Theorem itg1mulc 23471

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 23455	. . 3 |- ( S.1 ` ( RR X. { 0 } ) ) = 0
2		reex 10027	. . . . . 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 23443	. . . . . . 7 |- ( F e. dom S.1 -> F : RR --> RR )
6	4, 5	syl 17	. . . . . 6 |- ( ph -> F : RR --> RR )
7	6	adantr 481	. . . . 5 |- ( ( ph /\ A = 0 ) -> F : RR --> RR )
8		i1fmulc.3	. . . . . 6 |- ( ph -> A e. RR )
9	8	adantr 481	. . . . 5 |- ( ( ph /\ A = 0 ) -> A e. RR )
10		0red 10041	. . . . 5 |- ( ( ph /\ A = 0 ) -> 0 e. RR )
11		simplr 792	. . . . . . 7 |- ( ( ( ph /\ A = 0 ) /\ x e. RR ) -> A = 0 )
12	11	oveq1d 6665	. . . . . 6 |- ( ( ( ph /\ A = 0 ) /\ x e. RR ) -> ( A x. x ) = ( 0 x. x ) )
13		mul02lem2 10213	. . . . . . 7 |- ( x e. RR -> ( 0 x. x ) = 0 )
14	13	adantl 482	. . . . . 6 |- ( ( ( ph /\ A = 0 ) /\ x e. RR ) -> ( 0 x. x ) = 0 )
15	12, 14	eqtrd 2656	. . . . 5 |- ( ( ( ph /\ A = 0 ) /\ x e. RR ) -> ( A x. x ) = 0 )
16	3, 7, 9, 10, 15	caofid2 6928	. . . 4 |- ( ( ph /\ A = 0 ) -> ( ( RR X. { A } ) oF x. F ) = ( RR X. { 0 } ) )
17	16	fveq2d 6195	. . 3 |- ( ( ph /\ A = 0 ) -> ( S.1 ` ( ( RR X. { A } ) oF x. F ) ) = ( S.1 ` ( RR X. { 0 } ) ) )
18		simpr 477	. . . . 5 |- ( ( ph /\ A = 0 ) -> A = 0 )
19	18	oveq1d 6665	. . . 4 |- ( ( ph /\ A = 0 ) -> ( A x. ( S.1 ` F ) ) = ( 0 x. ( S.1 ` F ) ) )
20		itg1cl 23452	. . . . . . . 8 |- ( F e. dom S.1 -> ( S.1 ` F ) e. RR )
21	4, 20	syl 17	. . . . . . 7 |- ( ph -> ( S.1 ` F ) e. RR )
22	21	recnd 10068	. . . . . 6 |- ( ph -> ( S.1 ` F ) e. CC )
23	22	mul02d 10234	. . . . 5 |- ( ph -> ( 0 x. ( S.1 ` F ) ) = 0 )
24	23	adantr 481	. . . 4 |- ( ( ph /\ A = 0 ) -> ( 0 x. ( S.1 ` F ) ) = 0 )
25	19, 24	eqtrd 2656	. . 3 |- ( ( ph /\ A = 0 ) -> ( A x. ( S.1 ` F ) ) = 0 )
26	1, 17, 25	3eqtr4a 2682	. 2 |- ( ( ph /\ A = 0 ) -> ( S.1 ` ( ( RR X. { A } ) oF x. F ) ) = ( A x. ( S.1 ` F ) ) )
27	4, 8	i1fmulc 23470	. . . . . . . . . . . . . 14 |- ( ph -> ( ( RR X. { A } ) oF x. F ) e. dom S.1 )
28	27	adantr 481	. . . . . . . . . . . . 13 |- ( ( ph /\ A =/= 0 ) -> ( ( RR X. { A } ) oF x. F ) e. dom S.1 )
29		i1ff 23443	. . . . . . . . . . . . 13 |- ( ( ( RR X. { A } ) oF x. F ) e. dom S.1 -> ( ( RR X. { A } ) oF x. F ) : RR --> RR )
30	28, 29	syl 17	. . . . . . . . . . . 12 |- ( ( ph /\ A =/= 0 ) -> ( ( RR X. { A } ) oF x. F ) : RR --> RR )
31		frn 6053	. . . . . . . . . . . 12 |- ( ( ( RR X. { A } ) oF x. F ) : RR --> RR -> ran ( ( RR X. { A } ) oF x. F ) C_ RR )
32	30, 31	syl 17	. . . . . . . . . . 11 |- ( ( ph /\ A =/= 0 ) -> ran ( ( RR X. { A } ) oF x. F ) C_ RR )
33	32	ssdifssd 3748	. . . . . . . . . 10 |- ( ( ph /\ A =/= 0 ) -> ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) C_ RR )
34	33	sselda 3603	. . . . . . . . 9 |- ( ( ( ph /\ A =/= 0 ) /\ m e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) ) -> m e. RR )
35	34	recnd 10068	. . . . . . . 8 |- ( ( ( ph /\ A =/= 0 ) /\ m e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) ) -> m e. CC )
36	8	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ A =/= 0 ) -> A e. RR )
37	36	recnd 10068	. . . . . . . . 9 |- ( ( ph /\ A =/= 0 ) -> A e. CC )
38	37	adantr 481	. . . . . . . 8 |- ( ( ( ph /\ A =/= 0 ) /\ m e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) ) -> A e. CC )
39		simplr 792	. . . . . . . 8 |- ( ( ( ph /\ A =/= 0 ) /\ m e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) ) -> A =/= 0 )
40	35, 38, 39	divcan2d 10803	. . . . . . 7 |- ( ( ( ph /\ A =/= 0 ) /\ m e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) ) -> ( A x. ( m / A ) ) = m )
41	4, 8	i1fmulclem 23469	. . . . . . . . . 10 |- ( ( ( ph /\ A =/= 0 ) /\ m e. RR ) -> ( `' ( ( RR X. { A } ) oF x. F ) " { m } ) = ( `' F " { ( m / A ) } ) )
42	34, 41	syldan 487	. . . . . . . . 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 6195	. . . . . . . 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 2628	. . . . . . 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 6668	. . . . . 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 762	. . . . . . . . 9 |- ( ( ( ph /\ A =/= 0 ) /\ m e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) ) -> A e. RR )
47	34, 46, 39	redivcld 10853	. . . . . . . 8 |- ( ( ( ph /\ A =/= 0 ) /\ m e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) ) -> ( m / A ) e. RR )
48	47	recnd 10068	. . . . . . 7 |- ( ( ( ph /\ A =/= 0 ) /\ m e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) ) -> ( m / A ) e. CC )
49	4	ad2antrr 762	. . . . . . . . 9 |- ( ( ( ph /\ A =/= 0 ) /\ m e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) ) -> F e. dom S.1 )
50	46	recnd 10068	. . . . . . . . . . 11 |- ( ( ( ph /\ A =/= 0 ) /\ m e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) ) -> A e. CC )
51		eldifsni 4320	. . . . . . . . . . . 12 |- ( m e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) -> m =/= 0 )
52	51	adantl 482	. . . . . . . . . . 11 |- ( ( ( ph /\ A =/= 0 ) /\ m e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) ) -> m =/= 0 )
53	35, 50, 52, 39	divne0d 10817	. . . . . . . . . 10 |- ( ( ( ph /\ A =/= 0 ) /\ m e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) ) -> ( m / A ) =/= 0 )
54		eldifsn 4317	. . . . . . . . . 10 |- ( ( m / A ) e. ( RR \ { 0 } ) <-> ( ( m / A ) e. RR /\ ( m / A ) =/= 0 ) )
55	47, 53, 54	sylanbrc 698	. . . . . . . . 9 |- ( ( ( ph /\ A =/= 0 ) /\ m e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) ) -> ( m / A ) e. ( RR \ { 0 } ) )
56		i1fima2sn 23447	. . . . . . . . 9 |- ( ( F e. dom S.1 /\ ( m / A ) e. ( RR \ { 0 } ) ) -> ( vol ` ( `' F " { ( m / A ) } ) ) e. RR )
57	49, 55, 56	syl2anc 693	. . . . . . . 8 |- ( ( ( ph /\ A =/= 0 ) /\ m e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) ) -> ( vol ` ( `' F " { ( m / A ) } ) ) e. RR )
58	57	recnd 10068	. . . . . . 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 10063	. . . . . 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 2658	. . . . 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 14433	. . . 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 23444	. . . . . . 7 |- ( ( ( RR X. { A } ) oF x. F ) e. dom S.1 -> ran ( ( RR X. { A } ) oF x. F ) e. Fin )
63	28, 62	syl 17	. . . . . 6 |- ( ( ph /\ A =/= 0 ) -> ran ( ( RR X. { A } ) oF x. F ) e. Fin )
64		difss 3737	. . . . . 6 |- ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) C_ ran ( ( RR X. { A } ) oF x. F )
65		ssfi 8180	. . . . . 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 694	. . . . 5 |- ( ( ph /\ A =/= 0 ) -> ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) e. Fin )
67	48, 58	mulcld 10060	. . . . 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 14516	. . . 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 2659	. . 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 23450	. . . 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 17	. . 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 481	. . . . . 6 |- ( ( ph /\ A =/= 0 ) -> F e. dom S.1 )
73		itg1val 23450	. . . . . 6 |- ( F e. dom S.1 -> ( S.1 ` F ) = sum_ k e. ( ran F \ { 0 } ) ( k x. ( vol ` ( `' F " { k } ) ) ) )
74	72, 73	syl 17	. . . . 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 4187	. . . . . . . . 9 |- ( k = ( m / A ) -> { k } = { ( m / A ) } )
77	76	imaeq2d 5466	. . . . . . . 8 |- ( k = ( m / A ) -> ( `' F " { k } ) = ( `' F " { ( m / A ) } ) )
78	77	fveq2d 6195	. . . . . . 7 |- ( k = ( m / A ) -> ( vol ` ( `' F " { k } ) ) = ( vol ` ( `' F " { ( m / A ) } ) ) )
79	75, 78	oveq12d 6668	. . . . . 6 |- ( k = ( m / A ) -> ( k x. ( vol ` ( `' F " { k } ) ) ) = ( ( m / A ) x. ( vol ` ( `' F " { ( m / A ) } ) ) ) )
80		eqid 2622	. . . . . . 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 3732	. . . . . . . . 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 6045	. . . . . . . . . . . . . . . . . 18 |- ( F : RR --> RR -> F Fn RR )
84	6, 83	syl 17	. . . . . . . . . . . . . . . . 17 |- ( ph -> F Fn RR )
85		eqidd 2623	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ y e. RR ) -> ( F ` y ) = ( F ` y ) )
86	82, 8, 84, 85	ofc1 6920	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ y e. RR ) -> ( ( ( RR X. { A } ) oF x. F ) ` y ) = ( A x. ( F ` y ) ) )
87	86	adantlr 751	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ A =/= 0 ) /\ y e. RR ) -> ( ( ( RR X. { A } ) oF x. F ) ` y ) = ( A x. ( F ` y ) ) )
88	87	oveq1d 6665	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ A =/= 0 ) /\ y e. RR ) -> ( ( ( ( RR X. { A } ) oF x. F ) ` y ) / A ) = ( ( A x. ( F ` y ) ) / A ) )
89	6	adantr 481	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ A =/= 0 ) -> F : RR --> RR )
90	89	ffvelrnda 6359	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ A =/= 0 ) /\ y e. RR ) -> ( F ` y ) e. RR )
91	90	recnd 10068	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ A =/= 0 ) /\ y e. RR ) -> ( F ` y ) e. CC )
92	37	adantr 481	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ A =/= 0 ) /\ y e. RR ) -> A e. CC )
93		simplr 792	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ A =/= 0 ) /\ y e. RR ) -> A =/= 0 )
94	91, 92, 93	divcan3d 10806	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ A =/= 0 ) /\ y e. RR ) -> ( ( A x. ( F ` y ) ) / A ) = ( F ` y ) )
95	88, 94	eqtrd 2656	. . . . . . . . . . . . 13 |- ( ( ( ph /\ A =/= 0 ) /\ y e. RR ) -> ( ( ( ( RR X. { A } ) oF x. F ) ` y ) / A ) = ( F ` y ) )
96	89, 83	syl 17	. . . . . . . . . . . . . 14 |- ( ( ph /\ A =/= 0 ) -> F Fn RR )
97		fnfvelrn 6356	. . . . . . . . . . . . . 14 |- ( ( F Fn RR /\ y e. RR ) -> ( F ` y ) e. ran F )
98	96, 97	sylan 488	. . . . . . . . . . . . 13 |- ( ( ( ph /\ A =/= 0 ) /\ y e. RR ) -> ( F ` y ) e. ran F )
99	95, 98	eqeltrd 2701	. . . . . . . . . . . 12 |- ( ( ( ph /\ A =/= 0 ) /\ y e. RR ) -> ( ( ( ( RR X. { A } ) oF x. F ) ` y ) / A ) e. ran F )
100	99	ralrimiva 2966	. . . . . . . . . . 11 |- ( ( ph /\ A =/= 0 ) -> A. y e. RR ( ( ( ( RR X. { A } ) oF x. F ) ` y ) / A ) e. ran F )
101		ffn 6045	. . . . . . . . . . . . 13 |- ( ( ( RR X. { A } ) oF x. F ) : RR --> RR -> ( ( RR X. { A } ) oF x. F ) Fn RR )
102	30, 101	syl 17	. . . . . . . . . . . 12 |- ( ( ph /\ A =/= 0 ) -> ( ( RR X. { A } ) oF x. F ) Fn RR )
103		oveq1 6657	. . . . . . . . . . . . . 14 |- ( n = ( ( ( RR X. { A } ) oF x. F ) ` y ) -> ( n / A ) = ( ( ( ( RR X. { A } ) oF x. F ) ` y ) / A ) )
104	103	eleq1d 2686	. . . . . . . . . . . . 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 6362	. . . . . . . . . . . 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 17	. . . . . . . . . . 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 247	. . . . . . . . . 10 |- ( ( ph /\ A =/= 0 ) -> A. n e. ran ( ( RR X. { A } ) oF x. F ) ( n / A ) e. ran F )
108	107	r19.21bi 2932	. . . . . . . . 9 |- ( ( ( ph /\ A =/= 0 ) /\ n e. ran ( ( RR X. { A } ) oF x. F ) ) -> ( n / A ) e. ran F )
109	81, 108	sylan2 491	. . . . . . . 8 |- ( ( ( ph /\ A =/= 0 ) /\ n e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) ) -> ( n / A ) e. ran F )
110	33	sselda 3603	. . . . . . . . . 10 |- ( ( ( ph /\ A =/= 0 ) /\ n e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) ) -> n e. RR )
111	110	recnd 10068	. . . . . . . . 9 |- ( ( ( ph /\ A =/= 0 ) /\ n e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) ) -> n e. CC )
112	37	adantr 481	. . . . . . . . 9 |- ( ( ( ph /\ A =/= 0 ) /\ n e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) ) -> A e. CC )
113		eldifsni 4320	. . . . . . . . . 10 |- ( n e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) -> n =/= 0 )
114	113	adantl 482	. . . . . . . . 9 |- ( ( ( ph /\ A =/= 0 ) /\ n e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) ) -> n =/= 0 )
115		simplr 792	. . . . . . . . 9 |- ( ( ( ph /\ A =/= 0 ) /\ n e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) ) -> A =/= 0 )
116	111, 112, 114, 115	divne0d 10817	. . . . . . . 8 |- ( ( ( ph /\ A =/= 0 ) /\ n e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) ) -> ( n / A ) =/= 0 )
117		eldifsn 4317	. . . . . . . 8 |- ( ( n / A ) e. ( ran F \ { 0 } ) <-> ( ( n / A ) e. ran F /\ ( n / A ) =/= 0 ) )
118	109, 116, 117	sylanbrc 698	. . . . . . 7 |- ( ( ( ph /\ A =/= 0 ) /\ n e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) ) -> ( n / A ) e. ( ran F \ { 0 } ) )
119		eldifi 3732	. . . . . . . . 9 |- ( k e. ( ran F \ { 0 } ) -> k e. ran F )
120		fnfvelrn 6356	. . . . . . . . . . . . . 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 488	. . . . . . . . . . . . 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 2702	. . . . . . . . . . . 12 |- ( ( ( ph /\ A =/= 0 ) /\ y e. RR ) -> ( A x. ( F ` y ) ) e. ran ( ( RR X. { A } ) oF x. F ) )
123	122	ralrimiva 2966	. . . . . . . . . . 11 |- ( ( ph /\ A =/= 0 ) -> A. y e. RR ( A x. ( F ` y ) ) e. ran ( ( RR X. { A } ) oF x. F ) )
124		oveq2 6658	. . . . . . . . . . . . . 14 |- ( k = ( F ` y ) -> ( A x. k ) = ( A x. ( F ` y ) ) )
125	124	eleq1d 2686	. . . . . . . . . . . . 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 6362	. . . . . . . . . . . 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 17	. . . . . . . . . . 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 247	. . . . . . . . . 10 |- ( ( ph /\ A =/= 0 ) -> A. k e. ran F ( A x. k ) e. ran ( ( RR X. { A } ) oF x. F ) )
129	128	r19.21bi 2932	. . . . . . . . 9 |- ( ( ( ph /\ A =/= 0 ) /\ k e. ran F ) -> ( A x. k ) e. ran ( ( RR X. { A } ) oF x. F ) )
130	119, 129	sylan2 491	. . . . . . . 8 |- ( ( ( ph /\ A =/= 0 ) /\ k e. ( ran F \ { 0 } ) ) -> ( A x. k ) e. ran ( ( RR X. { A } ) oF x. F ) )
131	37	adantr 481	. . . . . . . . 9 |- ( ( ( ph /\ A =/= 0 ) /\ k e. ( ran F \ { 0 } ) ) -> A e. CC )
132		frn 6053	. . . . . . . . . . . . 13 |- ( F : RR --> RR -> ran F C_ RR )
133	89, 132	syl 17	. . . . . . . . . . . 12 |- ( ( ph /\ A =/= 0 ) -> ran F C_ RR )
134	133	ssdifssd 3748	. . . . . . . . . . 11 |- ( ( ph /\ A =/= 0 ) -> ( ran F \ { 0 } ) C_ RR )
135	134	sselda 3603	. . . . . . . . . 10 |- ( ( ( ph /\ A =/= 0 ) /\ k e. ( ran F \ { 0 } ) ) -> k e. RR )
136	135	recnd 10068	. . . . . . . . 9 |- ( ( ( ph /\ A =/= 0 ) /\ k e. ( ran F \ { 0 } ) ) -> k e. CC )
137		simplr 792	. . . . . . . . 9 |- ( ( ( ph /\ A =/= 0 ) /\ k e. ( ran F \ { 0 } ) ) -> A =/= 0 )
138		eldifsni 4320	. . . . . . . . . 10 |- ( k e. ( ran F \ { 0 } ) -> k =/= 0 )
139	138	adantl 482	. . . . . . . . 9 |- ( ( ( ph /\ A =/= 0 ) /\ k e. ( ran F \ { 0 } ) ) -> k =/= 0 )
140	131, 136, 137, 139	mulne0d 10679	. . . . . . . 8 |- ( ( ( ph /\ A =/= 0 ) /\ k e. ( ran F \ { 0 } ) ) -> ( A x. k ) =/= 0 )
141		eldifsn 4317	. . . . . . . 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 698	. . . . . . 7 |- ( ( ( ph /\ A =/= 0 ) /\ k e. ( ran F \ { 0 } ) ) -> ( A x. k ) e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) )
143		simpl 473	. . . . . . . . . . . 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 3598	. . . . . . . . . . . 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 494	. . . . . . . . . . 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 10068	. . . . . . . . . 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 762	. . . . . . . . . . 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 10068	. . . . . . . . . 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 752	. . . . . . . . . . 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 10068	. . . . . . . . . 10 |- ( ( ( ph /\ A =/= 0 ) /\ ( n e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) /\ k e. ( ran F \ { 0 } ) ) ) -> k e. CC )
151		simplr 792	. . . . . . . . . 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 10823	. . . . . . . . 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 213	. . . . . . . 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 2629	. . . . . . . 8 |- ( n = ( A x. k ) <-> ( A x. k ) = n )
155		eqcom 2629	. . . . . . . 8 |- ( k = ( n / A ) <-> ( n / A ) = k )
156	153, 154, 155	3bitr4g 303	. . . . . . 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 6887	. . . . . 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 6657	. . . . . . . 8 |- ( n = m -> ( n / A ) = ( m / A ) )
159		ovex 6678	. . . . . . . 8 |- ( m / A ) e. _V
160	158, 80, 159	fvmpt 6282	. . . . . . 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 482	. . . . . 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 23447	. . . . . . . . 9 |- ( ( F e. dom S.1 /\ k e. ( ran F \ { 0 } ) ) -> ( vol ` ( `' F " { k } ) ) e. RR )
163	72, 162	sylan 488	. . . . . . . 8 |- ( ( ( ph /\ A =/= 0 ) /\ k e. ( ran F \ { 0 } ) ) -> ( vol ` ( `' F " { k } ) ) e. RR )
164	135, 163	remulcld 10070	. . . . . . 7 |- ( ( ( ph /\ A =/= 0 ) /\ k e. ( ran F \ { 0 } ) ) -> ( k x. ( vol ` ( `' F " { k } ) ) ) e. RR )
165	164	recnd 10068	. . . . . 6 |- ( ( ( ph /\ A =/= 0 ) /\ k e. ( ran F \ { 0 } ) ) -> ( k x. ( vol ` ( `' F " { k } ) ) ) e. CC )
166	79, 66, 157, 161, 165	fsumf1o 14454	. . . . 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 2656	. . . 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 6666	. . 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 2666	. 2 |- ( ( ph /\ A =/= 0 ) -> ( S.1 ` ( ( RR X. { A } ) oF x. F ) ) = ( A x. ( S.1 ` F ) ) )
170	26, 169	pm2.61dane 2881	1 |- ( ph -> ( S.1 ` ( ( RR X. { A } ) oF x. F ) ) = ( A x. ( S.1 ` F ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   _Vcvv 3200   \ cdif 3571   C_ wss 3574   {csn 4177   |-> cmpt 4729   X. cxp 5112   `'ccnv 5113   dom cdm 5114   ran crn 5115   "cima 5117   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   oFcof 6895   Fincfn 7955   CCcc 9934   RRcr 9935   0cc0 9936   x. cmul 9941   / cdiv 10684   sum_csu 14416   volcvol 23232   S.1citg1 23384

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

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-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-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  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-n0 11293  df-z 11378  df-uz 11688  df-q 11789  df-rp 11833  df-xadd 11947  df-ioo 12179  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-fl 12593  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-sum 14417  df-xmet 19739  df-met 19740  df-ovol 23233  df-vol 23234  df-mbf 23388  df-itg1 23389

This theorem is referenced by:  itg1sub  23476  itg2const  23507  itg2mulclem  23513  itg2monolem1  23517  itg2addnclem  33461