Theorem itg2mulc 23514

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 481	. . . 4 |- ( ( ph /\ 0 < A ) -> F : RR --> ( 0 [,) +oo ) )
3		itg2mulc.3	. . . . 5 |- ( ph -> ( S.2 ` F ) e. RR )
4	3	adantr 481	. . . 4 |- ( ( ph /\ 0 < A ) -> ( S.2 ` F ) e. RR )
5		itg2mulc.4	. . . . . . . 8 |- ( ph -> A e. ( 0 [,) +oo ) )
6		elrege0 12278	. . . . . . . 8 |- ( A e. ( 0 [,) +oo ) <-> ( A e. RR /\ 0 <_ A ) )
7	5, 6	sylib 208	. . . . . . 7 |- ( ph -> ( A e. RR /\ 0 <_ A ) )
8	7	simpld 475	. . . . . 6 |- ( ph -> A e. RR )
9	8	anim1i 592	. . . . 5 |- ( ( ph /\ 0 < A ) -> ( A e. RR /\ 0 < A ) )
10		elrp 11834	. . . . 5 |- ( A e. RR+ <-> ( A e. RR /\ 0 < A ) )
11	9, 10	sylibr 224	. . . 4 |- ( ( ph /\ 0 < A ) -> A e. RR+ )
12	2, 4, 11	itg2mulclem 23513	. . 3 |- ( ( ph /\ 0 < A ) -> ( S.2 ` ( ( RR X. { A } ) oF x. F ) ) <_ ( A x. ( S.2 ` F ) ) )
13		ge0mulcl 12285	. . . . . . . . 9 |- ( ( x e. ( 0 [,) +oo ) /\ y e. ( 0 [,) +oo ) ) -> ( x x. y ) e. ( 0 [,) +oo ) )
14	13	adantl 482	. . . . . . . 8 |- ( ( ph /\ ( x e. ( 0 [,) +oo ) /\ y e. ( 0 [,) +oo ) ) ) -> ( x x. y ) e. ( 0 [,) +oo ) )
15		fconst6g 6094	. . . . . . . . 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 10027	. . . . . . . . 9 |- RR e. _V
18	17	a1i 11	. . . . . . . 8 |- ( ph -> RR e. _V )
19		inidm 3822	. . . . . . . 8 |- ( RR i^i RR ) = RR
20	14, 16, 1, 18, 18, 19	off 6912	. . . . . . 7 |- ( ph -> ( ( RR X. { A } ) oF x. F ) : RR --> ( 0 [,) +oo ) )
21	20	adantr 481	. . . . . 6 |- ( ( ph /\ 0 < A ) -> ( ( RR X. { A } ) oF x. F ) : RR --> ( 0 [,) +oo ) )
22		icossicc 12260	. . . . . . . . 9 |- ( 0 [,) +oo ) C_ ( 0 [,] +oo )
23		fss 6056	. . . . . . . . 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 694	. . . . . . . 8 |- ( ph -> ( ( RR X. { A } ) oF x. F ) : RR --> ( 0 [,] +oo ) )
25	24	adantr 481	. . . . . . 7 |- ( ( ph /\ 0 < A ) -> ( ( RR X. { A } ) oF x. F ) : RR --> ( 0 [,] +oo ) )
26	8, 3	remulcld 10070	. . . . . . . 8 |- ( ph -> ( A x. ( S.2 ` F ) ) e. RR )
27	26	adantr 481	. . . . . . 7 |- ( ( ph /\ 0 < A ) -> ( A x. ( S.2 ` F ) ) e. RR )
28		itg2lecl 23505	. . . . . . 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 1326	. . . . . 6 |- ( ( ph /\ 0 < A ) -> ( S.2 ` ( ( RR X. { A } ) oF x. F ) ) e. RR )
30	11	rpreccld 11882	. . . . . 6 |- ( ( ph /\ 0 < A ) -> ( 1 / A ) e. RR+ )
31	21, 29, 30	itg2mulclem 23513	. . . . 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 6249	. . . . . . . 8 |- ( ( ph /\ 0 < A ) -> F = ( y e. RR |-> ( F ` y ) ) )
33		rge0ssre 12280	. . . . . . . . . . . . . 14 |- ( 0 [,) +oo ) C_ RR
34		ax-resscn 9993	. . . . . . . . . . . . . 14 |- RR C_ CC
35	33, 34	sstri 3612	. . . . . . . . . . . . 13 |- ( 0 [,) +oo ) C_ CC
36		fss 6056	. . . . . . . . . . . . 13 |- ( ( F : RR --> ( 0 [,) +oo ) /\ ( 0 [,) +oo ) C_ CC ) -> F : RR --> CC )
37	1, 35, 36	sylancl 694	. . . . . . . . . . . 12 |- ( ph -> F : RR --> CC )
38	37	adantr 481	. . . . . . . . . . 11 |- ( ( ph /\ 0 < A ) -> F : RR --> CC )
39	38	ffvelrnda 6359	. . . . . . . . . 10 |- ( ( ( ph /\ 0 < A ) /\ y e. RR ) -> ( F ` y ) e. CC )
40	39	mulid2d 10058	. . . . . . . . 9 |- ( ( ( ph /\ 0 < A ) /\ y e. RR ) -> ( 1 x. ( F ` y ) ) = ( F ` y ) )
41	40	mpteq2dva 4744	. . . . . . . 8 |- ( ( ph /\ 0 < A ) -> ( y e. RR |-> ( 1 x. ( F ` y ) ) ) = ( y e. RR |-> ( F ` y ) ) )
42	32, 41	eqtr4d 2659	. . . . . . 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 10055	. . . . . . . 8 |- ( ( ( ph /\ 0 < A ) /\ y e. RR ) -> 1 e. RR )
45	43, 30, 11	ofc12 6922	. . . . . . . . . 10 |- ( ( ph /\ 0 < A ) -> ( ( RR X. { ( 1 / A ) } ) oF x. ( RR X. { A } ) ) = ( RR X. { ( ( 1 / A ) x. A ) } ) )
46		fconstmpt 5163	. . . . . . . . . 10 |- ( RR X. { ( ( 1 / A ) x. A ) } ) = ( y e. RR |-> ( ( 1 / A ) x. A ) )
47	45, 46	syl6eq 2672	. . . . . . . . 9 |- ( ( ph /\ 0 < A ) -> ( ( RR X. { ( 1 / A ) } ) oF x. ( RR X. { A } ) ) = ( y e. RR |-> ( ( 1 / A ) x. A ) ) )
48	8	recnd 10068	. . . . . . . . . . . 12 |- ( ph -> A e. CC )
49	48	adantr 481	. . . . . . . . . . 11 |- ( ( ph /\ 0 < A ) -> A e. CC )
50	11	rpne0d 11877	. . . . . . . . . . 11 |- ( ( ph /\ 0 < A ) -> A =/= 0 )
51	49, 50	recid2d 10797	. . . . . . . . . 10 |- ( ( ph /\ 0 < A ) -> ( ( 1 / A ) x. A ) = 1 )
52	51	mpteq2dv 4745	. . . . . . . . 9 |- ( ( ph /\ 0 < A ) -> ( y e. RR |-> ( ( 1 / A ) x. A ) ) = ( y e. RR |-> 1 ) )
53	47, 52	eqtrd 2656	. . . . . . . 8 |- ( ( ph /\ 0 < A ) -> ( ( RR X. { ( 1 / A ) } ) oF x. ( RR X. { A } ) ) = ( y e. RR |-> 1 ) )
54	43, 44, 39, 53, 32	offval2 6914	. . . . . . 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 11874	. . . . . . . . 9 |- ( ( ph /\ 0 < A ) -> ( 1 / A ) e. CC )
56		fconst6g 6094	. . . . . . . . 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 6094	. . . . . . . . 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 10024	. . . . . . . . 9 |- ( ( x e. CC /\ y e. CC /\ z e. CC ) -> ( ( x x. y ) x. z ) = ( x x. ( y x. z ) ) )
61	60	adantl 482	. . . . . . . 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 6931	. . . . . . 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 2662	. . . . . 6 |- ( ( ph /\ 0 < A ) -> F = ( ( RR X. { ( 1 / A ) } ) oF x. ( ( RR X. { A } ) oF x. F ) ) )
64	63	fveq2d 6195	. . . . 5 |- ( ( ph /\ 0 < A ) -> ( S.2 ` F ) = ( S.2 ` ( ( RR X. { ( 1 / A ) } ) oF x. ( ( RR X. { A } ) oF x. F ) ) ) )
65	29	recnd 10068	. . . . . 6 |- ( ( ph /\ 0 < A ) -> ( S.2 ` ( ( RR X. { A } ) oF x. F ) ) e. CC )
66	65, 49, 50	divrec2d 10805	. . . . 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 4685	. . . 4 |- ( ( ph /\ 0 < A ) -> ( S.2 ` F ) <_ ( ( S.2 ` ( ( RR X. { A } ) oF x. F ) ) / A ) )
68	4, 29, 11	lemuldiv2d 11922	. . . 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 247	. . 3 |- ( ( ph /\ 0 < A ) -> ( A x. ( S.2 ` F ) ) <_ ( S.2 ` ( ( RR X. { A } ) oF x. F ) ) )
70		itg2cl 23499	. . . . . 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 10089	. . . . 5 |- ( ph -> ( A x. ( S.2 ` F ) ) e. RR* )
73		xrletri3 11985	. . . . 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 693	. . . 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 481	. . 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 957	. 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 481	. . . . . 6 |- ( ( ph /\ 0 = A ) -> F : RR --> CC )
79	8	adantr 481	. . . . . 6 |- ( ( ph /\ 0 = A ) -> A e. RR )
80		0re 10040	. . . . . . 7 |- 0 e. RR
81	80	a1i 11	. . . . . 6 |- ( ( ph /\ 0 = A ) -> 0 e. RR )
82		simplr 792	. . . . . . . 8 |- ( ( ( ph /\ 0 = A ) /\ x e. CC ) -> 0 = A )
83	82	oveq1d 6665	. . . . . . 7 |- ( ( ( ph /\ 0 = A ) /\ x e. CC ) -> ( 0 x. x ) = ( A x. x ) )
84		mul02 10214	. . . . . . . 8 |- ( x e. CC -> ( 0 x. x ) = 0 )
85	84	adantl 482	. . . . . . 7 |- ( ( ( ph /\ 0 = A ) /\ x e. CC ) -> ( 0 x. x ) = 0 )
86	83, 85	eqtr3d 2658	. . . . . 6 |- ( ( ( ph /\ 0 = A ) /\ x e. CC ) -> ( A x. x ) = 0 )
87	77, 78, 79, 81, 86	caofid2 6928	. . . . 5 |- ( ( ph /\ 0 = A ) -> ( ( RR X. { A } ) oF x. F ) = ( RR X. { 0 } ) )
88	87	fveq2d 6195	. . . 4 |- ( ( ph /\ 0 = A ) -> ( S.2 ` ( ( RR X. { A } ) oF x. F ) ) = ( S.2 ` ( RR X. { 0 } ) ) )
89		itg20 23504	. . . 4 |- ( S.2 ` ( RR X. { 0 } ) ) = 0
90	88, 89	syl6eq 2672	. . 3 |- ( ( ph /\ 0 = A ) -> ( S.2 ` ( ( RR X. { A } ) oF x. F ) ) = 0 )
91	3	adantr 481	. . . . 5 |- ( ( ph /\ 0 = A ) -> ( S.2 ` F ) e. RR )
92	91	recnd 10068	. . . 4 |- ( ( ph /\ 0 = A ) -> ( S.2 ` F ) e. CC )
93	92	mul02d 10234	. . 3 |- ( ( ph /\ 0 = A ) -> ( 0 x. ( S.2 ` F ) ) = 0 )
94		simpr 477	. . . 4 |- ( ( ph /\ 0 = A ) -> 0 = A )
95	94	oveq1d 6665	. . 3 |- ( ( ph /\ 0 = A ) -> ( 0 x. ( S.2 ` F ) ) = ( A x. ( S.2 ` F ) ) )
96	90, 93, 95	3eqtr2d 2662	. 2 |- ( ( ph /\ 0 = A ) -> ( S.2 ` ( ( RR X. { A } ) oF x. F ) ) = ( A x. ( S.2 ` F ) ) )
97	7	simprd 479	. . 3 |- ( ph -> 0 <_ A )
98		leloe 10124	. . . 4 |- ( ( 0 e. RR /\ A e. RR ) -> ( 0 <_ A <-> ( 0 < A \/ 0 = A ) ) )
99	80, 8, 98	sylancr 695	. . 3 |- ( ph -> ( 0 <_ A <-> ( 0 < A \/ 0 = A ) ) )
100	97, 99	mpbid 222	. 2 |- ( ph -> ( 0 < A \/ 0 = A ) )
101	76, 96, 100	mpjaodan 827	1 |- ( ph -> ( S.2 ` ( ( RR X. { A } ) oF x. F ) ) = ( A x. ( S.2 ` F ) ) )

Syntax hints:   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   _Vcvv 3200   C_ wss 3574   {csn 4177   class class class wbr 4653   |-> cmpt 4729   X. cxp 5112   -->wf 5884   ` cfv 5888  (class class class)co 6650   oFcof 6895   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   x. cmul 9941   +oocpnf 10071   RR*cxr 10073   < clt 10074   <_ cle 10075   / cdiv 10684   RR+crp 11832   [,)cico 12177   [,]cicc 12178   S.2citg2 23385

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  ax-addf 10015

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-disj 4621  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-ofr 6898  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  df-itg2 23390  df-0p 23437

This theorem is referenced by:  iblmulc2  23597  itgmulc2lem1  23598  bddmulibl  23605  iblmulc2nc  33475  itgmulc2nclem1  33476