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Theorem itg2le 19634
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 9086 . . . . . . . . . 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 19571 . . . . . . . . . . 11  |-  ( h  e.  dom  S.1  ->  h : RR --> RR )
43adantl 454 . . . . . . . . . 10  |-  ( ( ( F : RR --> ( 0 [,]  +oo )  /\  G : RR --> ( 0 [,]  +oo ) )  /\  h  e.  dom  S.1 )  ->  h : RR --> RR )
5 ressxr 9134 . . . . . . . . . 10  |-  RR  C_  RR*
6 fss 5602 . . . . . . . . . 10  |-  ( ( h : RR --> RR  /\  RR  C_  RR* )  ->  h : RR --> RR* )
74, 5, 6sylancl 645 . . . . . . . . 9  |-  ( ( ( F : RR --> ( 0 [,]  +oo )  /\  G : RR --> ( 0 [,]  +oo ) )  /\  h  e.  dom  S.1 )  ->  h : RR --> RR* )
8 simpll 732 . . . . . . . . . 10  |-  ( ( ( F : RR --> ( 0 [,]  +oo )  /\  G : RR --> ( 0 [,]  +oo ) )  /\  h  e.  dom  S.1 )  ->  F : RR --> ( 0 [,] 
+oo ) )
9 iccssxr 10998 . . . . . . . . . 10  |-  ( 0 [,]  +oo )  C_  RR*
10 fss 5602 . . . . . . . . . 10  |-  ( ( F : RR --> ( 0 [,]  +oo )  /\  (
0 [,]  +oo )  C_  RR* )  ->  F : RR
--> RR* )
118, 9, 10sylancl 645 . . . . . . . . 9  |-  ( ( ( F : RR --> ( 0 [,]  +oo )  /\  G : RR --> ( 0 [,]  +oo ) )  /\  h  e.  dom  S.1 )  ->  F : RR --> RR* )
12 simplr 733 . . . . . . . . . 10  |-  ( ( ( F : RR --> ( 0 [,]  +oo )  /\  G : RR --> ( 0 [,]  +oo ) )  /\  h  e.  dom  S.1 )  ->  G : RR --> ( 0 [,] 
+oo ) )
13 fss 5602 . . . . . . . . . 10  |-  ( ( G : RR --> ( 0 [,]  +oo )  /\  (
0 [,]  +oo )  C_  RR* )  ->  G : RR
--> RR* )
1412, 9, 13sylancl 645 . . . . . . . . 9  |-  ( ( ( F : RR --> ( 0 [,]  +oo )  /\  G : RR --> ( 0 [,]  +oo ) )  /\  h  e.  dom  S.1 )  ->  G : RR --> RR* )
15 xrletr 10753 . . . . . . . . . 10  |-  ( ( x  e.  RR*  /\  y  e.  RR*  /\  z  e. 
RR* )  ->  (
( x  <_  y  /\  y  <_  z )  ->  x  <_  z
) )
1615adantl 454 . . . . . . . . 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 6342 . . . . . . . 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 733 . . . . . . . . . 10  |-  ( ( ( F : RR --> ( 0 [,]  +oo )  /\  G : RR --> ( 0 [,]  +oo ) )  /\  (
h  e.  dom  S.1  /\  h  o R  <_  G ) )  ->  G : RR --> ( 0 [,]  +oo ) )
19 simprl 734 . . . . . . . . . 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 735 . . . . . . . . . 10  |-  ( ( ( F : RR --> ( 0 [,]  +oo )  /\  G : RR --> ( 0 [,]  +oo ) )  /\  (
h  e.  dom  S.1  /\  h  o R  <_  G ) )  ->  h  o R  <_  G
)
21 itg2ub 19628 . . . . . . . . . 10  |-  ( ( G : RR --> ( 0 [,]  +oo )  /\  h  e.  dom  S.1  /\  h  o R  <_  G )  ->  ( S.1 `  h
)  <_  ( S.2 `  G ) )
2218, 19, 20, 21syl3anc 1185 . . . . . . . . 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 600 . . . . . . . 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 43 . . . . . . 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 442 . . . . . 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 592 . . . . 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 75 . . . 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 1151 . . 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 2790 . 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 958 . . 3  |-  ( ( F : RR --> ( 0 [,]  +oo )  /\  G : RR --> ( 0 [,] 
+oo )  /\  F  o R  <_  G )  ->  F : RR --> ( 0 [,]  +oo ) )
31 itg2cl 19627 . . . 4  |-  ( G : RR --> ( 0 [,]  +oo )  ->  ( S.2 `  G )  e. 
RR* )
32313ad2ant2 980 . . 3  |-  ( ( F : RR --> ( 0 [,]  +oo )  /\  G : RR --> ( 0 [,] 
+oo )  /\  F  o R  <_  G )  ->  ( S.2 `  G
)  e.  RR* )
33 itg2leub 19629 . . 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 644 . 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 225 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 178    /\ wa 360    /\ w3a 937    e. wcel 1726   A.wral 2707   _Vcvv 2958    C_ wss 3322   class class class wbr 4215   dom cdm 4881   -->wf 5453   ` cfv 5457  (class class class)co 6084    o Rcofr 6307   RRcr 8994   0cc0 8995    +oocpnf 9122   RR*cxr 9124    <_ cle 9126   [,]cicc 10924   S.1citg1 19512   S.2citg2 19513
This theorem is referenced by:  itg2const2  19636  itg2monolem1  19645  itg2mono  19648  itg2gt0  19655  itg2cnlem2  19657  iblss  19699  itgle  19704  ibladdlem  19714  iblabs  19723  iblabsr  19724  iblmulc2  19725  bddmulibl  19733  itg2gt0cn  26274  ibladdnclem  26275  iblabsnc  26283  iblmulc2nc  26284  bddiblnc  26289  ftc1anclem4  26297  ftc1anclem6  26299  ftc1anclem7  26300  ftc1anclem8  26301  ftc1anc  26302
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-inf2 7599  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072  ax-pre-sup 9073
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-se 4545  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-isom 5466  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-of 6308  df-ofr 6309  df-1st 6352  df-2nd 6353  df-riota 6552  df-recs 6636  df-rdg 6671  df-1o 6727  df-2o 6728  df-oadd 6731  df-er 6908  df-map 7023  df-pm 7024  df-en 7113  df-dom 7114  df-sdom 7115  df-fin 7116  df-sup 7449  df-oi 7482  df-card 7831  df-cda 8053  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-div 9683  df-nn 10006  df-2 10063  df-3 10064  df-n0 10227  df-z 10288  df-uz 10494  df-q 10580  df-rp 10618  df-xadd 10716  df-ioo 10925  df-ico 10927  df-icc 10928  df-fz 11049  df-fzo 11141  df-fl 11207  df-seq 11329  df-exp 11388  df-hash 11624  df-cj 11909  df-re 11910  df-im 11911  df-sqr 12045  df-abs 12046  df-clim 12287  df-sum 12485  df-xmet 16700  df-met 16701  df-ovol 19366  df-vol 19367  df-mbf 19516  df-itg1 19517  df-itg2 19518
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