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.

Step	Hyp	Ref	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 } ) ) ) }
5	3, 4	ramvalOLD 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
8	6, 7	syl6eq 2501	. . 3 |- ( T = (/) -> sup ( T , RR* , `' < ) = +oo )
9	5, 8	sylan9eq 2505	. 2 |- ( ( ( M e. NN0 /\ R e. V /\ F : R --> NN0 ) /\ T = (/) ) -> ( M Ramsey F ) = +oo )
10		df-ne 2624	. . 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 11440	. . . . . . 7 |- < Or RR*
13		cnvso 5375	. . . . . . 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 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
17	4, 16	eqsstri 3462	. . . . . . . 8 |- T C_ NN0
18		nn0ssre 10873	. . . . . . . 8 |- NN0 C_ RR
19	17, 18	sstri 3441	. . . . . . 7 |- T C_ RR
20		nn0uz 11193	. . . . . . . . . 10 |- NN0 = ( ZZ>= ` 0 )
21	17, 20	sseqtri 3464	. . . . . . . . 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 11247	. . . . . . . 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 3430	. . . . . 6 |- ( ( ( M e. NN0 /\ R e. V /\ F : R --> NN0 ) /\ T =/= (/) ) -> sup ( T , RR , `' < ) e. RR )
26	25	rexrd 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 )
29	21, 27, 28	sylancr 669	. . . . . . 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 3432	. . . . . . . 8 |- ( ( ( ( M e. NN0 /\ R e. V /\ F : R --> NN0 ) /\ T =/= (/) ) /\ z e. T ) -> z e. RR )
33	30, 32	lenltd 9781	. . . . . . 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 9715	. . . . . . . 8 |- `' < Or RR
36	35	supex 7977	. . . . . . 7 |- sup ( T , RR , `' < ) e. _V
37		vex 3048	. . . . . . 7 |- z e. _V
38	36, 37	brcnv 5017	. . . . . 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 7981	. . . 4 |- ( ( ( M e. NN0 /\ R e. V /\ F : R --> NN0 ) /\ T =/= (/) ) -> sup ( T , RR* , `' < ) = sup ( T , RR , `' < ) )
41	11, 40	eqtrd 2485	. . 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 3916	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 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