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Theorem erdsze2 23738
Description: Generalize the statement of the Erdős-Szekeres theorem erdsze 23735 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 2285 . . 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 23736 . 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 23737 . . . 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 1666 . 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 1530    e. wcel 1686   E.wrex 2546    C_ wss 3154   ~Pcpw 3627   class class class wbr 4025   `'ccnv 4690   ran crn 4692    |` cres 4693   "cima 4694   -1-1->wf1 5254   ` cfv 5257    Isom wiso 5258  (class class class)co 5860   RRcr 8738   1c1 8740    + caddc 8742    x. cmul 8744    < clt 8869    <_ cle 8870    - cmin 9039   NNcn 9748   ...cfz 10784   #chash 11339
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-se 4355  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-oi 7227  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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