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Theorem erdsze 23735
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 4952 . . . . . . . . . 10  |-  ( w  =  y  ->  ( F  |`  w )  =  ( F  |`  y
) )
4 isoeq1 5818 . . . . . . . . . 10  |-  ( ( F  |`  w )  =  ( F  |`  y )  ->  (
( F  |`  w
)  Isom  <  ,  <  ( w ,  ( F
" w ) )  <-> 
( F  |`  y
)  Isom  <  ,  <  ( w ,  ( F
" w ) ) ) )
53, 4syl 15 . . . . . . . . 9  |-  ( w  =  y  ->  (
( F  |`  w
)  Isom  <  ,  <  ( w ,  ( F
" w ) )  <-> 
( F  |`  y
)  Isom  <  ,  <  ( w ,  ( F
" w ) ) ) )
6 isoeq4 5821 . . . . . . . . 9  |-  ( w  =  y  ->  (
( F  |`  y
)  Isom  <  ,  <  ( w ,  ( F
" w ) )  <-> 
( F  |`  y
)  Isom  <  ,  <  ( y ,  ( F
" w ) ) ) )
7 imaeq2 5010 . . . . . . . . . 10  |-  ( w  =  y  ->  ( F " w )  =  ( F " y
) )
8 isoeq5 5822 . . . . . . . . . 10  |-  ( ( F " w )  =  ( F "
y )  ->  (
( F  |`  y
)  Isom  <  ,  <  ( y ,  ( F
" w ) )  <-> 
( F  |`  y
)  Isom  <  ,  <  ( y ,  ( F
" y ) ) ) )
97, 8syl 15 . . . . . . . . 9  |-  ( w  =  y  ->  (
( F  |`  y
)  Isom  <  ,  <  ( y ,  ( F
" w ) )  <-> 
( F  |`  y
)  Isom  <  ,  <  ( y ,  ( F
" y ) ) ) )
105, 6, 93bitrd 270 . . . . . . . 8  |-  ( w  =  y  ->  (
( F  |`  w
)  Isom  <  ,  <  ( w ,  ( F
" w ) )  <-> 
( F  |`  y
)  Isom  <  ,  <  ( y ,  ( F
" y ) ) ) )
11 elequ2 1691 . . . . . . . 8  |-  ( w  =  y  ->  (
z  e.  w  <->  z  e.  y ) )
1210, 11anbi12d 691 . . . . . . 7  |-  ( w  =  y  ->  (
( ( F  |`  w )  Isom  <  ,  <  ( w ,  ( F " w
) )  /\  z  e.  w )  <->  ( ( F  |`  y )  Isom  <  ,  <  ( y ,  ( F " y
) )  /\  z  e.  y ) ) )
1312cbvrabv 2789 . . . . . 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 5868 . . . . . . . 8  |-  ( z  =  x  ->  (
1 ... z )  =  ( 1 ... x
) )
1514pweqd 3632 . . . . . . 7  |-  ( z  =  x  ->  ~P ( 1 ... z
)  =  ~P (
1 ... x ) )
16 elequ1 1689 . . . . . . . 8  |-  ( z  =  x  ->  (
z  e.  y  <->  x  e.  y ) )
1716anbi2d 684 . . . . . . 7  |-  ( z  =  x  ->  (
( ( F  |`  y )  Isom  <  ,  <  ( y ,  ( F " y
) )  /\  z  e.  y )  <->  ( ( F  |`  y )  Isom  <  ,  <  ( y ,  ( F " y
) )  /\  x  e.  y ) ) )
1815, 17rabeqbidv 2785 . . . . . 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 2329 . . . . 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 5014 . . . 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 7201 . . 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 4113 . 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 5818 . . . . . . . . . 10  |-  ( ( F  |`  w )  =  ( F  |`  y )  ->  (
( F  |`  w
)  Isom  <  ,  `'  <  ( w ,  ( F " w ) )  <->  ( F  |`  y )  Isom  <  ,  `'  <  ( w ,  ( F " w
) ) ) )
243, 23syl 15 . . . . . . . . 9  |-  ( w  =  y  ->  (
( F  |`  w
)  Isom  <  ,  `'  <  ( w ,  ( F " w ) )  <->  ( F  |`  y )  Isom  <  ,  `'  <  ( w ,  ( F " w
) ) ) )
25 isoeq4 5821 . . . . . . . . 9  |-  ( w  =  y  ->  (
( F  |`  y
)  Isom  <  ,  `'  <  ( w ,  ( F " w ) )  <->  ( F  |`  y )  Isom  <  ,  `'  <  ( y ,  ( F " w
) ) ) )
26 isoeq5 5822 . . . . . . . . . 10  |-  ( ( F " w )  =  ( F "
y )  ->  (
( F  |`  y
)  Isom  <  ,  `'  <  ( y ,  ( F " w ) )  <->  ( F  |`  y )  Isom  <  ,  `'  <  ( y ,  ( F " y
) ) ) )
277, 26syl 15 . . . . . . . . 9  |-  ( w  =  y  ->  (
( F  |`  y
)  Isom  <  ,  `'  <  ( y ,  ( F " w ) )  <->  ( F  |`  y )  Isom  <  ,  `'  <  ( y ,  ( F " y
) ) ) )
2824, 25, 273bitrd 270 . . . . . . . 8  |-  ( w  =  y  ->  (
( F  |`  w
)  Isom  <  ,  `'  <  ( w ,  ( F " w ) )  <->  ( F  |`  y )  Isom  <  ,  `'  <  ( y ,  ( F " y
) ) ) )
2928, 11anbi12d 691 . . . . . . 7  |-  ( w  =  y  ->  (
( ( F  |`  w )  Isom  <  ,  `'  <  ( w ,  ( F " w
) )  /\  z  e.  w )  <->  ( ( F  |`  y )  Isom  <  ,  `'  <  ( y ,  ( F "
y ) )  /\  z  e.  y )
) )
3029cbvrabv 2789 . . . . . 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 684 . . . . . . 7  |-  ( z  =  x  ->  (
( ( F  |`  y )  Isom  <  ,  `'  <  ( y ,  ( F " y
) )  /\  z  e.  y )  <->  ( ( F  |`  y )  Isom  <  ,  `'  <  ( y ,  ( F "
y ) )  /\  x  e.  y )
) )
3215, 31rabeqbidv 2785 . . . . . 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 2329 . . . . 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 5014 . . . 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 7201 . . 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 4113 . 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 2285 . 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 23734 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 176    \/ wo 357    /\ wa 358    = wceq 1625    e. wcel 1686   E.wrex 2546   {crab 2549   ~Pcpw 3627   <.cop 3645   class class class wbr 4025    e. cmpt 4079   `'ccnv 4690    |` cres 4693   "cima 4694   -1-1->wf1 5254   ` cfv 5257    Isom wiso 5258  (class class class)co 5860   supcsup 7195   RRcr 8738   1c1 8740    x. cmul 8744    < clt 8869    <_ cle 8870    - cmin 9039   NNcn 9748   ...cfz 10784   #chash 11339
This theorem is referenced by:  erdsze2lem2  23737
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514  ax-cnex 8795  ax-resscn 8796  ax-1cn 8797  ax-icn 8798  ax-addcl 8799  ax-addrcl 8800  ax-mulcl 8801  ax-mulrcl 8802  ax-mulcom 8803  ax-addass 8804  ax-mulass 8805  ax-distr 8806  ax-i2m1 8807  ax-1ne0 8808  ax-1rid 8809  ax-rnegex 8810  ax-rrecex 8811  ax-cnre 8812  ax-pre-lttri 8813  ax-pre-lttrn 8814  ax-pre-ltadd 8815  ax-pre-mulgt0 8816  ax-pre-sup 8817
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rmo 2553  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-int 3865  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  df-eprel 4307  df-id 4311  df-po 4316  df-so 4317  df-fr 4354  df-we 4356  df-ord 4397  df-on 4398  df-lim 4399  df-suc 4400  df-om 4659  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-isom 5266  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-1st 6124  df-2nd 6125  df-riota 6306  df-recs 6390  df-rdg 6425  df-1o 6481  df-2o 6482  df-oadd 6485  df-er 6662  df-map 6776  df-en 6866  df-dom 6867  df-sdom 6868  df-fin 6869  df-sup 7196  df-card 7574  df-cda 7796  df-pnf 8871  df-mnf 8872  df-xr 8873  df-ltxr 8874  df-le 8875  df-sub 9041  df-neg 9042  df-nn 9749  df-n0 9968  df-z 10027  df-uz 10233  df-fz 10785  df-hash 11340
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