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Theorem ramcl2lem 14066
Description: Lemma for extended real closure of the Ramsey number function. (Contributed by Mario Carneiro, 20-Apr-2015.)
Hypotheses
Ref Expression
ramval.c  |-  C  =  ( a  e.  _V ,  i  e.  NN0  |->  { b  e.  ~P a  |  ( # `  b
)  =  i } )
ramval.t  |-  T  =  { n  e.  NN0  | 
A. s ( n  <_  ( # `  s
)  ->  A. f  e.  ( R  ^m  (
s C M ) ) E. c  e.  R  E. x  e. 
~P  s ( ( F `  c )  <_  ( # `  x
)  /\  ( x C M )  C_  ( `' f " {
c } ) ) ) }
Assertion
Ref Expression
ramcl2lem  |-  ( ( M  e.  NN0  /\  R  e.  V  /\  F : R --> NN0 )  ->  ( M Ramsey  F )  =  if ( T  =  (/) , +oo ,  sup ( T ,  RR ,  `'  <  ) ) )
Distinct variable groups:    f, c, x, C    n, c, s, F, f, x    a,
b, c, f, i, n, s, x, M    R, c, f, n, s, x    V, c, f, n, s, x
Allowed substitution hints:    C( i, n, s, a, b)    R( i, a, b)    T( x, f, i, n, s, a, b, c)    F( i, a, b)    V( i, a, b)

Proof of Theorem ramcl2lem
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 eqeq2 2450 . 2  |-  ( +oo  =  if ( T  =  (/) , +oo ,  sup ( T ,  RR ,  `'  <  ) )  -> 
( ( M Ramsey  F
)  = +oo  <->  ( M Ramsey  F )  =  if ( T  =  (/) , +oo ,  sup ( T ,  RR ,  `'  <  ) ) ) )
2 eqeq2 2450 . 2  |-  ( sup ( T ,  RR ,  `'  <  )  =  if ( T  =  (/) , +oo ,  sup ( T ,  RR ,  `'  <  ) )  -> 
( ( M Ramsey  F
)  =  sup ( T ,  RR ,  `'  <  )  <->  ( M Ramsey  F )  =  if ( T  =  (/) , +oo ,  sup ( T ,  RR ,  `'  <  ) ) ) )
3 ramval.c . . . 4  |-  C  =  ( a  e.  _V ,  i  e.  NN0  |->  { b  e.  ~P a  |  ( # `  b
)  =  i } )
4 ramval.t . . . 4  |-  T  =  { n  e.  NN0  | 
A. s ( n  <_  ( # `  s
)  ->  A. f  e.  ( R  ^m  (
s C M ) ) E. c  e.  R  E. x  e. 
~P  s ( ( F `  c )  <_  ( # `  x
)  /\  ( x C M )  C_  ( `' f " {
c } ) ) ) }
53, 4ramval 14065 . . 3  |-  ( ( M  e.  NN0  /\  R  e.  V  /\  F : R --> NN0 )  ->  ( M Ramsey  F )  =  sup ( T ,  RR* ,  `'  <  ) )
6 supeq1 7691 . . . 4  |-  ( T  =  (/)  ->  sup ( T ,  RR* ,  `'  <  )  =  sup ( (/)
,  RR* ,  `'  <  ) )
7 xrinfm0 11295 . . . 4  |-  sup ( (/)
,  RR* ,  `'  <  )  = +oo
86, 7syl6eq 2489 . . 3  |-  ( T  =  (/)  ->  sup ( T ,  RR* ,  `'  <  )  = +oo )
95, 8sylan9eq 2493 . 2  |-  ( ( ( M  e.  NN0  /\  R  e.  V  /\  F : R --> NN0 )  /\  T  =  (/) )  -> 
( M Ramsey  F )  = +oo )
10 df-ne 2606 . . 3  |-  ( T  =/=  (/)  <->  -.  T  =  (/) )
115adantr 462 . . . 4  |-  ( ( ( M  e.  NN0  /\  R  e.  V  /\  F : R --> NN0 )  /\  T  =/=  (/) )  -> 
( M Ramsey  F )  =  sup ( T ,  RR* ,  `'  <  )
)
12 xrltso 11114 . . . . . . 7  |-  <  Or  RR*
13 cnvso 5373 . . . . . . 7  |-  (  < 
Or  RR*  <->  `'  <  Or  RR* )
1412, 13mpbi 208 . . . . . 6  |-  `'  <  Or 
RR*
1514a1i 11 . . . . 5  |-  ( ( ( M  e.  NN0  /\  R  e.  V  /\  F : R --> NN0 )  /\  T  =/=  (/) )  ->  `'  <  Or  RR* )
16 ssrab2 3434 . . . . . . . . 9  |-  { n  e.  NN0  |  A. s
( n  <_  ( # `
 s )  ->  A. f  e.  ( R  ^m  ( s C M ) ) E. c  e.  R  E. x  e.  ~P  s
( ( F `  c )  <_  ( # `
 x )  /\  ( x C M )  C_  ( `' f " { c } ) ) ) } 
C_  NN0
174, 16eqsstri 3383 . . . . . . . 8  |-  T  C_  NN0
18 nn0ssre 10579 . . . . . . . 8  |-  NN0  C_  RR
1917, 18sstri 3362 . . . . . . 7  |-  T  C_  RR
20 nn0uz 10891 . . . . . . . . . 10  |-  NN0  =  ( ZZ>= `  0 )
2117, 20sseqtri 3385 . . . . . . . . 9  |-  T  C_  ( ZZ>= `  0 )
2221a1i 11 . . . . . . . 8  |-  ( ( M  e.  NN0  /\  R  e.  V  /\  F : R --> NN0 )  ->  T  C_  ( ZZ>= ` 
0 ) )
23 infmssuzcl 10934 . . . . . . . 8  |-  ( ( T  C_  ( ZZ>= ` 
0 )  /\  T  =/=  (/) )  ->  sup ( T ,  RR ,  `'  <  )  e.  T
)
2422, 23sylan 468 . . . . . . 7  |-  ( ( ( M  e.  NN0  /\  R  e.  V  /\  F : R --> NN0 )  /\  T  =/=  (/) )  ->  sup ( T ,  RR ,  `'  <  )  e.  T )
2519, 24sseldi 3351 . . . . . 6  |-  ( ( ( M  e.  NN0  /\  R  e.  V  /\  F : R --> NN0 )  /\  T  =/=  (/) )  ->  sup ( T ,  RR ,  `'  <  )  e.  RR )
2625rexrd 9429 . . . . 5  |-  ( ( ( M  e.  NN0  /\  R  e.  V  /\  F : R --> NN0 )  /\  T  =/=  (/) )  ->  sup ( T ,  RR ,  `'  <  )  e. 
RR* )
27 simpr 458 . . . . . . . 8  |-  ( ( ( ( M  e. 
NN0  /\  R  e.  V  /\  F : R --> NN0 )  /\  T  =/=  (/) )  /\  z  e.  T )  ->  z  e.  T )
28 infmssuzle 10933 . . . . . . . 8  |-  ( ( T  C_  ( ZZ>= ` 
0 )  /\  z  e.  T )  ->  sup ( T ,  RR ,  `'  <  )  <_  z
)
2921, 27, 28sylancr 658 . . . . . . 7  |-  ( ( ( ( M  e. 
NN0  /\  R  e.  V  /\  F : R --> NN0 )  /\  T  =/=  (/) )  /\  z  e.  T )  ->  sup ( T ,  RR ,  `'  <  )  <_  z
)
3025adantr 462 . . . . . . . 8  |-  ( ( ( ( M  e. 
NN0  /\  R  e.  V  /\  F : R --> NN0 )  /\  T  =/=  (/) )  /\  z  e.  T )  ->  sup ( T ,  RR ,  `'  <  )  e.  RR )
3119a1i 11 . . . . . . . . 9  |-  ( ( ( M  e.  NN0  /\  R  e.  V  /\  F : R --> NN0 )  /\  T  =/=  (/) )  ->  T  C_  RR )
3231sselda 3353 . . . . . . . 8  |-  ( ( ( ( M  e. 
NN0  /\  R  e.  V  /\  F : R --> NN0 )  /\  T  =/=  (/) )  /\  z  e.  T )  ->  z  e.  RR )
3330, 32lenltd 9516 . . . . . . 7  |-  ( ( ( ( M  e. 
NN0  /\  R  e.  V  /\  F : R --> NN0 )  /\  T  =/=  (/) )  /\  z  e.  T )  ->  ( sup ( T ,  RR ,  `'  <  )  <_ 
z  <->  -.  z  <  sup ( T ,  RR ,  `'  <  ) ) )
3429, 33mpbid 210 . . . . . 6  |-  ( ( ( ( M  e. 
NN0  /\  R  e.  V  /\  F : R --> NN0 )  /\  T  =/=  (/) )  /\  z  e.  T )  ->  -.  z  <  sup ( T ,  RR ,  `'  <  ) )
35 gtso 9452 . . . . . . . 8  |-  `'  <  Or  RR
3635supex 7709 . . . . . . 7  |-  sup ( T ,  RR ,  `'  <  )  e.  _V
37 vex 2973 . . . . . . 7  |-  z  e. 
_V
3836, 37brcnv 5018 . . . . . 6  |-  ( sup ( T ,  RR ,  `'  <  ) `'  <  z  <->  z  <  sup ( T ,  RR ,  `'  <  ) )
3934, 38sylnibr 305 . . . . 5  |-  ( ( ( ( M  e. 
NN0  /\  R  e.  V  /\  F : R --> NN0 )  /\  T  =/=  (/) )  /\  z  e.  T )  ->  -.  sup ( T ,  RR ,  `'  <  ) `'  <  z )
4015, 26, 24, 39supmax 7711 . . . 4  |-  ( ( ( M  e.  NN0  /\  R  e.  V  /\  F : R --> NN0 )  /\  T  =/=  (/) )  ->  sup ( T ,  RR* ,  `'  <  )  =  sup ( T ,  RR ,  `'  <  ) )
4111, 40eqtrd 2473 . . 3  |-  ( ( ( M  e.  NN0  /\  R  e.  V  /\  F : R --> NN0 )  /\  T  =/=  (/) )  -> 
( M Ramsey  F )  =  sup ( T ,  RR ,  `'  <  ) )
4210, 41sylan2br 473 . 2  |-  ( ( ( M  e.  NN0  /\  R  e.  V  /\  F : R --> NN0 )  /\  -.  T  =  (/) )  ->  ( M Ramsey  F
)  =  sup ( T ,  RR ,  `'  <  ) )
431, 2, 9, 42ifbothda 3821 1  |-  ( ( M  e.  NN0  /\  R  e.  V  /\  F : R --> NN0 )  ->  ( M Ramsey  F )  =  if ( T  =  (/) , +oo ,  sup ( T ,  RR ,  `'  <  ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 960   A.wal 1362    = wceq 1364    e. wcel 1761    =/= wne 2604   A.wral 2713   E.wrex 2714   {crab 2717   _Vcvv 2970    C_ wss 3325   (/)c0 3634   ifcif 3788   ~Pcpw 3857   {csn 3874   class class class wbr 4289    Or wor 4636   `'ccnv 4835   "cima 4839   -->wf 5411   ` cfv 5415  (class class class)co 6090    e. cmpt2 6092    ^m cmap 7210   supcsup 7686   RRcr 9277   0cc0 9278   +oocpnf 9411   RR*cxr 9413    < clt 9414    <_ cle 9415   NN0cn0 10575   ZZ>=cuz 10857   #chash 12099   Ramsey cram 14056
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  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 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-er 7097  df-map 7212  df-en 7307  df-dom 7308  df-sdom 7309  df-sup 7687  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-nn 10319  df-n0 10576  df-z 10643  df-uz 10858  df-ram 14058
This theorem is referenced by:  ramtcl  14067  ramtcl2  14068  ramtub  14069  ramcl2  14073
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