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Theorem itg2monolem2 21229
Description: Lemma for itg2mono 21231. (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 )  oR  <_  ( 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  oR  <_  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 9429 . . . . . . . . . . 11  |-  ( x  e.  RR  ->  x  e.  RR* )
43anim1i 568 . . . . . . . . . 10  |-  ( ( x  e.  RR  /\  0  <_  x )  -> 
( x  e.  RR*  /\  0  <_  x )
)
5 elrege0 11392 . . . . . . . . . 10  |-  ( x  e.  ( 0 [,) +oo )  <->  ( x  e.  RR  /\  0  <_  x ) )
6 elxrge0 11394 . . . . . . . . . 10  |-  ( x  e.  ( 0 [,] +oo )  <->  ( x  e. 
RR*  /\  0  <_  x ) )
74, 5, 63imtr4i 266 . . . . . . . . 9  |-  ( x  e.  ( 0 [,) +oo )  ->  x  e.  ( 0 [,] +oo ) )
87ssriv 3360 . . . . . . . 8  |-  ( 0 [,) +oo )  C_  ( 0 [,] +oo )
9 fss 5567 . . . . . . . 8  |-  ( ( ( F `  n
) : RR --> ( 0 [,) +oo )  /\  ( 0 [,) +oo )  C_  ( 0 [,] +oo ) )  ->  ( F `  n ) : RR --> ( 0 [,] +oo ) )
102, 8, 9sylancl 662 . . . . . . 7  |-  ( (
ph  /\  n  e.  NN )  ->  ( F `
 n ) : RR --> ( 0 [,] +oo ) )
11 itg2cl 21210 . . . . . . 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 2443 . . . . . 6  |-  ( n  e.  NN  |->  ( S.2 `  ( F `  n
) ) )  =  ( n  e.  NN  |->  ( S.2 `  ( F `
 n ) ) )
1412, 13fmptd 5867 . . . . 5  |-  ( ph  ->  ( n  e.  NN  |->  ( S.2 `  ( F `
 n ) ) ) : NN --> RR* )
15 frn 5565 . . . . 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 11277 . . . 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 2527 . 2  |-  ( ph  ->  S  e.  RR* )
20 itg2monolem2.7 . . 3  |-  ( ph  ->  P  e.  dom  S.1 )
21 itg1cl 21163 . . 3  |-  ( P  e.  dom  S.1  ->  ( S.1 `  P )  e.  RR )
2220, 21syl 16 . 2  |-  ( ph  ->  ( S.1 `  P
)  e.  RR )
23 mnfxr 11094 . . . 4  |- -oo  e.  RR*
2423a1i 11 . . 3  |-  ( ph  -> -oo  e.  RR* )
25 1nn 10333 . . . . 5  |-  1  e.  NN
2610ralrimiva 2799 . . . . 5  |-  ( ph  ->  A. n  e.  NN  ( F `  n ) : RR --> ( 0 [,] +oo ) )
27 fveq2 5691 . . . . . . 7  |-  ( n  =  1  ->  ( F `  n )  =  ( F ` 
1 ) )
2827feq1d 5546 . . . . . 6  |-  ( n  =  1  ->  (
( F `  n
) : RR --> ( 0 [,] +oo )  <->  ( F `  1 ) : RR --> ( 0 [,] +oo ) ) )
2928rspcv 3069 . . . . 5  |-  ( 1  e.  NN  ->  ( A. n  e.  NN  ( F `  n ) : RR --> ( 0 [,] +oo )  -> 
( F `  1
) : RR --> ( 0 [,] +oo ) ) )
3025, 26, 29mpsyl 63 . . . 4  |-  ( ph  ->  ( F `  1
) : RR --> ( 0 [,] +oo ) )
31 itg2cl 21210 . . . 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 21213 . . . . 5  |-  ( ( F `  1 ) : RR --> ( 0 [,] +oo )  -> 
0  <_  ( S.2 `  ( F `  1
) ) )
3430, 33syl 16 . . . 4  |-  ( ph  ->  0  <_  ( S.2 `  ( F `  1
) ) )
35 mnflt0 11105 . . . . 5  |- -oo  <  0
36 0xr 9430 . . . . . . 7  |-  0  e.  RR*
37 xrltletr 11131 . . . . . . 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 ) ) ) )
3823, 36, 37mp3an12 1304 . . . . . 6  |-  ( ( S.2 `  ( F `
 1 ) )  e.  RR*  ->  ( ( -oo  <  0  /\  0  <_  ( S.2 `  ( F `  1 )
) )  -> -oo  <  ( S.2 `  ( F `
 1 ) ) ) )
3932, 38syl 16 . . . . 5  |-  ( ph  ->  ( ( -oo  <  0  /\  0  <_  ( S.2 `  ( F ` 
1 ) ) )  -> -oo  <  ( S.2 `  ( F `  1
) ) ) )
4035, 39mpani 676 . . . 4  |-  ( ph  ->  ( 0  <_  ( S.2 `  ( F ` 
1 ) )  -> -oo  <  ( S.2 `  ( F `  1 )
) ) )
4134, 40mpd 15 . . 3  |-  ( ph  -> -oo  <  ( S.2 `  ( F `  1
) ) )
4227fveq2d 5695 . . . . . . . 8  |-  ( n  =  1  ->  ( S.2 `  ( F `  n ) )  =  ( S.2 `  ( F `  1 )
) )
43 fvex 5701 . . . . . . . 8  |-  ( S.2 `  ( F `  1
) )  e.  _V
4442, 13, 43fvmpt 5774 . . . . . . 7  |-  ( 1  e.  NN  ->  (
( n  e.  NN  |->  ( S.2 `  ( F `
 n ) ) ) `  1 )  =  ( S.2 `  ( F `  1 )
) )
4525, 44ax-mp 5 . . . . . 6  |-  ( ( n  e.  NN  |->  ( S.2 `  ( F `
 n ) ) ) `  1 )  =  ( S.2 `  ( F `  1 )
)
46 ffn 5559 . . . . . . . 8  |-  ( ( n  e.  NN  |->  ( S.2 `  ( F `
 n ) ) ) : NN --> RR*  ->  ( n  e.  NN  |->  ( S.2 `  ( F `
 n ) ) )  Fn  NN )
4714, 46syl 16 . . . . . . 7  |-  ( ph  ->  ( n  e.  NN  |->  ( S.2 `  ( F `
 n ) ) )  Fn  NN )
48 fnfvelrn 5840 . . . . . . 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 ) ) ) )
4947, 25, 48sylancl 662 . . . . . 6  |-  ( ph  ->  ( ( n  e.  NN  |->  ( S.2 `  ( F `  n )
) ) `  1
)  e.  ran  (
n  e.  NN  |->  ( S.2 `  ( F `
 n ) ) ) )
5045, 49syl5eqelr 2528 . . . . 5  |-  ( ph  ->  ( S.2 `  ( F `  1 )
)  e.  ran  (
n  e.  NN  |->  ( S.2 `  ( F `
 n ) ) ) )
51 supxrub 11287 . . . . 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* ,  <  ) )
5216, 50, 51syl2anc 661 . . . 4  |-  ( ph  ->  ( S.2 `  ( F `  1 )
)  <_  sup ( ran  ( n  e.  NN  |->  ( S.2 `  ( F `
 n ) ) ) ,  RR* ,  <  ) )
5352, 1syl6breqr 4332 . . 3  |-  ( ph  ->  ( S.2 `  ( F `  1 )
)  <_  S )
5424, 32, 19, 41, 53xrltletrd 11135 . 2  |-  ( ph  -> -oo  <  S )
55 itg2monolem2.9 . . . 4  |-  ( ph  ->  -.  ( S.1 `  P
)  <_  S )
5622rexrd 9433 . . . . 5  |-  ( ph  ->  ( S.1 `  P
)  e.  RR* )
57 xrltnle 9443 . . . . 5  |-  ( ( S  e.  RR*  /\  ( S.1 `  P )  e. 
RR* )  ->  ( S  <  ( S.1 `  P
)  <->  -.  ( S.1 `  P )  <_  S
) )
5819, 56, 57syl2anc 661 . . . 4  |-  ( ph  ->  ( S  <  ( S.1 `  P )  <->  -.  ( S.1 `  P )  <_  S ) )
5955, 58mpbird 232 . . 3  |-  ( ph  ->  S  <  ( S.1 `  P ) )
60 xrltle 11126 . . . 4  |-  ( ( S  e.  RR*  /\  ( S.1 `  P )  e. 
RR* )  ->  ( S  <  ( S.1 `  P
)  ->  S  <_  ( S.1 `  P ) ) )
6119, 56, 60syl2anc 661 . . 3  |-  ( ph  ->  ( S  <  ( S.1 `  P )  ->  S  <_  ( S.1 `  P
) ) )
6259, 61mpd 15 . 2  |-  ( ph  ->  S  <_  ( S.1 `  P ) )
63 xrre 11141 . 2  |-  ( ( ( S  e.  RR*  /\  ( S.1 `  P
)  e.  RR )  /\  ( -oo  <  S  /\  S  <_  ( S.1 `  P ) ) )  ->  S  e.  RR )
6419, 22, 54, 62, 63syl22anc 1219 1  |-  ( ph  ->  S  e.  RR )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2715   E.wrex 2716    C_ wss 3328   class class class wbr 4292    e. cmpt 4350   dom cdm 4840   ran crn 4841    Fn wfn 5413   -->wf 5414   ` cfv 5418  (class class class)co 6091    oRcofr 6319   supcsup 7690   RRcr 9281   0cc0 9282   1c1 9283    + caddc 9285   +oocpnf 9415   -oocmnf 9416   RR*cxr 9417    < clt 9418    <_ cle 9419   NNcn 10322   [,)cico 11302   [,]cicc 11303  MblFncmbf 21094   S.1citg1 21095   S.2citg2 21096
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 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-inf2 7847  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359  ax-pre-sup 9360
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 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-se 4680  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-isom 5427  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-of 6320  df-ofr 6321  df-om 6477  df-1st 6577  df-2nd 6578  df-recs 6832  df-rdg 6866  df-1o 6920  df-2o 6921  df-oadd 6924  df-er 7101  df-map 7216  df-pm 7217  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-sup 7691  df-oi 7724  df-card 8109  df-cda 8337  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-div 9994  df-nn 10323  df-2 10380  df-3 10381  df-n0 10580  df-z 10647  df-uz 10862  df-q 10954  df-rp 10992  df-xadd 11090  df-ioo 11304  df-ico 11306  df-icc 11307  df-fz 11438  df-fzo 11549  df-fl 11642  df-seq 11807  df-exp 11866  df-hash 12104  df-cj 12588  df-re 12589  df-im 12590  df-sqr 12724  df-abs 12725  df-clim 12966  df-sum 13164  df-xmet 17810  df-met 17811  df-ovol 20948  df-vol 20949  df-mbf 21099  df-itg1 21100  df-itg2 21101
This theorem is referenced by:  itg2monolem3  21230
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