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Theorem itg2monolem2 19122
Description: Lemma for itg2mono 19124. (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 8893 . . . . . . . . . . 11  |-  ( x  e.  RR  ->  x  e.  RR* )
43anim1i 551 . . . . . . . . . 10  |-  ( ( x  e.  RR  /\  0  <_  x )  -> 
( x  e.  RR*  /\  0  <_  x )
)
5 elrege0 10762 . . . . . . . . . 10  |-  ( x  e.  ( 0 [,) 
+oo )  <->  ( x  e.  RR  /\  0  <_  x ) )
6 elxrge0 10763 . . . . . . . . . 10  |-  ( x  e.  ( 0 [,] 
+oo )  <->  ( x  e.  RR*  /\  0  <_  x ) )
74, 5, 63imtr4i 257 . . . . . . . . 9  |-  ( x  e.  ( 0 [,) 
+oo )  ->  x  e.  ( 0 [,]  +oo ) )
87ssriv 3197 . . . . . . . 8  |-  ( 0 [,)  +oo )  C_  (
0 [,]  +oo )
9 fss 5413 . . . . . . . 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 19103 . . . . . . 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 2296 . . . . . 6  |-  ( n  e.  NN  |->  ( S.2 `  ( F `  n
) ) )  =  ( n  e.  NN  |->  ( S.2 `  ( F `
 n ) ) )
1412, 13fmptd 5700 . . . . 5  |-  ( ph  ->  ( n  e.  NN  |->  ( S.2 `  ( F `
 n ) ) ) : NN --> RR* )
15 frn 5411 . . . . 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 10649 . . . 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 2380 . 2  |-  ( ph  ->  S  e.  RR* )
20 itg2monolem2.7 . . 3  |-  ( ph  ->  P  e.  dom  S.1 )
21 itg1cl 19056 . . 3  |-  ( P  e.  dom  S.1  ->  ( S.1 `  P )  e.  RR )
2220, 21syl 15 . 2  |-  ( ph  ->  ( S.1 `  P
)  e.  RR )
23 mnfxr 10472 . . . 4  |-  -oo  e.  RR*
2423a1i 10 . . 3  |-  ( ph  ->  -oo  e.  RR* )
25 1nn 9773 . . . . 5  |-  1  e.  NN
2610ralrimiva 2639 . . . . 5  |-  ( ph  ->  A. n  e.  NN  ( F `  n ) : RR --> ( 0 [,]  +oo ) )
27 fveq2 5541 . . . . . . 7  |-  ( n  =  1  ->  ( F `  n )  =  ( F ` 
1 ) )
2827feq1d 5395 . . . . . 6  |-  ( n  =  1  ->  (
( F `  n
) : RR --> ( 0 [,]  +oo )  <->  ( F `  1 ) : RR --> ( 0 [,] 
+oo ) ) )
2928rspcv 2893 . . . . 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 19103 . . . 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 19106 . . . . 5  |-  ( ( F `  1 ) : RR --> ( 0 [,]  +oo )  ->  0  <_  ( S.2 `  ( F `  1 )
) )
3430, 33syl 15 . . . 4  |-  ( ph  ->  0  <_  ( S.2 `  ( F `  1
) ) )
35 0re 8854 . . . . . 6  |-  0  e.  RR
36 mnflt 10480 . . . . . 6  |-  ( 0  e.  RR  ->  -oo  <  0 )
3735, 36ax-mp 8 . . . . 5  |-  -oo  <  0
38 0xr 8894 . . . . . . 7  |-  0  e.  RR*
39 xrltletr 10504 . . . . . . 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 5545 . . . . . . . 8  |-  ( n  =  1  ->  ( S.2 `  ( F `  n ) )  =  ( S.2 `  ( F `  1 )
) )
45 fvex 5555 . . . . . . . 8  |-  ( S.2 `  ( F `  1
) )  e.  _V
4644, 13, 45fvmpt 5618 . . . . . . 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 5405 . . . . . . . 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 5678 . . . . . . 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 2381 . . . . 5  |-  ( ph  ->  ( S.2 `  ( F `  1 )
)  e.  ran  (
n  e.  NN  |->  ( S.2 `  ( F `
 n ) ) ) )
53 supxrub 10659 . . . . 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 4079 . . 3  |-  ( ph  ->  ( S.2 `  ( F `  1 )
)  <_  S )
5624, 32, 19, 43, 55xrltletrd 10508 . 2  |-  ( ph  ->  -oo  <  S )
57 itg2monolem2.9 . . . 4  |-  ( ph  ->  -.  ( S.1 `  P
)  <_  S )
5822rexrd 8897 . . . . 5  |-  ( ph  ->  ( S.1 `  P
)  e.  RR* )
59 xrltnle 8907 . . . . 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 10499 . . . 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 10514 . 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 1632    e. wcel 1696   A.wral 2556   E.wrex 2557    C_ wss 3165   class class class wbr 4039    e. cmpt 4093   dom cdm 4705   ran crn 4706    Fn wfn 5266   -->wf 5267   ` cfv 5271  (class class class)co 5874    o Rcofr 6093   supcsup 7209   RRcr 8752   0cc0 8753   1c1 8754    + caddc 8756    +oocpnf 8880    -oocmnf 8881   RR*cxr 8882    < clt 8883    <_ cle 8884   NNcn 9762   [,)cico 10674   [,]cicc 10675  MblFncmbf 18985   S.1citg1 18986   S.2citg2 18987
This theorem is referenced by:  itg2monolem3  19123
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-ofr 6095  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-er 6676  df-map 6790  df-pm 6791  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-oi 7241  df-card 7588  df-cda 7810  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-q 10333  df-rp 10371  df-xadd 10469  df-ioo 10676  df-ico 10678  df-icc 10679  df-fz 10799  df-fzo 10887  df-fl 10941  df-seq 11063  df-exp 11121  df-hash 11354  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-clim 11978  df-sum 12175  df-xmet 16389  df-met 16390  df-ovol 18840  df-vol 18841  df-mbf 18991  df-itg1 18992  df-itg2 18993
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