Theorem itg2mulc 22590

Description: The integral of a nonnegative constant times a function is the constant times the integral of the original function. (Contributed by Mario Carneiro, 28-Jun-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)

Hypotheses

Ref	Expression
itg2mulc.2	|- ( ph -> F : RR --> ( 0 [,) +oo ) )
itg2mulc.3	|- ( ph -> ( S.2 ` F ) e. RR )
itg2mulc.4	|- ( ph -> A e. ( 0 [,) +oo ) )

Assertion

Ref	Expression
itg2mulc	|- ( ph -> ( S.2 ` ( ( RR X. { A } ) oF x. F ) ) = ( A x. ( S.2 ` F ) ) )

Proof of Theorem itg2mulc

Dummy variables x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		itg2mulc.2	. . . . 5 |- ( ph -> F : RR --> ( 0 [,) +oo ) )
2	1	adantr 466	. . . 4 |- ( ( ph /\ 0 < A ) -> F : RR --> ( 0 [,) +oo ) )
3		itg2mulc.3	. . . . 5 |- ( ph -> ( S.2 ` F ) e. RR )
4	3	adantr 466	. . . 4 |- ( ( ph /\ 0 < A ) -> ( S.2 ` F ) e. RR )
5		itg2mulc.4	. . . . . . . 8 |- ( ph -> A e. ( 0 [,) +oo ) )
6		elrege0 11737	. . . . . . . 8 |- ( A e. ( 0 [,) +oo ) <-> ( A e. RR /\ 0 <_ A ) )
7	5, 6	sylib 199	. . . . . . 7 |- ( ph -> ( A e. RR /\ 0 <_ A ) )
8	7	simpld 460	. . . . . 6 |- ( ph -> A e. RR )
9	8	anim1i 570	. . . . 5 |- ( ( ph /\ 0 < A ) -> ( A e. RR /\ 0 < A ) )
10		elrp 11304	. . . . 5 |- ( A e. RR+ <-> ( A e. RR /\ 0 < A ) )
11	9, 10	sylibr 215	. . . 4 |- ( ( ph /\ 0 < A ) -> A e. RR+ )
12	2, 4, 11	itg2mulclem 22589	. . 3 |- ( ( ph /\ 0 < A ) -> ( S.2 ` ( ( RR X. { A } ) oF x. F ) ) <_ ( A x. ( S.2 ` F ) ) )
13		ge0mulcl 11743	. . . . . . . . 9 |- ( ( x e. ( 0 [,) +oo ) /\ y e. ( 0 [,) +oo ) ) -> ( x x. y ) e. ( 0 [,) +oo ) )
14	13	adantl 467	. . . . . . . 8 |- ( ( ph /\ ( x e. ( 0 [,) +oo ) /\ y e. ( 0 [,) +oo ) ) ) -> ( x x. y ) e. ( 0 [,) +oo ) )
15		fconst6g 5789	. . . . . . . . 9 |- ( A e. ( 0 [,) +oo ) -> ( RR X. { A } ) : RR --> ( 0 [,) +oo ) )
16	5, 15	syl 17	. . . . . . . 8 |- ( ph -> ( RR X. { A } ) : RR --> ( 0 [,) +oo ) )
17		reex 9629	. . . . . . . . 9 |- RR e. _V
18	17	a1i 11	. . . . . . . 8 |- ( ph -> RR e. _V )
19		inidm 3677	. . . . . . . 8 |- ( RR i^i RR ) = RR
20	14, 16, 1, 18, 18, 19	off 6560	. . . . . . 7 |- ( ph -> ( ( RR X. { A } ) oF x. F ) : RR --> ( 0 [,) +oo ) )
21	20	adantr 466	. . . . . 6 |- ( ( ph /\ 0 < A ) -> ( ( RR X. { A } ) oF x. F ) : RR --> ( 0 [,) +oo ) )
22		icossicc 11721	. . . . . . . . 9 |- ( 0 [,) +oo ) C_ ( 0 [,] +oo )
23		fss 5754	. . . . . . . . 9 |- ( ( ( ( RR X. { A } ) oF x. F ) : RR --> ( 0 [,) +oo ) /\ ( 0 [,) +oo ) C_ ( 0 [,] +oo ) ) -> ( ( RR X. { A } ) oF x. F ) : RR --> ( 0 [,] +oo ) )
24	20, 22, 23	sylancl 666	. . . . . . . 8 |- ( ph -> ( ( RR X. { A } ) oF x. F ) : RR --> ( 0 [,] +oo ) )
25	24	adantr 466	. . . . . . 7 |- ( ( ph /\ 0 < A ) -> ( ( RR X. { A } ) oF x. F ) : RR --> ( 0 [,] +oo ) )
26	8, 3	remulcld 9670	. . . . . . . 8 |- ( ph -> ( A x. ( S.2 ` F ) ) e. RR )
27	26	adantr 466	. . . . . . 7 |- ( ( ph /\ 0 < A ) -> ( A x. ( S.2 ` F ) ) e. RR )
28		itg2lecl 22581	. . . . . . 7 |- ( ( ( ( RR X. { A } ) oF x. F ) : RR --> ( 0 [,] +oo ) /\ ( A x. ( S.2 ` F ) ) e. RR /\ ( S.2 ` ( ( RR X. { A } ) oF x. F ) ) <_ ( A x. ( S.2 ` F ) ) ) -> ( S.2 ` ( ( RR X. { A } ) oF x. F ) ) e. RR )
29	25, 27, 12, 28	syl3anc 1264	. . . . . 6 |- ( ( ph /\ 0 < A ) -> ( S.2 ` ( ( RR X. { A } ) oF x. F ) ) e. RR )
30	11	rpreccld 11351	. . . . . 6 |- ( ( ph /\ 0 < A ) -> ( 1 / A ) e. RR+ )
31	21, 29, 30	itg2mulclem 22589	. . . . 5 |- ( ( ph /\ 0 < A ) -> ( S.2 ` ( ( RR X. { ( 1 / A ) } ) oF x. ( ( RR X. { A } ) oF x. F ) ) ) <_ ( ( 1 / A ) x. ( S.2 ` ( ( RR X. { A } ) oF x. F ) ) ) )
32	2	feqmptd 5934	. . . . . . . 8 |- ( ( ph /\ 0 < A ) -> F = ( y e. RR |-> ( F ` y ) ) )
33		rge0ssre 11738	. . . . . . . . . . . . . 14 |- ( 0 [,) +oo ) C_ RR
34		ax-resscn 9595	. . . . . . . . . . . . . 14 |- RR C_ CC
35	33, 34	sstri 3479	. . . . . . . . . . . . 13 |- ( 0 [,) +oo ) C_ CC
36		fss 5754	. . . . . . . . . . . . 13 |- ( ( F : RR --> ( 0 [,) +oo ) /\ ( 0 [,) +oo ) C_ CC ) -> F : RR --> CC )
37	1, 35, 36	sylancl 666	. . . . . . . . . . . 12 |- ( ph -> F : RR --> CC )
38	37	adantr 466	. . . . . . . . . . 11 |- ( ( ph /\ 0 < A ) -> F : RR --> CC )
39	38	ffvelrnda 6037	. . . . . . . . . 10 |- ( ( ( ph /\ 0 < A ) /\ y e. RR ) -> ( F ` y ) e. CC )
40	39	mulid2d 9660	. . . . . . . . 9 |- ( ( ( ph /\ 0 < A ) /\ y e. RR ) -> ( 1 x. ( F ` y ) ) = ( F ` y ) )
41	40	mpteq2dva 4512	. . . . . . . 8 |- ( ( ph /\ 0 < A ) -> ( y e. RR |-> ( 1 x. ( F ` y ) ) ) = ( y e. RR |-> ( F ` y ) ) )
42	32, 41	eqtr4d 2473	. . . . . . 7 |- ( ( ph /\ 0 < A ) -> F = ( y e. RR |-> ( 1 x. ( F ` y ) ) ) )
43	17	a1i 11	. . . . . . . 8 |- ( ( ph /\ 0 < A ) -> RR e. _V )
44		1red 9657	. . . . . . . 8 |- ( ( ( ph /\ 0 < A ) /\ y e. RR ) -> 1 e. RR )
45	43, 30, 11	ofc12 6570	. . . . . . . . . 10 |- ( ( ph /\ 0 < A ) -> ( ( RR X. { ( 1 / A ) } ) oF x. ( RR X. { A } ) ) = ( RR X. { ( ( 1 / A ) x. A ) } ) )
46		fconstmpt 4898	. . . . . . . . . 10 |- ( RR X. { ( ( 1 / A ) x. A ) } ) = ( y e. RR |-> ( ( 1 / A ) x. A ) )
47	45, 46	syl6eq 2486	. . . . . . . . 9 |- ( ( ph /\ 0 < A ) -> ( ( RR X. { ( 1 / A ) } ) oF x. ( RR X. { A } ) ) = ( y e. RR |-> ( ( 1 / A ) x. A ) ) )
48	8	recnd 9668	. . . . . . . . . . . 12 |- ( ph -> A e. CC )
49	48	adantr 466	. . . . . . . . . . 11 |- ( ( ph /\ 0 < A ) -> A e. CC )
50	11	rpne0d 11346	. . . . . . . . . . 11 |- ( ( ph /\ 0 < A ) -> A =/= 0 )
51	49, 50	recid2d 10378	. . . . . . . . . 10 |- ( ( ph /\ 0 < A ) -> ( ( 1 / A ) x. A ) = 1 )
52	51	mpteq2dv 4513	. . . . . . . . 9 |- ( ( ph /\ 0 < A ) -> ( y e. RR |-> ( ( 1 / A ) x. A ) ) = ( y e. RR |-> 1 ) )
53	47, 52	eqtrd 2470	. . . . . . . 8 |- ( ( ph /\ 0 < A ) -> ( ( RR X. { ( 1 / A ) } ) oF x. ( RR X. { A } ) ) = ( y e. RR |-> 1 ) )
54	43, 44, 39, 53, 32	offval2 6562	. . . . . . 7 |- ( ( ph /\ 0 < A ) -> ( ( ( RR X. { ( 1 / A ) } ) oF x. ( RR X. { A } ) ) oF x. F ) = ( y e. RR |-> ( 1 x. ( F ` y ) ) ) )
55	30	rpcnd 11343	. . . . . . . . 9 |- ( ( ph /\ 0 < A ) -> ( 1 / A ) e. CC )
56		fconst6g 5789	. . . . . . . . 9 |- ( ( 1 / A ) e. CC -> ( RR X. { ( 1 / A ) } ) : RR --> CC )
57	55, 56	syl 17	. . . . . . . 8 |- ( ( ph /\ 0 < A ) -> ( RR X. { ( 1 / A ) } ) : RR --> CC )
58		fconst6g 5789	. . . . . . . . 9 |- ( A e. CC -> ( RR X. { A } ) : RR --> CC )
59	49, 58	syl 17	. . . . . . . 8 |- ( ( ph /\ 0 < A ) -> ( RR X. { A } ) : RR --> CC )
60		mulass 9626	. . . . . . . . 9 |- ( ( x e. CC /\ y e. CC /\ z e. CC ) -> ( ( x x. y ) x. z ) = ( x x. ( y x. z ) ) )
61	60	adantl 467	. . . . . . . 8 |- ( ( ( ph /\ 0 < A ) /\ ( x e. CC /\ y e. CC /\ z e. CC ) ) -> ( ( x x. y ) x. z ) = ( x x. ( y x. z ) ) )
62	43, 57, 59, 38, 61	caofass 6579	. . . . . . 7 |- ( ( ph /\ 0 < A ) -> ( ( ( RR X. { ( 1 / A ) } ) oF x. ( RR X. { A } ) ) oF x. F ) = ( ( RR X. { ( 1 / A ) } ) oF x. ( ( RR X. { A } ) oF x. F ) ) )
63	42, 54, 62	3eqtr2d 2476	. . . . . 6 |- ( ( ph /\ 0 < A ) -> F = ( ( RR X. { ( 1 / A ) } ) oF x. ( ( RR X. { A } ) oF x. F ) ) )
64	63	fveq2d 5885	. . . . 5 |- ( ( ph /\ 0 < A ) -> ( S.2 ` F ) = ( S.2 ` ( ( RR X. { ( 1 / A ) } ) oF x. ( ( RR X. { A } ) oF x. F ) ) ) )
65	29	recnd 9668	. . . . . 6 |- ( ( ph /\ 0 < A ) -> ( S.2 ` ( ( RR X. { A } ) oF x. F ) ) e. CC )
66	65, 49, 50	divrec2d 10386	. . . . 5 |- ( ( ph /\ 0 < A ) -> ( ( S.2 ` ( ( RR X. { A } ) oF x. F ) ) / A ) = ( ( 1 / A ) x. ( S.2 ` ( ( RR X. { A } ) oF x. F ) ) ) )
67	31, 64, 66	3brtr4d 4456	. . . 4 |- ( ( ph /\ 0 < A ) -> ( S.2 ` F ) <_ ( ( S.2 ` ( ( RR X. { A } ) oF x. F ) ) / A ) )
68	4, 29, 11	lemuldiv2d 11388	. . . 4 |- ( ( ph /\ 0 < A ) -> ( ( A x. ( S.2 ` F ) ) <_ ( S.2 ` ( ( RR X. { A } ) oF x. F ) ) <-> ( S.2 ` F ) <_ ( ( S.2 ` ( ( RR X. { A } ) oF x. F ) ) / A ) ) )
69	67, 68	mpbird 235	. . 3 |- ( ( ph /\ 0 < A ) -> ( A x. ( S.2 ` F ) ) <_ ( S.2 ` ( ( RR X. { A } ) oF x. F ) ) )
70		itg2cl 22575	. . . . . 6 |- ( ( ( RR X. { A } ) oF x. F ) : RR --> ( 0 [,] +oo ) -> ( S.2 ` ( ( RR X. { A } ) oF x. F ) ) e. RR* )
71	24, 70	syl 17	. . . . 5 |- ( ph -> ( S.2 ` ( ( RR X. { A } ) oF x. F ) ) e. RR* )
72	26	rexrd 9689	. . . . 5 |- ( ph -> ( A x. ( S.2 ` F ) ) e. RR* )
73		xrletri3 11451	. . . . 5 |- ( ( ( S.2 ` ( ( RR X. { A } ) oF x. F ) ) e. RR* /\ ( A x. ( S.2 ` F ) ) e. RR* ) -> ( ( S.2 ` ( ( RR X. { A } ) oF x. F ) ) = ( A x. ( S.2 ` F ) ) <-> ( ( S.2 ` ( ( RR X. { A } ) oF x. F ) ) <_ ( A x. ( S.2 ` F ) ) /\ ( A x. ( S.2 ` F ) ) <_ ( S.2 ` ( ( RR X. { A } ) oF x. F ) ) ) ) )
74	71, 72, 73	syl2anc 665	. . . 4 |- ( ph -> ( ( S.2 ` ( ( RR X. { A } ) oF x. F ) ) = ( A x. ( S.2 ` F ) ) <-> ( ( S.2 ` ( ( RR X. { A } ) oF x. F ) ) <_ ( A x. ( S.2 ` F ) ) /\ ( A x. ( S.2 ` F ) ) <_ ( S.2 ` ( ( RR X. { A } ) oF x. F ) ) ) ) )
75	74	adantr 466	. . 3 |- ( ( ph /\ 0 < A ) -> ( ( S.2 ` ( ( RR X. { A } ) oF x. F ) ) = ( A x. ( S.2 ` F ) ) <-> ( ( S.2 ` ( ( RR X. { A } ) oF x. F ) ) <_ ( A x. ( S.2 ` F ) ) /\ ( A x. ( S.2 ` F ) ) <_ ( S.2 ` ( ( RR X. { A } ) oF x. F ) ) ) ) )
76	12, 69, 75	mpbir2and 930	. 2 |- ( ( ph /\ 0 < A ) -> ( S.2 ` ( ( RR X. { A } ) oF x. F ) ) = ( A x. ( S.2 ` F ) ) )
77	17	a1i 11	. . . . . 6 |- ( ( ph /\ 0 = A ) -> RR e. _V )
78	37	adantr 466	. . . . . 6 |- ( ( ph /\ 0 = A ) -> F : RR --> CC )
79	8	adantr 466	. . . . . 6 |- ( ( ph /\ 0 = A ) -> A e. RR )
80		0re 9642	. . . . . . 7 |- 0 e. RR
81	80	a1i 11	. . . . . 6 |- ( ( ph /\ 0 = A ) -> 0 e. RR )
82		simplr 760	. . . . . . . 8 |- ( ( ( ph /\ 0 = A ) /\ x e. CC ) -> 0 = A )
83	82	oveq1d 6320	. . . . . . 7 |- ( ( ( ph /\ 0 = A ) /\ x e. CC ) -> ( 0 x. x ) = ( A x. x ) )
84		mul02 9810	. . . . . . . 8 |- ( x e. CC -> ( 0 x. x ) = 0 )
85	84	adantl 467	. . . . . . 7 |- ( ( ( ph /\ 0 = A ) /\ x e. CC ) -> ( 0 x. x ) = 0 )
86	83, 85	eqtr3d 2472	. . . . . 6 |- ( ( ( ph /\ 0 = A ) /\ x e. CC ) -> ( A x. x ) = 0 )
87	77, 78, 79, 81, 86	caofid2 6576	. . . . 5 |- ( ( ph /\ 0 = A ) -> ( ( RR X. { A } ) oF x. F ) = ( RR X. { 0 } ) )
88	87	fveq2d 5885	. . . 4 |- ( ( ph /\ 0 = A ) -> ( S.2 ` ( ( RR X. { A } ) oF x. F ) ) = ( S.2 ` ( RR X. { 0 } ) ) )
89		itg20 22580	. . . 4 |- ( S.2 ` ( RR X. { 0 } ) ) = 0
90	88, 89	syl6eq 2486	. . 3 |- ( ( ph /\ 0 = A ) -> ( S.2 ` ( ( RR X. { A } ) oF x. F ) ) = 0 )
91	3	adantr 466	. . . . 5 |- ( ( ph /\ 0 = A ) -> ( S.2 ` F ) e. RR )
92	91	recnd 9668	. . . 4 |- ( ( ph /\ 0 = A ) -> ( S.2 ` F ) e. CC )
93	92	mul02d 9830	. . 3 |- ( ( ph /\ 0 = A ) -> ( 0 x. ( S.2 ` F ) ) = 0 )
94		simpr 462	. . . 4 |- ( ( ph /\ 0 = A ) -> 0 = A )
95	94	oveq1d 6320	. . 3 |- ( ( ph /\ 0 = A ) -> ( 0 x. ( S.2 ` F ) ) = ( A x. ( S.2 ` F ) ) )
96	90, 93, 95	3eqtr2d 2476	. 2 |- ( ( ph /\ 0 = A ) -> ( S.2 ` ( ( RR X. { A } ) oF x. F ) ) = ( A x. ( S.2 ` F ) ) )
97	7	simprd 464	. . 3 |- ( ph -> 0 <_ A )
98		leloe 9719	. . . 4 |- ( ( 0 e. RR /\ A e. RR ) -> ( 0 <_ A <-> ( 0 < A \/ 0 = A ) ) )
99	80, 8, 98	sylancr 667	. . 3 |- ( ph -> ( 0 <_ A <-> ( 0 < A \/ 0 = A ) ) )
100	97, 99	mpbid 213	. 2 |- ( ph -> ( 0 < A \/ 0 = A ) )
101	76, 96, 100	mpjaodan 793	1 |- ( ph -> ( S.2 ` ( ( RR X. { A } ) oF x. F ) ) = ( A x. ( S.2 ` F ) ) )

Syntax hints:   -> wi 4   <-> wb 187   \/ wo 369   /\ wa 370   /\ w3a 982   = wceq 1437   e. wcel 1870   _Vcvv 3087   C_ wss 3442   {csn 4002   class class class wbr 4426   |-> cmpt 4484   X. cxp 4852   -->wf 5597   ` cfv 5601  (class class class)co 6305   oFcof 6543   CCcc 9536   RRcr 9537   0cc0 9538   1c1 9539   x. cmul 9543   +oocpnf 9671   RR*cxr 9673   < clt 9674   <_ cle 9675   / cdiv 10268   RR+crp 11302   [,)cico 11637   [,]cicc 11638   S.2citg2 22459

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-inf2 8146  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615  ax-pre-sup 9616  ax-addf 9617

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-disj 4398  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-se 4814  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-isom 5610  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-of 6545  df-ofr 6546  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-2o 7191  df-oadd 7194  df-er 7371  df-map 7482  df-pm 7483  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-sup 7962  df-inf 7963  df-oi 8025  df-card 8372  df-cda 8596  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-div 10269  df-nn 10610  df-2 10668  df-3 10669  df-n0 10870  df-z 10938  df-uz 11160  df-q 11265  df-rp 11303  df-xadd 11410  df-ioo 11639  df-ico 11641  df-icc 11642  df-fz 11783  df-fzo 11914  df-fl 12025  df-seq 12211  df-exp 12270  df-hash 12513  df-cj 13141  df-re 13142  df-im 13143  df-sqrt 13277  df-abs 13278  df-clim 13530  df-sum 13731  df-xmet 18902  df-met 18903  df-ovol 22301  df-vol 22303  df-mbf 22462  df-itg1 22463  df-itg2 22464  df-0p 22513

This theorem is referenced by:  iblmulc2  22673  itgmulc2lem1  22674  bddmulibl  22681  iblmulc2nc  31722  itgmulc2nclem1  31723