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Theorem erdsze 24876
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 5132 . . . . . . . . . 10  |-  ( w  =  y  ->  ( F  |`  w )  =  ( F  |`  y
) )
4 isoeq1 6030 . . . . . . . . . 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 6033 . . . . . . . . 9  |-  ( w  =  y  ->  (
( F  |`  y
)  Isom  <  ,  <  ( w ,  ( F
" w ) )  <-> 
( F  |`  y
)  Isom  <  ,  <  ( y ,  ( F
" w ) ) ) )
7 imaeq2 5190 . . . . . . . . . 10  |-  ( w  =  y  ->  ( F " w )  =  ( F " y
) )
8 isoeq5 6034 . . . . . . . . . 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 1730 . . . . . . . 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 2947 . . . . . 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 6080 . . . . . . . 8  |-  ( z  =  x  ->  (
1 ... z )  =  ( 1 ... x
) )
1514pweqd 3796 . . . . . . 7  |-  ( z  =  x  ->  ~P ( 1 ... z
)  =  ~P (
1 ... x ) )
16 elequ1 1728 . . . . . . . 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 2943 . . . . . 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 2479 . . . . 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 5194 . . . 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 7442 . . 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 4292 . 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 6030 . . . . . . . . . 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 6033 . . . . . . . . 9  |-  ( w  =  y  ->  (
( F  |`  y
)  Isom  <  ,  `'  <  ( w ,  ( F " w ) )  <->  ( F  |`  y )  Isom  <  ,  `'  <  ( y ,  ( F " w
) ) ) )
26 isoeq5 6034 . . . . . . . . . 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 2947 . . . . . 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 2943 . . . . . 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 2479 . . . . 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 5194 . . . 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 7442 . . 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 4292 . 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 2435 . 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 24875 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 1652    e. wcel 1725   E.wrex 2698   {crab 2701   ~Pcpw 3791   <.cop 3809   class class class wbr 4204    e. cmpt 4258   `'ccnv 4868    |` cres 4871   "cima 4872   -1-1->wf1 5442   ` cfv 5445    Isom wiso 5446  (class class class)co 6072   supcsup 7436   RRcr 8978   1c1 8980    x. cmul 8984    < clt 9109    <_ cle 9110    - cmin 9280   NNcn 9989   ...cfz 11032   #chash 11606
This theorem is referenced by:  erdsze2lem2  24878
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4692  ax-cnex 9035  ax-resscn 9036  ax-1cn 9037  ax-icn 9038  ax-addcl 9039  ax-addrcl 9040  ax-mulcl 9041  ax-mulrcl 9042  ax-mulcom 9043  ax-addass 9044  ax-mulass 9045  ax-distr 9046  ax-i2m1 9047  ax-1ne0 9048  ax-1rid 9049  ax-rnegex 9050  ax-rrecex 9051  ax-cnre 9052  ax-pre-lttri 9053  ax-pre-lttrn 9054  ax-pre-ltadd 9055  ax-pre-mulgt0 9056  ax-pre-sup 9057
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4837  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-isom 5454  df-ov 6075  df-oprab 6076  df-mpt2 6077  df-1st 6340  df-2nd 6341  df-riota 6540  df-recs 6624  df-rdg 6659  df-1o 6715  df-2o 6716  df-oadd 6719  df-er 6896  df-map 7011  df-en 7101  df-dom 7102  df-sdom 7103  df-fin 7104  df-sup 7437  df-card 7815  df-cda 8037  df-pnf 9111  df-mnf 9112  df-xr 9113  df-ltxr 9114  df-le 9115  df-sub 9282  df-neg 9283  df-nn 9990  df-n0 10211  df-z 10272  df-uz 10478  df-fz 11033  df-hash 11607
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