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Theorem ramz 14630
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 10793 . . 3  |-  ( M  e.  NN0  <->  ( M  e.  NN  \/  M  =  0 ) )
2 n0 3793 . . . . . 6  |-  ( R  =/=  (/)  <->  E. c  c  e.  R )
3 simpll 751 . . . . . . . . 9  |-  ( ( ( M  e.  NN  /\  R  e.  V )  /\  c  e.  R
)  ->  M  e.  NN )
4 simplr 753 . . . . . . . . 9  |-  ( ( ( M  e.  NN  /\  R  e.  V )  /\  c  e.  R
)  ->  R  e.  V )
5 0nn0 10806 . . . . . . . . . . 11  |-  0  e.  NN0
65fconst6 5757 . . . . . . . . . 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 459 . . . . . . . . 9  |-  ( ( ( M  e.  NN  /\  R  e.  V )  /\  c  e.  R
)  ->  c  e.  R )
9 fvconst2g 6101 . . . . . . . . . 10  |-  ( ( 0  e.  NN0  /\  c  e.  R )  ->  ( ( R  X.  { 0 } ) `
 c )  =  0 )
105, 8, 9sylancr 661 . . . . . . . . 9  |-  ( ( ( M  e.  NN  /\  R  e.  V )  /\  c  e.  R
)  ->  ( ( R  X.  { 0 } ) `  c )  =  0 )
11 ramz2 14629 . . . . . . . . 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 1234 . . . . . . . 8  |-  ( ( ( M  e.  NN  /\  R  e.  V )  /\  c  e.  R
)  ->  ( M Ramsey  ( R  X.  { 0 } ) )  =  0 )
1312ex 432 . . . . . . 7  |-  ( ( M  e.  NN  /\  R  e.  V )  ->  ( c  e.  R  ->  ( M Ramsey  ( R  X.  { 0 } ) )  =  0 ) )
1413exlimdv 1729 . . . . . 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 601 . . . 4  |-  ( M  e.  NN  ->  (
( R  e.  V  /\  R  =/=  (/) )  -> 
( M Ramsey  ( R  X.  { 0 } ) )  =  0 ) )
17 simpl 455 . . . . . . 7  |-  ( ( R  e.  V  /\  R  =/=  (/) )  ->  R  e.  V )
18 simpr 459 . . . . . . 7  |-  ( ( R  e.  V  /\  R  =/=  (/) )  ->  R  =/=  (/) )
196a1i 11 . . . . . . 7  |-  ( ( R  e.  V  /\  R  =/=  (/) )  ->  ( R  X.  { 0 } ) : R --> NN0 )
20 0z 10871 . . . . . . . 8  |-  0  e.  ZZ
21 elsni 4041 . . . . . . . . . . 11  |-  ( y  e.  { 0 }  ->  y  =  0 )
22 0le0 10621 . . . . . . . . . . 11  |-  0  <_  0
2321, 22syl6eqbr 4476 . . . . . . . . . 10  |-  ( y  e.  { 0 }  ->  y  <_  0
)
2423rgen 2814 . . . . . . . . 9  |-  A. y  e.  { 0 } y  <_  0
25 rnxp 5422 . . . . . . . . . . 11  |-  ( R  =/=  (/)  ->  ran  ( R  X.  { 0 } )  =  { 0 } )
2625adantl 464 . . . . . . . . . 10  |-  ( ( R  e.  V  /\  R  =/=  (/) )  ->  ran  ( R  X.  { 0 } )  =  {
0 } )
2726raleqdv 3057 . . . . . . . . 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 4443 . . . . . . . . . 10  |-  ( x  =  0  ->  (
y  <_  x  <->  y  <_  0 ) )
3029ralbidv 2893 . . . . . . . . 9  |-  ( x  =  0  ->  ( A. y  e.  ran  ( R  X.  { 0 } ) y  <_  x 
<-> 
A. y  e.  ran  ( R  X.  { 0 } ) y  <_ 
0 ) )
3130rspcev 3207 . . . . . . . 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 661 . . . . . . 7  |-  ( ( R  e.  V  /\  R  =/=  (/) )  ->  E. x  e.  ZZ  A. y  e. 
ran  ( R  X.  { 0 } ) y  <_  x )
33 0ram 14625 . . . . . . 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 1229 . . . . . 6  |-  ( ( R  e.  V  /\  R  =/=  (/) )  ->  (
0 Ramsey  ( R  X.  {
0 } ) )  =  sup ( ran  ( R  X.  {
0 } ) ,  RR ,  <  )
)
3526supeq1d 7897 . . . . . 6  |-  ( ( R  e.  V  /\  R  =/=  (/) )  ->  sup ( ran  ( R  X.  { 0 } ) ,  RR ,  <  )  =  sup ( { 0 } ,  RR ,  <  ) )
36 ltso 9654 . . . . . . . 8  |-  <  Or  RR
37 0re 9585 . . . . . . . 8  |-  0  e.  RR
38 supsn 7922 . . . . . . . 8  |-  ( (  <  Or  RR  /\  0  e.  RR )  ->  sup ( { 0 } ,  RR ,  <  )  =  0 )
3936, 37, 38mp2an 670 . . . . . . 7  |-  sup ( { 0 } ,  RR ,  <  )  =  0
4039a1i 11 . . . . . 6  |-  ( ( R  e.  V  /\  R  =/=  (/) )  ->  sup ( { 0 } ,  RR ,  <  )  =  0 )
4134, 35, 403eqtrd 2499 . . . . 5  |-  ( ( R  e.  V  /\  R  =/=  (/) )  ->  (
0 Ramsey  ( R  X.  {
0 } ) )  =  0 )
42 oveq1 6277 . . . . . 6  |-  ( M  =  0  ->  ( M Ramsey  ( R  X.  {
0 } ) )  =  ( 0 Ramsey  ( R  X.  { 0 } ) ) )
4342eqeq1d 2456 . . . . 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 377 . . 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 1192 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 366    /\ wa 367    /\ w3a 971    = wceq 1398   E.wex 1617    e. wcel 1823    =/= wne 2649   A.wral 2804   E.wrex 2805   (/)c0 3783   {csn 4016   class class class wbr 4439    Or wor 4788    X. cxp 4986   ran crn 4989   -->wf 5566   ` cfv 5570  (class class class)co 6270   supcsup 7892   RRcr 9480   0cc0 9481    < clt 9617    <_ cle 9618   NNcn 10531   NN0cn0 10791   ZZcz 10860   Ramsey cram 14604
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-2o 7123  df-oadd 7126  df-er 7303  df-map 7414  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-sup 7893  df-card 8311  df-cda 8539  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-nn 10532  df-n0 10792  df-z 10861  df-uz 11083  df-rp 11222  df-fz 11676  df-seq 12093  df-fac 12339  df-bc 12366  df-hash 12391  df-ram 14606
This theorem is referenced by:  ramcl  14634
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