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Theorem ramz 13072
Description: The Ramsey number when  F is the zero function. (Contributed by Mario Carneiro, 22-Apr-2015.)
Assertion
Ref Expression
ramz  |-  ( ( M  e.  NN0  /\  R  e.  V  /\  R  =/=  (/) )  ->  ( M Ramsey  ( R  X.  {
0 } ) )  =  0 )

Proof of Theorem ramz
Dummy variables  c  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elnn0 9967 . . 3  |-  ( M  e.  NN0  <->  ( M  e.  NN  \/  M  =  0 ) )
2 n0 3464 . . . . . 6  |-  ( R  =/=  (/)  <->  E. c  c  e.  R )
3 simpll 730 . . . . . . . . 9  |-  ( ( ( M  e.  NN  /\  R  e.  V )  /\  c  e.  R
)  ->  M  e.  NN )
4 simplr 731 . . . . . . . . 9  |-  ( ( ( M  e.  NN  /\  R  e.  V )  /\  c  e.  R
)  ->  R  e.  V )
5 0nn0 9980 . . . . . . . . . . 11  |-  0  e.  NN0
65fconst6 5431 . . . . . . . . . 10  |-  ( R  X.  { 0 } ) : R --> NN0
76a1i 10 . . . . . . . . 9  |-  ( ( ( M  e.  NN  /\  R  e.  V )  /\  c  e.  R
)  ->  ( R  X.  { 0 } ) : R --> NN0 )
8 simpr 447 . . . . . . . . 9  |-  ( ( ( M  e.  NN  /\  R  e.  V )  /\  c  e.  R
)  ->  c  e.  R )
9 fvconst2g 5727 . . . . . . . . . 10  |-  ( ( 0  e.  NN0  /\  c  e.  R )  ->  ( ( R  X.  { 0 } ) `
 c )  =  0 )
105, 8, 9sylancr 644 . . . . . . . . 9  |-  ( ( ( M  e.  NN  /\  R  e.  V )  /\  c  e.  R
)  ->  ( ( R  X.  { 0 } ) `  c )  =  0 )
11 ramz2 13071 . . . . . . . . 9  |-  ( ( ( M  e.  NN  /\  R  e.  V  /\  ( R  X.  { 0 } ) : R --> NN0 )  /\  (
c  e.  R  /\  ( ( R  X.  { 0 } ) `
 c )  =  0 ) )  -> 
( M Ramsey  ( R  X.  { 0 } ) )  =  0 )
123, 4, 7, 8, 10, 11syl32anc 1190 . . . . . . . 8  |-  ( ( ( M  e.  NN  /\  R  e.  V )  /\  c  e.  R
)  ->  ( M Ramsey  ( R  X.  { 0 } ) )  =  0 )
1312ex 423 . . . . . . 7  |-  ( ( M  e.  NN  /\  R  e.  V )  ->  ( c  e.  R  ->  ( M Ramsey  ( R  X.  { 0 } ) )  =  0 ) )
1413exlimdv 1664 . . . . . 6  |-  ( ( M  e.  NN  /\  R  e.  V )  ->  ( E. c  c  e.  R  ->  ( M Ramsey  ( R  X.  {
0 } ) )  =  0 ) )
152, 14syl5bi 208 . . . . 5  |-  ( ( M  e.  NN  /\  R  e.  V )  ->  ( R  =/=  (/)  ->  ( M Ramsey  ( R  X.  {
0 } ) )  =  0 ) )
1615expimpd 586 . . . 4  |-  ( M  e.  NN  ->  (
( R  e.  V  /\  R  =/=  (/) )  -> 
( M Ramsey  ( R  X.  { 0 } ) )  =  0 ) )
17 simpl 443 . . . . . . 7  |-  ( ( R  e.  V  /\  R  =/=  (/) )  ->  R  e.  V )
18 simpr 447 . . . . . . 7  |-  ( ( R  e.  V  /\  R  =/=  (/) )  ->  R  =/=  (/) )
196a1i 10 . . . . . . 7  |-  ( ( R  e.  V  /\  R  =/=  (/) )  ->  ( R  X.  { 0 } ) : R --> NN0 )
20 0z 10035 . . . . . . . 8  |-  0  e.  ZZ
21 elsni 3664 . . . . . . . . . . 11  |-  ( y  e.  { 0 }  ->  y  =  0 )
22 0le0 9827 . . . . . . . . . . 11  |-  0  <_  0
2321, 22syl6eqbr 4060 . . . . . . . . . 10  |-  ( y  e.  { 0 }  ->  y  <_  0
)
2423rgen 2608 . . . . . . . . 9  |-  A. y  e.  { 0 } y  <_  0
25 rnxp 5106 . . . . . . . . . . 11  |-  ( R  =/=  (/)  ->  ran  ( R  X.  { 0 } )  =  { 0 } )
2625adantl 452 . . . . . . . . . 10  |-  ( ( R  e.  V  /\  R  =/=  (/) )  ->  ran  ( R  X.  { 0 } )  =  {
0 } )
2726raleqdv 2742 . . . . . . . . 9  |-  ( ( R  e.  V  /\  R  =/=  (/) )  ->  ( A. y  e.  ran  ( R  X.  { 0 } ) y  <_ 
0  <->  A. y  e.  {
0 } y  <_ 
0 ) )
2824, 27mpbiri 224 . . . . . . . 8  |-  ( ( R  e.  V  /\  R  =/=  (/) )  ->  A. y  e.  ran  ( R  X.  { 0 } ) y  <_  0 )
29 breq2 4027 . . . . . . . . . 10  |-  ( x  =  0  ->  (
y  <_  x  <->  y  <_  0 ) )
3029ralbidv 2563 . . . . . . . . 9  |-  ( x  =  0  ->  ( A. y  e.  ran  ( R  X.  { 0 } ) y  <_  x 
<-> 
A. y  e.  ran  ( R  X.  { 0 } ) y  <_ 
0 ) )
3130rspcev 2884 . . . . . . . 8  |-  ( ( 0  e.  ZZ  /\  A. y  e.  ran  ( R  X.  { 0 } ) y  <_  0
)  ->  E. x  e.  ZZ  A. y  e. 
ran  ( R  X.  { 0 } ) y  <_  x )
3220, 28, 31sylancr 644 . . . . . . 7  |-  ( ( R  e.  V  /\  R  =/=  (/) )  ->  E. x  e.  ZZ  A. y  e. 
ran  ( R  X.  { 0 } ) y  <_  x )
33 0ram 13067 . . . . . . 7  |-  ( ( ( R  e.  V  /\  R  =/=  (/)  /\  ( R  X.  { 0 } ) : R --> NN0 )  /\  E. x  e.  ZZ  A. y  e.  ran  ( R  X.  { 0 } ) y  <_  x
)  ->  ( 0 Ramsey 
( R  X.  {
0 } ) )  =  sup ( ran  ( R  X.  {
0 } ) ,  RR ,  <  )
)
3417, 18, 19, 32, 33syl31anc 1185 . . . . . 6  |-  ( ( R  e.  V  /\  R  =/=  (/) )  ->  (
0 Ramsey  ( R  X.  {
0 } ) )  =  sup ( ran  ( R  X.  {
0 } ) ,  RR ,  <  )
)
3526supeq1d 7199 . . . . . 6  |-  ( ( R  e.  V  /\  R  =/=  (/) )  ->  sup ( ran  ( R  X.  { 0 } ) ,  RR ,  <  )  =  sup ( { 0 } ,  RR ,  <  ) )
36 ltso 8903 . . . . . . . 8  |-  <  Or  RR
37 0re 8838 . . . . . . . 8  |-  0  e.  RR
38 supsn 7220 . . . . . . . 8  |-  ( (  <  Or  RR  /\  0  e.  RR )  ->  sup ( { 0 } ,  RR ,  <  )  =  0 )
3936, 37, 38mp2an 653 . . . . . . 7  |-  sup ( { 0 } ,  RR ,  <  )  =  0
4039a1i 10 . . . . . 6  |-  ( ( R  e.  V  /\  R  =/=  (/) )  ->  sup ( { 0 } ,  RR ,  <  )  =  0 )
4134, 35, 403eqtrd 2319 . . . . 5  |-  ( ( R  e.  V  /\  R  =/=  (/) )  ->  (
0 Ramsey  ( R  X.  {
0 } ) )  =  0 )
42 oveq1 5865 . . . . . 6  |-  ( M  =  0  ->  ( M Ramsey  ( R  X.  {
0 } ) )  =  ( 0 Ramsey  ( R  X.  { 0 } ) ) )
4342eqeq1d 2291 . . . . 5  |-  ( M  =  0  ->  (
( M Ramsey  ( R  X.  { 0 } ) )  =  0  <->  (
0 Ramsey  ( R  X.  {
0 } ) )  =  0 ) )
4441, 43syl5ibr 212 . . . 4  |-  ( M  =  0  ->  (
( R  e.  V  /\  R  =/=  (/) )  -> 
( M Ramsey  ( R  X.  { 0 } ) )  =  0 ) )
4516, 44jaoi 368 . . 3  |-  ( ( M  e.  NN  \/  M  =  0 )  ->  ( ( R  e.  V  /\  R  =/=  (/) )  ->  ( M Ramsey  ( R  X.  {
0 } ) )  =  0 ) )
461, 45sylbi 187 . 2  |-  ( M  e.  NN0  ->  ( ( R  e.  V  /\  R  =/=  (/) )  ->  ( M Ramsey  ( R  X.  {
0 } ) )  =  0 ) )
47463impib 1149 1  |-  ( ( M  e.  NN0  /\  R  e.  V  /\  R  =/=  (/) )  ->  ( M Ramsey  ( R  X.  {
0 } ) )  =  0 )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 357    /\ wa 358    /\ w3a 934   E.wex 1528    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   E.wrex 2544   (/)c0 3455   {csn 3640   class class class wbr 4023    Or wor 4313    X. cxp 4687   ran crn 4690   -->wf 5251   ` cfv 5255  (class class class)co 5858   supcsup 7193   RRcr 8736   0cc0 8737    < clt 8867    <_ cle 8868   NNcn 9746   NN0cn0 9965   ZZcz 10024   Ramsey cram 13046
This theorem is referenced by:  ramcl  13076
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-card 7572  df-cda 7794  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-fz 10783  df-seq 11047  df-fac 11289  df-bc 11316  df-hash 11338  df-ram 13048
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