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Theorem ramcl2lem 14386
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 2482 . 2  |-  ( +oo  =  if ( T  =  (/) , +oo ,  sup ( T ,  RR ,  `'  <  ) )  -> 
( ( M Ramsey  F
)  = +oo  <->  ( M Ramsey  F )  =  if ( T  =  (/) , +oo ,  sup ( T ,  RR ,  `'  <  ) ) ) )
2 eqeq2 2482 . 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 14385 . . 3  |-  ( ( M  e.  NN0  /\  R  e.  V  /\  F : R --> NN0 )  ->  ( M Ramsey  F )  =  sup ( T ,  RR* ,  `'  <  ) )
6 supeq1 7905 . . . 4  |-  ( T  =  (/)  ->  sup ( T ,  RR* ,  `'  <  )  =  sup ( (/)
,  RR* ,  `'  <  ) )
7 xrinfm0 11528 . . . 4  |-  sup ( (/)
,  RR* ,  `'  <  )  = +oo
86, 7syl6eq 2524 . . 3  |-  ( T  =  (/)  ->  sup ( T ,  RR* ,  `'  <  )  = +oo )
95, 8sylan9eq 2528 . 2  |-  ( ( ( M  e.  NN0  /\  R  e.  V  /\  F : R --> NN0 )  /\  T  =  (/) )  -> 
( M Ramsey  F )  = +oo )
10 df-ne 2664 . . 3  |-  ( T  =/=  (/)  <->  -.  T  =  (/) )
115adantr 465 . . . 4  |-  ( ( ( M  e.  NN0  /\  R  e.  V  /\  F : R --> NN0 )  /\  T  =/=  (/) )  -> 
( M Ramsey  F )  =  sup ( T ,  RR* ,  `'  <  )
)
12 xrltso 11347 . . . . . . 7  |-  <  Or  RR*
13 cnvso 5546 . . . . . . 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 3585 . . . . . . . . 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 3534 . . . . . . . 8  |-  T  C_  NN0
18 nn0ssre 10799 . . . . . . . 8  |-  NN0  C_  RR
1917, 18sstri 3513 . . . . . . 7  |-  T  C_  RR
20 nn0uz 11116 . . . . . . . . . 10  |-  NN0  =  ( ZZ>= `  0 )
2117, 20sseqtri 3536 . . . . . . . . 9  |-  T  C_  ( ZZ>= `  0 )
2221a1i 11 . . . . . . . 8  |-  ( ( M  e.  NN0  /\  R  e.  V  /\  F : R --> NN0 )  ->  T  C_  ( ZZ>= ` 
0 ) )
23 infmssuzcl 11165 . . . . . . . 8  |-  ( ( T  C_  ( ZZ>= ` 
0 )  /\  T  =/=  (/) )  ->  sup ( T ,  RR ,  `'  <  )  e.  T
)
2422, 23sylan 471 . . . . . . 7  |-  ( ( ( M  e.  NN0  /\  R  e.  V  /\  F : R --> NN0 )  /\  T  =/=  (/) )  ->  sup ( T ,  RR ,  `'  <  )  e.  T )
2519, 24sseldi 3502 . . . . . 6  |-  ( ( ( M  e.  NN0  /\  R  e.  V  /\  F : R --> NN0 )  /\  T  =/=  (/) )  ->  sup ( T ,  RR ,  `'  <  )  e.  RR )
2625rexrd 9643 . . . . 5  |-  ( ( ( M  e.  NN0  /\  R  e.  V  /\  F : R --> NN0 )  /\  T  =/=  (/) )  ->  sup ( T ,  RR ,  `'  <  )  e. 
RR* )
27 simpr 461 . . . . . . . 8  |-  ( ( ( ( M  e. 
NN0  /\  R  e.  V  /\  F : R --> NN0 )  /\  T  =/=  (/) )  /\  z  e.  T )  ->  z  e.  T )
28 infmssuzle 11164 . . . . . . . 8  |-  ( ( T  C_  ( ZZ>= ` 
0 )  /\  z  e.  T )  ->  sup ( T ,  RR ,  `'  <  )  <_  z
)
2921, 27, 28sylancr 663 . . . . . . 7  |-  ( ( ( ( M  e. 
NN0  /\  R  e.  V  /\  F : R --> NN0 )  /\  T  =/=  (/) )  /\  z  e.  T )  ->  sup ( T ,  RR ,  `'  <  )  <_  z
)
3025adantr 465 . . . . . . . 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 3504 . . . . . . . 8  |-  ( ( ( ( M  e. 
NN0  /\  R  e.  V  /\  F : R --> NN0 )  /\  T  =/=  (/) )  /\  z  e.  T )  ->  z  e.  RR )
3330, 32lenltd 9730 . . . . . . 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 9666 . . . . . . . 8  |-  `'  <  Or  RR
3635supex 7923 . . . . . . 7  |-  sup ( T ,  RR ,  `'  <  )  e.  _V
37 vex 3116 . . . . . . 7  |-  z  e. 
_V
3836, 37brcnv 5185 . . . . . 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 7925 . . . 4  |-  ( ( ( M  e.  NN0  /\  R  e.  V  /\  F : R --> NN0 )  /\  T  =/=  (/) )  ->  sup ( T ,  RR* ,  `'  <  )  =  sup ( T ,  RR ,  `'  <  ) )
4111, 40eqtrd 2508 . . 3  |-  ( ( ( M  e.  NN0  /\  R  e.  V  /\  F : R --> NN0 )  /\  T  =/=  (/) )  -> 
( M Ramsey  F )  =  sup ( T ,  RR ,  `'  <  ) )
4210, 41sylan2br 476 . 2  |-  ( ( ( M  e.  NN0  /\  R  e.  V  /\  F : R --> NN0 )  /\  -.  T  =  (/) )  ->  ( M Ramsey  F
)  =  sup ( T ,  RR ,  `'  <  ) )
431, 2, 9, 42ifbothda 3974 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 973   A.wal 1377    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2814   E.wrex 2815   {crab 2818   _Vcvv 3113    C_ wss 3476   (/)c0 3785   ifcif 3939   ~Pcpw 4010   {csn 4027   class class class wbr 4447    Or wor 4799   `'ccnv 4998   "cima 5002   -->wf 5584   ` cfv 5588  (class class class)co 6284    |-> cmpt2 6286    ^m cmap 7420   supcsup 7900   RRcr 9491   0cc0 9492   +oocpnf 9625   RR*cxr 9627    < clt 9628    <_ cle 9629   NN0cn0 10795   ZZ>=cuz 11082   #chash 12373   Ramsey cram 14376
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-om 6685  df-1st 6784  df-2nd 6785  df-recs 7042  df-rdg 7076  df-er 7311  df-map 7422  df-en 7517  df-dom 7518  df-sdom 7519  df-sup 7901  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-nn 10537  df-n0 10796  df-z 10865  df-uz 11083  df-ram 14378
This theorem is referenced by:  ramtcl  14387  ramtcl2  14388  ramtub  14389  ramcl2  14393
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