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Theorem ramcl2lem 14914
Description: Lemma for extended real closure of the Ramsey number function. (Contributed by Mario Carneiro, 20-Apr-2015.) (Revised by AV, 14-Sep-2020.)
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 , inf ( 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 2435 . 2  |-  ( +oo  =  if ( T  =  (/) , +oo , inf ( T ,  RR ,  <  ) )  ->  (
( M Ramsey  F )  = +oo  <->  ( M Ramsey  F
)  =  if ( T  =  (/) , +oo , inf ( T ,  RR ,  <  ) ) ) )
2 eqeq2 2435 . 2  |-  (inf ( T ,  RR ,  <  )  =  if ( T  =  (/) , +oo , inf ( T ,  RR ,  <  ) )  -> 
( ( M Ramsey  F
)  = inf ( T ,  RR ,  <  )  <-> 
( M Ramsey  F )  =  if ( T  =  (/) , +oo , inf ( 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 14912 . . 3  |-  ( ( M  e.  NN0  /\  R  e.  V  /\  F : R --> NN0 )  ->  ( M Ramsey  F )  = inf ( T ,  RR* ,  <  ) )
6 infeq1 7989 . . . 4  |-  ( T  =  (/)  -> inf ( T ,  RR* ,  <  )  = inf ( (/) ,  RR* ,  <  ) )
7 xrinf0 11612 . . . 4  |- inf ( (/) , 
RR* ,  <  )  = +oo
86, 7syl6eq 2477 . . 3  |-  ( T  =  (/)  -> inf ( T ,  RR* ,  <  )  = +oo )
95, 8sylan9eq 2481 . 2  |-  ( ( ( M  e.  NN0  /\  R  e.  V  /\  F : R --> NN0 )  /\  T  =  (/) )  -> 
( M Ramsey  F )  = +oo )
10 df-ne 2618 . . 3  |-  ( T  =/=  (/)  <->  -.  T  =  (/) )
115adantr 466 . . . 4  |-  ( ( ( M  e.  NN0  /\  R  e.  V  /\  F : R --> NN0 )  /\  T  =/=  (/) )  -> 
( M Ramsey  F )  = inf ( T ,  RR* ,  <  ) )
12 xrltso 11429 . . . . . 6  |-  <  Or  RR*
1312a1i 11 . . . . 5  |-  ( ( ( M  e.  NN0  /\  R  e.  V  /\  F : R --> NN0 )  /\  T  =/=  (/) )  ->  <  Or  RR* )
14 ssrab2 3543 . . . . . . . . 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
154, 14eqsstri 3491 . . . . . . . 8  |-  T  C_  NN0
16 nn0ssre 10862 . . . . . . . 8  |-  NN0  C_  RR
1715, 16sstri 3470 . . . . . . 7  |-  T  C_  RR
18 nn0uz 11182 . . . . . . . . . 10  |-  NN0  =  ( ZZ>= `  0 )
1915, 18sseqtri 3493 . . . . . . . . 9  |-  T  C_  ( ZZ>= `  0 )
2019a1i 11 . . . . . . . 8  |-  ( ( M  e.  NN0  /\  R  e.  V  /\  F : R --> NN0 )  ->  T  C_  ( ZZ>= ` 
0 ) )
21 infssuzcl 11234 . . . . . . . 8  |-  ( ( T  C_  ( ZZ>= ` 
0 )  /\  T  =/=  (/) )  -> inf ( T ,  RR ,  <  )  e.  T )
2220, 21sylan 473 . . . . . . 7  |-  ( ( ( M  e.  NN0  /\  R  e.  V  /\  F : R --> NN0 )  /\  T  =/=  (/) )  -> inf ( T ,  RR ,  <  )  e.  T )
2317, 22sseldi 3459 . . . . . 6  |-  ( ( ( M  e.  NN0  /\  R  e.  V  /\  F : R --> NN0 )  /\  T  =/=  (/) )  -> inf ( T ,  RR ,  <  )  e.  RR )
2423rexrd 9679 . . . . 5  |-  ( ( ( M  e.  NN0  /\  R  e.  V  /\  F : R --> NN0 )  /\  T  =/=  (/) )  -> inf ( T ,  RR ,  <  )  e.  RR* )
25 simpr 462 . . . . . . 7  |-  ( ( ( ( M  e. 
NN0  /\  R  e.  V  /\  F : R --> NN0 )  /\  T  =/=  (/) )  /\  z  e.  T )  ->  z  e.  T )
26 infssuzle 11233 . . . . . . 7  |-  ( ( T  C_  ( ZZ>= ` 
0 )  /\  z  e.  T )  -> inf ( T ,  RR ,  <  )  <_  z )
2719, 25, 26sylancr 667 . . . . . 6  |-  ( ( ( ( M  e. 
NN0  /\  R  e.  V  /\  F : R --> NN0 )  /\  T  =/=  (/) )  /\  z  e.  T )  -> inf ( T ,  RR ,  <  )  <_  z )
2823adantr 466 . . . . . . 7  |-  ( ( ( ( M  e. 
NN0  /\  R  e.  V  /\  F : R --> NN0 )  /\  T  =/=  (/) )  /\  z  e.  T )  -> inf ( T ,  RR ,  <  )  e.  RR )
2917a1i 11 . . . . . . . 8  |-  ( ( ( M  e.  NN0  /\  R  e.  V  /\  F : R --> NN0 )  /\  T  =/=  (/) )  ->  T  C_  RR )
3029sselda 3461 . . . . . . 7  |-  ( ( ( ( M  e. 
NN0  /\  R  e.  V  /\  F : R --> NN0 )  /\  T  =/=  (/) )  /\  z  e.  T )  ->  z  e.  RR )
3128, 30lenltd 9770 . . . . . 6  |-  ( ( ( ( M  e. 
NN0  /\  R  e.  V  /\  F : R --> NN0 )  /\  T  =/=  (/) )  /\  z  e.  T )  ->  (inf ( T ,  RR ,  <  )  <_  z  <->  -.  z  < inf ( T ,  RR ,  <  ) ) )
3227, 31mpbid 213 . . . . 5  |-  ( ( ( ( M  e. 
NN0  /\  R  e.  V  /\  F : R --> NN0 )  /\  T  =/=  (/) )  /\  z  e.  T )  ->  -.  z  < inf ( T ,  RR ,  <  ) )
3313, 24, 22, 32infmin 8007 . . . 4  |-  ( ( ( M  e.  NN0  /\  R  e.  V  /\  F : R --> NN0 )  /\  T  =/=  (/) )  -> inf ( T ,  RR* ,  <  )  = inf ( T ,  RR ,  <  ) )
3411, 33eqtrd 2461 . . 3  |-  ( ( ( M  e.  NN0  /\  R  e.  V  /\  F : R --> NN0 )  /\  T  =/=  (/) )  -> 
( M Ramsey  F )  = inf ( T ,  RR ,  <  ) )
3510, 34sylan2br 478 . 2  |-  ( ( ( M  e.  NN0  /\  R  e.  V  /\  F : R --> NN0 )  /\  -.  T  =  (/) )  ->  ( M Ramsey  F
)  = inf ( T ,  RR ,  <  ) )
361, 2, 9, 35ifbothda 3941 1  |-  ( ( M  e.  NN0  /\  R  e.  V  /\  F : R --> NN0 )  ->  ( M Ramsey  F )  =  if ( T  =  (/) , +oo , inf ( T ,  RR ,  <  ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 370    /\ w3a 982   A.wal 1435    = wceq 1437    e. wcel 1867    =/= wne 2616   A.wral 2773   E.wrex 2774   {crab 2777   _Vcvv 3078    C_ wss 3433   (/)c0 3758   ifcif 3906   ~Pcpw 3976   {csn 3993   class class class wbr 4417    Or wor 4765   `'ccnv 4844   "cima 4848   -->wf 5588   ` cfv 5592  (class class class)co 6296    |-> cmpt2 6298    ^m cmap 7471  infcinf 7952   RRcr 9527   0cc0 9528   +oocpnf 9661   RR*cxr 9663    < clt 9664    <_ cle 9665   NN0cn0 10858   ZZ>=cuz 11148   #chash 12501   Ramsey cram 14901
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4529  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588  ax-cnex 9584  ax-resscn 9585  ax-1cn 9586  ax-icn 9587  ax-addcl 9588  ax-addrcl 9589  ax-mulcl 9590  ax-mulrcl 9591  ax-mulcom 9592  ax-addass 9593  ax-mulass 9594  ax-distr 9595  ax-i2m1 9596  ax-1ne0 9597  ax-1rid 9598  ax-rnegex 9599  ax-rrecex 9600  ax-cnre 9601  ax-pre-lttri 9602  ax-pre-lttrn 9603  ax-pre-ltadd 9604  ax-pre-mulgt0 9605
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-nel 2619  df-ral 2778  df-rex 2779  df-reu 2780  df-rmo 2781  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-pss 3449  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-tp 3998  df-op 4000  df-uni 4214  df-iun 4295  df-br 4418  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4756  df-id 4760  df-po 4766  df-so 4767  df-fr 4804  df-we 4806  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-pred 5390  df-ord 5436  df-on 5437  df-lim 5438  df-suc 5439  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6698  df-1st 6798  df-2nd 6799  df-wrecs 7027  df-recs 7089  df-rdg 7127  df-er 7362  df-map 7473  df-en 7569  df-dom 7570  df-sdom 7571  df-sup 7953  df-inf 7954  df-pnf 9666  df-mnf 9667  df-xr 9668  df-ltxr 9669  df-le 9670  df-sub 9851  df-neg 9852  df-nn 10599  df-n0 10859  df-z 10927  df-uz 11149  df-ram 14904
This theorem is referenced by:  ramtcl  14916  ramtcl2  14918  ramtub  14920  ramcl2  14925
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