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Theorem ramub1 13388
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 2435 . 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 10266 . 2  |-  ( ph  ->  M  e.  NN0 )
4 ramub1.r . 2  |-  ( ph  ->  R  e.  Fin )
5 ramub1.f . . 3  |-  ( ph  ->  F : R --> NN )
6 nnssnn0 10216 . . 3  |-  NN  C_  NN0
7 fss 5591 . . 3  |-  ( ( F : R --> NN  /\  NN  C_  NN0 )  ->  F : R --> NN0 )
85, 6, 7sylancl 644 . 2  |-  ( ph  ->  F : R --> NN0 )
9 ramub1.2 . . 3  |-  ( ph  ->  ( ( M  - 
1 ) Ramsey  G )  e.  NN0 )
10 peano2nn0 10252 . . 3  |-  ( ( ( M  -  1 ) Ramsey  G )  e. 
NN0  ->  ( ( ( M  -  1 ) Ramsey  G )  +  1 )  e.  NN0 )
119, 10syl 16 . 2  |-  ( ph  ->  ( ( ( M  -  1 ) Ramsey  G
)  +  1 )  e.  NN0 )
12 simprl 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 ) )  ->  ( # `  s
)  =  ( ( ( M  -  1 ) Ramsey  G )  +  1 ) )
139adantr 452 . . . . . . 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 10251 . . . . . . 7  |-  ( ( ( M  -  1 ) Ramsey  G )  e. 
NN0  ->  ( ( ( M  -  1 ) Ramsey  G )  +  1 )  e.  NN )
1513, 14syl 16 . . . . . 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 2509 . . . . 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 10266 . . . . . . 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 2951 . . . . . . . 8  |-  s  e. 
_V
19 hashclb 11633 . . . . . . . 8  |-  ( s  e.  _V  ->  (
s  e.  Fin  <->  ( # `  s
)  e.  NN0 )
)
2018, 19ax-mp 8 . . . . . . 7  |-  ( s  e.  Fin  <->  ( # `  s
)  e.  NN0 )
2117, 20sylibr 204 . . . . . 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 11637 . . . . . 6  |-  ( s  e.  Fin  ->  (
( # `  s )  e.  NN  <->  s  =/=  (/) ) )
2321, 22syl 16 . . . . 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 202 . . . 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 3629 . . . 4  |-  ( s  =/=  (/)  <->  E. w  w  e.  s )
2624, 25sylib 189 . . 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 452 . . . . . 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 452 . . . . . 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 452 . . . . . 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 452 . . . . . 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 452 . . . . . 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 698 . . . . . 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 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 ) )  -> 
( # `  s )  =  ( ( ( M  -  1 ) Ramsey  G )  +  1 ) )
36 simprlr 740 . . . . . 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 734 . . . . . 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 3486 . . . . . . . 8  |-  ( v  =  u  ->  (
v  u.  { w } )  =  ( u  u.  { w } ) )
3938fveq2d 5724 . . . . . . 7  |-  ( v  =  u  ->  (
f `  ( v  u.  { w } ) )  =  ( f `
 ( u  u. 
{ w } ) ) )
4039cbvmptv 4292 . . . . . 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 13387 . . . . 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 599 . . . 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 1646 . . 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 15 . 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 13374 1  |-  ( ph  ->  ( M Ramsey  F )  <_  ( ( ( M  -  1 ) Ramsey  G )  +  1 ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359   E.wex 1550    = wceq 1652    e. wcel 1725    =/= wne 2598   E.wrex 2698   {crab 2701   _Vcvv 2948    \ cdif 3309    u. cun 3310    C_ wss 3312   (/)c0 3620   ifcif 3731   ~Pcpw 3791   {csn 3806   class class class wbr 4204    e. cmpt 4258   `'ccnv 4869   "cima 4873   -->wf 5442   ` cfv 5446  (class class class)co 6073    e. cmpt2 6075   Fincfn 7101   1c1 8983    + caddc 8985    <_ cle 9113    - cmin 9283   NNcn 9992   NN0cn0 10213   #chash 11610   Ramsey cram 13359
This theorem is referenced by:  ramcl  13389
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-map 7012  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-sup 7438  df-card 7818  df-cda 8040  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-n0 10214  df-z 10275  df-uz 10481  df-fz 11036  df-hash 11611  df-ram 13361
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