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Theorem itg1ge0a 19605
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.
StepHypRef Expression
1 itg10a.1 . . . . 5  |-  ( ph  ->  F  e.  dom  S.1 )
2 i1frn 19571 . . . . 5  |-  ( F  e.  dom  S.1  ->  ran 
F  e.  Fin )
31, 2syl 16 . . . 4  |-  ( ph  ->  ran  F  e.  Fin )
4 difss 3476 . . . 4  |-  ( ran 
F  \  { 0 } )  C_  ran  F
5 ssfi 7331 . . . 4  |-  ( ( ran  F  e.  Fin  /\  ( ran  F  \  { 0 } ) 
C_  ran  F )  ->  ( ran  F  \  { 0 } )  e.  Fin )
63, 4, 5sylancl 645 . . 3  |-  ( ph  ->  ( ran  F  \  { 0 } )  e.  Fin )
7 i1ff 19570 . . . . . . . 8  |-  ( F  e.  dom  S.1  ->  F : RR --> RR )
81, 7syl 16 . . . . . . 7  |-  ( ph  ->  F : RR --> RR )
9 frn 5599 . . . . . . 7  |-  ( F : RR --> RR  ->  ran 
F  C_  RR )
108, 9syl 16 . . . . . 6  |-  ( ph  ->  ran  F  C_  RR )
1110ssdifssd 3487 . . . . 5  |-  ( ph  ->  ( ran  F  \  { 0 } ) 
C_  RR )
1211sselda 3350 . . . 4  |-  ( (
ph  /\  k  e.  ( ran  F  \  {
0 } ) )  ->  k  e.  RR )
13 i1fima2sn 19574 . . . . 5  |-  ( ( F  e.  dom  S.1  /\  k  e.  ( ran 
F  \  { 0 } ) )  -> 
( vol `  ( `' F " { k } ) )  e.  RR )
141, 13sylan 459 . . . 4  |-  ( (
ph  /\  k  e.  ( ran  F  \  {
0 } ) )  ->  ( vol `  ( `' F " { k } ) )  e.  RR )
1512, 14remulcld 9118 . . 3  |-  ( (
ph  /\  k  e.  ( ran  F  \  {
0 } ) )  ->  ( k  x.  ( vol `  ( `' F " { k } ) ) )  e.  RR )
16 0le0 10083 . . . . 5  |-  0  <_  0
17 i1fima 19572 . . . . . . . . . . 11  |-  ( F  e.  dom  S.1  ->  ( `' F " { k } )  e.  dom  vol )
181, 17syl 16 . . . . . . . . . 10  |-  ( ph  ->  ( `' F " { k } )  e.  dom  vol )
19 mblvol 19428 . . . . . . . . . 10  |-  ( ( `' F " { k } )  e.  dom  vol 
->  ( vol `  ( `' F " { k } ) )  =  ( vol * `  ( `' F " { k } ) ) )
2018, 19syl 16 . . . . . . . . 9  |-  ( ph  ->  ( vol `  ( `' F " { k } ) )  =  ( vol * `  ( `' F " { k } ) ) )
2120ad2antrr 708 . . . . . . . 8  |-  ( ( ( ph  /\  k  e.  ( ran  F  \  { 0 } ) )  /\  k  <  0 )  ->  ( vol `  ( `' F " { k } ) )  =  ( vol
* `  ( `' F " { k } ) ) )
22 ffn 5593 . . . . . . . . . . . . . 14  |-  ( F : RR --> RR  ->  F  Fn  RR )
238, 22syl 16 . . . . . . . . . . . . 13  |-  ( ph  ->  F  Fn  RR )
24 fniniseg 5853 . . . . . . . . . . . . 13  |-  ( F  Fn  RR  ->  (
x  e.  ( `' F " { k } )  <->  ( x  e.  RR  /\  ( F `
 x )  =  k ) ) )
2523, 24syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  ( x  e.  ( `' F " { k } )  <->  ( x  e.  RR  /\  ( F `
 x )  =  k ) ) )
2625ad2antrr 708 . . . . . . . . . . 11  |-  ( ( ( ph  /\  k  e.  ( ran  F  \  { 0 } ) )  /\  k  <  0 )  ->  (
x  e.  ( `' F " { k } )  <->  ( x  e.  RR  /\  ( F `
 x )  =  k ) ) )
27 simprl 734 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  k  e.  ( ran  F  \  { 0 } ) )  /\  ( x  e.  RR  /\  ( F `  x )  =  k ) )  ->  x  e.  RR )
28 eldif 3332 . . . . . . . . . . . . . . 15  |-  ( x  e.  ( RR  \  A )  <->  ( x  e.  RR  /\  -.  x  e.  A ) )
29 itg1ge0a.4 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  x  e.  ( RR  \  A ) )  ->  0  <_  ( F `  x ) )
3029ex 425 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( x  e.  ( RR  \  A )  ->  0  <_  ( F `  x )
) )
3130ad2antrr 708 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  k  e.  ( ran  F  \  { 0 } ) )  /\  ( x  e.  RR  /\  ( F `  x )  =  k ) )  ->  ( x  e.  ( RR  \  A
)  ->  0  <_  ( F `  x ) ) )
32 simprr 735 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  k  e.  ( ran  F  \  { 0 } ) )  /\  ( x  e.  RR  /\  ( F `  x )  =  k ) )  ->  ( F `  x )  =  k )
3332breq2d 4226 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  k  e.  ( ran  F  \  { 0 } ) )  /\  ( x  e.  RR  /\  ( F `  x )  =  k ) )  ->  ( 0  <_ 
( F `  x
)  <->  0  <_  k
) )
34 0re 9093 . . . . . . . . . . . . . . . . . . 19  |-  0  e.  RR
3534a1i 11 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  k  e.  ( ran  F  \  { 0 } ) )  /\  ( x  e.  RR  /\  ( F `  x )  =  k ) )  ->  0  e.  RR )
3612adantr 453 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  k  e.  ( ran  F  \  { 0 } ) )  /\  ( x  e.  RR  /\  ( F `  x )  =  k ) )  ->  k  e.  RR )
3735, 36lenltd 9221 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  k  e.  ( ran  F  \  { 0 } ) )  /\  ( x  e.  RR  /\  ( F `  x )  =  k ) )  ->  ( 0  <_ 
k  <->  -.  k  <  0 ) )
3833, 37bitrd 246 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  k  e.  ( ran  F  \  { 0 } ) )  /\  ( x  e.  RR  /\  ( F `  x )  =  k ) )  ->  ( 0  <_ 
( F `  x
)  <->  -.  k  <  0 ) )
3931, 38sylibd 207 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  k  e.  ( ran  F  \  { 0 } ) )  /\  ( x  e.  RR  /\  ( F `  x )  =  k ) )  ->  ( x  e.  ( RR  \  A
)  ->  -.  k  <  0 ) )
4028, 39syl5bir 211 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  k  e.  ( ran  F  \  { 0 } ) )  /\  ( x  e.  RR  /\  ( F `  x )  =  k ) )  ->  ( ( x  e.  RR  /\  -.  x  e.  A )  ->  -.  k  <  0
) )
4127, 40mpand 658 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  k  e.  ( ran  F  \  { 0 } ) )  /\  ( x  e.  RR  /\  ( F `  x )  =  k ) )  ->  ( -.  x  e.  A  ->  -.  k  <  0 ) )
4241con4d 100 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  k  e.  ( ran  F  \  { 0 } ) )  /\  ( x  e.  RR  /\  ( F `  x )  =  k ) )  ->  ( k  <  0  ->  x  e.  A ) )
4342impancom 429 . . . . . . . . . . 11  |-  ( ( ( ph  /\  k  e.  ( ran  F  \  { 0 } ) )  /\  k  <  0 )  ->  (
( x  e.  RR  /\  ( F `  x
)  =  k )  ->  x  e.  A
) )
4426, 43sylbid 208 . . . . . . . . . 10  |-  ( ( ( ph  /\  k  e.  ( ran  F  \  { 0 } ) )  /\  k  <  0 )  ->  (
x  e.  ( `' F " { k } )  ->  x  e.  A ) )
4544ssrdv 3356 . . . . . . . . 9  |-  ( ( ( ph  /\  k  e.  ( ran  F  \  { 0 } ) )  /\  k  <  0 )  ->  ( `' F " { k } )  C_  A
)
46 itg10a.2 . . . . . . . . . 10  |-  ( ph  ->  A  C_  RR )
4746ad2antrr 708 . . . . . . . . 9  |-  ( ( ( ph  /\  k  e.  ( ran  F  \  { 0 } ) )  /\  k  <  0 )  ->  A  C_  RR )
48 itg10a.3 . . . . . . . . . 10  |-  ( ph  ->  ( vol * `  A )  =  0 )
4948ad2antrr 708 . . . . . . . . 9  |-  ( ( ( ph  /\  k  e.  ( ran  F  \  { 0 } ) )  /\  k  <  0 )  ->  ( vol * `  A )  =  0 )
50 ovolssnul 19385 . . . . . . . . 9  |-  ( ( ( `' F " { k } ) 
C_  A  /\  A  C_  RR  /\  ( vol
* `  A )  =  0 )  -> 
( vol * `  ( `' F " { k } ) )  =  0 )
5145, 47, 49, 50syl3anc 1185 . . . . . . . 8  |-  ( ( ( ph  /\  k  e.  ( ran  F  \  { 0 } ) )  /\  k  <  0 )  ->  ( vol * `  ( `' F " { k } ) )  =  0 )
5221, 51eqtrd 2470 . . . . . . 7  |-  ( ( ( ph  /\  k  e.  ( ran  F  \  { 0 } ) )  /\  k  <  0 )  ->  ( vol `  ( `' F " { k } ) )  =  0 )
5352oveq2d 6099 . . . . . 6  |-  ( ( ( ph  /\  k  e.  ( ran  F  \  { 0 } ) )  /\  k  <  0 )  ->  (
k  x.  ( vol `  ( `' F " { k } ) ) )  =  ( k  x.  0 ) )
5412recnd 9116 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( ran  F  \  {
0 } ) )  ->  k  e.  CC )
5554adantr 453 . . . . . . 7  |-  ( ( ( ph  /\  k  e.  ( ran  F  \  { 0 } ) )  /\  k  <  0 )  ->  k  e.  CC )
5655mul01d 9267 . . . . . 6  |-  ( ( ( ph  /\  k  e.  ( ran  F  \  { 0 } ) )  /\  k  <  0 )  ->  (
k  x.  0 )  =  0 )
5753, 56eqtrd 2470 . . . . 5  |-  ( ( ( ph  /\  k  e.  ( ran  F  \  { 0 } ) )  /\  k  <  0 )  ->  (
k  x.  ( vol `  ( `' F " { k } ) ) )  =  0 )
5816, 57syl5breqr 4250 . . . 4  |-  ( ( ( ph  /\  k  e.  ( ran  F  \  { 0 } ) )  /\  k  <  0 )  ->  0  <_  ( k  x.  ( vol `  ( `' F " { k } ) ) ) )
5912adantr 453 . . . . 5  |-  ( ( ( ph  /\  k  e.  ( ran  F  \  { 0 } ) )  /\  0  <_ 
k )  ->  k  e.  RR )
6014adantr 453 . . . . 5  |-  ( ( ( ph  /\  k  e.  ( ran  F  \  { 0 } ) )  /\  0  <_ 
k )  ->  ( vol `  ( `' F " { k } ) )  e.  RR )
61 simpr 449 . . . . 5  |-  ( ( ( ph  /\  k  e.  ( ran  F  \  { 0 } ) )  /\  0  <_ 
k )  ->  0  <_  k )
6218ad2antrr 708 . . . . . . . 8  |-  ( ( ( ph  /\  k  e.  ( ran  F  \  { 0 } ) )  /\  0  <_ 
k )  ->  ( `' F " { k } )  e.  dom  vol )
63 mblss 19429 . . . . . . . 8  |-  ( ( `' F " { k } )  e.  dom  vol 
->  ( `' F " { k } ) 
C_  RR )
6462, 63syl 16 . . . . . . 7  |-  ( ( ( ph  /\  k  e.  ( ran  F  \  { 0 } ) )  /\  0  <_ 
k )  ->  ( `' F " { k } )  C_  RR )
65 ovolge0 19379 . . . . . . 7  |-  ( ( `' F " { k } )  C_  RR  ->  0  <_  ( vol * `
 ( `' F " { k } ) ) )
6664, 65syl 16 . . . . . 6  |-  ( ( ( ph  /\  k  e.  ( ran  F  \  { 0 } ) )  /\  0  <_ 
k )  ->  0  <_  ( vol * `  ( `' F " { k } ) ) )
6720ad2antrr 708 . . . . . 6  |-  ( ( ( ph  /\  k  e.  ( ran  F  \  { 0 } ) )  /\  0  <_ 
k )  ->  ( vol `  ( `' F " { k } ) )  =  ( vol
* `  ( `' F " { k } ) ) )
6866, 67breqtrrd 4240 . . . . 5  |-  ( ( ( ph  /\  k  e.  ( ran  F  \  { 0 } ) )  /\  0  <_ 
k )  ->  0  <_  ( vol `  ( `' F " { k } ) ) )
6959, 60, 61, 68mulge0d 9605 . . . 4  |-  ( ( ( ph  /\  k  e.  ( ran  F  \  { 0 } ) )  /\  0  <_ 
k )  ->  0  <_  ( k  x.  ( vol `  ( `' F " { k } ) ) ) )
7034a1i 11 . . . 4  |-  ( (
ph  /\  k  e.  ( ran  F  \  {
0 } ) )  ->  0  e.  RR )
7158, 69, 12, 70ltlecasei 9183 . . 3  |-  ( (
ph  /\  k  e.  ( ran  F  \  {
0 } ) )  ->  0  <_  (
k  x.  ( vol `  ( `' F " { k } ) ) ) )
726, 15, 71fsumge0 12576 . 2  |-  ( ph  ->  0  <_  sum_ k  e.  ( ran  F  \  { 0 } ) ( k  x.  ( vol `  ( `' F " { k } ) ) ) )
73 itg1val 19577 . . 3  |-  ( F  e.  dom  S.1  ->  ( S.1 `  F )  =  sum_ k  e.  ( ran  F  \  {
0 } ) ( k  x.  ( vol `  ( `' F " { k } ) ) ) )
741, 73syl 16 . 2  |-  ( ph  ->  ( S.1 `  F
)  =  sum_ k  e.  ( ran  F  \  { 0 } ) ( k  x.  ( vol `  ( `' F " { k } ) ) ) )
7572, 74breqtrrd 4240 1  |-  ( ph  ->  0  <_  ( S.1 `  F ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726    \ cdif 3319    C_ wss 3322   {csn 3816   class class class wbr 4214   `'ccnv 4879   dom cdm 4880   ran crn 4881   "cima 4883    Fn wfn 5451   -->wf 5452   ` cfv 5456  (class class class)co 6083   Fincfn 7111   CCcc 8990   RRcr 8991   0cc0 8992    x. cmul 8997    < clt 9122    <_ cle 9123   sum_csu 12481   vol *covol 19361   volcvol 19362   S.1citg1 19509
This theorem is referenced by:  itg1lea  19606
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-inf2 7598  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069  ax-pre-sup 9070
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-se 4544  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-isom 5465  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-of 6307  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-1o 6726  df-2o 6727  df-oadd 6730  df-er 6907  df-map 7022  df-pm 7023  df-en 7112  df-dom 7113  df-sdom 7114  df-fin 7115  df-sup 7448  df-oi 7481  df-card 7828  df-cda 8050  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-div 9680  df-nn 10003  df-2 10060  df-3 10061  df-n0 10224  df-z 10285  df-uz 10491  df-q 10577  df-rp 10615  df-xadd 10713  df-ioo 10922  df-ico 10924  df-icc 10925  df-fz 11046  df-fzo 11138  df-fl 11204  df-seq 11326  df-exp 11385  df-hash 11621  df-cj 11906  df-re 11907  df-im 11908  df-sqr 12042  df-abs 12043  df-clim 12284  df-sum 12482  df-xmet 16697  df-met 16698  df-ovol 19363  df-vol 19364  df-mbf 19514  df-itg1 19515
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