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Theorem ramcl2lem 13340
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 2421 . 2  |-  (  +oo  =  if ( T  =  (/) ,  +oo ,  sup ( T ,  RR ,  `'  <  ) )  -> 
( ( M Ramsey  F
)  =  +oo  <->  ( M Ramsey  F )  =  if ( T  =  (/) ,  +oo ,  sup ( T ,  RR ,  `'  <  ) ) ) )
2 eqeq2 2421 . 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 13339 . . 3  |-  ( ( M  e.  NN0  /\  R  e.  V  /\  F : R --> NN0 )  ->  ( M Ramsey  F )  =  sup ( T ,  RR* ,  `'  <  ) )
6 supeq1 7416 . . . 4  |-  ( T  =  (/)  ->  sup ( T ,  RR* ,  `'  <  )  =  sup ( (/)
,  RR* ,  `'  <  ) )
7 xrinfm0 10879 . . . 4  |-  sup ( (/)
,  RR* ,  `'  <  )  =  +oo
86, 7syl6eq 2460 . . 3  |-  ( T  =  (/)  ->  sup ( T ,  RR* ,  `'  <  )  =  +oo )
95, 8sylan9eq 2464 . 2  |-  ( ( ( M  e.  NN0  /\  R  e.  V  /\  F : R --> NN0 )  /\  T  =  (/) )  -> 
( M Ramsey  F )  =  +oo )
10 df-ne 2577 . . 3  |-  ( T  =/=  (/)  <->  -.  T  =  (/) )
115adantr 452 . . . 4  |-  ( ( ( M  e.  NN0  /\  R  e.  V  /\  F : R --> NN0 )  /\  T  =/=  (/) )  -> 
( M Ramsey  F )  =  sup ( T ,  RR* ,  `'  <  )
)
12 xrltso 10698 . . . . . . 7  |-  <  Or  RR*
13 cnvso 5378 . . . . . . 7  |-  (  < 
Or  RR*  <->  `'  <  Or  RR* )
1412, 13mpbi 200 . . . . . 6  |-  `'  <  Or 
RR*
1514a1i 11 . . . . 5  |-  ( ( ( M  e.  NN0  /\  R  e.  V  /\  F : R --> NN0 )  /\  T  =/=  (/) )  ->  `'  <  Or  RR* )
16 ssrab2 3396 . . . . . . . . 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 3346 . . . . . . . 8  |-  T  C_  NN0
18 nn0ssre 10189 . . . . . . . 8  |-  NN0  C_  RR
1917, 18sstri 3325 . . . . . . 7  |-  T  C_  RR
20 nn0uz 10484 . . . . . . . . . 10  |-  NN0  =  ( ZZ>= `  0 )
2117, 20sseqtri 3348 . . . . . . . . 9  |-  T  C_  ( ZZ>= `  0 )
2221a1i 11 . . . . . . . 8  |-  ( ( M  e.  NN0  /\  R  e.  V  /\  F : R --> NN0 )  ->  T  C_  ( ZZ>= ` 
0 ) )
23 infmssuzcl 10523 . . . . . . . 8  |-  ( ( T  C_  ( ZZ>= ` 
0 )  /\  T  =/=  (/) )  ->  sup ( T ,  RR ,  `'  <  )  e.  T
)
2422, 23sylan 458 . . . . . . 7  |-  ( ( ( M  e.  NN0  /\  R  e.  V  /\  F : R --> NN0 )  /\  T  =/=  (/) )  ->  sup ( T ,  RR ,  `'  <  )  e.  T )
2519, 24sseldi 3314 . . . . . 6  |-  ( ( ( M  e.  NN0  /\  R  e.  V  /\  F : R --> NN0 )  /\  T  =/=  (/) )  ->  sup ( T ,  RR ,  `'  <  )  e.  RR )
2625rexrd 9098 . . . . 5  |-  ( ( ( M  e.  NN0  /\  R  e.  V  /\  F : R --> NN0 )  /\  T  =/=  (/) )  ->  sup ( T ,  RR ,  `'  <  )  e. 
RR* )
27 simpr 448 . . . . . . . 8  |-  ( ( ( ( M  e. 
NN0  /\  R  e.  V  /\  F : R --> NN0 )  /\  T  =/=  (/) )  /\  z  e.  T )  ->  z  e.  T )
28 infmssuzle 10522 . . . . . . . 8  |-  ( ( T  C_  ( ZZ>= ` 
0 )  /\  z  e.  T )  ->  sup ( T ,  RR ,  `'  <  )  <_  z
)
2921, 27, 28sylancr 645 . . . . . . 7  |-  ( ( ( ( M  e. 
NN0  /\  R  e.  V  /\  F : R --> NN0 )  /\  T  =/=  (/) )  /\  z  e.  T )  ->  sup ( T ,  RR ,  `'  <  )  <_  z
)
3025adantr 452 . . . . . . . 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 3316 . . . . . . . 8  |-  ( ( ( ( M  e. 
NN0  /\  R  e.  V  /\  F : R --> NN0 )  /\  T  =/=  (/) )  /\  z  e.  T )  ->  z  e.  RR )
3330, 32lenltd 9183 . . . . . . 7  |-  ( ( ( ( M  e. 
NN0  /\  R  e.  V  /\  F : R --> NN0 )  /\  T  =/=  (/) )  /\  z  e.  T )  ->  ( sup ( T ,  RR ,  `'  <  )  <_ 
z  <->  -.  z  <  sup ( T ,  RR ,  `'  <  ) ) )
3429, 33mpbid 202 . . . . . 6  |-  ( ( ( ( M  e. 
NN0  /\  R  e.  V  /\  F : R --> NN0 )  /\  T  =/=  (/) )  /\  z  e.  T )  ->  -.  z  <  sup ( T ,  RR ,  `'  <  ) )
35 ltso 9120 . . . . . . . . 9  |-  <  Or  RR
36 cnvso 5378 . . . . . . . . 9  |-  (  < 
Or  RR  <->  `'  <  Or  RR )
3735, 36mpbi 200 . . . . . . . 8  |-  `'  <  Or  RR
3837supex 7432 . . . . . . 7  |-  sup ( T ,  RR ,  `'  <  )  e.  _V
39 vex 2927 . . . . . . 7  |-  z  e. 
_V
4038, 39brcnv 5022 . . . . . 6  |-  ( sup ( T ,  RR ,  `'  <  ) `'  <  z  <->  z  <  sup ( T ,  RR ,  `'  <  ) )
4134, 40sylnibr 297 . . . . 5  |-  ( ( ( ( M  e. 
NN0  /\  R  e.  V  /\  F : R --> NN0 )  /\  T  =/=  (/) )  /\  z  e.  T )  ->  -.  sup ( T ,  RR ,  `'  <  ) `'  <  z )
4215, 26, 24, 41supmax 7434 . . . 4  |-  ( ( ( M  e.  NN0  /\  R  e.  V  /\  F : R --> NN0 )  /\  T  =/=  (/) )  ->  sup ( T ,  RR* ,  `'  <  )  =  sup ( T ,  RR ,  `'  <  ) )
4311, 42eqtrd 2444 . . 3  |-  ( ( ( M  e.  NN0  /\  R  e.  V  /\  F : R --> NN0 )  /\  T  =/=  (/) )  -> 
( M Ramsey  F )  =  sup ( T ,  RR ,  `'  <  ) )
4410, 43sylan2br 463 . 2  |-  ( ( ( M  e.  NN0  /\  R  e.  V  /\  F : R --> NN0 )  /\  -.  T  =  (/) )  ->  ( M Ramsey  F
)  =  sup ( T ,  RR ,  `'  <  ) )
451, 2, 9, 44ifbothda 3737 1  |-  ( ( M  e.  NN0  /\  R  e.  V  /\  F : R --> NN0 )  ->  ( M Ramsey  F )  =  if ( T  =  (/) ,  +oo ,  sup ( T ,  RR ,  `'  <  ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    /\ w3a 936   A.wal 1546    = wceq 1649    e. wcel 1721    =/= wne 2575   A.wral 2674   E.wrex 2675   {crab 2678   _Vcvv 2924    C_ wss 3288   (/)c0 3596   ifcif 3707   ~Pcpw 3767   {csn 3782   class class class wbr 4180    Or wor 4470   `'ccnv 4844   "cima 4848   -->wf 5417   ` cfv 5421  (class class class)co 6048    e. cmpt2 6050    ^m cmap 6985   supcsup 7411   RRcr 8953   0cc0 8954    +oocpnf 9081   RR*cxr 9083    < clt 9084    <_ cle 9085   NN0cn0 10185   ZZ>=cuz 10452   #chash 11581   Ramsey cram 13330
This theorem is referenced by:  ramtcl  13341  ramtcl2  13342  ramtub  13343  ramcl2  13347
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-cnex 9010  ax-resscn 9011  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-mulcom 9018  ax-addass 9019  ax-mulass 9020  ax-distr 9021  ax-i2m1 9022  ax-1ne0 9023  ax-1rid 9024  ax-rnegex 9025  ax-rrecex 9026  ax-cnre 9027  ax-pre-lttri 9028  ax-pre-lttrn 9029  ax-pre-ltadd 9030  ax-pre-mulgt0 9031
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-riota 6516  df-recs 6600  df-rdg 6635  df-er 6872  df-map 6987  df-en 7077  df-dom 7078  df-sdom 7079  df-sup 7412  df-pnf 9086  df-mnf 9087  df-xr 9088  df-ltxr 9089  df-le 9090  df-sub 9257  df-neg 9258  df-nn 9965  df-n0 10186  df-z 10247  df-uz 10453  df-ram 13332
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