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Theorem itg2le 19198
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 8918 . . . . . . . . . 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 19135 . . . . . . . . . . 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 8966 . . . . . . . . . 10  |-  RR  C_  RR*
6 fss 5480 . . . . . . . . . 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 10824 . . . . . . . . . 10  |-  ( 0 [,]  +oo )  C_  RR*
10 fss 5480 . . . . . . . . . 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 5480 . . . . . . . . . 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 10581 . . . . . . . . . 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 6199 . . . . . . . 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 19192 . . . . . . . . . 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 2701 . 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 19191 . . . 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 19193 . . 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 1710   A.wral 2619   _Vcvv 2864    C_ wss 3228   class class class wbr 4104   dom cdm 4771   -->wf 5333   ` cfv 5337  (class class class)co 5945    o Rcofr 6164   RRcr 8826   0cc0 8827    +oocpnf 8954   RR*cxr 8956    <_ cle 8958   [,]cicc 10751   S.1citg1 19074   S.2citg2 19075
This theorem is referenced by:  itg2const2  19200  itg2monolem1  19209  itg2mono  19212  itg2gt0  19219  itg2cnlem2  19221  iblss  19263  itgle  19268  ibladdlem  19278  iblabs  19287  iblabsr  19288  iblmulc2  19289  bddmulibl  19297  itg2gt0cn  25495  ibladdnclem  25496  iblabsnc  25504  iblmulc2nc  25505  bddiblnc  25510
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4212  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594  ax-inf2 7432  ax-cnex 8883  ax-resscn 8884  ax-1cn 8885  ax-icn 8886  ax-addcl 8887  ax-addrcl 8888  ax-mulcl 8889  ax-mulrcl 8890  ax-mulcom 8891  ax-addass 8892  ax-mulass 8893  ax-distr 8894  ax-i2m1 8895  ax-1ne0 8896  ax-1rid 8897  ax-rnegex 8898  ax-rrecex 8899  ax-cnre 8900  ax-pre-lttri 8901  ax-pre-lttrn 8902  ax-pre-ltadd 8903  ax-pre-mulgt0 8904  ax-pre-sup 8905
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3909  df-int 3944  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-tr 4195  df-eprel 4387  df-id 4391  df-po 4396  df-so 4397  df-fr 4434  df-se 4435  df-we 4436  df-ord 4477  df-on 4478  df-lim 4479  df-suc 4480  df-om 4739  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-isom 5346  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-of 6165  df-ofr 6166  df-1st 6209  df-2nd 6210  df-riota 6391  df-recs 6475  df-rdg 6510  df-1o 6566  df-2o 6567  df-oadd 6570  df-er 6747  df-map 6862  df-pm 6863  df-en 6952  df-dom 6953  df-sdom 6954  df-fin 6955  df-sup 7284  df-oi 7315  df-card 7662  df-cda 7884  df-pnf 8959  df-mnf 8960  df-xr 8961  df-ltxr 8962  df-le 8963  df-sub 9129  df-neg 9130  df-div 9514  df-nn 9837  df-2 9894  df-3 9895  df-n0 10058  df-z 10117  df-uz 10323  df-q 10409  df-rp 10447  df-xadd 10545  df-ioo 10752  df-ico 10754  df-icc 10755  df-fz 10875  df-fzo 10963  df-fl 11017  df-seq 11139  df-exp 11198  df-hash 11431  df-cj 11680  df-re 11681  df-im 11682  df-sqr 11816  df-abs 11817  df-clim 12058  df-sum 12256  df-xmet 16475  df-met 16476  df-ovol 18928  df-vol 18929  df-mbf 19079  df-itg1 19080  df-itg2 19081
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