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Theorem erdsze2 23736
Description: Generalize the statement of the Erdős-Szekeres theorem erdsze 23733 to "sequences" indexed by an arbitrary subset of  RR, which can be infinite. (Contributed by Mario Carneiro, 22-Jan-2015.)
Hypotheses
Ref Expression
erdsze2.r  |-  ( ph  ->  R  e.  NN )
erdsze2.s  |-  ( ph  ->  S  e.  NN )
erdsze2.f  |-  ( ph  ->  F : A -1-1-> RR )
erdsze2.a  |-  ( ph  ->  A  C_  RR )
erdsze2.l  |-  ( ph  ->  ( ( R  - 
1 )  x.  ( S  -  1 ) )  <  ( # `  A ) )
Assertion
Ref Expression
erdsze2  |-  ( ph  ->  E. s  e.  ~P  A ( ( R  <_  ( # `  s
)  /\  ( F  |`  s )  Isom  <  ,  <  ( s ,  ( F " s
) ) )  \/  ( S  <_  ( # `
 s )  /\  ( F  |`  s ) 
Isom  <  ,  `'  <  ( s ,  ( F
" s ) ) ) ) )
Distinct variable groups:    A, s    F, s    R, s    S, s    ph, s

Proof of Theorem erdsze2
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 erdsze2.r . . 3  |-  ( ph  ->  R  e.  NN )
2 erdsze2.s . . 3  |-  ( ph  ->  S  e.  NN )
3 erdsze2.f . . 3  |-  ( ph  ->  F : A -1-1-> RR )
4 erdsze2.a . . 3  |-  ( ph  ->  A  C_  RR )
5 eqid 2283 . . 3  |-  ( ( R  -  1 )  x.  ( S  - 
1 ) )  =  ( ( R  - 
1 )  x.  ( S  -  1 ) )
6 erdsze2.l . . 3  |-  ( ph  ->  ( ( R  - 
1 )  x.  ( S  -  1 ) )  <  ( # `  A ) )
71, 2, 3, 4, 5, 6erdsze2lem1 23734 . 2  |-  ( ph  ->  E. f ( f : ( 1 ... ( ( ( R  -  1 )  x.  ( S  -  1 ) )  +  1 ) ) -1-1-> A  /\  f  Isom  <  ,  <  ( ( 1 ... (
( ( R  - 
1 )  x.  ( S  -  1 ) )  +  1 ) ) ,  ran  f
) ) )
81adantr 451 . . . . 5  |-  ( (
ph  /\  ( f : ( 1 ... ( ( ( R  -  1 )  x.  ( S  -  1 ) )  +  1 ) ) -1-1-> A  /\  f  Isom  <  ,  <  ( ( 1 ... (
( ( R  - 
1 )  x.  ( S  -  1 ) )  +  1 ) ) ,  ran  f
) ) )  ->  R  e.  NN )
92adantr 451 . . . . 5  |-  ( (
ph  /\  ( f : ( 1 ... ( ( ( R  -  1 )  x.  ( S  -  1 ) )  +  1 ) ) -1-1-> A  /\  f  Isom  <  ,  <  ( ( 1 ... (
( ( R  - 
1 )  x.  ( S  -  1 ) )  +  1 ) ) ,  ran  f
) ) )  ->  S  e.  NN )
103adantr 451 . . . . 5  |-  ( (
ph  /\  ( f : ( 1 ... ( ( ( R  -  1 )  x.  ( S  -  1 ) )  +  1 ) ) -1-1-> A  /\  f  Isom  <  ,  <  ( ( 1 ... (
( ( R  - 
1 )  x.  ( S  -  1 ) )  +  1 ) ) ,  ran  f
) ) )  ->  F : A -1-1-> RR )
114adantr 451 . . . . 5  |-  ( (
ph  /\  ( f : ( 1 ... ( ( ( R  -  1 )  x.  ( S  -  1 ) )  +  1 ) ) -1-1-> A  /\  f  Isom  <  ,  <  ( ( 1 ... (
( ( R  - 
1 )  x.  ( S  -  1 ) )  +  1 ) ) ,  ran  f
) ) )  ->  A  C_  RR )
126adantr 451 . . . . 5  |-  ( (
ph  /\  ( f : ( 1 ... ( ( ( R  -  1 )  x.  ( S  -  1 ) )  +  1 ) ) -1-1-> A  /\  f  Isom  <  ,  <  ( ( 1 ... (
( ( R  - 
1 )  x.  ( S  -  1 ) )  +  1 ) ) ,  ran  f
) ) )  -> 
( ( R  - 
1 )  x.  ( S  -  1 ) )  <  ( # `  A ) )
13 simprl 732 . . . . 5  |-  ( (
ph  /\  ( f : ( 1 ... ( ( ( R  -  1 )  x.  ( S  -  1 ) )  +  1 ) ) -1-1-> A  /\  f  Isom  <  ,  <  ( ( 1 ... (
( ( R  - 
1 )  x.  ( S  -  1 ) )  +  1 ) ) ,  ran  f
) ) )  -> 
f : ( 1 ... ( ( ( R  -  1 )  x.  ( S  - 
1 ) )  +  1 ) ) -1-1-> A
)
14 simprr 733 . . . . 5  |-  ( (
ph  /\  ( f : ( 1 ... ( ( ( R  -  1 )  x.  ( S  -  1 ) )  +  1 ) ) -1-1-> A  /\  f  Isom  <  ,  <  ( ( 1 ... (
( ( R  - 
1 )  x.  ( S  -  1 ) )  +  1 ) ) ,  ran  f
) ) )  -> 
f  Isom  <  ,  <  ( ( 1 ... (
( ( R  - 
1 )  x.  ( S  -  1 ) )  +  1 ) ) ,  ran  f
) )
158, 9, 10, 11, 5, 12, 13, 14erdsze2lem2 23735 . . . 4  |-  ( (
ph  /\  ( f : ( 1 ... ( ( ( R  -  1 )  x.  ( S  -  1 ) )  +  1 ) ) -1-1-> A  /\  f  Isom  <  ,  <  ( ( 1 ... (
( ( R  - 
1 )  x.  ( S  -  1 ) )  +  1 ) ) ,  ran  f
) ) )  ->  E. s  e.  ~P  A ( ( R  <_  ( # `  s
)  /\  ( F  |`  s )  Isom  <  ,  <  ( s ,  ( F " s
) ) )  \/  ( S  <_  ( # `
 s )  /\  ( F  |`  s ) 
Isom  <  ,  `'  <  ( s ,  ( F
" s ) ) ) ) )
1615ex 423 . . 3  |-  ( ph  ->  ( ( f : ( 1 ... (
( ( R  - 
1 )  x.  ( S  -  1 ) )  +  1 ) ) -1-1-> A  /\  f  Isom  <  ,  <  (
( 1 ... (
( ( R  - 
1 )  x.  ( S  -  1 ) )  +  1 ) ) ,  ran  f
) )  ->  E. s  e.  ~P  A ( ( R  <_  ( # `  s
)  /\  ( F  |`  s )  Isom  <  ,  <  ( s ,  ( F " s
) ) )  \/  ( S  <_  ( # `
 s )  /\  ( F  |`  s ) 
Isom  <  ,  `'  <  ( s ,  ( F
" s ) ) ) ) ) )
1716exlimdv 1664 . 2  |-  ( ph  ->  ( E. f ( f : ( 1 ... ( ( ( R  -  1 )  x.  ( S  - 
1 ) )  +  1 ) ) -1-1-> A  /\  f  Isom  <  ,  <  ( ( 1 ... ( ( ( R  -  1 )  x.  ( S  -  1 ) )  +  1 ) ) ,  ran  f ) )  ->  E. s  e.  ~P  A ( ( R  <_  ( # `  s
)  /\  ( F  |`  s )  Isom  <  ,  <  ( s ,  ( F " s
) ) )  \/  ( S  <_  ( # `
 s )  /\  ( F  |`  s ) 
Isom  <  ,  `'  <  ( s ,  ( F
" s ) ) ) ) ) )
187, 17mpd 14 1  |-  ( ph  ->  E. s  e.  ~P  A ( ( 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    \/ wo 357    /\ wa 358   E.wex 1528    e. wcel 1684   E.wrex 2544    C_ wss 3152   ~Pcpw 3625   class class class wbr 4023   `'ccnv 4688   ran crn 4690    |` cres 4691   "cima 4692   -1-1->wf1 5252   ` cfv 5255    Isom wiso 5256  (class class class)co 5858   RRcr 8736   1c1 8738    + caddc 8740    x. cmul 8742    < clt 8867    <_ cle 8868    - cmin 9037   NNcn 9746   ...cfz 10782   #chash 11337
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-oi 7225  df-card 7572  df-cda 7794  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-n0 9966  df-z 10025  df-uz 10231  df-fz 10783  df-hash 11338
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