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Theorem ramlb 13066
Description: Establish a lower bound on a Ramsey number. (Contributed by Mario Carneiro, 22-Apr-2015.)
Hypotheses
Ref Expression
ramlb.c  |-  C  =  ( a  e.  _V ,  i  e.  NN0  |->  { b  e.  ~P a  |  ( # `  b
)  =  i } )
ramlb.m  |-  ( ph  ->  M  e.  NN0 )
ramlb.r  |-  ( ph  ->  R  e.  V )
ramlb.f  |-  ( ph  ->  F : R --> NN0 )
ramlb.s  |-  ( ph  ->  N  e.  NN0 )
ramlb.g  |-  ( ph  ->  G : ( ( 1 ... N ) C M ) --> R )
ramlb.i  |-  ( (
ph  /\  ( c  e.  R  /\  x  C_  ( 1 ... N
) ) )  -> 
( ( x C M )  C_  ( `' G " { c } )  ->  ( # `
 x )  < 
( F `  c
) ) )
Assertion
Ref Expression
ramlb  |-  ( ph  ->  N  <  ( M Ramsey  F ) )
Distinct variable groups:    x, c, C    F, c, x    G, c, x    a, b, c, i, x, M    ph, c, x    N, c, x    R, c, x    V, c, x
Allowed substitution hints:    ph( i, a, b)    C( i, a, b)    R( i, a, b)    F( i, a, b)    G( i, a, b)    N( i, a, b)    V( i, a, b)

Proof of Theorem ramlb
StepHypRef Expression
1 simpr 447 . . 3  |-  ( (
ph  /\  ( M Ramsey  F )  <_  N )  ->  ( M Ramsey  F )  <_  N )
2 ramlb.c . . . . 5  |-  C  =  ( a  e.  _V ,  i  e.  NN0  |->  { b  e.  ~P a  |  ( # `  b
)  =  i } )
3 ramlb.m . . . . . 6  |-  ( ph  ->  M  e.  NN0 )
43adantr 451 . . . . 5  |-  ( (
ph  /\  ( M Ramsey  F )  <_  N )  ->  M  e.  NN0 )
5 ramlb.r . . . . . 6  |-  ( ph  ->  R  e.  V )
65adantr 451 . . . . 5  |-  ( (
ph  /\  ( M Ramsey  F )  <_  N )  ->  R  e.  V )
7 ramlb.f . . . . . 6  |-  ( ph  ->  F : R --> NN0 )
87adantr 451 . . . . 5  |-  ( (
ph  /\  ( M Ramsey  F )  <_  N )  ->  F : R --> NN0 )
9 ramlb.s . . . . . . 7  |-  ( ph  ->  N  e.  NN0 )
109adantr 451 . . . . . 6  |-  ( (
ph  /\  ( M Ramsey  F )  <_  N )  ->  N  e.  NN0 )
11 ramubcl 13065 . . . . . 6  |-  ( ( ( M  e.  NN0  /\  R  e.  V  /\  F : R --> NN0 )  /\  ( N  e.  NN0  /\  ( M Ramsey  F )  <_  N ) )  ->  ( M Ramsey  F
)  e.  NN0 )
124, 6, 8, 10, 1, 11syl32anc 1190 . . . . 5  |-  ( (
ph  /\  ( M Ramsey  F )  <_  N )  ->  ( M Ramsey  F )  e.  NN0 )
13 fzfid 11035 . . . . 5  |-  ( (
ph  /\  ( M Ramsey  F )  <_  N )  ->  ( 1 ... N
)  e.  Fin )
14 hashfz1 11345 . . . . . . . 8  |-  ( N  e.  NN0  ->  ( # `  ( 1 ... N
) )  =  N )
159, 14syl 15 . . . . . . 7  |-  ( ph  ->  ( # `  (
1 ... N ) )  =  N )
1615breq2d 4035 . . . . . 6  |-  ( ph  ->  ( ( M Ramsey  F
)  <_  ( # `  (
1 ... N ) )  <-> 
( M Ramsey  F )  <_  N ) )
1716biimpar 471 . . . . 5  |-  ( (
ph  /\  ( M Ramsey  F )  <_  N )  ->  ( M Ramsey  F )  <_  ( # `  (
1 ... N ) ) )
18 ramlb.g . . . . . 6  |-  ( ph  ->  G : ( ( 1 ... N ) C M ) --> R )
1918adantr 451 . . . . 5  |-  ( (
ph  /\  ( M Ramsey  F )  <_  N )  ->  G : ( ( 1 ... N ) C M ) --> R )
202, 4, 6, 8, 12, 13, 17, 19rami 13062 . . . 4  |-  ( (
ph  /\  ( M Ramsey  F )  <_  N )  ->  E. c  e.  R  E. x  e.  ~P  ( 1 ... N
) ( ( F `
 c )  <_ 
( # `  x )  /\  ( x C M )  C_  ( `' G " { c } ) ) )
21 elpwi 3633 . . . . . . . . 9  |-  ( x  e.  ~P ( 1 ... N )  ->  x  C_  ( 1 ... N ) )
22 ramlb.i . . . . . . . . . . 11  |-  ( (
ph  /\  ( c  e.  R  /\  x  C_  ( 1 ... N
) ) )  -> 
( ( x C M )  C_  ( `' G " { c } )  ->  ( # `
 x )  < 
( F `  c
) ) )
2322adantlr 695 . . . . . . . . . 10  |-  ( ( ( ph  /\  ( M Ramsey  F )  <_  N
)  /\  ( c  e.  R  /\  x  C_  ( 1 ... N
) ) )  -> 
( ( x C M )  C_  ( `' G " { c } )  ->  ( # `
 x )  < 
( F `  c
) ) )
24 fzfid 11035 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  ( M Ramsey  F )  <_  N
)  /\  ( c  e.  R  /\  x  C_  ( 1 ... N
) ) )  -> 
( 1 ... N
)  e.  Fin )
25 simprr 733 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  ( M Ramsey  F )  <_  N
)  /\  ( c  e.  R  /\  x  C_  ( 1 ... N
) ) )  ->  x  C_  ( 1 ... N ) )
26 ssfi 7083 . . . . . . . . . . . . . 14  |-  ( ( ( 1 ... N
)  e.  Fin  /\  x  C_  ( 1 ... N ) )  ->  x  e.  Fin )
2724, 25, 26syl2anc 642 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  ( M Ramsey  F )  <_  N
)  /\  ( c  e.  R  /\  x  C_  ( 1 ... N
) ) )  ->  x  e.  Fin )
28 hashcl 11350 . . . . . . . . . . . . 13  |-  ( x  e.  Fin  ->  ( # `
 x )  e. 
NN0 )
2927, 28syl 15 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  ( M Ramsey  F )  <_  N
)  /\  ( c  e.  R  /\  x  C_  ( 1 ... N
) ) )  -> 
( # `  x )  e.  NN0 )
3029nn0red 10019 . . . . . . . . . . 11  |-  ( ( ( ph  /\  ( M Ramsey  F )  <_  N
)  /\  ( c  e.  R  /\  x  C_  ( 1 ... N
) ) )  -> 
( # `  x )  e.  RR )
31 simpl 443 . . . . . . . . . . . . 13  |-  ( ( c  e.  R  /\  x  C_  ( 1 ... N ) )  -> 
c  e.  R )
32 ffvelrn 5663 . . . . . . . . . . . . 13  |-  ( ( F : R --> NN0  /\  c  e.  R )  ->  ( F `  c
)  e.  NN0 )
338, 31, 32syl2an 463 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  ( M Ramsey  F )  <_  N
)  /\  ( c  e.  R  /\  x  C_  ( 1 ... N
) ) )  -> 
( F `  c
)  e.  NN0 )
3433nn0red 10019 . . . . . . . . . . 11  |-  ( ( ( ph  /\  ( M Ramsey  F )  <_  N
)  /\  ( c  e.  R  /\  x  C_  ( 1 ... N
) ) )  -> 
( F `  c
)  e.  RR )
3530, 34ltnled 8966 . . . . . . . . . 10  |-  ( ( ( ph  /\  ( M Ramsey  F )  <_  N
)  /\  ( c  e.  R  /\  x  C_  ( 1 ... N
) ) )  -> 
( ( # `  x
)  <  ( F `  c )  <->  -.  ( F `  c )  <_  ( # `  x
) ) )
3623, 35sylibd 205 . . . . . . . . 9  |-  ( ( ( ph  /\  ( M Ramsey  F )  <_  N
)  /\  ( c  e.  R  /\  x  C_  ( 1 ... N
) ) )  -> 
( ( x C M )  C_  ( `' G " { c } )  ->  -.  ( F `  c )  <_  ( # `  x
) ) )
3721, 36sylanr2 634 . . . . . . . 8  |-  ( ( ( ph  /\  ( M Ramsey  F )  <_  N
)  /\  ( c  e.  R  /\  x  e.  ~P ( 1 ... N ) ) )  ->  ( ( x C M )  C_  ( `' G " { c } )  ->  -.  ( F `  c )  <_  ( # `  x
) ) )
3837con2d 107 . . . . . . 7  |-  ( ( ( ph  /\  ( M Ramsey  F )  <_  N
)  /\  ( c  e.  R  /\  x  e.  ~P ( 1 ... N ) ) )  ->  ( ( F `
 c )  <_ 
( # `  x )  ->  -.  ( x C M )  C_  ( `' G " { c } ) ) )
39 imnan 411 . . . . . . 7  |-  ( ( ( F `  c
)  <_  ( # `  x
)  ->  -.  (
x C M ) 
C_  ( `' G " { c } ) )  <->  -.  ( ( F `  c )  <_  ( # `  x
)  /\  ( x C M )  C_  ( `' G " { c } ) ) )
4038, 39sylib 188 . . . . . 6  |-  ( ( ( ph  /\  ( M Ramsey  F )  <_  N
)  /\  ( c  e.  R  /\  x  e.  ~P ( 1 ... N ) ) )  ->  -.  ( ( F `  c )  <_  ( # `  x
)  /\  ( x C M )  C_  ( `' G " { c } ) ) )
4140pm2.21d 98 . . . . 5  |-  ( ( ( ph  /\  ( M Ramsey  F )  <_  N
)  /\  ( c  e.  R  /\  x  e.  ~P ( 1 ... N ) ) )  ->  ( ( ( F `  c )  <_  ( # `  x
)  /\  ( x C M )  C_  ( `' G " { c } ) )  ->  -.  ( M Ramsey  F )  <_  N ) )
4241rexlimdvva 2674 . . . 4  |-  ( (
ph  /\  ( M Ramsey  F )  <_  N )  ->  ( E. c  e.  R  E. x  e. 
~P  ( 1 ... N ) ( ( F `  c )  <_  ( # `  x
)  /\  ( x C M )  C_  ( `' G " { c } ) )  ->  -.  ( M Ramsey  F )  <_  N ) )
4320, 42mpd 14 . . 3  |-  ( (
ph  /\  ( M Ramsey  F )  <_  N )  ->  -.  ( M Ramsey  F
)  <_  N )
441, 43pm2.65da 559 . 2  |-  ( ph  ->  -.  ( M Ramsey  F
)  <_  N )
459nn0red 10019 . . . 4  |-  ( ph  ->  N  e.  RR )
4645rexrd 8881 . . 3  |-  ( ph  ->  N  e.  RR* )
47 ramxrcl 13064 . . . 4  |-  ( ( M  e.  NN0  /\  R  e.  V  /\  F : R --> NN0 )  ->  ( M Ramsey  F )  e.  RR* )
483, 5, 7, 47syl3anc 1182 . . 3  |-  ( ph  ->  ( M Ramsey  F )  e.  RR* )
49 xrltnle 8891 . . 3  |-  ( ( N  e.  RR*  /\  ( M Ramsey  F )  e.  RR* )  ->  ( N  < 
( M Ramsey  F )  <->  -.  ( M Ramsey  F )  <_  N ) )
5046, 48, 49syl2anc 642 . 2  |-  ( ph  ->  ( N  <  ( M Ramsey  F )  <->  -.  ( M Ramsey  F )  <_  N
) )
5144, 50mpbird 223 1  |-  ( ph  ->  N  <  ( M Ramsey  F ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   E.wrex 2544   {crab 2547   _Vcvv 2788    C_ wss 3152   ~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   Fincfn 6863   1c1 8738   RR*cxr 8866    < clt 8867    <_ cle 8868   NN0cn0 9965   ...cfz 10782   #chash 11337   Ramsey cram 13046
This theorem is referenced by:  0ram  13067  ram0  13069
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-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-n0 9966  df-z 10025  df-uz 10231  df-fz 10783  df-hash 11338  df-ram 13048
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