Theorem itg1mulc 21862

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 21846	. . 3 |- ( S.1 ` ( RR X. { 0 } ) ) = 0
2		reex 9582	. . . . . 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 21834	. . . . . . 7 |- ( F e. dom S.1 -> F : RR --> RR )
6	4, 5	syl 16	. . . . . 6 |- ( ph -> F : RR --> RR )
7	6	adantr 465	. . . . 5 |- ( ( ph /\ A = 0 ) -> F : RR --> RR )
8		i1fmulc.3	. . . . . 6 |- ( ph -> A e. RR )
9	8	adantr 465	. . . . 5 |- ( ( ph /\ A = 0 ) -> A e. RR )
10		0red 9596	. . . . 5 |- ( ( ph /\ A = 0 ) -> 0 e. RR )
11		simplr 754	. . . . . . 7 |- ( ( ( ph /\ A = 0 ) /\ x e. RR ) -> A = 0 )
12	11	oveq1d 6298	. . . . . 6 |- ( ( ( ph /\ A = 0 ) /\ x e. RR ) -> ( A x. x ) = ( 0 x. x ) )
13		mul02lem2 9755	. . . . . . 7 |- ( x e. RR -> ( 0 x. x ) = 0 )
14	13	adantl 466	. . . . . 6 |- ( ( ( ph /\ A = 0 ) /\ x e. RR ) -> ( 0 x. x ) = 0 )
15	12, 14	eqtrd 2508	. . . . 5 |- ( ( ( ph /\ A = 0 ) /\ x e. RR ) -> ( A x. x ) = 0 )
16	3, 7, 9, 10, 15	caofid2 6554	. . . 4 |- ( ( ph /\ A = 0 ) -> ( ( RR X. { A } ) oF x. F ) = ( RR X. { 0 } ) )
17	16	fveq2d 5869	. . 3 |- ( ( ph /\ A = 0 ) -> ( S.1 ` ( ( RR X. { A } ) oF x. F ) ) = ( S.1 ` ( RR X. { 0 } ) ) )
18		simpr 461	. . . . 5 |- ( ( ph /\ A = 0 ) -> A = 0 )
19	18	oveq1d 6298	. . . 4 |- ( ( ph /\ A = 0 ) -> ( A x. ( S.1 ` F ) ) = ( 0 x. ( S.1 ` F ) ) )
20		itg1cl 21843	. . . . . . . 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 9621	. . . . . 6 |- ( ph -> ( S.1 ` F ) e. CC )
23	22	mul02d 9776	. . . . 5 |- ( ph -> ( 0 x. ( S.1 ` F ) ) = 0 )
24	23	adantr 465	. . . 4 |- ( ( ph /\ A = 0 ) -> ( 0 x. ( S.1 ` F ) ) = 0 )
25	19, 24	eqtrd 2508	. . 3 |- ( ( ph /\ A = 0 ) -> ( A x. ( S.1 ` F ) ) = 0 )
26	1, 17, 25	3eqtr4a 2534	. 2 |- ( ( ph /\ A = 0 ) -> ( S.1 ` ( ( RR X. { A } ) oF x. F ) ) = ( A x. ( S.1 ` F ) ) )
27	4, 8	i1fmulc 21861	. . . . . . . . . . . . . 14 |- ( ph -> ( ( RR X. { A } ) oF x. F ) e. dom S.1 )
28	27	adantr 465	. . . . . . . . . . . . 13 |- ( ( ph /\ A =/= 0 ) -> ( ( RR X. { A } ) oF x. F ) e. dom S.1 )
29		i1ff 21834	. . . . . . . . . . . . 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 5736	. . . . . . . . . . . 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 3642	. . . . . . . . . 10 |- ( ( ph /\ A =/= 0 ) -> ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) C_ RR )
34	33	sselda 3504	. . . . . . . . 9 |- ( ( ( ph /\ A =/= 0 ) /\ m e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) ) -> m e. RR )
35	34	recnd 9621	. . . . . . . 8 |- ( ( ( ph /\ A =/= 0 ) /\ m e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) ) -> m e. CC )
36	8	adantr 465	. . . . . . . . . 10 |- ( ( ph /\ A =/= 0 ) -> A e. RR )
37	36	recnd 9621	. . . . . . . . 9 |- ( ( ph /\ A =/= 0 ) -> A e. CC )
38	37	adantr 465	. . . . . . . 8 |- ( ( ( ph /\ A =/= 0 ) /\ m e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) ) -> A e. CC )
39		simplr 754	. . . . . . . 8 |- ( ( ( ph /\ A =/= 0 ) /\ m e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) ) -> A =/= 0 )
40	35, 38, 39	divcan2d 10321	. . . . . . 7 |- ( ( ( ph /\ A =/= 0 ) /\ m e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) ) -> ( A x. ( m / A ) ) = m )
41	4, 8	i1fmulclem 21860	. . . . . . . . . 10 |- ( ( ( ph /\ A =/= 0 ) /\ m e. RR ) -> ( `' ( ( RR X. { A } ) oF x. F ) " { m } ) = ( `' F " { ( m / A ) } ) )
42	34, 41	syldan 470	. . . . . . . . 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 5869	. . . . . . . 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 2475	. . . . . . 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 6301	. . . . . 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 725	. . . . . . . . 9 |- ( ( ( ph /\ A =/= 0 ) /\ m e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) ) -> A e. RR )
47	34, 46, 39	redivcld 10371	. . . . . . . 8 |- ( ( ( ph /\ A =/= 0 ) /\ m e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) ) -> ( m / A ) e. RR )
48	47	recnd 9621	. . . . . . 7 |- ( ( ( ph /\ A =/= 0 ) /\ m e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) ) -> ( m / A ) e. CC )
49	4	ad2antrr 725	. . . . . . . . 9 |- ( ( ( ph /\ A =/= 0 ) /\ m e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) ) -> F e. dom S.1 )
50	46	recnd 9621	. . . . . . . . . . 11 |- ( ( ( ph /\ A =/= 0 ) /\ m e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) ) -> A e. CC )
51		eldifsni 4153	. . . . . . . . . . . 12 |- ( m e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) -> m =/= 0 )
52	51	adantl 466	. . . . . . . . . . 11 |- ( ( ( ph /\ A =/= 0 ) /\ m e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) ) -> m =/= 0 )
53	35, 50, 52, 39	divne0d 10335	. . . . . . . . . 10 |- ( ( ( ph /\ A =/= 0 ) /\ m e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) ) -> ( m / A ) =/= 0 )
54		eldifsn 4152	. . . . . . . . . 10 |- ( ( m / A ) e. ( RR \ { 0 } ) <-> ( ( m / A ) e. RR /\ ( m / A ) =/= 0 ) )
55	47, 53, 54	sylanbrc 664	. . . . . . . . 9 |- ( ( ( ph /\ A =/= 0 ) /\ m e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) ) -> ( m / A ) e. ( RR \ { 0 } ) )
56		i1fima2sn 21838	. . . . . . . . 9 |- ( ( F e. dom S.1 /\ ( m / A ) e. ( RR \ { 0 } ) ) -> ( vol ` ( `' F " { ( m / A ) } ) ) e. RR )
57	49, 55, 56	syl2anc 661	. . . . . . . 8 |- ( ( ( ph /\ A =/= 0 ) /\ m e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) ) -> ( vol ` ( `' F " { ( m / A ) } ) ) e. RR )
58	57	recnd 9621	. . . . . . 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 9618	. . . . . 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 2510	. . . . 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 13487	. . . 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 21835	. . . . . . 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 3631	. . . . . 6 |- ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) C_ ran ( ( RR X. { A } ) oF x. F )
65		ssfi 7740	. . . . . 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 662	. . . . 5 |- ( ( ph /\ A =/= 0 ) -> ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) e. Fin )
67	48, 58	mulcld 9615	. . . . 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 13561	. . . 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 2511	. . 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 21841	. . . 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 465	. . . . . 6 |- ( ( ph /\ A =/= 0 ) -> F e. dom S.1 )
73		itg1val 21841	. . . . . 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 4037	. . . . . . . . 9 |- ( k = ( m / A ) -> { k } = { ( m / A ) } )
77	76	imaeq2d 5336	. . . . . . . 8 |- ( k = ( m / A ) -> ( `' F " { k } ) = ( `' F " { ( m / A ) } ) )
78	77	fveq2d 5869	. . . . . . 7 |- ( k = ( m / A ) -> ( vol ` ( `' F " { k } ) ) = ( vol ` ( `' F " { ( m / A ) } ) ) )
79	75, 78	oveq12d 6301	. . . . . 6 |- ( k = ( m / A ) -> ( k x. ( vol ` ( `' F " { k } ) ) ) = ( ( m / A ) x. ( vol ` ( `' F " { ( m / A ) } ) ) ) )
80		eqid 2467	. . . . . . 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 3626	. . . . . . . . 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 5730	. . . . . . . . . . . . . . . . . 18 |- ( F : RR --> RR -> F Fn RR )
84	6, 83	syl 16	. . . . . . . . . . . . . . . . 17 |- ( ph -> F Fn RR )
85		eqidd 2468	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ y e. RR ) -> ( F ` y ) = ( F ` y ) )
86	82, 8, 84, 85	ofc1 6546	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ y e. RR ) -> ( ( ( RR X. { A } ) oF x. F ) ` y ) = ( A x. ( F ` y ) ) )
87	86	adantlr 714	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ A =/= 0 ) /\ y e. RR ) -> ( ( ( RR X. { A } ) oF x. F ) ` y ) = ( A x. ( F ` y ) ) )
88	87	oveq1d 6298	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ A =/= 0 ) /\ y e. RR ) -> ( ( ( ( RR X. { A } ) oF x. F ) ` y ) / A ) = ( ( A x. ( F ` y ) ) / A ) )
89	6	adantr 465	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ A =/= 0 ) -> F : RR --> RR )
90	89	ffvelrnda 6020	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ A =/= 0 ) /\ y e. RR ) -> ( F ` y ) e. RR )
91	90	recnd 9621	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ A =/= 0 ) /\ y e. RR ) -> ( F ` y ) e. CC )
92	37	adantr 465	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ A =/= 0 ) /\ y e. RR ) -> A e. CC )
93		simplr 754	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ A =/= 0 ) /\ y e. RR ) -> A =/= 0 )
94	91, 92, 93	divcan3d 10324	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ A =/= 0 ) /\ y e. RR ) -> ( ( A x. ( F ` y ) ) / A ) = ( F ` y ) )
95	88, 94	eqtrd 2508	. . . . . . . . . . . . 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 6017	. . . . . . . . . . . . . 14 |- ( ( F Fn RR /\ y e. RR ) -> ( F ` y ) e. ran F )
98	96, 97	sylan 471	. . . . . . . . . . . . 13 |- ( ( ( ph /\ A =/= 0 ) /\ y e. RR ) -> ( F ` y ) e. ran F )
99	95, 98	eqeltrd 2555	. . . . . . . . . . . 12 |- ( ( ( ph /\ A =/= 0 ) /\ y e. RR ) -> ( ( ( ( RR X. { A } ) oF x. F ) ` y ) / A ) e. ran F )
100	99	ralrimiva 2878	. . . . . . . . . . 11 |- ( ( ph /\ A =/= 0 ) -> A. y e. RR ( ( ( ( RR X. { A } ) oF x. F ) ` y ) / A ) e. ran F )
101		ffn 5730	. . . . . . . . . . . . 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 6290	. . . . . . . . . . . . . 14 |- ( n = ( ( ( RR X. { A } ) oF x. F ) ` y ) -> ( n / A ) = ( ( ( ( RR X. { A } ) oF x. F ) ` y ) / A ) )
104	103	eleq1d 2536	. . . . . . . . . . . . 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 6023	. . . . . . . . . . . 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 2833	. . . . . . . . 9 |- ( ( ( ph /\ A =/= 0 ) /\ n e. ran ( ( RR X. { A } ) oF x. F ) ) -> ( n / A ) e. ran F )
109	81, 108	sylan2 474	. . . . . . . 8 |- ( ( ( ph /\ A =/= 0 ) /\ n e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) ) -> ( n / A ) e. ran F )
110	33	sselda 3504	. . . . . . . . . 10 |- ( ( ( ph /\ A =/= 0 ) /\ n e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) ) -> n e. RR )
111	110	recnd 9621	. . . . . . . . 9 |- ( ( ( ph /\ A =/= 0 ) /\ n e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) ) -> n e. CC )
112	37	adantr 465	. . . . . . . . 9 |- ( ( ( ph /\ A =/= 0 ) /\ n e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) ) -> A e. CC )
113		eldifsni 4153	. . . . . . . . . 10 |- ( n e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) -> n =/= 0 )
114	113	adantl 466	. . . . . . . . 9 |- ( ( ( ph /\ A =/= 0 ) /\ n e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) ) -> n =/= 0 )
115		simplr 754	. . . . . . . . 9 |- ( ( ( ph /\ A =/= 0 ) /\ n e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) ) -> A =/= 0 )
116	111, 112, 114, 115	divne0d 10335	. . . . . . . 8 |- ( ( ( ph /\ A =/= 0 ) /\ n e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) ) -> ( n / A ) =/= 0 )
117		eldifsn 4152	. . . . . . . 8 |- ( ( n / A ) e. ( ran F \ { 0 } ) <-> ( ( n / A ) e. ran F /\ ( n / A ) =/= 0 ) )
118	109, 116, 117	sylanbrc 664	. . . . . . 7 |- ( ( ( ph /\ A =/= 0 ) /\ n e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) ) -> ( n / A ) e. ( ran F \ { 0 } ) )
119		eldifi 3626	. . . . . . . . 9 |- ( k e. ( ran F \ { 0 } ) -> k e. ran F )
120		fnfvelrn 6017	. . . . . . . . . . . . . 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 471	. . . . . . . . . . . . 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 2556	. . . . . . . . . . . 12 |- ( ( ( ph /\ A =/= 0 ) /\ y e. RR ) -> ( A x. ( F ` y ) ) e. ran ( ( RR X. { A } ) oF x. F ) )
123	122	ralrimiva 2878	. . . . . . . . . . 11 |- ( ( ph /\ A =/= 0 ) -> A. y e. RR ( A x. ( F ` y ) ) e. ran ( ( RR X. { A } ) oF x. F ) )
124		oveq2 6291	. . . . . . . . . . . . . 14 |- ( k = ( F ` y ) -> ( A x. k ) = ( A x. ( F ` y ) ) )
125	124	eleq1d 2536	. . . . . . . . . . . . 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 6023	. . . . . . . . . . . 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 2833	. . . . . . . . 9 |- ( ( ( ph /\ A =/= 0 ) /\ k e. ran F ) -> ( A x. k ) e. ran ( ( RR X. { A } ) oF x. F ) )
130	119, 129	sylan2 474	. . . . . . . 8 |- ( ( ( ph /\ A =/= 0 ) /\ k e. ( ran F \ { 0 } ) ) -> ( A x. k ) e. ran ( ( RR X. { A } ) oF x. F ) )
131	37	adantr 465	. . . . . . . . 9 |- ( ( ( ph /\ A =/= 0 ) /\ k e. ( ran F \ { 0 } ) ) -> A e. CC )
132		frn 5736	. . . . . . . . . . . . 13 |- ( F : RR --> RR -> ran F C_ RR )
133	89, 132	syl 16	. . . . . . . . . . . 12 |- ( ( ph /\ A =/= 0 ) -> ran F C_ RR )
134	133	ssdifssd 3642	. . . . . . . . . . 11 |- ( ( ph /\ A =/= 0 ) -> ( ran F \ { 0 } ) C_ RR )
135	134	sselda 3504	. . . . . . . . . 10 |- ( ( ( ph /\ A =/= 0 ) /\ k e. ( ran F \ { 0 } ) ) -> k e. RR )
136	135	recnd 9621	. . . . . . . . 9 |- ( ( ( ph /\ A =/= 0 ) /\ k e. ( ran F \ { 0 } ) ) -> k e. CC )
137		simplr 754	. . . . . . . . 9 |- ( ( ( ph /\ A =/= 0 ) /\ k e. ( ran F \ { 0 } ) ) -> A =/= 0 )
138		eldifsni 4153	. . . . . . . . . 10 |- ( k e. ( ran F \ { 0 } ) -> k =/= 0 )
139	138	adantl 466	. . . . . . . . 9 |- ( ( ( ph /\ A =/= 0 ) /\ k e. ( ran F \ { 0 } ) ) -> k =/= 0 )
140	131, 136, 137, 139	mulne0d 10200	. . . . . . . 8 |- ( ( ( ph /\ A =/= 0 ) /\ k e. ( ran F \ { 0 } ) ) -> ( A x. k ) =/= 0 )
141		eldifsn 4152	. . . . . . . 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 664	. . . . . . 7 |- ( ( ( ph /\ A =/= 0 ) /\ k e. ( ran F \ { 0 } ) ) -> ( A x. k ) e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) )
143		simpl 457	. . . . . . . . . . . 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 3499	. . . . . . . . . . . 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 477	. . . . . . . . . . 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 9621	. . . . . . . . . 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 725	. . . . . . . . . . 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 9621	. . . . . . . . . 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 715	. . . . . . . . . . 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 9621	. . . . . . . . . 10 |- ( ( ( ph /\ A =/= 0 ) /\ ( n e. ( ran ( ( RR X. { A } ) oF x. F ) \ { 0 } ) /\ k e. ( ran F \ { 0 } ) ) ) -> k e. CC )
151		simplr 754	. . . . . . . . . 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 10341	. . . . . . . . 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 2476	. . . . . . . 8 |- ( n = ( A x. k ) <-> ( A x. k ) = n )
155		eqcom 2476	. . . . . . . 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 6510	. . . . . 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 6290	. . . . . . . 8 |- ( n = m -> ( n / A ) = ( m / A ) )
159		ovex 6308	. . . . . . . 8 |- ( m / A ) e. _V
160	158, 80, 159	fvmpt 5949	. . . . . . 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 466	. . . . . 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 21838	. . . . . . . . 9 |- ( ( F e. dom S.1 /\ k e. ( ran F \ { 0 } ) ) -> ( vol ` ( `' F " { k } ) ) e. RR )
163	72, 162	sylan 471	. . . . . . . 8 |- ( ( ( ph /\ A =/= 0 ) /\ k e. ( ran F \ { 0 } ) ) -> ( vol ` ( `' F " { k } ) ) e. RR )
164	135, 163	remulcld 9623	. . . . . . 7 |- ( ( ( ph /\ A =/= 0 ) /\ k e. ( ran F \ { 0 } ) ) -> ( k x. ( vol ` ( `' F " { k } ) ) ) e. RR )
165	164	recnd 9621	. . . . . 6 |- ( ( ( ph /\ A =/= 0 ) /\ k e. ( ran F \ { 0 } ) ) -> ( k x. ( vol ` ( `' F " { k } ) ) ) e. CC )
166	79, 66, 157, 161, 165	fsumf1o 13507	. . . . 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 2508	. . . 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 6299	. . 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 2518	. 2 |- ( ( ph /\ A =/= 0 ) -> ( S.1 ` ( ( RR X. { A } ) oF x. F ) ) = ( A x. ( S.1 ` F ) ) )
170	26, 169	pm2.61dane 2785	1 |- ( ph -> ( S.1 ` ( ( RR X. { A } ) oF x. F ) ) = ( A x. ( S.1 ` F ) ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1379   e. wcel 1767   =/= wne 2662   A.wral 2814   _Vcvv 3113   \ cdif 3473   C_ wss 3476   {csn 4027   |-> cmpt 4505   X. cxp 4997   `'ccnv 4998   dom cdm 4999   ran crn 5000   "cima 5002   Fn wfn 5582   -->wf 5583   ` cfv 5587  (class class class)co 6283   oFcof 6521   Fincfn 7516   CCcc 9489   RRcr 9490   0cc0 9491   x. cmul 9496   / cdiv 10205   sum_csu 13470   volcvol 21626   S.1citg1 21775

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6575  ax-inf2 8057  ax-cnex 9547  ax-resscn 9548  ax-1cn 9549  ax-icn 9550  ax-addcl 9551  ax-addrcl 9552  ax-mulcl 9553  ax-mulrcl 9554  ax-mulcom 9555  ax-addass 9556  ax-mulass 9557  ax-distr 9558  ax-i2m1 9559  ax-1ne0 9560  ax-1rid 9561  ax-rnegex 9562  ax-rrecex 9563  ax-cnre 9564  ax-pre-lttri 9565  ax-pre-lttrn 9566  ax-pre-ltadd 9567  ax-pre-mulgt0 9568  ax-pre-sup 9569

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5550  df-fun 5589  df-fn 5590  df-f 5591  df-f1 5592  df-fo 5593  df-f1o 5594  df-fv 5595  df-isom 5596  df-riota 6244  df-ov 6286  df-oprab 6287  df-mpt2 6288  df-of 6523  df-om 6680  df-1st 6784  df-2nd 6785  df-recs 7042  df-rdg 7076  df-1o 7130  df-2o 7131  df-oadd 7134  df-er 7311  df-map 7422  df-pm 7423  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-sup 7900  df-oi 7934  df-card 8319  df-cda 8547  df-pnf 9629  df-mnf 9630  df-xr 9631  df-ltxr 9632  df-le 9633  df-sub 9806  df-neg 9807  df-div 10206  df-nn 10536  df-2 10593  df-3 10594  df-n0 10795  df-z 10864  df-uz 11082  df-q 11182  df-rp 11220  df-xadd 11318  df-ioo 11532  df-ico 11534  df-icc 11535  df-fz 11672  df-fzo 11792  df-fl 11896  df-seq 12075  df-exp 12134  df-hash 12373  df-cj 12894  df-re 12895  df-im 12896  df-sqrt 13030  df-abs 13031  df-clim 13273  df-sum 13471  df-xmet 18199  df-met 18200  df-ovol 21627  df-vol 21628  df-mbf 21779  df-itg1 21780

This theorem is referenced by:  itg1sub  21867  itg2const  21898  itg2mulclem  21904  itg2monolem1  21908  itg2addnclem  29659