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Theorem itg1ge0a 22287
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 22253 . . . . 5  |-  ( F  e.  dom  S.1  ->  ran 
F  e.  Fin )
31, 2syl 16 . . . 4  |-  ( ph  ->  ran  F  e.  Fin )
4 difss 3617 . . . 4  |-  ( ran 
F  \  { 0 } )  C_  ran  F
5 ssfi 7733 . . . 4  |-  ( ( ran  F  e.  Fin  /\  ( ran  F  \  { 0 } ) 
C_  ran  F )  ->  ( ran  F  \  { 0 } )  e.  Fin )
63, 4, 5sylancl 660 . . 3  |-  ( ph  ->  ( ran  F  \  { 0 } )  e.  Fin )
7 i1ff 22252 . . . . . . . 8  |-  ( F  e.  dom  S.1  ->  F : RR --> RR )
81, 7syl 16 . . . . . . 7  |-  ( ph  ->  F : RR --> RR )
9 frn 5719 . . . . . . 7  |-  ( F : RR --> RR  ->  ran 
F  C_  RR )
108, 9syl 16 . . . . . 6  |-  ( ph  ->  ran  F  C_  RR )
1110ssdifssd 3628 . . . . 5  |-  ( ph  ->  ( ran  F  \  { 0 } ) 
C_  RR )
1211sselda 3489 . . . 4  |-  ( (
ph  /\  k  e.  ( ran  F  \  {
0 } ) )  ->  k  e.  RR )
13 i1fima2sn 22256 . . . . 5  |-  ( ( F  e.  dom  S.1  /\  k  e.  ( ran 
F  \  { 0 } ) )  -> 
( vol `  ( `' F " { k } ) )  e.  RR )
141, 13sylan 469 . . . 4  |-  ( (
ph  /\  k  e.  ( ran  F  \  {
0 } ) )  ->  ( vol `  ( `' F " { k } ) )  e.  RR )
1512, 14remulcld 9613 . . 3  |-  ( (
ph  /\  k  e.  ( ran  F  \  {
0 } ) )  ->  ( k  x.  ( vol `  ( `' F " { k } ) ) )  e.  RR )
16 0le0 10621 . . . . 5  |-  0  <_  0
17 i1fima 22254 . . . . . . . . . . 11  |-  ( F  e.  dom  S.1  ->  ( `' F " { k } )  e.  dom  vol )
181, 17syl 16 . . . . . . . . . 10  |-  ( ph  ->  ( `' F " { k } )  e.  dom  vol )
19 mblvol 22110 . . . . . . . . . 10  |-  ( ( `' F " { k } )  e.  dom  vol 
->  ( vol `  ( `' F " { k } ) )  =  ( vol* `  ( `' F " { k } ) ) )
2018, 19syl 16 . . . . . . . . 9  |-  ( ph  ->  ( vol `  ( `' F " { k } ) )  =  ( vol* `  ( `' F " { k } ) ) )
2120ad2antrr 723 . . . . . . . 8  |-  ( ( ( ph  /\  k  e.  ( ran  F  \  { 0 } ) )  /\  k  <  0 )  ->  ( vol `  ( `' F " { k } ) )  =  ( vol* `  ( `' F " { k } ) ) )
22 ffn 5713 . . . . . . . . . . . . . 14  |-  ( F : RR --> RR  ->  F  Fn  RR )
238, 22syl 16 . . . . . . . . . . . . 13  |-  ( ph  ->  F  Fn  RR )
24 fniniseg 5984 . . . . . . . . . . . . 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 723 . . . . . . . . . . 11  |-  ( ( ( ph  /\  k  e.  ( ran  F  \  { 0 } ) )  /\  k  <  0 )  ->  (
x  e.  ( `' F " { k } )  <->  ( x  e.  RR  /\  ( F `
 x )  =  k ) ) )
27 simprl 754 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  k  e.  ( ran  F  \  { 0 } ) )  /\  ( x  e.  RR  /\  ( F `  x )  =  k ) )  ->  x  e.  RR )
28 eldif 3471 . . . . . . . . . . . . . . 15  |-  ( x  e.  ( RR  \  A )  <->  ( x  e.  RR  /\  -.  x  e.  A ) )
29 itg1ge0a.4 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  x  e.  ( RR  \  A ) )  ->  0  <_  ( F `  x ) )
3029ex 432 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( x  e.  ( RR  \  A )  ->  0  <_  ( F `  x )
) )
3130ad2antrr 723 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  k  e.  ( ran  F  \  { 0 } ) )  /\  ( x  e.  RR  /\  ( F `  x )  =  k ) )  ->  ( x  e.  ( RR  \  A
)  ->  0  <_  ( F `  x ) ) )
32 simprr 755 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  k  e.  ( ran  F  \  { 0 } ) )  /\  ( x  e.  RR  /\  ( F `  x )  =  k ) )  ->  ( F `  x )  =  k )
3332breq2d 4451 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  k  e.  ( ran  F  \  { 0 } ) )  /\  ( x  e.  RR  /\  ( F `  x )  =  k ) )  ->  ( 0  <_ 
( F `  x
)  <->  0  <_  k
) )
34 0red 9586 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  k  e.  ( ran  F  \  { 0 } ) )  /\  ( x  e.  RR  /\  ( F `  x )  =  k ) )  ->  0  e.  RR )
3512adantr 463 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  k  e.  ( ran  F  \  { 0 } ) )  /\  ( x  e.  RR  /\  ( F `  x )  =  k ) )  ->  k  e.  RR )
3634, 35lenltd 9720 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  k  e.  ( ran  F  \  { 0 } ) )  /\  ( x  e.  RR  /\  ( F `  x )  =  k ) )  ->  ( 0  <_ 
k  <->  -.  k  <  0 ) )
3733, 36bitrd 253 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  k  e.  ( ran  F  \  { 0 } ) )  /\  ( x  e.  RR  /\  ( F `  x )  =  k ) )  ->  ( 0  <_ 
( F `  x
)  <->  -.  k  <  0 ) )
3831, 37sylibd 214 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  k  e.  ( ran  F  \  { 0 } ) )  /\  ( x  e.  RR  /\  ( F `  x )  =  k ) )  ->  ( x  e.  ( RR  \  A
)  ->  -.  k  <  0 ) )
3928, 38syl5bir 218 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  k  e.  ( ran  F  \  { 0 } ) )  /\  ( x  e.  RR  /\  ( F `  x )  =  k ) )  ->  ( ( x  e.  RR  /\  -.  x  e.  A )  ->  -.  k  <  0
) )
4027, 39mpand 673 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  k  e.  ( ran  F  \  { 0 } ) )  /\  ( x  e.  RR  /\  ( F `  x )  =  k ) )  ->  ( -.  x  e.  A  ->  -.  k  <  0 ) )
4140con4d 105 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  k  e.  ( ran  F  \  { 0 } ) )  /\  ( x  e.  RR  /\  ( F `  x )  =  k ) )  ->  ( k  <  0  ->  x  e.  A ) )
4241impancom 438 . . . . . . . . . . 11  |-  ( ( ( ph  /\  k  e.  ( ran  F  \  { 0 } ) )  /\  k  <  0 )  ->  (
( x  e.  RR  /\  ( F `  x
)  =  k )  ->  x  e.  A
) )
4326, 42sylbid 215 . . . . . . . . . 10  |-  ( ( ( ph  /\  k  e.  ( ran  F  \  { 0 } ) )  /\  k  <  0 )  ->  (
x  e.  ( `' F " { k } )  ->  x  e.  A ) )
4443ssrdv 3495 . . . . . . . . 9  |-  ( ( ( ph  /\  k  e.  ( ran  F  \  { 0 } ) )  /\  k  <  0 )  ->  ( `' F " { k } )  C_  A
)
45 itg10a.2 . . . . . . . . . 10  |-  ( ph  ->  A  C_  RR )
4645ad2antrr 723 . . . . . . . . 9  |-  ( ( ( ph  /\  k  e.  ( ran  F  \  { 0 } ) )  /\  k  <  0 )  ->  A  C_  RR )
47 itg10a.3 . . . . . . . . . 10  |-  ( ph  ->  ( vol* `  A )  =  0 )
4847ad2antrr 723 . . . . . . . . 9  |-  ( ( ( ph  /\  k  e.  ( ran  F  \  { 0 } ) )  /\  k  <  0 )  ->  ( vol* `  A )  =  0 )
49 ovolssnul 22067 . . . . . . . . 9  |-  ( ( ( `' F " { k } ) 
C_  A  /\  A  C_  RR  /\  ( vol* `  A )  =  0 )  -> 
( vol* `  ( `' F " { k } ) )  =  0 )
5044, 46, 48, 49syl3anc 1226 . . . . . . . 8  |-  ( ( ( ph  /\  k  e.  ( ran  F  \  { 0 } ) )  /\  k  <  0 )  ->  ( vol* `  ( `' F " { k } ) )  =  0 )
5121, 50eqtrd 2495 . . . . . . 7  |-  ( ( ( ph  /\  k  e.  ( ran  F  \  { 0 } ) )  /\  k  <  0 )  ->  ( vol `  ( `' F " { k } ) )  =  0 )
5251oveq2d 6286 . . . . . 6  |-  ( ( ( ph  /\  k  e.  ( ran  F  \  { 0 } ) )  /\  k  <  0 )  ->  (
k  x.  ( vol `  ( `' F " { k } ) ) )  =  ( k  x.  0 ) )
5312recnd 9611 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( ran  F  \  {
0 } ) )  ->  k  e.  CC )
5453adantr 463 . . . . . . 7  |-  ( ( ( ph  /\  k  e.  ( ran  F  \  { 0 } ) )  /\  k  <  0 )  ->  k  e.  CC )
5554mul01d 9768 . . . . . 6  |-  ( ( ( ph  /\  k  e.  ( ran  F  \  { 0 } ) )  /\  k  <  0 )  ->  (
k  x.  0 )  =  0 )
5652, 55eqtrd 2495 . . . . 5  |-  ( ( ( ph  /\  k  e.  ( ran  F  \  { 0 } ) )  /\  k  <  0 )  ->  (
k  x.  ( vol `  ( `' F " { k } ) ) )  =  0 )
5716, 56syl5breqr 4475 . . . 4  |-  ( ( ( ph  /\  k  e.  ( ran  F  \  { 0 } ) )  /\  k  <  0 )  ->  0  <_  ( k  x.  ( vol `  ( `' F " { k } ) ) ) )
5812adantr 463 . . . . 5  |-  ( ( ( ph  /\  k  e.  ( ran  F  \  { 0 } ) )  /\  0  <_ 
k )  ->  k  e.  RR )
5914adantr 463 . . . . 5  |-  ( ( ( ph  /\  k  e.  ( ran  F  \  { 0 } ) )  /\  0  <_ 
k )  ->  ( vol `  ( `' F " { k } ) )  e.  RR )
60 simpr 459 . . . . 5  |-  ( ( ( ph  /\  k  e.  ( ran  F  \  { 0 } ) )  /\  0  <_ 
k )  ->  0  <_  k )
6118ad2antrr 723 . . . . . . . 8  |-  ( ( ( ph  /\  k  e.  ( ran  F  \  { 0 } ) )  /\  0  <_ 
k )  ->  ( `' F " { k } )  e.  dom  vol )
62 mblss 22111 . . . . . . . 8  |-  ( ( `' F " { k } )  e.  dom  vol 
->  ( `' F " { k } ) 
C_  RR )
6361, 62syl 16 . . . . . . 7  |-  ( ( ( ph  /\  k  e.  ( ran  F  \  { 0 } ) )  /\  0  <_ 
k )  ->  ( `' F " { k } )  C_  RR )
64 ovolge0 22061 . . . . . . 7  |-  ( ( `' F " { k } )  C_  RR  ->  0  <_  ( vol* `  ( `' F " { k } ) ) )
6563, 64syl 16 . . . . . 6  |-  ( ( ( ph  /\  k  e.  ( ran  F  \  { 0 } ) )  /\  0  <_ 
k )  ->  0  <_  ( vol* `  ( `' F " { k } ) ) )
6620ad2antrr 723 . . . . . 6  |-  ( ( ( ph  /\  k  e.  ( ran  F  \  { 0 } ) )  /\  0  <_ 
k )  ->  ( vol `  ( `' F " { k } ) )  =  ( vol* `  ( `' F " { k } ) ) )
6765, 66breqtrrd 4465 . . . . 5  |-  ( ( ( ph  /\  k  e.  ( ran  F  \  { 0 } ) )  /\  0  <_ 
k )  ->  0  <_  ( vol `  ( `' F " { k } ) ) )
6858, 59, 60, 67mulge0d 10125 . . . 4  |-  ( ( ( ph  /\  k  e.  ( ran  F  \  { 0 } ) )  /\  0  <_ 
k )  ->  0  <_  ( k  x.  ( vol `  ( `' F " { k } ) ) ) )
69 0red 9586 . . . 4  |-  ( (
ph  /\  k  e.  ( ran  F  \  {
0 } ) )  ->  0  e.  RR )
7057, 68, 12, 69ltlecasei 9681 . . 3  |-  ( (
ph  /\  k  e.  ( ran  F  \  {
0 } ) )  ->  0  <_  (
k  x.  ( vol `  ( `' F " { k } ) ) ) )
716, 15, 70fsumge0 13694 . 2  |-  ( ph  ->  0  <_  sum_ k  e.  ( ran  F  \  { 0 } ) ( k  x.  ( vol `  ( `' F " { k } ) ) ) )
72 itg1val 22259 . . 3  |-  ( F  e.  dom  S.1  ->  ( S.1 `  F )  =  sum_ k  e.  ( ran  F  \  {
0 } ) ( k  x.  ( vol `  ( `' F " { k } ) ) ) )
731, 72syl 16 . 2  |-  ( ph  ->  ( S.1 `  F
)  =  sum_ k  e.  ( ran  F  \  { 0 } ) ( k  x.  ( vol `  ( `' F " { k } ) ) ) )
7471, 73breqtrrd 4465 1  |-  ( ph  ->  0  <_  ( S.1 `  F ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1398    e. wcel 1823    \ cdif 3458    C_ wss 3461   {csn 4016   class class class wbr 4439   `'ccnv 4987   dom cdm 4988   ran crn 4989   "cima 4991    Fn wfn 5565   -->wf 5566   ` cfv 5570  (class class class)co 6270   Fincfn 7509   CCcc 9479   RRcr 9480   0cc0 9481    x. cmul 9486    < clt 9617    <_ cle 9618   sum_csu 13593   vol*covol 22043   volcvol 22044   S.1citg1 22193
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-inf2 8049  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-fal 1404  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-se 4828  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-isom 5579  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-of 6513  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-2o 7123  df-oadd 7126  df-er 7303  df-map 7414  df-pm 7415  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-sup 7893  df-oi 7927  df-card 8311  df-cda 8539  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-nn 10532  df-2 10590  df-3 10591  df-n0 10792  df-z 10861  df-uz 11083  df-q 11184  df-rp 11222  df-xadd 11322  df-ioo 11536  df-ico 11538  df-icc 11539  df-fz 11676  df-fzo 11800  df-fl 11910  df-seq 12093  df-exp 12152  df-hash 12391  df-cj 13017  df-re 13018  df-im 13019  df-sqrt 13153  df-abs 13154  df-clim 13396  df-sum 13594  df-xmet 18610  df-met 18611  df-ovol 22045  df-vol 22046  df-mbf 22197  df-itg1 22198
This theorem is referenced by:  itg1lea  22288
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