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Theorem ramz2 13071
Description: The Ramsey number when  F has value zero for some color  C. (Contributed by Mario Carneiro, 22-Apr-2015.)
Assertion
Ref Expression
ramz2  |-  ( ( ( M  e.  NN  /\  R  e.  V  /\  F : R --> NN0 )  /\  ( C  e.  R  /\  ( F `  C
)  =  0 ) )  ->  ( M Ramsey  F )  =  0 )

Proof of Theorem ramz2
Dummy variables  b 
f  c  s  x  a  i are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2283 . . 3  |-  ( a  e.  _V ,  i  e.  NN0  |->  { b  e.  ~P a  |  ( # `  b
)  =  i } )  =  ( a  e.  _V ,  i  e.  NN0  |->  { b  e.  ~P a  |  ( # `  b
)  =  i } )
2 simpl1 958 . . . 4  |-  ( ( ( M  e.  NN  /\  R  e.  V  /\  F : R --> NN0 )  /\  ( C  e.  R  /\  ( F `  C
)  =  0 ) )  ->  M  e.  NN )
32nnnn0d 10018 . . 3  |-  ( ( ( M  e.  NN  /\  R  e.  V  /\  F : R --> NN0 )  /\  ( C  e.  R  /\  ( F `  C
)  =  0 ) )  ->  M  e.  NN0 )
4 simpl2 959 . . 3  |-  ( ( ( M  e.  NN  /\  R  e.  V  /\  F : R --> NN0 )  /\  ( C  e.  R  /\  ( F `  C
)  =  0 ) )  ->  R  e.  V )
5 simpl3 960 . . 3  |-  ( ( ( M  e.  NN  /\  R  e.  V  /\  F : R --> NN0 )  /\  ( C  e.  R  /\  ( F `  C
)  =  0 ) )  ->  F : R
--> NN0 )
6 0nn0 9980 . . . 4  |-  0  e.  NN0
76a1i 10 . . 3  |-  ( ( ( M  e.  NN  /\  R  e.  V  /\  F : R --> NN0 )  /\  ( C  e.  R  /\  ( F `  C
)  =  0 ) )  ->  0  e.  NN0 )
8 simplrl 736 . . . 4  |-  ( ( ( ( M  e.  NN  /\  R  e.  V  /\  F : R
--> NN0 )  /\  ( C  e.  R  /\  ( F `  C )  =  0 ) )  /\  ( 0  <_ 
( # `  s )  /\  f : ( s ( a  e. 
_V ,  i  e. 
NN0  |->  { b  e. 
~P a  |  (
# `  b )  =  i } ) M ) --> R ) )  ->  C  e.  R )
9 0elpw 4180 . . . . 5  |-  (/)  e.  ~P s
109a1i 10 . . . 4  |-  ( ( ( ( M  e.  NN  /\  R  e.  V  /\  F : R
--> NN0 )  /\  ( C  e.  R  /\  ( F `  C )  =  0 ) )  /\  ( 0  <_ 
( # `  s )  /\  f : ( s ( a  e. 
_V ,  i  e. 
NN0  |->  { b  e. 
~P a  |  (
# `  b )  =  i } ) M ) --> R ) )  ->  (/)  e.  ~P s )
11 simplrr 737 . . . . 5  |-  ( ( ( ( M  e.  NN  /\  R  e.  V  /\  F : R
--> NN0 )  /\  ( C  e.  R  /\  ( F `  C )  =  0 ) )  /\  ( 0  <_ 
( # `  s )  /\  f : ( s ( a  e. 
_V ,  i  e. 
NN0  |->  { b  e. 
~P a  |  (
# `  b )  =  i } ) M ) --> R ) )  ->  ( F `  C )  =  0 )
12 0le0 9827 . . . . 5  |-  0  <_  0
1311, 12syl6eqbr 4060 . . . 4  |-  ( ( ( ( M  e.  NN  /\  R  e.  V  /\  F : R
--> NN0 )  /\  ( C  e.  R  /\  ( F `  C )  =  0 ) )  /\  ( 0  <_ 
( # `  s )  /\  f : ( s ( a  e. 
_V ,  i  e. 
NN0  |->  { b  e. 
~P a  |  (
# `  b )  =  i } ) M ) --> R ) )  ->  ( F `  C )  <_  0
)
14 simpll1 994 . . . . . 6  |-  ( ( ( ( M  e.  NN  /\  R  e.  V  /\  F : R
--> NN0 )  /\  ( C  e.  R  /\  ( F `  C )  =  0 ) )  /\  ( 0  <_ 
( # `  s )  /\  f : ( s ( a  e. 
_V ,  i  e. 
NN0  |->  { b  e. 
~P a  |  (
# `  b )  =  i } ) M ) --> R ) )  ->  M  e.  NN )
1510hashbc 13054 . . . . . 6  |-  ( M  e.  NN  ->  ( (/) ( a  e.  _V ,  i  e.  NN0  |->  { b  e.  ~P a  |  ( # `  b
)  =  i } ) M )  =  (/) )
1614, 15syl 15 . . . . 5  |-  ( ( ( ( M  e.  NN  /\  R  e.  V  /\  F : R
--> NN0 )  /\  ( C  e.  R  /\  ( F `  C )  =  0 ) )  /\  ( 0  <_ 
( # `  s )  /\  f : ( s ( a  e. 
_V ,  i  e. 
NN0  |->  { b  e. 
~P a  |  (
# `  b )  =  i } ) M ) --> R ) )  ->  ( (/) ( a  e.  _V ,  i  e.  NN0  |->  { b  e.  ~P a  |  ( # `  b
)  =  i } ) M )  =  (/) )
17 0ss 3483 . . . . . 6  |-  (/)  C_  ( `' f " { C } )
1817a1i 10 . . . . 5  |-  ( ( ( ( M  e.  NN  /\  R  e.  V  /\  F : R
--> NN0 )  /\  ( C  e.  R  /\  ( F `  C )  =  0 ) )  /\  ( 0  <_ 
( # `  s )  /\  f : ( s ( a  e. 
_V ,  i  e. 
NN0  |->  { b  e. 
~P a  |  (
# `  b )  =  i } ) M ) --> R ) )  ->  (/)  C_  ( `' f " { C } ) )
1916, 18eqsstrd 3212 . . . 4  |-  ( ( ( ( M  e.  NN  /\  R  e.  V  /\  F : R
--> NN0 )  /\  ( C  e.  R  /\  ( F `  C )  =  0 ) )  /\  ( 0  <_ 
( # `  s )  /\  f : ( s ( a  e. 
_V ,  i  e. 
NN0  |->  { b  e. 
~P a  |  (
# `  b )  =  i } ) M ) --> R ) )  ->  ( (/) ( a  e.  _V ,  i  e.  NN0  |->  { b  e.  ~P a  |  ( # `  b
)  =  i } ) M )  C_  ( `' f " { C } ) )
20 fveq2 5525 . . . . . . 7  |-  ( c  =  C  ->  ( F `  c )  =  ( F `  C ) )
2120breq1d 4033 . . . . . 6  |-  ( c  =  C  ->  (
( F `  c
)  <_  ( # `  x
)  <->  ( F `  C )  <_  ( # `
 x ) ) )
22 sneq 3651 . . . . . . . 8  |-  ( c  =  C  ->  { c }  =  { C } )
2322imaeq2d 5012 . . . . . . 7  |-  ( c  =  C  ->  ( `' f " {
c } )  =  ( `' f " { C } ) )
2423sseq2d 3206 . . . . . 6  |-  ( c  =  C  ->  (
( x ( a  e.  _V ,  i  e.  NN0  |->  { b  e.  ~P a  |  ( # `  b
)  =  i } ) M )  C_  ( `' f " {
c } )  <->  ( x
( a  e.  _V ,  i  e.  NN0  |->  { b  e.  ~P a  |  ( # `  b
)  =  i } ) M )  C_  ( `' f " { C } ) ) )
2521, 24anbi12d 691 . . . . 5  |-  ( c  =  C  ->  (
( ( F `  c )  <_  ( # `
 x )  /\  ( x ( a  e.  _V ,  i  e.  NN0  |->  { b  e.  ~P a  |  ( # `  b
)  =  i } ) M )  C_  ( `' f " {
c } ) )  <-> 
( ( F `  C )  <_  ( # `
 x )  /\  ( x ( a  e.  _V ,  i  e.  NN0  |->  { b  e.  ~P a  |  ( # `  b
)  =  i } ) M )  C_  ( `' f " { C } ) ) ) )
26 fveq2 5525 . . . . . . . 8  |-  ( x  =  (/)  ->  ( # `  x )  =  (
# `  (/) ) )
27 hash0 11355 . . . . . . . 8  |-  ( # `  (/) )  =  0
2826, 27syl6eq 2331 . . . . . . 7  |-  ( x  =  (/)  ->  ( # `  x )  =  0 )
2928breq2d 4035 . . . . . 6  |-  ( x  =  (/)  ->  ( ( F `  C )  <_  ( # `  x
)  <->  ( F `  C )  <_  0
) )
30 oveq1 5865 . . . . . . 7  |-  ( x  =  (/)  ->  ( x ( a  e.  _V ,  i  e.  NN0  |->  { b  e.  ~P a  |  ( # `  b
)  =  i } ) M )  =  ( (/) ( a  e. 
_V ,  i  e. 
NN0  |->  { b  e. 
~P a  |  (
# `  b )  =  i } ) M ) )
3130sseq1d 3205 . . . . . 6  |-  ( x  =  (/)  ->  ( ( x ( a  e. 
_V ,  i  e. 
NN0  |->  { b  e. 
~P a  |  (
# `  b )  =  i } ) M )  C_  ( `' f " { C } )  <->  ( (/) ( a  e.  _V ,  i  e.  NN0  |->  { b  e.  ~P a  |  ( # `  b
)  =  i } ) M )  C_  ( `' f " { C } ) ) )
3229, 31anbi12d 691 . . . . 5  |-  ( x  =  (/)  ->  ( ( ( F `  C
)  <_  ( # `  x
)  /\  ( x
( a  e.  _V ,  i  e.  NN0  |->  { b  e.  ~P a  |  ( # `  b
)  =  i } ) M )  C_  ( `' f " { C } ) )  <->  ( ( F `  C )  <_  0  /\  ( (/) ( a  e.  _V ,  i  e.  NN0  |->  { b  e.  ~P a  |  ( # `  b
)  =  i } ) M )  C_  ( `' f " { C } ) ) ) )
3325, 32rspc2ev 2892 . . . 4  |-  ( ( C  e.  R  /\  (/) 
e.  ~P s  /\  (
( F `  C
)  <_  0  /\  ( (/) ( a  e. 
_V ,  i  e. 
NN0  |->  { b  e. 
~P a  |  (
# `  b )  =  i } ) M )  C_  ( `' f " { C } ) ) )  ->  E. c  e.  R  E. x  e.  ~P  s ( ( F `
 c )  <_ 
( # `  x )  /\  ( x ( a  e.  _V , 
i  e.  NN0  |->  { b  e.  ~P a  |  ( # `  b
)  =  i } ) M )  C_  ( `' f " {
c } ) ) )
348, 10, 13, 19, 33syl112anc 1186 . . 3  |-  ( ( ( ( M  e.  NN  /\  R  e.  V  /\  F : R
--> NN0 )  /\  ( C  e.  R  /\  ( F `  C )  =  0 ) )  /\  ( 0  <_ 
( # `  s )  /\  f : ( s ( a  e. 
_V ,  i  e. 
NN0  |->  { b  e. 
~P a  |  (
# `  b )  =  i } ) M ) --> R ) )  ->  E. c  e.  R  E. x  e.  ~P  s ( ( F `  c )  <_  ( # `  x
)  /\  ( x
( a  e.  _V ,  i  e.  NN0  |->  { b  e.  ~P a  |  ( # `  b
)  =  i } ) M )  C_  ( `' f " {
c } ) ) )
351, 3, 4, 5, 7, 34ramub 13060 . 2  |-  ( ( ( M  e.  NN  /\  R  e.  V  /\  F : R --> NN0 )  /\  ( C  e.  R  /\  ( F `  C
)  =  0 ) )  ->  ( M Ramsey  F )  <_  0 )
36 ramubcl 13065 . . . 4  |-  ( ( ( M  e.  NN0  /\  R  e.  V  /\  F : R --> NN0 )  /\  ( 0  e.  NN0  /\  ( M Ramsey  F )  <_  0 ) )  ->  ( M Ramsey  F
)  e.  NN0 )
373, 4, 5, 7, 35, 36syl32anc 1190 . . 3  |-  ( ( ( M  e.  NN  /\  R  e.  V  /\  F : R --> NN0 )  /\  ( C  e.  R  /\  ( F `  C
)  =  0 ) )  ->  ( M Ramsey  F )  e.  NN0 )
38 nn0le0eq0 9994 . . 3  |-  ( ( M Ramsey  F )  e. 
NN0  ->  ( ( M Ramsey  F )  <_  0  <->  ( M Ramsey  F )  =  0 ) )
3937, 38syl 15 . 2  |-  ( ( ( M  e.  NN  /\  R  e.  V  /\  F : R --> NN0 )  /\  ( C  e.  R  /\  ( F `  C
)  =  0 ) )  ->  ( ( M Ramsey  F )  <_  0  <->  ( M Ramsey  F )  =  0 ) )
4035, 39mpbid 201 1  |-  ( ( ( M  e.  NN  /\  R  e.  V  /\  F : R --> NN0 )  /\  ( C  e.  R  /\  ( F `  C
)  =  0 ) )  ->  ( M Ramsey  F )  =  0 )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   E.wrex 2544   {crab 2547   _Vcvv 2788    C_ wss 3152   (/)c0 3455   ~Pcpw 3625   {csn 3640   class class class wbr 4023   `'ccnv 4688   "cima 4692   -->wf 5251   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   0cc0 8737    <_ cle 8868   NNcn 9746   NN0cn0 9965   #chash 11337   Ramsey cram 13046
This theorem is referenced by:  ramz  13072  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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