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Theorem itg2le 19094
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  o R  <_  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 8828 . . . . . . . . . 10  |-  RR  e.  _V
21a1i 10 . . . . . . . . 9  |-  ( ( ( F : RR --> ( 0 [,]  +oo )  /\  G : RR --> ( 0 [,]  +oo ) )  /\  h  e.  dom  S.1 )  ->  RR  e.  _V )
3 i1ff 19031 . . . . . . . . . . 11  |-  ( h  e.  dom  S.1  ->  h : RR --> RR )
43adantl 452 . . . . . . . . . 10  |-  ( ( ( F : RR --> ( 0 [,]  +oo )  /\  G : RR --> ( 0 [,]  +oo ) )  /\  h  e.  dom  S.1 )  ->  h : RR --> RR )
5 ressxr 8876 . . . . . . . . . 10  |-  RR  C_  RR*
6 fss 5397 . . . . . . . . . 10  |-  ( ( h : RR --> RR  /\  RR  C_  RR* )  ->  h : RR --> RR* )
74, 5, 6sylancl 643 . . . . . . . . 9  |-  ( ( ( F : RR --> ( 0 [,]  +oo )  /\  G : RR --> ( 0 [,]  +oo ) )  /\  h  e.  dom  S.1 )  ->  h : RR --> RR* )
8 simpll 730 . . . . . . . . . 10  |-  ( ( ( F : RR --> ( 0 [,]  +oo )  /\  G : RR --> ( 0 [,]  +oo ) )  /\  h  e.  dom  S.1 )  ->  F : RR --> ( 0 [,] 
+oo ) )
9 iccssxr 10732 . . . . . . . . . 10  |-  ( 0 [,]  +oo )  C_  RR*
10 fss 5397 . . . . . . . . . 10  |-  ( ( F : RR --> ( 0 [,]  +oo )  /\  (
0 [,]  +oo )  C_  RR* )  ->  F : RR
--> RR* )
118, 9, 10sylancl 643 . . . . . . . . 9  |-  ( ( ( F : RR --> ( 0 [,]  +oo )  /\  G : RR --> ( 0 [,]  +oo ) )  /\  h  e.  dom  S.1 )  ->  F : RR --> RR* )
12 simplr 731 . . . . . . . . . 10  |-  ( ( ( F : RR --> ( 0 [,]  +oo )  /\  G : RR --> ( 0 [,]  +oo ) )  /\  h  e.  dom  S.1 )  ->  G : RR --> ( 0 [,] 
+oo ) )
13 fss 5397 . . . . . . . . . 10  |-  ( ( G : RR --> ( 0 [,]  +oo )  /\  (
0 [,]  +oo )  C_  RR* )  ->  G : RR
--> RR* )
1412, 9, 13sylancl 643 . . . . . . . . 9  |-  ( ( ( F : RR --> ( 0 [,]  +oo )  /\  G : RR --> ( 0 [,]  +oo ) )  /\  h  e.  dom  S.1 )  ->  G : RR --> RR* )
15 xrletr 10489 . . . . . . . . . 10  |-  ( ( x  e.  RR*  /\  y  e.  RR*  /\  z  e. 
RR* )  ->  (
( x  <_  y  /\  y  <_  z )  ->  x  <_  z
) )
1615adantl 452 . . . . . . . . 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 6112 . . . . . . . 8  |-  ( ( ( F : RR --> ( 0 [,]  +oo )  /\  G : RR --> ( 0 [,]  +oo ) )  /\  h  e.  dom  S.1 )  ->  (
( h  o R  <_  F  /\  F  o R  <_  G )  ->  h  o R  <_  G ) )
18 simplr 731 . . . . . . . . . 10  |-  ( ( ( F : RR --> ( 0 [,]  +oo )  /\  G : RR --> ( 0 [,]  +oo ) )  /\  (
h  e.  dom  S.1  /\  h  o R  <_  G ) )  ->  G : RR --> ( 0 [,]  +oo ) )
19 simprl 732 . . . . . . . . . 10  |-  ( ( ( F : RR --> ( 0 [,]  +oo )  /\  G : RR --> ( 0 [,]  +oo ) )  /\  (
h  e.  dom  S.1  /\  h  o R  <_  G ) )  ->  h  e.  dom  S.1 )
20 simprr 733 . . . . . . . . . 10  |-  ( ( ( F : RR --> ( 0 [,]  +oo )  /\  G : RR --> ( 0 [,]  +oo ) )  /\  (
h  e.  dom  S.1  /\  h  o R  <_  G ) )  ->  h  o R  <_  G
)
21 itg2ub 19088 . . . . . . . . . 10  |-  ( ( G : RR --> ( 0 [,]  +oo )  /\  h  e.  dom  S.1  /\  h  o R  <_  G )  ->  ( S.1 `  h
)  <_  ( S.2 `  G ) )
2218, 19, 20, 21syl3anc 1182 . . . . . . . . 9  |-  ( ( ( F : RR --> ( 0 [,]  +oo )  /\  G : RR --> ( 0 [,]  +oo ) )  /\  (
h  e.  dom  S.1  /\  h  o R  <_  G ) )  -> 
( S.1 `  h )  <_  ( S.2 `  G
) )
2322expr 598 . . . . . . . 8  |-  ( ( ( F : RR --> ( 0 [,]  +oo )  /\  G : RR --> ( 0 [,]  +oo ) )  /\  h  e.  dom  S.1 )  ->  (
h  o R  <_  G  ->  ( S.1 `  h
)  <_  ( S.2 `  G ) ) )
2417, 23syld 40 . . . . . . 7  |-  ( ( ( F : RR --> ( 0 [,]  +oo )  /\  G : RR --> ( 0 [,]  +oo ) )  /\  h  e.  dom  S.1 )  ->  (
( h  o R  <_  F  /\  F  o R  <_  G )  ->  ( S.1 `  h
)  <_  ( S.2 `  G ) ) )
2524ancomsd 440 . . . . . 6  |-  ( ( ( F : RR --> ( 0 [,]  +oo )  /\  G : RR --> ( 0 [,]  +oo ) )  /\  h  e.  dom  S.1 )  ->  (
( F  o R  <_  G  /\  h  o R  <_  F )  ->  ( S.1 `  h
)  <_  ( S.2 `  G ) ) )
2625exp4b 590 . . . . 5  |-  ( ( F : RR --> ( 0 [,]  +oo )  /\  G : RR --> ( 0 [,] 
+oo ) )  -> 
( h  e.  dom  S.1 
->  ( F  o R  <_  G  ->  (
h  o R  <_  F  ->  ( S.1 `  h
)  <_  ( S.2 `  G ) ) ) ) )
2726com23 72 . . . 4  |-  ( ( F : RR --> ( 0 [,]  +oo )  /\  G : RR --> ( 0 [,] 
+oo ) )  -> 
( F  o R  <_  G  ->  (
h  e.  dom  S.1  ->  ( h  o R  <_  F  ->  ( S.1 `  h )  <_ 
( S.2 `  G ) ) ) ) )
28273impia 1148 . . 3  |-  ( ( F : RR --> ( 0 [,]  +oo )  /\  G : RR --> ( 0 [,] 
+oo )  /\  F  o R  <_  G )  ->  ( h  e. 
dom  S.1  ->  ( h  o R  <_  F  -> 
( S.1 `  h )  <_  ( S.2 `  G
) ) ) )
2928ralrimiv 2625 . 2  |-  ( ( F : RR --> ( 0 [,]  +oo )  /\  G : RR --> ( 0 [,] 
+oo )  /\  F  o R  <_  G )  ->  A. h  e.  dom  S.1 ( h  o R  <_  F  ->  ( S.1 `  h )  <_ 
( S.2 `  G ) ) )
30 simp1 955 . . 3  |-  ( ( F : RR --> ( 0 [,]  +oo )  /\  G : RR --> ( 0 [,] 
+oo )  /\  F  o R  <_  G )  ->  F : RR --> ( 0 [,]  +oo ) )
31 itg2cl 19087 . . . 4  |-  ( G : RR --> ( 0 [,]  +oo )  ->  ( S.2 `  G )  e. 
RR* )
32313ad2ant2 977 . . 3  |-  ( ( F : RR --> ( 0 [,]  +oo )  /\  G : RR --> ( 0 [,] 
+oo )  /\  F  o R  <_  G )  ->  ( S.2 `  G
)  e.  RR* )
33 itg2leub 19089 . . 3  |-  ( ( F : RR --> ( 0 [,]  +oo )  /\  ( S.2 `  G )  e. 
RR* )  ->  (
( S.2 `  F )  <_  ( S.2 `  G
)  <->  A. h  e.  dom  S.1 ( h  o R  <_  F  ->  ( S.1 `  h )  <_ 
( S.2 `  G ) ) ) )
3430, 32, 33syl2anc 642 . 2  |-  ( ( F : RR --> ( 0 [,]  +oo )  /\  G : RR --> ( 0 [,] 
+oo )  /\  F  o R  <_  G )  ->  ( ( S.2 `  F )  <_  ( S.2 `  G )  <->  A. h  e.  dom  S.1 ( h  o R  <_  F  ->  ( S.1 `  h )  <_  ( S.2 `  G
) ) ) )
3529, 34mpbird 223 1  |-  ( ( F : RR --> ( 0 [,]  +oo )  /\  G : RR --> ( 0 [,] 
+oo )  /\  F  o R  <_  G )  ->  ( S.2 `  F
)  <_  ( S.2 `  G ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    e. wcel 1684   A.wral 2543   _Vcvv 2788    C_ wss 3152   class class class wbr 4023   dom cdm 4689   -->wf 5251   ` cfv 5255  (class class class)co 5858    o Rcofr 6077   RRcr 8736   0cc0 8737    +oocpnf 8864   RR*cxr 8866    <_ cle 8868   [,]cicc 10659   S.1citg1 18970   S.2citg2 18971
This theorem is referenced by:  itg2const2  19096  itg2monolem1  19105  itg2mono  19108  itg2gt0  19115  itg2cnlem2  19117  iblss  19159  itgle  19164  ibladdlem  19174  iblabs  19183  iblabsr  19184  iblmulc2  19185  bddmulibl  19193
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-ofr 6079  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-map 6774  df-pm 6775  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-oi 7225  df-card 7572  df-cda 7794  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-q 10317  df-rp 10355  df-xadd 10453  df-ioo 10660  df-ico 10662  df-icc 10663  df-fz 10783  df-fzo 10871  df-fl 10925  df-seq 11047  df-exp 11105  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-clim 11962  df-sum 12159  df-xmet 16373  df-met 16374  df-ovol 18824  df-vol 18825  df-mbf 18975  df-itg1 18976  df-itg2 18977
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