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Theorem erdsze2 23073
Description: Generalize the statement of the Erdős-Szekeres theorem erdsze 23070 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
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 2256 . . 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 23071 . 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 453 . . . . 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 453 . . . . 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 453 . . . . 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 453 . . . . 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 453 . . . . 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 735 . . . . 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 736 . . . . 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 23072 . . . 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 425 . . 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 1933 . 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 16 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 6    \/ wo 359    /\ wa 360   E.wex 1537    e. wcel 1621   E.wrex 2517    C_ wss 3094   ~Pcpw 3566   class class class wbr 3963   `'ccnv 4625   ran crn 4627    |` cres 4628   "cima 4629   -1-1->wf1 4635   ` cfv 4638    Isom wiso 4639  (class class class)co 5757   RRcr 8669   1c1 8671    + caddc 8673    x. cmul 8675    < clt 8800    <_ cle 8801    - cmin 8970   NNcn 9679   ...cfz 10713   #chash 11268
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-se 4290  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-oi 7158  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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