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Theorem ramz2 14085
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 2443 . . 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 991 . . . 4  |-  ( ( ( M  e.  NN  /\  R  e.  V  /\  F : R --> NN0 )  /\  ( C  e.  R  /\  ( F `  C
)  =  0 ) )  ->  M  e.  NN )
32nnnn0d 10636 . . 3  |-  ( ( ( M  e.  NN  /\  R  e.  V  /\  F : R --> NN0 )  /\  ( C  e.  R  /\  ( F `  C
)  =  0 ) )  ->  M  e.  NN0 )
4 simpl2 992 . . 3  |-  ( ( ( M  e.  NN  /\  R  e.  V  /\  F : R --> NN0 )  /\  ( C  e.  R  /\  ( F `  C
)  =  0 ) )  ->  R  e.  V )
5 simpl3 993 . . 3  |-  ( ( ( M  e.  NN  /\  R  e.  V  /\  F : R --> NN0 )  /\  ( C  e.  R  /\  ( F `  C
)  =  0 ) )  ->  F : R
--> NN0 )
6 0nn0 10594 . . . 4  |-  0  e.  NN0
76a1i 11 . . 3  |-  ( ( ( M  e.  NN  /\  R  e.  V  /\  F : R --> NN0 )  /\  ( C  e.  R  /\  ( F `  C
)  =  0 ) )  ->  0  e.  NN0 )
8 simplrl 759 . . . 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 4461 . . . . 5  |-  (/)  e.  ~P s
109a1i 11 . . . 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 760 . . . . 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 10411 . . . . 5  |-  0  <_  0
1311, 12syl6eqbr 4329 . . . 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 1027 . . . . . 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 14068 . . . . . 6  |-  ( M  e.  NN  ->  ( (/) ( a  e.  _V ,  i  e.  NN0  |->  { b  e.  ~P a  |  ( # `  b
)  =  i } ) M )  =  (/) )
1614, 15syl 16 . . . . 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 3666 . . . . 5  |-  (/)  C_  ( `' f " { C } )
1816, 17syl6eqss 3406 . . . 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 } ) )
19 fveq2 5691 . . . . . . 7  |-  ( c  =  C  ->  ( F `  c )  =  ( F `  C ) )
2019breq1d 4302 . . . . . 6  |-  ( c  =  C  ->  (
( F `  c
)  <_  ( # `  x
)  <->  ( F `  C )  <_  ( # `
 x ) ) )
21 sneq 3887 . . . . . . . 8  |-  ( c  =  C  ->  { c }  =  { C } )
2221imaeq2d 5169 . . . . . . 7  |-  ( c  =  C  ->  ( `' f " {
c } )  =  ( `' f " { C } ) )
2322sseq2d 3384 . . . . . 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 } ) ) )
2420, 23anbi12d 710 . . . . 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 } ) ) ) )
25 fveq2 5691 . . . . . . . 8  |-  ( x  =  (/)  ->  ( # `  x )  =  (
# `  (/) ) )
26 hash0 12135 . . . . . . . 8  |-  ( # `  (/) )  =  0
2725, 26syl6eq 2491 . . . . . . 7  |-  ( x  =  (/)  ->  ( # `  x )  =  0 )
2827breq2d 4304 . . . . . 6  |-  ( x  =  (/)  ->  ( ( F `  C )  <_  ( # `  x
)  <->  ( F `  C )  <_  0
) )
29 oveq1 6098 . . . . . . 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 ) )
3029sseq1d 3383 . . . . . 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 } ) ) )
3128, 30anbi12d 710 . . . . 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 } ) ) ) )
3224, 31rspc2ev 3081 . . . 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 } ) ) )
338, 10, 13, 18, 32syl112anc 1222 . . 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 } ) ) )
341, 3, 4, 5, 7, 33ramub 14074 . 2  |-  ( ( ( M  e.  NN  /\  R  e.  V  /\  F : R --> NN0 )  /\  ( C  e.  R  /\  ( F `  C
)  =  0 ) )  ->  ( M Ramsey  F )  <_  0 )
35 ramubcl 14079 . . . 4  |-  ( ( ( M  e.  NN0  /\  R  e.  V  /\  F : R --> NN0 )  /\  ( 0  e.  NN0  /\  ( M Ramsey  F )  <_  0 ) )  ->  ( M Ramsey  F
)  e.  NN0 )
363, 4, 5, 7, 34, 35syl32anc 1226 . . 3  |-  ( ( ( M  e.  NN  /\  R  e.  V  /\  F : R --> NN0 )  /\  ( C  e.  R  /\  ( F `  C
)  =  0 ) )  ->  ( M Ramsey  F )  e.  NN0 )
37 nn0le0eq0 10608 . . 3  |-  ( ( M Ramsey  F )  e. 
NN0  ->  ( ( M Ramsey  F )  <_  0  <->  ( M Ramsey  F )  =  0 ) )
3836, 37syl 16 . 2  |-  ( ( ( M  e.  NN  /\  R  e.  V  /\  F : R --> NN0 )  /\  ( C  e.  R  /\  ( F `  C
)  =  0 ) )  ->  ( ( M Ramsey  F )  <_  0  <->  ( M Ramsey  F )  =  0 ) )
3934, 38mpbid 210 1  |-  ( ( ( M  e.  NN  /\  R  e.  V  /\  F : R --> NN0 )  /\  ( C  e.  R  /\  ( F `  C
)  =  0 ) )  ->  ( M Ramsey  F )  =  0 )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   E.wrex 2716   {crab 2719   _Vcvv 2972    C_ wss 3328   (/)c0 3637   ~Pcpw 3860   {csn 3877   class class class wbr 4292   `'ccnv 4839   "cima 4843   -->wf 5414   ` cfv 5418  (class class class)co 6091    e. cmpt2 6093   0cc0 9282    <_ cle 9419   NNcn 10322   NN0cn0 10579   #chash 12103   Ramsey cram 14060
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 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359
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 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-1st 6577  df-2nd 6578  df-recs 6832  df-rdg 6866  df-1o 6920  df-2o 6921  df-oadd 6924  df-er 7101  df-map 7216  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-sup 7691  df-card 8109  df-cda 8337  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-div 9994  df-nn 10323  df-n0 10580  df-z 10647  df-uz 10862  df-rp 10992  df-fz 11438  df-seq 11807  df-fac 12052  df-bc 12079  df-hash 12104  df-ram 14062
This theorem is referenced by:  ramz  14086  ramcl  14090
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