Theorem itg1mulc 21024

Description: The integral of a constant times a simple function is the constant times the original integral. (Contributed by Mario Carneiro, 25-Jun-2014.)

Hypotheses

Ref	Expression
i1fmulc.2	|- ( ph -> F e. dom S.1 )
i1fmulc.3	|- ( ph -> A e. RR )

Assertion

Ref	Expression
itg1mulc	|- ( ph -> ( S.1 ` ( ( RR X. { A } ) oF x. F ) ) = ( A x. ( S.1 ` F ) ) )

Proof of Theorem itg1mulc

Dummy variables k m n x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		itg10 21008	. . 3 |- ( S.1 ` ( RR X. { 0 } ) ) = 0
2		reex 9361	. . . . . 6 |- RR e. _V
3	2	a1i 11	. . . . 5 |- ( ( ph /\ A = 0 ) -> RR e. _V )
4		i1fmulc.2	. . . . . . 7 |- ( ph -> F e. dom S.1 )
5		i1ff 20996	. . . . . . 7 |- ( F e. dom S.1 -> F : RR --> RR )
6	4, 5	syl 16	. . . . . 6 |- ( ph -> F : RR --> RR )
7	6	adantr 462	. . . . 5 |- ( ( ph /\ A = 0 ) -> F : RR --> RR )
8		i1fmulc.3	. . . . . 6 |- ( ph -> A e. RR )
9	8	adantr 462	. . . . 5 |- ( ( ph /\ A = 0 ) -> A e. RR )
10		0red 9375	. . . . 5 |- ( ( ph /\ A = 0 ) -> 0 e. RR )
11		simplr 747	. . . . . . 7 |- ( ( ( ph /\ A = 0 ) /\ x e. RR ) -> A = 0 )
12	11	oveq1d 6095	. . . . . 6 |- ( ( ( ph /\ A = 0 ) /\ x e. RR ) -> ( A x. x ) = ( 0 x. x ) )
13		mul02lem2 9534	. . . . . . 7 |- ( x e. RR -> ( 0 x. x ) = 0 )
14	13	adantl 463	. . . . . 6 |- ( ( ( ph /\ A = 0 ) /\ x e. RR ) -> ( 0 x. x ) = 0 )
15	12, 14	eqtrd 2465	. . . . 5 |- ( ( ( ph /\ A = 0 ) /\ x e. RR ) -> ( A x. x ) = 0 )
16	3, 7, 9, 10, 15	caofid2 6340	. . . 4 |- ( ( ph /\ A = 0 ) -> ( ( RR X. { A } ) oF x. F ) = ( RR X. { 0 } ) )
17	16	fveq2d 5683	. . 3 |- ( ( ph /\ A = 0 ) -> ( S.1 ` ( ( RR X. { A } ) oF x. F ) ) = ( S.1 ` ( RR X. { 0 } ) ) )
18		simpr 458	. . . . 5 |- ( ( ph /\ A = 0 ) -> A = 0 )
19	18	oveq1d 6095	. . . 4 |- ( ( ph /\ A = 0 ) -> ( A x. ( S.1 ` F ) ) = ( 0 x. ( S.1 ` F ) ) )
20		itg1cl 21005	. . . . . . . 8 |- ( F e. dom S.1 -> ( S.1 ` F ) e. RR )
21	4, 20	syl 16	. . . . . . 7 |- ( ph -> ( S.1 ` F ) e. RR )
22	21	recnd 9400	. . . . . 6 |- ( ph -> ( S.1 ` F ) e. CC )
23	22	mul02d 9555	. . . . 5 |- ( ph -> ( 0 x. ( S.1 ` F ) ) = 0 )
24	23	adantr 462	. . . 4 |- ( ( ph /\ A = 0 ) -> ( 0 x. ( S.1 ` F ) ) = 0 )
25	19, 24	eqtrd 2465	. . 3 |- ( ( ph /\ A = 0 ) -> ( A x. ( S.1 ` F ) ) = 0 )
26	1, 17, 25	3eqtr4a 2491	. 2 |- ( ( ph /\ A = 0 ) -> ( S.1 ` ( ( RR X. { A } ) oF x. F ) ) = ( A x. ( S.1 ` F ) ) )
27	4, 8	i1fmulc 21023	. . . . . . . . . . . . . 14 |- ( ph -> ( ( RR X. { A } ) oF x. F ) e. dom S.1 )
28	27	adantr 462	. . . . . . . . . . . . 13 |- ( ( ph /\ A =/= 0 ) -> ( ( RR X. { A } ) oF x. F ) e. dom S.1 )
29		i1ff 20996	. . . . . . . . . . . . 13 |- ( ( ( RR X. { A } ) oF x. F ) e. dom S.1 -> ( ( RR X. { A } ) oF x. F ) : RR --> RR )
30	28, 29	syl 16	. . . . . . . . . . . 12 |- ( ( ph /\ A =/= 0 ) -> ( ( RR X. { A } ) oF x. F ) : RR --> RR )
31		frn 5553	. . . . . . . . . . . 12 |- ( ( ( RR X. { A } ) oF x. F ) : RR --> RR -> ran ( ( RR X. { A } ) oF x. F ) C_ RR )
32	30, 31	syl 16	. . . . . . . . . . 11 |- ( ( ph /\ A =/= 0 ) -> ran ( ( RR X. { A } ) oF x. F ) C_ RR )
33	32	ssdifssd 3482	. . . . . . . . . 10 |- ( ( ph /\ A =/= 0 ) -> ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) C_ RR )
34	33	sselda 3344	. . . . . . . . 9 |- ( ( ( ph /\ A =/= 0 ) /\ m e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) ) -> m e. RR )
35	34	recnd 9400	. . . . . . . 8 |- ( ( ( ph /\ A =/= 0 ) /\ m e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) ) -> m e. CC )
36	8	adantr 462	. . . . . . . . . 10 |- ( ( ph /\ A =/= 0 ) -> A e. RR )
37	36	recnd 9400	. . . . . . . . 9 |- ( ( ph /\ A =/= 0 ) -> A e. CC )
38	37	adantr 462	. . . . . . . 8 |- ( ( ( ph /\ A =/= 0 ) /\ m e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) ) -> A e. CC )
39		simplr 747	. . . . . . . 8 |- ( ( ( ph /\ A =/= 0 ) /\ m e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) ) -> A =/= 0 )
40	35, 38, 39	divcan2d 10097	. . . . . . 7 |- ( ( ( ph /\ A =/= 0 ) /\ m e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) ) -> ( A x. ( m / A ) ) = m )
41	4, 8	i1fmulclem 21022	. . . . . . . . . 10 |- ( ( ( ph /\ A =/= 0 ) /\ m e. RR ) -> ( `' ( ( RR X. { A } ) oF x. F ) " { m } ) = ( `' F " { ( m / A ) } ) )
42	34, 41	syldan 467	. . . . . . . . 9 |- ( ( ( ph /\ A =/= 0 ) /\ m e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) ) -> ( `' ( ( RR X. { A } ) oF x. F ) " { m } ) = ( `' F " { ( m / A ) } ) )
43	42	fveq2d 5683	. . . . . . . 8 |- ( ( ( ph /\ A =/= 0 ) /\ m e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) ) -> ( vol ` ( `' ( ( RR X. { A } ) oF x. F ) " { m } ) ) = ( vol ` ( `' F " { ( m / A ) } ) ) )
44	43	eqcomd 2438	. . . . . . 7 |- ( ( ( ph /\ A =/= 0 ) /\ m e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) ) -> ( vol ` ( `' F " { ( m / A ) } ) ) = ( vol ` ( `' ( ( RR X. { A } ) oF x. F ) " { m } ) ) )
45	40, 44	oveq12d 6098	. . . . . 6 |- ( ( ( ph /\ A =/= 0 ) /\ m e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) ) -> ( ( A x. ( m / A ) ) x. ( vol ` ( `' F " { ( m / A ) } ) ) ) = ( m x. ( vol ` ( `' ( ( RR X. { A } ) oF x. F ) " { m } ) ) ) )
46	8	ad2antrr 718	. . . . . . . . 9 |- ( ( ( ph /\ A =/= 0 ) /\ m e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) ) -> A e. RR )
47	34, 46, 39	redivcld 10147	. . . . . . . 8 |- ( ( ( ph /\ A =/= 0 ) /\ m e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) ) -> ( m / A ) e. RR )
48	47	recnd 9400	. . . . . . 7 |- ( ( ( ph /\ A =/= 0 ) /\ m e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) ) -> ( m / A ) e. CC )
49	4	ad2antrr 718	. . . . . . . . 9 |- ( ( ( ph /\ A =/= 0 ) /\ m e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) ) -> F e. dom S.1 )
50	46	recnd 9400	. . . . . . . . . . 11 |- ( ( ( ph /\ A =/= 0 ) /\ m e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) ) -> A e. CC )
51		eldifsni 3989	. . . . . . . . . . . 12 |- ( m e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) -> m =/= 0 )
52	51	adantl 463	. . . . . . . . . . 11 |- ( ( ( ph /\ A =/= 0 ) /\ m e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) ) -> m =/= 0 )
53	35, 50, 52, 39	divne0d 10111	. . . . . . . . . 10 |- ( ( ( ph /\ A =/= 0 ) /\ m e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) ) -> ( m / A ) =/= 0 )
54		eldifsn 3988	. . . . . . . . . 10 |- ( ( m / A ) e. ( RR \ { 0 } ) <-> ( ( m / A ) e. RR /\ ( m / A ) =/= 0 ) )
55	47, 53, 54	sylanbrc 657	. . . . . . . . 9 |- ( ( ( ph /\ A =/= 0 ) /\ m e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) ) -> ( m / A ) e. ( RR \ { 0 } ) )
56		i1fima2sn 21000	. . . . . . . . 9 |- ( ( F e. dom S.1 /\ ( m / A ) e. ( RR \ { 0 } ) ) -> ( vol ` ( `' F " { ( m / A ) } ) ) e. RR )
57	49, 55, 56	syl2anc 654	. . . . . . . 8 |- ( ( ( ph /\ A =/= 0 ) /\ m e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) ) -> ( vol ` ( `' F " { ( m / A ) } ) ) e. RR )
58	57	recnd 9400	. . . . . . 7 |- ( ( ( ph /\ A =/= 0 ) /\ m e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) ) -> ( vol ` ( `' F " { ( m / A ) } ) ) e. CC )
59	38, 48, 58	mulassd 9397	. . . . . 6 |- ( ( ( ph /\ A =/= 0 ) /\ m e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) ) -> ( ( A x. ( m / A ) ) x. ( vol ` ( `' F " { ( m / A ) } ) ) ) = ( A x. ( ( m / A ) x. ( vol ` ( `' F " { ( m / A ) } ) ) ) ) )
60	45, 59	eqtr3d 2467	. . . . 5 |- ( ( ( ph /\ A =/= 0 ) /\ m e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) ) -> ( m x. ( vol ` ( `' ( ( RR X. { A } ) oF x. F ) " { m } ) ) ) = ( A x. ( ( m / A ) x. ( vol ` ( `' F " { ( m / A ) } ) ) ) ) )
61	60	sumeq2dv 13164	. . . 4 |- ( ( ph /\ A =/= 0 ) -> sum_ m e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) ( m x. ( vol ` ( `' ( ( RR X. { A } ) oF x. F ) " { m } ) ) ) = sum_ m e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) ( A x. ( ( m / A ) x. ( vol ` ( `' F " { ( m / A ) } ) ) ) ) )
62		i1frn 20997	. . . . . . 7 |- ( ( ( RR X. { A } ) oF x. F ) e. dom S.1 -> ran ( ( RR X. { A } ) oF x. F ) e. Fin )
63	28, 62	syl 16	. . . . . 6 |- ( ( ph /\ A =/= 0 ) -> ran ( ( RR X. { A } ) oF x. F ) e. Fin )
64		difss 3471	. . . . . 6 |- ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) C_ ran ( ( RR X. { A } ) oF x. F )
65		ssfi 7521	. . . . . 6 |- ( ( ran ( ( RR X. { A } ) oF x. F ) e. Fin /\ ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) C_ ran ( ( RR X. { A } ) oF x. F ) ) -> ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) e. Fin )
66	63, 64, 65	sylancl 655	. . . . 5 |- ( ( ph /\ A =/= 0 ) -> ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) e. Fin )
67	48, 58	mulcld 9394	. . . . 5 |- ( ( ( ph /\ A =/= 0 ) /\ m e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) ) -> ( ( m / A ) x. ( vol ` ( `' F " { ( m / A ) } ) ) ) e. CC )
68	66, 37, 67	fsummulc2 13234	. . . 4 |- ( ( ph /\ A =/= 0 ) -> ( A x. sum_ m e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) ( ( m / A ) x. ( vol ` ( `' F " { ( m / A ) } ) ) ) ) = sum_ m e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) ( A x. ( ( m / A ) x. ( vol ` ( `' F " { ( m / A ) } ) ) ) ) )
69	61, 68	eqtr4d 2468	. . 3 |- ( ( ph /\ A =/= 0 ) -> sum_ m e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) ( m x. ( vol ` ( `' ( ( RR X. { A } ) oF x. F ) " { m } ) ) ) = ( A x. sum_ m e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) ( ( m / A ) x. ( vol ` ( `' F " { ( m / A ) } ) ) ) ) )
70		itg1val 21003	. . . 4 |- ( ( ( RR X. { A } ) oF x. F ) e. dom S.1 -> ( S.1 ` ( ( RR X. { A } ) oF x. F ) ) = sum_ m e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) ( m x. ( vol ` ( `' ( ( RR X. { A } ) oF x. F ) " { m } ) ) ) )
71	28, 70	syl 16	. . 3 |- ( ( ph /\ A =/= 0 ) -> ( S.1 ` ( ( RR X. { A } ) oF x. F ) ) = sum_ m e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) ( m x. ( vol ` ( `' ( ( RR X. { A } ) oF x. F ) " { m } ) ) ) )
72	4	adantr 462	. . . . . 6 |- ( ( ph /\ A =/= 0 ) -> F e. dom S.1 )
73		itg1val 21003	. . . . . 6 |- ( F e. dom S.1 -> ( S.1 ` F ) = sum_ k e. ( ran F \ { 0 } ) ( k x. ( vol ` ( `' F " { k } ) ) ) )
74	72, 73	syl 16	. . . . 5 |- ( ( ph /\ A =/= 0 ) -> ( S.1 ` F ) = sum_ k e. ( ran F \ { 0 } ) ( k x. ( vol ` ( `' F " { k } ) ) ) )
75		id 22	. . . . . . 7 |- ( k = ( m / A ) -> k = ( m / A ) )
76		sneq 3875	. . . . . . . . 9 |- ( k = ( m / A ) -> { k } = { ( m / A ) } )
77	76	imaeq2d 5157	. . . . . . . 8 |- ( k = ( m / A ) -> ( `' F " { k } ) = ( `' F " { ( m / A ) } ) )
78	77	fveq2d 5683	. . . . . . 7 |- ( k = ( m / A ) -> ( vol ` ( `' F " { k } ) ) = ( vol ` ( `' F " { ( m / A ) } ) ) )
79	75, 78	oveq12d 6098	. . . . . 6 |- ( k = ( m / A ) -> ( k x. ( vol ` ( `' F " { k } ) ) ) = ( ( m / A ) x. ( vol ` ( `' F " { ( m / A ) } ) ) ) )
80		eqid 2433	. . . . . . 7 |- ( n e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) |-> ( n / A ) ) = ( n e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) |-> ( n / A ) )
81		eldifi 3466	. . . . . . . . 9 |- ( n e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) -> n e. ran ( ( RR X. { A } ) oF x. F ) )
82	2	a1i 11	. . . . . . . . . . . . . . . . 17 |- ( ph -> RR e. _V )
83		ffn 5547	. . . . . . . . . . . . . . . . . 18 |- ( F : RR --> RR -> F Fn RR )
84	6, 83	syl 16	. . . . . . . . . . . . . . . . 17 |- ( ph -> F Fn RR )
85		eqidd 2434	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ y e. RR ) -> ( F ` y ) = ( F ` y ) )
86	82, 8, 84, 85	ofc1 6332	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ y e. RR ) -> ( ( ( RR X. { A } ) oF x. F ) ` y ) = ( A x. ( F ` y ) ) )
87	86	adantlr 707	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ A =/= 0 ) /\ y e. RR ) -> ( ( ( RR X. { A } ) oF x. F ) ` y ) = ( A x. ( F ` y ) ) )
88	87	oveq1d 6095	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ A =/= 0 ) /\ y e. RR ) -> ( ( ( ( RR X. { A } ) oF x. F ) ` y ) / A ) = ( ( A x. ( F ` y ) ) / A ) )
89	6	adantr 462	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ A =/= 0 ) -> F : RR --> RR )
90	89	ffvelrnda 5831	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ A =/= 0 ) /\ y e. RR ) -> ( F ` y ) e. RR )
91	90	recnd 9400	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ A =/= 0 ) /\ y e. RR ) -> ( F ` y ) e. CC )
92	37	adantr 462	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ A =/= 0 ) /\ y e. RR ) -> A e. CC )
93		simplr 747	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ A =/= 0 ) /\ y e. RR ) -> A =/= 0 )
94	91, 92, 93	divcan3d 10100	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ A =/= 0 ) /\ y e. RR ) -> ( ( A x. ( F ` y ) ) / A ) = ( F ` y ) )
95	88, 94	eqtrd 2465	. . . . . . . . . . . . 13 |- ( ( ( ph /\ A =/= 0 ) /\ y e. RR ) -> ( ( ( ( RR X. { A } ) oF x. F ) ` y ) / A ) = ( F ` y ) )
96	89, 83	syl 16	. . . . . . . . . . . . . 14 |- ( ( ph /\ A =/= 0 ) -> F Fn RR )
97		fnfvelrn 5828	. . . . . . . . . . . . . 14 |- ( ( F Fn RR /\ y e. RR ) -> ( F ` y ) e. ran F )
98	96, 97	sylan 468	. . . . . . . . . . . . 13 |- ( ( ( ph /\ A =/= 0 ) /\ y e. RR ) -> ( F ` y ) e. ran F )
99	95, 98	eqeltrd 2507	. . . . . . . . . . . 12 |- ( ( ( ph /\ A =/= 0 ) /\ y e. RR ) -> ( ( ( ( RR X. { A } ) oF x. F ) ` y ) / A ) e. ran F )
100	99	ralrimiva 2789	. . . . . . . . . . 11 |- ( ( ph /\ A =/= 0 ) -> A. y e. RR ( ( ( ( RR X. { A } ) oF x. F ) ` y ) / A ) e. ran F )
101		ffn 5547	. . . . . . . . . . . . 13 |- ( ( ( RR X. { A } ) oF x. F ) : RR --> RR -> ( ( RR X. { A } ) oF x. F ) Fn RR )
102	30, 101	syl 16	. . . . . . . . . . . 12 |- ( ( ph /\ A =/= 0 ) -> ( ( RR X. { A } ) oF x. F ) Fn RR )
103		oveq1 6087	. . . . . . . . . . . . . 14 |- ( n = ( ( ( RR X. { A } ) oF x. F ) ` y ) -> ( n / A ) = ( ( ( ( RR X. { A } ) oF x. F ) ` y ) / A ) )
104	103	eleq1d 2499	. . . . . . . . . . . . 13 |- ( n = ( ( ( RR X. { A } ) oF x. F ) ` y ) -> ( ( n / A ) e. ran F <-> ( ( ( ( RR X. { A } ) oF x. F ) ` y ) / A ) e. ran F ) )
105	104	ralrn 5834	. . . . . . . . . . . 12 |- ( ( ( RR X. { A } ) oF x. F ) Fn RR -> ( A. n e. ran ( ( RR X. { A } ) oF x. F ) ( n / A ) e. ran F <-> A. y e. RR ( ( ( ( RR X. { A } ) oF x. F ) ` y ) / A ) e. ran F ) )
106	102, 105	syl 16	. . . . . . . . . . 11 |- ( ( ph /\ A =/= 0 ) -> ( A. n e. ran ( ( RR X. { A } ) oF x. F ) ( n / A ) e. ran F <-> A. y e. RR ( ( ( ( RR X. { A } ) oF x. F ) ` y ) / A ) e. ran F ) )
107	100, 106	mpbird 232	. . . . . . . . . 10 |- ( ( ph /\ A =/= 0 ) -> A. n e. ran ( ( RR X. { A } ) oF x. F ) ( n / A ) e. ran F )
108	107	r19.21bi 2804	. . . . . . . . 9 |- ( ( ( ph /\ A =/= 0 ) /\ n e. ran ( ( RR X. { A } ) oF x. F ) ) -> ( n / A ) e. ran F )
109	81, 108	sylan2 471	. . . . . . . 8 |- ( ( ( ph /\ A =/= 0 ) /\ n e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) ) -> ( n / A ) e. ran F )
110	33	sselda 3344	. . . . . . . . . 10 |- ( ( ( ph /\ A =/= 0 ) /\ n e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) ) -> n e. RR )
111	110	recnd 9400	. . . . . . . . 9 |- ( ( ( ph /\ A =/= 0 ) /\ n e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) ) -> n e. CC )
112	37	adantr 462	. . . . . . . . 9 |- ( ( ( ph /\ A =/= 0 ) /\ n e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) ) -> A e. CC )
113		eldifsni 3989	. . . . . . . . . 10 |- ( n e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) -> n =/= 0 )
114	113	adantl 463	. . . . . . . . 9 |- ( ( ( ph /\ A =/= 0 ) /\ n e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) ) -> n =/= 0 )
115		simplr 747	. . . . . . . . 9 |- ( ( ( ph /\ A =/= 0 ) /\ n e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) ) -> A =/= 0 )
116	111, 112, 114, 115	divne0d 10111	. . . . . . . 8 |- ( ( ( ph /\ A =/= 0 ) /\ n e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) ) -> ( n / A ) =/= 0 )
117		eldifsn 3988	. . . . . . . 8 |- ( ( n / A ) e. ( ran F \ { 0 } ) <-> ( ( n / A ) e. ran F /\ ( n / A ) =/= 0 ) )
118	109, 116, 117	sylanbrc 657	. . . . . . 7 |- ( ( ( ph /\ A =/= 0 ) /\ n e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) ) -> ( n / A ) e. ( ran F \ { 0 } ) )
119		eldifi 3466	. . . . . . . . 9 |- ( k e. ( ran F \ { 0 } ) -> k e. ran F )
120		fnfvelrn 5828	. . . . . . . . . . . . . 14 |- ( ( ( ( RR X. { A } ) oF x. F ) Fn RR /\ y e. RR ) -> ( ( ( RR X. { A } ) oF x. F ) ` y ) e. ran ( ( RR X. { A } ) oF x. F ) )
121	102, 120	sylan 468	. . . . . . . . . . . . 13 |- ( ( ( ph /\ A =/= 0 ) /\ y e. RR ) -> ( ( ( RR X. { A } ) oF x. F ) ` y ) e. ran ( ( RR X. { A } ) oF x. F ) )
122	87, 121	eqeltrrd 2508	. . . . . . . . . . . 12 |- ( ( ( ph /\ A =/= 0 ) /\ y e. RR ) -> ( A x. ( F ` y ) ) e. ran ( ( RR X. { A } ) oF x. F ) )
123	122	ralrimiva 2789	. . . . . . . . . . 11 |- ( ( ph /\ A =/= 0 ) -> A. y e. RR ( A x. ( F ` y ) ) e. ran ( ( RR X. { A } ) oF x. F ) )
124		oveq2 6088	. . . . . . . . . . . . . 14 |- ( k = ( F ` y ) -> ( A x. k ) = ( A x. ( F ` y ) ) )
125	124	eleq1d 2499	. . . . . . . . . . . . 13 |- ( k = ( F ` y ) -> ( ( A x. k ) e. ran ( ( RR X. { A } ) oF x. F ) <-> ( A x. ( F ` y ) ) e. ran ( ( RR X. { A } ) oF x. F ) ) )
126	125	ralrn 5834	. . . . . . . . . . . 12 |- ( F Fn RR -> ( A. k e. ran F ( A x. k ) e. ran ( ( RR X. { A } ) oF x. F ) <-> A. y e. RR ( A x. ( F ` y ) ) e. ran ( ( RR X. { A } ) oF x. F ) ) )
127	96, 126	syl 16	. . . . . . . . . . 11 |- ( ( ph /\ A =/= 0 ) -> ( A. k e. ran F ( A x. k ) e. ran ( ( RR X. { A } ) oF x. F ) <-> A. y e. RR ( A x. ( F ` y ) ) e. ran ( ( RR X. { A } ) oF x. F ) ) )
128	123, 127	mpbird 232	. . . . . . . . . 10 |- ( ( ph /\ A =/= 0 ) -> A. k e. ran F ( A x. k ) e. ran ( ( RR X. { A } ) oF x. F ) )
129	128	r19.21bi 2804	. . . . . . . . 9 |- ( ( ( ph /\ A =/= 0 ) /\ k e. ran F ) -> ( A x. k ) e. ran ( ( RR X. { A } ) oF x. F ) )
130	119, 129	sylan2 471	. . . . . . . 8 |- ( ( ( ph /\ A =/= 0 ) /\ k e. ( ran F \ { 0 } ) ) -> ( A x. k ) e. ran ( ( RR X. { A } ) oF x. F ) )
131	37	adantr 462	. . . . . . . . 9 |- ( ( ( ph /\ A =/= 0 ) /\ k e. ( ran F \ { 0 } ) ) -> A e. CC )
132		frn 5553	. . . . . . . . . . . . 13 |- ( F : RR --> RR -> ran F C_ RR )
133	89, 132	syl 16	. . . . . . . . . . . 12 |- ( ( ph /\ A =/= 0 ) -> ran F C_ RR )
134	133	ssdifssd 3482	. . . . . . . . . . 11 |- ( ( ph /\ A =/= 0 ) -> ( ran F \ { 0 } ) C_ RR )
135	134	sselda 3344	. . . . . . . . . 10 |- ( ( ( ph /\ A =/= 0 ) /\ k e. ( ran F \ { 0 } ) ) -> k e. RR )
136	135	recnd 9400	. . . . . . . . 9 |- ( ( ( ph /\ A =/= 0 ) /\ k e. ( ran F \ { 0 } ) ) -> k e. CC )
137		simplr 747	. . . . . . . . 9 |- ( ( ( ph /\ A =/= 0 ) /\ k e. ( ran F \ { 0 } ) ) -> A =/= 0 )
138		eldifsni 3989	. . . . . . . . . 10 |- ( k e. ( ran F \ { 0 } ) -> k =/= 0 )
139	138	adantl 463	. . . . . . . . 9 |- ( ( ( ph /\ A =/= 0 ) /\ k e. ( ran F \ { 0 } ) ) -> k =/= 0 )
140	131, 136, 137, 139	mulne0d 9976	. . . . . . . 8 |- ( ( ( ph /\ A =/= 0 ) /\ k e. ( ran F \ { 0 } ) ) -> ( A x. k ) =/= 0 )
141		eldifsn 3988	. . . . . . . 8 |- ( ( A x. k ) e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) <-> ( ( A x. k ) e. ran ( ( RR X. { A } ) oF x. F ) /\ ( A x. k ) =/= 0 ) )
142	130, 140, 141	sylanbrc 657	. . . . . . 7 |- ( ( ( ph /\ A =/= 0 ) /\ k e. ( ran F \ { 0 } ) ) -> ( A x. k ) e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) )
143		simpl 454	. . . . . . . . . . . 12 |- ( ( n e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) /\ k e. ( ran F \ { 0 } ) ) -> n e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) )
144		ssel2 3339	. . . . . . . . . . . 12 |- ( ( ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) C_ RR /\ n e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) ) -> n e. RR )
145	33, 143, 144	syl2an 474	. . . . . . . . . . 11 |- ( ( ( ph /\ A =/= 0 ) /\ ( n e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) /\ k e. ( ran F \ { 0 } ) ) ) -> n e. RR )
146	145	recnd 9400	. . . . . . . . . 10 |- ( ( ( ph /\ A =/= 0 ) /\ ( n e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) /\ k e. ( ran F \ { 0 } ) ) ) -> n e. CC )
147	8	ad2antrr 718	. . . . . . . . . . 11 |- ( ( ( ph /\ A =/= 0 ) /\ ( n e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) /\ k e. ( ran F \ { 0 } ) ) ) -> A e. RR )
148	147	recnd 9400	. . . . . . . . . 10 |- ( ( ( ph /\ A =/= 0 ) /\ ( n e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) /\ k e. ( ran F \ { 0 } ) ) ) -> A e. CC )
149	135	adantrl 708	. . . . . . . . . . 11 |- ( ( ( ph /\ A =/= 0 ) /\ ( n e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) /\ k e. ( ran F \ { 0 } ) ) ) -> k e. RR )
150	149	recnd 9400	. . . . . . . . . 10 |- ( ( ( ph /\ A =/= 0 ) /\ ( n e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) /\ k e. ( ran F \ { 0 } ) ) ) -> k e. CC )
151		simplr 747	. . . . . . . . . 10 |- ( ( ( ph /\ A =/= 0 ) /\ ( n e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) /\ k e. ( ran F \ { 0 } ) ) ) -> A =/= 0 )
152	146, 148, 150, 151	divmuld 10117	. . . . . . . . 9 |- ( ( ( ph /\ A =/= 0 ) /\ ( n e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) /\ k e. ( ran F \ { 0 } ) ) ) -> ( ( n / A ) = k <-> ( A x. k ) = n ) )
153	152	bicomd 201	. . . . . . . 8 |- ( ( ( ph /\ A =/= 0 ) /\ ( n e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) /\ k e. ( ran F \ { 0 } ) ) ) -> ( ( A x. k ) = n <-> ( n / A ) = k ) )
154		eqcom 2435	. . . . . . . 8 |- ( n = ( A x. k ) <-> ( A x. k ) = n )
155		eqcom 2435	. . . . . . . 8 |- ( k = ( n / A ) <-> ( n / A ) = k )
156	153, 154, 155	3bitr4g 288	. . . . . . 7 |- ( ( ( ph /\ A =/= 0 ) /\ ( n e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) /\ k e. ( ran F \ { 0 } ) ) ) -> ( n = ( A x. k ) <-> k = ( n / A ) ) )
157	80, 118, 142, 156	f1o2d 6301	. . . . . 6 |- ( ( ph /\ A =/= 0 ) -> ( n e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) |-> ( n / A ) ) : ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) -1-1-onto-> ( ran F \ { 0 } ) )
158		oveq1 6087	. . . . . . . 8 |- ( n = m -> ( n / A ) = ( m / A ) )
159		ovex 6105	. . . . . . . 8 |- ( m / A ) e. _V
160	158, 80, 159	fvmpt 5762	. . . . . . 7 |- ( m e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) -> ( ( n e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) |-> ( n / A ) ) ` m ) = ( m / A ) )
161	160	adantl 463	. . . . . 6 |- ( ( ( ph /\ A =/= 0 ) /\ m e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) ) -> ( ( n e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) |-> ( n / A ) ) ` m ) = ( m / A ) )
162		i1fima2sn 21000	. . . . . . . . 9 |- ( ( F e. dom S.1 /\ k e. ( ran F \ { 0 } ) ) -> ( vol ` ( `' F " { k } ) ) e. RR )
163	72, 162	sylan 468	. . . . . . . 8 |- ( ( ( ph /\ A =/= 0 ) /\ k e. ( ran F \ { 0 } ) ) -> ( vol ` ( `' F " { k } ) ) e. RR )
164	135, 163	remulcld 9402	. . . . . . 7 |- ( ( ( ph /\ A =/= 0 ) /\ k e. ( ran F \ { 0 } ) ) -> ( k x. ( vol ` ( `' F " { k } ) ) ) e. RR )
165	164	recnd 9400	. . . . . 6 |- ( ( ( ph /\ A =/= 0 ) /\ k e. ( ran F \ { 0 } ) ) -> ( k x. ( vol ` ( `' F " { k } ) ) ) e. CC )
166	79, 66, 157, 161, 165	fsumf1o 13184	. . . . 5 |- ( ( ph /\ A =/= 0 ) -> sum_ k e. ( ran F \ { 0 } ) ( k x. ( vol ` ( `' F " { k } ) ) ) = sum_ m e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) ( ( m / A ) x. ( vol ` ( `' F " { ( m / A ) } ) ) ) )
167	74, 166	eqtrd 2465	. . . 4 |- ( ( ph /\ A =/= 0 ) -> ( S.1 ` F ) = sum_ m e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) ( ( m / A ) x. ( vol ` ( `' F " { ( m / A ) } ) ) ) )
168	167	oveq2d 6096	. . 3 |- ( ( ph /\ A =/= 0 ) -> ( A x. ( S.1 ` F ) ) = ( A x. sum_ m e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) ( ( m / A ) x. ( vol ` ( `' F " { ( m / A ) } ) ) ) ) )
169	69, 71, 168	3eqtr4d 2475	. 2 |- ( ( ph /\ A =/= 0 ) -> ( S.1 ` ( ( RR X. { A } ) oF x. F ) ) = ( A x. ( S.1 ` F ) ) )
170	26, 169	pm2.61dane 2679	1 |- ( ph -> ( S.1 ` ( ( RR X. { A } ) oF x. F ) ) = ( A x. ( S.1 ` F ) ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1362   e. wcel 1755   =/= wne 2596   A.wral 2705   _Vcvv 2962   \ cdif 3313   C_ wss 3316   {csn 3865   e. cmpt 4338   X. cxp 4825   `'ccnv 4826   dom cdm 4827   ran crn 4828   "cima 4830   Fn wfn 5401   -->wf 5402   ` cfv 5406  (class class class)co 6080   oFcof 6307   Fincfn 7298   CCcc 9268   RRcr 9269   0cc0 9270   x. cmul 9275   / cdiv 9981   sum_csu 13147   volcvol 20789   S.1citg1 20937

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-inf2 7835  ax-cnex 9326  ax-resscn 9327  ax-1cn 9328  ax-icn 9329  ax-addcl 9330  ax-addrcl 9331  ax-mulcl 9332  ax-mulrcl 9333  ax-mulcom 9334  ax-addass 9335  ax-mulass 9336  ax-distr 9337  ax-i2m1 9338  ax-1ne0 9339  ax-1rid 9340  ax-rnegex 9341  ax-rrecex 9342  ax-cnre 9343  ax-pre-lttri 9344  ax-pre-lttrn 9345  ax-pre-ltadd 9346  ax-pre-mulgt0 9347  ax-pre-sup 9348

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-fal 1368  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-uni 4080  df-int 4117  df-iun 4161  df-br 4281  df-opab 4339  df-mpt 4340  df-tr 4374  df-eprel 4619  df-id 4623  df-po 4628  df-so 4629  df-fr 4666  df-se 4667  df-we 4668  df-ord 4709  df-on 4710  df-lim 4711  df-suc 4712  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-isom 5415  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-of 6309  df-om 6466  df-1st 6566  df-2nd 6567  df-recs 6818  df-rdg 6852  df-1o 6908  df-2o 6909  df-oadd 6912  df-er 7089  df-map 7204  df-pm 7205  df-en 7299  df-dom 7300  df-sdom 7301  df-fin 7302  df-sup 7679  df-oi 7712  df-card 8097  df-cda 8325  df-pnf 9408  df-mnf 9409  df-xr 9410  df-ltxr 9411  df-le 9412  df-sub 9585  df-neg 9586  df-div 9982  df-nn 10311  df-2 10368  df-3 10369  df-n0 10568  df-z 10635  df-uz 10850  df-q 10942  df-rp 10980  df-xadd 11078  df-ioo 11292  df-ico 11294  df-icc 11295  df-fz 11425  df-fzo 11533  df-fl 11626  df-seq 11791  df-exp 11850  df-hash 12088  df-cj 12572  df-re 12573  df-im 12574  df-sqr 12708  df-abs 12709  df-clim 12950  df-sum 13148  df-xmet 17654  df-met 17655  df-ovol 20790  df-vol 20791  df-mbf 20941  df-itg1 20942

This theorem is referenced by:  itg1sub  21029  itg2const  21060  itg2mulclem  21066  itg2monolem1  21070  itg2addnclem  28287