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Theorem ramz2 13394
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 2438 . . 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 961 . . . 4  |-  ( ( ( M  e.  NN  /\  R  e.  V  /\  F : R --> NN0 )  /\  ( C  e.  R  /\  ( F `  C
)  =  0 ) )  ->  M  e.  NN )
32nnnn0d 10276 . . 3  |-  ( ( ( M  e.  NN  /\  R  e.  V  /\  F : R --> NN0 )  /\  ( C  e.  R  /\  ( F `  C
)  =  0 ) )  ->  M  e.  NN0 )
4 simpl2 962 . . 3  |-  ( ( ( M  e.  NN  /\  R  e.  V  /\  F : R --> NN0 )  /\  ( C  e.  R  /\  ( F `  C
)  =  0 ) )  ->  R  e.  V )
5 simpl3 963 . . 3  |-  ( ( ( M  e.  NN  /\  R  e.  V  /\  F : R --> NN0 )  /\  ( C  e.  R  /\  ( F `  C
)  =  0 ) )  ->  F : R
--> NN0 )
6 0nn0 10238 . . . 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 738 . . . 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 4371 . . . . 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 739 . . . . 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 10083 . . . . 5  |-  0  <_  0
1311, 12syl6eqbr 4251 . . . 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 997 . . . . . 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 13377 . . . . . 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 3658 . . . . 5  |-  (/)  C_  ( `' f " { C } )
1816, 17syl6eqss 3400 . . . 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 5730 . . . . . . 7  |-  ( c  =  C  ->  ( F `  c )  =  ( F `  C ) )
2019breq1d 4224 . . . . . 6  |-  ( c  =  C  ->  (
( F `  c
)  <_  ( # `  x
)  <->  ( F `  C )  <_  ( # `
 x ) ) )
21 sneq 3827 . . . . . . . 8  |-  ( c  =  C  ->  { c }  =  { C } )
2221imaeq2d 5205 . . . . . . 7  |-  ( c  =  C  ->  ( `' f " {
c } )  =  ( `' f " { C } ) )
2322sseq2d 3378 . . . . . 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 693 . . . . 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 5730 . . . . . . . 8  |-  ( x  =  (/)  ->  ( # `  x )  =  (
# `  (/) ) )
26 hash0 11648 . . . . . . . 8  |-  ( # `  (/) )  =  0
2725, 26syl6eq 2486 . . . . . . 7  |-  ( x  =  (/)  ->  ( # `  x )  =  0 )
2827breq2d 4226 . . . . . 6  |-  ( x  =  (/)  ->  ( ( F `  C )  <_  ( # `  x
)  <->  ( F `  C )  <_  0
) )
29 oveq1 6090 . . . . . . 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 3377 . . . . . 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 693 . . . . 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 3062 . . . 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 1189 . . 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 13383 . 2  |-  ( ( ( M  e.  NN  /\  R  e.  V  /\  F : R --> NN0 )  /\  ( C  e.  R  /\  ( F `  C
)  =  0 ) )  ->  ( M Ramsey  F )  <_  0 )
35 ramubcl 13388 . . . 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 1193 . . 3  |-  ( ( ( M  e.  NN  /\  R  e.  V  /\  F : R --> NN0 )  /\  ( C  e.  R  /\  ( F `  C
)  =  0 ) )  ->  ( M Ramsey  F )  e.  NN0 )
37 nn0le0eq0 10252 . . 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 203 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 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   E.wrex 2708   {crab 2711   _Vcvv 2958    C_ wss 3322   (/)c0 3630   ~Pcpw 3801   {csn 3816   class class class wbr 4214   `'ccnv 4879   "cima 4883   -->wf 5452   ` cfv 5456  (class class class)co 6083    e. cmpt2 6085   0cc0 8992    <_ cle 9123   NNcn 10002   NN0cn0 10223   #chash 11620   Ramsey cram 13369
This theorem is referenced by:  ramz  13395  ramcl  13399
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-1o 6726  df-2o 6727  df-oadd 6730  df-er 6907  df-map 7022  df-en 7112  df-dom 7113  df-sdom 7114  df-fin 7115  df-sup 7448  df-card 7828  df-cda 8050  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-div 9680  df-nn 10003  df-n0 10224  df-z 10285  df-uz 10491  df-rp 10615  df-fz 11046  df-seq 11326  df-fac 11569  df-bc 11596  df-hash 11621  df-ram 13371
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