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Theorem itg1ge0a 19556
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 19522 . . . . 5  |-  ( F  e.  dom  S.1  ->  ran 
F  e.  Fin )
31, 2syl 16 . . . 4  |-  ( ph  ->  ran  F  e.  Fin )
4 difss 3434 . . . 4  |-  ( ran 
F  \  { 0 } )  C_  ran  F
5 ssfi 7288 . . . 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 19521 . . . . . . . 8  |-  ( F  e.  dom  S.1  ->  F : RR --> RR )
81, 7syl 16 . . . . . . 7  |-  ( ph  ->  F : RR --> RR )
9 frn 5556 . . . . . . 7  |-  ( F : RR --> RR  ->  ran 
F  C_  RR )
108, 9syl 16 . . . . . 6  |-  ( ph  ->  ran  F  C_  RR )
1110ssdifssd 3445 . . . . 5  |-  ( ph  ->  ( ran  F  \  { 0 } ) 
C_  RR )
1211sselda 3308 . . . 4  |-  ( (
ph  /\  k  e.  ( ran  F  \  {
0 } ) )  ->  k  e.  RR )
13 i1fima2sn 19525 . . . . 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 9072 . . 3  |-  ( (
ph  /\  k  e.  ( ran  F  \  {
0 } ) )  ->  ( k  x.  ( vol `  ( `' F " { k } ) ) )  e.  RR )
16 0le0 10037 . . . . 5  |-  0  <_  0
17 i1fima 19523 . . . . . . . . . . 11  |-  ( F  e.  dom  S.1  ->  ( `' F " { k } )  e.  dom  vol )
181, 17syl 16 . . . . . . . . . 10  |-  ( ph  ->  ( `' F " { k } )  e.  dom  vol )
19 mblvol 19379 . . . . . . . . . 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 5550 . . . . . . . . . . . . . 14  |-  ( F : RR --> RR  ->  F  Fn  RR )
238, 22syl 16 . . . . . . . . . . . . 13  |-  ( ph  ->  F  Fn  RR )
24 fniniseg 5810 . . . . . . . . . . . . 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 3290 . . . . . . . . . . . . . . 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 4184 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  k  e.  ( ran  F  \  { 0 } ) )  /\  ( x  e.  RR  /\  ( F `  x )  =  k ) )  ->  ( 0  <_ 
( F `  x
)  <->  0  <_  k
) )
34 0re 9047 . . . . . . . . . . . . . . . . . . 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 9175 . . . . . . . . . . . . . . . . 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 3314 . . . . . . . . 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 19336 . . . . . . . . 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 2436 . . . . . . 7  |-  ( ( ( ph  /\  k  e.  ( ran  F  \  { 0 } ) )  /\  k  <  0 )  ->  ( vol `  ( `' F " { k } ) )  =  0 )
5352oveq2d 6056 . . . . . 6  |-  ( ( ( ph  /\  k  e.  ( ran  F  \  { 0 } ) )  /\  k  <  0 )  ->  (
k  x.  ( vol `  ( `' F " { k } ) ) )  =  ( k  x.  0 ) )
5412recnd 9070 . . . . . . . 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 9221 . . . . . 6  |-  ( ( ( ph  /\  k  e.  ( ran  F  \  { 0 } ) )  /\  k  <  0 )  ->  (
k  x.  0 )  =  0 )
5753, 56eqtrd 2436 . . . . 5  |-  ( ( ( ph  /\  k  e.  ( ran  F  \  { 0 } ) )  /\  k  <  0 )  ->  (
k  x.  ( vol `  ( `' F " { k } ) ) )  =  0 )
5816, 57syl5breqr 4208 . . . 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 19380 . . . . . . . 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 19330 . . . . . . 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 4198 . . . . 5  |-  ( ( ( ph  /\  k  e.  ( ran  F  \  { 0 } ) )  /\  0  <_ 
k )  ->  0  <_  ( vol `  ( `' F " { k } ) ) )
6959, 60, 61, 68mulge0d 9559 . . . 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 9137 . . 3  |-  ( (
ph  /\  k  e.  ( ran  F  \  {
0 } ) )  ->  0  <_  (
k  x.  ( vol `  ( `' F " { k } ) ) ) )
726, 15, 71fsumge0 12529 . 2  |-  ( ph  ->  0  <_  sum_ k  e.  ( ran  F  \  { 0 } ) ( k  x.  ( vol `  ( `' F " { k } ) ) ) )
73 itg1val 19528 . . 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 4198 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 1721    \ cdif 3277    C_ wss 3280   {csn 3774   class class class wbr 4172   `'ccnv 4836   dom cdm 4837   ran crn 4838   "cima 4840    Fn wfn 5408   -->wf 5409   ` cfv 5413  (class class class)co 6040   Fincfn 7068   CCcc 8944   RRcr 8945   0cc0 8946    x. cmul 8951    < clt 9076    <_ cle 9077   sum_csu 12434   vol *covol 19312   volcvol 19313   S.1citg1 19460
This theorem is referenced by:  itg1lea  19557
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-er 6864  df-map 6979  df-pm 6980  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-sup 7404  df-oi 7435  df-card 7782  df-cda 8004  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-n0 10178  df-z 10239  df-uz 10445  df-q 10531  df-rp 10569  df-xadd 10667  df-ioo 10876  df-ico 10878  df-icc 10879  df-fz 11000  df-fzo 11091  df-fl 11157  df-seq 11279  df-exp 11338  df-hash 11574  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-clim 12237  df-sum 12435  df-xmet 16650  df-met 16651  df-ovol 19314  df-vol 19315  df-mbf 19465  df-itg1 19466
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