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Theorem ramub1 14087
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 2441 . 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 10634 . 2  |-  ( ph  ->  M  e.  NN0 )
4 ramub1.r . 2  |-  ( ph  ->  R  e.  Fin )
5 ramub1.f . . 3  |-  ( ph  ->  F : R --> NN )
6 nnssnn0 10580 . . 3  |-  NN  C_  NN0
7 fss 5565 . . 3  |-  ( ( F : R --> NN  /\  NN  C_  NN0 )  ->  F : R --> NN0 )
85, 6, 7sylancl 662 . 2  |-  ( ph  ->  F : R --> NN0 )
9 ramub1.2 . . 3  |-  ( ph  ->  ( ( M  - 
1 ) Ramsey  G )  e.  NN0 )
10 peano2nn0 10618 . . 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 755 . . . . . 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 465 . . . . . . 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 10617 . . . . . . 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 2515 . . . . 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 10634 . . . . . . 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 2973 . . . . . . . 8  |-  s  e. 
_V
19 hashclb 12126 . . . . . . . 8  |-  ( s  e.  _V  ->  (
s  e.  Fin  <->  ( # `  s
)  e.  NN0 )
)
2018, 19ax-mp 5 . . . . . . 7  |-  ( s  e.  Fin  <->  ( # `  s
)  e.  NN0 )
2117, 20sylibr 212 . . . . . 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 12132 . . . . . 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 210 . . . 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 3644 . . . 4  |-  ( s  =/=  (/)  <->  E. w  w  e.  s )
2624, 25sylib 196 . . 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 465 . . . . . 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 465 . . . . . 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 465 . . . . . 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 465 . . . . . 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 465 . . . . . 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 716 . . . . . 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 761 . . . . . 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 762 . . . . . 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 756 . . . . . 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 3501 . . . . . . . 8  |-  ( v  =  u  ->  (
v  u.  { w } )  =  ( u  u.  { w } ) )
3938fveq2d 5693 . . . . . . 7  |-  ( v  =  u  ->  (
f `  ( v  u.  { w } ) )  =  ( f `
 ( u  u. 
{ w } ) ) )
4039cbvmptv 4381 . . . . . 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 14086 . . . . 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 615 . . . 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 1690 . . 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 14073 1  |-  ( ph  ->  ( M Ramsey  F )  <_  ( ( ( M  -  1 ) Ramsey  G )  +  1 ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369   E.wex 1586    e. wcel 1756    =/= wne 2604   E.wrex 2714   {crab 2717   _Vcvv 2970    \ cdif 3323    u. cun 3324    C_ wss 3326   (/)c0 3635   ifcif 3789   ~Pcpw 3858   {csn 3875   class class class wbr 4290    e. cmpt 4348   `'ccnv 4837   "cima 4841   -->wf 5412   ` cfv 5416  (class class class)co 6089    e. cmpt2 6091   Fincfn 7308   1c1 9281    + caddc 9283    <_ cle 9417    - cmin 9593   NNcn 10320   NN0cn0 10577   #chash 12101   Ramsey cram 14058
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-rep 4401  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370  ax-cnex 9336  ax-resscn 9337  ax-1cn 9338  ax-icn 9339  ax-addcl 9340  ax-addrcl 9341  ax-mulcl 9342  ax-mulrcl 9343  ax-mulcom 9344  ax-addass 9345  ax-mulass 9346  ax-distr 9347  ax-i2m1 9348  ax-1ne0 9349  ax-1rid 9350  ax-rnegex 9351  ax-rrecex 9352  ax-cnre 9353  ax-pre-lttri 9354  ax-pre-lttrn 9355  ax-pre-ltadd 9356  ax-pre-mulgt0 9357
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-pss 3342  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-tp 3880  df-op 3882  df-uni 4090  df-int 4127  df-iun 4171  df-br 4291  df-opab 4349  df-mpt 4350  df-tr 4384  df-eprel 4630  df-id 4634  df-po 4639  df-so 4640  df-fr 4677  df-we 4679  df-ord 4720  df-on 4721  df-lim 4722  df-suc 4723  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-riota 6050  df-ov 6092  df-oprab 6093  df-mpt2 6094  df-om 6475  df-1st 6575  df-2nd 6576  df-recs 6830  df-rdg 6864  df-1o 6918  df-oadd 6922  df-er 7099  df-map 7214  df-en 7309  df-dom 7310  df-sdom 7311  df-fin 7312  df-sup 7689  df-card 8107  df-cda 8335  df-pnf 9418  df-mnf 9419  df-xr 9420  df-ltxr 9421  df-le 9422  df-sub 9595  df-neg 9596  df-nn 10321  df-n0 10578  df-z 10645  df-uz 10860  df-fz 11436  df-hash 12102  df-ram 14060
This theorem is referenced by:  ramcl  14088
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