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Theorem erdsze 23070
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
StepHypRef Expression
1 erdsze.n . 2  |-  ( ph  ->  N  e.  NN )
2 erdsze.f . 2  |-  ( ph  ->  F : ( 1 ... N ) -1-1-> RR )
3 reseq2 4903 . . . . . . . . . 10  |-  ( w  =  y  ->  ( F  |`  w )  =  ( F  |`  y
) )
4 isoeq1 5715 . . . . . . . . . 10  |-  ( ( F  |`  w )  =  ( F  |`  y )  ->  (
( F  |`  w
)  Isom  <  ,  <  ( w ,  ( F
" w ) )  <-> 
( F  |`  y
)  Isom  <  ,  <  ( w ,  ( F
" w ) ) ) )
53, 4syl 17 . . . . . . . . 9  |-  ( w  =  y  ->  (
( F  |`  w
)  Isom  <  ,  <  ( w ,  ( F
" w ) )  <-> 
( F  |`  y
)  Isom  <  ,  <  ( w ,  ( F
" w ) ) ) )
6 isoeq4 5718 . . . . . . . . 9  |-  ( w  =  y  ->  (
( F  |`  y
)  Isom  <  ,  <  ( w ,  ( F
" w ) )  <-> 
( F  |`  y
)  Isom  <  ,  <  ( y ,  ( F
" w ) ) ) )
7 imaeq2 4961 . . . . . . . . . 10  |-  ( w  =  y  ->  ( F " w )  =  ( F " y
) )
8 isoeq5 5719 . . . . . . . . . 10  |-  ( ( F " w )  =  ( F "
y )  ->  (
( F  |`  y
)  Isom  <  ,  <  ( y ,  ( F
" w ) )  <-> 
( F  |`  y
)  Isom  <  ,  <  ( y ,  ( F
" y ) ) ) )
97, 8syl 17 . . . . . . . . 9  |-  ( w  =  y  ->  (
( F  |`  y
)  Isom  <  ,  <  ( y ,  ( F
" w ) )  <-> 
( F  |`  y
)  Isom  <  ,  <  ( y ,  ( F
" y ) ) ) )
105, 6, 93bitrd 272 . . . . . . . 8  |-  ( w  =  y  ->  (
( F  |`  w
)  Isom  <  ,  <  ( w ,  ( F
" w ) )  <-> 
( F  |`  y
)  Isom  <  ,  <  ( y ,  ( F
" y ) ) ) )
11 elequ2 1832 . . . . . . . 8  |-  ( w  =  y  ->  (
z  e.  w  <->  z  e.  y ) )
1210, 11anbi12d 694 . . . . . . 7  |-  ( w  =  y  ->  (
( ( F  |`  w )  Isom  <  ,  <  ( w ,  ( F " w
) )  /\  z  e.  w )  <->  ( ( F  |`  y )  Isom  <  ,  <  ( y ,  ( F " y
) )  /\  z  e.  y ) ) )
1312cbvrabv 2739 . . . . . 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 5765 . . . . . . . 8  |-  ( z  =  x  ->  (
1 ... z )  =  ( 1 ... x
) )
1514pweqd 3571 . . . . . . 7  |-  ( z  =  x  ->  ~P ( 1 ... z
)  =  ~P (
1 ... x ) )
16 elequ1 1831 . . . . . . . 8  |-  ( z  =  x  ->  (
z  e.  y  <->  x  e.  y ) )
1716anbi2d 687 . . . . . . 7  |-  ( z  =  x  ->  (
( ( F  |`  y )  Isom  <  ,  <  ( y ,  ( F " y
) )  /\  z  e.  y )  <->  ( ( F  |`  y )  Isom  <  ,  <  ( y ,  ( F " y
) )  /\  x  e.  y ) ) )
1815, 17rabeqbidv 2735 . . . . . 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 2300 . . . . 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 4965 . . . 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 7132 . . 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 4051 . 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 5715 . . . . . . . . . 10  |-  ( ( F  |`  w )  =  ( F  |`  y )  ->  (
( F  |`  w
)  Isom  <  ,  `'  <  ( w ,  ( F " w ) )  <->  ( F  |`  y )  Isom  <  ,  `'  <  ( w ,  ( F " w
) ) ) )
243, 23syl 17 . . . . . . . . 9  |-  ( w  =  y  ->  (
( F  |`  w
)  Isom  <  ,  `'  <  ( w ,  ( F " w ) )  <->  ( F  |`  y )  Isom  <  ,  `'  <  ( w ,  ( F " w
) ) ) )
25 isoeq4 5718 . . . . . . . . 9  |-  ( w  =  y  ->  (
( F  |`  y
)  Isom  <  ,  `'  <  ( w ,  ( F " w ) )  <->  ( F  |`  y )  Isom  <  ,  `'  <  ( y ,  ( F " w
) ) ) )
26 isoeq5 5719 . . . . . . . . . 10  |-  ( ( F " w )  =  ( F "
y )  ->  (
( F  |`  y
)  Isom  <  ,  `'  <  ( y ,  ( F " w ) )  <->  ( F  |`  y )  Isom  <  ,  `'  <  ( y ,  ( F " y
) ) ) )
277, 26syl 17 . . . . . . . . 9  |-  ( w  =  y  ->  (
( F  |`  y
)  Isom  <  ,  `'  <  ( y ,  ( F " w ) )  <->  ( F  |`  y )  Isom  <  ,  `'  <  ( y ,  ( F " y
) ) ) )
2824, 25, 273bitrd 272 . . . . . . . 8  |-  ( w  =  y  ->  (
( F  |`  w
)  Isom  <  ,  `'  <  ( w ,  ( F " w ) )  <->  ( F  |`  y )  Isom  <  ,  `'  <  ( y ,  ( F " y
) ) ) )
2928, 11anbi12d 694 . . . . . . 7  |-  ( w  =  y  ->  (
( ( F  |`  w )  Isom  <  ,  `'  <  ( w ,  ( F " w
) )  /\  z  e.  w )  <->  ( ( F  |`  y )  Isom  <  ,  `'  <  ( y ,  ( F "
y ) )  /\  z  e.  y )
) )
3029cbvrabv 2739 . . . . . 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 687 . . . . . . 7  |-  ( z  =  x  ->  (
( ( F  |`  y )  Isom  <  ,  `'  <  ( y ,  ( F " y
) )  /\  z  e.  y )  <->  ( ( F  |`  y )  Isom  <  ,  `'  <  ( y ,  ( F "
y ) )  /\  x  e.  y )
) )
3215, 31rabeqbidv 2735 . . . . . 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 2300 . . . . 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 4965 . . . 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 7132 . . 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 4051 . 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 2256 . 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 23069 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 6    <-> wb 178    \/ wo 359    /\ wa 360    = wceq 1619    e. wcel 1621   E.wrex 2517   {crab 2519   ~Pcpw 3566   <.cop 3584   class class class wbr 3963    e. cmpt 4017   `'ccnv 4625    |` cres 4628   "cima 4629   -1-1->wf1 4635   ` cfv 4638    Isom wiso 4639  (class class class)co 5757   supcsup 7126   RRcr 8669   1c1 8671    x. cmul 8675    < clt 8800    <_ cle 8801    - cmin 8970   NNcn 9679   ...cfz 10713   #chash 11268
This theorem is referenced by:  erdsze2lem2  23072
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-rep 4071  ax-sep 4081  ax-nul 4089  ax-pow 4126  ax-pr 4152  ax-un 4449  ax-cnex 8726  ax-resscn 8727  ax-1cn 8728  ax-icn 8729  ax-addcl 8730  ax-addrcl 8731  ax-mulcl 8732  ax-mulrcl 8733  ax-mulcom 8734  ax-addass 8735  ax-mulass 8736  ax-distr 8737  ax-i2m1 8738  ax-1ne0 8739  ax-1rid 8740  ax-rnegex 8741  ax-rrecex 8742  ax-cnre 8743  ax-pre-lttri 8744  ax-pre-lttrn 8745  ax-pre-ltadd 8746  ax-pre-mulgt0 8747  ax-pre-sup 8748
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-nel 2422  df-ral 2520  df-rex 2521  df-reu 2522  df-rab 2523  df-v 2742  df-sbc 2936  df-csb 3024  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-pss 3110  df-nul 3398  df-if 3507  df-pw 3568  df-sn 3587  df-pr 3588  df-tp 3589  df-op 3590  df-uni 3769  df-int 3804  df-iun 3848  df-br 3964  df-opab 4018  df-mpt 4019  df-tr 4054  df-eprel 4242  df-id 4246  df-po 4251  df-so 4252  df-fr 4289  df-we 4291  df-ord 4332  df-on 4333  df-lim 4334  df-suc 4335  df-om 4594  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fn 4649  df-f 4650  df-f1 4651  df-fo 4652  df-f1o 4653  df-fv 4654  df-isom 4655  df-ov 5760  df-oprab 5761  df-mpt2 5762  df-1st 6021  df-2nd 6022  df-iota 6190  df-riota 6237  df-recs 6321  df-rdg 6356  df-1o 6412  df-2o 6413  df-oadd 6416  df-er 6593  df-map 6707  df-en 6797  df-dom 6798  df-sdom 6799  df-fin 6800  df-sup 7127  df-card 7505  df-cda 7727  df-pnf 8802  df-mnf 8803  df-xr 8804  df-ltxr 8805  df-le 8806  df-sub 8972  df-neg 8973  df-n 9680  df-n0 9898  df-z 9957  df-uz 10163  df-fz 10714  df-hash 11269
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