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Theorem erdsze 24841
Description: The Erdős-Szekeres theorem. For any injective sequence  F on the reals of length at least 
( R  -  1 )  x.  ( S  -  1 )  +  1, there is either a subsequence of length at least  R on which  F is increasing (i.e. a  <  ,  < order isomorphism) or a subsequence of length at least  S on which  F is decreasing (i.e. a  <  ,  `'  < order isomorphism, recalling that  `'  < is the greater-than relation). (Contributed by Mario Carneiro, 22-Jan-2015.)
Hypotheses
Ref Expression
erdsze.n  |-  ( ph  ->  N  e.  NN )
erdsze.f  |-  ( ph  ->  F : ( 1 ... N ) -1-1-> RR )
erdsze.r  |-  ( ph  ->  R  e.  NN )
erdsze.s  |-  ( ph  ->  S  e.  NN )
erdsze.l  |-  ( ph  ->  ( ( R  - 
1 )  x.  ( S  -  1 ) )  <  N )
Assertion
Ref Expression
erdsze  |-  ( ph  ->  E. s  e.  ~P  ( 1 ... N
) ( ( R  <_  ( # `  s
)  /\  ( F  |`  s )  Isom  <  ,  <  ( s ,  ( F " s
) ) )  \/  ( S  <_  ( # `
 s )  /\  ( F  |`  s ) 
Isom  <  ,  `'  <  ( s ,  ( F
" s ) ) ) ) )
Distinct variable groups:    F, s    R, s    N, s    ph, s    S, s

Proof of Theorem erdsze
Dummy variables  w  x  y  z  n are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 erdsze.n . 2  |-  ( ph  ->  N  e.  NN )
2 erdsze.f . 2  |-  ( ph  ->  F : ( 1 ... N ) -1-1-> RR )
3 reseq2 5100 . . . . . . . . . 10  |-  ( w  =  y  ->  ( F  |`  w )  =  ( F  |`  y
) )
4 isoeq1 5998 . . . . . . . . . 10  |-  ( ( F  |`  w )  =  ( F  |`  y )  ->  (
( F  |`  w
)  Isom  <  ,  <  ( w ,  ( F
" w ) )  <-> 
( F  |`  y
)  Isom  <  ,  <  ( w ,  ( F
" w ) ) ) )
53, 4syl 16 . . . . . . . . 9  |-  ( w  =  y  ->  (
( F  |`  w
)  Isom  <  ,  <  ( w ,  ( F
" w ) )  <-> 
( F  |`  y
)  Isom  <  ,  <  ( w ,  ( F
" w ) ) ) )
6 isoeq4 6001 . . . . . . . . 9  |-  ( w  =  y  ->  (
( F  |`  y
)  Isom  <  ,  <  ( w ,  ( F
" w ) )  <-> 
( F  |`  y
)  Isom  <  ,  <  ( y ,  ( F
" w ) ) ) )
7 imaeq2 5158 . . . . . . . . . 10  |-  ( w  =  y  ->  ( F " w )  =  ( F " y
) )
8 isoeq5 6002 . . . . . . . . . 10  |-  ( ( F " w )  =  ( F "
y )  ->  (
( F  |`  y
)  Isom  <  ,  <  ( y ,  ( F
" w ) )  <-> 
( F  |`  y
)  Isom  <  ,  <  ( y ,  ( F
" y ) ) ) )
97, 8syl 16 . . . . . . . . 9  |-  ( w  =  y  ->  (
( F  |`  y
)  Isom  <  ,  <  ( y ,  ( F
" w ) )  <-> 
( F  |`  y
)  Isom  <  ,  <  ( y ,  ( F
" y ) ) ) )
105, 6, 93bitrd 271 . . . . . . . 8  |-  ( w  =  y  ->  (
( F  |`  w
)  Isom  <  ,  <  ( w ,  ( F
" w ) )  <-> 
( F  |`  y
)  Isom  <  ,  <  ( y ,  ( F
" y ) ) ) )
11 elequ2 1726 . . . . . . . 8  |-  ( w  =  y  ->  (
z  e.  w  <->  z  e.  y ) )
1210, 11anbi12d 692 . . . . . . 7  |-  ( w  =  y  ->  (
( ( F  |`  w )  Isom  <  ,  <  ( w ,  ( F " w
) )  /\  z  e.  w )  <->  ( ( F  |`  y )  Isom  <  ,  <  ( y ,  ( F " y
) )  /\  z  e.  y ) ) )
1312cbvrabv 2915 . . . . . 6  |-  { w  e.  ~P ( 1 ... z )  |  ( ( F  |`  w
)  Isom  <  ,  <  ( w ,  ( F
" w ) )  /\  z  e.  w
) }  =  {
y  e.  ~P (
1 ... z )  |  ( ( F  |`  y )  Isom  <  ,  <  ( y ,  ( F " y
) )  /\  z  e.  y ) }
14 oveq2 6048 . . . . . . . 8  |-  ( z  =  x  ->  (
1 ... z )  =  ( 1 ... x
) )
1514pweqd 3764 . . . . . . 7  |-  ( z  =  x  ->  ~P ( 1 ... z
)  =  ~P (
1 ... x ) )
16 elequ1 1724 . . . . . . . 8  |-  ( z  =  x  ->  (
z  e.  y  <->  x  e.  y ) )
1716anbi2d 685 . . . . . . 7  |-  ( z  =  x  ->  (
( ( F  |`  y )  Isom  <  ,  <  ( y ,  ( F " y
) )  /\  z  e.  y )  <->  ( ( F  |`  y )  Isom  <  ,  <  ( y ,  ( F " y
) )  /\  x  e.  y ) ) )
1815, 17rabeqbidv 2911 . . . . . 6  |-  ( z  =  x  ->  { y  e.  ~P ( 1 ... z )  |  ( ( F  |`  y )  Isom  <  ,  <  ( y ,  ( F " y
) )  /\  z  e.  y ) }  =  { y  e.  ~P ( 1 ... x
)  |  ( ( F  |`  y )  Isom  <  ,  <  (
y ,  ( F
" y ) )  /\  x  e.  y ) } )
1913, 18syl5eq 2448 . . . . 5  |-  ( z  =  x  ->  { w  e.  ~P ( 1 ... z )  |  ( ( F  |`  w
)  Isom  <  ,  <  ( w ,  ( F
" w ) )  /\  z  e.  w
) }  =  {
y  e.  ~P (
1 ... x )  |  ( ( F  |`  y )  Isom  <  ,  <  ( y ,  ( F " y
) )  /\  x  e.  y ) } )
2019imaeq2d 5162 . . . 4  |-  ( z  =  x  ->  ( #
" { w  e. 
~P ( 1 ... z )  |  ( ( F  |`  w
)  Isom  <  ,  <  ( w ,  ( F
" w ) )  /\  z  e.  w
) } )  =  ( # " {
y  e.  ~P (
1 ... x )  |  ( ( F  |`  y )  Isom  <  ,  <  ( y ,  ( F " y
) )  /\  x  e.  y ) } ) )
2120supeq1d 7409 . . 3  |-  ( z  =  x  ->  sup ( ( # " {
w  e.  ~P (
1 ... z )  |  ( ( F  |`  w )  Isom  <  ,  <  ( w ,  ( F " w
) )  /\  z  e.  w ) } ) ,  RR ,  <  )  =  sup ( (
# " { y  e.  ~P ( 1 ... x )  |  ( ( F  |`  y )  Isom  <  ,  <  ( y ,  ( F " y
) )  /\  x  e.  y ) } ) ,  RR ,  <  ) )
2221cbvmptv 4260 . 2  |-  ( z  e.  ( 1 ... N )  |->  sup (
( # " { w  e.  ~P ( 1 ... z )  |  ( ( F  |`  w
)  Isom  <  ,  <  ( w ,  ( F
" w ) )  /\  z  e.  w
) } ) ,  RR ,  <  )
)  =  ( x  e.  ( 1 ... N )  |->  sup (
( # " { y  e.  ~P ( 1 ... x )  |  ( ( F  |`  y )  Isom  <  ,  <  ( y ,  ( F " y
) )  /\  x  e.  y ) } ) ,  RR ,  <  ) )
23 isoeq1 5998 . . . . . . . . . 10  |-  ( ( F  |`  w )  =  ( F  |`  y )  ->  (
( F  |`  w
)  Isom  <  ,  `'  <  ( w ,  ( F " w ) )  <->  ( F  |`  y )  Isom  <  ,  `'  <  ( w ,  ( F " w
) ) ) )
243, 23syl 16 . . . . . . . . 9  |-  ( w  =  y  ->  (
( F  |`  w
)  Isom  <  ,  `'  <  ( w ,  ( F " w ) )  <->  ( F  |`  y )  Isom  <  ,  `'  <  ( w ,  ( F " w
) ) ) )
25 isoeq4 6001 . . . . . . . . 9  |-  ( w  =  y  ->  (
( F  |`  y
)  Isom  <  ,  `'  <  ( w ,  ( F " w ) )  <->  ( F  |`  y )  Isom  <  ,  `'  <  ( y ,  ( F " w
) ) ) )
26 isoeq5 6002 . . . . . . . . . 10  |-  ( ( F " w )  =  ( F "
y )  ->  (
( F  |`  y
)  Isom  <  ,  `'  <  ( y ,  ( F " w ) )  <->  ( F  |`  y )  Isom  <  ,  `'  <  ( y ,  ( F " y
) ) ) )
277, 26syl 16 . . . . . . . . 9  |-  ( w  =  y  ->  (
( F  |`  y
)  Isom  <  ,  `'  <  ( y ,  ( F " w ) )  <->  ( F  |`  y )  Isom  <  ,  `'  <  ( y ,  ( F " y
) ) ) )
2824, 25, 273bitrd 271 . . . . . . . 8  |-  ( w  =  y  ->  (
( F  |`  w
)  Isom  <  ,  `'  <  ( w ,  ( F " w ) )  <->  ( F  |`  y )  Isom  <  ,  `'  <  ( y ,  ( F " y
) ) ) )
2928, 11anbi12d 692 . . . . . . 7  |-  ( w  =  y  ->  (
( ( F  |`  w )  Isom  <  ,  `'  <  ( w ,  ( F " w
) )  /\  z  e.  w )  <->  ( ( F  |`  y )  Isom  <  ,  `'  <  ( y ,  ( F "
y ) )  /\  z  e.  y )
) )
3029cbvrabv 2915 . . . . . 6  |-  { w  e.  ~P ( 1 ... z )  |  ( ( F  |`  w
)  Isom  <  ,  `'  <  ( w ,  ( F " w ) )  /\  z  e.  w ) }  =  { y  e.  ~P ( 1 ... z
)  |  ( ( F  |`  y )  Isom  <  ,  `'  <  ( y ,  ( F
" y ) )  /\  z  e.  y ) }
3116anbi2d 685 . . . . . . 7  |-  ( z  =  x  ->  (
( ( F  |`  y )  Isom  <  ,  `'  <  ( y ,  ( F " y
) )  /\  z  e.  y )  <->  ( ( F  |`  y )  Isom  <  ,  `'  <  ( y ,  ( F "
y ) )  /\  x  e.  y )
) )
3215, 31rabeqbidv 2911 . . . . . 6  |-  ( z  =  x  ->  { y  e.  ~P ( 1 ... z )  |  ( ( F  |`  y )  Isom  <  ,  `'  <  ( y ,  ( F " y
) )  /\  z  e.  y ) }  =  { y  e.  ~P ( 1 ... x
)  |  ( ( F  |`  y )  Isom  <  ,  `'  <  ( y ,  ( F
" y ) )  /\  x  e.  y ) } )
3330, 32syl5eq 2448 . . . . 5  |-  ( z  =  x  ->  { w  e.  ~P ( 1 ... z )  |  ( ( F  |`  w
)  Isom  <  ,  `'  <  ( w ,  ( F " w ) )  /\  z  e.  w ) }  =  { y  e.  ~P ( 1 ... x
)  |  ( ( F  |`  y )  Isom  <  ,  `'  <  ( y ,  ( F
" y ) )  /\  x  e.  y ) } )
3433imaeq2d 5162 . . . 4  |-  ( z  =  x  ->  ( #
" { w  e. 
~P ( 1 ... z )  |  ( ( F  |`  w
)  Isom  <  ,  `'  <  ( w ,  ( F " w ) )  /\  z  e.  w ) } )  =  ( # " {
y  e.  ~P (
1 ... x )  |  ( ( F  |`  y )  Isom  <  ,  `'  <  ( y ,  ( F " y
) )  /\  x  e.  y ) } ) )
3534supeq1d 7409 . . 3  |-  ( z  =  x  ->  sup ( ( # " {
w  e.  ~P (
1 ... z )  |  ( ( F  |`  w )  Isom  <  ,  `'  <  ( w ,  ( F " w
) )  /\  z  e.  w ) } ) ,  RR ,  <  )  =  sup ( (
# " { y  e.  ~P ( 1 ... x )  |  ( ( F  |`  y )  Isom  <  ,  `'  <  ( y ,  ( F " y
) )  /\  x  e.  y ) } ) ,  RR ,  <  ) )
3635cbvmptv 4260 . 2  |-  ( z  e.  ( 1 ... N )  |->  sup (
( # " { w  e.  ~P ( 1 ... z )  |  ( ( F  |`  w
)  Isom  <  ,  `'  <  ( w ,  ( F " w ) )  /\  z  e.  w ) } ) ,  RR ,  <  ) )  =  ( x  e.  ( 1 ... N )  |->  sup (
( # " { y  e.  ~P ( 1 ... x )  |  ( ( F  |`  y )  Isom  <  ,  `'  <  ( y ,  ( F " y
) )  /\  x  e.  y ) } ) ,  RR ,  <  ) )
37 eqid 2404 . 2  |-  ( n  e.  ( 1 ... N )  |->  <. (
( z  e.  ( 1 ... N ) 
|->  sup ( ( # " { w  e.  ~P ( 1 ... z
)  |  ( ( F  |`  w )  Isom  <  ,  <  (
w ,  ( F
" w ) )  /\  z  e.  w
) } ) ,  RR ,  <  )
) `  n ) ,  ( ( z  e.  ( 1 ... N )  |->  sup (
( # " { w  e.  ~P ( 1 ... z )  |  ( ( F  |`  w
)  Isom  <  ,  `'  <  ( w ,  ( F " w ) )  /\  z  e.  w ) } ) ,  RR ,  <  ) ) `  n )
>. )  =  (
n  e.  ( 1 ... N )  |->  <.
( ( z  e.  ( 1 ... N
)  |->  sup ( ( # " { w  e.  ~P ( 1 ... z
)  |  ( ( F  |`  w )  Isom  <  ,  <  (
w ,  ( F
" w ) )  /\  z  e.  w
) } ) ,  RR ,  <  )
) `  n ) ,  ( ( z  e.  ( 1 ... N )  |->  sup (
( # " { w  e.  ~P ( 1 ... z )  |  ( ( F  |`  w
)  Isom  <  ,  `'  <  ( w ,  ( F " w ) )  /\  z  e.  w ) } ) ,  RR ,  <  ) ) `  n )
>. )
38 erdsze.r . 2  |-  ( ph  ->  R  e.  NN )
39 erdsze.s . 2  |-  ( ph  ->  S  e.  NN )
40 erdsze.l . 2  |-  ( ph  ->  ( ( R  - 
1 )  x.  ( S  -  1 ) )  <  N )
411, 2, 22, 36, 37, 38, 39, 40erdszelem11 24840 1  |-  ( ph  ->  E. s  e.  ~P  ( 1 ... N
) ( ( R  <_  ( # `  s
)  /\  ( F  |`  s )  Isom  <  ,  <  ( s ,  ( F " s
) ) )  \/  ( S  <_  ( # `
 s )  /\  ( F  |`  s ) 
Isom  <  ,  `'  <  ( s ,  ( F
" s ) ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    = wceq 1649    e. wcel 1721   E.wrex 2667   {crab 2670   ~Pcpw 3759   <.cop 3777   class class class wbr 4172    e. cmpt 4226   `'ccnv 4836    |` cres 4839   "cima 4840   -1-1->wf1 5410   ` cfv 5413    Isom wiso 5414  (class class class)co 6040   supcsup 7403   RRcr 8945   1c1 8947    x. cmul 8951    < clt 9076    <_ cle 9077    - cmin 9247   NNcn 9956   ...cfz 10999   #chash 11573
This theorem is referenced by:  erdsze2lem2  24843
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-er 6864  df-map 6979  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-sup 7404  df-card 7782  df-cda 8004  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-n0 10178  df-z 10239  df-uz 10445  df-fz 11000  df-hash 11574
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