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Theorem ramcl2lemOLD 14963
Description: Lemma for extended real closure of the Ramsey number function. (Contributed by Mario Carneiro, 20-Apr-2015.) Obsolete version of ramcl2lem 14962 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 2462 . 2  |-  ( +oo  =  if ( T  =  (/) , +oo ,  sup ( T ,  RR ,  `'  <  ) )  -> 
( ( M Ramsey  F
)  = +oo  <->  ( M Ramsey  F )  =  if ( T  =  (/) , +oo ,  sup ( T ,  RR ,  `'  <  ) ) ) )
2 eqeq2 2462 . 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 14961 . . 3  |-  ( ( M  e.  NN0  /\  R  e.  V  /\  F : R --> NN0 )  ->  ( M Ramsey  F )  =  sup ( T ,  RR* ,  `'  <  ) )
6 supeq1 7959 . . . 4  |-  ( T  =  (/)  ->  sup ( T ,  RR* ,  `'  <  )  =  sup ( (/)
,  RR* ,  `'  <  ) )
7 xrinfm0OLD 11627 . . . 4  |-  sup ( (/)
,  RR* ,  `'  <  )  = +oo
86, 7syl6eq 2501 . . 3  |-  ( T  =  (/)  ->  sup ( T ,  RR* ,  `'  <  )  = +oo )
95, 8sylan9eq 2505 . 2  |-  ( ( ( M  e.  NN0  /\  R  e.  V  /\  F : R --> NN0 )  /\  T  =  (/) )  -> 
( M Ramsey  F )  = +oo )
10 df-ne 2624 . . 3  |-  ( T  =/=  (/)  <->  -.  T  =  (/) )
115adantr 467 . . . 4  |-  ( ( ( M  e.  NN0  /\  R  e.  V  /\  F : R --> NN0 )  /\  T  =/=  (/) )  -> 
( M Ramsey  F )  =  sup ( T ,  RR* ,  `'  <  )
)
12 xrltso 11440 . . . . . . 7  |-  <  Or  RR*
13 cnvso 5375 . . . . . . 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 3514 . . . . . . . . 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 3462 . . . . . . . 8  |-  T  C_  NN0
18 nn0ssre 10873 . . . . . . . 8  |-  NN0  C_  RR
1917, 18sstri 3441 . . . . . . 7  |-  T  C_  RR
20 nn0uz 11193 . . . . . . . . . 10  |-  NN0  =  ( ZZ>= `  0 )
2117, 20sseqtri 3464 . . . . . . . . 9  |-  T  C_  ( ZZ>= `  0 )
2221a1i 11 . . . . . . . 8  |-  ( ( M  e.  NN0  /\  R  e.  V  /\  F : R --> NN0 )  ->  T  C_  ( ZZ>= ` 
0 ) )
23 infmssuzclOLD 11247 . . . . . . . 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 3430 . . . . . 6  |-  ( ( ( M  e.  NN0  /\  R  e.  V  /\  F : R --> NN0 )  /\  T  =/=  (/) )  ->  sup ( T ,  RR ,  `'  <  )  e.  RR )
2625rexrd 9690 . . . . 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 11246 . . . . . . . 8  |-  ( ( T  C_  ( ZZ>= ` 
0 )  /\  z  e.  T )  ->  sup ( T ,  RR ,  `'  <  )  <_  z
)
2921, 27, 28sylancr 669 . . . . . . 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 3432 . . . . . . . 8  |-  ( ( ( ( M  e. 
NN0  /\  R  e.  V  /\  F : R --> NN0 )  /\  T  =/=  (/) )  /\  z  e.  T )  ->  z  e.  RR )
3330, 32lenltd 9781 . . . . . . 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 9715 . . . . . . . 8  |-  `'  <  Or  RR
3635supex 7977 . . . . . . 7  |-  sup ( T ,  RR ,  `'  <  )  e.  _V
37 vex 3048 . . . . . . 7  |-  z  e. 
_V
3836, 37brcnv 5017 . . . . . 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 7981 . . . 4  |-  ( ( ( M  e.  NN0  /\  R  e.  V  /\  F : R --> NN0 )  /\  T  =/=  (/) )  ->  sup ( T ,  RR* ,  `'  <  )  =  sup ( T ,  RR ,  `'  <  ) )
4111, 40eqtrd 2485 . . 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 3916 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 985   A.wal 1442    = wceq 1444    e. wcel 1887    =/= wne 2622   A.wral 2737   E.wrex 2738   {crab 2741   _Vcvv 3045    C_ wss 3404   (/)c0 3731   ifcif 3881   ~Pcpw 3951   {csn 3968   class class class wbr 4402    Or wor 4754   `'ccnv 4833   "cima 4837   -->wf 5578   ` cfv 5582  (class class class)co 6290    |-> cmpt2 6292    ^m cmap 7472   supcsup 7954   RRcr 9538   0cc0 9539   +oocpnf 9672   RR*cxr 9674    < clt 9675    <_ cle 9676   NN0cn0 10869   ZZ>=cuz 11159   #chash 12515   Ramsey cramold 14950
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-1st 6793  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-er 7363  df-map 7474  df-en 7570  df-dom 7571  df-sdom 7572  df-sup 7956  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-nn 10610  df-n0 10870  df-z 10938  df-uz 11160  df-ramOLD 14953
This theorem is referenced by:  ramtclOLD  14965  ramtcl2OLD  14967  ramtubOLD  14969  ramcl2OLD  14974
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