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Theorem itg2monolem2 19635
Description: Lemma for itg2mono 19637. (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 9122 . . . . . . . . . . 11  |-  ( x  e.  RR  ->  x  e.  RR* )
43anim1i 552 . . . . . . . . . 10  |-  ( ( x  e.  RR  /\  0  <_  x )  -> 
( x  e.  RR*  /\  0  <_  x )
)
5 elrege0 10999 . . . . . . . . . 10  |-  ( x  e.  ( 0 [,) 
+oo )  <->  ( x  e.  RR  /\  0  <_  x ) )
6 elxrge0 11000 . . . . . . . . . 10  |-  ( x  e.  ( 0 [,] 
+oo )  <->  ( x  e.  RR*  /\  0  <_  x ) )
74, 5, 63imtr4i 258 . . . . . . . . 9  |-  ( x  e.  ( 0 [,) 
+oo )  ->  x  e.  ( 0 [,]  +oo ) )
87ssriv 3344 . . . . . . . 8  |-  ( 0 [,)  +oo )  C_  (
0 [,]  +oo )
9 fss 5591 . . . . . . . 8  |-  ( ( ( F `  n
) : RR --> ( 0 [,)  +oo )  /\  (
0 [,)  +oo )  C_  ( 0 [,]  +oo ) )  ->  ( F `  n ) : RR --> ( 0 [,] 
+oo ) )
102, 8, 9sylancl 644 . . . . . . 7  |-  ( (
ph  /\  n  e.  NN )  ->  ( F `
 n ) : RR --> ( 0 [,] 
+oo ) )
11 itg2cl 19616 . . . . . . 7  |-  ( ( F `  n ) : RR --> ( 0 [,]  +oo )  ->  ( S.2 `  ( F `  n ) )  e. 
RR* )
1210, 11syl 16 . . . . . 6  |-  ( (
ph  /\  n  e.  NN )  ->  ( S.2 `  ( F `  n
) )  e.  RR* )
13 eqid 2435 . . . . . 6  |-  ( n  e.  NN  |->  ( S.2 `  ( F `  n
) ) )  =  ( n  e.  NN  |->  ( S.2 `  ( F `
 n ) ) )
1412, 13fmptd 5885 . . . . 5  |-  ( ph  ->  ( n  e.  NN  |->  ( S.2 `  ( F `
 n ) ) ) : NN --> RR* )
15 frn 5589 . . . . 5  |-  ( ( n  e.  NN  |->  ( S.2 `  ( F `
 n ) ) ) : NN --> RR*  ->  ran  ( n  e.  NN  |->  ( S.2 `  ( F `
 n ) ) )  C_  RR* )
1614, 15syl 16 . . . 4  |-  ( ph  ->  ran  ( n  e.  NN  |->  ( S.2 `  ( F `  n )
) )  C_  RR* )
17 supxrcl 10885 . . . 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 16 . . 3  |-  ( ph  ->  sup ( ran  (
n  e.  NN  |->  ( S.2 `  ( F `
 n ) ) ) ,  RR* ,  <  )  e.  RR* )
191, 18syl5eqel 2519 . 2  |-  ( ph  ->  S  e.  RR* )
20 itg2monolem2.7 . . 3  |-  ( ph  ->  P  e.  dom  S.1 )
21 itg1cl 19569 . . 3  |-  ( P  e.  dom  S.1  ->  ( S.1 `  P )  e.  RR )
2220, 21syl 16 . 2  |-  ( ph  ->  ( S.1 `  P
)  e.  RR )
23 mnfxr 10706 . . . 4  |-  -oo  e.  RR*
2423a1i 11 . . 3  |-  ( ph  ->  -oo  e.  RR* )
25 1nn 10003 . . . . 5  |-  1  e.  NN
2610ralrimiva 2781 . . . . 5  |-  ( ph  ->  A. n  e.  NN  ( F `  n ) : RR --> ( 0 [,]  +oo ) )
27 fveq2 5720 . . . . . . 7  |-  ( n  =  1  ->  ( F `  n )  =  ( F ` 
1 ) )
2827feq1d 5572 . . . . . 6  |-  ( n  =  1  ->  (
( F `  n
) : RR --> ( 0 [,]  +oo )  <->  ( F `  1 ) : RR --> ( 0 [,] 
+oo ) ) )
2928rspcv 3040 . . . . 5  |-  ( 1  e.  NN  ->  ( A. n  e.  NN  ( F `  n ) : RR --> ( 0 [,]  +oo )  ->  ( F `  1 ) : RR --> ( 0 [,] 
+oo ) ) )
3025, 26, 29mpsyl 61 . . . 4  |-  ( ph  ->  ( F `  1
) : RR --> ( 0 [,]  +oo ) )
31 itg2cl 19616 . . . 4  |-  ( ( F `  1 ) : RR --> ( 0 [,]  +oo )  ->  ( S.2 `  ( F ` 
1 ) )  e. 
RR* )
3230, 31syl 16 . . 3  |-  ( ph  ->  ( S.2 `  ( F `  1 )
)  e.  RR* )
33 itg2ge0 19619 . . . . 5  |-  ( ( F `  1 ) : RR --> ( 0 [,]  +oo )  ->  0  <_  ( S.2 `  ( F `  1 )
) )
3430, 33syl 16 . . . 4  |-  ( ph  ->  0  <_  ( S.2 `  ( F `  1
) ) )
35 0re 9083 . . . . . 6  |-  0  e.  RR
36 mnflt 10714 . . . . . 6  |-  ( 0  e.  RR  ->  -oo  <  0 )
3735, 36ax-mp 8 . . . . 5  |-  -oo  <  0
38 0xr 9123 . . . . . . 7  |-  0  e.  RR*
39 xrltletr 10739 . . . . . . 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 1269 . . . . . 6  |-  ( ( S.2 `  ( F `
 1 ) )  e.  RR*  ->  ( ( 
-oo  <  0  /\  0  <_  ( S.2 `  ( F `  1 )
) )  ->  -oo  <  ( S.2 `  ( F `
 1 ) ) ) )
4132, 40syl 16 . . . . 5  |-  ( ph  ->  ( (  -oo  <  0  /\  0  <_  ( S.2 `  ( F ` 
1 ) ) )  ->  -oo  <  ( S.2 `  ( F `  1
) ) ) )
4237, 41mpani 658 . . . 4  |-  ( ph  ->  ( 0  <_  ( S.2 `  ( F ` 
1 ) )  ->  -oo  <  ( S.2 `  ( F `  1 )
) ) )
4334, 42mpd 15 . . 3  |-  ( ph  ->  -oo  <  ( S.2 `  ( F `  1
) ) )
4427fveq2d 5724 . . . . . . . 8  |-  ( n  =  1  ->  ( S.2 `  ( F `  n ) )  =  ( S.2 `  ( F `  1 )
) )
45 fvex 5734 . . . . . . . 8  |-  ( S.2 `  ( F `  1
) )  e.  _V
4644, 13, 45fvmpt 5798 . . . . . . 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 5583 . . . . . . . 8  |-  ( ( n  e.  NN  |->  ( S.2 `  ( F `
 n ) ) ) : NN --> RR*  ->  ( n  e.  NN  |->  ( S.2 `  ( F `
 n ) ) )  Fn  NN )
4914, 48syl 16 . . . . . . 7  |-  ( ph  ->  ( n  e.  NN  |->  ( S.2 `  ( F `
 n ) ) )  Fn  NN )
50 fnfvelrn 5859 . . . . . . 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 644 . . . . . 6  |-  ( ph  ->  ( ( n  e.  NN  |->  ( S.2 `  ( F `  n )
) ) `  1
)  e.  ran  (
n  e.  NN  |->  ( S.2 `  ( F `
 n ) ) ) )
5247, 51syl5eqelr 2520 . . . . 5  |-  ( ph  ->  ( S.2 `  ( F `  1 )
)  e.  ran  (
n  e.  NN  |->  ( S.2 `  ( F `
 n ) ) ) )
53 supxrub 10895 . . . . 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 643 . . . 4  |-  ( ph  ->  ( S.2 `  ( F `  1 )
)  <_  sup ( ran  ( n  e.  NN  |->  ( S.2 `  ( F `
 n ) ) ) ,  RR* ,  <  ) )
5554, 1syl6breqr 4244 . . 3  |-  ( ph  ->  ( S.2 `  ( F `  1 )
)  <_  S )
5624, 32, 19, 43, 55xrltletrd 10743 . 2  |-  ( ph  ->  -oo  <  S )
57 itg2monolem2.9 . . . 4  |-  ( ph  ->  -.  ( S.1 `  P
)  <_  S )
5822rexrd 9126 . . . . 5  |-  ( ph  ->  ( S.1 `  P
)  e.  RR* )
59 xrltnle 9136 . . . . 5  |-  ( ( S  e.  RR*  /\  ( S.1 `  P )  e. 
RR* )  ->  ( S  <  ( S.1 `  P
)  <->  -.  ( S.1 `  P )  <_  S
) )
6019, 58, 59syl2anc 643 . . . 4  |-  ( ph  ->  ( S  <  ( S.1 `  P )  <->  -.  ( S.1 `  P )  <_  S ) )
6157, 60mpbird 224 . . 3  |-  ( ph  ->  S  <  ( S.1 `  P ) )
62 xrltle 10734 . . . 4  |-  ( ( S  e.  RR*  /\  ( S.1 `  P )  e. 
RR* )  ->  ( S  <  ( S.1 `  P
)  ->  S  <_  ( S.1 `  P ) ) )
6319, 58, 62syl2anc 643 . . 3  |-  ( ph  ->  ( S  <  ( S.1 `  P )  ->  S  <_  ( S.1 `  P
) ) )
6461, 63mpd 15 . 2  |-  ( ph  ->  S  <_  ( S.1 `  P ) )
65 xrre 10749 . 2  |-  ( ( ( S  e.  RR*  /\  ( S.1 `  P
)  e.  RR )  /\  (  -oo  <  S  /\  S  <_  ( S.1 `  P ) ) )  ->  S  e.  RR )
6619, 22, 56, 64, 65syl22anc 1185 1  |-  ( ph  ->  S  e.  RR )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2697   E.wrex 2698    C_ wss 3312   class class class wbr 4204    e. cmpt 4258   dom cdm 4870   ran crn 4871    Fn wfn 5441   -->wf 5442   ` cfv 5446  (class class class)co 6073    o Rcofr 6296   supcsup 7437   RRcr 8981   0cc0 8982   1c1 8983    + caddc 8985    +oocpnf 9109    -oocmnf 9110   RR*cxr 9111    < clt 9112    <_ cle 9113   NNcn 9992   [,)cico 10910   [,]cicc 10911  MblFncmbf 19498   S.1citg1 19499   S.2citg2 19500
This theorem is referenced by:  itg2monolem3  19636
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7588  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-of 6297  df-ofr 6298  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-2o 6717  df-oadd 6720  df-er 6897  df-map 7012  df-pm 7013  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-sup 7438  df-oi 7471  df-card 7818  df-cda 8040  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-n0 10214  df-z 10275  df-uz 10481  df-q 10567  df-rp 10605  df-xadd 10703  df-ioo 10912  df-ico 10914  df-icc 10915  df-fz 11036  df-fzo 11128  df-fl 11194  df-seq 11316  df-exp 11375  df-hash 11611  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033  df-clim 12274  df-sum 12472  df-xmet 16687  df-met 16688  df-ovol 19353  df-vol 19354  df-mbf 19504  df-itg1 19505  df-itg2 19506
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