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Theorem itg2le 19592
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 9045 . . . . . . . . . 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 19529 . . . . . . . . . . 11  |-  ( h  e.  dom  S.1  ->  h : RR --> RR )
43adantl 453 . . . . . . . . . 10  |-  ( ( ( F : RR --> ( 0 [,]  +oo )  /\  G : RR --> ( 0 [,]  +oo ) )  /\  h  e.  dom  S.1 )  ->  h : RR --> RR )
5 ressxr 9093 . . . . . . . . . 10  |-  RR  C_  RR*
6 fss 5566 . . . . . . . . . 10  |-  ( ( h : RR --> RR  /\  RR  C_  RR* )  ->  h : RR --> RR* )
74, 5, 6sylancl 644 . . . . . . . . 9  |-  ( ( ( F : RR --> ( 0 [,]  +oo )  /\  G : RR --> ( 0 [,]  +oo ) )  /\  h  e.  dom  S.1 )  ->  h : RR --> RR* )
8 simpll 731 . . . . . . . . . 10  |-  ( ( ( F : RR --> ( 0 [,]  +oo )  /\  G : RR --> ( 0 [,]  +oo ) )  /\  h  e.  dom  S.1 )  ->  F : RR --> ( 0 [,] 
+oo ) )
9 iccssxr 10957 . . . . . . . . . 10  |-  ( 0 [,]  +oo )  C_  RR*
10 fss 5566 . . . . . . . . . 10  |-  ( ( F : RR --> ( 0 [,]  +oo )  /\  (
0 [,]  +oo )  C_  RR* )  ->  F : RR
--> RR* )
118, 9, 10sylancl 644 . . . . . . . . 9  |-  ( ( ( F : RR --> ( 0 [,]  +oo )  /\  G : RR --> ( 0 [,]  +oo ) )  /\  h  e.  dom  S.1 )  ->  F : RR --> RR* )
12 simplr 732 . . . . . . . . . 10  |-  ( ( ( F : RR --> ( 0 [,]  +oo )  /\  G : RR --> ( 0 [,]  +oo ) )  /\  h  e.  dom  S.1 )  ->  G : RR --> ( 0 [,] 
+oo ) )
13 fss 5566 . . . . . . . . . 10  |-  ( ( G : RR --> ( 0 [,]  +oo )  /\  (
0 [,]  +oo )  C_  RR* )  ->  G : RR
--> RR* )
1412, 9, 13sylancl 644 . . . . . . . . 9  |-  ( ( ( F : RR --> ( 0 [,]  +oo )  /\  G : RR --> ( 0 [,]  +oo ) )  /\  h  e.  dom  S.1 )  ->  G : RR --> RR* )
15 xrletr 10712 . . . . . . . . . 10  |-  ( ( x  e.  RR*  /\  y  e.  RR*  /\  z  e. 
RR* )  ->  (
( x  <_  y  /\  y  <_  z )  ->  x  <_  z
) )
1615adantl 453 . . . . . . . . 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 6306 . . . . . . . 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 732 . . . . . . . . . 10  |-  ( ( ( F : RR --> ( 0 [,]  +oo )  /\  G : RR --> ( 0 [,]  +oo ) )  /\  (
h  e.  dom  S.1  /\  h  o R  <_  G ) )  ->  G : RR --> ( 0 [,]  +oo ) )
19 simprl 733 . . . . . . . . . 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 734 . . . . . . . . . 10  |-  ( ( ( F : RR --> ( 0 [,]  +oo )  /\  G : RR --> ( 0 [,]  +oo ) )  /\  (
h  e.  dom  S.1  /\  h  o R  <_  G ) )  ->  h  o R  <_  G
)
21 itg2ub 19586 . . . . . . . . . 10  |-  ( ( G : RR --> ( 0 [,]  +oo )  /\  h  e.  dom  S.1  /\  h  o R  <_  G )  ->  ( S.1 `  h
)  <_  ( S.2 `  G ) )
2218, 19, 20, 21syl3anc 1184 . . . . . . . . 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 599 . . . . . . . 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 42 . . . . . . 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 441 . . . . . 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 591 . . . . 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 74 . . . 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 1150 . . 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 2756 . 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 957 . . 3  |-  ( ( F : RR --> ( 0 [,]  +oo )  /\  G : RR --> ( 0 [,] 
+oo )  /\  F  o R  <_  G )  ->  F : RR --> ( 0 [,]  +oo ) )
31 itg2cl 19585 . . . 4  |-  ( G : RR --> ( 0 [,]  +oo )  ->  ( S.2 `  G )  e. 
RR* )
32313ad2ant2 979 . . 3  |-  ( ( F : RR --> ( 0 [,]  +oo )  /\  G : RR --> ( 0 [,] 
+oo )  /\  F  o R  <_  G )  ->  ( S.2 `  G
)  e.  RR* )
33 itg2leub 19587 . . 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 643 . 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 224 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 177    /\ wa 359    /\ w3a 936    e. wcel 1721   A.wral 2674   _Vcvv 2924    C_ wss 3288   class class class wbr 4180   dom cdm 4845   -->wf 5417   ` cfv 5421  (class class class)co 6048    o Rcofr 6271   RRcr 8953   0cc0 8954    +oocpnf 9081   RR*cxr 9083    <_ cle 9085   [,]cicc 10883   S.1citg1 19468   S.2citg2 19469
This theorem is referenced by:  itg2const2  19594  itg2monolem1  19603  itg2mono  19606  itg2gt0  19613  itg2cnlem2  19615  iblss  19657  itgle  19662  ibladdlem  19672  iblabs  19681  iblabsr  19682  iblmulc2  19683  bddmulibl  19691  itg2gt0cn  26167  ibladdnclem  26168  iblabsnc  26176  iblmulc2nc  26177  bddiblnc  26182
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 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-inf2 7560  ax-cnex 9010  ax-resscn 9011  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-mulcom 9018  ax-addass 9019  ax-mulass 9020  ax-distr 9021  ax-i2m1 9022  ax-1ne0 9023  ax-1rid 9024  ax-rnegex 9025  ax-rrecex 9026  ax-cnre 9027  ax-pre-lttri 9028  ax-pre-lttrn 9029  ax-pre-ltadd 9030  ax-pre-mulgt0 9031  ax-pre-sup 9032
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 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-int 4019  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-se 4510  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-isom 5430  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-of 6272  df-ofr 6273  df-1st 6316  df-2nd 6317  df-riota 6516  df-recs 6600  df-rdg 6635  df-1o 6691  df-2o 6692  df-oadd 6695  df-er 6872  df-map 6987  df-pm 6988  df-en 7077  df-dom 7078  df-sdom 7079  df-fin 7080  df-sup 7412  df-oi 7443  df-card 7790  df-cda 8012  df-pnf 9086  df-mnf 9087  df-xr 9088  df-ltxr 9089  df-le 9090  df-sub 9257  df-neg 9258  df-div 9642  df-nn 9965  df-2 10022  df-3 10023  df-n0 10186  df-z 10247  df-uz 10453  df-q 10539  df-rp 10577  df-xadd 10675  df-ioo 10884  df-ico 10886  df-icc 10887  df-fz 11008  df-fzo 11099  df-fl 11165  df-seq 11287  df-exp 11346  df-hash 11582  df-cj 11867  df-re 11868  df-im 11869  df-sqr 12003  df-abs 12004  df-clim 12245  df-sum 12443  df-xmet 16658  df-met 16659  df-ovol 19322  df-vol 19323  df-mbf 19473  df-itg1 19474  df-itg2 19475
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