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.

Step	Hyp	Ref	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 } ) ) ) }
5	3, 4	ramvalOLD 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
8	6, 7	syl6eq 2480	. . 3 |- ( T = (/) -> sup ( T , RR* , `' < ) = +oo )
9	5, 8	sylan9eq 2484	. 2 |- ( ( ( M e. NN0 /\ R e. V /\ F : R --> NN0 ) /\ T = (/) ) -> ( M Ramsey F ) = +oo )
10		df-ne 2621	. . 3 |- ( T =/= (/) <-> -. T = (/) )
11	5	adantr 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* )
14	12, 13	mpbi 212	. . . . . 6 |- `' < Or RR*
15	14	a1i 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
17	4, 16	eqsstri 3495	. . . . . . . 8 |- T C_ NN0
18		nn0ssre 10875	. . . . . . . 8 |- NN0 C_ RR
19	17, 18	sstri 3474	. . . . . . 7 |- T C_ RR
20		nn0uz 11195	. . . . . . . . . 10 |- NN0 = ( ZZ>= ` 0 )
21	17, 20	sseqtri 3497	. . . . . . . . 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 11249	. . . . . . . 8 |- ( ( T C_ ( ZZ>= ` 0 ) /\ T =/= (/) ) -> sup ( T , RR , `' < ) e. T )
24	22, 23	sylan 474	. . . . . . 7 |- ( ( ( M e. NN0 /\ R e. V /\ F : R --> NN0 ) /\ T =/= (/) ) -> sup ( T , RR , `' < ) e. T )
25	19, 24	sseldi 3463	. . . . . 6 |- ( ( ( M e. NN0 /\ R e. V /\ F : R --> NN0 ) /\ T =/= (/) ) -> sup ( T , RR , `' < ) e. RR )
26	25	rexrd 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 )
29	21, 27, 28	sylancr 668	. . . . . . 7 |- ( ( ( ( M e. NN0 /\ R e. V /\ F : R --> NN0 ) /\ T =/= (/) ) /\ z e. T ) -> sup ( T , RR , `' < ) <_ z )
30	25	adantr 467	. . . . . . . 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 3465	. . . . . . . 8 |- ( ( ( ( M e. NN0 /\ R e. V /\ F : R --> NN0 ) /\ T =/= (/) ) /\ z e. T ) -> z e. RR )
33	30, 32	lenltd 9783	. . . . . . 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 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
36	35	supex 7981	. . . . . . 7 |- sup ( T , RR , `' < ) e. _V
37		vex 3085	. . . . . . 7 |- z e. _V
38	36, 37	brcnv 5034	. . . . . 6 |- ( sup ( T , RR , `' < ) `' < z <-> z < sup ( T , RR , `' < ) )
39	34, 38	sylnibr 307	. . . . 5 |- ( ( ( ( M e. NN0 /\ R e. V /\ F : R --> NN0 ) /\ T =/= (/) ) /\ z e. T ) -> -. sup ( T , RR , `' < ) `' < z )
40	15, 26, 24, 39	supmax 7985	. . . 4 |- ( ( ( M e. NN0 /\ R e. V /\ F : R --> NN0 ) /\ T =/= (/) ) -> sup ( T , RR* , `' < ) = sup ( T , RR , `' < ) )
41	11, 40	eqtrd 2464	. . 3 |- ( ( ( M e. NN0 /\ R e. V /\ F : R --> NN0 ) /\ T =/= (/) ) -> ( M Ramsey F ) = sup ( T , RR , `' < ) )
42	10, 41	sylan2br 479	. 2 |- ( ( ( M e. NN0 /\ R e. V /\ F : R --> NN0 ) /\ -. T = (/) ) -> ( M Ramsey F ) = sup ( T , RR , `' < ) )
43	1, 2, 9, 42	ifbothda 3945	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 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