Theorem itg11 23458

Description: The integral of an indicator function is the volume of the set. (Contributed by Mario Carneiro, 18-Jun-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)

Hypothesis

Ref	Expression
i1f1.1	|- F = ( x e. RR |-> if ( x e. A , 1 , 0 ) )

Assertion

Ref	Expression
itg11	|- ( ( A e. dom vol /\ ( vol ` A ) e. RR ) -> ( S.1 ` F ) = ( vol ` A ) )

Distinct variable group:   x, A

Allowed substitution hint:   F( x)

Proof of Theorem itg11

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

Step	Hyp	Ref	Expression
1		ovol0 23261	. . . . 5 |- ( vol* ` (/) ) = 0
2		0mbl 23307	. . . . . 6 |- (/) e. dom vol
3		mblvol 23298	. . . . . 6 |- ( (/) e. dom vol -> ( vol ` (/) ) = ( vol* ` (/) ) )
4	2, 3	ax-mp 5	. . . . 5 |- ( vol ` (/) ) = ( vol* ` (/) )
5		itg10 23455	. . . . 5 |- ( S.1 ` ( RR X. { 0 } ) ) = 0
6	1, 4, 5	3eqtr4ri 2655	. . . 4 |- ( S.1 ` ( RR X. { 0 } ) ) = ( vol ` (/) )
7		noel 3919	. . . . . . . . 9 |- -. x e. (/)
8		eleq2 2690	. . . . . . . . 9 |- ( A = (/) -> ( x e. A <-> x e. (/) ) )
9	7, 8	mtbiri 317	. . . . . . . 8 |- ( A = (/) -> -. x e. A )
10	9	iffalsed 4097	. . . . . . 7 |- ( A = (/) -> if ( x e. A , 1 , 0 ) = 0 )
11	10	mpteq2dv 4745	. . . . . 6 |- ( A = (/) -> ( x e. RR |-> if ( x e. A , 1 , 0 ) ) = ( x e. RR |-> 0 ) )
12		i1f1.1	. . . . . 6 |- F = ( x e. RR |-> if ( x e. A , 1 , 0 ) )
13		fconstmpt 5163	. . . . . 6 |- ( RR X. { 0 } ) = ( x e. RR |-> 0 )
14	11, 12, 13	3eqtr4g 2681	. . . . 5 |- ( A = (/) -> F = ( RR X. { 0 } ) )
15	14	fveq2d 6195	. . . 4 |- ( A = (/) -> ( S.1 ` F ) = ( S.1 ` ( RR X. { 0 } ) ) )
16		fveq2 6191	. . . 4 |- ( A = (/) -> ( vol ` A ) = ( vol ` (/) ) )
17	6, 15, 16	3eqtr4a 2682	. . 3 |- ( A = (/) -> ( S.1 ` F ) = ( vol ` A ) )
18	17	a1i 11	. 2 |- ( ( A e. dom vol /\ ( vol ` A ) e. RR ) -> ( A = (/) -> ( S.1 ` F ) = ( vol ` A ) ) )
19		n0 3931	. . 3 |- ( A =/= (/) <-> E. y y e. A )
20	12	i1f1 23457	. . . . . . . 8 |- ( ( A e. dom vol /\ ( vol ` A ) e. RR ) -> F e. dom S.1 )
21	20	adantr 481	. . . . . . 7 |- ( ( ( A e. dom vol /\ ( vol ` A ) e. RR ) /\ y e. A ) -> F e. dom S.1 )
22		itg1val 23450	. . . . . . 7 |- ( F e. dom S.1 -> ( S.1 ` F ) = sum_ z e. ( ran F \ { 0 } ) ( z x. ( vol ` ( `' F " { z } ) ) ) )
23	21, 22	syl 17	. . . . . 6 |- ( ( ( A e. dom vol /\ ( vol ` A ) e. RR ) /\ y e. A ) -> ( S.1 ` F ) = sum_ z e. ( ran F \ { 0 } ) ( z x. ( vol ` ( `' F " { z } ) ) ) )
24	12	i1f1lem 23456	. . . . . . . . . . . . . 14 |- ( F : RR --> { 0 , 1 } /\ ( A e. dom vol -> ( `' F " { 1 } ) = A ) )
25	24	simpli 474	. . . . . . . . . . . . 13 |- F : RR --> { 0 , 1 }
26		frn 6053	. . . . . . . . . . . . 13 |- ( F : RR --> { 0 , 1 } -> ran F C_ { 0 , 1 } )
27	25, 26	ax-mp 5	. . . . . . . . . . . 12 |- ran F C_ { 0 , 1 }
28		ssdif 3745	. . . . . . . . . . . 12 |- ( ran F C_ { 0 , 1 } -> ( ran F \ { 0 } ) C_ ( { 0 , 1 } \ { 0 } ) )
29	27, 28	ax-mp 5	. . . . . . . . . . 11 |- ( ran F \ { 0 } ) C_ ( { 0 , 1 } \ { 0 } )
30		difprsnss 4329	. . . . . . . . . . 11 |- ( { 0 , 1 } \ { 0 } ) C_ { 1 }
31	29, 30	sstri 3612	. . . . . . . . . 10 |- ( ran F \ { 0 } ) C_ { 1 }
32	31	a1i 11	. . . . . . . . 9 |- ( ( ( A e. dom vol /\ ( vol ` A ) e. RR ) /\ y e. A ) -> ( ran F \ { 0 } ) C_ { 1 } )
33		mblss 23299	. . . . . . . . . . . . . . . 16 |- ( A e. dom vol -> A C_ RR )
34	33	adantr 481	. . . . . . . . . . . . . . 15 |- ( ( A e. dom vol /\ ( vol ` A ) e. RR ) -> A C_ RR )
35	34	sselda 3603	. . . . . . . . . . . . . 14 |- ( ( ( A e. dom vol /\ ( vol ` A ) e. RR ) /\ y e. A ) -> y e. RR )
36		eleq1 2689	. . . . . . . . . . . . . . . 16 |- ( x = y -> ( x e. A <-> y e. A ) )
37	36	ifbid 4108	. . . . . . . . . . . . . . 15 |- ( x = y -> if ( x e. A , 1 , 0 ) = if ( y e. A , 1 , 0 ) )
38		1ex 10035	. . . . . . . . . . . . . . . 16 |- 1 e. _V
39		c0ex 10034	. . . . . . . . . . . . . . . 16 |- 0 e. _V
40	38, 39	ifex 4156	. . . . . . . . . . . . . . 15 |- if ( y e. A , 1 , 0 ) e. _V
41	37, 12, 40	fvmpt 6282	. . . . . . . . . . . . . 14 |- ( y e. RR -> ( F ` y ) = if ( y e. A , 1 , 0 ) )
42	35, 41	syl 17	. . . . . . . . . . . . 13 |- ( ( ( A e. dom vol /\ ( vol ` A ) e. RR ) /\ y e. A ) -> ( F ` y ) = if ( y e. A , 1 , 0 ) )
43		iftrue 4092	. . . . . . . . . . . . . 14 |- ( y e. A -> if ( y e. A , 1 , 0 ) = 1 )
44	43	adantl 482	. . . . . . . . . . . . 13 |- ( ( ( A e. dom vol /\ ( vol ` A ) e. RR ) /\ y e. A ) -> if ( y e. A , 1 , 0 ) = 1 )
45	42, 44	eqtrd 2656	. . . . . . . . . . . 12 |- ( ( ( A e. dom vol /\ ( vol ` A ) e. RR ) /\ y e. A ) -> ( F ` y ) = 1 )
46		ffn 6045	. . . . . . . . . . . . . 14 |- ( F : RR --> { 0 , 1 } -> F Fn RR )
47	25, 46	ax-mp 5	. . . . . . . . . . . . 13 |- F Fn RR
48		fnfvelrn 6356	. . . . . . . . . . . . 13 |- ( ( F Fn RR /\ y e. RR ) -> ( F ` y ) e. ran F )
49	47, 35, 48	sylancr 695	. . . . . . . . . . . 12 |- ( ( ( A e. dom vol /\ ( vol ` A ) e. RR ) /\ y e. A ) -> ( F ` y ) e. ran F )
50	45, 49	eqeltrrd 2702	. . . . . . . . . . 11 |- ( ( ( A e. dom vol /\ ( vol ` A ) e. RR ) /\ y e. A ) -> 1 e. ran F )
51		ax-1ne0 10005	. . . . . . . . . . . 12 |- 1 =/= 0
52	51	a1i 11	. . . . . . . . . . 11 |- ( ( ( A e. dom vol /\ ( vol ` A ) e. RR ) /\ y e. A ) -> 1 =/= 0 )
53		eldifsn 4317	. . . . . . . . . . 11 |- ( 1 e. ( ran F \ { 0 } ) <-> ( 1 e. ran F /\ 1 =/= 0 ) )
54	50, 52, 53	sylanbrc 698	. . . . . . . . . 10 |- ( ( ( A e. dom vol /\ ( vol ` A ) e. RR ) /\ y e. A ) -> 1 e. ( ran F \ { 0 } ) )
55	54	snssd 4340	. . . . . . . . 9 |- ( ( ( A e. dom vol /\ ( vol ` A ) e. RR ) /\ y e. A ) -> { 1 } C_ ( ran F \ { 0 } ) )
56	32, 55	eqssd 3620	. . . . . . . 8 |- ( ( ( A e. dom vol /\ ( vol ` A ) e. RR ) /\ y e. A ) -> ( ran F \ { 0 } ) = { 1 } )
57	56	sumeq1d 14431	. . . . . . 7 |- ( ( ( A e. dom vol /\ ( vol ` A ) e. RR ) /\ y e. A ) -> sum_ z e. ( ran F \ { 0 } ) ( z x. ( vol ` ( `' F " { z } ) ) ) = sum_ z e. { 1 } ( z x. ( vol ` ( `' F " { z } ) ) ) )
58		1re 10039	. . . . . . . . 9 |- 1 e. RR
59	24	simpri 478	. . . . . . . . . . . . . 14 |- ( A e. dom vol -> ( `' F " { 1 } ) = A )
60	59	ad2antrr 762	. . . . . . . . . . . . 13 |- ( ( ( A e. dom vol /\ ( vol ` A ) e. RR ) /\ y e. A ) -> ( `' F " { 1 } ) = A )
61	60	fveq2d 6195	. . . . . . . . . . . 12 |- ( ( ( A e. dom vol /\ ( vol ` A ) e. RR ) /\ y e. A ) -> ( vol ` ( `' F " { 1 } ) ) = ( vol ` A ) )
62	61	oveq2d 6666	. . . . . . . . . . 11 |- ( ( ( A e. dom vol /\ ( vol ` A ) e. RR ) /\ y e. A ) -> ( 1 x. ( vol ` ( `' F " { 1 } ) ) ) = ( 1 x. ( vol ` A ) ) )
63		simplr 792	. . . . . . . . . . . . 13 |- ( ( ( A e. dom vol /\ ( vol ` A ) e. RR ) /\ y e. A ) -> ( vol ` A ) e. RR )
64	63	recnd 10068	. . . . . . . . . . . 12 |- ( ( ( A e. dom vol /\ ( vol ` A ) e. RR ) /\ y e. A ) -> ( vol ` A ) e. CC )
65	64	mulid2d 10058	. . . . . . . . . . 11 |- ( ( ( A e. dom vol /\ ( vol ` A ) e. RR ) /\ y e. A ) -> ( 1 x. ( vol ` A ) ) = ( vol ` A ) )
66	62, 65	eqtrd 2656	. . . . . . . . . 10 |- ( ( ( A e. dom vol /\ ( vol ` A ) e. RR ) /\ y e. A ) -> ( 1 x. ( vol ` ( `' F " { 1 } ) ) ) = ( vol ` A ) )
67	66, 64	eqeltrd 2701	. . . . . . . . 9 |- ( ( ( A e. dom vol /\ ( vol ` A ) e. RR ) /\ y e. A ) -> ( 1 x. ( vol ` ( `' F " { 1 } ) ) ) e. CC )
68		id 22	. . . . . . . . . . 11 |- ( z = 1 -> z = 1 )
69		sneq 4187	. . . . . . . . . . . . 13 |- ( z = 1 -> { z } = { 1 } )
70	69	imaeq2d 5466	. . . . . . . . . . . 12 |- ( z = 1 -> ( `' F " { z } ) = ( `' F " { 1 } ) )
71	70	fveq2d 6195	. . . . . . . . . . 11 |- ( z = 1 -> ( vol ` ( `' F " { z } ) ) = ( vol ` ( `' F " { 1 } ) ) )
72	68, 71	oveq12d 6668	. . . . . . . . . 10 |- ( z = 1 -> ( z x. ( vol ` ( `' F " { z } ) ) ) = ( 1 x. ( vol ` ( `' F " { 1 } ) ) ) )
73	72	sumsn 14475	. . . . . . . . 9 |- ( ( 1 e. RR /\ ( 1 x. ( vol ` ( `' F " { 1 } ) ) ) e. CC ) -> sum_ z e. { 1 } ( z x. ( vol ` ( `' F " { z } ) ) ) = ( 1 x. ( vol ` ( `' F " { 1 } ) ) ) )
74	58, 67, 73	sylancr 695	. . . . . . . 8 |- ( ( ( A e. dom vol /\ ( vol ` A ) e. RR ) /\ y e. A ) -> sum_ z e. { 1 } ( z x. ( vol ` ( `' F " { z } ) ) ) = ( 1 x. ( vol ` ( `' F " { 1 } ) ) ) )
75	74, 66	eqtrd 2656	. . . . . . 7 |- ( ( ( A e. dom vol /\ ( vol ` A ) e. RR ) /\ y e. A ) -> sum_ z e. { 1 } ( z x. ( vol ` ( `' F " { z } ) ) ) = ( vol ` A ) )
76	57, 75	eqtrd 2656	. . . . . 6 |- ( ( ( A e. dom vol /\ ( vol ` A ) e. RR ) /\ y e. A ) -> sum_ z e. ( ran F \ { 0 } ) ( z x. ( vol ` ( `' F " { z } ) ) ) = ( vol ` A ) )
77	23, 76	eqtrd 2656	. . . . 5 |- ( ( ( A e. dom vol /\ ( vol ` A ) e. RR ) /\ y e. A ) -> ( S.1 ` F ) = ( vol ` A ) )
78	77	ex 450	. . . 4 |- ( ( A e. dom vol /\ ( vol ` A ) e. RR ) -> ( y e. A -> ( S.1 ` F ) = ( vol ` A ) ) )
79	78	exlimdv 1861	. . 3 |- ( ( A e. dom vol /\ ( vol ` A ) e. RR ) -> ( E. y y e. A -> ( S.1 ` F ) = ( vol ` A ) ) )
80	19, 79	syl5bi 232	. 2 |- ( ( A e. dom vol /\ ( vol ` A ) e. RR ) -> ( A =/= (/) -> ( S.1 ` F ) = ( vol ` A ) ) )
81	18, 80	pm2.61dne 2880	1 |- ( ( A e. dom vol /\ ( vol ` A ) e. RR ) -> ( S.1 ` F ) = ( vol ` A ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   =/= wne 2794   \ cdif 3571   C_ wss 3574   (/)c0 3915   ifcif 4086   {csn 4177   {cpr 4179   |-> cmpt 4729   X. cxp 5112   `'ccnv 5113   dom cdm 5114   ran crn 5115   "cima 5117   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   x. cmul 9941   sum_csu 14416   vol*covol 23231   volcvol 23232   S.1citg1 23384

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

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-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-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

This theorem is referenced by:  itg2const  23507  itg2addnclem  33461