Theorem ftc2nc 33494

Description: Choice-free proof of ftc2 23807. (Contributed by Brendan Leahy, 19-Jun-2018.)

Hypotheses

Ref	Expression
ftc2nc.a	|- ( ph -> A e. RR )
ftc2nc.b	|- ( ph -> B e. RR )
ftc2nc.le	|- ( ph -> A <_ B )
ftc2nc.c	|- ( ph -> ( RR _D F ) e. ( ( A (,) B ) -cn-> CC ) )
ftc2nc.i	|- ( ph -> ( RR _D F ) e. L^1 )
ftc2nc.f	|- ( ph -> F e. ( ( A [,] B ) -cn-> CC ) )

Assertion

Ref	Expression
ftc2nc	|- ( 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 ftc2nc

Dummy variables s x are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ftc2nc.a	. . . . . . 7 |- ( ph -> A e. RR )
2	1	rexrd 10089	. . . . . 6 |- ( ph -> A e. RR* )
3		ftc2nc.b	. . . . . . 7 |- ( ph -> B e. RR )
4	3	rexrd 10089	. . . . . 6 |- ( ph -> B e. RR* )
5		ftc2nc.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		ftc2nc.i	. . . . . . . . 9 |- ( ph -> ( RR _D F ) e. L^1 )
20		ftc2nc.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		ioof 12271	. . . . . . . . . . . . 13 |- (,) : ( RR* X. RR* ) --> ~P RR
24		ffun 6048	. . . . . . . . . . . . 13 |- ( (,) : ( RR* X. RR* ) --> ~P RR -> Fun (,) )
25	23, 24	ax-mp 5	. . . . . . . . . . . 12 |- Fun (,)
26		fvelima 6248	. . . . . . . . . . . 12 |- ( ( Fun (,) /\ s e. ( (,) " ( ( A [,] B ) X. ( A [,] B ) ) ) ) -> E. x e. ( ( A [,] B ) X. ( A [,] B ) ) ( (,) ` x ) = s )
27	25, 26	mpan 706	. . . . . . . . . . 11 |- ( s e. ( (,) " ( ( A [,] B ) X. ( A [,] B ) ) ) -> E. x e. ( ( A [,] B ) X. ( A [,] B ) ) ( (,) ` x ) = s )
28		1st2nd2 7205	. . . . . . . . . . . . . . . . 17 |- ( x e. ( ( A [,] B ) X. ( A [,] B ) ) -> x = <. ( 1st ` x ) , ( 2nd ` x ) >. )
29	28	fveq2d 6195	. . . . . . . . . . . . . . . 16 |- ( x e. ( ( A [,] B ) X. ( A [,] B ) ) -> ( (,) ` x ) = ( (,) ` <. ( 1st ` x ) , ( 2nd ` x ) >. ) )
30		df-ov 6653	. . . . . . . . . . . . . . . 16 |- ( ( 1st ` x ) (,) ( 2nd ` x ) ) = ( (,) ` <. ( 1st ` x ) , ( 2nd ` x ) >. )
31	29, 30	syl6eqr 2674	. . . . . . . . . . . . . . 15 |- ( x e. ( ( A [,] B ) X. ( A [,] B ) ) -> ( (,) ` x ) = ( ( 1st ` x ) (,) ( 2nd ` x ) ) )
32	31	eqeq1d 2624	. . . . . . . . . . . . . 14 |- ( x e. ( ( A [,] B ) X. ( A [,] B ) ) -> ( ( (,) ` x ) = s <-> ( ( 1st ` x ) (,) ( 2nd ` x ) ) = s ) )
33	32	adantl 482	. . . . . . . . . . . . 13 |- ( ( ph /\ x e. ( ( A [,] B ) X. ( A [,] B ) ) ) -> ( ( (,) ` x ) = s <-> ( ( 1st ` x ) (,) ( 2nd ` x ) ) = s ) )
34	2, 4	jca 554	. . . . . . . . . . . . . . . . . . . 20 |- ( ph -> ( A e. RR* /\ B e. RR* ) )
35	34	adantr 481	. . . . . . . . . . . . . . . . . . 19 |- ( ( ph /\ x e. ( ( A [,] B ) X. ( A [,] B ) ) ) -> ( A e. RR* /\ B e. RR* ) )
36		xp1st 7198	. . . . . . . . . . . . . . . . . . . 20 |- ( x e. ( ( A [,] B ) X. ( A [,] B ) ) -> ( 1st ` x ) e. ( A [,] B ) )
37		elicc1 12219	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( A e. RR* /\ B e. RR* ) -> ( ( 1st ` x ) e. ( A [,] B ) <-> ( ( 1st ` x ) e. RR* /\ A <_ ( 1st ` x ) /\ ( 1st ` x ) <_ B ) ) )
38	2, 4, 37	syl2anc 693	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ph -> ( ( 1st ` x ) e. ( A [,] B ) <-> ( ( 1st ` x ) e. RR* /\ A <_ ( 1st ` x ) /\ ( 1st ` x ) <_ B ) ) )
39	38	biimpa 501	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ph /\ ( 1st ` x ) e. ( A [,] B ) ) -> ( ( 1st ` x ) e. RR* /\ A <_ ( 1st ` x ) /\ ( 1st ` x ) <_ B ) )
40	39	simp2d 1074	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ph /\ ( 1st ` x ) e. ( A [,] B ) ) -> A <_ ( 1st ` x ) )
41	36, 40	sylan2 491	. . . . . . . . . . . . . . . . . . 19 |- ( ( ph /\ x e. ( ( A [,] B ) X. ( A [,] B ) ) ) -> A <_ ( 1st ` x ) )
42		xp2nd 7199	. . . . . . . . . . . . . . . . . . . 20 |- ( x e. ( ( A [,] B ) X. ( A [,] B ) ) -> ( 2nd ` x ) e. ( A [,] B ) )
43		iccleub 12229	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( A e. RR* /\ B e. RR* /\ ( 2nd ` x ) e. ( A [,] B ) ) -> ( 2nd ` x ) <_ B )
44	43	3expa 1265	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( 2nd ` x ) e. ( A [,] B ) ) -> ( 2nd ` x ) <_ B )
45	34, 42, 44	syl2an 494	. . . . . . . . . . . . . . . . . . 19 |- ( ( ph /\ x e. ( ( A [,] B ) X. ( A [,] B ) ) ) -> ( 2nd ` x ) <_ B )
46		ioossioo 12265	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( A <_ ( 1st ` x ) /\ ( 2nd ` x ) <_ B ) ) -> ( ( 1st ` x ) (,) ( 2nd ` x ) ) C_ ( A (,) B ) )
47	35, 41, 45, 46	syl12anc 1324	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ x e. ( ( A [,] B ) X. ( A [,] B ) ) ) -> ( ( 1st ` x ) (,) ( 2nd ` x ) ) C_ ( A (,) B ) )
48	47	sselda 3603	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ x e. ( ( A [,] B ) X. ( A [,] B ) ) ) /\ t e. ( ( 1st ` x ) (,) ( 2nd ` x ) ) ) -> t e. ( A (,) B ) )
49	22	ffvelrnda 6359	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ t e. ( A (,) B ) ) -> ( ( RR _D F ) ` t ) e. CC )
50	49	adantlr 751	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ x e. ( ( A [,] B ) X. ( A [,] B ) ) ) /\ t e. ( A (,) B ) ) -> ( ( RR _D F ) ` t ) e. CC )
51	48, 50	syldan 487	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ x e. ( ( A [,] B ) X. ( A [,] B ) ) ) /\ t e. ( ( 1st ` x ) (,) ( 2nd ` x ) ) ) -> ( ( RR _D F ) ` t ) e. CC )
52		ioombl 23333	. . . . . . . . . . . . . . . . . 18 |- ( ( 1st ` x ) (,) ( 2nd ` x ) ) e. dom vol
53	52	a1i 11	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ x e. ( ( A [,] B ) X. ( A [,] B ) ) ) -> ( ( 1st ` x ) (,) ( 2nd ` x ) ) e. dom vol )
54		fvexd 6203	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ x e. ( ( A [,] B ) X. ( A [,] B ) ) ) /\ t e. ( A (,) B ) ) -> ( ( RR _D F ) ` t ) e. _V )
55	22	feqmptd 6249	. . . . . . . . . . . . . . . . . . 19 |- ( ph -> ( RR _D F ) = ( t e. ( A (,) B ) |-> ( ( RR _D F ) ` t ) ) )
56	55, 19	eqeltrrd 2702	. . . . . . . . . . . . . . . . . 18 |- ( ph -> ( t e. ( A (,) B ) |-> ( ( RR _D F ) ` t ) ) e. L^1 )
57	56	adantr 481	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ x e. ( ( A [,] B ) X. ( A [,] B ) ) ) -> ( t e. ( A (,) B ) |-> ( ( RR _D F ) ` t ) ) e. L^1 )
58	47, 53, 54, 57	iblss 23571	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ x e. ( ( A [,] B ) X. ( A [,] B ) ) ) -> ( t e. ( ( 1st ` x ) (,) ( 2nd ` x ) ) |-> ( ( RR _D F ) ` t ) ) e. L^1 )
59		ax-resscn 9993	. . . . . . . . . . . . . . . . . . . . 21 |- RR C_ CC
60		ssid 3624	. . . . . . . . . . . . . . . . . . . . 21 |- CC C_ CC
61		cncfss 22702	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( RR C_ CC /\ CC C_ CC ) -> ( CC -cn-> RR ) C_ ( CC -cn-> CC ) )
62	59, 60, 61	mp2an 708	. . . . . . . . . . . . . . . . . . . 20 |- ( CC -cn-> RR ) C_ ( CC -cn-> CC )
63		abscncf 22704	. . . . . . . . . . . . . . . . . . . 20 |- abs e. ( CC -cn-> RR )
64	62, 63	sselii 3600	. . . . . . . . . . . . . . . . . . 19 |- abs e. ( CC -cn-> CC )
65	64	a1i 11	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ x e. ( ( A [,] B ) X. ( A [,] B ) ) ) -> abs e. ( CC -cn-> CC ) )
66	55	reseq1d 5395	. . . . . . . . . . . . . . . . . . . . 21 |- ( ph -> ( ( RR _D F ) |` ( ( 1st ` x ) (,) ( 2nd ` x ) ) ) = ( ( t e. ( A (,) B ) |-> ( ( RR _D F ) ` t ) ) |` ( ( 1st ` x ) (,) ( 2nd ` x ) ) ) )
67	66	adantr 481	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ph /\ x e. ( ( A [,] B ) X. ( A [,] B ) ) ) -> ( ( RR _D F ) |` ( ( 1st ` x ) (,) ( 2nd ` x ) ) ) = ( ( t e. ( A (,) B ) |-> ( ( RR _D F ) ` t ) ) |` ( ( 1st ` x ) (,) ( 2nd ` x ) ) ) )
68	47	resmptd 5452	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ph /\ x e. ( ( A [,] B ) X. ( A [,] B ) ) ) -> ( ( t e. ( A (,) B ) |-> ( ( RR _D F ) ` t ) ) |` ( ( 1st ` x ) (,) ( 2nd ` x ) ) ) = ( t e. ( ( 1st ` x ) (,) ( 2nd ` x ) ) |-> ( ( RR _D F ) ` t ) ) )
69	67, 68	eqtrd 2656	. . . . . . . . . . . . . . . . . . 19 |- ( ( ph /\ x e. ( ( A [,] B ) X. ( A [,] B ) ) ) -> ( ( RR _D F ) |` ( ( 1st ` x ) (,) ( 2nd ` x ) ) ) = ( t e. ( ( 1st ` x ) (,) ( 2nd ` x ) ) |-> ( ( RR _D F ) ` t ) ) )
70	20	adantr 481	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ph /\ x e. ( ( A [,] B ) X. ( A [,] B ) ) ) -> ( RR _D F ) e. ( ( A (,) B ) -cn-> CC ) )
71		rescncf 22700	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( 1st ` x ) (,) ( 2nd ` x ) ) C_ ( A (,) B ) -> ( ( RR _D F ) e. ( ( A (,) B ) -cn-> CC ) -> ( ( RR _D F ) |` ( ( 1st ` x ) (,) ( 2nd ` x ) ) ) e. ( ( ( 1st ` x ) (,) ( 2nd ` x ) ) -cn-> CC ) ) )
72	47, 70, 71	sylc 65	. . . . . . . . . . . . . . . . . . 19 |- ( ( ph /\ x e. ( ( A [,] B ) X. ( A [,] B ) ) ) -> ( ( RR _D F ) |` ( ( 1st ` x ) (,) ( 2nd ` x ) ) ) e. ( ( ( 1st ` x ) (,) ( 2nd ` x ) ) -cn-> CC ) )
73	69, 72	eqeltrrd 2702	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ x e. ( ( A [,] B ) X. ( A [,] B ) ) ) -> ( t e. ( ( 1st ` x ) (,) ( 2nd ` x ) ) |-> ( ( RR _D F ) ` t ) ) e. ( ( ( 1st ` x ) (,) ( 2nd ` x ) ) -cn-> CC ) )
74	65, 73	cncfmpt1f 22716	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ x e. ( ( A [,] B ) X. ( A [,] B ) ) ) -> ( t e. ( ( 1st ` x ) (,) ( 2nd ` x ) ) |-> ( abs ` ( ( RR _D F ) ` t ) ) ) e. ( ( ( 1st ` x ) (,) ( 2nd ` x ) ) -cn-> CC ) )
75		cnmbf 23426	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( 1st ` x ) (,) ( 2nd ` x ) ) e. dom vol /\ ( t e. ( ( 1st ` x ) (,) ( 2nd ` x ) ) |-> ( abs ` ( ( RR _D F ) ` t ) ) ) e. ( ( ( 1st ` x ) (,) ( 2nd ` x ) ) -cn-> CC ) ) -> ( t e. ( ( 1st ` x ) (,) ( 2nd ` x ) ) |-> ( abs ` ( ( RR _D F ) ` t ) ) ) e. MblFn )
76	52, 74, 75	sylancr 695	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ x e. ( ( A [,] B ) X. ( A [,] B ) ) ) -> ( t e. ( ( 1st ` x ) (,) ( 2nd ` x ) ) |-> ( abs ` ( ( RR _D F ) ` t ) ) ) e. MblFn )
77	51, 58	itgcl 23550	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ph /\ x e. ( ( A [,] B ) X. ( A [,] B ) ) ) -> S. ( ( 1st ` x ) (,) ( 2nd ` x ) ) ( ( RR _D F ) ` t ) _d t e. CC )
78	77	cjcld 13936	. . . . . . . . . . . . . . . . . . 19 |- ( ( ph /\ x e. ( ( A [,] B ) X. ( A [,] B ) ) ) -> ( * ` S. ( ( 1st ` x ) (,) ( 2nd ` x ) ) ( ( RR _D F ) ` t ) _d t ) e. CC )
79		ioossre 12235	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( 1st ` x ) (,) ( 2nd ` x ) ) C_ RR
80	79, 59	sstri 3612	. . . . . . . . . . . . . . . . . . . 20 |- ( ( 1st ` x ) (,) ( 2nd ` x ) ) C_ CC
81		cncfmptc 22714	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( * ` S. ( ( 1st ` x ) (,) ( 2nd ` x ) ) ( ( RR _D F ) ` t ) _d t ) e. CC /\ ( ( 1st ` x ) (,) ( 2nd ` x ) ) C_ CC /\ CC C_ CC ) -> ( s e. ( ( 1st ` x ) (,) ( 2nd ` x ) ) |-> ( * ` S. ( ( 1st ` x ) (,) ( 2nd ` x ) ) ( ( RR _D F ) ` t ) _d t ) ) e. ( ( ( 1st ` x ) (,) ( 2nd ` x ) ) -cn-> CC ) )
82	80, 60, 81	mp3an23 1416	. . . . . . . . . . . . . . . . . . 19 |- ( ( * ` S. ( ( 1st ` x ) (,) ( 2nd ` x ) ) ( ( RR _D F ) ` t ) _d t ) e. CC -> ( s e. ( ( 1st ` x ) (,) ( 2nd ` x ) ) |-> ( * ` S. ( ( 1st ` x ) (,) ( 2nd ` x ) ) ( ( RR _D F ) ` t ) _d t ) ) e. ( ( ( 1st ` x ) (,) ( 2nd ` x ) ) -cn-> CC ) )
83	78, 82	syl 17	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ x e. ( ( A [,] B ) X. ( A [,] B ) ) ) -> ( s e. ( ( 1st ` x ) (,) ( 2nd ` x ) ) |-> ( * ` S. ( ( 1st ` x ) (,) ( 2nd ` x ) ) ( ( RR _D F ) ` t ) _d t ) ) e. ( ( ( 1st ` x ) (,) ( 2nd ` x ) ) -cn-> CC ) )
84		nfcv 2764	. . . . . . . . . . . . . . . . . . . 20 |- F/_ s ( ( RR _D F ) ` t )
85		nfcsb1v 3549	. . . . . . . . . . . . . . . . . . . 20 |- F/_ t [_ s / t ]_ ( ( RR _D F ) ` t )
86		csbeq1a 3542	. . . . . . . . . . . . . . . . . . . 20 |- ( t = s -> ( ( RR _D F ) ` t ) = [_ s / t ]_ ( ( RR _D F ) ` t ) )
87	84, 85, 86	cbvmpt 4749	. . . . . . . . . . . . . . . . . . 19 |- ( t e. ( ( 1st ` x ) (,) ( 2nd ` x ) ) |-> ( ( RR _D F ) ` t ) ) = ( s e. ( ( 1st ` x ) (,) ( 2nd ` x ) ) |-> [_ s / t ]_ ( ( RR _D F ) ` t ) )
88	87, 73	syl5eqelr 2706	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ x e. ( ( A [,] B ) X. ( A [,] B ) ) ) -> ( s e. ( ( 1st ` x ) (,) ( 2nd ` x ) ) |-> [_ s / t ]_ ( ( RR _D F ) ` t ) ) e. ( ( ( 1st ` x ) (,) ( 2nd ` x ) ) -cn-> CC ) )
89	83, 88	mulcncf 23215	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ x e. ( ( A [,] B ) X. ( A [,] B ) ) ) -> ( s e. ( ( 1st ` x ) (,) ( 2nd ` x ) ) |-> ( ( * ` S. ( ( 1st ` x ) (,) ( 2nd ` x ) ) ( ( RR _D F ) ` t ) _d t ) x. [_ s / t ]_ ( ( RR _D F ) ` t ) ) ) e. ( ( ( 1st ` x ) (,) ( 2nd ` x ) ) -cn-> CC ) )
90		cnmbf 23426	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( 1st ` x ) (,) ( 2nd ` x ) ) e. dom vol /\ ( s e. ( ( 1st ` x ) (,) ( 2nd ` x ) ) |-> ( ( * ` S. ( ( 1st ` x ) (,) ( 2nd ` x ) ) ( ( RR _D F ) ` t ) _d t ) x. [_ s / t ]_ ( ( RR _D F ) ` t ) ) ) e. ( ( ( 1st ` x ) (,) ( 2nd ` x ) ) -cn-> CC ) ) -> ( s e. ( ( 1st ` x ) (,) ( 2nd ` x ) ) |-> ( ( * ` S. ( ( 1st ` x ) (,) ( 2nd ` x ) ) ( ( RR _D F ) ` t ) _d t ) x. [_ s / t ]_ ( ( RR _D F ) ` t ) ) ) e. MblFn )
91	52, 89, 90	sylancr 695	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ x e. ( ( A [,] B ) X. ( A [,] B ) ) ) -> ( s e. ( ( 1st ` x ) (,) ( 2nd ` x ) ) |-> ( ( * ` S. ( ( 1st ` x ) (,) ( 2nd ` x ) ) ( ( RR _D F ) ` t ) _d t ) x. [_ s / t ]_ ( ( RR _D F ) ` t ) ) ) e. MblFn )
92	51, 58, 76, 91	itgabsnc 33479	. . . . . . . . . . . . . . 15 |- ( ( ph /\ x e. ( ( A [,] B ) X. ( A [,] B ) ) ) -> ( abs ` S. ( ( 1st ` x ) (,) ( 2nd ` x ) ) ( ( RR _D F ) ` t ) _d t ) <_ S. ( ( 1st ` x ) (,) ( 2nd ` x ) ) ( abs ` ( ( RR _D F ) ` t ) ) _d t )
93	51	abscld 14175	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ x e. ( ( A [,] B ) X. ( A [,] B ) ) ) /\ t e. ( ( 1st ` x ) (,) ( 2nd ` x ) ) ) -> ( abs ` ( ( RR _D F ) ` t ) ) e. RR )
94		fvexd 6203	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ x e. ( ( A [,] B ) X. ( A [,] B ) ) ) /\ t e. ( ( 1st ` x ) (,) ( 2nd ` x ) ) ) -> ( ( RR _D F ) ` t ) e. _V )
95	94, 58, 76	iblabsnc 33474	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ x e. ( ( A [,] B ) X. ( A [,] B ) ) ) -> ( t e. ( ( 1st ` x ) (,) ( 2nd ` x ) ) |-> ( abs ` ( ( RR _D F ) ` t ) ) ) e. L^1 )
96	51	absge0d 14183	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ x e. ( ( A [,] B ) X. ( A [,] B ) ) ) /\ t e. ( ( 1st ` x ) (,) ( 2nd ` x ) ) ) -> 0 <_ ( abs ` ( ( RR _D F ) ` t ) ) )
97	93, 95, 96	itgposval 23562	. . . . . . . . . . . . . . 15 |- ( ( ph /\ x e. ( ( A [,] B ) X. ( A [,] B ) ) ) -> S. ( ( 1st ` x ) (,) ( 2nd ` x ) ) ( abs ` ( ( RR _D F ) ` t ) ) _d t = ( S.2 ` ( t e. RR |-> if ( t e. ( ( 1st ` x ) (,) ( 2nd ` x ) ) , ( abs ` ( ( RR _D F ) ` t ) ) , 0 ) ) ) )
98	92, 97	breqtrd 4679	. . . . . . . . . . . . . 14 |- ( ( ph /\ x e. ( ( A [,] B ) X. ( A [,] B ) ) ) -> ( abs ` S. ( ( 1st ` x ) (,) ( 2nd ` x ) ) ( ( RR _D F ) ` t ) _d t ) <_ ( S.2 ` ( t e. RR |-> if ( t e. ( ( 1st ` x ) (,) ( 2nd ` x ) ) , ( abs ` ( ( RR _D F ) ` t ) ) , 0 ) ) ) )
99		itgeq1 23539	. . . . . . . . . . . . . . . 16 |- ( ( ( 1st ` x ) (,) ( 2nd ` x ) ) = s -> S. ( ( 1st ` x ) (,) ( 2nd ` x ) ) ( ( RR _D F ) ` t ) _d t = S. s ( ( RR _D F ) ` t ) _d t )
100	99	fveq2d 6195	. . . . . . . . . . . . . . 15 |- ( ( ( 1st ` x ) (,) ( 2nd ` x ) ) = s -> ( abs ` S. ( ( 1st ` x ) (,) ( 2nd ` x ) ) ( ( RR _D F ) ` t ) _d t ) = ( abs ` S. s ( ( RR _D F ) ` t ) _d t ) )
101		eleq2 2690	. . . . . . . . . . . . . . . . . 18 |- ( ( ( 1st ` x ) (,) ( 2nd ` x ) ) = s -> ( t e. ( ( 1st ` x ) (,) ( 2nd ` x ) ) <-> t e. s ) )
102	101	ifbid 4108	. . . . . . . . . . . . . . . . 17 |- ( ( ( 1st ` x ) (,) ( 2nd ` x ) ) = s -> if ( t e. ( ( 1st ` x ) (,) ( 2nd ` x ) ) , ( abs ` ( ( RR _D F ) ` t ) ) , 0 ) = if ( t e. s , ( abs ` ( ( RR _D F ) ` t ) ) , 0 ) )
103	102	mpteq2dv 4745	. . . . . . . . . . . . . . . 16 |- ( ( ( 1st ` x ) (,) ( 2nd ` x ) ) = s -> ( t e. RR |-> if ( t e. ( ( 1st ` x ) (,) ( 2nd ` x ) ) , ( abs ` ( ( RR _D F ) ` t ) ) , 0 ) ) = ( t e. RR |-> if ( t e. s , ( abs ` ( ( RR _D F ) ` t ) ) , 0 ) ) )
104	103	fveq2d 6195	. . . . . . . . . . . . . . 15 |- ( ( ( 1st ` x ) (,) ( 2nd ` x ) ) = s -> ( S.2 ` ( t e. RR |-> if ( t e. ( ( 1st ` x ) (,) ( 2nd ` x ) ) , ( abs ` ( ( RR _D F ) ` t ) ) , 0 ) ) ) = ( S.2 ` ( t e. RR |-> if ( t e. s , ( abs ` ( ( RR _D F ) ` t ) ) , 0 ) ) ) )
105	100, 104	breq12d 4666	. . . . . . . . . . . . . 14 |- ( ( ( 1st ` x ) (,) ( 2nd ` x ) ) = s -> ( ( abs ` S. ( ( 1st ` x ) (,) ( 2nd ` x ) ) ( ( RR _D F ) ` t ) _d t ) <_ ( S.2 ` ( t e. RR |-> if ( t e. ( ( 1st ` x ) (,) ( 2nd ` x ) ) , ( abs ` ( ( RR _D F ) ` t ) ) , 0 ) ) ) <-> ( abs ` S. s ( ( RR _D F ) ` t ) _d t ) <_ ( S.2 ` ( t e. RR |-> if ( t e. s , ( abs ` ( ( RR _D F ) ` t ) ) , 0 ) ) ) ) )
106	98, 105	syl5ibcom 235	. . . . . . . . . . . . 13 |- ( ( ph /\ x e. ( ( A [,] B ) X. ( A [,] B ) ) ) -> ( ( ( 1st ` x ) (,) ( 2nd ` x ) ) = s -> ( abs ` S. s ( ( RR _D F ) ` t ) _d t ) <_ ( S.2 ` ( t e. RR |-> if ( t e. s , ( abs ` ( ( RR _D F ) ` t ) ) , 0 ) ) ) ) )
107	33, 106	sylbid 230	. . . . . . . . . . . 12 |- ( ( ph /\ x e. ( ( A [,] B ) X. ( A [,] B ) ) ) -> ( ( (,) ` x ) = s -> ( abs ` S. s ( ( RR _D F ) ` t ) _d t ) <_ ( S.2 ` ( t e. RR |-> if ( t e. s , ( abs ` ( ( RR _D F ) ` t ) ) , 0 ) ) ) ) )
108	107	rexlimdva 3031	. . . . . . . . . . 11 |- ( ph -> ( E. x e. ( ( A [,] B ) X. ( A [,] B ) ) ( (,) ` x ) = s -> ( abs ` S. s ( ( RR _D F ) ` t ) _d t ) <_ ( S.2 ` ( t e. RR |-> if ( t e. s , ( abs ` ( ( RR _D F ) ` t ) ) , 0 ) ) ) ) )
109	27, 108	syl5 34	. . . . . . . . . 10 |- ( ph -> ( s e. ( (,) " ( ( A [,] B ) X. ( A [,] B ) ) ) -> ( abs ` S. s ( ( RR _D F ) ` t ) _d t ) <_ ( S.2 ` ( t e. RR |-> if ( t e. s , ( abs ` ( ( RR _D F ) ` t ) ) , 0 ) ) ) ) )
110	109	ralrimiv 2965	. . . . . . . . 9 |- ( ph -> A. s e. ( (,) " ( ( A [,] B ) X. ( A [,] B ) ) ) ( abs ` S. s ( ( RR _D F ) ` t ) _d t ) <_ ( S.2 ` ( t e. RR |-> if ( t e. s , ( abs ` ( ( RR _D F ) ` t ) ) , 0 ) ) ) )
111	14, 1, 3, 5, 16, 18, 19, 22, 110	ftc1anc 33493	. . . . . . . 8 |- ( ph -> ( x e. ( A [,] B ) |-> S. ( A (,) x ) ( ( RR _D F ) ` t ) _d t ) e. ( ( A [,] B ) -cn-> CC ) )
112		ftc2nc.f	. . . . . . . . . . 11 |- ( ph -> F e. ( ( A [,] B ) -cn-> CC ) )
113		cncff 22696	. . . . . . . . . . 11 |- ( F e. ( ( A [,] B ) -cn-> CC ) -> F : ( A [,] B ) --> CC )
114	112, 113	syl 17	. . . . . . . . . 10 |- ( ph -> F : ( A [,] B ) --> CC )
115	114	feqmptd 6249	. . . . . . . . 9 |- ( ph -> F = ( x e. ( A [,] B ) |-> ( F ` x ) ) )
116	115, 112	eqeltrrd 2702	. . . . . . . 8 |- ( ph -> ( x e. ( A [,] B ) |-> ( F ` x ) ) e. ( ( A [,] B ) -cn-> CC ) )
117	11, 13, 111, 116	cncfmpt2f 22717	. . . . . . 7 |- ( ph -> ( x e. ( A [,] B ) |-> ( S. ( A (,) x ) ( ( RR _D F ) ` t ) _d t - ( F ` x ) ) ) e. ( ( A [,] B ) -cn-> CC ) )
118	59	a1i 11	. . . . . . . . . 10 |- ( ph -> RR C_ CC )
119		iccssre 12255	. . . . . . . . . . 11 |- ( ( A e. RR /\ B e. RR ) -> ( A [,] B ) C_ RR )
120	1, 3, 119	syl2anc 693	. . . . . . . . . 10 |- ( ph -> ( A [,] B ) C_ RR )
121		fvexd 6203	. . . . . . . . . . . 12 |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ t e. ( A (,) x ) ) -> ( ( RR _D F ) ` t ) e. _V )
122	3	adantr 481	. . . . . . . . . . . . . . 15 |- ( ( ph /\ x e. ( A [,] B ) ) -> B e. RR )
123	122	rexrd 10089	. . . . . . . . . . . . . 14 |- ( ( ph /\ x e. ( A [,] B ) ) -> B e. RR* )
124		elicc2 12238	. . . . . . . . . . . . . . . . 17 |- ( ( A e. RR /\ B e. RR ) -> ( x e. ( A [,] B ) <-> ( x e. RR /\ A <_ x /\ x <_ B ) ) )
125	1, 3, 124	syl2anc 693	. . . . . . . . . . . . . . . 16 |- ( ph -> ( x e. ( A [,] B ) <-> ( x e. RR /\ A <_ x /\ x <_ B ) ) )
126	125	biimpa 501	. . . . . . . . . . . . . . 15 |- ( ( ph /\ x e. ( A [,] B ) ) -> ( x e. RR /\ A <_ x /\ x <_ B ) )
127	126	simp3d 1075	. . . . . . . . . . . . . 14 |- ( ( ph /\ x e. ( A [,] B ) ) -> x <_ B )
128		iooss2 12211	. . . . . . . . . . . . . 14 |- ( ( B e. RR* /\ x <_ B ) -> ( A (,) x ) C_ ( A (,) B ) )
129	123, 127, 128	syl2anc 693	. . . . . . . . . . . . 13 |- ( ( ph /\ x e. ( A [,] B ) ) -> ( A (,) x ) C_ ( A (,) B ) )
130		ioombl 23333	. . . . . . . . . . . . . 14 |- ( A (,) x ) e. dom vol
131	130	a1i 11	. . . . . . . . . . . . 13 |- ( ( ph /\ x e. ( A [,] B ) ) -> ( A (,) x ) e. dom vol )
132		fvexd 6203	. . . . . . . . . . . . 13 |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ t e. ( A (,) B ) ) -> ( ( RR _D F ) ` t ) e. _V )
133	56	adantr 481	. . . . . . . . . . . . 13 |- ( ( ph /\ x e. ( A [,] B ) ) -> ( t e. ( A (,) B ) |-> ( ( RR _D F ) ` t ) ) e. L^1 )
134	129, 131, 132, 133	iblss 23571	. . . . . . . . . . . 12 |- ( ( ph /\ x e. ( A [,] B ) ) -> ( t e. ( A (,) x ) |-> ( ( RR _D F ) ` t ) ) e. L^1 )
135	121, 134	itgcl 23550	. . . . . . . . . . 11 |- ( ( ph /\ x e. ( A [,] B ) ) -> S. ( A (,) x ) ( ( RR _D F ) ` t ) _d t e. CC )
136	114	ffvelrnda 6359	. . . . . . . . . . 11 |- ( ( ph /\ x e. ( A [,] B ) ) -> ( F ` x ) e. CC )
137	135, 136	subcld 10392	. . . . . . . . . 10 |- ( ( ph /\ x e. ( A [,] B ) ) -> ( S. ( A (,) x ) ( ( RR _D F ) ` t ) _d t - ( F ` x ) ) e. CC )
138	11	tgioo2 22606	. . . . . . . . . 10 |- ( topGen ` ran (,) ) = ( ( TopOpen ` ℂfld ) ↾t RR )
139		iccntr 22624	. . . . . . . . . . 11 |- ( ( A e. RR /\ B e. RR ) -> ( ( int ` ( topGen ` ran (,) ) ) ` ( A [,] B ) ) = ( A (,) B ) )
140	1, 3, 139	syl2anc 693	. . . . . . . . . 10 |- ( ph -> ( ( int ` ( topGen ` ran (,) ) ) ` ( A [,] B ) ) = ( A (,) B ) )
141	118, 120, 137, 138, 11, 140	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 ) ) ) ) )
142		reelprrecn 10028	. . . . . . . . . . 11 |- RR e. { RR , CC }
143	142	a1i 11	. . . . . . . . . 10 |- ( ph -> RR e. { RR , CC } )
144		ioossicc 12259	. . . . . . . . . . . 12 |- ( A (,) B ) C_ ( A [,] B )
145	144	sseli 3599	. . . . . . . . . . 11 |- ( x e. ( A (,) B ) -> x e. ( A [,] B ) )
146	145, 135	sylan2 491	. . . . . . . . . 10 |- ( ( ph /\ x e. ( A (,) B ) ) -> S. ( A (,) x ) ( ( RR _D F ) ` t ) _d t e. CC )
147	22	ffvelrnda 6359	. . . . . . . . . 10 |- ( ( ph /\ x e. ( A (,) B ) ) -> ( ( RR _D F ) ` x ) e. CC )
148	14, 1, 3, 5, 20, 19	ftc1cnnc 33484	. . . . . . . . . . 11 |- ( ph -> ( RR _D ( x e. ( A [,] B ) |-> S. ( A (,) x ) ( ( RR _D F ) ` t ) _d t ) ) = ( RR _D F ) )
149	118, 120, 135, 138, 11, 140	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 ) ) )
150	22	feqmptd 6249	. . . . . . . . . . 11 |- ( ph -> ( RR _D F ) = ( x e. ( A (,) B ) |-> ( ( RR _D F ) ` x ) ) )
151	148, 149, 150	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 ) ) )
152	145, 136	sylan2 491	. . . . . . . . . 10 |- ( ( ph /\ x e. ( A (,) B ) ) -> ( F ` x ) e. CC )
153	115	oveq2d 6666	. . . . . . . . . . 11 |- ( ph -> ( RR _D F ) = ( RR _D ( x e. ( A [,] B ) |-> ( F ` x ) ) ) )
154	118, 120, 136, 138, 11, 140	dvmptntr 23734	. . . . . . . . . . 11 |- ( ph -> ( RR _D ( x e. ( A [,] B ) |-> ( F ` x ) ) ) = ( RR _D ( x e. ( A (,) B ) |-> ( F ` x ) ) ) )
155	153, 150, 154	3eqtr3rd 2665	. . . . . . . . . 10 |- ( ph -> ( RR _D ( x e. ( A (,) B ) |-> ( F ` x ) ) ) = ( x e. ( A (,) B ) |-> ( ( RR _D F ) ` x ) ) )
156	143, 146, 147, 151, 152, 147, 155	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 ) ) ) )
157	147	subidd 10380	. . . . . . . . . 10 |- ( ( ph /\ x e. ( A (,) B ) ) -> ( ( ( RR _D F ) ` x ) - ( ( RR _D F ) ` x ) ) = 0 )
158	157	mpteq2dva 4744	. . . . . . . . 9 |- ( ph -> ( x e. ( A (,) B ) |-> ( ( ( RR _D F ) ` x ) - ( ( RR _D F ) ` x ) ) ) = ( x e. ( A (,) B ) |-> 0 ) )
159	141, 156, 158	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 ) )
160		fconstmpt 5163	. . . . . . . 8 |- ( ( A (,) B ) X. { 0 } ) = ( x e. ( A (,) B ) |-> 0 )
161	159, 160	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 } ) )
162	1, 3, 117, 161	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 ) } ) )
163	162	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 ) )
164		oveq2 6658	. . . . . . . . 9 |- ( x = B -> ( A (,) x ) = ( A (,) B ) )
165		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 )
166	164, 165	syl 17	. . . . . . . 8 |- ( x = B -> S. ( A (,) x ) ( ( RR _D F ) ` t ) _d t = S. ( A (,) B ) ( ( RR _D F ) ` t ) _d t )
167		fveq2 6191	. . . . . . . 8 |- ( x = B -> ( F ` x ) = ( F ` B ) )
168	166, 167	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 ) ) )
169		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 ) ) )
170		ovex 6678	. . . . . . 7 |- ( S. ( A (,) B ) ( ( RR _D F ) ` t ) _d t - ( F ` B ) ) e. _V
171	168, 169, 170	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 ) ) )
172	7, 171	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 ) ) )
173	163, 172	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 ) ) )
174		lbicc2 12288	. . . . . 6 |- ( ( A e. RR* /\ B e. RR* /\ A <_ B ) -> A e. ( A [,] B ) )
175	2, 4, 5, 174	syl3anc 1326	. . . . 5 |- ( ph -> A e. ( A [,] B ) )
176		oveq2 6658	. . . . . . . . . . 11 |- ( x = A -> ( A (,) x ) = ( A (,) A ) )
177		iooid 12203	. . . . . . . . . . 11 |- ( A (,) A ) = (/)
178	176, 177	syl6eq 2672	. . . . . . . . . 10 |- ( x = A -> ( A (,) x ) = (/) )
179		itgeq1 23539	. . . . . . . . . 10 |- ( ( A (,) x ) = (/) -> S. ( A (,) x ) ( ( RR _D F ) ` t ) _d t = S. (/) ( ( RR _D F ) ` t ) _d t )
180	178, 179	syl 17	. . . . . . . . 9 |- ( x = A -> S. ( A (,) x ) ( ( RR _D F ) ` t ) _d t = S. (/) ( ( RR _D F ) ` t ) _d t )
181		itg0 23546	. . . . . . . . 9 |- S. (/) ( ( RR _D F ) ` t ) _d t = 0
182	180, 181	syl6eq 2672	. . . . . . . 8 |- ( x = A -> S. ( A (,) x ) ( ( RR _D F ) ` t ) _d t = 0 )
183		fveq2 6191	. . . . . . . 8 |- ( x = A -> ( F ` x ) = ( F ` A ) )
184	182, 183	oveq12d 6668	. . . . . . 7 |- ( x = A -> ( S. ( A (,) x ) ( ( RR _D F ) ` t ) _d t - ( F ` x ) ) = ( 0 - ( F ` A ) ) )
185		df-neg 10269	. . . . . . 7 |- -u ( F ` A ) = ( 0 - ( F ` A ) )
186	184, 185	syl6eqr 2674	. . . . . 6 |- ( x = A -> ( S. ( A (,) x ) ( ( RR _D F ) ` t ) _d t - ( F ` x ) ) = -u ( F ` A ) )
187		negex 10279	. . . . . 6 |- -u ( F ` A ) e. _V
188	186, 169, 187	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 ) )
189	175, 188	syl 17	. . . 4 |- ( ph -> ( ( x e. ( A [,] B ) |-> ( S. ( A (,) x ) ( ( RR _D F ) ` t ) _d t - ( F ` x ) ) ) ` A ) = -u ( F ` A ) )
190	10, 173, 189	3eqtr3d 2664	. . 3 |- ( ph -> ( S. ( A (,) B ) ( ( RR _D F ) ` t ) _d t - ( F ` B ) ) = -u ( F ` A ) )
191	190	oveq2d 6666	. 2 |- ( ph -> ( ( F ` B ) + ( S. ( A (,) B ) ( ( RR _D F ) ` t ) _d t - ( F ` B ) ) ) = ( ( F ` B ) + -u ( F ` A ) ) )
192	114, 7	ffvelrnd 6360	. . 3 |- ( ph -> ( F ` B ) e. CC )
193		fvexd 6203	. . . 4 |- ( ( ph /\ t e. ( A (,) B ) ) -> ( ( RR _D F ) ` t ) e. _V )
194	193, 56	itgcl 23550	. . 3 |- ( ph -> S. ( A (,) B ) ( ( RR _D F ) ` t ) _d t e. CC )
195	192, 194	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 )
196	114, 175	ffvelrnd 6360	. . 3 |- ( ph -> ( F ` A ) e. CC )
197	192, 196	negsubd 10398	. 2 |- ( ph -> ( ( F ` B ) + -u ( F ` A ) ) = ( ( F ` B ) - ( F ` A ) ) )
198	191, 195, 197	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   E.wrex 2913   _Vcvv 3200   [_csb 3533   C_ wss 3574   (/)c0 3915   ifcif 4086   ~Pcpw 4158   {csn 4177   {cpr 4179   <.cop 4183   class class class wbr 4653   |-> cmpt 4729   X. cxp 5112   dom cdm 5114   ran crn 5115   |` cres 5116   "cima 5117   Fun wfun 5882   -->wf 5884   ` cfv 5888  (class class class)co 6650   1stc1st 7166   2ndc2nd 7167   CCcc 9934   RRcr 9935   0cc0 9936   + caddc 9939   x. cmul 9941   RR*cxr 10073   <_ cle 10075   - cmin 10266   -ucneg 10267   (,)cioo 12175   [,]cicc 12178   *ccj 13836   abscabs 13974   TopOpenctopn 16082   topGenctg 16098  ℂ_fldccnfld 19746   intcnt 20821   Cn ccn 21028   tX ctx 21363   -cn->ccncf 22679   volcvol 23232  MblFncmbf 23383   S.2citg2 23385   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-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-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:  areacirc  33505