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Theorem ramub1 13075
Description: Inductive step for Ramsey's theorem, in the form of an explicit upper bound. (Contributed by Mario Carneiro, 23-Apr-2015.)
Hypotheses
Ref Expression
ramub1.m  |-  ( ph  ->  M  e.  NN )
ramub1.r  |-  ( ph  ->  R  e.  Fin )
ramub1.f  |-  ( ph  ->  F : R --> NN )
ramub1.g  |-  G  =  ( x  e.  R  |->  ( M Ramsey  ( y  e.  R  |->  if ( y  =  x ,  ( ( F `  x )  -  1 ) ,  ( F `
 y ) ) ) ) )
ramub1.1  |-  ( ph  ->  G : R --> NN0 )
ramub1.2  |-  ( ph  ->  ( ( M  - 
1 ) Ramsey  G )  e.  NN0 )
Assertion
Ref Expression
ramub1  |-  ( ph  ->  ( M Ramsey  F )  <_  ( ( ( M  -  1 ) Ramsey  G )  +  1 ) )
Distinct variable groups:    x, y, F    x, M, y    x, G, y    x, R, y    ph, x, y

Proof of Theorem ramub1
Dummy variables  u  c  f  s  v  w  z  a  b 
i are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2283 . 2  |-  ( 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 ramub1.m . . 3  |-  ( ph  ->  M  e.  NN )
32nnnn0d 10018 . 2  |-  ( ph  ->  M  e.  NN0 )
4 ramub1.r . 2  |-  ( ph  ->  R  e.  Fin )
5 ramub1.f . . 3  |-  ( ph  ->  F : R --> NN )
6 nnssnn0 9968 . . 3  |-  NN  C_  NN0
7 fss 5397 . . 3  |-  ( ( F : R --> NN  /\  NN  C_  NN0 )  ->  F : R --> NN0 )
85, 6, 7sylancl 643 . 2  |-  ( ph  ->  F : R --> NN0 )
9 ramub1.2 . . 3  |-  ( ph  ->  ( ( M  - 
1 ) Ramsey  G )  e.  NN0 )
10 peano2nn0 10004 . . 3  |-  ( ( ( M  -  1 ) Ramsey  G )  e. 
NN0  ->  ( ( ( M  -  1 ) Ramsey  G )  +  1 )  e.  NN0 )
119, 10syl 15 . 2  |-  ( ph  ->  ( ( ( M  -  1 ) Ramsey  G
)  +  1 )  e.  NN0 )
12 simprl 732 . . . . . 6  |-  ( (
ph  /\  ( ( # `
 s )  =  ( ( ( M  -  1 ) Ramsey  G
)  +  1 )  /\  f : ( s ( a  e. 
_V ,  i  e. 
NN0  |->  { b  e. 
~P a  |  (
# `  b )  =  i } ) M ) --> R ) )  ->  ( # `  s
)  =  ( ( ( M  -  1 ) Ramsey  G )  +  1 ) )
139adantr 451 . . . . . . 7  |-  ( (
ph  /\  ( ( # `
 s )  =  ( ( ( M  -  1 ) Ramsey  G
)  +  1 )  /\  f : ( s ( a  e. 
_V ,  i  e. 
NN0  |->  { b  e. 
~P a  |  (
# `  b )  =  i } ) M ) --> R ) )  ->  ( ( M  -  1 ) Ramsey  G )  e.  NN0 )
14 nn0p1nn 10003 . . . . . . 7  |-  ( ( ( M  -  1 ) Ramsey  G )  e. 
NN0  ->  ( ( ( M  -  1 ) Ramsey  G )  +  1 )  e.  NN )
1513, 14syl 15 . . . . . 6  |-  ( (
ph  /\  ( ( # `
 s )  =  ( ( ( M  -  1 ) Ramsey  G
)  +  1 )  /\  f : ( s ( a  e. 
_V ,  i  e. 
NN0  |->  { b  e. 
~P a  |  (
# `  b )  =  i } ) M ) --> R ) )  ->  ( (
( M  -  1 ) Ramsey  G )  +  1 )  e.  NN )
1612, 15eqeltrd 2357 . . . . 5  |-  ( (
ph  /\  ( ( # `
 s )  =  ( ( ( M  -  1 ) Ramsey  G
)  +  1 )  /\  f : ( s ( a  e. 
_V ,  i  e. 
NN0  |->  { b  e. 
~P a  |  (
# `  b )  =  i } ) M ) --> R ) )  ->  ( # `  s
)  e.  NN )
1716nnnn0d 10018 . . . . . . 7  |-  ( (
ph  /\  ( ( # `
 s )  =  ( ( ( M  -  1 ) Ramsey  G
)  +  1 )  /\  f : ( s ( a  e. 
_V ,  i  e. 
NN0  |->  { b  e. 
~P a  |  (
# `  b )  =  i } ) M ) --> R ) )  ->  ( # `  s
)  e.  NN0 )
18 vex 2791 . . . . . . . 8  |-  s  e. 
_V
19 hashclb 11352 . . . . . . . 8  |-  ( s  e.  _V  ->  (
s  e.  Fin  <->  ( # `  s
)  e.  NN0 )
)
2018, 19ax-mp 8 . . . . . . 7  |-  ( s  e.  Fin  <->  ( # `  s
)  e.  NN0 )
2117, 20sylibr 203 . . . . . 6  |-  ( (
ph  /\  ( ( # `
 s )  =  ( ( ( M  -  1 ) Ramsey  G
)  +  1 )  /\  f : ( s ( a  e. 
_V ,  i  e. 
NN0  |->  { b  e. 
~P a  |  (
# `  b )  =  i } ) M ) --> R ) )  ->  s  e.  Fin )
22 hashnncl 11354 . . . . . 6  |-  ( s  e.  Fin  ->  (
( # `  s )  e.  NN  <->  s  =/=  (/) ) )
2321, 22syl 15 . . . . 5  |-  ( (
ph  /\  ( ( # `
 s )  =  ( ( ( M  -  1 ) Ramsey  G
)  +  1 )  /\  f : ( s ( a  e. 
_V ,  i  e. 
NN0  |->  { b  e. 
~P a  |  (
# `  b )  =  i } ) M ) --> R ) )  ->  ( ( # `
 s )  e.  NN  <->  s  =/=  (/) ) )
2416, 23mpbid 201 . . . 4  |-  ( (
ph  /\  ( ( # `
 s )  =  ( ( ( M  -  1 ) Ramsey  G
)  +  1 )  /\  f : ( s ( a  e. 
_V ,  i  e. 
NN0  |->  { b  e. 
~P a  |  (
# `  b )  =  i } ) M ) --> R ) )  ->  s  =/=  (/) )
25 n0 3464 . . . 4  |-  ( s  =/=  (/)  <->  E. w  w  e.  s )
2624, 25sylib 188 . . 3  |-  ( (
ph  /\  ( ( # `
 s )  =  ( ( ( M  -  1 ) Ramsey  G
)  +  1 )  /\  f : ( s ( a  e. 
_V ,  i  e. 
NN0  |->  { b  e. 
~P a  |  (
# `  b )  =  i } ) M ) --> R ) )  ->  E. w  w  e.  s )
272adantr 451 . . . . . 6  |-  ( (
ph  /\  ( (
( # `  s )  =  ( ( ( M  -  1 ) Ramsey  G )  +  1 )  /\  f : ( s ( a  e.  _V ,  i  e.  NN0  |->  { b  e.  ~P a  |  ( # `  b
)  =  i } ) M ) --> R )  /\  w  e.  s ) )  ->  M  e.  NN )
284adantr 451 . . . . . 6  |-  ( (
ph  /\  ( (
( # `  s )  =  ( ( ( M  -  1 ) Ramsey  G )  +  1 )  /\  f : ( s ( a  e.  _V ,  i  e.  NN0  |->  { b  e.  ~P a  |  ( # `  b
)  =  i } ) M ) --> R )  /\  w  e.  s ) )  ->  R  e.  Fin )
295adantr 451 . . . . . 6  |-  ( (
ph  /\  ( (
( # `  s )  =  ( ( ( M  -  1 ) Ramsey  G )  +  1 )  /\  f : ( s ( a  e.  _V ,  i  e.  NN0  |->  { b  e.  ~P a  |  ( # `  b
)  =  i } ) M ) --> R )  /\  w  e.  s ) )  ->  F : R --> NN )
30 ramub1.g . . . . . 6  |-  G  =  ( x  e.  R  |->  ( M Ramsey  ( y  e.  R  |->  if ( y  =  x ,  ( ( F `  x )  -  1 ) ,  ( F `
 y ) ) ) ) )
31 ramub1.1 . . . . . . 7  |-  ( ph  ->  G : R --> NN0 )
3231adantr 451 . . . . . 6  |-  ( (
ph  /\  ( (
( # `  s )  =  ( ( ( M  -  1 ) Ramsey  G )  +  1 )  /\  f : ( s ( a  e.  _V ,  i  e.  NN0  |->  { b  e.  ~P a  |  ( # `  b
)  =  i } ) M ) --> R )  /\  w  e.  s ) )  ->  G : R --> NN0 )
339adantr 451 . . . . . 6  |-  ( (
ph  /\  ( (
( # `  s )  =  ( ( ( M  -  1 ) Ramsey  G )  +  1 )  /\  f : ( s ( a  e.  _V ,  i  e.  NN0  |->  { b  e.  ~P a  |  ( # `  b
)  =  i } ) M ) --> R )  /\  w  e.  s ) )  -> 
( ( M  - 
1 ) Ramsey  G )  e.  NN0 )
3421adantrr 697 . . . . . 6  |-  ( (
ph  /\  ( (
( # `  s )  =  ( ( ( M  -  1 ) Ramsey  G )  +  1 )  /\  f : ( s ( a  e.  _V ,  i  e.  NN0  |->  { b  e.  ~P a  |  ( # `  b
)  =  i } ) M ) --> R )  /\  w  e.  s ) )  -> 
s  e.  Fin )
35 simprll 738 . . . . . 6  |-  ( (
ph  /\  ( (
( # `  s )  =  ( ( ( M  -  1 ) Ramsey  G )  +  1 )  /\  f : ( s ( a  e.  _V ,  i  e.  NN0  |->  { b  e.  ~P a  |  ( # `  b
)  =  i } ) M ) --> R )  /\  w  e.  s ) )  -> 
( # `  s )  =  ( ( ( M  -  1 ) Ramsey  G )  +  1 ) )
36 simprlr 739 . . . . . 6  |-  ( (
ph  /\  ( (
( # `  s )  =  ( ( ( M  -  1 ) Ramsey  G )  +  1 )  /\  f : ( s ( a  e.  _V ,  i  e.  NN0  |->  { b  e.  ~P a  |  ( # `  b
)  =  i } ) M ) --> R )  /\  w  e.  s ) )  -> 
f : ( s ( a  e.  _V ,  i  e.  NN0  |->  { b  e.  ~P a  |  ( # `  b
)  =  i } ) M ) --> R )
37 simprr 733 . . . . . 6  |-  ( (
ph  /\  ( (
( # `  s )  =  ( ( ( M  -  1 ) Ramsey  G )  +  1 )  /\  f : ( s ( a  e.  _V ,  i  e.  NN0  |->  { b  e.  ~P a  |  ( # `  b
)  =  i } ) M ) --> R )  /\  w  e.  s ) )  ->  w  e.  s )
38 uneq1 3322 . . . . . . . 8  |-  ( v  =  u  ->  (
v  u.  { w } )  =  ( u  u.  { w } ) )
3938fveq2d 5529 . . . . . . 7  |-  ( v  =  u  ->  (
f `  ( v  u.  { w } ) )  =  ( f `
 ( u  u. 
{ w } ) ) )
4039cbvmptv 4111 . . . . . 6  |-  ( v  e.  ( ( s 
\  { w }
) ( a  e. 
_V ,  i  e. 
NN0  |->  { b  e. 
~P a  |  (
# `  b )  =  i } ) ( M  -  1 ) )  |->  ( f `
 ( v  u. 
{ w } ) ) )  =  ( u  e.  ( ( s  \  { w } ) ( a  e.  _V ,  i  e.  NN0  |->  { b  e.  ~P a  |  ( # `  b
)  =  i } ) ( M  - 
1 ) )  |->  ( f `  ( u  u.  { w }
) ) )
4127, 28, 29, 30, 32, 33, 1, 34, 35, 36, 37, 40ramub1lem2 13074 . . . . 5  |-  ( (
ph  /\  ( (
( # `  s )  =  ( ( ( M  -  1 ) Ramsey  G )  +  1 )  /\  f : ( s ( a  e.  _V ,  i  e.  NN0  |->  { b  e.  ~P a  |  ( # `  b
)  =  i } ) M ) --> R )  /\  w  e.  s ) )  ->  E. c  e.  R  E. z  e.  ~P  s ( ( F `
 c )  <_ 
( # `  z )  /\  ( z ( a  e.  _V , 
i  e.  NN0  |->  { b  e.  ~P a  |  ( # `  b
)  =  i } ) M )  C_  ( `' f " {
c } ) ) )
4241expr 598 . . . 4  |-  ( (
ph  /\  ( ( # `
 s )  =  ( ( ( M  -  1 ) Ramsey  G
)  +  1 )  /\  f : ( s ( a  e. 
_V ,  i  e. 
NN0  |->  { b  e. 
~P a  |  (
# `  b )  =  i } ) M ) --> R ) )  ->  ( w  e.  s  ->  E. c  e.  R  E. z  e.  ~P  s ( ( F `  c )  <_  ( # `  z
)  /\  ( z
( a  e.  _V ,  i  e.  NN0  |->  { b  e.  ~P a  |  ( # `  b
)  =  i } ) M )  C_  ( `' f " {
c } ) ) ) )
4342exlimdv 1664 . . 3  |-  ( (
ph  /\  ( ( # `
 s )  =  ( ( ( M  -  1 ) Ramsey  G
)  +  1 )  /\  f : ( s ( a  e. 
_V ,  i  e. 
NN0  |->  { b  e. 
~P a  |  (
# `  b )  =  i } ) M ) --> R ) )  ->  ( E. w  w  e.  s  ->  E. c  e.  R  E. z  e.  ~P  s ( ( F `
 c )  <_ 
( # `  z )  /\  ( z ( a  e.  _V , 
i  e.  NN0  |->  { b  e.  ~P a  |  ( # `  b
)  =  i } ) M )  C_  ( `' f " {
c } ) ) ) )
4426, 43mpd 14 . 2  |-  ( (
ph  /\  ( ( # `
 s )  =  ( ( ( M  -  1 ) Ramsey  G
)  +  1 )  /\  f : ( s ( a  e. 
_V ,  i  e. 
NN0  |->  { b  e. 
~P a  |  (
# `  b )  =  i } ) M ) --> R ) )  ->  E. c  e.  R  E. z  e.  ~P  s ( ( F `  c )  <_  ( # `  z
)  /\  ( z
( a  e.  _V ,  i  e.  NN0  |->  { b  e.  ~P a  |  ( # `  b
)  =  i } ) M )  C_  ( `' f " {
c } ) ) )
451, 3, 4, 8, 11, 44ramub2 13061 1  |-  ( ph  ->  ( M Ramsey  F )  <_  ( ( ( M  -  1 ) Ramsey  G )  +  1 ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1684    =/= wne 2446   E.wrex 2544   {crab 2547   _Vcvv 2788    \ cdif 3149    u. cun 3150    C_ wss 3152   (/)c0 3455   ifcif 3565   ~Pcpw 3625   {csn 3640   class class class wbr 4023    e. cmpt 4077   `'ccnv 4688   "cima 4692   -->wf 5251   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   Fincfn 6863   1c1 8738    + caddc 8740    <_ cle 8868    - cmin 9037   NNcn 9746   NN0cn0 9965   #chash 11337   Ramsey cram 13046
This theorem is referenced by:  ramcl  13076
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-oadd 6483  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-cda 7794  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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