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Theorem ramlb 13342
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 452 . . . . 5  |-  ( (
ph  /\  ( M Ramsey  F )  <_  N )  ->  M  e.  NN0 )
4 ramlb.r . . . . . 6  |-  ( ph  ->  R  e.  V )
54adantr 452 . . . . 5  |-  ( (
ph  /\  ( M Ramsey  F )  <_  N )  ->  R  e.  V )
6 ramlb.f . . . . . 6  |-  ( ph  ->  F : R --> NN0 )
76adantr 452 . . . . 5  |-  ( (
ph  /\  ( M Ramsey  F )  <_  N )  ->  F : R --> NN0 )
8 ramlb.s . . . . . . 7  |-  ( ph  ->  N  e.  NN0 )
98adantr 452 . . . . . 6  |-  ( (
ph  /\  ( M Ramsey  F )  <_  N )  ->  N  e.  NN0 )
10 simpr 448 . . . . . 6  |-  ( (
ph  /\  ( M Ramsey  F )  <_  N )  ->  ( M Ramsey  F )  <_  N )
11 ramubcl 13341 . . . . . 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 1192 . . . . 5  |-  ( (
ph  /\  ( M Ramsey  F )  <_  N )  ->  ( M Ramsey  F )  e.  NN0 )
13 fzfid 11267 . . . . 5  |-  ( (
ph  /\  ( M Ramsey  F )  <_  N )  ->  ( 1 ... N
)  e.  Fin )
14 hashfz1 11585 . . . . . . . 8  |-  ( N  e.  NN0  ->  ( # `  ( 1 ... N
) )  =  N )
158, 14syl 16 . . . . . . 7  |-  ( ph  ->  ( # `  (
1 ... N ) )  =  N )
1615breq2d 4184 . . . . . 6  |-  ( ph  ->  ( ( M Ramsey  F
)  <_  ( # `  (
1 ... N ) )  <-> 
( M Ramsey  F )  <_  N ) )
1716biimpar 472 . . . . 5  |-  ( (
ph  /\  ( M Ramsey  F )  <_  N )  ->  ( M Ramsey  F )  <_  ( # `  (
1 ... N ) ) )
18 ramlb.g . . . . . 6  |-  ( ph  ->  G : ( ( 1 ... N ) C M ) --> R )
1918adantr 452 . . . . 5  |-  ( (
ph  /\  ( M Ramsey  F )  <_  N )  ->  G : ( ( 1 ... N ) C M ) --> R )
201, 3, 5, 7, 12, 13, 17, 19rami 13338 . . . 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 3767 . . . . . . . . 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 696 . . . . . . . . . 10  |-  ( ( ( ph  /\  ( M Ramsey  F )  <_  N
)  /\  ( c  e.  R  /\  x  C_  ( 1 ... N
) ) )  -> 
( ( x C M )  C_  ( `' G " { c } )  ->  ( # `
 x )  < 
( F `  c
) ) )
24 fzfid 11267 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  ( M Ramsey  F )  <_  N
)  /\  ( c  e.  R  /\  x  C_  ( 1 ... N
) ) )  -> 
( 1 ... N
)  e.  Fin )
25 simprr 734 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  ( M Ramsey  F )  <_  N
)  /\  ( c  e.  R  /\  x  C_  ( 1 ... N
) ) )  ->  x  C_  ( 1 ... N ) )
26 ssfi 7288 . . . . . . . . . . . . . 14  |-  ( ( ( 1 ... N
)  e.  Fin  /\  x  C_  ( 1 ... N ) )  ->  x  e.  Fin )
2724, 25, 26syl2anc 643 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  ( M Ramsey  F )  <_  N
)  /\  ( c  e.  R  /\  x  C_  ( 1 ... N
) ) )  ->  x  e.  Fin )
28 hashcl 11594 . . . . . . . . . . . . 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 10231 . . . . . . . . . . 11  |-  ( ( ( ph  /\  ( M Ramsey  F )  <_  N
)  /\  ( c  e.  R  /\  x  C_  ( 1 ... N
) ) )  -> 
( # `  x )  e.  RR )
31 simpl 444 . . . . . . . . . . . . 13  |-  ( ( c  e.  R  /\  x  C_  ( 1 ... N ) )  -> 
c  e.  R )
32 ffvelrn 5827 . . . . . . . . . . . . 13  |-  ( ( F : R --> NN0  /\  c  e.  R )  ->  ( F `  c
)  e.  NN0 )
337, 31, 32syl2an 464 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  ( M Ramsey  F )  <_  N
)  /\  ( c  e.  R  /\  x  C_  ( 1 ... N
) ) )  -> 
( F `  c
)  e.  NN0 )
3433nn0red 10231 . . . . . . . . . . 11  |-  ( ( ( ph  /\  ( M Ramsey  F )  <_  N
)  /\  ( c  e.  R  /\  x  C_  ( 1 ... N
) ) )  -> 
( F `  c
)  e.  RR )
3530, 34ltnled 9176 . . . . . . . . . 10  |-  ( ( ( ph  /\  ( M Ramsey  F )  <_  N
)  /\  ( c  e.  R  /\  x  C_  ( 1 ... N
) ) )  -> 
( ( # `  x
)  <  ( F `  c )  <->  -.  ( F `  c )  <_  ( # `  x
) ) )
3623, 35sylibd 206 . . . . . . . . 9  |-  ( ( ( ph  /\  ( M Ramsey  F )  <_  N
)  /\  ( c  e.  R  /\  x  C_  ( 1 ... N
) ) )  -> 
( ( x C M )  C_  ( `' G " { c } )  ->  -.  ( F `  c )  <_  ( # `  x
) ) )
3721, 36sylanr2 635 . . . . . . . 8  |-  ( ( ( ph  /\  ( M Ramsey  F )  <_  N
)  /\  ( c  e.  R  /\  x  e.  ~P ( 1 ... N ) ) )  ->  ( ( x C M )  C_  ( `' G " { c } )  ->  -.  ( F `  c )  <_  ( # `  x
) ) )
3837con2d 109 . . . . . . 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 412 . . . . . . 7  |-  ( ( ( F `  c
)  <_  ( # `  x
)  ->  -.  (
x C M ) 
C_  ( `' G " { c } ) )  <->  -.  ( ( F `  c )  <_  ( # `  x
)  /\  ( x C M )  C_  ( `' G " { c } ) ) )
4038, 39sylib 189 . . . . . 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 100 . . . . 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 2797 . . . 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 430 . 2  |-  ( ph  ->  -.  ( M Ramsey  F
)  <_  N )
458nn0red 10231 . . . 4  |-  ( ph  ->  N  e.  RR )
4645rexrd 9090 . . 3  |-  ( ph  ->  N  e.  RR* )
47 ramxrcl 13340 . . . 4  |-  ( ( M  e.  NN0  /\  R  e.  V  /\  F : R --> NN0 )  ->  ( M Ramsey  F )  e.  RR* )
482, 4, 6, 47syl3anc 1184 . . 3  |-  ( ph  ->  ( M Ramsey  F )  e.  RR* )
49 xrltnle 9100 . . 3  |-  ( ( N  e.  RR*  /\  ( M Ramsey  F )  e.  RR* )  ->  ( N  < 
( M Ramsey  F )  <->  -.  ( M Ramsey  F )  <_  N ) )
5046, 48, 49syl2anc 643 . 2  |-  ( ph  ->  ( N  <  ( M Ramsey  F )  <->  -.  ( M Ramsey  F )  <_  N
) )
5144, 50mpbird 224 1  |-  ( ph  ->  N  <  ( M Ramsey  F ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721   E.wrex 2667   {crab 2670   _Vcvv 2916    C_ wss 3280   ~Pcpw 3759   {csn 3774   class class class wbr 4172   `'ccnv 4836   "cima 4840   -->wf 5409   ` cfv 5413  (class class class)co 6040    e. cmpt2 6042   Fincfn 7068   1c1 8947   RR*cxr 9075    < clt 9076    <_ cle 9077   NN0cn0 10177   ...cfz 10999   #chash 11573   Ramsey cram 13322
This theorem is referenced by:  0ram  13343  ram0  13345
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-er 6864  df-map 6979  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-sup 7404  df-card 7782  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-n0 10178  df-z 10239  df-uz 10445  df-fz 11000  df-hash 11574  df-ram 13324
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