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Theorem itg2monolem2 21909
Description: Lemma for itg2mono 21911. (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 9638 . . . . . . . . . . 11  |-  ( x  e.  RR  ->  x  e.  RR* )
43anim1i 568 . . . . . . . . . 10  |-  ( ( x  e.  RR  /\  0  <_  x )  -> 
( x  e.  RR*  /\  0  <_  x )
)
5 elrege0 11626 . . . . . . . . . 10  |-  ( x  e.  ( 0 [,) +oo )  <->  ( x  e.  RR  /\  0  <_  x ) )
6 elxrge0 11628 . . . . . . . . . 10  |-  ( x  e.  ( 0 [,] +oo )  <->  ( x  e. 
RR*  /\  0  <_  x ) )
74, 5, 63imtr4i 266 . . . . . . . . 9  |-  ( x  e.  ( 0 [,) +oo )  ->  x  e.  ( 0 [,] +oo ) )
87ssriv 3508 . . . . . . . 8  |-  ( 0 [,) +oo )  C_  ( 0 [,] +oo )
9 fss 5738 . . . . . . . 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 21890 . . . . . . 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 2467 . . . . . 6  |-  ( n  e.  NN  |->  ( S.2 `  ( F `  n
) ) )  =  ( n  e.  NN  |->  ( S.2 `  ( F `
 n ) ) )
1412, 13fmptd 6044 . . . . 5  |-  ( ph  ->  ( n  e.  NN  |->  ( S.2 `  ( F `
 n ) ) ) : NN --> RR* )
15 frn 5736 . . . . 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 11505 . . . 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 2559 . 2  |-  ( ph  ->  S  e.  RR* )
20 itg2monolem2.7 . . 3  |-  ( ph  ->  P  e.  dom  S.1 )
21 itg1cl 21843 . . 3  |-  ( P  e.  dom  S.1  ->  ( S.1 `  P )  e.  RR )
2220, 21syl 16 . 2  |-  ( ph  ->  ( S.1 `  P
)  e.  RR )
23 mnfxr 11322 . . . 4  |- -oo  e.  RR*
2423a1i 11 . . 3  |-  ( ph  -> -oo  e.  RR* )
25 1nn 10546 . . . . 5  |-  1  e.  NN
2610ralrimiva 2878 . . . . 5  |-  ( ph  ->  A. n  e.  NN  ( F `  n ) : RR --> ( 0 [,] +oo ) )
27 fveq2 5865 . . . . . . 7  |-  ( n  =  1  ->  ( F `  n )  =  ( F ` 
1 ) )
2827feq1d 5716 . . . . . 6  |-  ( n  =  1  ->  (
( F `  n
) : RR --> ( 0 [,] +oo )  <->  ( F `  1 ) : RR --> ( 0 [,] +oo ) ) )
2928rspcv 3210 . . . . 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 21890 . . . 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 21893 . . . . 5  |-  ( ( F `  1 ) : RR --> ( 0 [,] +oo )  -> 
0  <_  ( S.2 `  ( F `  1
) ) )
3430, 33syl 16 . . . 4  |-  ( ph  ->  0  <_  ( S.2 `  ( F `  1
) ) )
35 mnflt0 11333 . . . . 5  |- -oo  <  0
36 0xr 9639 . . . . . . 7  |-  0  e.  RR*
37 xrltletr 11359 . . . . . . 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 1314 . . . . . 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 5869 . . . . . . . 8  |-  ( n  =  1  ->  ( S.2 `  ( F `  n ) )  =  ( S.2 `  ( F `  1 )
) )
43 fvex 5875 . . . . . . . 8  |-  ( S.2 `  ( F `  1
) )  e.  _V
4442, 13, 43fvmpt 5949 . . . . . . 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 5730 . . . . . . . 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 6017 . . . . . . 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 2560 . . . . 5  |-  ( ph  ->  ( S.2 `  ( F `  1 )
)  e.  ran  (
n  e.  NN  |->  ( S.2 `  ( F `
 n ) ) ) )
51 supxrub 11515 . . . . 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 4487 . . 3  |-  ( ph  ->  ( S.2 `  ( F `  1 )
)  <_  S )
5424, 32, 19, 41, 53xrltletrd 11363 . 2  |-  ( ph  -> -oo  <  S )
55 itg2monolem2.9 . . . 4  |-  ( ph  ->  -.  ( S.1 `  P
)  <_  S )
5622rexrd 9642 . . . . 5  |-  ( ph  ->  ( S.1 `  P
)  e.  RR* )
57 xrltnle 9652 . . . . 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 11354 . . . 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 11369 . 2  |-  ( ( ( S  e.  RR*  /\  ( S.1 `  P
)  e.  RR )  /\  ( -oo  <  S  /\  S  <_  ( S.1 `  P ) ) )  ->  S  e.  RR )
6419, 22, 54, 62, 63syl22anc 1229 1  |-  ( ph  ->  S  e.  RR )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2814   E.wrex 2815    C_ wss 3476   class class class wbr 4447    |-> cmpt 4505   dom cdm 4999   ran crn 5000    Fn wfn 5582   -->wf 5583   ` cfv 5587  (class class class)co 6283    oRcofr 6522   supcsup 7899   RRcr 9490   0cc0 9491   1c1 9492    + caddc 9494   +oocpnf 9624   -oocmnf 9625   RR*cxr 9626    < clt 9627    <_ cle 9628   NNcn 10535   [,)cico 11530   [,]cicc 11531  MblFncmbf 21774   S.1citg1 21775   S.2citg2 21776
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6575  ax-inf2 8057  ax-cnex 9547  ax-resscn 9548  ax-1cn 9549  ax-icn 9550  ax-addcl 9551  ax-addrcl 9552  ax-mulcl 9553  ax-mulrcl 9554  ax-mulcom 9555  ax-addass 9556  ax-mulass 9557  ax-distr 9558  ax-i2m1 9559  ax-1ne0 9560  ax-1rid 9561  ax-rnegex 9562  ax-rrecex 9563  ax-cnre 9564  ax-pre-lttri 9565  ax-pre-lttrn 9566  ax-pre-ltadd 9567  ax-pre-mulgt0 9568  ax-pre-sup 9569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5550  df-fun 5589  df-fn 5590  df-f 5591  df-f1 5592  df-fo 5593  df-f1o 5594  df-fv 5595  df-isom 5596  df-riota 6244  df-ov 6286  df-oprab 6287  df-mpt2 6288  df-of 6523  df-ofr 6524  df-om 6680  df-1st 6784  df-2nd 6785  df-recs 7042  df-rdg 7076  df-1o 7130  df-2o 7131  df-oadd 7134  df-er 7311  df-map 7422  df-pm 7423  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-sup 7900  df-oi 7934  df-card 8319  df-cda 8547  df-pnf 9629  df-mnf 9630  df-xr 9631  df-ltxr 9632  df-le 9633  df-sub 9806  df-neg 9807  df-div 10206  df-nn 10536  df-2 10593  df-3 10594  df-n0 10795  df-z 10864  df-uz 11082  df-q 11182  df-rp 11220  df-xadd 11318  df-ioo 11532  df-ico 11534  df-icc 11535  df-fz 11672  df-fzo 11792  df-fl 11896  df-seq 12075  df-exp 12134  df-hash 12373  df-cj 12894  df-re 12895  df-im 12896  df-sqrt 13030  df-abs 13031  df-clim 13273  df-sum 13471  df-xmet 18199  df-met 18200  df-ovol 21627  df-vol 21628  df-mbf 21779  df-itg1 21780  df-itg2 21781
This theorem is referenced by:  itg2monolem3  21910
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