MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  0ram2 Structured version   Unicode version

Theorem 0ram2 14978
Description: The Ramsey number when  M  = 
0. (Contributed by Mario Carneiro, 22-Apr-2015.)
Assertion
Ref Expression
0ram2  |-  ( ( R  e.  Fin  /\  R  =/=  (/)  /\  F : R
--> NN0 )  ->  (
0 Ramsey  F )  =  sup ( ran  F ,  RR ,  <  ) )

Proof of Theorem 0ram2
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 frn 5752 . . . . 5  |-  ( F : R --> NN0  ->  ran 
F  C_  NN0 )
213ad2ant3 1028 . . . 4  |-  ( ( R  e.  Fin  /\  R  =/=  (/)  /\  F : R
--> NN0 )  ->  ran  F 
C_  NN0 )
3 nn0ssz 10965 . . . 4  |-  NN0  C_  ZZ
42, 3syl6ss 3476 . . 3  |-  ( ( R  e.  Fin  /\  R  =/=  (/)  /\  F : R
--> NN0 )  ->  ran  F 
C_  ZZ )
5 nn0ssre 10880 . . . . 5  |-  NN0  C_  RR
62, 5syl6ss 3476 . . . 4  |-  ( ( R  e.  Fin  /\  R  =/=  (/)  /\  F : R
--> NN0 )  ->  ran  F 
C_  RR )
7 simp1 1005 . . . . 5  |-  ( ( R  e.  Fin  /\  R  =/=  (/)  /\  F : R
--> NN0 )  ->  R  e.  Fin )
8 ffn 5746 . . . . . . 7  |-  ( F : R --> NN0  ->  F  Fn  R )
983ad2ant3 1028 . . . . . 6  |-  ( ( R  e.  Fin  /\  R  =/=  (/)  /\  F : R
--> NN0 )  ->  F  Fn  R )
10 dffn4 5816 . . . . . 6  |-  ( F  Fn  R  <->  F : R -onto-> ran  F )
119, 10sylib 199 . . . . 5  |-  ( ( R  e.  Fin  /\  R  =/=  (/)  /\  F : R
--> NN0 )  ->  F : R -onto-> ran  F )
12 fofi 7869 . . . . 5  |-  ( ( R  e.  Fin  /\  F : R -onto-> ran  F
)  ->  ran  F  e. 
Fin )
137, 11, 12syl2anc 665 . . . 4  |-  ( ( R  e.  Fin  /\  R  =/=  (/)  /\  F : R
--> NN0 )  ->  ran  F  e.  Fin )
14 fdm 5750 . . . . . . 7  |-  ( F : R --> NN0  ->  dom 
F  =  R )
15143ad2ant3 1028 . . . . . 6  |-  ( ( R  e.  Fin  /\  R  =/=  (/)  /\  F : R
--> NN0 )  ->  dom  F  =  R )
16 simp2 1006 . . . . . 6  |-  ( ( R  e.  Fin  /\  R  =/=  (/)  /\  F : R
--> NN0 )  ->  R  =/=  (/) )
1715, 16eqnetrd 2713 . . . . 5  |-  ( ( R  e.  Fin  /\  R  =/=  (/)  /\  F : R
--> NN0 )  ->  dom  F  =/=  (/) )
18 dm0rn0 5070 . . . . . 6  |-  ( dom 
F  =  (/)  <->  ran  F  =  (/) )
1918necon3bii 2688 . . . . 5  |-  ( dom 
F  =/=  (/)  <->  ran  F  =/=  (/) )
2017, 19sylib 199 . . . 4  |-  ( ( R  e.  Fin  /\  R  =/=  (/)  /\  F : R
--> NN0 )  ->  ran  F  =/=  (/) )
21 fimaxre 10558 . . . 4  |-  ( ( ran  F  C_  RR  /\ 
ran  F  e.  Fin  /\ 
ran  F  =/=  (/) )  ->  E. x  e.  ran  F A. y  e.  ran  F  y  <_  x )
226, 13, 20, 21syl3anc 1264 . . 3  |-  ( ( R  e.  Fin  /\  R  =/=  (/)  /\  F : R
--> NN0 )  ->  E. x  e.  ran  F A. y  e.  ran  F  y  <_  x )
23 ssrexv 3526 . . 3  |-  ( ran 
F  C_  ZZ  ->  ( E. x  e.  ran  F A. y  e.  ran  F  y  <_  x  ->  E. x  e.  ZZ  A. y  e.  ran  F  y  <_  x ) )
244, 22, 23sylc 62 . 2  |-  ( ( R  e.  Fin  /\  R  =/=  (/)  /\  F : R
--> NN0 )  ->  E. x  e.  ZZ  A. y  e. 
ran  F  y  <_  x )
25 0ram 14977 . 2  |-  ( ( ( R  e.  Fin  /\  R  =/=  (/)  /\  F : R --> NN0 )  /\  E. x  e.  ZZ  A. y  e.  ran  F  y  <_  x )  ->  (
0 Ramsey  F )  =  sup ( ran  F ,  RR ,  <  ) )
2624, 25mpdan 672 1  |-  ( ( R  e.  Fin  /\  R  =/=  (/)  /\  F : R
--> NN0 )  ->  (
0 Ramsey  F )  =  sup ( ran  F ,  RR ,  <  ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 982    = wceq 1437    e. wcel 1872    =/= wne 2614   A.wral 2771   E.wrex 2772    C_ wss 3436   (/)c0 3761   class class class wbr 4423   dom cdm 4853   ran crn 4854    Fn wfn 5596   -->wf 5597   -onto->wfo 5599  (class class class)co 6305   Fincfn 7580   supcsup 7963   RRcr 9545   0cc0 9546    < clt 9682    <_ cle 9683   NN0cn0 10876   ZZcz 10944   Ramsey cram 14948
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-rep 4536  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6597  ax-cnex 9602  ax-resscn 9603  ax-1cn 9604  ax-icn 9605  ax-addcl 9606  ax-addrcl 9607  ax-mulcl 9608  ax-mulrcl 9609  ax-mulcom 9610  ax-addass 9611  ax-mulass 9612  ax-distr 9613  ax-i2m1 9614  ax-1ne0 9615  ax-1rid 9616  ax-rnegex 9617  ax-rrecex 9618  ax-cnre 9619  ax-pre-lttri 9620  ax-pre-lttrn 9621  ax-pre-ltadd 9622  ax-pre-mulgt0 9623
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-nel 2617  df-ral 2776  df-rex 2777  df-reu 2778  df-rmo 2779  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-tp 4003  df-op 4005  df-uni 4220  df-int 4256  df-iun 4301  df-br 4424  df-opab 4483  df-mpt 4484  df-tr 4519  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7039  df-recs 7101  df-rdg 7139  df-1o 7193  df-oadd 7197  df-er 7374  df-map 7485  df-en 7581  df-dom 7582  df-sdom 7583  df-fin 7584  df-sup 7965  df-inf 7966  df-card 8381  df-pnf 9684  df-mnf 9685  df-xr 9686  df-ltxr 9687  df-le 9688  df-sub 9869  df-neg 9870  df-nn 10617  df-n0 10877  df-z 10945  df-uz 11167  df-fz 11792  df-hash 12522  df-ram 14951
This theorem is referenced by:  0ramcl  14980
  Copyright terms: Public domain W3C validator