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Theorem ramlb 14549
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 465 . . . . 5  |-  ( (
ph  /\  ( M Ramsey  F )  <_  N )  ->  M  e.  NN0 )
4 ramlb.r . . . . . 6  |-  ( ph  ->  R  e.  V )
54adantr 465 . . . . 5  |-  ( (
ph  /\  ( M Ramsey  F )  <_  N )  ->  R  e.  V )
6 ramlb.f . . . . . 6  |-  ( ph  ->  F : R --> NN0 )
76adantr 465 . . . . 5  |-  ( (
ph  /\  ( M Ramsey  F )  <_  N )  ->  F : R --> NN0 )
8 ramlb.s . . . . . . 7  |-  ( ph  ->  N  e.  NN0 )
98adantr 465 . . . . . 6  |-  ( (
ph  /\  ( M Ramsey  F )  <_  N )  ->  N  e.  NN0 )
10 simpr 461 . . . . . 6  |-  ( (
ph  /\  ( M Ramsey  F )  <_  N )  ->  ( M Ramsey  F )  <_  N )
11 ramubcl 14548 . . . . . 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 1236 . . . . 5  |-  ( (
ph  /\  ( M Ramsey  F )  <_  N )  ->  ( M Ramsey  F )  e.  NN0 )
13 fzfid 12086 . . . . 5  |-  ( (
ph  /\  ( M Ramsey  F )  <_  N )  ->  ( 1 ... N
)  e.  Fin )
14 hashfz1 12422 . . . . . . . 8  |-  ( N  e.  NN0  ->  ( # `  ( 1 ... N
) )  =  N )
158, 14syl 16 . . . . . . 7  |-  ( ph  ->  ( # `  (
1 ... N ) )  =  N )
1615breq2d 4468 . . . . . 6  |-  ( ph  ->  ( ( M Ramsey  F
)  <_  ( # `  (
1 ... N ) )  <-> 
( M Ramsey  F )  <_  N ) )
1716biimpar 485 . . . . 5  |-  ( (
ph  /\  ( M Ramsey  F )  <_  N )  ->  ( M Ramsey  F )  <_  ( # `  (
1 ... N ) ) )
18 ramlb.g . . . . . 6  |-  ( ph  ->  G : ( ( 1 ... N ) C M ) --> R )
1918adantr 465 . . . . 5  |-  ( (
ph  /\  ( M Ramsey  F )  <_  N )  ->  G : ( ( 1 ... N ) C M ) --> R )
201, 3, 5, 7, 12, 13, 17, 19rami 14545 . . . 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 4024 . . . . . . . . 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 714 . . . . . . . . . 10  |-  ( ( ( ph  /\  ( M Ramsey  F )  <_  N
)  /\  ( c  e.  R  /\  x  C_  ( 1 ... N
) ) )  -> 
( ( x C M )  C_  ( `' G " { c } )  ->  ( # `
 x )  < 
( F `  c
) ) )
24 fzfid 12086 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  ( M Ramsey  F )  <_  N
)  /\  ( c  e.  R  /\  x  C_  ( 1 ... N
) ) )  -> 
( 1 ... N
)  e.  Fin )
25 simprr 757 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  ( M Ramsey  F )  <_  N
)  /\  ( c  e.  R  /\  x  C_  ( 1 ... N
) ) )  ->  x  C_  ( 1 ... N ) )
26 ssfi 7759 . . . . . . . . . . . . . 14  |-  ( ( ( 1 ... N
)  e.  Fin  /\  x  C_  ( 1 ... N ) )  ->  x  e.  Fin )
2724, 25, 26syl2anc 661 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  ( M Ramsey  F )  <_  N
)  /\  ( c  e.  R  /\  x  C_  ( 1 ... N
) ) )  ->  x  e.  Fin )
28 hashcl 12431 . . . . . . . . . . . . 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 10874 . . . . . . . . . . 11  |-  ( ( ( ph  /\  ( M Ramsey  F )  <_  N
)  /\  ( c  e.  R  /\  x  C_  ( 1 ... N
) ) )  -> 
( # `  x )  e.  RR )
31 simpl 457 . . . . . . . . . . . . 13  |-  ( ( c  e.  R  /\  x  C_  ( 1 ... N ) )  -> 
c  e.  R )
32 ffvelrn 6030 . . . . . . . . . . . . 13  |-  ( ( F : R --> NN0  /\  c  e.  R )  ->  ( F `  c
)  e.  NN0 )
337, 31, 32syl2an 477 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  ( M Ramsey  F )  <_  N
)  /\  ( c  e.  R  /\  x  C_  ( 1 ... N
) ) )  -> 
( F `  c
)  e.  NN0 )
3433nn0red 10874 . . . . . . . . . . 11  |-  ( ( ( ph  /\  ( M Ramsey  F )  <_  N
)  /\  ( c  e.  R  /\  x  C_  ( 1 ... N
) ) )  -> 
( F `  c
)  e.  RR )
3530, 34ltnled 9749 . . . . . . . . . 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 653 . . . . . . . 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 2956 . . . 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 442 . 2  |-  ( ph  ->  -.  ( M Ramsey  F
)  <_  N )
458nn0red 10874 . . . 4  |-  ( ph  ->  N  e.  RR )
4645rexrd 9660 . . 3  |-  ( ph  ->  N  e.  RR* )
47 ramxrcl 14547 . . . 4  |-  ( ( M  e.  NN0  /\  R  e.  V  /\  F : R --> NN0 )  ->  ( M Ramsey  F )  e.  RR* )
482, 4, 6, 47syl3anc 1228 . . 3  |-  ( ph  ->  ( M Ramsey  F )  e.  RR* )
49 xrltnle 9670 . . 3  |-  ( ( N  e.  RR*  /\  ( M Ramsey  F )  e.  RR* )  ->  ( N  < 
( M Ramsey  F )  <->  -.  ( M Ramsey  F )  <_  N ) )
5046, 48, 49syl2anc 661 . 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 1395    e. wcel 1819   E.wrex 2808   {crab 2811   _Vcvv 3109    C_ wss 3471   ~Pcpw 4015   {csn 4032   class class class wbr 4456   `'ccnv 5007   "cima 5011   -->wf 5590   ` cfv 5594  (class class class)co 6296    |-> cmpt2 6298   Fincfn 7535   1c1 9510   RR*cxr 9644    < clt 9645    <_ cle 9646   NN0cn0 10816   ...cfz 11697   #chash 12408   Ramsey cram 14529
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-er 7329  df-map 7440  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-sup 7919  df-card 8337  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-n0 10817  df-z 10886  df-uz 11107  df-fz 11698  df-hash 12409  df-ram 14531
This theorem is referenced by:  0ram  14550  ram0  14552
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