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Theorem itg1ge0a 19471
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 19437 . . . . 5  |-  ( F  e.  dom  S.1  ->  ran 
F  e.  Fin )
31, 2syl 16 . . . 4  |-  ( ph  ->  ran  F  e.  Fin )
4 difss 3418 . . . 4  |-  ( ran 
F  \  { 0 } )  C_  ran  F
5 ssfi 7266 . . . 4  |-  ( ( ran  F  e.  Fin  /\  ( ran  F  \  { 0 } ) 
C_  ran  F )  ->  ( ran  F  \  { 0 } )  e.  Fin )
63, 4, 5sylancl 644 . . 3  |-  ( ph  ->  ( ran  F  \  { 0 } )  e.  Fin )
7 i1ff 19436 . . . . . . . 8  |-  ( F  e.  dom  S.1  ->  F : RR --> RR )
81, 7syl 16 . . . . . . 7  |-  ( ph  ->  F : RR --> RR )
9 frn 5538 . . . . . . 7  |-  ( F : RR --> RR  ->  ran 
F  C_  RR )
108, 9syl 16 . . . . . 6  |-  ( ph  ->  ran  F  C_  RR )
1110ssdifssd 3429 . . . . 5  |-  ( ph  ->  ( ran  F  \  { 0 } ) 
C_  RR )
1211sselda 3292 . . . 4  |-  ( (
ph  /\  k  e.  ( ran  F  \  {
0 } ) )  ->  k  e.  RR )
13 i1fima2sn 19440 . . . . 5  |-  ( ( F  e.  dom  S.1  /\  k  e.  ( ran 
F  \  { 0 } ) )  -> 
( vol `  ( `' F " { k } ) )  e.  RR )
141, 13sylan 458 . . . 4  |-  ( (
ph  /\  k  e.  ( ran  F  \  {
0 } ) )  ->  ( vol `  ( `' F " { k } ) )  e.  RR )
1512, 14remulcld 9050 . . 3  |-  ( (
ph  /\  k  e.  ( ran  F  \  {
0 } ) )  ->  ( k  x.  ( vol `  ( `' F " { k } ) ) )  e.  RR )
16 0le0 10014 . . . . 5  |-  0  <_  0
17 i1fima 19438 . . . . . . . . . . 11  |-  ( F  e.  dom  S.1  ->  ( `' F " { k } )  e.  dom  vol )
181, 17syl 16 . . . . . . . . . 10  |-  ( ph  ->  ( `' F " { k } )  e.  dom  vol )
19 mblvol 19294 . . . . . . . . . 10  |-  ( ( `' F " { k } )  e.  dom  vol 
->  ( vol `  ( `' F " { k } ) )  =  ( vol * `  ( `' F " { k } ) ) )
2018, 19syl 16 . . . . . . . . 9  |-  ( ph  ->  ( vol `  ( `' F " { k } ) )  =  ( vol * `  ( `' F " { k } ) ) )
2120ad2antrr 707 . . . . . . . 8  |-  ( ( ( ph  /\  k  e.  ( ran  F  \  { 0 } ) )  /\  k  <  0 )  ->  ( vol `  ( `' F " { k } ) )  =  ( vol
* `  ( `' F " { k } ) ) )
22 ffn 5532 . . . . . . . . . . . . . 14  |-  ( F : RR --> RR  ->  F  Fn  RR )
238, 22syl 16 . . . . . . . . . . . . 13  |-  ( ph  ->  F  Fn  RR )
24 fniniseg 5791 . . . . . . . . . . . . 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 707 . . . . . . . . . . 11  |-  ( ( ( ph  /\  k  e.  ( ran  F  \  { 0 } ) )  /\  k  <  0 )  ->  (
x  e.  ( `' F " { k } )  <->  ( x  e.  RR  /\  ( F `
 x )  =  k ) ) )
27 simprl 733 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  k  e.  ( ran  F  \  { 0 } ) )  /\  ( x  e.  RR  /\  ( F `  x )  =  k ) )  ->  x  e.  RR )
28 eldif 3274 . . . . . . . . . . . . . . 15  |-  ( x  e.  ( RR  \  A )  <->  ( x  e.  RR  /\  -.  x  e.  A ) )
29 itg1ge0a.4 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  x  e.  ( RR  \  A ) )  ->  0  <_  ( F `  x ) )
3029ex 424 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( x  e.  ( RR  \  A )  ->  0  <_  ( F `  x )
) )
3130ad2antrr 707 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  k  e.  ( ran  F  \  { 0 } ) )  /\  ( x  e.  RR  /\  ( F `  x )  =  k ) )  ->  ( x  e.  ( RR  \  A
)  ->  0  <_  ( F `  x ) ) )
32 simprr 734 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  k  e.  ( ran  F  \  { 0 } ) )  /\  ( x  e.  RR  /\  ( F `  x )  =  k ) )  ->  ( F `  x )  =  k )
3332breq2d 4166 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  k  e.  ( ran  F  \  { 0 } ) )  /\  ( x  e.  RR  /\  ( F `  x )  =  k ) )  ->  ( 0  <_ 
( F `  x
)  <->  0  <_  k
) )
34 0re 9025 . . . . . . . . . . . . . . . . . . 19  |-  0  e.  RR
3534a1i 11 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  k  e.  ( ran  F  \  { 0 } ) )  /\  ( x  e.  RR  /\  ( F `  x )  =  k ) )  ->  0  e.  RR )
3612adantr 452 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  k  e.  ( ran  F  \  { 0 } ) )  /\  ( x  e.  RR  /\  ( F `  x )  =  k ) )  ->  k  e.  RR )
3735, 36lenltd 9152 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  k  e.  ( ran  F  \  { 0 } ) )  /\  ( x  e.  RR  /\  ( F `  x )  =  k ) )  ->  ( 0  <_ 
k  <->  -.  k  <  0 ) )
3833, 37bitrd 245 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  k  e.  ( ran  F  \  { 0 } ) )  /\  ( x  e.  RR  /\  ( F `  x )  =  k ) )  ->  ( 0  <_ 
( F `  x
)  <->  -.  k  <  0 ) )
3931, 38sylibd 206 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  k  e.  ( ran  F  \  { 0 } ) )  /\  ( x  e.  RR  /\  ( F `  x )  =  k ) )  ->  ( x  e.  ( RR  \  A
)  ->  -.  k  <  0 ) )
4028, 39syl5bir 210 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  k  e.  ( ran  F  \  { 0 } ) )  /\  ( x  e.  RR  /\  ( F `  x )  =  k ) )  ->  ( ( x  e.  RR  /\  -.  x  e.  A )  ->  -.  k  <  0
) )
4127, 40mpand 657 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  k  e.  ( ran  F  \  { 0 } ) )  /\  ( x  e.  RR  /\  ( F `  x )  =  k ) )  ->  ( -.  x  e.  A  ->  -.  k  <  0 ) )
4241con4d 99 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  k  e.  ( ran  F  \  { 0 } ) )  /\  ( x  e.  RR  /\  ( F `  x )  =  k ) )  ->  ( k  <  0  ->  x  e.  A ) )
4342impancom 428 . . . . . . . . . . 11  |-  ( ( ( ph  /\  k  e.  ( ran  F  \  { 0 } ) )  /\  k  <  0 )  ->  (
( x  e.  RR  /\  ( F `  x
)  =  k )  ->  x  e.  A
) )
4426, 43sylbid 207 . . . . . . . . . 10  |-  ( ( ( ph  /\  k  e.  ( ran  F  \  { 0 } ) )  /\  k  <  0 )  ->  (
x  e.  ( `' F " { k } )  ->  x  e.  A ) )
4544ssrdv 3298 . . . . . . . . 9  |-  ( ( ( ph  /\  k  e.  ( ran  F  \  { 0 } ) )  /\  k  <  0 )  ->  ( `' F " { k } )  C_  A
)
46 itg10a.2 . . . . . . . . . 10  |-  ( ph  ->  A  C_  RR )
4746ad2antrr 707 . . . . . . . . 9  |-  ( ( ( ph  /\  k  e.  ( ran  F  \  { 0 } ) )  /\  k  <  0 )  ->  A  C_  RR )
48 itg10a.3 . . . . . . . . . 10  |-  ( ph  ->  ( vol * `  A )  =  0 )
4948ad2antrr 707 . . . . . . . . 9  |-  ( ( ( ph  /\  k  e.  ( ran  F  \  { 0 } ) )  /\  k  <  0 )  ->  ( vol * `  A )  =  0 )
50 ovolssnul 19251 . . . . . . . . 9  |-  ( ( ( `' F " { k } ) 
C_  A  /\  A  C_  RR  /\  ( vol
* `  A )  =  0 )  -> 
( vol * `  ( `' F " { k } ) )  =  0 )
5145, 47, 49, 50syl3anc 1184 . . . . . . . 8  |-  ( ( ( ph  /\  k  e.  ( ran  F  \  { 0 } ) )  /\  k  <  0 )  ->  ( vol * `  ( `' F " { k } ) )  =  0 )
5221, 51eqtrd 2420 . . . . . . 7  |-  ( ( ( ph  /\  k  e.  ( ran  F  \  { 0 } ) )  /\  k  <  0 )  ->  ( vol `  ( `' F " { k } ) )  =  0 )
5352oveq2d 6037 . . . . . 6  |-  ( ( ( ph  /\  k  e.  ( ran  F  \  { 0 } ) )  /\  k  <  0 )  ->  (
k  x.  ( vol `  ( `' F " { k } ) ) )  =  ( k  x.  0 ) )
5412recnd 9048 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( ran  F  \  {
0 } ) )  ->  k  e.  CC )
5554adantr 452 . . . . . . 7  |-  ( ( ( ph  /\  k  e.  ( ran  F  \  { 0 } ) )  /\  k  <  0 )  ->  k  e.  CC )
5655mul01d 9198 . . . . . 6  |-  ( ( ( ph  /\  k  e.  ( ran  F  \  { 0 } ) )  /\  k  <  0 )  ->  (
k  x.  0 )  =  0 )
5753, 56eqtrd 2420 . . . . 5  |-  ( ( ( ph  /\  k  e.  ( ran  F  \  { 0 } ) )  /\  k  <  0 )  ->  (
k  x.  ( vol `  ( `' F " { k } ) ) )  =  0 )
5816, 57syl5breqr 4190 . . . 4  |-  ( ( ( ph  /\  k  e.  ( ran  F  \  { 0 } ) )  /\  k  <  0 )  ->  0  <_  ( k  x.  ( vol `  ( `' F " { k } ) ) ) )
5912adantr 452 . . . . 5  |-  ( ( ( ph  /\  k  e.  ( ran  F  \  { 0 } ) )  /\  0  <_ 
k )  ->  k  e.  RR )
6014adantr 452 . . . . 5  |-  ( ( ( ph  /\  k  e.  ( ran  F  \  { 0 } ) )  /\  0  <_ 
k )  ->  ( vol `  ( `' F " { k } ) )  e.  RR )
61 simpr 448 . . . . 5  |-  ( ( ( ph  /\  k  e.  ( ran  F  \  { 0 } ) )  /\  0  <_ 
k )  ->  0  <_  k )
6218ad2antrr 707 . . . . . . . 8  |-  ( ( ( ph  /\  k  e.  ( ran  F  \  { 0 } ) )  /\  0  <_ 
k )  ->  ( `' F " { k } )  e.  dom  vol )
63 mblss 19295 . . . . . . . 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 19245 . . . . . . 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 707 . . . . . 6  |-  ( ( ( ph  /\  k  e.  ( ran  F  \  { 0 } ) )  /\  0  <_ 
k )  ->  ( vol `  ( `' F " { k } ) )  =  ( vol
* `  ( `' F " { k } ) ) )
6866, 67breqtrrd 4180 . . . . 5  |-  ( ( ( ph  /\  k  e.  ( ran  F  \  { 0 } ) )  /\  0  <_ 
k )  ->  0  <_  ( vol `  ( `' F " { k } ) ) )
6959, 60, 61, 68mulge0d 9536 . . . 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 9115 . . 3  |-  ( (
ph  /\  k  e.  ( ran  F  \  {
0 } ) )  ->  0  <_  (
k  x.  ( vol `  ( `' F " { k } ) ) ) )
726, 15, 71fsumge0 12502 . 2  |-  ( ph  ->  0  <_  sum_ k  e.  ( ran  F  \  { 0 } ) ( k  x.  ( vol `  ( `' F " { k } ) ) ) )
73 itg1val 19443 . . 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 4180 1  |-  ( ph  ->  0  <_  ( S.1 `  F ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717    \ cdif 3261    C_ wss 3264   {csn 3758   class class class wbr 4154   `'ccnv 4818   dom cdm 4819   ran crn 4820   "cima 4822    Fn wfn 5390   -->wf 5391   ` cfv 5395  (class class class)co 6021   Fincfn 7046   CCcc 8922   RRcr 8923   0cc0 8924    x. cmul 8929    < clt 9054    <_ cle 9055   sum_csu 12407   vol *covol 19227   volcvol 19228   S.1citg1 19375
This theorem is referenced by:  itg1lea  19472
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-inf2 7530  ax-cnex 8980  ax-resscn 8981  ax-1cn 8982  ax-icn 8983  ax-addcl 8984  ax-addrcl 8985  ax-mulcl 8986  ax-mulrcl 8987  ax-mulcom 8988  ax-addass 8989  ax-mulass 8990  ax-distr 8991  ax-i2m1 8992  ax-1ne0 8993  ax-1rid 8994  ax-rnegex 8995  ax-rrecex 8996  ax-cnre 8997  ax-pre-lttri 8998  ax-pre-lttrn 8999  ax-pre-ltadd 9000  ax-pre-mulgt0 9001  ax-pre-sup 9002
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rmo 2658  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-int 3994  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-se 4484  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-isom 5404  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-of 6245  df-1st 6289  df-2nd 6290  df-riota 6486  df-recs 6570  df-rdg 6605  df-1o 6661  df-2o 6662  df-oadd 6665  df-er 6842  df-map 6957  df-pm 6958  df-en 7047  df-dom 7048  df-sdom 7049  df-fin 7050  df-sup 7382  df-oi 7413  df-card 7760  df-cda 7982  df-pnf 9056  df-mnf 9057  df-xr 9058  df-ltxr 9059  df-le 9060  df-sub 9226  df-neg 9227  df-div 9611  df-nn 9934  df-2 9991  df-3 9992  df-n0 10155  df-z 10216  df-uz 10422  df-q 10508  df-rp 10546  df-xadd 10644  df-ioo 10853  df-ico 10855  df-icc 10856  df-fz 10977  df-fzo 11067  df-fl 11130  df-seq 11252  df-exp 11311  df-hash 11547  df-cj 11832  df-re 11833  df-im 11834  df-sqr 11968  df-abs 11969  df-clim 12210  df-sum 12408  df-xmet 16620  df-met 16621  df-ovol 19229  df-vol 19230  df-mbf 19380  df-itg1 19381
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