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Theorem ramz2 13087
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 2296 . . 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 10034 . . 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 9996 . . . 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 4196 . . . . 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 9843 . . . . 5  |-  0  <_  0
1311, 12syl6eqbr 4076 . . . 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 13070 . . . . . 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 3496 . . . . . 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 3225 . . . 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 5541 . . . . . . 7  |-  ( c  =  C  ->  ( F `  c )  =  ( F `  C ) )
2120breq1d 4049 . . . . . 6  |-  ( c  =  C  ->  (
( F `  c
)  <_  ( # `  x
)  <->  ( F `  C )  <_  ( # `
 x ) ) )
22 sneq 3664 . . . . . . . 8  |-  ( c  =  C  ->  { c }  =  { C } )
2322imaeq2d 5028 . . . . . . 7  |-  ( c  =  C  ->  ( `' f " {
c } )  =  ( `' f " { C } ) )
2423sseq2d 3219 . . . . . 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 5541 . . . . . . . 8  |-  ( x  =  (/)  ->  ( # `  x )  =  (
# `  (/) ) )
27 hash0 11371 . . . . . . . 8  |-  ( # `  (/) )  =  0
2826, 27syl6eq 2344 . . . . . . 7  |-  ( x  =  (/)  ->  ( # `  x )  =  0 )
2928breq2d 4051 . . . . . 6  |-  ( x  =  (/)  ->  ( ( F `  C )  <_  ( # `  x
)  <->  ( F `  C )  <_  0
) )
30 oveq1 5881 . . . . . . 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 3218 . . . . . 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 2905 . . . 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 13076 . 2  |-  ( ( ( M  e.  NN  /\  R  e.  V  /\  F : R --> NN0 )  /\  ( C  e.  R  /\  ( F `  C
)  =  0 ) )  ->  ( M Ramsey  F )  <_  0 )
36 ramubcl 13081 . . . 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 10010 . . 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 1632    e. wcel 1696   E.wrex 2557   {crab 2560   _Vcvv 2801    C_ wss 3165   (/)c0 3468   ~Pcpw 3638   {csn 3653   class class class wbr 4039   `'ccnv 4704   "cima 4708   -->wf 5267   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876   0cc0 8753    <_ cle 8884   NNcn 9762   NN0cn0 9981   #chash 11353   Ramsey cram 13062
This theorem is referenced by:  ramz  13088  ramcl  13092
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-card 7588  df-cda 7810  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-n0 9982  df-z 10041  df-uz 10247  df-rp 10371  df-fz 10799  df-seq 11063  df-fac 11305  df-bc 11332  df-hash 11354  df-ram 13064
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