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Theorem ramz 13385
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 10215 . . 3  |-  ( M  e.  NN0  <->  ( M  e.  NN  \/  M  =  0 ) )
2 n0 3629 . . . . . 6  |-  ( R  =/=  (/)  <->  E. c  c  e.  R )
3 simpll 731 . . . . . . . . 9  |-  ( ( ( M  e.  NN  /\  R  e.  V )  /\  c  e.  R
)  ->  M  e.  NN )
4 simplr 732 . . . . . . . . 9  |-  ( ( ( M  e.  NN  /\  R  e.  V )  /\  c  e.  R
)  ->  R  e.  V )
5 0nn0 10228 . . . . . . . . . . 11  |-  0  e.  NN0
65fconst6 5625 . . . . . . . . . 10  |-  ( R  X.  { 0 } ) : R --> NN0
76a1i 11 . . . . . . . . 9  |-  ( ( ( M  e.  NN  /\  R  e.  V )  /\  c  e.  R
)  ->  ( R  X.  { 0 } ) : R --> NN0 )
8 simpr 448 . . . . . . . . 9  |-  ( ( ( M  e.  NN  /\  R  e.  V )  /\  c  e.  R
)  ->  c  e.  R )
9 fvconst2g 5937 . . . . . . . . . 10  |-  ( ( 0  e.  NN0  /\  c  e.  R )  ->  ( ( R  X.  { 0 } ) `
 c )  =  0 )
105, 8, 9sylancr 645 . . . . . . . . 9  |-  ( ( ( M  e.  NN  /\  R  e.  V )  /\  c  e.  R
)  ->  ( ( R  X.  { 0 } ) `  c )  =  0 )
11 ramz2 13384 . . . . . . . . 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 1192 . . . . . . . 8  |-  ( ( ( M  e.  NN  /\  R  e.  V )  /\  c  e.  R
)  ->  ( M Ramsey  ( R  X.  { 0 } ) )  =  0 )
1312ex 424 . . . . . . 7  |-  ( ( M  e.  NN  /\  R  e.  V )  ->  ( c  e.  R  ->  ( M Ramsey  ( R  X.  { 0 } ) )  =  0 ) )
1413exlimdv 1646 . . . . . 6  |-  ( ( M  e.  NN  /\  R  e.  V )  ->  ( E. c  c  e.  R  ->  ( M Ramsey  ( R  X.  {
0 } ) )  =  0 ) )
152, 14syl5bi 209 . . . . 5  |-  ( ( M  e.  NN  /\  R  e.  V )  ->  ( R  =/=  (/)  ->  ( M Ramsey  ( R  X.  {
0 } ) )  =  0 ) )
1615expimpd 587 . . . 4  |-  ( M  e.  NN  ->  (
( R  e.  V  /\  R  =/=  (/) )  -> 
( M Ramsey  ( R  X.  { 0 } ) )  =  0 ) )
17 simpl 444 . . . . . . 7  |-  ( ( R  e.  V  /\  R  =/=  (/) )  ->  R  e.  V )
18 simpr 448 . . . . . . 7  |-  ( ( R  e.  V  /\  R  =/=  (/) )  ->  R  =/=  (/) )
196a1i 11 . . . . . . 7  |-  ( ( R  e.  V  /\  R  =/=  (/) )  ->  ( R  X.  { 0 } ) : R --> NN0 )
20 0z 10285 . . . . . . . 8  |-  0  e.  ZZ
21 elsni 3830 . . . . . . . . . . 11  |-  ( y  e.  { 0 }  ->  y  =  0 )
22 0le0 10073 . . . . . . . . . . 11  |-  0  <_  0
2321, 22syl6eqbr 4241 . . . . . . . . . 10  |-  ( y  e.  { 0 }  ->  y  <_  0
)
2423rgen 2763 . . . . . . . . 9  |-  A. y  e.  { 0 } y  <_  0
25 rnxp 5291 . . . . . . . . . . 11  |-  ( R  =/=  (/)  ->  ran  ( R  X.  { 0 } )  =  { 0 } )
2625adantl 453 . . . . . . . . . 10  |-  ( ( R  e.  V  /\  R  =/=  (/) )  ->  ran  ( R  X.  { 0 } )  =  {
0 } )
2726raleqdv 2902 . . . . . . . . 9  |-  ( ( R  e.  V  /\  R  =/=  (/) )  ->  ( A. y  e.  ran  ( R  X.  { 0 } ) y  <_ 
0  <->  A. y  e.  {
0 } y  <_ 
0 ) )
2824, 27mpbiri 225 . . . . . . . 8  |-  ( ( R  e.  V  /\  R  =/=  (/) )  ->  A. y  e.  ran  ( R  X.  { 0 } ) y  <_  0 )
29 breq2 4208 . . . . . . . . . 10  |-  ( x  =  0  ->  (
y  <_  x  <->  y  <_  0 ) )
3029ralbidv 2717 . . . . . . . . 9  |-  ( x  =  0  ->  ( A. y  e.  ran  ( R  X.  { 0 } ) y  <_  x 
<-> 
A. y  e.  ran  ( R  X.  { 0 } ) y  <_ 
0 ) )
3130rspcev 3044 . . . . . . . 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 645 . . . . . . 7  |-  ( ( R  e.  V  /\  R  =/=  (/) )  ->  E. x  e.  ZZ  A. y  e. 
ran  ( R  X.  { 0 } ) y  <_  x )
33 0ram 13380 . . . . . . 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 1187 . . . . . 6  |-  ( ( R  e.  V  /\  R  =/=  (/) )  ->  (
0 Ramsey  ( R  X.  {
0 } ) )  =  sup ( ran  ( R  X.  {
0 } ) ,  RR ,  <  )
)
3526supeq1d 7443 . . . . . 6  |-  ( ( R  e.  V  /\  R  =/=  (/) )  ->  sup ( ran  ( R  X.  { 0 } ) ,  RR ,  <  )  =  sup ( { 0 } ,  RR ,  <  ) )
36 ltso 9148 . . . . . . . 8  |-  <  Or  RR
37 0re 9083 . . . . . . . 8  |-  0  e.  RR
38 supsn 7466 . . . . . . . 8  |-  ( (  <  Or  RR  /\  0  e.  RR )  ->  sup ( { 0 } ,  RR ,  <  )  =  0 )
3936, 37, 38mp2an 654 . . . . . . 7  |-  sup ( { 0 } ,  RR ,  <  )  =  0
4039a1i 11 . . . . . 6  |-  ( ( R  e.  V  /\  R  =/=  (/) )  ->  sup ( { 0 } ,  RR ,  <  )  =  0 )
4134, 35, 403eqtrd 2471 . . . . 5  |-  ( ( R  e.  V  /\  R  =/=  (/) )  ->  (
0 Ramsey  ( R  X.  {
0 } ) )  =  0 )
42 oveq1 6080 . . . . . 6  |-  ( M  =  0  ->  ( M Ramsey  ( R  X.  {
0 } ) )  =  ( 0 Ramsey  ( R  X.  { 0 } ) ) )
4342eqeq1d 2443 . . . . 5  |-  ( M  =  0  ->  (
( M Ramsey  ( R  X.  { 0 } ) )  =  0  <->  (
0 Ramsey  ( R  X.  {
0 } ) )  =  0 ) )
4441, 43syl5ibr 213 . . . 4  |-  ( M  =  0  ->  (
( R  e.  V  /\  R  =/=  (/) )  -> 
( M Ramsey  ( R  X.  { 0 } ) )  =  0 ) )
4516, 44jaoi 369 . . 3  |-  ( ( M  e.  NN  \/  M  =  0 )  ->  ( ( R  e.  V  /\  R  =/=  (/) )  ->  ( M Ramsey  ( R  X.  {
0 } ) )  =  0 ) )
461, 45sylbi 188 . 2  |-  ( M  e.  NN0  ->  ( ( R  e.  V  /\  R  =/=  (/) )  ->  ( M Ramsey  ( R  X.  {
0 } ) )  =  0 ) )
47463impib 1151 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 358    /\ wa 359    /\ w3a 936   E.wex 1550    = wceq 1652    e. wcel 1725    =/= wne 2598   A.wral 2697   E.wrex 2698   (/)c0 3620   {csn 3806   class class class wbr 4204    Or wor 4494    X. cxp 4868   ran crn 4871   -->wf 5442   ` cfv 5446  (class class class)co 6073   supcsup 7437   RRcr 8981   0cc0 8982    < clt 9112    <_ cle 9113   NNcn 9992   NN0cn0 10213   ZZcz 10274   Ramsey cram 13359
This theorem is referenced by:  ramcl  13389
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
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-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-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-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-div 9670  df-nn 9993  df-n0 10214  df-z 10275  df-uz 10481  df-rp 10605  df-fz 11036  df-seq 11316  df-fac 11559  df-bc 11586  df-hash 11611  df-ram 13361
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