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Theorem erdsze2 24460
Description: Generalize the statement of the Erdős-Szekeres theorem erdsze 24457 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 2366 . . 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 24458 . 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 24459 . . . 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 1641 . 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 1546    e. wcel 1715   E.wrex 2629    C_ wss 3238   ~Pcpw 3714   class class class wbr 4125   `'ccnv 4791   ran crn 4793    |` cres 4794   "cima 4795   -1-1->wf1 5355   ` cfv 5358    Isom wiso 5359  (class class class)co 5981   RRcr 8883   1c1 8885    + caddc 8887    x. cmul 8889    < clt 9014    <_ cle 9015    - cmin 9184   NNcn 9893   ...cfz 10935   #chash 11505
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615  ax-cnex 8940  ax-resscn 8941  ax-1cn 8942  ax-icn 8943  ax-addcl 8944  ax-addrcl 8945  ax-mulcl 8946  ax-mulrcl 8947  ax-mulcom 8948  ax-addass 8949  ax-mulass 8950  ax-distr 8951  ax-i2m1 8952  ax-1ne0 8953  ax-1rid 8954  ax-rnegex 8955  ax-rrecex 8956  ax-cnre 8957  ax-pre-lttri 8958  ax-pre-lttrn 8959  ax-pre-ltadd 8960  ax-pre-mulgt0 8961  ax-pre-sup 8962
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 936  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-nel 2532  df-ral 2633  df-rex 2634  df-reu 2635  df-rmo 2636  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-pss 3254  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-tp 3737  df-op 3738  df-uni 3930  df-int 3965  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-tr 4216  df-eprel 4408  df-id 4412  df-po 4417  df-so 4418  df-fr 4455  df-se 4456  df-we 4457  df-ord 4498  df-on 4499  df-lim 4500  df-suc 4501  df-om 4760  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-isom 5367  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-1st 6249  df-2nd 6250  df-riota 6446  df-recs 6530  df-rdg 6565  df-1o 6621  df-2o 6622  df-oadd 6625  df-er 6802  df-map 6917  df-en 7007  df-dom 7008  df-sdom 7009  df-fin 7010  df-sup 7341  df-oi 7372  df-card 7719  df-cda 7941  df-pnf 9016  df-mnf 9017  df-xr 9018  df-ltxr 9019  df-le 9020  df-sub 9186  df-neg 9187  df-nn 9894  df-n0 10115  df-z 10176  df-uz 10382  df-fz 10936  df-hash 11506
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