . 5 |- ( ph -> B e. ( A [,] B ) )
8		fvex 6201	. . . . . 6 |- ( ( x e. ( A [,] B ) |-> ( S. ( A (,) x ) ( ( RR _D F ) ` t ) _d t - ( F ` x ) ) ) ` A ) e. _V
9	8	fvconst2 6469	. . . . 5 |- ( B e. ( A [,] B ) -> ( ( ( A [,] B ) X. { ( ( x e. ( A [,] B ) |-> ( S. ( A (,) x ) ( ( RR _D F ) ` t ) _d t - ( F ` x ) ) ) ` A ) } ) ` B ) = ( ( x e. ( A [,] B ) |-> ( S. ( A (,) x ) ( ( RR _D F ) ` t ) _d t - ( F ` x ) ) ) ` A ) )
10	7, 9	syl 17	. . . 4 |- ( ph -> ( ( ( A [,] B ) X. { ( ( x e. ( A [,] B ) |-> ( S. ( A (,) x ) ( ( RR _D F ) ` t ) _d t - ( F ` x ) ) ) ` A ) } ) ` B ) = ( ( x e. ( A [,] B ) |-> ( S. ( A (,) x ) ( ( RR _D F ) ` t ) _d t - ( F ` x ) ) ) ` A ) )
11		eqid 2622	. . . . . . . 8 |- ( TopOpen ` ℂfld ) = ( TopOpen ` ℂfld )
12	11	subcn 22669	. . . . . . . . 9 |- - e. ( ( ( TopOpen ` ℂfld ) tX ( TopOpen ` ℂfld ) ) Cn ( TopOpen ` ℂfld ) )
13	12	a1i 11	. . . . . . . 8 |- ( ph -> - e. ( ( ( TopOpen ` ℂfld ) tX ( TopOpen ` ℂfld ) ) Cn ( TopOpen ` ℂfld ) ) )
14		eqid 2622	. . . . . . . . 9 |- ( x e. ( A [,] B ) |-> S. ( A (,) x ) ( ( RR _D F ) ` t ) _d t ) = ( x e. ( A [,] B ) |-> S. ( A (,) x ) ( ( RR _D F ) ` t ) _d t )
15		ssid 3624	. . . . . . . . . 10 |- ( A (,) B ) C_ ( A (,) B )
16	15	a1i 11	. . . . . . . . 9 |- ( ph -> ( A (,) B ) C_ ( A (,) B ) )
17		ioossre 12235	. . . . . . . . . 10 |- ( A (,) B ) C_ RR
18	17	a1i 11	. . . . . . . . 9 |- ( ph -> ( A (,) B ) C_ RR )
19		ftc2.i	. . . . . . . . 9 |- ( ph -> ( RR _D F ) e. L^1 )
20		ftc2.c	. . . . . . . . . 10 |- ( ph -> ( RR _D F ) e. ( ( A (,) B ) -cn-> CC ) )
21		cncff 22696	. . . . . . . . . 10 |- ( ( RR _D F ) e. ( ( A (,) B ) -cn-> CC ) -> ( RR _D F ) : ( A (,) B ) --> CC )
22	20, 21	syl 17	. . . . . . . . 9 |- ( ph -> ( RR _D F ) : ( A (,) B ) --> CC )
23	14, 1, 3, 5, 16, 18, 19, 22	ftc1a 23800	. . . . . . . 8 |- ( ph -> ( x e. ( A [,] B ) |-> S. ( A (,) x ) ( ( RR _D F ) ` t ) _d t ) e. ( ( A [,] B ) -cn-> CC ) )
24		ftc2.f	. . . . . . . . . . 11 |- ( ph -> F e. ( ( A [,] B ) -cn-> CC ) )
25		cncff 22696	. . . . . . . . . . 11 |- ( F e. ( ( A [,] B ) -cn-> CC ) -> F : ( A [,] B ) --> CC )
26	24, 25	syl 17	. . . . . . . . . 10 |- ( ph -> F : ( A [,] B ) --> CC )
27	26	feqmptd 6249	. . . . . . . . 9 |- ( ph -> F = ( x e. ( A [,] B ) |-> ( F ` x ) ) )
28	27, 24	eqeltrrd 2702	. . . . . . . 8 |- ( ph -> ( x e. ( A [,] B ) |-> ( F ` x ) ) e. ( ( A [,] B ) -cn-> CC ) )
29	11, 13, 23, 28	cncfmpt2f 22717	. . . . . . 7 |- ( ph -> ( x e. ( A [,] B ) |-> ( S. ( A (,) x ) ( ( RR _D F ) ` t ) _d t - ( F ` x ) ) ) e. ( ( A [,] B ) -cn-> CC ) )
30		ax-resscn 9993	. . . . . . . . . . 11 |- RR C_ CC
31	30	a1i 11	. . . . . . . . . 10 |- ( ph -> RR C_ CC )
32		iccssre 12255	. . . . . . . . . . 11 |- ( ( A e. RR /\ B e. RR ) -> ( A [,] B ) C_ RR )
33	1, 3, 32	syl2anc 693	. . . . . . . . . 10 |- ( ph -> ( A [,] B ) C_ RR )
34		fvex 6201	. . . . . . . . . . . . 13 |- ( ( RR _D F ) ` t ) e. _V
35	34	a1i 11	. . . . . . . . . . . 12 |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ t e. ( A (,) x ) ) -> ( ( RR _D F ) ` t ) e. _V )
36	3	adantr 481	. . . . . . . . . . . . . . 15 |- ( ( ph /\ x e. ( A [,] B ) ) -> B e. RR )
37	36	rexrd 10089	. . . . . . . . . . . . . 14 |- ( ( ph /\ x e. ( A [,] B ) ) -> B e. RR* )
38		elicc2 12238	. . . . . . . . . . . . . . . . 17 |- ( ( A e. RR /\ B e. RR ) -> ( x e. ( A [,] B ) <-> ( x e. RR /\ A <_ x /\ x <_ B ) ) )
39	1, 3, 38	syl2anc 693	. . . . . . . . . . . . . . . 16 |- ( ph -> ( x e. ( A [,] B ) <-> ( x e. RR /\ A <_ x /\ x <_ B ) ) )
40	39	biimpa 501	. . . . . . . . . . . . . . 15 |- ( ( ph /\ x e. ( A [,] B ) ) -> ( x e. RR /\ A <_ x /\ x <_ B ) )
41	40	simp3d 1075	. . . . . . . . . . . . . 14 |- ( ( ph /\ x e. ( A [,] B ) ) -> x <_ B )
42		iooss2 12211	. . . . . . . . . . . . . 14 |- ( ( B e. RR* /\ x <_ B ) -> ( A (,) x ) C_ ( A (,) B ) )
43	37, 41, 42	syl2anc 693	. . . . . . . . . . . . 13 |- ( ( ph /\ x e. ( A [,] B ) ) -> ( A (,) x ) C_ ( A (,) B ) )
44		ioombl 23333	. . . . . . . . . . . . . 14 |- ( A (,) x ) e. dom vol
45	44	a1i 11	. . . . . . . . . . . . 13 |- ( ( ph /\ x e. ( A [,] B ) ) -> ( A (,) x ) e. dom vol )
46	34	a1i 11	. . . . . . . . . . . . 13 |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ t e. ( A (,) B ) ) -> ( ( RR _D F ) ` t ) e. _V )
47	22	feqmptd 6249	. . . . . . . . . . . . . . 15 |- ( ph -> ( RR _D F ) = ( t e. ( A (,) B ) |-> ( ( RR _D F ) ` t ) ) )
48	47, 19	eqeltrrd 2702	. . . . . . . . . . . . . 14 |- ( ph -> ( t e. ( A (,) B ) |-> ( ( RR _D F ) ` t ) ) e. L^1 )
49	48	adantr 481	. . . . . . . . . . . . 13 |- ( ( ph /\ x e. ( A [,] B ) ) -> ( t e. ( A (,) B ) |-> ( ( RR _D F ) ` t ) ) e. L^1 )
50	43, 45, 46, 49	iblss 23571	. . . . . . . . . . . 12 |- ( ( ph /\ x e. ( A [,] B ) ) -> ( t e. ( A (,) x ) |-> ( ( RR _D F ) ` t ) ) e. L^1 )
51	35, 50	itgcl 23550	. . . . . . . . . . 11 |- ( ( ph /\ x e. ( A [,] B ) ) -> S. ( A (,) x ) ( ( RR _D F ) ` t ) _d t e. CC )
52	26	ffvelrnda 6359	. . . . . . . . . . 11 |- ( ( ph /\ x e. ( A [,] B ) ) -> ( F ` x ) e. CC )
53	51, 52	subcld 10392	. . . . . . . . . 10 |- ( ( ph /\ x e. ( A [,] B ) ) -> ( S. ( A (,) x ) ( ( RR _D F ) ` t ) _d t - ( F ` x ) ) e. CC )
54	11	tgioo2 22606	. . . . . . . . . 10 |- ( topGen ` ran (,) ) = ( ( TopOpen ` ℂfld ) ↾t RR )
55		iccntr 22624	. . . . . . . . . . 11 |- ( ( A e. RR /\ B e. RR ) -> ( ( int ` ( topGen ` ran (,) ) ) ` ( A [,] B ) ) = ( A (,) B ) )
56	1, 3, 55	syl2anc 693	. . . . . . . . . 10 |- ( ph -> ( ( int ` ( topGen ` ran (,) ) ) ` ( A [,] B ) ) = ( A (,) B ) )
57	31, 33, 53, 54, 11, 56	dvmptntr 23734	. . . . . . . . 9 |- ( ph -> ( RR _D ( x e. ( A [,] B ) |-> ( S. ( A (,) x ) ( ( RR _D F ) ` t ) _d t - ( F ` x ) ) ) ) = ( RR _D ( x e. ( A (,) B ) |-> ( S. ( A (,) x ) ( ( RR _D F ) ` t ) _d t - ( F ` x ) ) ) ) )
58		reelprrecn 10028	. . . . . . . . . . 11 |- RR e. { RR , CC }
59	58	a1i 11	. . . . . . . . . 10 |- ( ph -> RR e. { RR , CC } )
60		ioossicc 12259	. . . . . . . . . . . 12 |- ( A (,) B ) C_ ( A [,] B )
61	60	sseli 3599	. . . . . . . . . . 11 |- ( x e. ( A (,) B ) -> x e. ( A [,] B ) )
62	61, 51	sylan2 491	. . . . . . . . . 10 |- ( ( ph /\ x e. ( A (,) B ) ) -> S. ( A (,) x ) ( ( RR _D F ) ` t ) _d t e. CC )
63	22	ffvelrnda 6359	. . . . . . . . . 10 |- ( ( ph /\ x e. ( A (,) B ) ) -> ( ( RR _D F ) ` x ) e. CC )
64	14, 1, 3, 5, 20, 19	ftc1cn 23806	. . . . . . . . . . 11 |- ( ph -> ( RR _D ( x e. ( A [,] B ) |-> S. ( A (,) x ) ( ( RR _D F ) ` t ) _d t ) ) = ( RR _D F ) )
65	31, 33, 51, 54, 11, 56	dvmptntr 23734	. . . . . . . . . . 11 |- ( ph -> ( RR _D ( x e. ( A [,] B ) |-> S. ( A (,) x ) ( ( RR _D F ) ` t ) _d t ) ) = ( RR _D ( x e. ( A (,) B ) |-> S. ( A (,) x ) ( ( RR _D F ) ` t ) _d t ) ) )
66	22	feqmptd 6249	. . . . . . . . . . 11 |- ( ph -> ( RR _D F ) = ( x e. ( A (,) B ) |-> ( ( RR _D F ) ` x ) ) )
67	64, 65, 66	3eqtr3d 2664	. . . . . . . . . 10 |- ( ph -> ( RR _D ( x e. ( A (,) B ) |-> S. ( A (,) x ) ( ( RR _D F ) ` t ) _d t ) ) = ( x e. ( A (,) B ) |-> ( ( RR _D F ) ` x ) ) )
68	61, 52	sylan2 491	. . . . . . . . . 10 |- ( ( ph /\ x e. ( A (,) B ) ) -> ( F ` x ) e. CC )
69	31, 33, 52, 54, 11, 56	dvmptntr 23734	. . . . . . . . . . 11 |- ( ph -> ( RR _D ( x e. ( A [,] B ) |-> ( F ` x ) ) ) = ( RR _D ( x e. ( A (,) B ) |-> ( F ` x ) ) ) )
70	27	oveq2d 6666	. . . . . . . . . . . 12 |- ( ph -> ( RR _D F ) = ( RR _D ( x e. ( A [,] B ) |-> ( F ` x ) ) ) )
71	70, 66	eqtr3d 2658	. . . . . . . . . . 11 |- ( ph -> ( RR _D ( x e. ( A [,] B ) |-> ( F ` x ) ) ) = ( x e. ( A (,) B ) |-> ( ( RR _D F ) ` x ) ) )
72	69, 71	eqtr3d 2658	. . . . . . . . . 10 |- ( ph -> ( RR _D ( x e. ( A (,) B ) |-> ( F ` x ) ) ) = ( x e. ( A (,) B ) |-> ( ( RR _D F ) ` x ) ) )
73	59, 62, 63, 67, 68, 63, 72	dvmptsub 23730	. . . . . . . . 9 |- ( ph -> ( RR _D ( x e. ( A (,) B ) |-> ( S. ( A (,) x ) ( ( RR _D F ) ` t ) _d t - ( F ` x ) ) ) ) = ( x e. ( A (,) B ) |-> ( ( ( RR _D F ) ` x ) - ( ( RR _D F ) ` x ) ) ) )
74	63	subidd 10380	. . . . . . . . . 10 |- ( ( ph /\ x e. ( A (,) B ) ) -> ( ( ( RR _D F ) ` x ) - ( ( RR _D F ) ` x ) ) = 0 )
75	74	mpteq2dva 4744	. . . . . . . . 9 |- ( ph -> ( x e. ( A (,) B ) |-> ( ( ( RR _D F ) ` x ) - ( ( RR _D F ) ` x ) ) ) = ( x e. ( A (,) B ) |-> 0 ) )
76	57, 73, 75	3eqtrd 2660	. . . . . . . 8 |- ( ph -> ( RR _D ( x e. ( A [,] B ) |-> ( S. ( A (,) x ) ( ( RR _D F ) ` t ) _d t - ( F ` x ) ) ) ) = ( x e. ( A (,) B ) |-> 0 ) )
77		fconstmpt 5163	. . . . . . . 8 |- ( ( A (,) B ) X. { 0 } ) = ( x e. ( A (,) B ) |-> 0 )
78	76, 77	syl6eqr 2674	. . . . . . 7 |- ( ph -> ( RR _D ( x e. ( A [,] B ) |-> ( S. ( A (,) x ) ( ( RR _D F ) ` t ) _d t - ( F ` x ) ) ) ) = ( ( A (,) B ) X. { 0 } ) )
79	1, 3, 29, 78	dveq0 23763	. . . . . 6 |- ( ph -> ( x e. ( A [,] B ) |-> ( S. ( A (,) x ) ( ( RR _D F ) ` t ) _d t - ( F ` x ) ) ) = ( ( A [,] B ) X. { ( ( x e. ( A [,] B ) |-> ( S. ( A (,) x ) ( ( RR _D F ) ` t ) _d t - ( F ` x ) ) ) ` A ) } ) )
80	79	fveq1d 6193	. . . . 5 |- ( ph -> ( ( x e. ( A [,] B ) |-> ( S. ( A (,) x ) ( ( RR _D F ) ` t ) _d t - ( F ` x ) ) ) ` B ) = ( ( ( A [,] B ) X. { ( ( x e. ( A [,] B ) |-> ( S. ( A (,) x ) ( ( RR _D F ) ` t ) _d t - ( F ` x ) ) ) ` A ) } ) ` B ) )
81		oveq2 6658	. . . . . . . . 9 |- ( x = B -> ( A (,) x ) = ( A (,) B ) )
82		itgeq1 23539	. . . . . . . . 9 |- ( ( A (,) x ) = ( A (,) B ) -> S. ( A (,) x ) ( ( RR _D F ) ` t ) _d t = S. ( A (,) B ) ( ( RR _D F ) ` t ) _d t )
83	81, 82	syl 17	. . . . . . . 8 |- ( x = B -> S. ( A (,) x ) ( ( RR _D F ) ` t ) _d t = S. ( A (,) B ) ( ( RR _D F ) ` t ) _d t )
84		fveq2 6191	. . . . . . . 8 |- ( x = B -> ( F ` x ) = ( F ` B ) )
85	83, 84	oveq12d 6668	. . . . . . 7 |- ( x = B -> ( S. ( A (,) x ) ( ( RR _D F ) ` t ) _d t - ( F ` x ) ) = ( S. ( A (,) B ) ( ( RR _D F ) ` t ) _d t - ( F ` B ) ) )
86		eqid 2622	. . . . . . 7 |- ( x e. ( A [,] B ) |-> ( S. ( A (,) x ) ( ( RR _D F ) ` t ) _d t - ( F ` x ) ) ) = ( x e. ( A [,] B ) |-> ( S. ( A (,) x ) ( ( RR _D F ) ` t ) _d t - ( F ` x ) ) )
87		ovex 6678	. . . . . . 7 |- ( S. ( A (,) B ) ( ( RR _D F ) ` t ) _d t - ( F ` B ) ) e. _V
88	85, 86, 87	fvmpt 6282	. . . . . 6 |- ( B e. ( A [,] B ) -> ( ( x e. ( A [,] B ) |-> ( S. ( A (,) x ) ( ( RR _D F ) ` t ) _d t - ( F ` x ) ) ) ` B ) = ( S. ( A (,) B ) ( ( RR _D F ) ` t ) _d t - ( F ` B ) ) )
89	7, 88	syl 17	. . . . 5 |- ( ph -> ( ( x e. ( A [,] B ) |-> ( S. ( A (,) x ) ( ( RR _D F ) ` t ) _d t - ( F ` x ) ) ) ` B ) = ( S. ( A (,) B ) ( ( RR _D F ) ` t ) _d t - ( F ` B ) ) )
90	80, 89	eqtr3d 2658	. . . 4 |- ( ph -> ( ( ( A [,] B ) X. { ( ( x e. ( A [,] B ) |-> ( S. ( A (,) x ) ( ( RR _D F ) ` t ) _d t - ( F ` x ) ) ) ` A ) } ) ` B ) = ( S. ( A (,) B ) ( ( RR _D F ) ` t ) _d t - ( F ` B ) ) )
91		lbicc2 12288	. . . . . 6 |- ( ( A e. RR* /\ B e. RR* /\ A <_ B ) -> A e. ( A [,] B ) )
92	2, 4, 5, 91	syl3anc 1326	. . . . 5 |- ( ph -> A e. ( A [,] B ) )
93		oveq2 6658	. . . . . . . . . . 11 |- ( x = A -> ( A (,) x ) = ( A (,) A ) )
94		iooid 12203	. . . . . . . . . . 11 |- ( A (,) A ) = (/)
95	93, 94	syl6eq 2672	. . . . . . . . . 10 |- ( x = A -> ( A (,) x ) = (/) )
96		itgeq1 23539	. . . . . . . . . 10 |- ( ( A (,) x ) = (/) -> S. ( A (,) x ) ( ( RR _D F ) ` t ) _d t = S. (/) ( ( RR _D F ) ` t ) _d t )
97	95, 96	syl 17	. . . . . . . . 9 |- ( x = A -> S. ( A (,) x ) ( ( RR _D F ) ` t ) _d t = S. (/) ( ( RR _D F ) ` t ) _d t )
98		itg0 23546	. . . . . . . . 9 |- S. (/) ( ( RR _D F ) ` t ) _d t = 0
99	97, 98	syl6eq 2672	. . . . . . . 8 |- ( x = A -> S. ( A (,) x ) ( ( RR _D F ) ` t ) _d t = 0 )
100		fveq2 6191	. . . . . . . 8 |- ( x = A -> ( F ` x ) = ( F ` A ) )
101	99, 100	oveq12d 6668	. . . . . . 7 |- ( x = A -> ( S. ( A (,) x ) ( ( RR _D F ) ` t ) _d t - ( F ` x ) ) = ( 0 - ( F ` A ) ) )
102		df-neg 10269	. . . . . . 7 |- -u ( F ` A ) = ( 0 - ( F ` A ) )
103	101, 102	syl6eqr 2674	. . . . . 6 |- ( x = A -> ( S. ( A (,) x ) ( ( RR _D F ) ` t ) _d t - ( F ` x ) ) = -u ( F ` A ) )
104		negex 10279	. . . . . 6 |- -u ( F ` A ) e. _V
105	103, 86, 104	fvmpt 6282	. . . . 5 |- ( A e. ( A [,] B ) -> ( ( x e. ( A [,] B ) |-> ( S. ( A (,) x ) ( ( RR _D F ) ` t ) _d t - ( F ` x ) ) ) ` A ) = -u ( F ` A ) )
106	92, 105	syl 17	. . . 4 |- ( ph -> ( ( x e. ( A [,] B ) |-> ( S. ( A (,) x ) ( ( RR _D F ) ` t ) _d t - ( F ` x ) ) ) ` A ) = -u ( F ` A ) )
107	10, 90, 106	3eqtr3d 2664	. . 3 |- ( ph -> ( S. ( A (,) B ) ( ( RR _D F ) ` t ) _d t - ( F ` B ) ) = -u ( F ` A ) )
108	107	oveq2d 6666	. 2 |- ( ph -> ( ( F ` B ) + ( S. ( A (,) B ) ( ( RR _D F ) ` t ) _d t - ( F ` B ) ) ) = ( ( F ` B ) + -u ( F ` A ) ) )
109	26, 7	ffvelrnd 6360	. . 3 |- ( ph -> ( F ` B ) e. CC )
110	34	a1i 11	. . . 4 |- ( ( ph /\ t e. ( A (,) B ) ) -> ( ( RR _D F ) ` t ) e. _V )
111	110, 48	itgcl 23550	. . 3 |- ( ph -> S. ( A (,) B ) ( ( RR _D F ) ` t ) _d t e. CC )
112	109, 111	pncan3d 10395	. 2 |- ( ph -> ( ( F ` B ) + ( S. ( A (,) B ) ( ( RR _D F ) ` t ) _d t - ( F ` B ) ) ) = S. ( A (,) B ) ( ( RR _D F ) ` t ) _d t )
113	26, 92	ffvelrnd 6360	. . 3 |- ( ph -> ( F ` A ) e. CC )
114	109, 113	negsubd 10398	. 2 |- ( ph -> ( ( F ` B ) + -u ( F ` A ) ) = ( ( F ` B ) - ( F ` A ) ) )
115	108, 112, 114	3eqtr3d 2664	1 |- ( ph -> S. ( A (,) B ) ( ( RR _D F ) ` t ) _d t = ( ( F ` B ) - ( F ` A ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   _Vcvv 3200   C_ wss 3574   (/)c0 3915   {csn 4177   {cpr 4179   class class class wbr 4653   |-> cmpt 4729   X. cxp 5112   dom cdm 5114   ran crn 5115   -->wf 5884   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   0cc0 9936   + caddc 9939   RR*cxr 10073   <_ cle 10075   - cmin 10266   -ucneg 10267   (,)cioo 12175   [,]cicc 12178   TopOpenctopn 16082   topGenctg 16098  ℂ_fldccnfld 19746   intcnt 20821   Cn ccn 21028   tX ctx 21363   -cn->ccncf 22679   volcvol 23232   L^1cibl 23386   S.citg 23387   _D cdv 23627

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-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-fbas 19743  df-fg 19744  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-nei 20902  df-lp 20940  df-perf 20941  df-cn 21031  df-cnp 21032  df-haus 21119  df-cmp 21190  df-tx 21365  df-hmeo 21558  df-fil 21650  df-fm 21742  df-flim 21743  df-flf 21744  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  df-dv 23631

This theorem is referenced by:  ftc2ditglem  23808  itgparts  23810  itgsubstlem  23811  ftc2re  30676  itgpowd  37800  lhe4.4ex1a  38528  itgsin0pilem1  40165  itgcoscmulx  40185  itgsincmulx  40190  dirkeritg  40319  etransclem46  40497