MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ovolshftlem2 Structured version   Unicode version

Theorem ovolshftlem2 21012
Description: Lemma for ovolshft 21013. (Contributed by Mario Carneiro, 22-Mar-2014.)
Hypotheses
Ref Expression
ovolshft.1  |-  ( ph  ->  A  C_  RR )
ovolshft.2  |-  ( ph  ->  C  e.  RR )
ovolshft.3  |-  ( ph  ->  B  =  { x  e.  RR  |  ( x  -  C )  e.  A } )
ovolshft.4  |-  M  =  { y  e.  RR*  |  E. f  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN ) ( B  C_  U. ran  ( (,)  o.  f )  /\  y  =  sup ( ran  seq 1 (  +  ,  ( ( abs 
o.  -  )  o.  f ) ) , 
RR* ,  <  ) ) }
Assertion
Ref Expression
ovolshftlem2  |-  ( ph  ->  { z  e.  RR*  |  E. g  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN ) ( A  C_  U. ran  ( (,)  o.  g )  /\  z  =  sup ( ran  seq 1 (  +  ,  ( ( abs 
o.  -  )  o.  g ) ) , 
RR* ,  <  ) ) }  C_  M )
Distinct variable groups:    f, g, x, y, z, A    C, f, g, x, y, z    B, f, g, y, z   
g, M, z    ph, f,
g, y, z
Allowed substitution hints:    ph( x)    B( x)    M( x, y, f)

Proof of Theorem ovolshftlem2
Dummy variables  n  m are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ovolshft.1 . . . . . . . 8  |-  ( ph  ->  A  C_  RR )
21ad3antrrr 729 . . . . . . 7  |-  ( ( ( ( ph  /\  z  e.  RR* )  /\  g  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN ) )  /\  A  C_ 
U. ran  ( (,)  o.  g ) )  ->  A  C_  RR )
3 ovolshft.2 . . . . . . . 8  |-  ( ph  ->  C  e.  RR )
43ad3antrrr 729 . . . . . . 7  |-  ( ( ( ( ph  /\  z  e.  RR* )  /\  g  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN ) )  /\  A  C_ 
U. ran  ( (,)  o.  g ) )  ->  C  e.  RR )
5 ovolshft.3 . . . . . . . 8  |-  ( ph  ->  B  =  { x  e.  RR  |  ( x  -  C )  e.  A } )
65ad3antrrr 729 . . . . . . 7  |-  ( ( ( ( ph  /\  z  e.  RR* )  /\  g  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN ) )  /\  A  C_ 
U. ran  ( (,)  o.  g ) )  ->  B  =  { x  e.  RR  |  ( x  -  C )  e.  A } )
7 ovolshft.4 . . . . . . 7  |-  M  =  { y  e.  RR*  |  E. f  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN ) ( B  C_  U. ran  ( (,)  o.  f )  /\  y  =  sup ( ran  seq 1 (  +  ,  ( ( abs 
o.  -  )  o.  f ) ) , 
RR* ,  <  ) ) }
8 eqid 2443 . . . . . . 7  |-  seq 1
(  +  ,  ( ( abs  o.  -  )  o.  g )
)  =  seq 1
(  +  ,  ( ( abs  o.  -  )  o.  g )
)
9 fveq2 5710 . . . . . . . . . . 11  |-  ( m  =  n  ->  (
g `  m )  =  ( g `  n ) )
109fveq2d 5714 . . . . . . . . . 10  |-  ( m  =  n  ->  ( 1st `  ( g `  m ) )  =  ( 1st `  (
g `  n )
) )
1110oveq1d 6125 . . . . . . . . 9  |-  ( m  =  n  ->  (
( 1st `  (
g `  m )
)  +  C )  =  ( ( 1st `  ( g `  n
) )  +  C
) )
129fveq2d 5714 . . . . . . . . . 10  |-  ( m  =  n  ->  ( 2nd `  ( g `  m ) )  =  ( 2nd `  (
g `  n )
) )
1312oveq1d 6125 . . . . . . . . 9  |-  ( m  =  n  ->  (
( 2nd `  (
g `  m )
)  +  C )  =  ( ( 2nd `  ( g `  n
) )  +  C
) )
1411, 13opeq12d 4086 . . . . . . . 8  |-  ( m  =  n  ->  <. (
( 1st `  (
g `  m )
)  +  C ) ,  ( ( 2nd `  ( g `  m
) )  +  C
) >.  =  <. (
( 1st `  (
g `  n )
)  +  C ) ,  ( ( 2nd `  ( g `  n
) )  +  C
) >. )
1514cbvmptv 4402 . . . . . . 7  |-  ( m  e.  NN  |->  <. (
( 1st `  (
g `  m )
)  +  C ) ,  ( ( 2nd `  ( g `  m
) )  +  C
) >. )  =  ( n  e.  NN  |->  <.
( ( 1st `  (
g `  n )
)  +  C ) ,  ( ( 2nd `  ( g `  n
) )  +  C
) >. )
16 simplr 754 . . . . . . . 8  |-  ( ( ( ( ph  /\  z  e.  RR* )  /\  g  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN ) )  /\  A  C_ 
U. ran  ( (,)  o.  g ) )  -> 
g  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN ) )
17 reex 9392 . . . . . . . . . . 11  |-  RR  e.  _V
1817, 17xpex 6527 . . . . . . . . . 10  |-  ( RR 
X.  RR )  e. 
_V
1918inex2 4453 . . . . . . . . 9  |-  (  <_  i^i  ( RR  X.  RR ) )  e.  _V
20 nnex 10347 . . . . . . . . 9  |-  NN  e.  _V
2119, 20elmap 7260 . . . . . . . 8  |-  ( g  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN ) 
<->  g : NN --> (  <_  i^i  ( RR  X.  RR ) ) )
2216, 21sylib 196 . . . . . . 7  |-  ( ( ( ( ph  /\  z  e.  RR* )  /\  g  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN ) )  /\  A  C_ 
U. ran  ( (,)  o.  g ) )  -> 
g : NN --> (  <_  i^i  ( RR  X.  RR ) ) )
23 simpr 461 . . . . . . 7  |-  ( ( ( ( ph  /\  z  e.  RR* )  /\  g  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN ) )  /\  A  C_ 
U. ran  ( (,)  o.  g ) )  ->  A  C_  U. ran  ( (,)  o.  g ) )
242, 4, 6, 7, 8, 15, 22, 23ovolshftlem1 21011 . . . . . 6  |-  ( ( ( ( ph  /\  z  e.  RR* )  /\  g  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN ) )  /\  A  C_ 
U. ran  ( (,)  o.  g ) )  ->  sup ( ran  seq 1
(  +  ,  ( ( abs  o.  -  )  o.  g )
) ,  RR* ,  <  )  e.  M )
25 eleq1a 2512 . . . . . 6  |-  ( sup ( ran  seq 1
(  +  ,  ( ( abs  o.  -  )  o.  g )
) ,  RR* ,  <  )  e.  M  ->  (
z  =  sup ( ran  seq 1 (  +  ,  ( ( abs 
o.  -  )  o.  g ) ) , 
RR* ,  <  )  -> 
z  e.  M ) )
2624, 25syl 16 . . . . 5  |-  ( ( ( ( ph  /\  z  e.  RR* )  /\  g  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN ) )  /\  A  C_ 
U. ran  ( (,)  o.  g ) )  -> 
( z  =  sup ( ran  seq 1 (  +  ,  ( ( abs  o.  -  )  o.  g ) ) , 
RR* ,  <  )  -> 
z  e.  M ) )
2726expimpd 603 . . . 4  |-  ( ( ( ph  /\  z  e.  RR* )  /\  g  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN ) )  ->  (
( A  C_  U. ran  ( (,)  o.  g )  /\  z  =  sup ( ran  seq 1 (  +  ,  ( ( abs  o.  -  )  o.  g ) ) , 
RR* ,  <  ) )  ->  z  e.  M
) )
2827rexlimdva 2860 . . 3  |-  ( (
ph  /\  z  e.  RR* )  ->  ( E. g  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN ) ( A  C_  U.
ran  ( (,)  o.  g )  /\  z  =  sup ( ran  seq 1 (  +  , 
( ( abs  o.  -  )  o.  g
) ) ,  RR* ,  <  ) )  -> 
z  e.  M ) )
2928ralrimiva 2818 . 2  |-  ( ph  ->  A. z  e.  RR*  ( E. g  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN ) ( A  C_  U. ran  ( (,)  o.  g )  /\  z  =  sup ( ran  seq 1 (  +  ,  ( ( abs 
o.  -  )  o.  g ) ) , 
RR* ,  <  ) )  ->  z  e.  M
) )
30 rabss 3448 . 2  |-  ( { z  e.  RR*  |  E. g  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN ) ( A  C_  U.
ran  ( (,)  o.  g )  /\  z  =  sup ( ran  seq 1 (  +  , 
( ( abs  o.  -  )  o.  g
) ) ,  RR* ,  <  ) ) } 
C_  M  <->  A. z  e.  RR*  ( E. g  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN ) ( A  C_  U.
ran  ( (,)  o.  g )  /\  z  =  sup ( ran  seq 1 (  +  , 
( ( abs  o.  -  )  o.  g
) ) ,  RR* ,  <  ) )  -> 
z  e.  M ) )
3129, 30sylibr 212 1  |-  ( ph  ->  { z  e.  RR*  |  E. g  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN ) ( A  C_  U. ran  ( (,)  o.  g )  /\  z  =  sup ( ran  seq 1 (  +  ,  ( ( abs 
o.  -  )  o.  g ) ) , 
RR* ,  <  ) ) }  C_  M )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2734   E.wrex 2735   {crab 2738    i^i cin 3346    C_ wss 3347   <.cop 3902   U.cuni 4110    e. cmpt 4369    X. cxp 4857   ran crn 4860    o. ccom 4863   -->wf 5433   ` cfv 5437  (class class class)co 6110   1stc1st 6594   2ndc2nd 6595    ^m cmap 7233   supcsup 7709   RRcr 9300   1c1 9302    + caddc 9304   RR*cxr 9436    < clt 9437    <_ cle 9438    - cmin 9614   NNcn 10341   (,)cioo 11319    seqcseq 11825   abscabs 12742
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-sep 4432  ax-nul 4440  ax-pow 4489  ax-pr 4550  ax-un 6391  ax-cnex 9357  ax-resscn 9358  ax-1cn 9359  ax-icn 9360  ax-addcl 9361  ax-addrcl 9362  ax-mulcl 9363  ax-mulrcl 9364  ax-mulcom 9365  ax-addass 9366  ax-mulass 9367  ax-distr 9368  ax-i2m1 9369  ax-1ne0 9370  ax-1rid 9371  ax-rnegex 9372  ax-rrecex 9373  ax-cnre 9374  ax-pre-lttri 9375  ax-pre-lttrn 9376  ax-pre-ltadd 9377  ax-pre-mulgt0 9378  ax-pre-sup 9379
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  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 2577  df-ne 2622  df-nel 2623  df-ral 2739  df-rex 2740  df-reu 2741  df-rmo 2742  df-rab 2743  df-v 2993  df-sbc 3206  df-csb 3308  df-dif 3350  df-un 3352  df-in 3354  df-ss 3361  df-pss 3363  df-nul 3657  df-if 3811  df-pw 3881  df-sn 3897  df-pr 3899  df-tp 3901  df-op 3903  df-uni 4111  df-iun 4192  df-br 4312  df-opab 4370  df-mpt 4371  df-tr 4405  df-eprel 4651  df-id 4655  df-po 4660  df-so 4661  df-fr 4698  df-we 4700  df-ord 4741  df-on 4742  df-lim 4743  df-suc 4744  df-xp 4865  df-rel 4866  df-cnv 4867  df-co 4868  df-dm 4869  df-rn 4870  df-res 4871  df-ima 4872  df-iota 5400  df-fun 5439  df-fn 5440  df-f 5441  df-f1 5442  df-fo 5443  df-f1o 5444  df-fv 5445  df-riota 6071  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-om 6496  df-1st 6596  df-2nd 6597  df-recs 6851  df-rdg 6885  df-er 7120  df-map 7235  df-en 7330  df-dom 7331  df-sdom 7332  df-sup 7710  df-pnf 9439  df-mnf 9440  df-xr 9441  df-ltxr 9442  df-le 9443  df-sub 9616  df-neg 9617  df-div 10013  df-nn 10342  df-2 10399  df-3 10400  df-n0 10599  df-z 10666  df-uz 10881  df-rp 11011  df-ioo 11323  df-ico 11325  df-fz 11457  df-seq 11826  df-exp 11885  df-cj 12607  df-re 12608  df-im 12609  df-sqr 12743  df-abs 12744
This theorem is referenced by:  ovolshft  21013
  Copyright terms: Public domain W3C validator