Theorem itg1mulc 22277

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 22261	. . 3 |- ( S.1 ` ( RR X. { 0 } ) ) = 0
2		reex 9572	. . . . . 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 22249	. . . . . . 7 |- ( F e. dom S.1 -> F : RR --> RR )
6	4, 5	syl 16	. . . . . 6 |- ( ph -> F : RR --> RR )
7	6	adantr 463	. . . . 5 |- ( ( ph /\ A = 0 ) -> F : RR --> RR )
8		i1fmulc.3	. . . . . 6 |- ( ph -> A e. RR )
9	8	adantr 463	. . . . 5 |- ( ( ph /\ A = 0 ) -> A e. RR )
10		0red 9586	. . . . 5 |- ( ( ph /\ A = 0 ) -> 0 e. RR )
11		simplr 753	. . . . . . 7 |- ( ( ( ph /\ A = 0 ) /\ x e. RR ) -> A = 0 )
12	11	oveq1d 6285	. . . . . 6 |- ( ( ( ph /\ A = 0 ) /\ x e. RR ) -> ( A x. x ) = ( 0 x. x ) )
13		mul02lem2 9746	. . . . . . 7 |- ( x e. RR -> ( 0 x. x ) = 0 )
14	13	adantl 464	. . . . . 6 |- ( ( ( ph /\ A = 0 ) /\ x e. RR ) -> ( 0 x. x ) = 0 )
15	12, 14	eqtrd 2495	. . . . 5 |- ( ( ( ph /\ A = 0 ) /\ x e. RR ) -> ( A x. x ) = 0 )
16	3, 7, 9, 10, 15	caofid2 6544	. . . 4 |- ( ( ph /\ A = 0 ) -> ( ( RR X. { A } ) oF x. F ) = ( RR X. { 0 } ) )
17	16	fveq2d 5852	. . 3 |- ( ( ph /\ A = 0 ) -> ( S.1 ` ( ( RR X. { A } ) oF x. F ) ) = ( S.1 ` ( RR X. { 0 } ) ) )
18		simpr 459	. . . . 5 |- ( ( ph /\ A = 0 ) -> A = 0 )
19	18	oveq1d 6285	. . . 4 |- ( ( ph /\ A = 0 ) -> ( A x. ( S.1 ` F ) ) = ( 0 x. ( S.1 ` F ) ) )
20		itg1cl 22258	. . . . . . . 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 9611	. . . . . 6 |- ( ph -> ( S.1 ` F ) e. CC )
23	22	mul02d 9767	. . . . 5 |- ( ph -> ( 0 x. ( S.1 ` F ) ) = 0 )
24	23	adantr 463	. . . 4 |- ( ( ph /\ A = 0 ) -> ( 0 x. ( S.1 ` F ) ) = 0 )
25	19, 24	eqtrd 2495	. . 3 |- ( ( ph /\ A = 0 ) -> ( A x. ( S.1 ` F ) ) = 0 )
26	1, 17, 25	3eqtr4a 2521	. 2 |- ( ( ph /\ A = 0 ) -> ( S.1 ` ( ( RR X. { A } ) oF x. F ) ) = ( A x. ( S.1 ` F ) ) )
27	4, 8	i1fmulc 22276	. . . . . . . . . . . . . 14 |- ( ph -> ( ( RR X. { A } ) oF x. F ) e. dom S.1 )
28	27	adantr 463	. . . . . . . . . . . . 13 |- ( ( ph /\ A =/= 0 ) -> ( ( RR X. { A } ) oF x. F ) e. dom S.1 )
29		i1ff 22249	. . . . . . . . . . . . 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 5719	. . . . . . . . . . . 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 3628	. . . . . . . . . 10 |- ( ( ph /\ A =/= 0 ) -> ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) C_ RR )
34	33	sselda 3489	. . . . . . . . 9 |- ( ( ( ph /\ A =/= 0 ) /\ m e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) ) -> m e. RR )
35	34	recnd 9611	. . . . . . . 8 |- ( ( ( ph /\ A =/= 0 ) /\ m e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) ) -> m e. CC )
36	8	adantr 463	. . . . . . . . . 10 |- ( ( ph /\ A =/= 0 ) -> A e. RR )
37	36	recnd 9611	. . . . . . . . 9 |- ( ( ph /\ A =/= 0 ) -> A e. CC )
38	37	adantr 463	. . . . . . . 8 |- ( ( ( ph /\ A =/= 0 ) /\ m e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) ) -> A e. CC )
39		simplr 753	. . . . . . . 8 |- ( ( ( ph /\ A =/= 0 ) /\ m e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) ) -> A =/= 0 )
40	35, 38, 39	divcan2d 10318	. . . . . . 7 |- ( ( ( ph /\ A =/= 0 ) /\ m e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) ) -> ( A x. ( m / A ) ) = m )
41	4, 8	i1fmulclem 22275	. . . . . . . . . 10 |- ( ( ( ph /\ A =/= 0 ) /\ m e. RR ) -> ( `' ( ( RR X. { A } ) oF x. F ) " { m } ) = ( `' F " { ( m / A ) } ) )
42	34, 41	syldan 468	. . . . . . . . 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 5852	. . . . . . . 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 2462	. . . . . . 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 6288	. . . . . 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 723	. . . . . . . . 9 |- ( ( ( ph /\ A =/= 0 ) /\ m e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) ) -> A e. RR )
47	34, 46, 39	redivcld 10368	. . . . . . . 8 |- ( ( ( ph /\ A =/= 0 ) /\ m e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) ) -> ( m / A ) e. RR )
48	47	recnd 9611	. . . . . . 7 |- ( ( ( ph /\ A =/= 0 ) /\ m e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) ) -> ( m / A ) e. CC )
49	4	ad2antrr 723	. . . . . . . . 9 |- ( ( ( ph /\ A =/= 0 ) /\ m e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) ) -> F e. dom S.1 )
50	46	recnd 9611	. . . . . . . . . . 11 |- ( ( ( ph /\ A =/= 0 ) /\ m e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) ) -> A e. CC )
51		eldifsni 4142	. . . . . . . . . . . 12 |- ( m e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) -> m =/= 0 )
52	51	adantl 464	. . . . . . . . . . 11 |- ( ( ( ph /\ A =/= 0 ) /\ m e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) ) -> m =/= 0 )
53	35, 50, 52, 39	divne0d 10332	. . . . . . . . . 10 |- ( ( ( ph /\ A =/= 0 ) /\ m e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) ) -> ( m / A ) =/= 0 )
54		eldifsn 4141	. . . . . . . . . 10 |- ( ( m / A ) e. ( RR \ { 0 } ) <-> ( ( m / A ) e. RR /\ ( m / A ) =/= 0 ) )
55	47, 53, 54	sylanbrc 662	. . . . . . . . 9 |- ( ( ( ph /\ A =/= 0 ) /\ m e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) ) -> ( m / A ) e. ( RR \ { 0 } ) )
56		i1fima2sn 22253	. . . . . . . . 9 |- ( ( F e. dom S.1 /\ ( m / A ) e. ( RR \ { 0 } ) ) -> ( vol ` ( `' F " { ( m / A ) } ) ) e. RR )
57	49, 55, 56	syl2anc 659	. . . . . . . 8 |- ( ( ( ph /\ A =/= 0 ) /\ m e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) ) -> ( vol ` ( `' F " { ( m / A ) } ) ) e. RR )
58	57	recnd 9611	. . . . . . 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 9608	. . . . . 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 2497	. . . . 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 13607	. . . 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 22250	. . . . . . 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 3617	. . . . . 6 |- ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) C_ ran ( ( RR X. { A } ) oF x. F )
65		ssfi 7733	. . . . . 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 660	. . . . 5 |- ( ( ph /\ A =/= 0 ) -> ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) e. Fin )
67	48, 58	mulcld 9605	. . . . 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 13681	. . . 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 2498	. . 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 22256	. . . 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 463	. . . . . 6 |- ( ( ph /\ A =/= 0 ) -> F e. dom S.1 )
73		itg1val 22256	. . . . . 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 4026	. . . . . . . . 9 |- ( k = ( m / A ) -> { k } = { ( m / A ) } )
77	76	imaeq2d 5325	. . . . . . . 8 |- ( k = ( m / A ) -> ( `' F " { k } ) = ( `' F " { ( m / A ) } ) )
78	77	fveq2d 5852	. . . . . . 7 |- ( k = ( m / A ) -> ( vol ` ( `' F " { k } ) ) = ( vol ` ( `' F " { ( m / A ) } ) ) )
79	75, 78	oveq12d 6288	. . . . . 6 |- ( k = ( m / A ) -> ( k x. ( vol ` ( `' F " { k } ) ) ) = ( ( m / A ) x. ( vol ` ( `' F " { ( m / A ) } ) ) ) )
80		eqid 2454	. . . . . . 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 3612	. . . . . . . . 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 5713	. . . . . . . . . . . . . . . . . 18 |- ( F : RR --> RR -> F Fn RR )
84	6, 83	syl 16	. . . . . . . . . . . . . . . . 17 |- ( ph -> F Fn RR )
85		eqidd 2455	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ y e. RR ) -> ( F ` y ) = ( F ` y ) )
86	82, 8, 84, 85	ofc1 6536	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ y e. RR ) -> ( ( ( RR X. { A } ) oF x. F ) ` y ) = ( A x. ( F ` y ) ) )
87	86	adantlr 712	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ A =/= 0 ) /\ y e. RR ) -> ( ( ( RR X. { A } ) oF x. F ) ` y ) = ( A x. ( F ` y ) ) )
88	87	oveq1d 6285	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ A =/= 0 ) /\ y e. RR ) -> ( ( ( ( RR X. { A } ) oF x. F ) ` y ) / A ) = ( ( A x. ( F ` y ) ) / A ) )
89	6	adantr 463	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ A =/= 0 ) -> F : RR --> RR )
90	89	ffvelrnda 6007	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ A =/= 0 ) /\ y e. RR ) -> ( F ` y ) e. RR )
91	90	recnd 9611	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ A =/= 0 ) /\ y e. RR ) -> ( F ` y ) e. CC )
92	37	adantr 463	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ A =/= 0 ) /\ y e. RR ) -> A e. CC )
93		simplr 753	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ A =/= 0 ) /\ y e. RR ) -> A =/= 0 )
94	91, 92, 93	divcan3d 10321	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ A =/= 0 ) /\ y e. RR ) -> ( ( A x. ( F ` y ) ) / A ) = ( F ` y ) )
95	88, 94	eqtrd 2495	. . . . . . . . . . . . 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 6004	. . . . . . . . . . . . . 14 |- ( ( F Fn RR /\ y e. RR ) -> ( F ` y ) e. ran F )
98	96, 97	sylan 469	. . . . . . . . . . . . 13 |- ( ( ( ph /\ A =/= 0 ) /\ y e. RR ) -> ( F ` y ) e. ran F )
99	95, 98	eqeltrd 2542	. . . . . . . . . . . 12 |- ( ( ( ph /\ A =/= 0 ) /\ y e. RR ) -> ( ( ( ( RR X. { A } ) oF x. F ) ` y ) / A ) e. ran F )
100	99	ralrimiva 2868	. . . . . . . . . . 11 |- ( ( ph /\ A =/= 0 ) -> A. y e. RR ( ( ( ( RR X. { A } ) oF x. F ) ` y ) / A ) e. ran F )
101		ffn 5713	. . . . . . . . . . . . 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 6277	. . . . . . . . . . . . . 14 |- ( n = ( ( ( RR X. { A } ) oF x. F ) ` y ) -> ( n / A ) = ( ( ( ( RR X. { A } ) oF x. F ) ` y ) / A ) )
104	103	eleq1d 2523	. . . . . . . . . . . . 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 6010	. . . . . . . . . . . 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 2823	. . . . . . . . 9 |- ( ( ( ph /\ A =/= 0 ) /\ n e. ran ( ( RR X. { A } ) oF x. F ) ) -> ( n / A ) e. ran F )
109	81, 108	sylan2 472	. . . . . . . 8 |- ( ( ( ph /\ A =/= 0 ) /\ n e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) ) -> ( n / A ) e. ran F )
110	33	sselda 3489	. . . . . . . . . 10 |- ( ( ( ph /\ A =/= 0 ) /\ n e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) ) -> n e. RR )
111	110	recnd 9611	. . . . . . . . 9 |- ( ( ( ph /\ A =/= 0 ) /\ n e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) ) -> n e. CC )
112	37	adantr 463	. . . . . . . . 9 |- ( ( ( ph /\ A =/= 0 ) /\ n e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) ) -> A e. CC )
113		eldifsni 4142	. . . . . . . . . 10 |- ( n e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) -> n =/= 0 )
114	113	adantl 464	. . . . . . . . 9 |- ( ( ( ph /\ A =/= 0 ) /\ n e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) ) -> n =/= 0 )
115		simplr 753	. . . . . . . . 9 |- ( ( ( ph /\ A =/= 0 ) /\ n e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) ) -> A =/= 0 )
116	111, 112, 114, 115	divne0d 10332	. . . . . . . 8 |- ( ( ( ph /\ A =/= 0 ) /\ n e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) ) -> ( n / A ) =/= 0 )
117		eldifsn 4141	. . . . . . . 8 |- ( ( n / A ) e. ( ran F \ { 0 } ) <-> ( ( n / A ) e. ran F /\ ( n / A ) =/= 0 ) )
118	109, 116, 117	sylanbrc 662	. . . . . . 7 |- ( ( ( ph /\ A =/= 0 ) /\ n e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) ) -> ( n / A ) e. ( ran F \ { 0 } ) )
119		eldifi 3612	. . . . . . . . 9 |- ( k e. ( ran F \ { 0 } ) -> k e. ran F )
120		fnfvelrn 6004	. . . . . . . . . . . . . 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 469	. . . . . . . . . . . . 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 2543	. . . . . . . . . . . 12 |- ( ( ( ph /\ A =/= 0 ) /\ y e. RR ) -> ( A x. ( F ` y ) ) e. ran ( ( RR X. { A } ) oF x. F ) )
123	122	ralrimiva 2868	. . . . . . . . . . 11 |- ( ( ph /\ A =/= 0 ) -> A. y e. RR ( A x. ( F ` y ) ) e. ran ( ( RR X. { A } ) oF x. F ) )
124		oveq2 6278	. . . . . . . . . . . . . 14 |- ( k = ( F ` y ) -> ( A x. k ) = ( A x. ( F ` y ) ) )
125	124	eleq1d 2523	. . . . . . . . . . . . 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 6010	. . . . . . . . . . . 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 2823	. . . . . . . . 9 |- ( ( ( ph /\ A =/= 0 ) /\ k e. ran F ) -> ( A x. k ) e. ran ( ( RR X. { A } ) oF x. F ) )
130	119, 129	sylan2 472	. . . . . . . 8 |- ( ( ( ph /\ A =/= 0 ) /\ k e. ( ran F \ { 0 } ) ) -> ( A x. k ) e. ran ( ( RR X. { A } ) oF x. F ) )
131	37	adantr 463	. . . . . . . . 9 |- ( ( ( ph /\ A =/= 0 ) /\ k e. ( ran F \ { 0 } ) ) -> A e. CC )
132		frn 5719	. . . . . . . . . . . . 13 |- ( F : RR --> RR -> ran F C_ RR )
133	89, 132	syl 16	. . . . . . . . . . . 12 |- ( ( ph /\ A =/= 0 ) -> ran F C_ RR )
134	133	ssdifssd 3628	. . . . . . . . . . 11 |- ( ( ph /\ A =/= 0 ) -> ( ran F \ { 0 } ) C_ RR )
135	134	sselda 3489	. . . . . . . . . 10 |- ( ( ( ph /\ A =/= 0 ) /\ k e. ( ran F \ { 0 } ) ) -> k e. RR )
136	135	recnd 9611	. . . . . . . . 9 |- ( ( ( ph /\ A =/= 0 ) /\ k e. ( ran F \ { 0 } ) ) -> k e. CC )
137		simplr 753	. . . . . . . . 9 |- ( ( ( ph /\ A =/= 0 ) /\ k e. ( ran F \ { 0 } ) ) -> A =/= 0 )
138		eldifsni 4142	. . . . . . . . . 10 |- ( k e. ( ran F \ { 0 } ) -> k =/= 0 )
139	138	adantl 464	. . . . . . . . 9 |- ( ( ( ph /\ A =/= 0 ) /\ k e. ( ran F \ { 0 } ) ) -> k =/= 0 )
140	131, 136, 137, 139	mulne0d 10197	. . . . . . . 8 |- ( ( ( ph /\ A =/= 0 ) /\ k e. ( ran F \ { 0 } ) ) -> ( A x. k ) =/= 0 )
141		eldifsn 4141	. . . . . . . 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 662	. . . . . . 7 |- ( ( ( ph /\ A =/= 0 ) /\ k e. ( ran F \ { 0 } ) ) -> ( A x. k ) e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) )
143		simpl 455	. . . . . . . . . . . 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 3484	. . . . . . . . . . . 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 475	. . . . . . . . . . 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 9611	. . . . . . . . . 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 723	. . . . . . . . . . 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 9611	. . . . . . . . . 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 713	. . . . . . . . . . 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 9611	. . . . . . . . . 10 |- ( ( ( ph /\ A =/= 0 ) /\ ( n e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) /\ k e. ( ran F \ { 0 } ) ) ) -> k e. CC )
151		simplr 753	. . . . . . . . . 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 10338	. . . . . . . . 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 2463	. . . . . . . 8 |- ( n = ( A x. k ) <-> ( A x. k ) = n )
155		eqcom 2463	. . . . . . . 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 6500	. . . . . 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 6277	. . . . . . . 8 |- ( n = m -> ( n / A ) = ( m / A ) )
159		ovex 6298	. . . . . . . 8 |- ( m / A ) e. _V
160	158, 80, 159	fvmpt 5931	. . . . . . 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 464	. . . . . 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 22253	. . . . . . . . 9 |- ( ( F e. dom S.1 /\ k e. ( ran F \ { 0 } ) ) -> ( vol ` ( `' F " { k } ) ) e. RR )
163	72, 162	sylan 469	. . . . . . . 8 |- ( ( ( ph /\ A =/= 0 ) /\ k e. ( ran F \ { 0 } ) ) -> ( vol ` ( `' F " { k } ) ) e. RR )
164	135, 163	remulcld 9613	. . . . . . 7 |- ( ( ( ph /\ A =/= 0 ) /\ k e. ( ran F \ { 0 } ) ) -> ( k x. ( vol ` ( `' F " { k } ) ) ) e. RR )
165	164	recnd 9611	. . . . . 6 |- ( ( ( ph /\ A =/= 0 ) /\ k e. ( ran F \ { 0 } ) ) -> ( k x. ( vol ` ( `' F " { k } ) ) ) e. CC )
166	79, 66, 157, 161, 165	fsumf1o 13627	. . . . 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 2495	. . . 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 6286	. . 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 2505	. 2 |- ( ( ph /\ A =/= 0 ) -> ( S.1 ` ( ( RR X. { A } ) oF x. F ) ) = ( A x. ( S.1 ` F ) ) )
170	26, 169	pm2.61dane 2772	1 |- ( ph -> ( S.1 ` ( ( RR X. { A } ) oF x. F ) ) = ( A x. ( S.1 ` F ) ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 367   = wceq 1398   e. wcel 1823   =/= wne 2649   A.wral 2804   _Vcvv 3106   \ cdif 3458   C_ wss 3461   {csn 4016   |-> cmpt 4497   X. cxp 4986   `'ccnv 4987   dom cdm 4988   ran crn 4989   "cima 4991   Fn wfn 5565   -->wf 5566   ` cfv 5570  (class class class)co 6270   oFcof 6511   Fincfn 7509   CCcc 9479   RRcr 9480   0cc0 9481   x. cmul 9486   / cdiv 10202   sum_csu 13590   volcvol 22041   S.1citg1 22190

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-inf2 8049  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-fal 1404  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-se 4828  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-isom 5579  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-of 6513  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-2o 7123  df-oadd 7126  df-er 7303  df-map 7414  df-pm 7415  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-sup 7893  df-oi 7927  df-card 8311  df-cda 8539  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-nn 10532  df-2 10590  df-3 10591  df-n0 10792  df-z 10861  df-uz 11083  df-q 11184  df-rp 11222  df-xadd 11322  df-ioo 11536  df-ico 11538  df-icc 11539  df-fz 11676  df-fzo 11800  df-fl 11910  df-seq 12090  df-exp 12149  df-hash 12388  df-cj 13014  df-re 13015  df-im 13016  df-sqrt 13150  df-abs 13151  df-clim 13393  df-sum 13591  df-xmet 18607  df-met 18608  df-ovol 22042  df-vol 22043  df-mbf 22194  df-itg1 22195

This theorem is referenced by:  itg1sub  22282  itg2const  22313  itg2mulclem  22319  itg2monolem1  22323  itg2addnclem  30306