Theorem itg2const 23507

Description: Integral of a constant function. (Contributed by Mario Carneiro, 12-Aug-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)

Assertion

Ref	Expression
itg2const	|- ( ( A e. dom vol /\ ( vol ` A ) e. RR /\ B e. ( 0 [,) +oo ) ) -> ( S.2 ` ( x e. RR |-> if ( x e. A , B , 0 ) ) ) = ( B x. ( vol ` A ) ) )

Distinct variable groups:   x, A   x, B

Proof of Theorem itg2const

Step	Hyp	Ref	Expression
1		reex 10027	. . . . . . 7 |- RR e. _V
2	1	a1i 11	. . . . . 6 |- ( ( A e. dom vol /\ ( vol ` A ) e. RR /\ B e. ( 0 [,) +oo ) ) -> RR e. _V )
3		simpl3 1066	. . . . . 6 |- ( ( ( A e. dom vol /\ ( vol ` A ) e. RR /\ B e. ( 0 [,) +oo ) ) /\ x e. RR ) -> B e. ( 0 [,) +oo ) )
4		1re 10039	. . . . . . . 8 |- 1 e. RR
5		0re 10040	. . . . . . . 8 |- 0 e. RR
6	4, 5	keepel 4155	. . . . . . 7 |- if ( x e. A , 1 , 0 ) e. RR
7	6	a1i 11	. . . . . 6 |- ( ( ( A e. dom vol /\ ( vol ` A ) e. RR /\ B e. ( 0 [,) +oo ) ) /\ x e. RR ) -> if ( x e. A , 1 , 0 ) e. RR )
8		fconstmpt 5163	. . . . . . 7 |- ( RR X. { B } ) = ( x e. RR |-> B )
9	8	a1i 11	. . . . . 6 |- ( ( A e. dom vol /\ ( vol ` A ) e. RR /\ B e. ( 0 [,) +oo ) ) -> ( RR X. { B } ) = ( x e. RR |-> B ) )
10		eqidd 2623	. . . . . 6 |- ( ( A e. dom vol /\ ( vol ` A ) e. RR /\ B e. ( 0 [,) +oo ) ) -> ( x e. RR |-> if ( x e. A , 1 , 0 ) ) = ( x e. RR |-> if ( x e. A , 1 , 0 ) ) )
11	2, 3, 7, 9, 10	offval2 6914	. . . . 5 |- ( ( A e. dom vol /\ ( vol ` A ) e. RR /\ B e. ( 0 [,) +oo ) ) -> ( ( RR X. { B } ) oF x. ( x e. RR |-> if ( x e. A , 1 , 0 ) ) ) = ( x e. RR |-> ( B x. if ( x e. A , 1 , 0 ) ) ) )
12		ovif2 6738	. . . . . . 7 |- ( B x. if ( x e. A , 1 , 0 ) ) = if ( x e. A , ( B x. 1 ) , ( B x. 0 ) )
13		simp3 1063	. . . . . . . . . . . 12 |- ( ( A e. dom vol /\ ( vol ` A ) e. RR /\ B e. ( 0 [,) +oo ) ) -> B e. ( 0 [,) +oo ) )
14		elrege0 12278	. . . . . . . . . . . 12 |- ( B e. ( 0 [,) +oo ) <-> ( B e. RR /\ 0 <_ B ) )
15	13, 14	sylib 208	. . . . . . . . . . 11 |- ( ( A e. dom vol /\ ( vol ` A ) e. RR /\ B e. ( 0 [,) +oo ) ) -> ( B e. RR /\ 0 <_ B ) )
16	15	simpld 475	. . . . . . . . . 10 |- ( ( A e. dom vol /\ ( vol ` A ) e. RR /\ B e. ( 0 [,) +oo ) ) -> B e. RR )
17	16	recnd 10068	. . . . . . . . 9 |- ( ( A e. dom vol /\ ( vol ` A ) e. RR /\ B e. ( 0 [,) +oo ) ) -> B e. CC )
18	17	mulid1d 10057	. . . . . . . 8 |- ( ( A e. dom vol /\ ( vol ` A ) e. RR /\ B e. ( 0 [,) +oo ) ) -> ( B x. 1 ) = B )
19	17	mul01d 10235	. . . . . . . 8 |- ( ( A e. dom vol /\ ( vol ` A ) e. RR /\ B e. ( 0 [,) +oo ) ) -> ( B x. 0 ) = 0 )
20	18, 19	ifeq12d 4106	. . . . . . 7 |- ( ( A e. dom vol /\ ( vol ` A ) e. RR /\ B e. ( 0 [,) +oo ) ) -> if ( x e. A , ( B x. 1 ) , ( B x. 0 ) ) = if ( x e. A , B , 0 ) )
21	12, 20	syl5eq 2668	. . . . . 6 |- ( ( A e. dom vol /\ ( vol ` A ) e. RR /\ B e. ( 0 [,) +oo ) ) -> ( B x. if ( x e. A , 1 , 0 ) ) = if ( x e. A , B , 0 ) )
22	21	mpteq2dv 4745	. . . . 5 |- ( ( A e. dom vol /\ ( vol ` A ) e. RR /\ B e. ( 0 [,) +oo ) ) -> ( x e. RR |-> ( B x. if ( x e. A , 1 , 0 ) ) ) = ( x e. RR |-> if ( x e. A , B , 0 ) ) )
23	11, 22	eqtrd 2656	. . . 4 |- ( ( A e. dom vol /\ ( vol ` A ) e. RR /\ B e. ( 0 [,) +oo ) ) -> ( ( RR X. { B } ) oF x. ( x e. RR |-> if ( x e. A , 1 , 0 ) ) ) = ( x e. RR |-> if ( x e. A , B , 0 ) ) )
24		eqid 2622	. . . . . . 7 |- ( x e. RR |-> if ( x e. A , 1 , 0 ) ) = ( x e. RR |-> if ( x e. A , 1 , 0 ) )
25	24	i1f1 23457	. . . . . 6 |- ( ( A e. dom vol /\ ( vol ` A ) e. RR ) -> ( x e. RR |-> if ( x e. A , 1 , 0 ) ) e. dom S.1 )
26	25	3adant3 1081	. . . . 5 |- ( ( A e. dom vol /\ ( vol ` A ) e. RR /\ B e. ( 0 [,) +oo ) ) -> ( x e. RR |-> if ( x e. A , 1 , 0 ) ) e. dom S.1 )
27	26, 16	i1fmulc 23470	. . . 4 |- ( ( A e. dom vol /\ ( vol ` A ) e. RR /\ B e. ( 0 [,) +oo ) ) -> ( ( RR X. { B } ) oF x. ( x e. RR |-> if ( x e. A , 1 , 0 ) ) ) e. dom S.1 )
28	23, 27	eqeltrrd 2702	. . 3 |- ( ( A e. dom vol /\ ( vol ` A ) e. RR /\ B e. ( 0 [,) +oo ) ) -> ( x e. RR |-> if ( x e. A , B , 0 ) ) e. dom S.1 )
29	15	simprd 479	. . . . . 6 |- ( ( A e. dom vol /\ ( vol ` A ) e. RR /\ B e. ( 0 [,) +oo ) ) -> 0 <_ B )
30		0le0 11110	. . . . . 6 |- 0 <_ 0
31		breq2 4657	. . . . . . 7 |- ( B = if ( x e. A , B , 0 ) -> ( 0 <_ B <-> 0 <_ if ( x e. A , B , 0 ) ) )
32		breq2 4657	. . . . . . 7 |- ( 0 = if ( x e. A , B , 0 ) -> ( 0 <_ 0 <-> 0 <_ if ( x e. A , B , 0 ) ) )
33	31, 32	ifboth 4124	. . . . . 6 |- ( ( 0 <_ B /\ 0 <_ 0 ) -> 0 <_ if ( x e. A , B , 0 ) )
34	29, 30, 33	sylancl 694	. . . . 5 |- ( ( A e. dom vol /\ ( vol ` A ) e. RR /\ B e. ( 0 [,) +oo ) ) -> 0 <_ if ( x e. A , B , 0 ) )
35	34	ralrimivw 2967	. . . 4 |- ( ( A e. dom vol /\ ( vol ` A ) e. RR /\ B e. ( 0 [,) +oo ) ) -> A. x e. RR 0 <_ if ( x e. A , B , 0 ) )
36		ax-resscn 9993	. . . . . . 7 |- RR C_ CC
37	36	a1i 11	. . . . . 6 |- ( ( A e. dom vol /\ ( vol ` A ) e. RR /\ B e. ( 0 [,) +oo ) ) -> RR C_ CC )
38	16	adantr 481	. . . . . . . . 9 |- ( ( ( A e. dom vol /\ ( vol ` A ) e. RR /\ B e. ( 0 [,) +oo ) ) /\ x e. RR ) -> B e. RR )
39		ifcl 4130	. . . . . . . . 9 |- ( ( B e. RR /\ 0 e. RR ) -> if ( x e. A , B , 0 ) e. RR )
40	38, 5, 39	sylancl 694	. . . . . . . 8 |- ( ( ( A e. dom vol /\ ( vol ` A ) e. RR /\ B e. ( 0 [,) +oo ) ) /\ x e. RR ) -> if ( x e. A , B , 0 ) e. RR )
41	40	ralrimiva 2966	. . . . . . 7 |- ( ( A e. dom vol /\ ( vol ` A ) e. RR /\ B e. ( 0 [,) +oo ) ) -> A. x e. RR if ( x e. A , B , 0 ) e. RR )
42		eqid 2622	. . . . . . . 8 |- ( x e. RR |-> if ( x e. A , B , 0 ) ) = ( x e. RR |-> if ( x e. A , B , 0 ) )
43	42	fnmpt 6020	. . . . . . 7 |- ( A. x e. RR if ( x e. A , B , 0 ) e. RR -> ( x e. RR |-> if ( x e. A , B , 0 ) ) Fn RR )
44	41, 43	syl 17	. . . . . 6 |- ( ( A e. dom vol /\ ( vol ` A ) e. RR /\ B e. ( 0 [,) +oo ) ) -> ( x e. RR |-> if ( x e. A , B , 0 ) ) Fn RR )
45	37, 44	0pledm 23440	. . . . 5 |- ( ( A e. dom vol /\ ( vol ` A ) e. RR /\ B e. ( 0 [,) +oo ) ) -> ( 0p oR <_ ( x e. RR |-> if ( x e. A , B , 0 ) ) <-> ( RR X. { 0 } ) oR <_ ( x e. RR |-> if ( x e. A , B , 0 ) ) ) )
46	5	a1i 11	. . . . . 6 |- ( ( ( A e. dom vol /\ ( vol ` A ) e. RR /\ B e. ( 0 [,) +oo ) ) /\ x e. RR ) -> 0 e. RR )
47		fconstmpt 5163	. . . . . . 7 |- ( RR X. { 0 } ) = ( x e. RR |-> 0 )
48	47	a1i 11	. . . . . 6 |- ( ( A e. dom vol /\ ( vol ` A ) e. RR /\ B e. ( 0 [,) +oo ) ) -> ( RR X. { 0 } ) = ( x e. RR |-> 0 ) )
49		eqidd 2623	. . . . . 6 |- ( ( A e. dom vol /\ ( vol ` A ) e. RR /\ B e. ( 0 [,) +oo ) ) -> ( x e. RR |-> if ( x e. A , B , 0 ) ) = ( x e. RR |-> if ( x e. A , B , 0 ) ) )
50	2, 46, 40, 48, 49	ofrfval2 6915	. . . . 5 |- ( ( A e. dom vol /\ ( vol ` A ) e. RR /\ B e. ( 0 [,) +oo ) ) -> ( ( RR X. { 0 } ) oR <_ ( x e. RR |-> if ( x e. A , B , 0 ) ) <-> A. x e. RR 0 <_ if ( x e. A , B , 0 ) ) )
51	45, 50	bitrd 268	. . . 4 |- ( ( A e. dom vol /\ ( vol ` A ) e. RR /\ B e. ( 0 [,) +oo ) ) -> ( 0p oR <_ ( x e. RR |-> if ( x e. A , B , 0 ) ) <-> A. x e. RR 0 <_ if ( x e. A , B , 0 ) ) )
52	35, 51	mpbird 247	. . 3 |- ( ( A e. dom vol /\ ( vol ` A ) e. RR /\ B e. ( 0 [,) +oo ) ) -> 0p oR <_ ( x e. RR |-> if ( x e. A , B , 0 ) ) )
53		itg2itg1 23503	. . 3 |- ( ( ( x e. RR |-> if ( x e. A , B , 0 ) ) e. dom S.1 /\ 0p oR <_ ( x e. RR |-> if ( x e. A , B , 0 ) ) ) -> ( S.2 ` ( x e. RR |-> if ( x e. A , B , 0 ) ) ) = ( S.1 ` ( x e. RR |-> if ( x e. A , B , 0 ) ) ) )
54	28, 52, 53	syl2anc 693	. 2 |- ( ( A e. dom vol /\ ( vol ` A ) e. RR /\ B e. ( 0 [,) +oo ) ) -> ( S.2 ` ( x e. RR |-> if ( x e. A , B , 0 ) ) ) = ( S.1 ` ( x e. RR |-> if ( x e. A , B , 0 ) ) ) )
55	26, 16	itg1mulc 23471	. . 3 |- ( ( A e. dom vol /\ ( vol ` A ) e. RR /\ B e. ( 0 [,) +oo ) ) -> ( S.1 ` ( ( RR X. { B } ) oF x. ( x e. RR |-> if ( x e. A , 1 , 0 ) ) ) ) = ( B x. ( S.1 ` ( x e. RR |-> if ( x e. A , 1 , 0 ) ) ) ) )
56	23	fveq2d 6195	. . 3 |- ( ( A e. dom vol /\ ( vol ` A ) e. RR /\ B e. ( 0 [,) +oo ) ) -> ( S.1 ` ( ( RR X. { B } ) oF x. ( x e. RR |-> if ( x e. A , 1 , 0 ) ) ) ) = ( S.1 ` ( x e. RR |-> if ( x e. A , B , 0 ) ) ) )
57	24	itg11 23458	. . . . 5 |- ( ( A e. dom vol /\ ( vol ` A ) e. RR ) -> ( S.1 ` ( x e. RR |-> if ( x e. A , 1 , 0 ) ) ) = ( vol ` A ) )
58	57	3adant3 1081	. . . 4 |- ( ( A e. dom vol /\ ( vol ` A ) e. RR /\ B e. ( 0 [,) +oo ) ) -> ( S.1 ` ( x e. RR |-> if ( x e. A , 1 , 0 ) ) ) = ( vol ` A ) )
59	58	oveq2d 6666	. . 3 |- ( ( A e. dom vol /\ ( vol ` A ) e. RR /\ B e. ( 0 [,) +oo ) ) -> ( B x. ( S.1 ` ( x e. RR |-> if ( x e. A , 1 , 0 ) ) ) ) = ( B x. ( vol ` A ) ) )
60	55, 56, 59	3eqtr3d 2664	. 2 |- ( ( A e. dom vol /\ ( vol ` A ) e. RR /\ B e. ( 0 [,) +oo ) ) -> ( S.1 ` ( x e. RR |-> if ( x e. A , B , 0 ) ) ) = ( B x. ( vol ` A ) ) )
61	54, 60	eqtrd 2656	1 |- ( ( A e. dom vol /\ ( vol ` A ) e. RR /\ B e. ( 0 [,) +oo ) ) -> ( S.2 ` ( x e. RR |-> if ( x e. A , B , 0 ) ) ) = ( B x. ( vol ` A ) ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   C_ wss 3574   ifcif 4086   {csn 4177   class class class wbr 4653   |-> cmpt 4729   X. cxp 5112   dom cdm 5114   Fn wfn 5883   ` cfv 5888  (class class class)co 6650   oFcof 6895   oRcofr 6896   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   x. cmul 9941   +oocpnf 10071   <_ cle 10075   [,)cico 12177   volcvol 23232   S.1citg1 23384   S.2citg2 23385   0pc0p 23436

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:  itg2const2  23508  itg2gt0  23527  itg2cnlem2  23529  iblconst  23584  itgconst  23585  itg2gt0cn  33465  bddiblnc  33480  ftc1anclem7  33491