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Theorem ramub1 13091
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 2296 . 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 10034 . 2  |-  ( ph  ->  M  e.  NN0 )
4 ramub1.r . 2  |-  ( ph  ->  R  e.  Fin )
5 ramub1.f . . 3  |-  ( ph  ->  F : R --> NN )
6 nnssnn0 9984 . . 3  |-  NN  C_  NN0
7 fss 5413 . . 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 10020 . . 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 10019 . . . . . . 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 2370 . . . . 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 10034 . . . . . . 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 2804 . . . . . . . 8  |-  s  e. 
_V
19 hashclb 11368 . . . . . . . 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 11370 . . . . . 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 3477 . . . 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 3335 . . . . . . . 8  |-  ( v  =  u  ->  (
v  u.  { w } )  =  ( u  u.  { w } ) )
3938fveq2d 5545 . . . . . . 7  |-  ( v  =  u  ->  (
f `  ( v  u.  { w } ) )  =  ( f `
 ( u  u. 
{ w } ) ) )
4039cbvmptv 4127 . . . . . 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 13090 . . . . 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 1626 . . 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 13077 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 1531    = wceq 1632    e. wcel 1696    =/= wne 2459   E.wrex 2557   {crab 2560   _Vcvv 2801    \ cdif 3162    u. cun 3163    C_ wss 3165   (/)c0 3468   ifcif 3578   ~Pcpw 3638   {csn 3653   class class class wbr 4039    e. cmpt 4093   `'ccnv 4704   "cima 4708   -->wf 5267   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876   Fincfn 6879   1c1 8754    + caddc 8756    <_ cle 8884    - cmin 9053   NNcn 9762   NN0cn0 9981   #chash 11353   Ramsey cram 13062
This theorem is referenced by:  ramcl  13092
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-card 7588  df-cda 7810  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-n0 9982  df-z 10041  df-uz 10247  df-fz 10799  df-hash 11354  df-ram 13064
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