Theorem itgsbtaddcnst 40198

Description: Integral substitution, adding a constant to the function's argument. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Hypotheses

Ref	Expression
itgsbtaddcnst.a	|- ( ph -> A e. RR )
itgsbtaddcnst.b	|- ( ph -> B e. RR )
itgsbtaddcnst.aleb	|- ( ph -> A <_ B )
itgsbtaddcnst.x	|- ( ph -> X e. RR )
itgsbtaddcnst.f	|- ( ph -> F e. ( ( A [,] B ) -cn-> CC ) )

Assertion

Ref	Expression
itgsbtaddcnst	|- ( ph -> S__ [ ( A - X ) -> ( B - X ) ] ( F ` ( X + s ) ) _d s = S__ [ A -> B ] ( F ` t ) _d t )

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

Proof of Theorem itgsbtaddcnst

Step	Hyp	Ref	Expression
1		itgsbtaddcnst.a	. . 3 |- ( ph -> A e. RR )
2		itgsbtaddcnst.b	. . 3 |- ( ph -> B e. RR )
3		itgsbtaddcnst.aleb	. . 3 |- ( ph -> A <_ B )
4	1, 2	iccssred 39727	. . . . . . . . 9 |- ( ph -> ( A [,] B ) C_ RR )
5	4	sselda 3603	. . . . . . . 8 |- ( ( ph /\ t e. ( A [,] B ) ) -> t e. RR )
6	5	recnd 10068	. . . . . . 7 |- ( ( ph /\ t e. ( A [,] B ) ) -> t e. CC )
7		itgsbtaddcnst.x	. . . . . . . . 9 |- ( ph -> X e. RR )
8	7	recnd 10068	. . . . . . . 8 |- ( ph -> X e. CC )
9	8	adantr 481	. . . . . . 7 |- ( ( ph /\ t e. ( A [,] B ) ) -> X e. CC )
10	6, 9	negsubd 10398	. . . . . 6 |- ( ( ph /\ t e. ( A [,] B ) ) -> ( t + -u X ) = ( t - X ) )
11	10	eqcomd 2628	. . . . 5 |- ( ( ph /\ t e. ( A [,] B ) ) -> ( t - X ) = ( t + -u X ) )
12	11	mpteq2dva 4744	. . . 4 |- ( ph -> ( t e. ( A [,] B ) |-> ( t - X ) ) = ( t e. ( A [,] B ) |-> ( t + -u X ) ) )
13	1	adantr 481	. . . . . . . . 9 |- ( ( ph /\ t e. ( A [,] B ) ) -> A e. RR )
14	7	adantr 481	. . . . . . . . 9 |- ( ( ph /\ t e. ( A [,] B ) ) -> X e. RR )
15	13, 14	resubcld 10458	. . . . . . . 8 |- ( ( ph /\ t e. ( A [,] B ) ) -> ( A - X ) e. RR )
16	2	adantr 481	. . . . . . . . 9 |- ( ( ph /\ t e. ( A [,] B ) ) -> B e. RR )
17	16, 14	resubcld 10458	. . . . . . . 8 |- ( ( ph /\ t e. ( A [,] B ) ) -> ( B - X ) e. RR )
18	5, 14	resubcld 10458	. . . . . . . 8 |- ( ( ph /\ t e. ( A [,] B ) ) -> ( t - X ) e. RR )
19		simpr 477	. . . . . . . . . . 11 |- ( ( ph /\ t e. ( A [,] B ) ) -> t e. ( A [,] B ) )
20	1, 2	jca 554	. . . . . . . . . . . . 13 |- ( ph -> ( A e. RR /\ B e. RR ) )
21	20	adantr 481	. . . . . . . . . . . 12 |- ( ( ph /\ t e. ( A [,] B ) ) -> ( A e. RR /\ B e. RR ) )
22		elicc2 12238	. . . . . . . . . . . 12 |- ( ( A e. RR /\ B e. RR ) -> ( t e. ( A [,] B ) <-> ( t e. RR /\ A <_ t /\ t <_ B ) ) )
23	21, 22	syl 17	. . . . . . . . . . 11 |- ( ( ph /\ t e. ( A [,] B ) ) -> ( t e. ( A [,] B ) <-> ( t e. RR /\ A <_ t /\ t <_ B ) ) )
24	19, 23	mpbid 222	. . . . . . . . . 10 |- ( ( ph /\ t e. ( A [,] B ) ) -> ( t e. RR /\ A <_ t /\ t <_ B ) )
25	24	simp2d 1074	. . . . . . . . 9 |- ( ( ph /\ t e. ( A [,] B ) ) -> A <_ t )
26	13, 5, 14, 25	lesub1dd 10643	. . . . . . . 8 |- ( ( ph /\ t e. ( A [,] B ) ) -> ( A - X ) <_ ( t - X ) )
27	24	simp3d 1075	. . . . . . . . 9 |- ( ( ph /\ t e. ( A [,] B ) ) -> t <_ B )
28	5, 16, 14, 27	lesub1dd 10643	. . . . . . . 8 |- ( ( ph /\ t e. ( A [,] B ) ) -> ( t - X ) <_ ( B - X ) )
29	15, 17, 18, 26, 28	eliccd 39726	. . . . . . 7 |- ( ( ph /\ t e. ( A [,] B ) ) -> ( t - X ) e. ( ( A - X ) [,] ( B - X ) ) )
30		eqid 2622	. . . . . . 7 |- ( t e. ( A [,] B ) |-> ( t - X ) ) = ( t e. ( A [,] B ) |-> ( t - X ) )
31	29, 30	fmptd 6385	. . . . . 6 |- ( ph -> ( t e. ( A [,] B ) |-> ( t - X ) ) : ( A [,] B ) --> ( ( A - X ) [,] ( B - X ) ) )
32	12, 31	feq1dd 39347	. . . . 5 |- ( ph -> ( t e. ( A [,] B ) |-> ( t + -u X ) ) : ( A [,] B ) --> ( ( A - X ) [,] ( B - X ) ) )
33	1, 7	resubcld 10458	. . . . . . . 8 |- ( ph -> ( A - X ) e. RR )
34	2, 7	resubcld 10458	. . . . . . . 8 |- ( ph -> ( B - X ) e. RR )
35	33, 34	iccssred 39727	. . . . . . 7 |- ( ph -> ( ( A - X ) [,] ( B - X ) ) C_ RR )
36		ax-resscn 9993	. . . . . . 7 |- RR C_ CC
37	35, 36	syl6ss 3615	. . . . . 6 |- ( ph -> ( ( A - X ) [,] ( B - X ) ) C_ CC )
38	4, 36	syl6ss 3615	. . . . . . . . 9 |- ( ph -> ( A [,] B ) C_ CC )
39	38	resmptd 5452	. . . . . . . 8 |- ( ph -> ( ( t e. CC |-> ( t - X ) ) |` ( A [,] B ) ) = ( t e. ( A [,] B ) |-> ( t - X ) ) )
40		ssid 3624	. . . . . . . . . . . . 13 |- CC C_ CC
41		cncfmptid 22715	. . . . . . . . . . . . 13 |- ( ( CC C_ CC /\ CC C_ CC ) -> ( t e. CC |-> t ) e. ( CC -cn-> CC ) )
42	40, 40, 41	mp2an 708	. . . . . . . . . . . 12 |- ( t e. CC |-> t ) e. ( CC -cn-> CC )
43	42	a1i 11	. . . . . . . . . . 11 |- ( X e. CC -> ( t e. CC |-> t ) e. ( CC -cn-> CC ) )
44	40	a1i 11	. . . . . . . . . . . 12 |- ( X e. CC -> CC C_ CC )
45		id 22	. . . . . . . . . . . 12 |- ( X e. CC -> X e. CC )
46	44, 45, 44	constcncfg 40084	. . . . . . . . . . 11 |- ( X e. CC -> ( t e. CC |-> X ) e. ( CC -cn-> CC ) )
47	43, 46	subcncf 40082	. . . . . . . . . 10 |- ( X e. CC -> ( t e. CC |-> ( t - X ) ) e. ( CC -cn-> CC ) )
48	8, 47	syl 17	. . . . . . . . 9 |- ( ph -> ( t e. CC |-> ( t - X ) ) e. ( CC -cn-> CC ) )
49		rescncf 22700	. . . . . . . . 9 |- ( ( A [,] B ) C_ CC -> ( ( t e. CC |-> ( t - X ) ) e. ( CC -cn-> CC ) -> ( ( t e. CC |-> ( t - X ) ) |` ( A [,] B ) ) e. ( ( A [,] B ) -cn-> CC ) ) )
50	38, 48, 49	sylc 65	. . . . . . . 8 |- ( ph -> ( ( t e. CC |-> ( t - X ) ) |` ( A [,] B ) ) e. ( ( A [,] B ) -cn-> CC ) )
51	39, 50	eqeltrrd 2702	. . . . . . 7 |- ( ph -> ( t e. ( A [,] B ) |-> ( t - X ) ) e. ( ( A [,] B ) -cn-> CC ) )
52	12, 51	eqeltrrd 2702	. . . . . 6 |- ( ph -> ( t e. ( A [,] B ) |-> ( t + -u X ) ) e. ( ( A [,] B ) -cn-> CC ) )
53		cncffvrn 22701	. . . . . 6 |- ( ( ( ( A - X ) [,] ( B - X ) ) C_ CC /\ ( t e. ( A [,] B ) |-> ( t + -u X ) ) e. ( ( A [,] B ) -cn-> CC ) ) -> ( ( t e. ( A [,] B ) |-> ( t + -u X ) ) e. ( ( A [,] B ) -cn-> ( ( A - X ) [,] ( B - X ) ) ) <-> ( t e. ( A [,] B ) |-> ( t + -u X ) ) : ( A [,] B ) --> ( ( A - X ) [,] ( B - X ) ) ) )
54	37, 52, 53	syl2anc 693	. . . . 5 |- ( ph -> ( ( t e. ( A [,] B ) |-> ( t + -u X ) ) e. ( ( A [,] B ) -cn-> ( ( A - X ) [,] ( B - X ) ) ) <-> ( t e. ( A [,] B ) |-> ( t + -u X ) ) : ( A [,] B ) --> ( ( A - X ) [,] ( B - X ) ) ) )
55	32, 54	mpbird 247	. . . 4 |- ( ph -> ( t e. ( A [,] B ) |-> ( t + -u X ) ) e. ( ( A [,] B ) -cn-> ( ( A - X ) [,] ( B - X ) ) ) )
56	12, 55	eqeltrd 2701	. . 3 |- ( ph -> ( t e. ( A [,] B ) |-> ( t - X ) ) e. ( ( A [,] B ) -cn-> ( ( A - X ) [,] ( B - X ) ) ) )
57		eqid 2622	. . . . 5 |- ( s e. CC |-> ( X + s ) ) = ( s e. CC |-> ( X + s ) )
58	8	adantr 481	. . . . . . . 8 |- ( ( ph /\ s e. CC ) -> X e. CC )
59		simpr 477	. . . . . . . 8 |- ( ( ph /\ s e. CC ) -> s e. CC )
60	58, 59	addcomd 10238	. . . . . . 7 |- ( ( ph /\ s e. CC ) -> ( X + s ) = ( s + X ) )
61	60	mpteq2dva 4744	. . . . . 6 |- ( ph -> ( s e. CC |-> ( X + s ) ) = ( s e. CC |-> ( s + X ) ) )
62		eqid 2622	. . . . . . . 8 |- ( s e. CC |-> ( s + X ) ) = ( s e. CC |-> ( s + X ) )
63	62	addccncf 22719	. . . . . . 7 |- ( X e. CC -> ( s e. CC |-> ( s + X ) ) e. ( CC -cn-> CC ) )
64	8, 63	syl 17	. . . . . 6 |- ( ph -> ( s e. CC |-> ( s + X ) ) e. ( CC -cn-> CC ) )
65	61, 64	eqeltrd 2701	. . . . 5 |- ( ph -> ( s e. CC |-> ( X + s ) ) e. ( CC -cn-> CC ) )
66	1	adantr 481	. . . . . 6 |- ( ( ph /\ s e. ( ( A - X ) [,] ( B - X ) ) ) -> A e. RR )
67	2	adantr 481	. . . . . 6 |- ( ( ph /\ s e. ( ( A - X ) [,] ( B - X ) ) ) -> B e. RR )
68	7	adantr 481	. . . . . . 7 |- ( ( ph /\ s e. ( ( A - X ) [,] ( B - X ) ) ) -> X e. RR )
69	35	sselda 3603	. . . . . . 7 |- ( ( ph /\ s e. ( ( A - X ) [,] ( B - X ) ) ) -> s e. RR )
70	68, 69	readdcld 10069	. . . . . 6 |- ( ( ph /\ s e. ( ( A - X ) [,] ( B - X ) ) ) -> ( X + s ) e. RR )
71		simpr 477	. . . . . . . . 9 |- ( ( ph /\ s e. ( ( A - X ) [,] ( B - X ) ) ) -> s e. ( ( A - X ) [,] ( B - X ) ) )
72	33	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ s e. ( ( A - X ) [,] ( B - X ) ) ) -> ( A - X ) e. RR )
73	34	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ s e. ( ( A - X ) [,] ( B - X ) ) ) -> ( B - X ) e. RR )
74		elicc2 12238	. . . . . . . . . 10 |- ( ( ( A - X ) e. RR /\ ( B - X ) e. RR ) -> ( s e. ( ( A - X ) [,] ( B - X ) ) <-> ( s e. RR /\ ( A - X ) <_ s /\ s <_ ( B - X ) ) ) )
75	72, 73, 74	syl2anc 693	. . . . . . . . 9 |- ( ( ph /\ s e. ( ( A - X ) [,] ( B - X ) ) ) -> ( s e. ( ( A - X ) [,] ( B - X ) ) <-> ( s e. RR /\ ( A - X ) <_ s /\ s <_ ( B - X ) ) ) )
76	71, 75	mpbid 222	. . . . . . . 8 |- ( ( ph /\ s e. ( ( A - X ) [,] ( B - X ) ) ) -> ( s e. RR /\ ( A - X ) <_ s /\ s <_ ( B - X ) ) )
77	76	simp2d 1074	. . . . . . 7 |- ( ( ph /\ s e. ( ( A - X ) [,] ( B - X ) ) ) -> ( A - X ) <_ s )
78	66, 68, 69	lesubadd2d 10626	. . . . . . 7 |- ( ( ph /\ s e. ( ( A - X ) [,] ( B - X ) ) ) -> ( ( A - X ) <_ s <-> A <_ ( X + s ) ) )
79	77, 78	mpbid 222	. . . . . 6 |- ( ( ph /\ s e. ( ( A - X ) [,] ( B - X ) ) ) -> A <_ ( X + s ) )
80	76	simp3d 1075	. . . . . . 7 |- ( ( ph /\ s e. ( ( A - X ) [,] ( B - X ) ) ) -> s <_ ( B - X ) )
81	68, 69, 67	leaddsub2d 10629	. . . . . . 7 |- ( ( ph /\ s e. ( ( A - X ) [,] ( B - X ) ) ) -> ( ( X + s ) <_ B <-> s <_ ( B - X ) ) )
82	80, 81	mpbird 247	. . . . . 6 |- ( ( ph /\ s e. ( ( A - X ) [,] ( B - X ) ) ) -> ( X + s ) <_ B )
83	66, 67, 70, 79, 82	eliccd 39726	. . . . 5 |- ( ( ph /\ s e. ( ( A - X ) [,] ( B - X ) ) ) -> ( X + s ) e. ( A [,] B ) )
84	57, 65, 37, 38, 83	cncfmptssg 40083	. . . 4 |- ( ph -> ( s e. ( ( A - X ) [,] ( B - X ) ) |-> ( X + s ) ) e. ( ( ( A - X ) [,] ( B - X ) ) -cn-> ( A [,] B ) ) )
85		itgsbtaddcnst.f	. . . 4 |- ( ph -> F e. ( ( A [,] B ) -cn-> CC ) )
86	84, 85	cncfcompt 40096	. . 3 |- ( ph -> ( s e. ( ( A - X ) [,] ( B - X ) ) |-> ( F ` ( X + s ) ) ) e. ( ( ( A - X ) [,] ( B - X ) ) -cn-> CC ) )
87		ax-1cn 9994	. . . . . 6 |- 1 e. CC
88		ioosscn 39716	. . . . . 6 |- ( A (,) B ) C_ CC
89		cncfmptc 22714	. . . . . 6 |- ( ( 1 e. CC /\ ( A (,) B ) C_ CC /\ CC C_ CC ) -> ( t e. ( A (,) B ) |-> 1 ) e. ( ( A (,) B ) -cn-> CC ) )
90	87, 88, 40, 89	mp3an 1424	. . . . 5 |- ( t e. ( A (,) B ) |-> 1 ) e. ( ( A (,) B ) -cn-> CC )
91	90	a1i 11	. . . 4 |- ( ph -> ( t e. ( A (,) B ) |-> 1 ) e. ( ( A (,) B ) -cn-> CC ) )
92		fconstmpt 5163	. . . . 5 |- ( ( A (,) B ) X. { 1 } ) = ( t e. ( A (,) B ) |-> 1 )
93		ioombl 23333	. . . . . . 7 |- ( A (,) B ) e. dom vol
94	93	a1i 11	. . . . . 6 |- ( ph -> ( A (,) B ) e. dom vol )
95		volioo 23337	. . . . . . . 8 |- ( ( A e. RR /\ B e. RR /\ A <_ B ) -> ( vol ` ( A (,) B ) ) = ( B - A ) )
96	1, 2, 3, 95	syl3anc 1326	. . . . . . 7 |- ( ph -> ( vol ` ( A (,) B ) ) = ( B - A ) )
97	2, 1	resubcld 10458	. . . . . . 7 |- ( ph -> ( B - A ) e. RR )
98	96, 97	eqeltrd 2701	. . . . . 6 |- ( ph -> ( vol ` ( A (,) B ) ) e. RR )
99		1cnd 10056	. . . . . 6 |- ( ph -> 1 e. CC )
100		iblconst 23584	. . . . . 6 |- ( ( ( A (,) B ) e. dom vol /\ ( vol ` ( A (,) B ) ) e. RR /\ 1 e. CC ) -> ( ( A (,) B ) X. { 1 } ) e. L^1 )
101	94, 98, 99, 100	syl3anc 1326	. . . . 5 |- ( ph -> ( ( A (,) B ) X. { 1 } ) e. L^1 )
102	92, 101	syl5eqelr 2706	. . . 4 |- ( ph -> ( t e. ( A (,) B ) |-> 1 ) e. L^1 )
103	91, 102	elind 3798	. . 3 |- ( ph -> ( t e. ( A (,) B ) |-> 1 ) e. ( ( ( A (,) B ) -cn-> CC ) i^i L^1 ) )
104	36	a1i 11	. . . . 5 |- ( ph -> RR C_ CC )
105	18	recnd 10068	. . . . 5 |- ( ( ph /\ t e. ( A [,] B ) ) -> ( t - X ) e. CC )
106		eqid 2622	. . . . . 6 |- ( TopOpen ` ℂfld ) = ( TopOpen ` ℂfld )
107	106	tgioo2 22606	. . . . 5 |- ( topGen ` ran (,) ) = ( ( TopOpen ` ℂfld ) ↾t RR )
108		iccntr 22624	. . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> ( ( int ` ( topGen ` ran (,) ) ) ` ( A [,] B ) ) = ( A (,) B ) )
109	20, 108	syl 17	. . . . 5 |- ( ph -> ( ( int ` ( topGen ` ran (,) ) ) ` ( A [,] B ) ) = ( A (,) B ) )
110	104, 4, 105, 107, 106, 109	dvmptntr 23734	. . . 4 |- ( ph -> ( RR _D ( t e. ( A [,] B ) |-> ( t - X ) ) ) = ( RR _D ( t e. ( A (,) B ) |-> ( t - X ) ) ) )
111		reelprrecn 10028	. . . . . 6 |- RR e. { RR , CC }
112	111	a1i 11	. . . . 5 |- ( ph -> RR e. { RR , CC } )
113		ioossre 12235	. . . . . . . 8 |- ( A (,) B ) C_ RR
114	113	sseli 3599	. . . . . . 7 |- ( t e. ( A (,) B ) -> t e. RR )
115	114	adantl 482	. . . . . 6 |- ( ( ph /\ t e. ( A (,) B ) ) -> t e. RR )
116	115	recnd 10068	. . . . 5 |- ( ( ph /\ t e. ( A (,) B ) ) -> t e. CC )
117		1cnd 10056	. . . . 5 |- ( ( ph /\ t e. ( A (,) B ) ) -> 1 e. CC )
118	104	sselda 3603	. . . . . 6 |- ( ( ph /\ t e. RR ) -> t e. CC )
119		1cnd 10056	. . . . . 6 |- ( ( ph /\ t e. RR ) -> 1 e. CC )
120	112	dvmptid 23720	. . . . . 6 |- ( ph -> ( RR _D ( t e. RR |-> t ) ) = ( t e. RR |-> 1 ) )
121	113	a1i 11	. . . . . 6 |- ( ph -> ( A (,) B ) C_ RR )
122		iooretop 22569	. . . . . . 7 |- ( A (,) B ) e. ( topGen ` ran (,) )
123	122	a1i 11	. . . . . 6 |- ( ph -> ( A (,) B ) e. ( topGen ` ran (,) ) )
124	112, 118, 119, 120, 121, 107, 106, 123	dvmptres 23726	. . . . 5 |- ( ph -> ( RR _D ( t e. ( A (,) B ) |-> t ) ) = ( t e. ( A (,) B ) |-> 1 ) )
125	8	adantr 481	. . . . 5 |- ( ( ph /\ t e. ( A (,) B ) ) -> X e. CC )
126		0cnd 10033	. . . . 5 |- ( ( ph /\ t e. ( A (,) B ) ) -> 0 e. CC )
127	8	adantr 481	. . . . . 6 |- ( ( ph /\ t e. RR ) -> X e. CC )
128		0cnd 10033	. . . . . 6 |- ( ( ph /\ t e. RR ) -> 0 e. CC )
129	112, 8	dvmptc 23721	. . . . . 6 |- ( ph -> ( RR _D ( t e. RR |-> X ) ) = ( t e. RR |-> 0 ) )
130	112, 127, 128, 129, 121, 107, 106, 123	dvmptres 23726	. . . . 5 |- ( ph -> ( RR _D ( t e. ( A (,) B ) |-> X ) ) = ( t e. ( A (,) B ) |-> 0 ) )
131	112, 116, 117, 124, 125, 126, 130	dvmptsub 23730	. . . 4 |- ( ph -> ( RR _D ( t e. ( A (,) B ) |-> ( t - X ) ) ) = ( t e. ( A (,) B ) |-> ( 1 - 0 ) ) )
132	117	subid1d 10381	. . . . 5 |- ( ( ph /\ t e. ( A (,) B ) ) -> ( 1 - 0 ) = 1 )
133	132	mpteq2dva 4744	. . . 4 |- ( ph -> ( t e. ( A (,) B ) |-> ( 1 - 0 ) ) = ( t e. ( A (,) B ) |-> 1 ) )
134	110, 131, 133	3eqtrd 2660	. . 3 |- ( ph -> ( RR _D ( t e. ( A [,] B ) |-> ( t - X ) ) ) = ( t e. ( A (,) B ) |-> 1 ) )
135		oveq2 6658	. . . 4 |- ( s = ( t - X ) -> ( X + s ) = ( X + ( t - X ) ) )
136	135	fveq2d 6195	. . 3 |- ( s = ( t - X ) -> ( F ` ( X + s ) ) = ( F ` ( X + ( t - X ) ) ) )
137		oveq1 6657	. . 3 |- ( t = A -> ( t - X ) = ( A - X ) )
138		oveq1 6657	. . 3 |- ( t = B -> ( t - X ) = ( B - X ) )
139	1, 2, 3, 56, 86, 103, 134, 136, 137, 138, 33, 34	itgsubsticc 40192	. 2 |- ( ph -> S__ [ ( A - X ) -> ( B - X ) ] ( F ` ( X + s ) ) _d s = S__ [ A -> B ] ( ( F ` ( X + ( t - X ) ) ) x. 1 ) _d t )
140	125, 116	pncan3d 10395	. . . . . 6 |- ( ( ph /\ t e. ( A (,) B ) ) -> ( X + ( t - X ) ) = t )
141	140	fveq2d 6195	. . . . 5 |- ( ( ph /\ t e. ( A (,) B ) ) -> ( F ` ( X + ( t - X ) ) ) = ( F ` t ) )
142	141	oveq1d 6665	. . . 4 |- ( ( ph /\ t e. ( A (,) B ) ) -> ( ( F ` ( X + ( t - X ) ) ) x. 1 ) = ( ( F ` t ) x. 1 ) )
143		cncff 22696	. . . . . . . 8 |- ( F e. ( ( A [,] B ) -cn-> CC ) -> F : ( A [,] B ) --> CC )
144	85, 143	syl 17	. . . . . . 7 |- ( ph -> F : ( A [,] B ) --> CC )
145	144	adantr 481	. . . . . 6 |- ( ( ph /\ t e. ( A (,) B ) ) -> F : ( A [,] B ) --> CC )
146		ioossicc 12259	. . . . . . . 8 |- ( A (,) B ) C_ ( A [,] B )
147	146	sseli 3599	. . . . . . 7 |- ( t e. ( A (,) B ) -> t e. ( A [,] B ) )
148	147	adantl 482	. . . . . 6 |- ( ( ph /\ t e. ( A (,) B ) ) -> t e. ( A [,] B ) )
149	145, 148	ffvelrnd 6360	. . . . 5 |- ( ( ph /\ t e. ( A (,) B ) ) -> ( F ` t ) e. CC )
150	149	mulid1d 10057	. . . 4 |- ( ( ph /\ t e. ( A (,) B ) ) -> ( ( F ` t ) x. 1 ) = ( F ` t ) )
151	142, 150	eqtrd 2656	. . 3 |- ( ( ph /\ t e. ( A (,) B ) ) -> ( ( F ` ( X + ( t - X ) ) ) x. 1 ) = ( F ` t ) )
152	3, 151	ditgeq3d 40180	. 2 |- ( ph -> S__ [ A -> B ] ( ( F ` ( X + ( t - X ) ) ) x. 1 ) _d t = S__ [ A -> B ] ( F ` t ) _d t )
153	139, 152	eqtrd 2656	1 |- ( ph -> S__ [ ( A - X ) -> ( B - X ) ] ( F ` ( X + s ) ) _d s = S__ [ A -> B ] ( F ` t ) _d t )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   C_ wss 3574   {csn 4177   {cpr 4179   class class class wbr 4653   |-> cmpt 4729   X. cxp 5112   dom cdm 5114   ran crn 5115   |` cres 5116   -->wf 5884   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   <_ cle 10075   - cmin 10266   -ucneg 10267   (,)cioo 12175   [,]cicc 12178   TopOpenctopn 16082   topGenctg 16098  ℂ_fldccnfld 19746   intcnt 20821   -cn->ccncf 22679   volcvol 23232   L^1cibl 23386   S__cdit 23610   _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-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-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-ditg 23611  df-limc 23630  df-dv 23631

This theorem is referenced by:  fourierdlem82  40405