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Theorem itg2le 21222
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 9378 . . . . . . . . . 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 21159 . . . . . . . . . . 11  |-  ( h  e.  dom  S.1  ->  h : RR --> RR )
43adantl 466 . . . . . . . . . 10  |-  ( ( ( F : RR --> ( 0 [,] +oo )  /\  G : RR --> ( 0 [,] +oo ) )  /\  h  e.  dom  S.1 )  ->  h : RR --> RR )
5 ressxr 9432 . . . . . . . . . 10  |-  RR  C_  RR*
6 fss 5572 . . . . . . . . . 10  |-  ( ( h : RR --> RR  /\  RR  C_  RR* )  ->  h : RR --> RR* )
74, 5, 6sylancl 662 . . . . . . . . 9  |-  ( ( ( F : RR --> ( 0 [,] +oo )  /\  G : RR --> ( 0 [,] +oo ) )  /\  h  e.  dom  S.1 )  ->  h : RR --> RR* )
8 simpll 753 . . . . . . . . . 10  |-  ( ( ( F : RR --> ( 0 [,] +oo )  /\  G : RR --> ( 0 [,] +oo ) )  /\  h  e.  dom  S.1 )  ->  F : RR --> ( 0 [,] +oo ) )
9 iccssxr 11383 . . . . . . . . . 10  |-  ( 0 [,] +oo )  C_  RR*
10 fss 5572 . . . . . . . . . 10  |-  ( ( F : RR --> ( 0 [,] +oo )  /\  ( 0 [,] +oo )  C_  RR* )  ->  F : RR --> RR* )
118, 9, 10sylancl 662 . . . . . . . . 9  |-  ( ( ( F : RR --> ( 0 [,] +oo )  /\  G : RR --> ( 0 [,] +oo ) )  /\  h  e.  dom  S.1 )  ->  F : RR --> RR* )
12 simplr 754 . . . . . . . . . 10  |-  ( ( ( F : RR --> ( 0 [,] +oo )  /\  G : RR --> ( 0 [,] +oo ) )  /\  h  e.  dom  S.1 )  ->  G : RR --> ( 0 [,] +oo ) )
13 fss 5572 . . . . . . . . . 10  |-  ( ( G : RR --> ( 0 [,] +oo )  /\  ( 0 [,] +oo )  C_  RR* )  ->  G : RR --> RR* )
1412, 9, 13sylancl 662 . . . . . . . . 9  |-  ( ( ( F : RR --> ( 0 [,] +oo )  /\  G : RR --> ( 0 [,] +oo ) )  /\  h  e.  dom  S.1 )  ->  G : RR --> RR* )
15 xrletr 11137 . . . . . . . . . 10  |-  ( ( x  e.  RR*  /\  y  e.  RR*  /\  z  e. 
RR* )  ->  (
( x  <_  y  /\  y  <_  z )  ->  x  <_  z
) )
1615adantl 466 . . . . . . . . 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 6360 . . . . . . . 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 754 . . . . . . . . . 10  |-  ( ( ( F : RR --> ( 0 [,] +oo )  /\  G : RR --> ( 0 [,] +oo ) )  /\  (
h  e.  dom  S.1  /\  h  oR  <_  G ) )  ->  G : RR --> ( 0 [,] +oo ) )
19 simprl 755 . . . . . . . . . 10  |-  ( ( ( F : RR --> ( 0 [,] +oo )  /\  G : RR --> ( 0 [,] +oo ) )  /\  (
h  e.  dom  S.1  /\  h  oR  <_  G ) )  ->  h  e.  dom  S.1 )
20 simprr 756 . . . . . . . . . 10  |-  ( ( ( F : RR --> ( 0 [,] +oo )  /\  G : RR --> ( 0 [,] +oo ) )  /\  (
h  e.  dom  S.1  /\  h  oR  <_  G ) )  ->  h  oR  <_  G
)
21 itg2ub 21216 . . . . . . . . . 10  |-  ( ( G : RR --> ( 0 [,] +oo )  /\  h  e.  dom  S.1  /\  h  oR  <_  G
)  ->  ( S.1 `  h )  <_  ( S.2 `  G ) )
2218, 19, 20, 21syl3anc 1218 . . . . . . . . 9  |-  ( ( ( F : RR --> ( 0 [,] +oo )  /\  G : RR --> ( 0 [,] +oo ) )  /\  (
h  e.  dom  S.1  /\  h  oR  <_  G ) )  -> 
( S.1 `  h )  <_  ( S.2 `  G
) )
2322expr 615 . . . . . . . 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 44 . . . . . . 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 454 . . . . . 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 607 . . . . 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 78 . . . 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 1184 . . 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 2803 . 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 988 . . 3  |-  ( ( F : RR --> ( 0 [,] +oo )  /\  G : RR --> ( 0 [,] +oo )  /\  F  oR  <_  G
)  ->  F : RR
--> ( 0 [,] +oo ) )
31 itg2cl 21215 . . . 4  |-  ( G : RR --> ( 0 [,] +oo )  -> 
( S.2 `  G )  e.  RR* )
32313ad2ant2 1010 . . 3  |-  ( ( F : RR --> ( 0 [,] +oo )  /\  G : RR --> ( 0 [,] +oo )  /\  F  oR  <_  G
)  ->  ( S.2 `  G )  e.  RR* )
33 itg2leub 21217 . . 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 661 . 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 232 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 184    /\ wa 369    /\ w3a 965    e. wcel 1756   A.wral 2720   _Vcvv 2977    C_ wss 3333   class class class wbr 4297   dom cdm 4845   -->wf 5419   ` cfv 5423  (class class class)co 6096    oRcofr 6324   RRcr 9286   0cc0 9287   +oocpnf 9420   RR*cxr 9422    <_ cle 9424   [,]cicc 11308   S.1citg1 21100   S.2citg2 21101
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-inf2 7852  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364  ax-pre-sup 9365
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-se 4685  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  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 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-isom 5432  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-of 6325  df-ofr 6326  df-om 6482  df-1st 6582  df-2nd 6583  df-recs 6837  df-rdg 6871  df-1o 6925  df-2o 6926  df-oadd 6929  df-er 7106  df-map 7221  df-pm 7222  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-sup 7696  df-oi 7729  df-card 8114  df-cda 8342  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-div 9999  df-nn 10328  df-2 10385  df-3 10386  df-n0 10585  df-z 10652  df-uz 10867  df-q 10959  df-rp 10997  df-xadd 11095  df-ioo 11309  df-ico 11311  df-icc 11312  df-fz 11443  df-fzo 11554  df-fl 11647  df-seq 11812  df-exp 11871  df-hash 12109  df-cj 12593  df-re 12594  df-im 12595  df-sqr 12729  df-abs 12730  df-clim 12971  df-sum 13169  df-xmet 17815  df-met 17816  df-ovol 20953  df-vol 20954  df-mbf 21104  df-itg1 21105  df-itg2 21106
This theorem is referenced by:  itg2const2  21224  itg2monolem1  21233  itg2mono  21236  itg2gt0  21243  itg2cnlem2  21245  iblss  21287  itgle  21292  ibladdlem  21302  iblabs  21311  iblabsr  21312  iblmulc2  21313  bddmulibl  21321  itg2gt0cn  28452  ibladdnclem  28453  iblabsnc  28461  iblmulc2nc  28462  bddiblnc  28467  ftc1anclem4  28475  ftc1anclem6  28477  ftc1anclem7  28478  ftc1anclem8  28479  ftc1anc  28480
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