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Theorem ramlb 14076
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 ramlb.c . . . . 5  |-  C  =  ( a  e.  _V ,  i  e.  NN0  |->  { b  e.  ~P a  |  ( # `  b
)  =  i } )
2 ramlb.m . . . . . 6  |-  ( ph  ->  M  e.  NN0 )
32adantr 462 . . . . 5  |-  ( (
ph  /\  ( M Ramsey  F )  <_  N )  ->  M  e.  NN0 )
4 ramlb.r . . . . . 6  |-  ( ph  ->  R  e.  V )
54adantr 462 . . . . 5  |-  ( (
ph  /\  ( M Ramsey  F )  <_  N )  ->  R  e.  V )
6 ramlb.f . . . . . 6  |-  ( ph  ->  F : R --> NN0 )
76adantr 462 . . . . 5  |-  ( (
ph  /\  ( M Ramsey  F )  <_  N )  ->  F : R --> NN0 )
8 ramlb.s . . . . . . 7  |-  ( ph  ->  N  e.  NN0 )
98adantr 462 . . . . . 6  |-  ( (
ph  /\  ( M Ramsey  F )  <_  N )  ->  N  e.  NN0 )
10 simpr 458 . . . . . 6  |-  ( (
ph  /\  ( M Ramsey  F )  <_  N )  ->  ( M Ramsey  F )  <_  N )
11 ramubcl 14075 . . . . . 6  |-  ( ( ( M  e.  NN0  /\  R  e.  V  /\  F : R --> NN0 )  /\  ( N  e.  NN0  /\  ( M Ramsey  F )  <_  N ) )  ->  ( M Ramsey  F
)  e.  NN0 )
123, 5, 7, 9, 10, 11syl32anc 1221 . . . . 5  |-  ( (
ph  /\  ( M Ramsey  F )  <_  N )  ->  ( M Ramsey  F )  e.  NN0 )
13 fzfid 11791 . . . . 5  |-  ( (
ph  /\  ( M Ramsey  F )  <_  N )  ->  ( 1 ... N
)  e.  Fin )
14 hashfz1 12113 . . . . . . . 8  |-  ( N  e.  NN0  ->  ( # `  ( 1 ... N
) )  =  N )
158, 14syl 16 . . . . . . 7  |-  ( ph  ->  ( # `  (
1 ... N ) )  =  N )
1615breq2d 4301 . . . . . 6  |-  ( ph  ->  ( ( M Ramsey  F
)  <_  ( # `  (
1 ... N ) )  <-> 
( M Ramsey  F )  <_  N ) )
1716biimpar 482 . . . . 5  |-  ( (
ph  /\  ( M Ramsey  F )  <_  N )  ->  ( M Ramsey  F )  <_  ( # `  (
1 ... N ) ) )
18 ramlb.g . . . . . 6  |-  ( ph  ->  G : ( ( 1 ... N ) C M ) --> R )
1918adantr 462 . . . . 5  |-  ( (
ph  /\  ( M Ramsey  F )  <_  N )  ->  G : ( ( 1 ... N ) C M ) --> R )
201, 3, 5, 7, 12, 13, 17, 19rami 14072 . . . 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 3866 . . . . . . . . 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 709 . . . . . . . . . 10  |-  ( ( ( ph  /\  ( M Ramsey  F )  <_  N
)  /\  ( c  e.  R  /\  x  C_  ( 1 ... N
) ) )  -> 
( ( x C M )  C_  ( `' G " { c } )  ->  ( # `
 x )  < 
( F `  c
) ) )
24 fzfid 11791 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  ( M Ramsey  F )  <_  N
)  /\  ( c  e.  R  /\  x  C_  ( 1 ... N
) ) )  -> 
( 1 ... N
)  e.  Fin )
25 simprr 751 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  ( M Ramsey  F )  <_  N
)  /\  ( c  e.  R  /\  x  C_  ( 1 ... N
) ) )  ->  x  C_  ( 1 ... N ) )
26 ssfi 7529 . . . . . . . . . . . . . 14  |-  ( ( ( 1 ... N
)  e.  Fin  /\  x  C_  ( 1 ... N ) )  ->  x  e.  Fin )
2724, 25, 26syl2anc 656 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  ( M Ramsey  F )  <_  N
)  /\  ( c  e.  R  /\  x  C_  ( 1 ... N
) ) )  ->  x  e.  Fin )
28 hashcl 12122 . . . . . . . . . . . . 13  |-  ( x  e.  Fin  ->  ( # `
 x )  e. 
NN0 )
2927, 28syl 16 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  ( M Ramsey  F )  <_  N
)  /\  ( c  e.  R  /\  x  C_  ( 1 ... N
) ) )  -> 
( # `  x )  e.  NN0 )
3029nn0red 10633 . . . . . . . . . . 11  |-  ( ( ( ph  /\  ( M Ramsey  F )  <_  N
)  /\  ( c  e.  R  /\  x  C_  ( 1 ... N
) ) )  -> 
( # `  x )  e.  RR )
31 simpl 454 . . . . . . . . . . . . 13  |-  ( ( c  e.  R  /\  x  C_  ( 1 ... N ) )  -> 
c  e.  R )
32 ffvelrn 5838 . . . . . . . . . . . . 13  |-  ( ( F : R --> NN0  /\  c  e.  R )  ->  ( F `  c
)  e.  NN0 )
337, 31, 32syl2an 474 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  ( M Ramsey  F )  <_  N
)  /\  ( c  e.  R  /\  x  C_  ( 1 ... N
) ) )  -> 
( F `  c
)  e.  NN0 )
3433nn0red 10633 . . . . . . . . . . 11  |-  ( ( ( ph  /\  ( M Ramsey  F )  <_  N
)  /\  ( c  e.  R  /\  x  C_  ( 1 ... N
) ) )  -> 
( F `  c
)  e.  RR )
3530, 34ltnled 9517 . . . . . . . . . 10  |-  ( ( ( ph  /\  ( M Ramsey  F )  <_  N
)  /\  ( c  e.  R  /\  x  C_  ( 1 ... N
) ) )  -> 
( ( # `  x
)  <  ( F `  c )  <->  -.  ( F `  c )  <_  ( # `  x
) ) )
3623, 35sylibd 214 . . . . . . . . 9  |-  ( ( ( ph  /\  ( M Ramsey  F )  <_  N
)  /\  ( c  e.  R  /\  x  C_  ( 1 ... N
) ) )  -> 
( ( x C M )  C_  ( `' G " { c } )  ->  -.  ( F `  c )  <_  ( # `  x
) ) )
3721, 36sylanr2 648 . . . . . . . 8  |-  ( ( ( ph  /\  ( M Ramsey  F )  <_  N
)  /\  ( c  e.  R  /\  x  e.  ~P ( 1 ... N ) ) )  ->  ( ( x C M )  C_  ( `' G " { c } )  ->  -.  ( F `  c )  <_  ( # `  x
) ) )
3837con2d 115 . . . . . . 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 422 . . . . . . 7  |-  ( ( ( F `  c
)  <_  ( # `  x
)  ->  -.  (
x C M ) 
C_  ( `' G " { c } ) )  <->  -.  ( ( F `  c )  <_  ( # `  x
)  /\  ( x C M )  C_  ( `' G " { c } ) ) )
4038, 39sylib 196 . . . . . 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 106 . . . . 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 2846 . . . 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 15 . . 3  |-  ( (
ph  /\  ( M Ramsey  F )  <_  N )  ->  -.  ( M Ramsey  F
)  <_  N )
4443pm2.01da 440 . 2  |-  ( ph  ->  -.  ( M Ramsey  F
)  <_  N )
458nn0red 10633 . . . 4  |-  ( ph  ->  N  e.  RR )
4645rexrd 9429 . . 3  |-  ( ph  ->  N  e.  RR* )
47 ramxrcl 14074 . . . 4  |-  ( ( M  e.  NN0  /\  R  e.  V  /\  F : R --> NN0 )  ->  ( M Ramsey  F )  e.  RR* )
482, 4, 6, 47syl3anc 1213 . . 3  |-  ( ph  ->  ( M Ramsey  F )  e.  RR* )
49 xrltnle 9439 . . 3  |-  ( ( N  e.  RR*  /\  ( M Ramsey  F )  e.  RR* )  ->  ( N  < 
( M Ramsey  F )  <->  -.  ( M Ramsey  F )  <_  N ) )
5046, 48, 49syl2anc 656 . 2  |-  ( ph  ->  ( N  <  ( M Ramsey  F )  <->  -.  ( M Ramsey  F )  <_  N
) )
5144, 50mpbird 232 1  |-  ( ph  ->  N  <  ( M Ramsey  F ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1364    e. wcel 1761   E.wrex 2714   {crab 2717   _Vcvv 2970    C_ wss 3325   ~Pcpw 3857   {csn 3874   class class class wbr 4289   `'ccnv 4835   "cima 4839   -->wf 5411   ` cfv 5415  (class class class)co 6090    e. cmpt2 6092   Fincfn 7306   1c1 9279   RR*cxr 9413    < clt 9414    <_ cle 9415   NN0cn0 10575   ...cfz 11433   #chash 12099   Ramsey cram 14056
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-1o 6916  df-er 7097  df-map 7212  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-sup 7687  df-card 8105  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-nn 10319  df-n0 10576  df-z 10643  df-uz 10858  df-fz 11434  df-hash 12100  df-ram 14058
This theorem is referenced by:  0ram  14077  ram0  14079
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