Theorem itg2mulc 22705

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 467	. . . 4 |- ( ( ph /\ 0 < A ) -> F : RR --> ( 0 [,) +oo ) )
3		itg2mulc.3	. . . . 5 |- ( ph -> ( S.2 ` F ) e. RR )
4	3	adantr 467	. . . 4 |- ( ( ph /\ 0 < A ) -> ( S.2 ` F ) e. RR )
5		itg2mulc.4	. . . . . . . 8 |- ( ph -> A e. ( 0 [,) +oo ) )
6		elrege0 11738	. . . . . . . 8 |- ( A e. ( 0 [,) +oo ) <-> ( A e. RR /\ 0 <_ A ) )
7	5, 6	sylib 200	. . . . . . 7 |- ( ph -> ( A e. RR /\ 0 <_ A ) )
8	7	simpld 461	. . . . . 6 |- ( ph -> A e. RR )
9	8	anim1i 572	. . . . 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 216	. . . 4 |- ( ( ph /\ 0 < A ) -> A e. RR+ )
12	2, 4, 11	itg2mulclem 22704	. . 3 |- ( ( ph /\ 0 < A ) -> ( S.2 ` ( ( RR X. { A } ) oF x. F ) ) <_ ( A x. ( S.2 ` F ) ) )
13		ge0mulcl 11745	. . . . . . . . 9 |- ( ( x e. ( 0 [,) +oo ) /\ y e. ( 0 [,) +oo ) ) -> ( x x. y ) e. ( 0 [,) +oo ) )
14	13	adantl 468	. . . . . . . 8 |- ( ( ph /\ ( x e. ( 0 [,) +oo ) /\ y e. ( 0 [,) +oo ) ) ) -> ( x x. y ) e. ( 0 [,) +oo ) )
15		fconst6g 5772	. . . . . . . . 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 9630	. . . . . . . . 9 |- RR e. _V
18	17	a1i 11	. . . . . . . 8 |- ( ph -> RR e. _V )
19		inidm 3641	. . . . . . . 8 |- ( RR i^i RR ) = RR
20	14, 16, 1, 18, 18, 19	off 6546	. . . . . . 7 |- ( ph -> ( ( RR X. { A } ) oF x. F ) : RR --> ( 0 [,) +oo ) )
21	20	adantr 467	. . . . . 6 |- ( ( ph /\ 0 < A ) -> ( ( RR X. { A } ) oF x. F ) : RR --> ( 0 [,) +oo ) )
22		icossicc 11721	. . . . . . . . 9 |- ( 0 [,) +oo ) C_ ( 0 [,] +oo )
23		fss 5737	. . . . . . . . 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 668	. . . . . . . 8 |- ( ph -> ( ( RR X. { A } ) oF x. F ) : RR --> ( 0 [,] +oo ) )
25	24	adantr 467	. . . . . . 7 |- ( ( ph /\ 0 < A ) -> ( ( RR X. { A } ) oF x. F ) : RR --> ( 0 [,] +oo ) )
26	8, 3	remulcld 9671	. . . . . . . 8 |- ( ph -> ( A x. ( S.2 ` F ) ) e. RR )
27	26	adantr 467	. . . . . . 7 |- ( ( ph /\ 0 < A ) -> ( A x. ( S.2 ` F ) ) e. RR )
28		itg2lecl 22696	. . . . . . 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 1268	. . . . . 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 22704	. . . . 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 5918	. . . . . . . 8 |- ( ( ph /\ 0 < A ) -> F = ( y e. RR |-> ( F ` y ) ) )
33		rge0ssre 11740	. . . . . . . . . . . . . 14 |- ( 0 [,) +oo ) C_ RR
34		ax-resscn 9596	. . . . . . . . . . . . . 14 |- RR C_ CC
35	33, 34	sstri 3441	. . . . . . . . . . . . 13 |- ( 0 [,) +oo ) C_ CC
36		fss 5737	. . . . . . . . . . . . 13 |- ( ( F : RR --> ( 0 [,) +oo ) /\ ( 0 [,) +oo ) C_ CC ) -> F : RR --> CC )
37	1, 35, 36	sylancl 668	. . . . . . . . . . . 12 |- ( ph -> F : RR --> CC )
38	37	adantr 467	. . . . . . . . . . 11 |- ( ( ph /\ 0 < A ) -> F : RR --> CC )
39	38	ffvelrnda 6022	. . . . . . . . . 10 |- ( ( ( ph /\ 0 < A ) /\ y e. RR ) -> ( F ` y ) e. CC )
40	39	mulid2d 9661	. . . . . . . . 9 |- ( ( ( ph /\ 0 < A ) /\ y e. RR ) -> ( 1 x. ( F ` y ) ) = ( F ` y ) )
41	40	mpteq2dva 4489	. . . . . . . 8 |- ( ( ph /\ 0 < A ) -> ( y e. RR |-> ( 1 x. ( F ` y ) ) ) = ( y e. RR |-> ( F ` y ) ) )
42	32, 41	eqtr4d 2488	. . . . . . 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 9658	. . . . . . . 8 |- ( ( ( ph /\ 0 < A ) /\ y e. RR ) -> 1 e. RR )
45	43, 30, 11	ofc12 6556	. . . . . . . . . 10 |- ( ( ph /\ 0 < A ) -> ( ( RR X. { ( 1 / A ) } ) oF x. ( RR X. { A } ) ) = ( RR X. { ( ( 1 / A ) x. A ) } ) )
46		fconstmpt 4878	. . . . . . . . . 10 |- ( RR X. { ( ( 1 / A ) x. A ) } ) = ( y e. RR |-> ( ( 1 / A ) x. A ) )
47	45, 46	syl6eq 2501	. . . . . . . . 9 |- ( ( ph /\ 0 < A ) -> ( ( RR X. { ( 1 / A ) } ) oF x. ( RR X. { A } ) ) = ( y e. RR |-> ( ( 1 / A ) x. A ) ) )
48	8	recnd 9669	. . . . . . . . . . . 12 |- ( ph -> A e. CC )
49	48	adantr 467	. . . . . . . . . . 11 |- ( ( ph /\ 0 < A ) -> A e. CC )
50	11	rpne0d 11346	. . . . . . . . . . 11 |- ( ( ph /\ 0 < A ) -> A =/= 0 )
51	49, 50	recid2d 10379	. . . . . . . . . 10 |- ( ( ph /\ 0 < A ) -> ( ( 1 / A ) x. A ) = 1 )
52	51	mpteq2dv 4490	. . . . . . . . 9 |- ( ( ph /\ 0 < A ) -> ( y e. RR |-> ( ( 1 / A ) x. A ) ) = ( y e. RR |-> 1 ) )
53	47, 52	eqtrd 2485	. . . . . . . 8 |- ( ( ph /\ 0 < A ) -> ( ( RR X. { ( 1 / A ) } ) oF x. ( RR X. { A } ) ) = ( y e. RR |-> 1 ) )
54	43, 44, 39, 53, 32	offval2 6548	. . . . . . 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 5772	. . . . . . . . 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 5772	. . . . . . . . 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 9627	. . . . . . . . 9 |- ( ( x e. CC /\ y e. CC /\ z e. CC ) -> ( ( x x. y ) x. z ) = ( x x. ( y x. z ) ) )
61	60	adantl 468	. . . . . . . 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 6565	. . . . . . 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 2491	. . . . . 6 |- ( ( ph /\ 0 < A ) -> F = ( ( RR X. { ( 1 / A ) } ) oF x. ( ( RR X. { A } ) oF x. F ) ) )
64	63	fveq2d 5869	. . . . 5 |- ( ( ph /\ 0 < A ) -> ( S.2 ` F ) = ( S.2 ` ( ( RR X. { ( 1 / A ) } ) oF x. ( ( RR X. { A } ) oF x. F ) ) ) )
65	29	recnd 9669	. . . . . 6 |- ( ( ph /\ 0 < A ) -> ( S.2 ` ( ( RR X. { A } ) oF x. F ) ) e. CC )
66	65, 49, 50	divrec2d 10387	. . . . 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 4433	. . . 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 236	. . 3 |- ( ( ph /\ 0 < A ) -> ( A x. ( S.2 ` F ) ) <_ ( S.2 ` ( ( RR X. { A } ) oF x. F ) ) )
70		itg2cl 22690	. . . . . 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 9690	. . . . 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 667	. . . 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 467	. . 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 933	. 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 467	. . . . . 6 |- ( ( ph /\ 0 = A ) -> F : RR --> CC )
79	8	adantr 467	. . . . . 6 |- ( ( ph /\ 0 = A ) -> A e. RR )
80		0re 9643	. . . . . . 7 |- 0 e. RR
81	80	a1i 11	. . . . . 6 |- ( ( ph /\ 0 = A ) -> 0 e. RR )
82		simplr 762	. . . . . . . 8 |- ( ( ( ph /\ 0 = A ) /\ x e. CC ) -> 0 = A )
83	82	oveq1d 6305	. . . . . . 7 |- ( ( ( ph /\ 0 = A ) /\ x e. CC ) -> ( 0 x. x ) = ( A x. x ) )
84		mul02 9811	. . . . . . . 8 |- ( x e. CC -> ( 0 x. x ) = 0 )
85	84	adantl 468	. . . . . . 7 |- ( ( ( ph /\ 0 = A ) /\ x e. CC ) -> ( 0 x. x ) = 0 )
86	83, 85	eqtr3d 2487	. . . . . 6 |- ( ( ( ph /\ 0 = A ) /\ x e. CC ) -> ( A x. x ) = 0 )
87	77, 78, 79, 81, 86	caofid2 6562	. . . . 5 |- ( ( ph /\ 0 = A ) -> ( ( RR X. { A } ) oF x. F ) = ( RR X. { 0 } ) )
88	87	fveq2d 5869	. . . 4 |- ( ( ph /\ 0 = A ) -> ( S.2 ` ( ( RR X. { A } ) oF x. F ) ) = ( S.2 ` ( RR X. { 0 } ) ) )
89		itg20 22695	. . . 4 |- ( S.2 ` ( RR X. { 0 } ) ) = 0
90	88, 89	syl6eq 2501	. . 3 |- ( ( ph /\ 0 = A ) -> ( S.2 ` ( ( RR X. { A } ) oF x. F ) ) = 0 )
91	3	adantr 467	. . . . 5 |- ( ( ph /\ 0 = A ) -> ( S.2 ` F ) e. RR )
92	91	recnd 9669	. . . 4 |- ( ( ph /\ 0 = A ) -> ( S.2 ` F ) e. CC )
93	92	mul02d 9831	. . 3 |- ( ( ph /\ 0 = A ) -> ( 0 x. ( S.2 ` F ) ) = 0 )
94		simpr 463	. . . 4 |- ( ( ph /\ 0 = A ) -> 0 = A )
95	94	oveq1d 6305	. . 3 |- ( ( ph /\ 0 = A ) -> ( 0 x. ( S.2 ` F ) ) = ( A x. ( S.2 ` F ) ) )
96	90, 93, 95	3eqtr2d 2491	. 2 |- ( ( ph /\ 0 = A ) -> ( S.2 ` ( ( RR X. { A } ) oF x. F ) ) = ( A x. ( S.2 ` F ) ) )
97	7	simprd 465	. . 3 |- ( ph -> 0 <_ A )
98		leloe 9720	. . . 4 |- ( ( 0 e. RR /\ A e. RR ) -> ( 0 <_ A <-> ( 0 < A \/ 0 = A ) ) )
99	80, 8, 98	sylancr 669	. . 3 |- ( ph -> ( 0 <_ A <-> ( 0 < A \/ 0 = A ) ) )
100	97, 99	mpbid 214	. 2 |- ( ph -> ( 0 < A \/ 0 = A ) )
101	76, 96, 100	mpjaodan 795	1 |- ( ph -> ( S.2 ` ( ( RR X. { A } ) oF x. F ) ) = ( A x. ( S.2 ` F ) ) )

Syntax hints:   -> wi 4   <-> wb 188   \/ wo 370   /\ wa 371   /\ w3a 985   = wceq 1444   e. wcel 1887   _Vcvv 3045   C_ wss 3404   {csn 3968   class class class wbr 4402   |-> cmpt 4461   X. cxp 4832   -->wf 5578   ` cfv 5582  (class class class)co 6290   oFcof 6529   CCcc 9537   RRcr 9538   0cc0 9539   1c1 9540   x. cmul 9544   +oocpnf 9672   RR*cxr 9674   < clt 9675   <_ cle 9676   / cdiv 10269   RR+crp 11302   [,)cico 11637   [,]cicc 11638   S.2citg2 22574

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-inf2 8146  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-pre-sup 9617  ax-addf 9618

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-fal 1450  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-disj 4374  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-se 4794  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-isom 5591  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-of 6531  df-ofr 6532  df-om 6693  df-1st 6793  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-2o 7183  df-oadd 7186  df-er 7363  df-map 7474  df-pm 7475  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-sup 7956  df-inf 7957  df-oi 8025  df-card 8373  df-cda 8598  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  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 11785  df-fzo 11916  df-fl 12028  df-seq 12214  df-exp 12273  df-hash 12516  df-cj 13162  df-re 13163  df-im 13164  df-sqrt 13298  df-abs 13299  df-clim 13552  df-sum 13753  df-xmet 18963  df-met 18964  df-ovol 22416  df-vol 22418  df-mbf 22577  df-itg1 22578  df-itg2 22579  df-0p 22628

This theorem is referenced by:  iblmulc2  22788  itgmulc2lem1  22789  bddmulibl  22796  iblmulc2nc  32007  itgmulc2nclem1  32008