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Theorem ramcl2lemOLD 14956
Description: Lemma for extended real closure of the Ramsey number function. (Contributed by Mario Carneiro, 20-Apr-2015.) Obsolete version of ramcl2lem 14955 as of 14-Sep-2020. (New usage is discouraged.) (Proof modification is discouraged.)
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
ramcl2lemOLD  |-  ( ( 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 ramcl2lemOLD
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 eqeq2 2438 . 2  |-  ( +oo  =  if ( T  =  (/) , +oo ,  sup ( T ,  RR ,  `'  <  ) )  -> 
( ( M Ramsey  F
)  = +oo  <->  ( M Ramsey  F )  =  if ( T  =  (/) , +oo ,  sup ( T ,  RR ,  `'  <  ) ) ) )
2 eqeq2 2438 . 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, 4ramvalOLD 14954 . . 3  |-  ( ( M  e.  NN0  /\  R  e.  V  /\  F : R --> NN0 )  ->  ( M Ramsey  F )  =  sup ( T ,  RR* ,  `'  <  ) )
6 supeq1 7963 . . . 4  |-  ( T  =  (/)  ->  sup ( T ,  RR* ,  `'  <  )  =  sup ( (/)
,  RR* ,  `'  <  ) )
7 xrinfm0OLD 11629 . . . 4  |-  sup ( (/)
,  RR* ,  `'  <  )  = +oo
86, 7syl6eq 2480 . . 3  |-  ( T  =  (/)  ->  sup ( T ,  RR* ,  `'  <  )  = +oo )
95, 8sylan9eq 2484 . 2  |-  ( ( ( M  e.  NN0  /\  R  e.  V  /\  F : R --> NN0 )  /\  T  =  (/) )  -> 
( M Ramsey  F )  = +oo )
10 df-ne 2621 . . 3  |-  ( T  =/=  (/)  <->  -.  T  =  (/) )
115adantr 467 . . . 4  |-  ( ( ( M  e.  NN0  /\  R  e.  V  /\  F : R --> NN0 )  /\  T  =/=  (/) )  -> 
( M Ramsey  F )  =  sup ( T ,  RR* ,  `'  <  )
)
12 xrltso 11442 . . . . . . 7  |-  <  Or  RR*
13 cnvso 5392 . . . . . . 7  |-  (  < 
Or  RR*  <->  `'  <  Or  RR* )
1412, 13mpbi 212 . . . . . 6  |-  `'  <  Or 
RR*
1514a1i 11 . . . . 5  |-  ( ( ( M  e.  NN0  /\  R  e.  V  /\  F : R --> NN0 )  /\  T  =/=  (/) )  ->  `'  <  Or  RR* )
16 ssrab2 3547 . . . . . . . . 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 3495 . . . . . . . 8  |-  T  C_  NN0
18 nn0ssre 10875 . . . . . . . 8  |-  NN0  C_  RR
1917, 18sstri 3474 . . . . . . 7  |-  T  C_  RR
20 nn0uz 11195 . . . . . . . . . 10  |-  NN0  =  ( ZZ>= `  0 )
2117, 20sseqtri 3497 . . . . . . . . 9  |-  T  C_  ( ZZ>= `  0 )
2221a1i 11 . . . . . . . 8  |-  ( ( M  e.  NN0  /\  R  e.  V  /\  F : R --> NN0 )  ->  T  C_  ( ZZ>= ` 
0 ) )
23 infmssuzclOLD 11249 . . . . . . . 8  |-  ( ( T  C_  ( ZZ>= ` 
0 )  /\  T  =/=  (/) )  ->  sup ( T ,  RR ,  `'  <  )  e.  T
)
2422, 23sylan 474 . . . . . . 7  |-  ( ( ( M  e.  NN0  /\  R  e.  V  /\  F : R --> NN0 )  /\  T  =/=  (/) )  ->  sup ( T ,  RR ,  `'  <  )  e.  T )
2519, 24sseldi 3463 . . . . . 6  |-  ( ( ( M  e.  NN0  /\  R  e.  V  /\  F : R --> NN0 )  /\  T  =/=  (/) )  ->  sup ( T ,  RR ,  `'  <  )  e.  RR )
2625rexrd 9692 . . . . 5  |-  ( ( ( M  e.  NN0  /\  R  e.  V  /\  F : R --> NN0 )  /\  T  =/=  (/) )  ->  sup ( T ,  RR ,  `'  <  )  e. 
RR* )
27 simpr 463 . . . . . . . 8  |-  ( ( ( ( M  e. 
NN0  /\  R  e.  V  /\  F : R --> NN0 )  /\  T  =/=  (/) )  /\  z  e.  T )  ->  z  e.  T )
28 infmssuzleOLD 11248 . . . . . . . 8  |-  ( ( T  C_  ( ZZ>= ` 
0 )  /\  z  e.  T )  ->  sup ( T ,  RR ,  `'  <  )  <_  z
)
2921, 27, 28sylancr 668 . . . . . . 7  |-  ( ( ( ( M  e. 
NN0  /\  R  e.  V  /\  F : R --> NN0 )  /\  T  =/=  (/) )  /\  z  e.  T )  ->  sup ( T ,  RR ,  `'  <  )  <_  z
)
3025adantr 467 . . . . . . . 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 3465 . . . . . . . 8  |-  ( ( ( ( M  e. 
NN0  /\  R  e.  V  /\  F : R --> NN0 )  /\  T  =/=  (/) )  /\  z  e.  T )  ->  z  e.  RR )
3330, 32lenltd 9783 . . . . . . 7  |-  ( ( ( ( M  e. 
NN0  /\  R  e.  V  /\  F : R --> NN0 )  /\  T  =/=  (/) )  /\  z  e.  T )  ->  ( sup ( T ,  RR ,  `'  <  )  <_ 
z  <->  -.  z  <  sup ( T ,  RR ,  `'  <  ) ) )
3429, 33mpbid 214 . . . . . 6  |-  ( ( ( ( M  e. 
NN0  /\  R  e.  V  /\  F : R --> NN0 )  /\  T  =/=  (/) )  /\  z  e.  T )  ->  -.  z  <  sup ( T ,  RR ,  `'  <  ) )
35 gtso 9717 . . . . . . . 8  |-  `'  <  Or  RR
3635supex 7981 . . . . . . 7  |-  sup ( T ,  RR ,  `'  <  )  e.  _V
37 vex 3085 . . . . . . 7  |-  z  e. 
_V
3836, 37brcnv 5034 . . . . . 6  |-  ( sup ( T ,  RR ,  `'  <  ) `'  <  z  <->  z  <  sup ( T ,  RR ,  `'  <  ) )
3934, 38sylnibr 307 . . . . 5  |-  ( ( ( ( M  e. 
NN0  /\  R  e.  V  /\  F : R --> NN0 )  /\  T  =/=  (/) )  /\  z  e.  T )  ->  -.  sup ( T ,  RR ,  `'  <  ) `'  <  z )
4015, 26, 24, 39supmax 7985 . . . 4  |-  ( ( ( M  e.  NN0  /\  R  e.  V  /\  F : R --> NN0 )  /\  T  =/=  (/) )  ->  sup ( T ,  RR* ,  `'  <  )  =  sup ( T ,  RR ,  `'  <  ) )
4111, 40eqtrd 2464 . . 3  |-  ( ( ( M  e.  NN0  /\  R  e.  V  /\  F : R --> NN0 )  /\  T  =/=  (/) )  -> 
( M Ramsey  F )  =  sup ( T ,  RR ,  `'  <  ) )
4210, 41sylan2br 479 . 2  |-  ( ( ( M  e.  NN0  /\  R  e.  V  /\  F : R --> NN0 )  /\  -.  T  =  (/) )  ->  ( M Ramsey  F
)  =  sup ( T ,  RR ,  `'  <  ) )
431, 2, 9, 42ifbothda 3945 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 371    /\ w3a 983   A.wal 1436    = wceq 1438    e. wcel 1869    =/= wne 2619   A.wral 2776   E.wrex 2777   {crab 2780   _Vcvv 3082    C_ wss 3437   (/)c0 3762   ifcif 3910   ~Pcpw 3980   {csn 3997   class class class wbr 4421    Or wor 4771   `'ccnv 4850   "cima 4854   -->wf 5595   ` cfv 5599  (class class class)co 6303    |-> cmpt2 6305    ^m cmap 7478   supcsup 7958   RRcr 9540   0cc0 9541   +oocpnf 9674   RR*cxr 9676    < clt 9677    <_ cle 9678   NN0cn0 10871   ZZ>=cuz 11161   #chash 12516   Ramsey cramold 14943
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-rep 4534  ax-sep 4544  ax-nul 4553  ax-pow 4600  ax-pr 4658  ax-un 6595  ax-cnex 9597  ax-resscn 9598  ax-1cn 9599  ax-icn 9600  ax-addcl 9601  ax-addrcl 9602  ax-mulcl 9603  ax-mulrcl 9604  ax-mulcom 9605  ax-addass 9606  ax-mulass 9607  ax-distr 9608  ax-i2m1 9609  ax-1ne0 9610  ax-1rid 9611  ax-rnegex 9612  ax-rrecex 9613  ax-cnre 9614  ax-pre-lttri 9615  ax-pre-lttrn 9616  ax-pre-ltadd 9617  ax-pre-mulgt0 9618
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 984  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-nel 2622  df-ral 2781  df-rex 2782  df-reu 2783  df-rmo 2784  df-rab 2785  df-v 3084  df-sbc 3301  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3763  df-if 3911  df-pw 3982  df-sn 3998  df-pr 4000  df-tp 4002  df-op 4004  df-uni 4218  df-iun 4299  df-br 4422  df-opab 4481  df-mpt 4482  df-tr 4517  df-eprel 4762  df-id 4766  df-po 4772  df-so 4773  df-fr 4810  df-we 4812  df-xp 4857  df-rel 4858  df-cnv 4859  df-co 4860  df-dm 4861  df-rn 4862  df-res 4863  df-ima 4864  df-pred 5397  df-ord 5443  df-on 5444  df-lim 5445  df-suc 5446  df-iota 5563  df-fun 5601  df-fn 5602  df-f 5603  df-f1 5604  df-fo 5605  df-f1o 5606  df-fv 5607  df-riota 6265  df-ov 6306  df-oprab 6307  df-mpt2 6308  df-om 6705  df-1st 6805  df-2nd 6806  df-wrecs 7034  df-recs 7096  df-rdg 7134  df-er 7369  df-map 7480  df-en 7576  df-dom 7577  df-sdom 7578  df-sup 7960  df-pnf 9679  df-mnf 9680  df-xr 9681  df-ltxr 9682  df-le 9683  df-sub 9864  df-neg 9865  df-nn 10612  df-n0 10872  df-z 10940  df-uz 11162  df-ramOLD 14946
This theorem is referenced by:  ramtclOLD  14958  ramtcl2OLD  14960  ramtubOLD  14962  ramcl2OLD  14967
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