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Theorem ramz2 14626
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 2454 . . 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 997 . . . 4  |-  ( ( ( M  e.  NN  /\  R  e.  V  /\  F : R --> NN0 )  /\  ( C  e.  R  /\  ( F `  C
)  =  0 ) )  ->  M  e.  NN )
32nnnn0d 10848 . . 3  |-  ( ( ( M  e.  NN  /\  R  e.  V  /\  F : R --> NN0 )  /\  ( C  e.  R  /\  ( F `  C
)  =  0 ) )  ->  M  e.  NN0 )
4 simpl2 998 . . 3  |-  ( ( ( M  e.  NN  /\  R  e.  V  /\  F : R --> NN0 )  /\  ( C  e.  R  /\  ( F `  C
)  =  0 ) )  ->  R  e.  V )
5 simpl3 999 . . 3  |-  ( ( ( M  e.  NN  /\  R  e.  V  /\  F : R --> NN0 )  /\  ( C  e.  R  /\  ( F `  C
)  =  0 ) )  ->  F : R
--> NN0 )
6 0nn0 10806 . . . 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 4606 . . . . 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 10621 . . . . 5  |-  0  <_  0
1311, 12syl6eqbr 4476 . . . 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 1033 . . . . . 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 14609 . . . . . 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 3813 . . . . 5  |-  (/)  C_  ( `' f " { C } )
1816, 17syl6eqss 3539 . . . 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 5848 . . . . . . 7  |-  ( c  =  C  ->  ( F `  c )  =  ( F `  C ) )
2019breq1d 4449 . . . . . 6  |-  ( c  =  C  ->  (
( F `  c
)  <_  ( # `  x
)  <->  ( F `  C )  <_  ( # `
 x ) ) )
21 sneq 4026 . . . . . . . 8  |-  ( c  =  C  ->  { c }  =  { C } )
2221imaeq2d 5325 . . . . . . 7  |-  ( c  =  C  ->  ( `' f " {
c } )  =  ( `' f " { C } ) )
2322sseq2d 3517 . . . . . 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 708 . . . . 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 5848 . . . . . . . 8  |-  ( x  =  (/)  ->  ( # `  x )  =  (
# `  (/) ) )
26 hash0 12420 . . . . . . . 8  |-  ( # `  (/) )  =  0
2725, 26syl6eq 2511 . . . . . . 7  |-  ( x  =  (/)  ->  ( # `  x )  =  0 )
2827breq2d 4451 . . . . . 6  |-  ( x  =  (/)  ->  ( ( F `  C )  <_  ( # `  x
)  <->  ( F `  C )  <_  0
) )
29 oveq1 6277 . . . . . . 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 3516 . . . . . 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 708 . . . . 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 3218 . . . 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 1230 . . 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 14615 . 2  |-  ( ( ( M  e.  NN  /\  R  e.  V  /\  F : R --> NN0 )  /\  ( C  e.  R  /\  ( F `  C
)  =  0 ) )  ->  ( M Ramsey  F )  <_  0 )
35 ramubcl 14620 . . . 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 1234 . . 3  |-  ( ( ( M  e.  NN  /\  R  e.  V  /\  F : R --> NN0 )  /\  ( C  e.  R  /\  ( F `  C
)  =  0 ) )  ->  ( M Ramsey  F )  e.  NN0 )
37 nn0le0eq0 10820 . . 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 367    /\ w3a 971    = wceq 1398    e. wcel 1823   E.wrex 2805   {crab 2808   _Vcvv 3106    C_ wss 3461   (/)c0 3783   ~Pcpw 3999   {csn 4016   class class class wbr 4439   `'ccnv 4987   "cima 4991   -->wf 5566   ` cfv 5570  (class class class)co 6270    |-> cmpt2 6272   0cc0 9481    <_ cle 9618   NNcn 10531   NN0cn0 10791   #chash 12387   Ramsey cram 14601
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 12090  df-fac 12336  df-bc 12363  df-hash 12388  df-ram 14603
This theorem is referenced by:  ramz  14627  ramcl  14631
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