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Theorem itg2monolem2 19106
Description: Lemma for itg2mono 19108. (Contributed by Mario Carneiro, 16-Aug-2014.)
Hypotheses
Ref Expression
itg2mono.1  |-  G  =  ( x  e.  RR  |->  sup ( ran  ( n  e.  NN  |->  ( ( F `  n ) `
 x ) ) ,  RR ,  <  ) )
itg2mono.2  |-  ( (
ph  /\  n  e.  NN )  ->  ( F `
 n )  e. MblFn
)
itg2mono.3  |-  ( (
ph  /\  n  e.  NN )  ->  ( F `
 n ) : RR --> ( 0 [,) 
+oo ) )
itg2mono.4  |-  ( (
ph  /\  n  e.  NN )  ->  ( F `
 n )  o R  <_  ( F `  ( n  +  1 ) ) )
itg2mono.5  |-  ( (
ph  /\  x  e.  RR )  ->  E. y  e.  RR  A. n  e.  NN  ( ( F `
 n ) `  x )  <_  y
)
itg2mono.6  |-  S  =  sup ( ran  (
n  e.  NN  |->  ( S.2 `  ( F `
 n ) ) ) ,  RR* ,  <  )
itg2monolem2.7  |-  ( ph  ->  P  e.  dom  S.1 )
itg2monolem2.8  |-  ( ph  ->  P  o R  <_  G )
itg2monolem2.9  |-  ( ph  ->  -.  ( S.1 `  P
)  <_  S )
Assertion
Ref Expression
itg2monolem2  |-  ( ph  ->  S  e.  RR )
Distinct variable groups:    x, n, y, G    P, n, x, y    n, F, x, y    ph, n, x, y    S, n, x, y

Proof of Theorem itg2monolem2
StepHypRef Expression
1 itg2mono.6 . . 3  |-  S  =  sup ( ran  (
n  e.  NN  |->  ( S.2 `  ( F `
 n ) ) ) ,  RR* ,  <  )
2 itg2mono.3 . . . . . . . 8  |-  ( (
ph  /\  n  e.  NN )  ->  ( F `
 n ) : RR --> ( 0 [,) 
+oo ) )
3 rexr 8877 . . . . . . . . . . 11  |-  ( x  e.  RR  ->  x  e.  RR* )
43anim1i 551 . . . . . . . . . 10  |-  ( ( x  e.  RR  /\  0  <_  x )  -> 
( x  e.  RR*  /\  0  <_  x )
)
5 elrege0 10746 . . . . . . . . . 10  |-  ( x  e.  ( 0 [,) 
+oo )  <->  ( x  e.  RR  /\  0  <_  x ) )
6 elxrge0 10747 . . . . . . . . . 10  |-  ( x  e.  ( 0 [,] 
+oo )  <->  ( x  e.  RR*  /\  0  <_  x ) )
74, 5, 63imtr4i 257 . . . . . . . . 9  |-  ( x  e.  ( 0 [,) 
+oo )  ->  x  e.  ( 0 [,]  +oo ) )
87ssriv 3184 . . . . . . . 8  |-  ( 0 [,)  +oo )  C_  (
0 [,]  +oo )
9 fss 5397 . . . . . . . 8  |-  ( ( ( F `  n
) : RR --> ( 0 [,)  +oo )  /\  (
0 [,)  +oo )  C_  ( 0 [,]  +oo ) )  ->  ( F `  n ) : RR --> ( 0 [,] 
+oo ) )
102, 8, 9sylancl 643 . . . . . . 7  |-  ( (
ph  /\  n  e.  NN )  ->  ( F `
 n ) : RR --> ( 0 [,] 
+oo ) )
11 itg2cl 19087 . . . . . . 7  |-  ( ( F `  n ) : RR --> ( 0 [,]  +oo )  ->  ( S.2 `  ( F `  n ) )  e. 
RR* )
1210, 11syl 15 . . . . . 6  |-  ( (
ph  /\  n  e.  NN )  ->  ( S.2 `  ( F `  n
) )  e.  RR* )
13 eqid 2283 . . . . . 6  |-  ( n  e.  NN  |->  ( S.2 `  ( F `  n
) ) )  =  ( n  e.  NN  |->  ( S.2 `  ( F `
 n ) ) )
1412, 13fmptd 5684 . . . . 5  |-  ( ph  ->  ( n  e.  NN  |->  ( S.2 `  ( F `
 n ) ) ) : NN --> RR* )
15 frn 5395 . . . . 5  |-  ( ( n  e.  NN  |->  ( S.2 `  ( F `
 n ) ) ) : NN --> RR*  ->  ran  ( n  e.  NN  |->  ( S.2 `  ( F `
 n ) ) )  C_  RR* )
1614, 15syl 15 . . . 4  |-  ( ph  ->  ran  ( n  e.  NN  |->  ( S.2 `  ( F `  n )
) )  C_  RR* )
17 supxrcl 10633 . . . 4  |-  ( ran  ( n  e.  NN  |->  ( S.2 `  ( F `
 n ) ) )  C_  RR*  ->  sup ( ran  ( n  e.  NN  |->  ( S.2 `  ( F `  n )
) ) ,  RR* ,  <  )  e.  RR* )
1816, 17syl 15 . . 3  |-  ( ph  ->  sup ( ran  (
n  e.  NN  |->  ( S.2 `  ( F `
 n ) ) ) ,  RR* ,  <  )  e.  RR* )
191, 18syl5eqel 2367 . 2  |-  ( ph  ->  S  e.  RR* )
20 itg2monolem2.7 . . 3  |-  ( ph  ->  P  e.  dom  S.1 )
21 itg1cl 19040 . . 3  |-  ( P  e.  dom  S.1  ->  ( S.1 `  P )  e.  RR )
2220, 21syl 15 . 2  |-  ( ph  ->  ( S.1 `  P
)  e.  RR )
23 mnfxr 10456 . . . 4  |-  -oo  e.  RR*
2423a1i 10 . . 3  |-  ( ph  ->  -oo  e.  RR* )
25 1nn 9757 . . . . 5  |-  1  e.  NN
2610ralrimiva 2626 . . . . 5  |-  ( ph  ->  A. n  e.  NN  ( F `  n ) : RR --> ( 0 [,]  +oo ) )
27 fveq2 5525 . . . . . . 7  |-  ( n  =  1  ->  ( F `  n )  =  ( F ` 
1 ) )
2827feq1d 5379 . . . . . 6  |-  ( n  =  1  ->  (
( F `  n
) : RR --> ( 0 [,]  +oo )  <->  ( F `  1 ) : RR --> ( 0 [,] 
+oo ) ) )
2928rspcv 2880 . . . . 5  |-  ( 1  e.  NN  ->  ( A. n  e.  NN  ( F `  n ) : RR --> ( 0 [,]  +oo )  ->  ( F `  1 ) : RR --> ( 0 [,] 
+oo ) ) )
3025, 26, 29mpsyl 59 . . . 4  |-  ( ph  ->  ( F `  1
) : RR --> ( 0 [,]  +oo ) )
31 itg2cl 19087 . . . 4  |-  ( ( F `  1 ) : RR --> ( 0 [,]  +oo )  ->  ( S.2 `  ( F ` 
1 ) )  e. 
RR* )
3230, 31syl 15 . . 3  |-  ( ph  ->  ( S.2 `  ( F `  1 )
)  e.  RR* )
33 itg2ge0 19090 . . . . 5  |-  ( ( F `  1 ) : RR --> ( 0 [,]  +oo )  ->  0  <_  ( S.2 `  ( F `  1 )
) )
3430, 33syl 15 . . . 4  |-  ( ph  ->  0  <_  ( S.2 `  ( F `  1
) ) )
35 0re 8838 . . . . . 6  |-  0  e.  RR
36 mnflt 10464 . . . . . 6  |-  ( 0  e.  RR  ->  -oo  <  0 )
3735, 36ax-mp 8 . . . . 5  |-  -oo  <  0
38 0xr 8878 . . . . . . 7  |-  0  e.  RR*
39 xrltletr 10488 . . . . . . 7  |-  ( ( 
-oo  e.  RR*  /\  0  e.  RR*  /\  ( S.2 `  ( F `  1
) )  e.  RR* )  ->  ( (  -oo  <  0  /\  0  <_ 
( S.2 `  ( F `
 1 ) ) )  ->  -oo  <  ( S.2 `  ( F ` 
1 ) ) ) )
4023, 38, 39mp3an12 1267 . . . . . 6  |-  ( ( S.2 `  ( F `
 1 ) )  e.  RR*  ->  ( ( 
-oo  <  0  /\  0  <_  ( S.2 `  ( F `  1 )
) )  ->  -oo  <  ( S.2 `  ( F `
 1 ) ) ) )
4132, 40syl 15 . . . . 5  |-  ( ph  ->  ( (  -oo  <  0  /\  0  <_  ( S.2 `  ( F ` 
1 ) ) )  ->  -oo  <  ( S.2 `  ( F `  1
) ) ) )
4237, 41mpani 657 . . . 4  |-  ( ph  ->  ( 0  <_  ( S.2 `  ( F ` 
1 ) )  ->  -oo  <  ( S.2 `  ( F `  1 )
) ) )
4334, 42mpd 14 . . 3  |-  ( ph  ->  -oo  <  ( S.2 `  ( F `  1
) ) )
4427fveq2d 5529 . . . . . . . 8  |-  ( n  =  1  ->  ( S.2 `  ( F `  n ) )  =  ( S.2 `  ( F `  1 )
) )
45 fvex 5539 . . . . . . . 8  |-  ( S.2 `  ( F `  1
) )  e.  _V
4644, 13, 45fvmpt 5602 . . . . . . 7  |-  ( 1  e.  NN  ->  (
( n  e.  NN  |->  ( S.2 `  ( F `
 n ) ) ) `  1 )  =  ( S.2 `  ( F `  1 )
) )
4725, 46ax-mp 8 . . . . . 6  |-  ( ( n  e.  NN  |->  ( S.2 `  ( F `
 n ) ) ) `  1 )  =  ( S.2 `  ( F `  1 )
)
48 ffn 5389 . . . . . . . 8  |-  ( ( n  e.  NN  |->  ( S.2 `  ( F `
 n ) ) ) : NN --> RR*  ->  ( n  e.  NN  |->  ( S.2 `  ( F `
 n ) ) )  Fn  NN )
4914, 48syl 15 . . . . . . 7  |-  ( ph  ->  ( n  e.  NN  |->  ( S.2 `  ( F `
 n ) ) )  Fn  NN )
50 fnfvelrn 5662 . . . . . . 7  |-  ( ( ( n  e.  NN  |->  ( S.2 `  ( F `
 n ) ) )  Fn  NN  /\  1  e.  NN )  ->  ( ( n  e.  NN  |->  ( S.2 `  ( F `  n )
) ) `  1
)  e.  ran  (
n  e.  NN  |->  ( S.2 `  ( F `
 n ) ) ) )
5149, 25, 50sylancl 643 . . . . . 6  |-  ( ph  ->  ( ( n  e.  NN  |->  ( S.2 `  ( F `  n )
) ) `  1
)  e.  ran  (
n  e.  NN  |->  ( S.2 `  ( F `
 n ) ) ) )
5247, 51syl5eqelr 2368 . . . . 5  |-  ( ph  ->  ( S.2 `  ( F `  1 )
)  e.  ran  (
n  e.  NN  |->  ( S.2 `  ( F `
 n ) ) ) )
53 supxrub 10643 . . . . 5  |-  ( ( ran  ( n  e.  NN  |->  ( S.2 `  ( F `  n )
) )  C_  RR*  /\  ( S.2 `  ( F ` 
1 ) )  e. 
ran  ( n  e.  NN  |->  ( S.2 `  ( F `  n )
) ) )  -> 
( S.2 `  ( F `
 1 ) )  <_  sup ( ran  (
n  e.  NN  |->  ( S.2 `  ( F `
 n ) ) ) ,  RR* ,  <  ) )
5416, 52, 53syl2anc 642 . . . 4  |-  ( ph  ->  ( S.2 `  ( F `  1 )
)  <_  sup ( ran  ( n  e.  NN  |->  ( S.2 `  ( F `
 n ) ) ) ,  RR* ,  <  ) )
5554, 1syl6breqr 4063 . . 3  |-  ( ph  ->  ( S.2 `  ( F `  1 )
)  <_  S )
5624, 32, 19, 43, 55xrltletrd 10492 . 2  |-  ( ph  ->  -oo  <  S )
57 itg2monolem2.9 . . . 4  |-  ( ph  ->  -.  ( S.1 `  P
)  <_  S )
5822rexrd 8881 . . . . 5  |-  ( ph  ->  ( S.1 `  P
)  e.  RR* )
59 xrltnle 8891 . . . . 5  |-  ( ( S  e.  RR*  /\  ( S.1 `  P )  e. 
RR* )  ->  ( S  <  ( S.1 `  P
)  <->  -.  ( S.1 `  P )  <_  S
) )
6019, 58, 59syl2anc 642 . . . 4  |-  ( ph  ->  ( S  <  ( S.1 `  P )  <->  -.  ( S.1 `  P )  <_  S ) )
6157, 60mpbird 223 . . 3  |-  ( ph  ->  S  <  ( S.1 `  P ) )
62 xrltle 10483 . . . 4  |-  ( ( S  e.  RR*  /\  ( S.1 `  P )  e. 
RR* )  ->  ( S  <  ( S.1 `  P
)  ->  S  <_  ( S.1 `  P ) ) )
6319, 58, 62syl2anc 642 . . 3  |-  ( ph  ->  ( S  <  ( S.1 `  P )  ->  S  <_  ( S.1 `  P
) ) )
6461, 63mpd 14 . 2  |-  ( ph  ->  S  <_  ( S.1 `  P ) )
65 xrre 10498 . 2  |-  ( ( ( S  e.  RR*  /\  ( S.1 `  P
)  e.  RR )  /\  (  -oo  <  S  /\  S  <_  ( S.1 `  P ) ) )  ->  S  e.  RR )
6619, 22, 56, 64, 65syl22anc 1183 1  |-  ( ph  ->  S  e.  RR )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544    C_ wss 3152   class class class wbr 4023    e. cmpt 4077   dom cdm 4689   ran crn 4690    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858    o Rcofr 6077   supcsup 7193   RRcr 8736   0cc0 8737   1c1 8738    + caddc 8740    +oocpnf 8864    -oocmnf 8865   RR*cxr 8866    < clt 8867    <_ cle 8868   NNcn 9746   [,)cico 10658   [,]cicc 10659  MblFncmbf 18969   S.1citg1 18970   S.2citg2 18971
This theorem is referenced by:  itg2monolem3  19107
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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