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Theorem erdsze2 24883
Description: Generalize the statement of the Erdős-Szekeres theorem erdsze 24880 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 2435 . . 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 24881 . 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 452 . . 3  |-  ( (
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 452 . . 3  |-  ( (
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 452 . . 3  |-  ( (
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 452 . . 3  |-  ( (
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 452 . . 3  |-  ( (
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 733 . . 3  |-  ( (
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 734 . . 3  |-  ( (
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 24882 . 2  |-  ( (
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 ) ) ) ) )
167, 15exlimddv 1648 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 358    /\ wa 359    e. wcel 1725   E.wrex 2698    C_ wss 3312   ~Pcpw 3791   class class class wbr 4204   `'ccnv 4869   ran crn 4871    |` cres 4872   "cima 4873   -1-1->wf1 5443   ` cfv 5446    Isom wiso 5447  (class class class)co 6073   RRcr 8981   1c1 8983    + caddc 8985    x. cmul 8987    < clt 9112    <_ cle 9113    - cmin 9283   NNcn 9992   ...cfz 11035   #chash 11610
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-2o 6717  df-oadd 6720  df-er 6897  df-map 7012  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-sup 7438  df-oi 7471  df-card 7818  df-cda 8040  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-n0 10214  df-z 10275  df-uz 10481  df-fz 11036  df-hash 11611
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