Theorem ramcl2lemOLD 15042

Description: Lemma for extended real closure of the Ramsey number function. (Contributed by Mario Carneiro, 20-Apr-2015.) Obsolete version of ramcl2lem 15041 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.

Step	Hyp	Ref	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 } ) ) ) }
5	3, 4	ramvalOLD 15040	. . 3 |- ( ( M e. NN0 /\ R e. V /\ F : R --> NN0 ) -> ( M Ramsey F ) = sup ( T , RR* , `' < ) )
6		supeq1 7977	. . . 4 |- ( T = (/) -> sup ( T , RR* , `' < ) = sup ( (/) , RR* , `' < ) )
7		xrinfm0OLD 11652	. . . 4 |- sup ( (/) , RR* , `' < ) = +oo
8	6, 7	syl6eq 2521	. . 3 |- ( T = (/) -> sup ( T , RR* , `' < ) = +oo )
9	5, 8	sylan9eq 2525	. 2 |- ( ( ( M e. NN0 /\ R e. V /\ F : R --> NN0 ) /\ T = (/) ) -> ( M Ramsey F ) = +oo )
10		df-ne 2643	. . 3 |- ( T =/= (/) <-> -. T = (/) )
11	5	adantr 472	. . . 4 |- ( ( ( M e. NN0 /\ R e. V /\ F : R --> NN0 ) /\ T =/= (/) ) -> ( M Ramsey F ) = sup ( T , RR* , `' < ) )
12		xrltso 11463	. . . . . . 7 |- < Or RR*
13		cnvso 5382	. . . . . . 7 |- ( < Or RR* <-> `' < Or RR* )
14	12, 13	mpbi 213	. . . . . 6 |- `' < Or RR*
15	14	a1i 11	. . . . 5 |- ( ( ( M e. NN0 /\ R e. V /\ F : R --> NN0 ) /\ T =/= (/) ) -> `' < Or RR* )
16		ssrab2 3500	. . . . . . . . 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
17	4, 16	eqsstri 3448	. . . . . . . 8 |- T C_ NN0
18		nn0ssre 10897	. . . . . . . 8 |- NN0 C_ RR
19	17, 18	sstri 3427	. . . . . . 7 |- T C_ RR
20		nn0uz 11217	. . . . . . . . . 10 |- NN0 = ( ZZ>= ` 0 )
21	17, 20	sseqtri 3450	. . . . . . . . 9 |- T C_ ( ZZ>= ` 0 )
22	21	a1i 11	. . . . . . . 8 |- ( ( M e. NN0 /\ R e. V /\ F : R --> NN0 ) -> T C_ ( ZZ>= ` 0 ) )
23		infmssuzclOLD 11270	. . . . . . . 8 |- ( ( T C_ ( ZZ>= ` 0 ) /\ T =/= (/) ) -> sup ( T , RR , `' < ) e. T )
24	22, 23	sylan 479	. . . . . . 7 |- ( ( ( M e. NN0 /\ R e. V /\ F : R --> NN0 ) /\ T =/= (/) ) -> sup ( T , RR , `' < ) e. T )
25	19, 24	sseldi 3416	. . . . . 6 |- ( ( ( M e. NN0 /\ R e. V /\ F : R --> NN0 ) /\ T =/= (/) ) -> sup ( T , RR , `' < ) e. RR )
26	25	rexrd 9708	. . . . 5 |- ( ( ( M e. NN0 /\ R e. V /\ F : R --> NN0 ) /\ T =/= (/) ) -> sup ( T , RR , `' < ) e. RR* )
27		simpr 468	. . . . . . . 8 |- ( ( ( ( M e. NN0 /\ R e. V /\ F : R --> NN0 ) /\ T =/= (/) ) /\ z e. T ) -> z e. T )
28		infmssuzleOLD 11269	. . . . . . . 8 |- ( ( T C_ ( ZZ>= ` 0 ) /\ z e. T ) -> sup ( T , RR , `' < ) <_ z )
29	21, 27, 28	sylancr 676	. . . . . . 7 |- ( ( ( ( M e. NN0 /\ R e. V /\ F : R --> NN0 ) /\ T =/= (/) ) /\ z e. T ) -> sup ( T , RR , `' < ) <_ z )
30	25	adantr 472	. . . . . . . 8 |- ( ( ( ( M e. NN0 /\ R e. V /\ F : R --> NN0 ) /\ T =/= (/) ) /\ z e. T ) -> sup ( T , RR , `' < ) e. RR )
31	19	a1i 11	. . . . . . . . 9 |- ( ( ( M e. NN0 /\ R e. V /\ F : R --> NN0 ) /\ T =/= (/) ) -> T C_ RR )
32	31	sselda 3418	. . . . . . . 8 |- ( ( ( ( M e. NN0 /\ R e. V /\ F : R --> NN0 ) /\ T =/= (/) ) /\ z e. T ) -> z e. RR )
33	30, 32	lenltd 9798	. . . . . . 7 |- ( ( ( ( M e. NN0 /\ R e. V /\ F : R --> NN0 ) /\ T =/= (/) ) /\ z e. T ) -> ( sup ( T , RR , `' < ) <_ z <-> -. z < sup ( T , RR , `' < ) ) )
34	29, 33	mpbid 215	. . . . . 6 |- ( ( ( ( M e. NN0 /\ R e. V /\ F : R --> NN0 ) /\ T =/= (/) ) /\ z e. T ) -> -. z < sup ( T , RR , `' < ) )
35		gtso 9733	. . . . . . . 8 |- `' < Or RR
36	35	supex 7995	. . . . . . 7 |- sup ( T , RR , `' < ) e. _V
37		vex 3034	. . . . . . 7 |- z e. _V
38	36, 37	brcnv 5022	. . . . . 6 |- ( sup ( T , RR , `' < ) `' < z <-> z < sup ( T , RR , `' < ) )
39	34, 38	sylnibr 312	. . . . 5 |- ( ( ( ( M e. NN0 /\ R e. V /\ F : R --> NN0 ) /\ T =/= (/) ) /\ z e. T ) -> -. sup ( T , RR , `' < ) `' < z )
40	15, 26, 24, 39	supmax 7999	. . . 4 |- ( ( ( M e. NN0 /\ R e. V /\ F : R --> NN0 ) /\ T =/= (/) ) -> sup ( T , RR* , `' < ) = sup ( T , RR , `' < ) )
41	11, 40	eqtrd 2505	. . 3 |- ( ( ( M e. NN0 /\ R e. V /\ F : R --> NN0 ) /\ T =/= (/) ) -> ( M Ramsey F ) = sup ( T , RR , `' < ) )
42	10, 41	sylan2br 484	. 2 |- ( ( ( M e. NN0 /\ R e. V /\ F : R --> NN0 ) /\ -. T = (/) ) -> ( M Ramsey F ) = sup ( T , RR , `' < ) )
43	1, 2, 9, 42	ifbothda 3907	1 |- ( ( M e. NN0 /\ R e. V /\ F : R --> NN0 ) -> ( M Ramsey F ) = if ( T = (/) , +oo , sup ( T , RR , `' < ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 376   /\ w3a 1007   A.wal 1450   = wceq 1452   e. wcel 1904   =/= wne 2641   A.wral 2756   E.wrex 2757   {crab 2760   _Vcvv 3031   C_ wss 3390   (/)c0 3722   ifcif 3872   ~Pcpw 3942   {csn 3959   class class class wbr 4395   Or wor 4759   `'ccnv 4838   "cima 4842   -->wf 5585   ` cfv 5589  (class class class)co 6308   |-> cmpt2 6310   ^m cmap 7490   supcsup 7972   RRcr 9556   0cc0 9557   +oocpnf 9690   RR*cxr 9692   < clt 9693   <_ cle 9694   NN0cn0 10893   ZZ>=cuz 11182   #chash 12553   Ramsey cramold 15029

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-er 7381  df-map 7492  df-en 7588  df-dom 7589  df-sdom 7590  df-sup 7974  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-n0 10894  df-z 10962  df-uz 11183  df-ramOLD 15032

This theorem is referenced by:  ramtclOLD  15044  ramtcl2OLD  15046  ramtubOLD  15048  ramcl2OLD  15053