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Theorem erdsze 27227
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). This is part of Metamath 100 proof #73. (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 5206 . . . . . . . . . 10  |-  ( w  =  y  ->  ( F  |`  w )  =  ( F  |`  y
) )
4 isoeq1 6112 . . . . . . . . . 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 6115 . . . . . . . . 9  |-  ( w  =  y  ->  (
( F  |`  y
)  Isom  <  ,  <  ( w ,  ( F
" w ) )  <-> 
( F  |`  y
)  Isom  <  ,  <  ( y ,  ( F
" w ) ) ) )
7 imaeq2 5266 . . . . . . . . . 10  |-  ( w  =  y  ->  ( F " w )  =  ( F " y
) )
8 isoeq5 6116 . . . . . . . . . 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 279 . . . . . . . 8  |-  ( w  =  y  ->  (
( F  |`  w
)  Isom  <  ,  <  ( w ,  ( F
" w ) )  <-> 
( F  |`  y
)  Isom  <  ,  <  ( y ,  ( F
" y ) ) ) )
11 elequ2 1763 . . . . . . . 8  |-  ( w  =  y  ->  (
z  e.  w  <->  z  e.  y ) )
1210, 11anbi12d 710 . . . . . . 7  |-  ( w  =  y  ->  (
( ( F  |`  w )  Isom  <  ,  <  ( w ,  ( F " w
) )  /\  z  e.  w )  <->  ( ( F  |`  y )  Isom  <  ,  <  ( y ,  ( F " y
) )  /\  z  e.  y ) ) )
1312cbvrabv 3070 . . . . . 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 6201 . . . . . . . 8  |-  ( z  =  x  ->  (
1 ... z )  =  ( 1 ... x
) )
1514pweqd 3966 . . . . . . 7  |-  ( z  =  x  ->  ~P ( 1 ... z
)  =  ~P (
1 ... x ) )
16 elequ1 1761 . . . . . . . 8  |-  ( z  =  x  ->  (
z  e.  y  <->  x  e.  y ) )
1716anbi2d 703 . . . . . . 7  |-  ( z  =  x  ->  (
( ( F  |`  y )  Isom  <  ,  <  ( y ,  ( F " y
) )  /\  z  e.  y )  <->  ( ( F  |`  y )  Isom  <  ,  <  ( y ,  ( F " y
) )  /\  x  e.  y ) ) )
1815, 17rabeqbidv 3066 . . . . . 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 2504 . . . . 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 5270 . . . 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 7800 . . 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 4484 . 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 6112 . . . . . . . . . 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 6115 . . . . . . . . 9  |-  ( w  =  y  ->  (
( F  |`  y
)  Isom  <  ,  `'  <  ( w ,  ( F " w ) )  <->  ( F  |`  y )  Isom  <  ,  `'  <  ( y ,  ( F " w
) ) ) )
26 isoeq5 6116 . . . . . . . . . 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 279 . . . . . . . 8  |-  ( w  =  y  ->  (
( F  |`  w
)  Isom  <  ,  `'  <  ( w ,  ( F " w ) )  <->  ( F  |`  y )  Isom  <  ,  `'  <  ( y ,  ( F " y
) ) ) )
2928, 11anbi12d 710 . . . . . . 7  |-  ( w  =  y  ->  (
( ( F  |`  w )  Isom  <  ,  `'  <  ( w ,  ( F " w
) )  /\  z  e.  w )  <->  ( ( F  |`  y )  Isom  <  ,  `'  <  ( y ,  ( F "
y ) )  /\  z  e.  y )
) )
3029cbvrabv 3070 . . . . . 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 703 . . . . . . 7  |-  ( z  =  x  ->  (
( ( F  |`  y )  Isom  <  ,  `'  <  ( y ,  ( F " y
) )  /\  z  e.  y )  <->  ( ( F  |`  y )  Isom  <  ,  `'  <  ( y ,  ( F "
y ) )  /\  x  e.  y )
) )
3215, 31rabeqbidv 3066 . . . . . 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 2504 . . . . 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 5270 . . . 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 7800 . . 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 4484 . 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 2451 . 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 27226 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1370    e. wcel 1758   E.wrex 2796   {crab 2799   ~Pcpw 3961   <.cop 3984   class class class wbr 4393    |-> cmpt 4451   `'ccnv 4940    |` cres 4943   "cima 4944   -1-1->wf1 5516   ` cfv 5519    Isom wiso 5520  (class class class)co 6193   supcsup 7794   RRcr 9385   1c1 9387    x. cmul 9391    < clt 9522    <_ cle 9523    - cmin 9699   NNcn 10426   ...cfz 11547   #chash 12213
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475  ax-cnex 9442  ax-resscn 9443  ax-1cn 9444  ax-icn 9445  ax-addcl 9446  ax-addrcl 9447  ax-mulcl 9448  ax-mulrcl 9449  ax-mulcom 9450  ax-addass 9451  ax-mulass 9452  ax-distr 9453  ax-i2m1 9454  ax-1ne0 9455  ax-1rid 9456  ax-rnegex 9457  ax-rrecex 9458  ax-cnre 9459  ax-pre-lttri 9460  ax-pre-lttrn 9461  ax-pre-ltadd 9462  ax-pre-mulgt0 9463  ax-pre-sup 9464
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-pss 3445  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-tp 3983  df-op 3985  df-uni 4193  df-int 4230  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-tr 4487  df-eprel 4733  df-id 4737  df-po 4742  df-so 4743  df-fr 4780  df-we 4782  df-ord 4823  df-on 4824  df-lim 4825  df-suc 4826  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-isom 5528  df-riota 6154  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-om 6580  df-1st 6680  df-2nd 6681  df-recs 6935  df-rdg 6969  df-1o 7023  df-2o 7024  df-oadd 7027  df-er 7204  df-map 7319  df-en 7414  df-dom 7415  df-sdom 7416  df-fin 7417  df-sup 7795  df-card 8213  df-cda 8441  df-pnf 9524  df-mnf 9525  df-xr 9526  df-ltxr 9527  df-le 9528  df-sub 9701  df-neg 9702  df-nn 10427  df-n0 10684  df-z 10751  df-uz 10966  df-fz 11548  df-hash 12214
This theorem is referenced by:  erdsze2lem2  27229
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