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Theorem 0ramcl 13391
Description: Lemma for ramcl 13397: Existence of the Ramsey number when  M  =  0. (Contributed by Mario Carneiro, 23-Apr-2015.)
Assertion
Ref Expression
0ramcl  |-  ( ( R  e.  Fin  /\  F : R --> NN0 )  ->  ( 0 Ramsey  F )  e.  NN0 )

Proof of Theorem 0ramcl
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ffn 5591 . . . . . . . 8  |-  ( F : R --> NN0  ->  F  Fn  R )
2 dffn4 5659 . . . . . . . 8  |-  ( F  Fn  R  <->  F : R -onto-> ran  F )
31, 2sylib 189 . . . . . . 7  |-  ( F : R --> NN0  ->  F : R -onto-> ran  F
)
43ad2antlr 708 . . . . . 6  |-  ( ( ( R  e.  Fin  /\  F : R --> NN0 )  /\  R  =  (/) )  ->  F : R -onto-> ran  F
)
5 foeq2 5650 . . . . . . 7  |-  ( R  =  (/)  ->  ( F : R -onto-> ran  F  <->  F : (/) -onto-> ran  F ) )
65adantl 453 . . . . . 6  |-  ( ( ( R  e.  Fin  /\  F : R --> NN0 )  /\  R  =  (/) )  -> 
( F : R -onto-> ran  F  <->  F : (/) -onto-> ran  F
) )
74, 6mpbid 202 . . . . 5  |-  ( ( ( R  e.  Fin  /\  F : R --> NN0 )  /\  R  =  (/) )  ->  F : (/) -onto-> ran  F )
8 fo00 5711 . . . . . 6  |-  ( F : (/) -onto-> ran  F  <->  ( F  =  (/)  /\  ran  F  =  (/) ) )
98simplbi 447 . . . . 5  |-  ( F : (/) -onto-> ran  F  ->  F  =  (/) )
107, 9syl 16 . . . 4  |-  ( ( ( R  e.  Fin  /\  F : R --> NN0 )  /\  R  =  (/) )  ->  F  =  (/) )
1110oveq2d 6097 . . 3  |-  ( ( ( R  e.  Fin  /\  F : R --> NN0 )  /\  R  =  (/) )  -> 
( 0 Ramsey  F )  =  ( 0 Ramsey  (/) ) )
12 0nn0 10236 . . . . 5  |-  0  e.  NN0
13 ram0 13390 . . . . 5  |-  ( 0  e.  NN0  ->  ( 0 Ramsey  (/) )  =  0 )
1412, 13ax-mp 8 . . . 4  |-  ( 0 Ramsey  (/) )  =  0
1514, 12eqeltri 2506 . . 3  |-  ( 0 Ramsey  (/) )  e.  NN0
1611, 15syl6eqel 2524 . 2  |-  ( ( ( R  e.  Fin  /\  F : R --> NN0 )  /\  R  =  (/) )  -> 
( 0 Ramsey  F )  e.  NN0 )
17 0ram2 13389 . . . . 5  |-  ( ( R  e.  Fin  /\  R  =/=  (/)  /\  F : R
--> NN0 )  ->  (
0 Ramsey  F )  =  sup ( ran  F ,  RR ,  <  ) )
18 frn 5597 . . . . . . 7  |-  ( F : R --> NN0  ->  ran 
F  C_  NN0 )
19183ad2ant3 980 . . . . . 6  |-  ( ( R  e.  Fin  /\  R  =/=  (/)  /\  F : R
--> NN0 )  ->  ran  F 
C_  NN0 )
20 nn0ssz 10302 . . . . . . . 8  |-  NN0  C_  ZZ
2119, 20syl6ss 3360 . . . . . . 7  |-  ( ( R  e.  Fin  /\  R  =/=  (/)  /\  F : R
--> NN0 )  ->  ran  F 
C_  ZZ )
22 fdm 5595 . . . . . . . . . 10  |-  ( F : R --> NN0  ->  dom 
F  =  R )
23223ad2ant3 980 . . . . . . . . 9  |-  ( ( R  e.  Fin  /\  R  =/=  (/)  /\  F : R
--> NN0 )  ->  dom  F  =  R )
24 simp2 958 . . . . . . . . 9  |-  ( ( R  e.  Fin  /\  R  =/=  (/)  /\  F : R
--> NN0 )  ->  R  =/=  (/) )
2523, 24eqnetrd 2619 . . . . . . . 8  |-  ( ( R  e.  Fin  /\  R  =/=  (/)  /\  F : R
--> NN0 )  ->  dom  F  =/=  (/) )
26 dm0rn0 5086 . . . . . . . . 9  |-  ( dom 
F  =  (/)  <->  ran  F  =  (/) )
2726necon3bii 2633 . . . . . . . 8  |-  ( dom 
F  =/=  (/)  <->  ran  F  =/=  (/) )
2825, 27sylib 189 . . . . . . 7  |-  ( ( R  e.  Fin  /\  R  =/=  (/)  /\  F : R
--> NN0 )  ->  ran  F  =/=  (/) )
29 nn0ssre 10225 . . . . . . . . . 10  |-  NN0  C_  RR
3019, 29syl6ss 3360 . . . . . . . . 9  |-  ( ( R  e.  Fin  /\  R  =/=  (/)  /\  F : R
--> NN0 )  ->  ran  F 
C_  RR )
31 simp1 957 . . . . . . . . . 10  |-  ( ( R  e.  Fin  /\  R  =/=  (/)  /\  F : R
--> NN0 )  ->  R  e.  Fin )
3233ad2ant3 980 . . . . . . . . . 10  |-  ( ( R  e.  Fin  /\  R  =/=  (/)  /\  F : R
--> NN0 )  ->  F : R -onto-> ran  F )
33 fofi 7392 . . . . . . . . . 10  |-  ( ( R  e.  Fin  /\  F : R -onto-> ran  F
)  ->  ran  F  e. 
Fin )
3431, 32, 33syl2anc 643 . . . . . . . . 9  |-  ( ( R  e.  Fin  /\  R  =/=  (/)  /\  F : R
--> NN0 )  ->  ran  F  e.  Fin )
35 fimaxre 9955 . . . . . . . . 9  |-  ( ( ran  F  C_  RR  /\ 
ran  F  e.  Fin  /\ 
ran  F  =/=  (/) )  ->  E. x  e.  ran  F A. y  e.  ran  F  y  <_  x )
3630, 34, 28, 35syl3anc 1184 . . . . . . . 8  |-  ( ( R  e.  Fin  /\  R  =/=  (/)  /\  F : R
--> NN0 )  ->  E. x  e.  ran  F A. y  e.  ran  F  y  <_  x )
37 ssrexv 3408 . . . . . . . 8  |-  ( 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 ) )
3821, 36, 37sylc 58 . . . . . . 7  |-  ( ( R  e.  Fin  /\  R  =/=  (/)  /\  F : R
--> NN0 )  ->  E. x  e.  ZZ  A. y  e. 
ran  F  y  <_  x )
39 suprzcl2 10566 . . . . . . 7  |-  ( ( ran  F  C_  ZZ  /\ 
ran  F  =/=  (/)  /\  E. x  e.  ZZ  A. y  e.  ran  F  y  <_  x )  ->  sup ( ran  F ,  RR ,  <  )  e.  ran  F )
4021, 28, 38, 39syl3anc 1184 . . . . . 6  |-  ( ( R  e.  Fin  /\  R  =/=  (/)  /\  F : R
--> NN0 )  ->  sup ( ran  F ,  RR ,  <  )  e.  ran  F )
4119, 40sseldd 3349 . . . . 5  |-  ( ( R  e.  Fin  /\  R  =/=  (/)  /\  F : R
--> NN0 )  ->  sup ( ran  F ,  RR ,  <  )  e.  NN0 )
4217, 41eqeltrd 2510 . . . 4  |-  ( ( R  e.  Fin  /\  R  =/=  (/)  /\  F : R
--> NN0 )  ->  (
0 Ramsey  F )  e.  NN0 )
43423expa 1153 . . 3  |-  ( ( ( R  e.  Fin  /\  R  =/=  (/) )  /\  F : R --> NN0 )  ->  ( 0 Ramsey  F )  e.  NN0 )
4443an32s 780 . 2  |-  ( ( ( R  e.  Fin  /\  F : R --> NN0 )  /\  R  =/=  (/) )  -> 
( 0 Ramsey  F )  e.  NN0 )
4516, 44pm2.61dane 2682 1  |-  ( ( R  e.  Fin  /\  F : R --> NN0 )  ->  ( 0 Ramsey  F )  e.  NN0 )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2599   A.wral 2705   E.wrex 2706    C_ wss 3320   (/)c0 3628   class class class wbr 4212   dom cdm 4878   ran crn 4879    Fn wfn 5449   -->wf 5450   -onto->wfo 5452  (class class class)co 6081   Fincfn 7109   supcsup 7445   RRcr 8989   0cc0 8990    < clt 9120    <_ cle 9121   NN0cn0 10221   ZZcz 10282   Ramsey cram 13367
This theorem is referenced by:  ramcl  13397
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 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-2o 6725  df-oadd 6728  df-er 6905  df-map 7020  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-sup 7446  df-card 7826  df-cda 8048  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-n0 10222  df-z 10283  df-uz 10489  df-rp 10613  df-fz 11044  df-seq 11324  df-fac 11567  df-bc 11594  df-hash 11619  df-ram 13369
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