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Theorem itg2monolem2 22586
Description: Lemma for itg2mono 22588. (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 icossicc 11721 . . . . . . . 8  |-  ( 0 [,) +oo )  C_  ( 0 [,] +oo )
4 fss 5754 . . . . . . . 8  |-  ( ( ( F `  n
) : RR --> ( 0 [,) +oo )  /\  ( 0 [,) +oo )  C_  ( 0 [,] +oo ) )  ->  ( F `  n ) : RR --> ( 0 [,] +oo ) )
52, 3, 4sylancl 666 . . . . . . 7  |-  ( (
ph  /\  n  e.  NN )  ->  ( F `
 n ) : RR --> ( 0 [,] +oo ) )
6 itg2cl 22567 . . . . . . 7  |-  ( ( F `  n ) : RR --> ( 0 [,] +oo )  -> 
( S.2 `  ( F `
 n ) )  e.  RR* )
75, 6syl 17 . . . . . 6  |-  ( (
ph  /\  n  e.  NN )  ->  ( S.2 `  ( F `  n
) )  e.  RR* )
8 eqid 2429 . . . . . 6  |-  ( n  e.  NN  |->  ( S.2 `  ( F `  n
) ) )  =  ( n  e.  NN  |->  ( S.2 `  ( F `
 n ) ) )
97, 8fmptd 6061 . . . . 5  |-  ( ph  ->  ( n  e.  NN  |->  ( S.2 `  ( F `
 n ) ) ) : NN --> RR* )
10 frn 5752 . . . . 5  |-  ( ( n  e.  NN  |->  ( S.2 `  ( F `
 n ) ) ) : NN --> RR*  ->  ran  ( n  e.  NN  |->  ( S.2 `  ( F `
 n ) ) )  C_  RR* )
119, 10syl 17 . . . 4  |-  ( ph  ->  ran  ( n  e.  NN  |->  ( S.2 `  ( F `  n )
) )  C_  RR* )
12 supxrcl 11600 . . . 4  |-  ( ran  ( n  e.  NN  |->  ( S.2 `  ( F `
 n ) ) )  C_  RR*  ->  sup ( ran  ( n  e.  NN  |->  ( S.2 `  ( F `  n )
) ) ,  RR* ,  <  )  e.  RR* )
1311, 12syl 17 . . 3  |-  ( ph  ->  sup ( ran  (
n  e.  NN  |->  ( S.2 `  ( F `
 n ) ) ) ,  RR* ,  <  )  e.  RR* )
141, 13syl5eqel 2521 . 2  |-  ( ph  ->  S  e.  RR* )
15 itg2monolem2.7 . . 3  |-  ( ph  ->  P  e.  dom  S.1 )
16 itg1cl 22520 . . 3  |-  ( P  e.  dom  S.1  ->  ( S.1 `  P )  e.  RR )
1715, 16syl 17 . 2  |-  ( ph  ->  ( S.1 `  P
)  e.  RR )
18 mnfxr 11414 . . . 4  |- -oo  e.  RR*
1918a1i 11 . . 3  |-  ( ph  -> -oo  e.  RR* )
20 1nn 10620 . . . . 5  |-  1  e.  NN
215ralrimiva 2846 . . . . 5  |-  ( ph  ->  A. n  e.  NN  ( F `  n ) : RR --> ( 0 [,] +oo ) )
22 fveq2 5881 . . . . . . 7  |-  ( n  =  1  ->  ( F `  n )  =  ( F ` 
1 ) )
2322feq1d 5732 . . . . . 6  |-  ( n  =  1  ->  (
( F `  n
) : RR --> ( 0 [,] +oo )  <->  ( F `  1 ) : RR --> ( 0 [,] +oo ) ) )
2423rspcv 3184 . . . . 5  |-  ( 1  e.  NN  ->  ( A. n  e.  NN  ( F `  n ) : RR --> ( 0 [,] +oo )  -> 
( F `  1
) : RR --> ( 0 [,] +oo ) ) )
2520, 21, 24mpsyl 65 . . . 4  |-  ( ph  ->  ( F `  1
) : RR --> ( 0 [,] +oo ) )
26 itg2cl 22567 . . . 4  |-  ( ( F `  1 ) : RR --> ( 0 [,] +oo )  -> 
( S.2 `  ( F `
 1 ) )  e.  RR* )
2725, 26syl 17 . . 3  |-  ( ph  ->  ( S.2 `  ( F `  1 )
)  e.  RR* )
28 itg2ge0 22570 . . . . 5  |-  ( ( F `  1 ) : RR --> ( 0 [,] +oo )  -> 
0  <_  ( S.2 `  ( F `  1
) ) )
2925, 28syl 17 . . . 4  |-  ( ph  ->  0  <_  ( S.2 `  ( F `  1
) ) )
30 mnflt0 11427 . . . . 5  |- -oo  <  0
31 0xr 9686 . . . . . . 7  |-  0  e.  RR*
32 xrltletr 11454 . . . . . . 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 ) ) ) )
3318, 31, 32mp3an12 1350 . . . . . 6  |-  ( ( S.2 `  ( F `
 1 ) )  e.  RR*  ->  ( ( -oo  <  0  /\  0  <_  ( S.2 `  ( F `  1 )
) )  -> -oo  <  ( S.2 `  ( F `
 1 ) ) ) )
3427, 33syl 17 . . . . 5  |-  ( ph  ->  ( ( -oo  <  0  /\  0  <_  ( S.2 `  ( F ` 
1 ) ) )  -> -oo  <  ( S.2 `  ( F `  1
) ) ) )
3530, 34mpani 680 . . . 4  |-  ( ph  ->  ( 0  <_  ( S.2 `  ( F ` 
1 ) )  -> -oo  <  ( S.2 `  ( F `  1 )
) ) )
3629, 35mpd 15 . . 3  |-  ( ph  -> -oo  <  ( S.2 `  ( F `  1
) ) )
3722fveq2d 5885 . . . . . . . 8  |-  ( n  =  1  ->  ( S.2 `  ( F `  n ) )  =  ( S.2 `  ( F `  1 )
) )
38 fvex 5891 . . . . . . . 8  |-  ( S.2 `  ( F `  1
) )  e.  _V
3937, 8, 38fvmpt 5964 . . . . . . 7  |-  ( 1  e.  NN  ->  (
( n  e.  NN  |->  ( S.2 `  ( F `
 n ) ) ) `  1 )  =  ( S.2 `  ( F `  1 )
) )
4020, 39ax-mp 5 . . . . . 6  |-  ( ( n  e.  NN  |->  ( S.2 `  ( F `
 n ) ) ) `  1 )  =  ( S.2 `  ( F `  1 )
)
41 ffn 5746 . . . . . . . 8  |-  ( ( n  e.  NN  |->  ( S.2 `  ( F `
 n ) ) ) : NN --> RR*  ->  ( n  e.  NN  |->  ( S.2 `  ( F `
 n ) ) )  Fn  NN )
429, 41syl 17 . . . . . . 7  |-  ( ph  ->  ( n  e.  NN  |->  ( S.2 `  ( F `
 n ) ) )  Fn  NN )
43 fnfvelrn 6034 . . . . . . 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 ) ) ) )
4442, 20, 43sylancl 666 . . . . . 6  |-  ( ph  ->  ( ( n  e.  NN  |->  ( S.2 `  ( F `  n )
) ) `  1
)  e.  ran  (
n  e.  NN  |->  ( S.2 `  ( F `
 n ) ) ) )
4540, 44syl5eqelr 2522 . . . . 5  |-  ( ph  ->  ( S.2 `  ( F `  1 )
)  e.  ran  (
n  e.  NN  |->  ( S.2 `  ( F `
 n ) ) ) )
46 supxrub 11610 . . . . 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* ,  <  ) )
4711, 45, 46syl2anc 665 . . . 4  |-  ( ph  ->  ( S.2 `  ( F `  1 )
)  <_  sup ( ran  ( n  e.  NN  |->  ( S.2 `  ( F `
 n ) ) ) ,  RR* ,  <  ) )
4847, 1syl6breqr 4466 . . 3  |-  ( ph  ->  ( S.2 `  ( F `  1 )
)  <_  S )
4919, 27, 14, 36, 48xrltletrd 11458 . 2  |-  ( ph  -> -oo  <  S )
50 itg2monolem2.9 . . . 4  |-  ( ph  ->  -.  ( S.1 `  P
)  <_  S )
5117rexrd 9689 . . . . 5  |-  ( ph  ->  ( S.1 `  P
)  e.  RR* )
52 xrltnle 9700 . . . . 5  |-  ( ( S  e.  RR*  /\  ( S.1 `  P )  e. 
RR* )  ->  ( S  <  ( S.1 `  P
)  <->  -.  ( S.1 `  P )  <_  S
) )
5314, 51, 52syl2anc 665 . . . 4  |-  ( ph  ->  ( S  <  ( S.1 `  P )  <->  -.  ( S.1 `  P )  <_  S ) )
5450, 53mpbird 235 . . 3  |-  ( ph  ->  S  <  ( S.1 `  P ) )
55 xrltle 11448 . . . 4  |-  ( ( S  e.  RR*  /\  ( S.1 `  P )  e. 
RR* )  ->  ( S  <  ( S.1 `  P
)  ->  S  <_  ( S.1 `  P ) ) )
5614, 51, 55syl2anc 665 . . 3  |-  ( ph  ->  ( S  <  ( S.1 `  P )  ->  S  <_  ( S.1 `  P
) ) )
5754, 56mpd 15 . 2  |-  ( ph  ->  S  <_  ( S.1 `  P ) )
58 xrre 11464 . 2  |-  ( ( ( S  e.  RR*  /\  ( S.1 `  P
)  e.  RR )  /\  ( -oo  <  S  /\  S  <_  ( S.1 `  P ) ) )  ->  S  e.  RR )
5914, 17, 49, 57, 58syl22anc 1265 1  |-  ( ph  ->  S  e.  RR )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1870   A.wral 2782   E.wrex 2783    C_ wss 3442   class class class wbr 4426    |-> cmpt 4484   dom cdm 4854   ran crn 4855    Fn wfn 5596   -->wf 5597   ` cfv 5601  (class class class)co 6305    oRcofr 6544   supcsup 7960   RRcr 9537   0cc0 9538   1c1 9539    + caddc 9541   +oocpnf 9671   -oocmnf 9672   RR*cxr 9673    < clt 9674    <_ cle 9675   NNcn 10609   [,)cico 11637   [,]cicc 11638  MblFncmbf 22449   S.1citg1 22450   S.2citg2 22451
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-inf2 8146  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615  ax-pre-sup 9616
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-se 4814  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-isom 5610  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-of 6545  df-ofr 6546  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-2o 7191  df-oadd 7194  df-er 7371  df-map 7482  df-pm 7483  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-sup 7962  df-oi 8025  df-card 8372  df-cda 8596  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-div 10269  df-nn 10610  df-2 10668  df-3 10669  df-n0 10870  df-z 10938  df-uz 11160  df-q 11265  df-rp 11303  df-xadd 11410  df-ioo 11639  df-ico 11641  df-icc 11642  df-fz 11783  df-fzo 11914  df-fl 12025  df-seq 12211  df-exp 12270  df-hash 12513  df-cj 13141  df-re 13142  df-im 13143  df-sqrt 13277  df-abs 13278  df-clim 13530  df-sum 13731  df-xmet 18898  df-met 18899  df-ovol 22296  df-vol 22297  df-mbf 22454  df-itg1 22455  df-itg2 22456
This theorem is referenced by:  itg2monolem3  22587
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