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Theorem ramub1 14557
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 2457 . 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 10873 . 2  |-  ( ph  ->  M  e.  NN0 )
4 ramub1.r . 2  |-  ( ph  ->  R  e.  Fin )
5 ramub1.f . . 3  |-  ( ph  ->  F : R --> NN )
6 nnssnn0 10819 . . 3  |-  NN  C_  NN0
7 fss 5745 . . 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 10857 . . 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 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 ) )  ->  ( # `  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 10856 . . . . . . 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 2545 . . . . 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 10873 . . . . . . 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 3112 . . . . . . . 8  |-  s  e. 
_V
19 hashclb 12432 . . . . . . . 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 12438 . . . . . 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 3803 . . . 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 763 . . . . . 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 764 . . . . . 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 757 . . . . . 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 3647 . . . . . . . 8  |-  ( v  =  u  ->  (
v  u.  { w } )  =  ( u  u.  { w } ) )
3938fveq2d 5876 . . . . . . 7  |-  ( v  =  u  ->  (
f `  ( v  u.  { w } ) )  =  ( f `
 ( u  u. 
{ w } ) ) )
4039cbvmptv 4548 . . . . . 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 14556 . . . . 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 1725 . . 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 14543 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 1395   E.wex 1613    e. wcel 1819    =/= wne 2652   E.wrex 2808   {crab 2811   _Vcvv 3109    \ cdif 3468    u. cun 3469    C_ wss 3471   (/)c0 3793   ifcif 3944   ~Pcpw 4015   {csn 4032   class class class wbr 4456    |-> cmpt 4515   `'ccnv 5007   "cima 5011   -->wf 5590   ` cfv 5594  (class class class)co 6296    |-> cmpt2 6298   Fincfn 7535   1c1 9510    + caddc 9512    <_ cle 9646    - cmin 9824   NNcn 10556   NN0cn0 10816   #chash 12407   Ramsey cram 14528
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-oadd 7152  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-cda 8565  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 12408  df-ram 14530
This theorem is referenced by:  ramcl  14558
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