MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  itg2le Structured version   Unicode version

Theorem itg2le 22684
Description: If one function dominates another, then the integral of the larger is also larger. (Contributed by Mario Carneiro, 28-Jun-2014.)
Assertion
Ref Expression
itg2le  |-  ( ( F : RR --> ( 0 [,] +oo )  /\  G : RR --> ( 0 [,] +oo )  /\  F  oR  <_  G
)  ->  ( S.2 `  F )  <_  ( S.2 `  G ) )

Proof of Theorem itg2le
Dummy variables  x  z  h  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 reex 9631 . . . . . . . . . 10  |-  RR  e.  _V
21a1i 11 . . . . . . . . 9  |-  ( ( ( F : RR --> ( 0 [,] +oo )  /\  G : RR --> ( 0 [,] +oo ) )  /\  h  e.  dom  S.1 )  ->  RR  e.  _V )
3 i1ff 22621 . . . . . . . . . . 11  |-  ( h  e.  dom  S.1  ->  h : RR --> RR )
43adantl 467 . . . . . . . . . 10  |-  ( ( ( F : RR --> ( 0 [,] +oo )  /\  G : RR --> ( 0 [,] +oo ) )  /\  h  e.  dom  S.1 )  ->  h : RR --> RR )
5 ressxr 9685 . . . . . . . . . 10  |-  RR  C_  RR*
6 fss 5751 . . . . . . . . . 10  |-  ( ( h : RR --> RR  /\  RR  C_  RR* )  ->  h : RR --> RR* )
74, 5, 6sylancl 666 . . . . . . . . 9  |-  ( ( ( F : RR --> ( 0 [,] +oo )  /\  G : RR --> ( 0 [,] +oo ) )  /\  h  e.  dom  S.1 )  ->  h : RR --> RR* )
8 simpll 758 . . . . . . . . . 10  |-  ( ( ( F : RR --> ( 0 [,] +oo )  /\  G : RR --> ( 0 [,] +oo ) )  /\  h  e.  dom  S.1 )  ->  F : RR --> ( 0 [,] +oo ) )
9 iccssxr 11718 . . . . . . . . . 10  |-  ( 0 [,] +oo )  C_  RR*
10 fss 5751 . . . . . . . . . 10  |-  ( ( F : RR --> ( 0 [,] +oo )  /\  ( 0 [,] +oo )  C_  RR* )  ->  F : RR --> RR* )
118, 9, 10sylancl 666 . . . . . . . . 9  |-  ( ( ( F : RR --> ( 0 [,] +oo )  /\  G : RR --> ( 0 [,] +oo ) )  /\  h  e.  dom  S.1 )  ->  F : RR --> RR* )
12 simplr 760 . . . . . . . . . 10  |-  ( ( ( F : RR --> ( 0 [,] +oo )  /\  G : RR --> ( 0 [,] +oo ) )  /\  h  e.  dom  S.1 )  ->  G : RR --> ( 0 [,] +oo ) )
13 fss 5751 . . . . . . . . . 10  |-  ( ( G : RR --> ( 0 [,] +oo )  /\  ( 0 [,] +oo )  C_  RR* )  ->  G : RR --> RR* )
1412, 9, 13sylancl 666 . . . . . . . . 9  |-  ( ( ( F : RR --> ( 0 [,] +oo )  /\  G : RR --> ( 0 [,] +oo ) )  /\  h  e.  dom  S.1 )  ->  G : RR --> RR* )
15 xrletr 11456 . . . . . . . . . 10  |-  ( ( x  e.  RR*  /\  y  e.  RR*  /\  z  e. 
RR* )  ->  (
( x  <_  y  /\  y  <_  z )  ->  x  <_  z
) )
1615adantl 467 . . . . . . . . 9  |-  ( ( ( ( F : RR
--> ( 0 [,] +oo )  /\  G : RR --> ( 0 [,] +oo ) )  /\  h  e.  dom  S.1 )  /\  (
x  e.  RR*  /\  y  e.  RR*  /\  z  e. 
RR* ) )  -> 
( ( x  <_ 
y  /\  y  <_  z )  ->  x  <_  z ) )
172, 7, 11, 14, 16caoftrn 6577 . . . . . . . 8  |-  ( ( ( F : RR --> ( 0 [,] +oo )  /\  G : RR --> ( 0 [,] +oo ) )  /\  h  e.  dom  S.1 )  ->  (
( h  oR  <_  F  /\  F  oR  <_  G )  ->  h  oR  <_  G ) )
18 simplr 760 . . . . . . . . . 10  |-  ( ( ( F : RR --> ( 0 [,] +oo )  /\  G : RR --> ( 0 [,] +oo ) )  /\  (
h  e.  dom  S.1  /\  h  oR  <_  G ) )  ->  G : RR --> ( 0 [,] +oo ) )
19 simprl 762 . . . . . . . . . 10  |-  ( ( ( F : RR --> ( 0 [,] +oo )  /\  G : RR --> ( 0 [,] +oo ) )  /\  (
h  e.  dom  S.1  /\  h  oR  <_  G ) )  ->  h  e.  dom  S.1 )
20 simprr 764 . . . . . . . . . 10  |-  ( ( ( F : RR --> ( 0 [,] +oo )  /\  G : RR --> ( 0 [,] +oo ) )  /\  (
h  e.  dom  S.1  /\  h  oR  <_  G ) )  ->  h  oR  <_  G
)
21 itg2ub 22678 . . . . . . . . . 10  |-  ( ( G : RR --> ( 0 [,] +oo )  /\  h  e.  dom  S.1  /\  h  oR  <_  G
)  ->  ( S.1 `  h )  <_  ( S.2 `  G ) )
2218, 19, 20, 21syl3anc 1264 . . . . . . . . 9  |-  ( ( ( F : RR --> ( 0 [,] +oo )  /\  G : RR --> ( 0 [,] +oo ) )  /\  (
h  e.  dom  S.1  /\  h  oR  <_  G ) )  -> 
( S.1 `  h )  <_  ( S.2 `  G
) )
2322expr 618 . . . . . . . 8  |-  ( ( ( F : RR --> ( 0 [,] +oo )  /\  G : RR --> ( 0 [,] +oo ) )  /\  h  e.  dom  S.1 )  ->  (
h  oR  <_  G  ->  ( S.1 `  h
)  <_  ( S.2 `  G ) ) )
2417, 23syld 45 . . . . . . 7  |-  ( ( ( F : RR --> ( 0 [,] +oo )  /\  G : RR --> ( 0 [,] +oo ) )  /\  h  e.  dom  S.1 )  ->  (
( h  oR  <_  F  /\  F  oR  <_  G )  ->  ( S.1 `  h
)  <_  ( S.2 `  G ) ) )
2524ancomsd 455 . . . . . 6  |-  ( ( ( F : RR --> ( 0 [,] +oo )  /\  G : RR --> ( 0 [,] +oo ) )  /\  h  e.  dom  S.1 )  ->  (
( F  oR  <_  G  /\  h  oR  <_  F )  ->  ( S.1 `  h
)  <_  ( S.2 `  G ) ) )
2625exp4b 610 . . . . 5  |-  ( ( F : RR --> ( 0 [,] +oo )  /\  G : RR --> ( 0 [,] +oo ) )  ->  ( h  e. 
dom  S.1  ->  ( F  oR  <_  G  -> 
( h  oR  <_  F  ->  ( S.1 `  h )  <_ 
( S.2 `  G ) ) ) ) )
2726com23 81 . . . 4  |-  ( ( F : RR --> ( 0 [,] +oo )  /\  G : RR --> ( 0 [,] +oo ) )  ->  ( F  oR  <_  G  ->  (
h  e.  dom  S.1  ->  ( h  oR  <_  F  ->  ( S.1 `  h )  <_ 
( S.2 `  G ) ) ) ) )
28273impia 1202 . . 3  |-  ( ( F : RR --> ( 0 [,] +oo )  /\  G : RR --> ( 0 [,] +oo )  /\  F  oR  <_  G
)  ->  ( h  e.  dom  S.1  ->  ( h  oR  <_  F  ->  ( S.1 `  h
)  <_  ( S.2 `  G ) ) ) )
2928ralrimiv 2837 . 2  |-  ( ( F : RR --> ( 0 [,] +oo )  /\  G : RR --> ( 0 [,] +oo )  /\  F  oR  <_  G
)  ->  A. h  e.  dom  S.1 ( h  oR  <_  F  ->  ( S.1 `  h )  <_  ( S.2 `  G
) ) )
30 simp1 1005 . . 3  |-  ( ( F : RR --> ( 0 [,] +oo )  /\  G : RR --> ( 0 [,] +oo )  /\  F  oR  <_  G
)  ->  F : RR
--> ( 0 [,] +oo ) )
31 itg2cl 22677 . . . 4  |-  ( G : RR --> ( 0 [,] +oo )  -> 
( S.2 `  G )  e.  RR* )
32313ad2ant2 1027 . . 3  |-  ( ( F : RR --> ( 0 [,] +oo )  /\  G : RR --> ( 0 [,] +oo )  /\  F  oR  <_  G
)  ->  ( S.2 `  G )  e.  RR* )
33 itg2leub 22679 . . 3  |-  ( ( F : RR --> ( 0 [,] +oo )  /\  ( S.2 `  G )  e.  RR* )  ->  (
( S.2 `  F )  <_  ( S.2 `  G
)  <->  A. h  e.  dom  S.1 ( h  oR  <_  F  ->  ( S.1 `  h )  <_ 
( S.2 `  G ) ) ) )
3430, 32, 33syl2anc 665 . 2  |-  ( ( F : RR --> ( 0 [,] +oo )  /\  G : RR --> ( 0 [,] +oo )  /\  F  oR  <_  G
)  ->  ( ( S.2 `  F )  <_ 
( S.2 `  G )  <->  A. h  e.  dom  S.1 ( h  oR  <_  F  ->  ( S.1 `  h )  <_ 
( S.2 `  G ) ) ) )
3529, 34mpbird 235 1  |-  ( ( F : RR --> ( 0 [,] +oo )  /\  G : RR --> ( 0 [,] +oo )  /\  F  oR  <_  G
)  ->  ( S.2 `  F )  <_  ( S.2 `  G ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    e. wcel 1868   A.wral 2775   _Vcvv 3081    C_ wss 3436   class class class wbr 4420   dom cdm 4850   -->wf 5594   ` cfv 5598  (class class class)co 6302    oRcofr 6541   RRcr 9539   0cc0 9540   +oocpnf 9673   RR*cxr 9675    <_ cle 9677   [,]cicc 11639   S.1citg1 22560   S.2citg2 22561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-rep 4533  ax-sep 4543  ax-nul 4552  ax-pow 4599  ax-pr 4657  ax-un 6594  ax-inf2 8149  ax-cnex 9596  ax-resscn 9597  ax-1cn 9598  ax-icn 9599  ax-addcl 9600  ax-addrcl 9601  ax-mulcl 9602  ax-mulrcl 9603  ax-mulcom 9604  ax-addass 9605  ax-mulass 9606  ax-distr 9607  ax-i2m1 9608  ax-1ne0 9609  ax-1rid 9610  ax-rnegex 9611  ax-rrecex 9612  ax-cnre 9613  ax-pre-lttri 9614  ax-pre-lttrn 9615  ax-pre-ltadd 9616  ax-pre-mulgt0 9617  ax-pre-sup 9618
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-nel 2621  df-ral 2780  df-rex 2781  df-reu 2782  df-rmo 2783  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-tp 4001  df-op 4003  df-uni 4217  df-int 4253  df-iun 4298  df-br 4421  df-opab 4480  df-mpt 4481  df-tr 4516  df-eprel 4761  df-id 4765  df-po 4771  df-so 4772  df-fr 4809  df-se 4810  df-we 4811  df-xp 4856  df-rel 4857  df-cnv 4858  df-co 4859  df-dm 4860  df-rn 4861  df-res 4862  df-ima 4863  df-pred 5396  df-ord 5442  df-on 5443  df-lim 5444  df-suc 5445  df-iota 5562  df-fun 5600  df-fn 5601  df-f 5602  df-f1 5603  df-fo 5604  df-f1o 5605  df-fv 5606  df-isom 5607  df-riota 6264  df-ov 6305  df-oprab 6306  df-mpt2 6307  df-of 6542  df-ofr 6543  df-om 6704  df-1st 6804  df-2nd 6805  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-1o 7187  df-2o 7188  df-oadd 7191  df-er 7368  df-map 7479  df-pm 7480  df-en 7575  df-dom 7576  df-sdom 7577  df-fin 7578  df-sup 7959  df-inf 7960  df-oi 8028  df-card 8375  df-cda 8599  df-pnf 9678  df-mnf 9679  df-xr 9680  df-ltxr 9681  df-le 9682  df-sub 9863  df-neg 9864  df-div 10271  df-nn 10611  df-2 10669  df-3 10670  df-n0 10871  df-z 10939  df-uz 11161  df-q 11266  df-rp 11304  df-xadd 11411  df-ioo 11640  df-ico 11642  df-icc 11643  df-fz 11786  df-fzo 11917  df-fl 12028  df-seq 12214  df-exp 12273  df-hash 12516  df-cj 13151  df-re 13152  df-im 13153  df-sqrt 13287  df-abs 13288  df-clim 13540  df-sum 13741  df-xmet 18951  df-met 18952  df-ovol 22403  df-vol 22405  df-mbf 22564  df-itg1 22565  df-itg2 22566
This theorem is referenced by:  itg2const2  22686  itg2monolem1  22695  itg2mono  22698  itg2gt0  22705  itg2cnlem2  22707  iblss  22749  itgle  22754  ibladdlem  22764  iblabs  22773  iblabsr  22774  iblmulc2  22775  bddmulibl  22783  itg2gt0cn  31911  ibladdnclem  31912  iblabsnc  31920  iblmulc2nc  31921  bddiblnc  31926  ftc1anclem4  31934  ftc1anclem6  31936  ftc1anclem7  31937  ftc1anclem8  31938  ftc1anc  31939
  Copyright terms: Public domain W3C validator