Theorem itg1mulc 19549

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 } ) o F 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 19533	. . 3 |- ( S.1 ` ( RR X. { 0 } ) ) = 0
2		reex 9037	. . . . . 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 19521	. . . . . . 7 |- ( F e. dom S.1 -> F : RR --> RR )
6	4, 5	syl 16	. . . . . 6 |- ( ph -> F : RR --> RR )
7	6	adantr 452	. . . . 5 |- ( ( ph /\ A = 0 ) -> F : RR --> RR )
8		i1fmulc.3	. . . . . 6 |- ( ph -> A e. RR )
9	8	adantr 452	. . . . 5 |- ( ( ph /\ A = 0 ) -> A e. RR )
10		0re 9047	. . . . . 6 |- 0 e. RR
11	10	a1i 11	. . . . 5 |- ( ( ph /\ A = 0 ) -> 0 e. RR )
12		simplr 732	. . . . . . 7 |- ( ( ( ph /\ A = 0 ) /\ x e. RR ) -> A = 0 )
13	12	oveq1d 6055	. . . . . 6 |- ( ( ( ph /\ A = 0 ) /\ x e. RR ) -> ( A x. x ) = ( 0 x. x ) )
14		mul02lem2 9199	. . . . . . 7 |- ( x e. RR -> ( 0 x. x ) = 0 )
15	14	adantl 453	. . . . . 6 |- ( ( ( ph /\ A = 0 ) /\ x e. RR ) -> ( 0 x. x ) = 0 )
16	13, 15	eqtrd 2436	. . . . 5 |- ( ( ( ph /\ A = 0 ) /\ x e. RR ) -> ( A x. x ) = 0 )
17	3, 7, 9, 11, 16	caofid2 6294	. . . 4 |- ( ( ph /\ A = 0 ) -> ( ( RR X. { A } ) o F x. F ) = ( RR X. { 0 } ) )
18	17	fveq2d 5691	. . 3 |- ( ( ph /\ A = 0 ) -> ( S.1 ` ( ( RR X. { A } ) o F x. F ) ) = ( S.1 ` ( RR X. { 0 } ) ) )
19		simpr 448	. . . . 5 |- ( ( ph /\ A = 0 ) -> A = 0 )
20	19	oveq1d 6055	. . . 4 |- ( ( ph /\ A = 0 ) -> ( A x. ( S.1 ` F ) ) = ( 0 x. ( S.1 ` F ) ) )
21		itg1cl 19530	. . . . . . . 8 |- ( F e. dom S.1 -> ( S.1 ` F ) e. RR )
22	4, 21	syl 16	. . . . . . 7 |- ( ph -> ( S.1 ` F ) e. RR )
23	22	recnd 9070	. . . . . 6 |- ( ph -> ( S.1 ` F ) e. CC )
24	23	mul02d 9220	. . . . 5 |- ( ph -> ( 0 x. ( S.1 ` F ) ) = 0 )
25	24	adantr 452	. . . 4 |- ( ( ph /\ A = 0 ) -> ( 0 x. ( S.1 ` F ) ) = 0 )
26	20, 25	eqtrd 2436	. . 3 |- ( ( ph /\ A = 0 ) -> ( A x. ( S.1 ` F ) ) = 0 )
27	1, 18, 26	3eqtr4a 2462	. 2 |- ( ( ph /\ A = 0 ) -> ( S.1 ` ( ( RR X. { A } ) o F x. F ) ) = ( A x. ( S.1 ` F ) ) )
28	4, 8	i1fmulc 19548	. . . . . . . . . . . . . 14 |- ( ph -> ( ( RR X. { A } ) o F x. F ) e. dom S.1 )
29	28	adantr 452	. . . . . . . . . . . . 13 |- ( ( ph /\ A =/= 0 ) -> ( ( RR X. { A } ) o F x. F ) e. dom S.1 )
30		i1ff 19521	. . . . . . . . . . . . 13 |- ( ( ( RR X. { A } ) o F x. F ) e. dom S.1 -> ( ( RR X. { A } ) o F x. F ) : RR --> RR )
31	29, 30	syl 16	. . . . . . . . . . . 12 |- ( ( ph /\ A =/= 0 ) -> ( ( RR X. { A } ) o F x. F ) : RR --> RR )
32		frn 5556	. . . . . . . . . . . 12 |- ( ( ( RR X. { A } ) o F x. F ) : RR --> RR -> ran ( ( RR X. { A } ) o F x. F ) C_ RR )
33	31, 32	syl 16	. . . . . . . . . . 11 |- ( ( ph /\ A =/= 0 ) -> ran ( ( RR X. { A } ) o F x. F ) C_ RR )
34	33	ssdifssd 3445	. . . . . . . . . 10 |- ( ( ph /\ A =/= 0 ) -> ( ran ( ( RR X. { A } ) o F x. F ) \ { 0 } ) C_ RR )
35	34	sselda 3308	. . . . . . . . 9 |- ( ( ( ph /\ A =/= 0 ) /\ m e. ( ran ( ( RR X. { A } ) o F x. F ) \ { 0 } ) ) -> m e. RR )
36	35	recnd 9070	. . . . . . . 8 |- ( ( ( ph /\ A =/= 0 ) /\ m e. ( ran ( ( RR X. { A } ) o F x. F ) \ { 0 } ) ) -> m e. CC )
37	8	adantr 452	. . . . . . . . . 10 |- ( ( ph /\ A =/= 0 ) -> A e. RR )
38	37	recnd 9070	. . . . . . . . 9 |- ( ( ph /\ A =/= 0 ) -> A e. CC )
39	38	adantr 452	. . . . . . . 8 |- ( ( ( ph /\ A =/= 0 ) /\ m e. ( ran ( ( RR X. { A } ) o F x. F ) \ { 0 } ) ) -> A e. CC )
40		simplr 732	. . . . . . . 8 |- ( ( ( ph /\ A =/= 0 ) /\ m e. ( ran ( ( RR X. { A } ) o F x. F ) \ { 0 } ) ) -> A =/= 0 )
41	36, 39, 40	divcan2d 9748	. . . . . . 7 |- ( ( ( ph /\ A =/= 0 ) /\ m e. ( ran ( ( RR X. { A } ) o F x. F ) \ { 0 } ) ) -> ( A x. ( m / A ) ) = m )
42	4, 8	i1fmulclem 19547	. . . . . . . . . 10 |- ( ( ( ph /\ A =/= 0 ) /\ m e. RR ) -> ( `' ( ( RR X. { A } ) o F x. F ) " { m } ) = ( `' F " { ( m / A ) } ) )
43	35, 42	syldan 457	. . . . . . . . 9 |- ( ( ( ph /\ A =/= 0 ) /\ m e. ( ran ( ( RR X. { A } ) o F x. F ) \ { 0 } ) ) -> ( `' ( ( RR X. { A } ) o F x. F ) " { m } ) = ( `' F " { ( m / A ) } ) )
44	43	fveq2d 5691	. . . . . . . 8 |- ( ( ( ph /\ A =/= 0 ) /\ m e. ( ran ( ( RR X. { A } ) o F x. F ) \ { 0 } ) ) -> ( vol ` ( `' ( ( RR X. { A } ) o F x. F ) " { m } ) ) = ( vol ` ( `' F " { ( m / A ) } ) ) )
45	44	eqcomd 2409	. . . . . . 7 |- ( ( ( ph /\ A =/= 0 ) /\ m e. ( ran ( ( RR X. { A } ) o F x. F ) \ { 0 } ) ) -> ( vol ` ( `' F " { ( m / A ) } ) ) = ( vol ` ( `' ( ( RR X. { A } ) o F x. F ) " { m } ) ) )
46	41, 45	oveq12d 6058	. . . . . 6 |- ( ( ( ph /\ A =/= 0 ) /\ m e. ( ran ( ( RR X. { A } ) o F x. F ) \ { 0 } ) ) -> ( ( A x. ( m / A ) ) x. ( vol ` ( `' F " { ( m / A ) } ) ) ) = ( m x. ( vol ` ( `' ( ( RR X. { A } ) o F x. F ) " { m } ) ) ) )
47	8	ad2antrr 707	. . . . . . . . 9 |- ( ( ( ph /\ A =/= 0 ) /\ m e. ( ran ( ( RR X. { A } ) o F x. F ) \ { 0 } ) ) -> A e. RR )
48	35, 47, 40	redivcld 9798	. . . . . . . 8 |- ( ( ( ph /\ A =/= 0 ) /\ m e. ( ran ( ( RR X. { A } ) o F x. F ) \ { 0 } ) ) -> ( m / A ) e. RR )
49	48	recnd 9070	. . . . . . 7 |- ( ( ( ph /\ A =/= 0 ) /\ m e. ( ran ( ( RR X. { A } ) o F x. F ) \ { 0 } ) ) -> ( m / A ) e. CC )
50	4	ad2antrr 707	. . . . . . . . 9 |- ( ( ( ph /\ A =/= 0 ) /\ m e. ( ran ( ( RR X. { A } ) o F x. F ) \ { 0 } ) ) -> F e. dom S.1 )
51	47	recnd 9070	. . . . . . . . . . 11 |- ( ( ( ph /\ A =/= 0 ) /\ m e. ( ran ( ( RR X. { A } ) o F x. F ) \ { 0 } ) ) -> A e. CC )
52		eldifsni 3888	. . . . . . . . . . . 12 |- ( m e. ( ran ( ( RR X. { A } ) o F x. F ) \ { 0 } ) -> m =/= 0 )
53	52	adantl 453	. . . . . . . . . . 11 |- ( ( ( ph /\ A =/= 0 ) /\ m e. ( ran ( ( RR X. { A } ) o F x. F ) \ { 0 } ) ) -> m =/= 0 )
54	36, 51, 53, 40	divne0d 9762	. . . . . . . . . 10 |- ( ( ( ph /\ A =/= 0 ) /\ m e. ( ran ( ( RR X. { A } ) o F x. F ) \ { 0 } ) ) -> ( m / A ) =/= 0 )
55		eldifsn 3887	. . . . . . . . . 10 |- ( ( m / A ) e. ( RR \ { 0 } ) <-> ( ( m / A ) e. RR /\ ( m / A ) =/= 0 ) )
56	48, 54, 55	sylanbrc 646	. . . . . . . . 9 |- ( ( ( ph /\ A =/= 0 ) /\ m e. ( ran ( ( RR X. { A } ) o F x. F ) \ { 0 } ) ) -> ( m / A ) e. ( RR \ { 0 } ) )
57		i1fima2sn 19525	. . . . . . . . 9 |- ( ( F e. dom S.1 /\ ( m / A ) e. ( RR \ { 0 } ) ) -> ( vol ` ( `' F " { ( m / A ) } ) ) e. RR )
58	50, 56, 57	syl2anc 643	. . . . . . . 8 |- ( ( ( ph /\ A =/= 0 ) /\ m e. ( ran ( ( RR X. { A } ) o F x. F ) \ { 0 } ) ) -> ( vol ` ( `' F " { ( m / A ) } ) ) e. RR )
59	58	recnd 9070	. . . . . . 7 |- ( ( ( ph /\ A =/= 0 ) /\ m e. ( ran ( ( RR X. { A } ) o F x. F ) \ { 0 } ) ) -> ( vol ` ( `' F " { ( m / A ) } ) ) e. CC )
60	39, 49, 59	mulassd 9067	. . . . . 6 |- ( ( ( ph /\ A =/= 0 ) /\ m e. ( ran ( ( RR X. { A } ) o F x. F ) \ { 0 } ) ) -> ( ( A x. ( m / A ) ) x. ( vol ` ( `' F " { ( m / A ) } ) ) ) = ( A x. ( ( m / A ) x. ( vol ` ( `' F " { ( m / A ) } ) ) ) ) )
61	46, 60	eqtr3d 2438	. . . . 5 |- ( ( ( ph /\ A =/= 0 ) /\ m e. ( ran ( ( RR X. { A } ) o F x. F ) \ { 0 } ) ) -> ( m x. ( vol ` ( `' ( ( RR X. { A } ) o F x. F ) " { m } ) ) ) = ( A x. ( ( m / A ) x. ( vol ` ( `' F " { ( m / A ) } ) ) ) ) )
62	61	sumeq2dv 12452	. . . 4 |- ( ( ph /\ A =/= 0 ) -> sum_ m e. ( ran ( ( RR X. { A } ) o F x. F ) \ { 0 } ) ( m x. ( vol ` ( `' ( ( RR X. { A } ) o F x. F ) " { m } ) ) ) = sum_ m e. ( ran ( ( RR X. { A } ) o F x. F ) \ { 0 } ) ( A x. ( ( m / A ) x. ( vol ` ( `' F " { ( m / A ) } ) ) ) ) )
63		i1frn 19522	. . . . . . 7 |- ( ( ( RR X. { A } ) o F x. F ) e. dom S.1 -> ran ( ( RR X. { A } ) o F x. F ) e. Fin )
64	29, 63	syl 16	. . . . . 6 |- ( ( ph /\ A =/= 0 ) -> ran ( ( RR X. { A } ) o F x. F ) e. Fin )
65		difss 3434	. . . . . 6 |- ( ran ( ( RR X. { A } ) o F x. F ) \ { 0 } ) C_ ran ( ( RR X. { A } ) o F x. F )
66		ssfi 7288	. . . . . 6 |- ( ( ran ( ( RR X. { A } ) o F x. F ) e. Fin /\ ( ran ( ( RR X. { A } ) o F x. F ) \ { 0 } ) C_ ran ( ( RR X. { A } ) o F x. F ) ) -> ( ran ( ( RR X. { A } ) o F x. F ) \ { 0 } ) e. Fin )
67	64, 65, 66	sylancl 644	. . . . 5 |- ( ( ph /\ A =/= 0 ) -> ( ran ( ( RR X. { A } ) o F x. F ) \ { 0 } ) e. Fin )
68	49, 59	mulcld 9064	. . . . 5 |- ( ( ( ph /\ A =/= 0 ) /\ m e. ( ran ( ( RR X. { A } ) o F x. F ) \ { 0 } ) ) -> ( ( m / A ) x. ( vol ` ( `' F " { ( m / A ) } ) ) ) e. CC )
69	67, 38, 68	fsummulc2 12522	. . . 4 |- ( ( ph /\ A =/= 0 ) -> ( A x. sum_ m e. ( ran ( ( RR X. { A } ) o F x. F ) \ { 0 } ) ( ( m / A ) x. ( vol ` ( `' F " { ( m / A ) } ) ) ) ) = sum_ m e. ( ran ( ( RR X. { A } ) o F x. F ) \ { 0 } ) ( A x. ( ( m / A ) x. ( vol ` ( `' F " { ( m / A ) } ) ) ) ) )
70	62, 69	eqtr4d 2439	. . 3 |- ( ( ph /\ A =/= 0 ) -> sum_ m e. ( ran ( ( RR X. { A } ) o F x. F ) \ { 0 } ) ( m x. ( vol ` ( `' ( ( RR X. { A } ) o F x. F ) " { m } ) ) ) = ( A x. sum_ m e. ( ran ( ( RR X. { A } ) o F x. F ) \ { 0 } ) ( ( m / A ) x. ( vol ` ( `' F " { ( m / A ) } ) ) ) ) )
71		itg1val 19528	. . . 4 |- ( ( ( RR X. { A } ) o F x. F ) e. dom S.1 -> ( S.1 ` ( ( RR X. { A } ) o F x. F ) ) = sum_ m e. ( ran ( ( RR X. { A } ) o F x. F ) \ { 0 } ) ( m x. ( vol ` ( `' ( ( RR X. { A } ) o F x. F ) " { m } ) ) ) )
72	29, 71	syl 16	. . 3 |- ( ( ph /\ A =/= 0 ) -> ( S.1 ` ( ( RR X. { A } ) o F x. F ) ) = sum_ m e. ( ran ( ( RR X. { A } ) o F x. F ) \ { 0 } ) ( m x. ( vol ` ( `' ( ( RR X. { A } ) o F x. F ) " { m } ) ) ) )
73	4	adantr 452	. . . . . 6 |- ( ( ph /\ A =/= 0 ) -> F e. dom S.1 )
74		itg1val 19528	. . . . . 6 |- ( F e. dom S.1 -> ( S.1 ` F ) = sum_ k e. ( ran F \ { 0 } ) ( k x. ( vol ` ( `' F " { k } ) ) ) )
75	73, 74	syl 16	. . . . 5 |- ( ( ph /\ A =/= 0 ) -> ( S.1 ` F ) = sum_ k e. ( ran F \ { 0 } ) ( k x. ( vol ` ( `' F " { k } ) ) ) )
76		id 20	. . . . . . 7 |- ( k = ( m / A ) -> k = ( m / A ) )
77		sneq 3785	. . . . . . . . 9 |- ( k = ( m / A ) -> { k } = { ( m / A ) } )
78	77	imaeq2d 5162	. . . . . . . 8 |- ( k = ( m / A ) -> ( `' F " { k } ) = ( `' F " { ( m / A ) } ) )
79	78	fveq2d 5691	. . . . . . 7 |- ( k = ( m / A ) -> ( vol ` ( `' F " { k } ) ) = ( vol ` ( `' F " { ( m / A ) } ) ) )
80	76, 79	oveq12d 6058	. . . . . 6 |- ( k = ( m / A ) -> ( k x. ( vol ` ( `' F " { k } ) ) ) = ( ( m / A ) x. ( vol ` ( `' F " { ( m / A ) } ) ) ) )
81		eqid 2404	. . . . . . 7 |- ( n e. ( ran ( ( RR X. { A } ) o F x. F ) \ { 0 } ) |-> ( n / A ) ) = ( n e. ( ran ( ( RR X. { A } ) o F x. F ) \ { 0 } ) |-> ( n / A ) )
82		eldifi 3429	. . . . . . . . 9 |- ( n e. ( ran ( ( RR X. { A } ) o F x. F ) \ { 0 } ) -> n e. ran ( ( RR X. { A } ) o F x. F ) )
83	2	a1i 11	. . . . . . . . . . . . . . . . 17 |- ( ph -> RR e. _V )
84		ffn 5550	. . . . . . . . . . . . . . . . . 18 |- ( F : RR --> RR -> F Fn RR )
85	6, 84	syl 16	. . . . . . . . . . . . . . . . 17 |- ( ph -> F Fn RR )
86		eqidd 2405	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ y e. RR ) -> ( F ` y ) = ( F ` y ) )
87	83, 8, 85, 86	ofc1 6286	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ y e. RR ) -> ( ( ( RR X. { A } ) o F x. F ) ` y ) = ( A x. ( F ` y ) ) )
88	87	adantlr 696	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ A =/= 0 ) /\ y e. RR ) -> ( ( ( RR X. { A } ) o F x. F ) ` y ) = ( A x. ( F ` y ) ) )
89	88	oveq1d 6055	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ A =/= 0 ) /\ y e. RR ) -> ( ( ( ( RR X. { A } ) o F x. F ) ` y ) / A ) = ( ( A x. ( F ` y ) ) / A ) )
90	6	adantr 452	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ A =/= 0 ) -> F : RR --> RR )
91	90	ffvelrnda 5829	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ A =/= 0 ) /\ y e. RR ) -> ( F ` y ) e. RR )
92	91	recnd 9070	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ A =/= 0 ) /\ y e. RR ) -> ( F ` y ) e. CC )
93	38	adantr 452	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ A =/= 0 ) /\ y e. RR ) -> A e. CC )
94		simplr 732	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ A =/= 0 ) /\ y e. RR ) -> A =/= 0 )
95	92, 93, 94	divcan3d 9751	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ A =/= 0 ) /\ y e. RR ) -> ( ( A x. ( F ` y ) ) / A ) = ( F ` y ) )
96	89, 95	eqtrd 2436	. . . . . . . . . . . . 13 |- ( ( ( ph /\ A =/= 0 ) /\ y e. RR ) -> ( ( ( ( RR X. { A } ) o F x. F ) ` y ) / A ) = ( F ` y ) )
97	90, 84	syl 16	. . . . . . . . . . . . . 14 |- ( ( ph /\ A =/= 0 ) -> F Fn RR )
98		fnfvelrn 5826	. . . . . . . . . . . . . 14 |- ( ( F Fn RR /\ y e. RR ) -> ( F ` y ) e. ran F )
99	97, 98	sylan 458	. . . . . . . . . . . . 13 |- ( ( ( ph /\ A =/= 0 ) /\ y e. RR ) -> ( F ` y ) e. ran F )
100	96, 99	eqeltrd 2478	. . . . . . . . . . . 12 |- ( ( ( ph /\ A =/= 0 ) /\ y e. RR ) -> ( ( ( ( RR X. { A } ) o F x. F ) ` y ) / A ) e. ran F )
101	100	ralrimiva 2749	. . . . . . . . . . 11 |- ( ( ph /\ A =/= 0 ) -> A. y e. RR ( ( ( ( RR X. { A } ) o F x. F ) ` y ) / A ) e. ran F )
102		ffn 5550	. . . . . . . . . . . . 13 |- ( ( ( RR X. { A } ) o F x. F ) : RR --> RR -> ( ( RR X. { A } ) o F x. F ) Fn RR )
103	31, 102	syl 16	. . . . . . . . . . . 12 |- ( ( ph /\ A =/= 0 ) -> ( ( RR X. { A } ) o F x. F ) Fn RR )
104		oveq1 6047	. . . . . . . . . . . . . 14 |- ( n = ( ( ( RR X. { A } ) o F x. F ) ` y ) -> ( n / A ) = ( ( ( ( RR X. { A } ) o F x. F ) ` y ) / A ) )
105	104	eleq1d 2470	. . . . . . . . . . . . 13 |- ( n = ( ( ( RR X. { A } ) o F x. F ) ` y ) -> ( ( n / A ) e. ran F <-> ( ( ( ( RR X. { A } ) o F x. F ) ` y ) / A ) e. ran F ) )
106	105	ralrn 5832	. . . . . . . . . . . 12 |- ( ( ( RR X. { A } ) o F x. F ) Fn RR -> ( A. n e. ran ( ( RR X. { A } ) o F x. F ) ( n / A ) e. ran F <-> A. y e. RR ( ( ( ( RR X. { A } ) o F x. F ) ` y ) / A ) e. ran F ) )
107	103, 106	syl 16	. . . . . . . . . . 11 |- ( ( ph /\ A =/= 0 ) -> ( A. n e. ran ( ( RR X. { A } ) o F x. F ) ( n / A ) e. ran F <-> A. y e. RR ( ( ( ( RR X. { A } ) o F x. F ) ` y ) / A ) e. ran F ) )
108	101, 107	mpbird 224	. . . . . . . . . 10 |- ( ( ph /\ A =/= 0 ) -> A. n e. ran ( ( RR X. { A } ) o F x. F ) ( n / A ) e. ran F )
109	108	r19.21bi 2764	. . . . . . . . 9 |- ( ( ( ph /\ A =/= 0 ) /\ n e. ran ( ( RR X. { A } ) o F x. F ) ) -> ( n / A ) e. ran F )
110	82, 109	sylan2 461	. . . . . . . 8 |- ( ( ( ph /\ A =/= 0 ) /\ n e. ( ran ( ( RR X. { A } ) o F x. F ) \ { 0 } ) ) -> ( n / A ) e. ran F )
111	34	sselda 3308	. . . . . . . . . 10 |- ( ( ( ph /\ A =/= 0 ) /\ n e. ( ran ( ( RR X. { A } ) o F x. F ) \ { 0 } ) ) -> n e. RR )
112	111	recnd 9070	. . . . . . . . 9 |- ( ( ( ph /\ A =/= 0 ) /\ n e. ( ran ( ( RR X. { A } ) o F x. F ) \ { 0 } ) ) -> n e. CC )
113	38	adantr 452	. . . . . . . . 9 |- ( ( ( ph /\ A =/= 0 ) /\ n e. ( ran ( ( RR X. { A } ) o F x. F ) \ { 0 } ) ) -> A e. CC )
114		eldifsni 3888	. . . . . . . . . 10 |- ( n e. ( ran ( ( RR X. { A } ) o F x. F ) \ { 0 } ) -> n =/= 0 )
115	114	adantl 453	. . . . . . . . 9 |- ( ( ( ph /\ A =/= 0 ) /\ n e. ( ran ( ( RR X. { A } ) o F x. F ) \ { 0 } ) ) -> n =/= 0 )
116		simplr 732	. . . . . . . . 9 |- ( ( ( ph /\ A =/= 0 ) /\ n e. ( ran ( ( RR X. { A } ) o F x. F ) \ { 0 } ) ) -> A =/= 0 )
117	112, 113, 115, 116	divne0d 9762	. . . . . . . 8 |- ( ( ( ph /\ A =/= 0 ) /\ n e. ( ran ( ( RR X. { A } ) o F x. F ) \ { 0 } ) ) -> ( n / A ) =/= 0 )
118		eldifsn 3887	. . . . . . . 8 |- ( ( n / A ) e. ( ran F \ { 0 } ) <-> ( ( n / A ) e. ran F /\ ( n / A ) =/= 0 ) )
119	110, 117, 118	sylanbrc 646	. . . . . . 7 |- ( ( ( ph /\ A =/= 0 ) /\ n e. ( ran ( ( RR X. { A } ) o F x. F ) \ { 0 } ) ) -> ( n / A ) e. ( ran F \ { 0 } ) )
120		eldifi 3429	. . . . . . . . 9 |- ( k e. ( ran F \ { 0 } ) -> k e. ran F )
121		fnfvelrn 5826	. . . . . . . . . . . . . 14 |- ( ( ( ( RR X. { A } ) o F x. F ) Fn RR /\ y e. RR ) -> ( ( ( RR X. { A } ) o F x. F ) ` y ) e. ran ( ( RR X. { A } ) o F x. F ) )
122	103, 121	sylan 458	. . . . . . . . . . . . 13 |- ( ( ( ph /\ A =/= 0 ) /\ y e. RR ) -> ( ( ( RR X. { A } ) o F x. F ) ` y ) e. ran ( ( RR X. { A } ) o F x. F ) )
123	88, 122	eqeltrrd 2479	. . . . . . . . . . . 12 |- ( ( ( ph /\ A =/= 0 ) /\ y e. RR ) -> ( A x. ( F ` y ) ) e. ran ( ( RR X. { A } ) o F x. F ) )
124	123	ralrimiva 2749	. . . . . . . . . . 11 |- ( ( ph /\ A =/= 0 ) -> A. y e. RR ( A x. ( F ` y ) ) e. ran ( ( RR X. { A } ) o F x. F ) )
125		oveq2 6048	. . . . . . . . . . . . . 14 |- ( k = ( F ` y ) -> ( A x. k ) = ( A x. ( F ` y ) ) )
126	125	eleq1d 2470	. . . . . . . . . . . . 13 |- ( k = ( F ` y ) -> ( ( A x. k ) e. ran ( ( RR X. { A } ) o F x. F ) <-> ( A x. ( F ` y ) ) e. ran ( ( RR X. { A } ) o F x. F ) ) )
127	126	ralrn 5832	. . . . . . . . . . . 12 |- ( F Fn RR -> ( A. k e. ran F ( A x. k ) e. ran ( ( RR X. { A } ) o F x. F ) <-> A. y e. RR ( A x. ( F ` y ) ) e. ran ( ( RR X. { A } ) o F x. F ) ) )
128	97, 127	syl 16	. . . . . . . . . . 11 |- ( ( ph /\ A =/= 0 ) -> ( A. k e. ran F ( A x. k ) e. ran ( ( RR X. { A } ) o F x. F ) <-> A. y e. RR ( A x. ( F ` y ) ) e. ran ( ( RR X. { A } ) o F x. F ) ) )
129	124, 128	mpbird 224	. . . . . . . . . 10 |- ( ( ph /\ A =/= 0 ) -> A. k e. ran F ( A x. k ) e. ran ( ( RR X. { A } ) o F x. F ) )
130	129	r19.21bi 2764	. . . . . . . . 9 |- ( ( ( ph /\ A =/= 0 ) /\ k e. ran F ) -> ( A x. k ) e. ran ( ( RR X. { A } ) o F x. F ) )
131	120, 130	sylan2 461	. . . . . . . 8 |- ( ( ( ph /\ A =/= 0 ) /\ k e. ( ran F \ { 0 } ) ) -> ( A x. k ) e. ran ( ( RR X. { A } ) o F x. F ) )
132	38	adantr 452	. . . . . . . . 9 |- ( ( ( ph /\ A =/= 0 ) /\ k e. ( ran F \ { 0 } ) ) -> A e. CC )
133		frn 5556	. . . . . . . . . . . . 13 |- ( F : RR --> RR -> ran F C_ RR )
134	90, 133	syl 16	. . . . . . . . . . . 12 |- ( ( ph /\ A =/= 0 ) -> ran F C_ RR )
135	134	ssdifssd 3445	. . . . . . . . . . 11 |- ( ( ph /\ A =/= 0 ) -> ( ran F \ { 0 } ) C_ RR )
136	135	sselda 3308	. . . . . . . . . 10 |- ( ( ( ph /\ A =/= 0 ) /\ k e. ( ran F \ { 0 } ) ) -> k e. RR )
137	136	recnd 9070	. . . . . . . . 9 |- ( ( ( ph /\ A =/= 0 ) /\ k e. ( ran F \ { 0 } ) ) -> k e. CC )
138		simplr 732	. . . . . . . . 9 |- ( ( ( ph /\ A =/= 0 ) /\ k e. ( ran F \ { 0 } ) ) -> A =/= 0 )
139		eldifsni 3888	. . . . . . . . . 10 |- ( k e. ( ran F \ { 0 } ) -> k =/= 0 )
140	139	adantl 453	. . . . . . . . 9 |- ( ( ( ph /\ A =/= 0 ) /\ k e. ( ran F \ { 0 } ) ) -> k =/= 0 )
141	132, 137, 138, 140	mulne0d 9630	. . . . . . . 8 |- ( ( ( ph /\ A =/= 0 ) /\ k e. ( ran F \ { 0 } ) ) -> ( A x. k ) =/= 0 )
142		eldifsn 3887	. . . . . . . 8 |- ( ( A x. k ) e. ( ran ( ( RR X. { A } ) o F x. F ) \ { 0 } ) <-> ( ( A x. k ) e. ran ( ( RR X. { A } ) o F x. F ) /\ ( A x. k ) =/= 0 ) )
143	131, 141, 142	sylanbrc 646	. . . . . . 7 |- ( ( ( ph /\ A =/= 0 ) /\ k e. ( ran F \ { 0 } ) ) -> ( A x. k ) e. ( ran ( ( RR X. { A } ) o F x. F ) \ { 0 } ) )
144		simpl 444	. . . . . . . . . . . 12 |- ( ( n e. ( ran ( ( RR X. { A } ) o F x. F ) \ { 0 } ) /\ k e. ( ran F \ { 0 } ) ) -> n e. ( ran ( ( RR X. { A } ) o F x. F ) \ { 0 } ) )
145		ssel2 3303	. . . . . . . . . . . 12 |- ( ( ( ran ( ( RR X. { A } ) o F x. F ) \ { 0 } ) C_ RR /\ n e. ( ran ( ( RR X. { A } ) o F x. F ) \ { 0 } ) ) -> n e. RR )
146	34, 144, 145	syl2an 464	. . . . . . . . . . 11 |- ( ( ( ph /\ A =/= 0 ) /\ ( n e. ( ran ( ( RR X. { A } ) o F x. F ) \ { 0 } ) /\ k e. ( ran F \ { 0 } ) ) ) -> n e. RR )
147	146	recnd 9070	. . . . . . . . . 10 |- ( ( ( ph /\ A =/= 0 ) /\ ( n e. ( ran ( ( RR X. { A } ) o F x. F ) \ { 0 } ) /\ k e. ( ran F \ { 0 } ) ) ) -> n e. CC )
148	8	ad2antrr 707	. . . . . . . . . . 11 |- ( ( ( ph /\ A =/= 0 ) /\ ( n e. ( ran ( ( RR X. { A } ) o F x. F ) \ { 0 } ) /\ k e. ( ran F \ { 0 } ) ) ) -> A e. RR )
149	148	recnd 9070	. . . . . . . . . 10 |- ( ( ( ph /\ A =/= 0 ) /\ ( n e. ( ran ( ( RR X. { A } ) o F x. F ) \ { 0 } ) /\ k e. ( ran F \ { 0 } ) ) ) -> A e. CC )
150	136	adantrl 697	. . . . . . . . . . 11 |- ( ( ( ph /\ A =/= 0 ) /\ ( n e. ( ran ( ( RR X. { A } ) o F x. F ) \ { 0 } ) /\ k e. ( ran F \ { 0 } ) ) ) -> k e. RR )
151	150	recnd 9070	. . . . . . . . . 10 |- ( ( ( ph /\ A =/= 0 ) /\ ( n e. ( ran ( ( RR X. { A } ) o F x. F ) \ { 0 } ) /\ k e. ( ran F \ { 0 } ) ) ) -> k e. CC )
152		simplr 732	. . . . . . . . . 10 |- ( ( ( ph /\ A =/= 0 ) /\ ( n e. ( ran ( ( RR X. { A } ) o F x. F ) \ { 0 } ) /\ k e. ( ran F \ { 0 } ) ) ) -> A =/= 0 )
153	147, 149, 151, 152	divmuld 9768	. . . . . . . . 9 |- ( ( ( ph /\ A =/= 0 ) /\ ( n e. ( ran ( ( RR X. { A } ) o F x. F ) \ { 0 } ) /\ k e. ( ran F \ { 0 } ) ) ) -> ( ( n / A ) = k <-> ( A x. k ) = n ) )
154	153	bicomd 193	. . . . . . . 8 |- ( ( ( ph /\ A =/= 0 ) /\ ( n e. ( ran ( ( RR X. { A } ) o F x. F ) \ { 0 } ) /\ k e. ( ran F \ { 0 } ) ) ) -> ( ( A x. k ) = n <-> ( n / A ) = k ) )
155		eqcom 2406	. . . . . . . 8 |- ( n = ( A x. k ) <-> ( A x. k ) = n )
156		eqcom 2406	. . . . . . . 8 |- ( k = ( n / A ) <-> ( n / A ) = k )
157	154, 155, 156	3bitr4g 280	. . . . . . 7 |- ( ( ( ph /\ A =/= 0 ) /\ ( n e. ( ran ( ( RR X. { A } ) o F x. F ) \ { 0 } ) /\ k e. ( ran F \ { 0 } ) ) ) -> ( n = ( A x. k ) <-> k = ( n / A ) ) )
158	81, 119, 143, 157	f1o2d 6255	. . . . . 6 |- ( ( ph /\ A =/= 0 ) -> ( n e. ( ran ( ( RR X. { A } ) o F x. F ) \ { 0 } ) |-> ( n / A ) ) : ( ran ( ( RR X. { A } ) o F x. F ) \ { 0 } ) -1-1-onto-> ( ran F \ { 0 } ) )
159		oveq1 6047	. . . . . . . 8 |- ( n = m -> ( n / A ) = ( m / A ) )
160		ovex 6065	. . . . . . . 8 |- ( m / A ) e. _V
161	159, 81, 160	fvmpt 5765	. . . . . . 7 |- ( m e. ( ran ( ( RR X. { A } ) o F x. F ) \ { 0 } ) -> ( ( n e. ( ran ( ( RR X. { A } ) o F x. F ) \ { 0 } ) |-> ( n / A ) ) ` m ) = ( m / A ) )
162	161	adantl 453	. . . . . 6 |- ( ( ( ph /\ A =/= 0 ) /\ m e. ( ran ( ( RR X. { A } ) o F x. F ) \ { 0 } ) ) -> ( ( n e. ( ran ( ( RR X. { A } ) o F x. F ) \ { 0 } ) |-> ( n / A ) ) ` m ) = ( m / A ) )
163		i1fima2sn 19525	. . . . . . . . 9 |- ( ( F e. dom S.1 /\ k e. ( ran F \ { 0 } ) ) -> ( vol ` ( `' F " { k } ) ) e. RR )
164	73, 163	sylan 458	. . . . . . . 8 |- ( ( ( ph /\ A =/= 0 ) /\ k e. ( ran F \ { 0 } ) ) -> ( vol ` ( `' F " { k } ) ) e. RR )
165	136, 164	remulcld 9072	. . . . . . 7 |- ( ( ( ph /\ A =/= 0 ) /\ k e. ( ran F \ { 0 } ) ) -> ( k x. ( vol ` ( `' F " { k } ) ) ) e. RR )
166	165	recnd 9070	. . . . . 6 |- ( ( ( ph /\ A =/= 0 ) /\ k e. ( ran F \ { 0 } ) ) -> ( k x. ( vol ` ( `' F " { k } ) ) ) e. CC )
167	80, 67, 158, 162, 166	fsumf1o 12472	. . . . 5 |- ( ( ph /\ A =/= 0 ) -> sum_ k e. ( ran F \ { 0 } ) ( k x. ( vol ` ( `' F " { k } ) ) ) = sum_ m e. ( ran ( ( RR X. { A } ) o F x. F ) \ { 0 } ) ( ( m / A ) x. ( vol ` ( `' F " { ( m / A ) } ) ) ) )
168	75, 167	eqtrd 2436	. . . 4 |- ( ( ph /\ A =/= 0 ) -> ( S.1 ` F ) = sum_ m e. ( ran ( ( RR X. { A } ) o F x. F ) \ { 0 } ) ( ( m / A ) x. ( vol ` ( `' F " { ( m / A ) } ) ) ) )
169	168	oveq2d 6056	. . 3 |- ( ( ph /\ A =/= 0 ) -> ( A x. ( S.1 ` F ) ) = ( A x. sum_ m e. ( ran ( ( RR X. { A } ) o F x. F ) \ { 0 } ) ( ( m / A ) x. ( vol ` ( `' F " { ( m / A ) } ) ) ) ) )
170	70, 72, 169	3eqtr4d 2446	. 2 |- ( ( ph /\ A =/= 0 ) -> ( S.1 ` ( ( RR X. { A } ) o F x. F ) ) = ( A x. ( S.1 ` F ) ) )
171	27, 170	pm2.61dane 2645	1 |- ( ph -> ( S.1 ` ( ( RR X. { A } ) o F x. F ) ) = ( A x. ( S.1 ` F ) ) )

Syntax hints:   -> wi 4   <-> wb 177   /\ wa 359   = wceq 1649   e. wcel 1721   =/= wne 2567   A.wral 2666   _Vcvv 2916   \ cdif 3277   C_ wss 3280   {csn 3774   e. cmpt 4226   X. cxp 4835   `'ccnv 4836   dom cdm 4837   ran crn 4838   "cima 4840   Fn wfn 5408   -->wf 5409   ` cfv 5413  (class class class)co 6040   o Fcof 6262   Fincfn 7068   CCcc 8944   RRcr 8945   0cc0 8946   x. cmul 8951   / cdiv 9633   sum_csu 12434   volcvol 19313   S.1citg1 19460

This theorem is referenced by:  itg1sub  19554  itg2const  19585  itg2mulclem  19591  itg2monolem1  19595  itg2addnclem  26155

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-er 6864  df-map 6979  df-pm 6980  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-sup 7404  df-oi 7435  df-card 7782  df-cda 8004  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-n0 10178  df-z 10239  df-uz 10445  df-q 10531  df-rp 10569  df-xadd 10667  df-ioo 10876  df-ico 10878  df-icc 10879  df-fz 11000  df-fzo 11091  df-fl 11157  df-seq 11279  df-exp 11338  df-hash 11574  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-clim 12237  df-sum 12435  df-xmet 16650  df-met 16651  df-ovol 19314  df-vol 19315  df-mbf 19465  df-itg1 19466