Theorem itg1ge0a 23478

Description: The integral of an almost positive simple function is positive. (Contributed by Mario Carneiro, 11-Aug-2014.)

Hypotheses

Ref	Expression
itg10a.1	|- ( ph -> F e. dom S.1 )
itg10a.2	|- ( ph -> A C_ RR )
itg10a.3	|- ( ph -> ( vol* ` A ) = 0 )
itg1ge0a.4	|- ( ( ph /\ x e. ( RR \ A ) ) -> 0 <_ ( F ` x ) )

Assertion

Ref	Expression
itg1ge0a	|- ( ph -> 0 <_ ( S.1 ` F ) )

Distinct variable groups:   x, A   x, F   ph, x

Proof of Theorem itg1ge0a

Dummy variable k is distinct from all other variables.

Step	Hyp	Ref	Expression
1		itg10a.1	. . . . 5 |- ( ph -> F e. dom S.1 )
2		i1frn 23444	. . . . 5 |- ( F e. dom S.1 -> ran F e. Fin )
3	1, 2	syl 17	. . . 4 |- ( ph -> ran F e. Fin )
4		difss 3737	. . . 4 |- ( ran F \ { 0 } ) C_ ran F
5		ssfi 8180	. . . 4 |- ( ( ran F e. Fin /\ ( ran F \ { 0 } ) C_ ran F ) -> ( ran F \ { 0 } ) e. Fin )
6	3, 4, 5	sylancl 694	. . 3 |- ( ph -> ( ran F \ { 0 } ) e. Fin )
7		i1ff 23443	. . . . . . . 8 |- ( F e. dom S.1 -> F : RR --> RR )
8	1, 7	syl 17	. . . . . . 7 |- ( ph -> F : RR --> RR )
9		frn 6053	. . . . . . 7 |- ( F : RR --> RR -> ran F C_ RR )
10	8, 9	syl 17	. . . . . 6 |- ( ph -> ran F C_ RR )
11	10	ssdifssd 3748	. . . . 5 |- ( ph -> ( ran F \ { 0 } ) C_ RR )
12	11	sselda 3603	. . . 4 |- ( ( ph /\ k e. ( ran F \ { 0 } ) ) -> k e. RR )
13		i1fima2sn 23447	. . . . 5 |- ( ( F e. dom S.1 /\ k e. ( ran F \ { 0 } ) ) -> ( vol ` ( `' F " { k } ) ) e. RR )
14	1, 13	sylan 488	. . . 4 |- ( ( ph /\ k e. ( ran F \ { 0 } ) ) -> ( vol ` ( `' F " { k } ) ) e. RR )
15	12, 14	remulcld 10070	. . 3 |- ( ( ph /\ k e. ( ran F \ { 0 } ) ) -> ( k x. ( vol ` ( `' F " { k } ) ) ) e. RR )
16		0le0 11110	. . . . 5 |- 0 <_ 0
17		i1fima 23445	. . . . . . . . . . 11 |- ( F e. dom S.1 -> ( `' F " { k } ) e. dom vol )
18	1, 17	syl 17	. . . . . . . . . 10 |- ( ph -> ( `' F " { k } ) e. dom vol )
19		mblvol 23298	. . . . . . . . . 10 |- ( ( `' F " { k } ) e. dom vol -> ( vol ` ( `' F " { k } ) ) = ( vol* ` ( `' F " { k } ) ) )
20	18, 19	syl 17	. . . . . . . . 9 |- ( ph -> ( vol ` ( `' F " { k } ) ) = ( vol* ` ( `' F " { k } ) ) )
21	20	ad2antrr 762	. . . . . . . 8 |- ( ( ( ph /\ k e. ( ran F \ { 0 } ) ) /\ k < 0 ) -> ( vol ` ( `' F " { k } ) ) = ( vol* ` ( `' F " { k } ) ) )
22		ffn 6045	. . . . . . . . . . . . . 14 |- ( F : RR --> RR -> F Fn RR )
23	8, 22	syl 17	. . . . . . . . . . . . 13 |- ( ph -> F Fn RR )
24		fniniseg 6338	. . . . . . . . . . . . 13 |- ( F Fn RR -> ( x e. ( `' F " { k } ) <-> ( x e. RR /\ ( F ` x ) = k ) ) )
25	23, 24	syl 17	. . . . . . . . . . . 12 |- ( ph -> ( x e. ( `' F " { k } ) <-> ( x e. RR /\ ( F ` x ) = k ) ) )
26	25	ad2antrr 762	. . . . . . . . . . 11 |- ( ( ( ph /\ k e. ( ran F \ { 0 } ) ) /\ k < 0 ) -> ( x e. ( `' F " { k } ) <-> ( x e. RR /\ ( F ` x ) = k ) ) )
27		simprl 794	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ k e. ( ran F \ { 0 } ) ) /\ ( x e. RR /\ ( F ` x ) = k ) ) -> x e. RR )
28		eldif 3584	. . . . . . . . . . . . . . 15 |- ( x e. ( RR \ A ) <-> ( x e. RR /\ -. x e. A ) )
29		itg1ge0a.4	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ x e. ( RR \ A ) ) -> 0 <_ ( F ` x ) )
30	29	ex 450	. . . . . . . . . . . . . . . . 17 |- ( ph -> ( x e. ( RR \ A ) -> 0 <_ ( F ` x ) ) )
31	30	ad2antrr 762	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ k e. ( ran F \ { 0 } ) ) /\ ( x e. RR /\ ( F ` x ) = k ) ) -> ( x e. ( RR \ A ) -> 0 <_ ( F ` x ) ) )
32		simprr 796	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ph /\ k e. ( ran F \ { 0 } ) ) /\ ( x e. RR /\ ( F ` x ) = k ) ) -> ( F ` x ) = k )
33	32	breq2d 4665	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ k e. ( ran F \ { 0 } ) ) /\ ( x e. RR /\ ( F ` x ) = k ) ) -> ( 0 <_ ( F ` x ) <-> 0 <_ k ) )
34		0red 10041	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ph /\ k e. ( ran F \ { 0 } ) ) /\ ( x e. RR /\ ( F ` x ) = k ) ) -> 0 e. RR )
35	12	adantr 481	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ph /\ k e. ( ran F \ { 0 } ) ) /\ ( x e. RR /\ ( F ` x ) = k ) ) -> k e. RR )
36	34, 35	lenltd 10183	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ k e. ( ran F \ { 0 } ) ) /\ ( x e. RR /\ ( F ` x ) = k ) ) -> ( 0 <_ k <-> -. k < 0 ) )
37	33, 36	bitrd 268	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ k e. ( ran F \ { 0 } ) ) /\ ( x e. RR /\ ( F ` x ) = k ) ) -> ( 0 <_ ( F ` x ) <-> -. k < 0 ) )
38	31, 37	sylibd 229	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ k e. ( ran F \ { 0 } ) ) /\ ( x e. RR /\ ( F ` x ) = k ) ) -> ( x e. ( RR \ A ) -> -. k < 0 ) )
39	28, 38	syl5bir 233	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ k e. ( ran F \ { 0 } ) ) /\ ( x e. RR /\ ( F ` x ) = k ) ) -> ( ( x e. RR /\ -. x e. A ) -> -. k < 0 ) )
40	27, 39	mpand 711	. . . . . . . . . . . . 13 |- ( ( ( ph /\ k e. ( ran F \ { 0 } ) ) /\ ( x e. RR /\ ( F ` x ) = k ) ) -> ( -. x e. A -> -. k < 0 ) )
41	40	con4d 114	. . . . . . . . . . . 12 |- ( ( ( ph /\ k e. ( ran F \ { 0 } ) ) /\ ( x e. RR /\ ( F ` x ) = k ) ) -> ( k < 0 -> x e. A ) )
42	41	impancom 456	. . . . . . . . . . 11 |- ( ( ( ph /\ k e. ( ran F \ { 0 } ) ) /\ k < 0 ) -> ( ( x e. RR /\ ( F ` x ) = k ) -> x e. A ) )
43	26, 42	sylbid 230	. . . . . . . . . 10 |- ( ( ( ph /\ k e. ( ran F \ { 0 } ) ) /\ k < 0 ) -> ( x e. ( `' F " { k } ) -> x e. A ) )
44	43	ssrdv 3609	. . . . . . . . 9 |- ( ( ( ph /\ k e. ( ran F \ { 0 } ) ) /\ k < 0 ) -> ( `' F " { k } ) C_ A )
45		itg10a.2	. . . . . . . . . 10 |- ( ph -> A C_ RR )
46	45	ad2antrr 762	. . . . . . . . 9 |- ( ( ( ph /\ k e. ( ran F \ { 0 } ) ) /\ k < 0 ) -> A C_ RR )
47		itg10a.3	. . . . . . . . . 10 |- ( ph -> ( vol* ` A ) = 0 )
48	47	ad2antrr 762	. . . . . . . . 9 |- ( ( ( ph /\ k e. ( ran F \ { 0 } ) ) /\ k < 0 ) -> ( vol* ` A ) = 0 )
49		ovolssnul 23255	. . . . . . . . 9 |- ( ( ( `' F " { k } ) C_ A /\ A C_ RR /\ ( vol* ` A ) = 0 ) -> ( vol* ` ( `' F " { k } ) ) = 0 )
50	44, 46, 48, 49	syl3anc 1326	. . . . . . . 8 |- ( ( ( ph /\ k e. ( ran F \ { 0 } ) ) /\ k < 0 ) -> ( vol* ` ( `' F " { k } ) ) = 0 )
51	21, 50	eqtrd 2656	. . . . . . 7 |- ( ( ( ph /\ k e. ( ran F \ { 0 } ) ) /\ k < 0 ) -> ( vol ` ( `' F " { k } ) ) = 0 )
52	51	oveq2d 6666	. . . . . 6 |- ( ( ( ph /\ k e. ( ran F \ { 0 } ) ) /\ k < 0 ) -> ( k x. ( vol ` ( `' F " { k } ) ) ) = ( k x. 0 ) )
53	12	recnd 10068	. . . . . . . 8 |- ( ( ph /\ k e. ( ran F \ { 0 } ) ) -> k e. CC )
54	53	adantr 481	. . . . . . 7 |- ( ( ( ph /\ k e. ( ran F \ { 0 } ) ) /\ k < 0 ) -> k e. CC )
55	54	mul01d 10235	. . . . . 6 |- ( ( ( ph /\ k e. ( ran F \ { 0 } ) ) /\ k < 0 ) -> ( k x. 0 ) = 0 )
56	52, 55	eqtrd 2656	. . . . 5 |- ( ( ( ph /\ k e. ( ran F \ { 0 } ) ) /\ k < 0 ) -> ( k x. ( vol ` ( `' F " { k } ) ) ) = 0 )
57	16, 56	syl5breqr 4691	. . . 4 |- ( ( ( ph /\ k e. ( ran F \ { 0 } ) ) /\ k < 0 ) -> 0 <_ ( k x. ( vol ` ( `' F " { k } ) ) ) )
58	12	adantr 481	. . . . 5 |- ( ( ( ph /\ k e. ( ran F \ { 0 } ) ) /\ 0 <_ k ) -> k e. RR )
59	14	adantr 481	. . . . 5 |- ( ( ( ph /\ k e. ( ran F \ { 0 } ) ) /\ 0 <_ k ) -> ( vol ` ( `' F " { k } ) ) e. RR )
60		simpr 477	. . . . 5 |- ( ( ( ph /\ k e. ( ran F \ { 0 } ) ) /\ 0 <_ k ) -> 0 <_ k )
61	18	ad2antrr 762	. . . . . . . 8 |- ( ( ( ph /\ k e. ( ran F \ { 0 } ) ) /\ 0 <_ k ) -> ( `' F " { k } ) e. dom vol )
62		mblss 23299	. . . . . . . 8 |- ( ( `' F " { k } ) e. dom vol -> ( `' F " { k } ) C_ RR )
63	61, 62	syl 17	. . . . . . 7 |- ( ( ( ph /\ k e. ( ran F \ { 0 } ) ) /\ 0 <_ k ) -> ( `' F " { k } ) C_ RR )
64		ovolge0 23249	. . . . . . 7 |- ( ( `' F " { k } ) C_ RR -> 0 <_ ( vol* ` ( `' F " { k } ) ) )
65	63, 64	syl 17	. . . . . 6 |- ( ( ( ph /\ k e. ( ran F \ { 0 } ) ) /\ 0 <_ k ) -> 0 <_ ( vol* ` ( `' F " { k } ) ) )
66	20	ad2antrr 762	. . . . . 6 |- ( ( ( ph /\ k e. ( ran F \ { 0 } ) ) /\ 0 <_ k ) -> ( vol ` ( `' F " { k } ) ) = ( vol* ` ( `' F " { k } ) ) )
67	65, 66	breqtrrd 4681	. . . . 5 |- ( ( ( ph /\ k e. ( ran F \ { 0 } ) ) /\ 0 <_ k ) -> 0 <_ ( vol ` ( `' F " { k } ) ) )
68	58, 59, 60, 67	mulge0d 10604	. . . 4 |- ( ( ( ph /\ k e. ( ran F \ { 0 } ) ) /\ 0 <_ k ) -> 0 <_ ( k x. ( vol ` ( `' F " { k } ) ) ) )
69		0red 10041	. . . 4 |- ( ( ph /\ k e. ( ran F \ { 0 } ) ) -> 0 e. RR )
70	57, 68, 12, 69	ltlecasei 10145	. . 3 |- ( ( ph /\ k e. ( ran F \ { 0 } ) ) -> 0 <_ ( k x. ( vol ` ( `' F " { k } ) ) ) )
71	6, 15, 70	fsumge0 14527	. 2 |- ( ph -> 0 <_ sum_ k e. ( ran F \ { 0 } ) ( k x. ( vol ` ( `' F " { k } ) ) ) )
72		itg1val 23450	. . 3 |- ( F e. dom S.1 -> ( S.1 ` F ) = sum_ k e. ( ran F \ { 0 } ) ( k x. ( vol ` ( `' F " { k } ) ) ) )
73	1, 72	syl 17	. 2 |- ( ph -> ( S.1 ` F ) = sum_ k e. ( ran F \ { 0 } ) ( k x. ( vol ` ( `' F " { k } ) ) ) )
74	71, 73	breqtrrd 4681	1 |- ( ph -> 0 <_ ( S.1 ` F ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   \ cdif 3571   C_ wss 3574   {csn 4177   class class class wbr 4653   `'ccnv 5113   dom cdm 5114   ran crn 5115   "cima 5117   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   Fincfn 7955   CCcc 9934   RRcr 9935   0cc0 9936   x. cmul 9941   < clt 10074   <_ cle 10075   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:  itg1lea  23479