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Theorem ramz 14084
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 10579 . . 3  |-  ( M  e.  NN0  <->  ( M  e.  NN  \/  M  =  0 ) )
2 n0 3644 . . . . . 6  |-  ( R  =/=  (/)  <->  E. c  c  e.  R )
3 simpll 753 . . . . . . . . 9  |-  ( ( ( M  e.  NN  /\  R  e.  V )  /\  c  e.  R
)  ->  M  e.  NN )
4 simplr 754 . . . . . . . . 9  |-  ( ( ( M  e.  NN  /\  R  e.  V )  /\  c  e.  R
)  ->  R  e.  V )
5 0nn0 10592 . . . . . . . . . . 11  |-  0  e.  NN0
65fconst6 5598 . . . . . . . . . 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 461 . . . . . . . . 9  |-  ( ( ( M  e.  NN  /\  R  e.  V )  /\  c  e.  R
)  ->  c  e.  R )
9 fvconst2g 5929 . . . . . . . . . 10  |-  ( ( 0  e.  NN0  /\  c  e.  R )  ->  ( ( R  X.  { 0 } ) `
 c )  =  0 )
105, 8, 9sylancr 663 . . . . . . . . 9  |-  ( ( ( M  e.  NN  /\  R  e.  V )  /\  c  e.  R
)  ->  ( ( R  X.  { 0 } ) `  c )  =  0 )
11 ramz2 14083 . . . . . . . . 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 1226 . . . . . . . 8  |-  ( ( ( M  e.  NN  /\  R  e.  V )  /\  c  e.  R
)  ->  ( M Ramsey  ( R  X.  { 0 } ) )  =  0 )
1312ex 434 . . . . . . 7  |-  ( ( M  e.  NN  /\  R  e.  V )  ->  ( c  e.  R  ->  ( M Ramsey  ( R  X.  { 0 } ) )  =  0 ) )
1413exlimdv 1690 . . . . . 6  |-  ( ( M  e.  NN  /\  R  e.  V )  ->  ( E. c  c  e.  R  ->  ( M Ramsey  ( R  X.  {
0 } ) )  =  0 ) )
152, 14syl5bi 217 . . . . 5  |-  ( ( M  e.  NN  /\  R  e.  V )  ->  ( R  =/=  (/)  ->  ( M Ramsey  ( R  X.  {
0 } ) )  =  0 ) )
1615expimpd 603 . . . 4  |-  ( M  e.  NN  ->  (
( R  e.  V  /\  R  =/=  (/) )  -> 
( M Ramsey  ( R  X.  { 0 } ) )  =  0 ) )
17 simpl 457 . . . . . . 7  |-  ( ( R  e.  V  /\  R  =/=  (/) )  ->  R  e.  V )
18 simpr 461 . . . . . . 7  |-  ( ( R  e.  V  /\  R  =/=  (/) )  ->  R  =/=  (/) )
196a1i 11 . . . . . . 7  |-  ( ( R  e.  V  /\  R  =/=  (/) )  ->  ( R  X.  { 0 } ) : R --> NN0 )
20 0z 10655 . . . . . . . 8  |-  0  e.  ZZ
21 elsni 3900 . . . . . . . . . . 11  |-  ( y  e.  { 0 }  ->  y  =  0 )
22 0le0 10409 . . . . . . . . . . 11  |-  0  <_  0
2321, 22syl6eqbr 4327 . . . . . . . . . 10  |-  ( y  e.  { 0 }  ->  y  <_  0
)
2423rgen 2779 . . . . . . . . 9  |-  A. y  e.  { 0 } y  <_  0
25 rnxp 5266 . . . . . . . . . . 11  |-  ( R  =/=  (/)  ->  ran  ( R  X.  { 0 } )  =  { 0 } )
2625adantl 466 . . . . . . . . . 10  |-  ( ( R  e.  V  /\  R  =/=  (/) )  ->  ran  ( R  X.  { 0 } )  =  {
0 } )
2726raleqdv 2921 . . . . . . . . 9  |-  ( ( R  e.  V  /\  R  =/=  (/) )  ->  ( A. y  e.  ran  ( R  X.  { 0 } ) y  <_ 
0  <->  A. y  e.  {
0 } y  <_ 
0 ) )
2824, 27mpbiri 233 . . . . . . . 8  |-  ( ( R  e.  V  /\  R  =/=  (/) )  ->  A. y  e.  ran  ( R  X.  { 0 } ) y  <_  0 )
29 breq2 4294 . . . . . . . . . 10  |-  ( x  =  0  ->  (
y  <_  x  <->  y  <_  0 ) )
3029ralbidv 2733 . . . . . . . . 9  |-  ( x  =  0  ->  ( A. y  e.  ran  ( R  X.  { 0 } ) y  <_  x 
<-> 
A. y  e.  ran  ( R  X.  { 0 } ) y  <_ 
0 ) )
3130rspcev 3071 . . . . . . . 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 663 . . . . . . 7  |-  ( ( R  e.  V  /\  R  =/=  (/) )  ->  E. x  e.  ZZ  A. y  e. 
ran  ( R  X.  { 0 } ) y  <_  x )
33 0ram 14079 . . . . . . 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 1221 . . . . . 6  |-  ( ( R  e.  V  /\  R  =/=  (/) )  ->  (
0 Ramsey  ( R  X.  {
0 } ) )  =  sup ( ran  ( R  X.  {
0 } ) ,  RR ,  <  )
)
3526supeq1d 7694 . . . . . 6  |-  ( ( R  e.  V  /\  R  =/=  (/) )  ->  sup ( ran  ( R  X.  { 0 } ) ,  RR ,  <  )  =  sup ( { 0 } ,  RR ,  <  ) )
36 ltso 9453 . . . . . . . 8  |-  <  Or  RR
37 0re 9384 . . . . . . . 8  |-  0  e.  RR
38 supsn 7717 . . . . . . . 8  |-  ( (  <  Or  RR  /\  0  e.  RR )  ->  sup ( { 0 } ,  RR ,  <  )  =  0 )
3936, 37, 38mp2an 672 . . . . . . 7  |-  sup ( { 0 } ,  RR ,  <  )  =  0
4039a1i 11 . . . . . 6  |-  ( ( R  e.  V  /\  R  =/=  (/) )  ->  sup ( { 0 } ,  RR ,  <  )  =  0 )
4134, 35, 403eqtrd 2477 . . . . 5  |-  ( ( R  e.  V  /\  R  =/=  (/) )  ->  (
0 Ramsey  ( R  X.  {
0 } ) )  =  0 )
42 oveq1 6096 . . . . . 6  |-  ( M  =  0  ->  ( M Ramsey  ( R  X.  {
0 } ) )  =  ( 0 Ramsey  ( R  X.  { 0 } ) ) )
4342eqeq1d 2449 . . . . 5  |-  ( M  =  0  ->  (
( M Ramsey  ( R  X.  { 0 } ) )  =  0  <->  (
0 Ramsey  ( R  X.  {
0 } ) )  =  0 ) )
4441, 43syl5ibr 221 . . . 4  |-  ( M  =  0  ->  (
( R  e.  V  /\  R  =/=  (/) )  -> 
( M Ramsey  ( R  X.  { 0 } ) )  =  0 ) )
4516, 44jaoi 379 . . 3  |-  ( ( M  e.  NN  \/  M  =  0 )  ->  ( ( R  e.  V  /\  R  =/=  (/) )  ->  ( M Ramsey  ( R  X.  {
0 } ) )  =  0 ) )
461, 45sylbi 195 . 2  |-  ( M  e.  NN0  ->  ( ( R  e.  V  /\  R  =/=  (/) )  ->  ( M Ramsey  ( R  X.  {
0 } ) )  =  0 ) )
47463impib 1185 1  |-  ( ( M  e.  NN0  /\  R  e.  V  /\  R  =/=  (/) )  ->  ( M Ramsey  ( R  X.  {
0 } ) )  =  0 )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 368    /\ wa 369    /\ w3a 965    = wceq 1369   E.wex 1586    e. wcel 1756    =/= wne 2604   A.wral 2713   E.wrex 2714   (/)c0 3635   {csn 3875   class class class wbr 4290    Or wor 4638    X. cxp 4836   ran crn 4839   -->wf 5412   ` cfv 5416  (class class class)co 6089   supcsup 7688   RRcr 9279   0cc0 9280    < clt 9416    <_ cle 9417   NNcn 10320   NN0cn0 10577   ZZcz 10644   Ramsey cram 14058
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-rep 4401  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370  ax-cnex 9336  ax-resscn 9337  ax-1cn 9338  ax-icn 9339  ax-addcl 9340  ax-addrcl 9341  ax-mulcl 9342  ax-mulrcl 9343  ax-mulcom 9344  ax-addass 9345  ax-mulass 9346  ax-distr 9347  ax-i2m1 9348  ax-1ne0 9349  ax-1rid 9350  ax-rnegex 9351  ax-rrecex 9352  ax-cnre 9353  ax-pre-lttri 9354  ax-pre-lttrn 9355  ax-pre-ltadd 9356  ax-pre-mulgt0 9357
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-pss 3342  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-tp 3880  df-op 3882  df-uni 4090  df-int 4127  df-iun 4171  df-br 4291  df-opab 4349  df-mpt 4350  df-tr 4384  df-eprel 4630  df-id 4634  df-po 4639  df-so 4640  df-fr 4677  df-we 4679  df-ord 4720  df-on 4721  df-lim 4722  df-suc 4723  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-riota 6050  df-ov 6092  df-oprab 6093  df-mpt2 6094  df-om 6475  df-1st 6575  df-2nd 6576  df-recs 6830  df-rdg 6864  df-1o 6918  df-2o 6919  df-oadd 6922  df-er 7099  df-map 7214  df-en 7309  df-dom 7310  df-sdom 7311  df-fin 7312  df-sup 7689  df-card 8107  df-cda 8335  df-pnf 9418  df-mnf 9419  df-xr 9420  df-ltxr 9421  df-le 9422  df-sub 9595  df-neg 9596  df-div 9992  df-nn 10321  df-n0 10578  df-z 10645  df-uz 10860  df-rp 10990  df-fz 11436  df-seq 11805  df-fac 12050  df-bc 12077  df-hash 12102  df-ram 14060
This theorem is referenced by:  ramcl  14088
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