Theorem ramcl2lem 15713

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.

Step	Hyp	Ref	Expression
1		eqeq2 2633	. 2 |- ( +oo = if ( T = (/) , +oo , inf ( T , RR , < ) ) -> ( ( M Ramsey F ) = +oo <-> ( M Ramsey F ) = if ( T = (/) , +oo , inf ( T , RR , < ) ) ) )
2		eqeq2 2633	. 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 } ) ) ) }
5	3, 4	ramval 15712	. . 3 |- ( ( M e. NN0 /\ R e. V /\ F : R --> NN0 ) -> ( M Ramsey F ) = inf ( T , RR* , < ) )
6		infeq1 8382	. . . 4 |- ( T = (/) -> inf ( T , RR* , < ) = inf ( (/) , RR* , < ) )
7		xrinf0 12168	. . . 4 |- inf ( (/) , RR* , < ) = +oo
8	6, 7	syl6eq 2672	. . 3 |- ( T = (/) -> inf ( T , RR* , < ) = +oo )
9	5, 8	sylan9eq 2676	. 2 |- ( ( ( M e. NN0 /\ R e. V /\ F : R --> NN0 ) /\ T = (/) ) -> ( M Ramsey F ) = +oo )
10		df-ne 2795	. . 3 |- ( T =/= (/) <-> -. T = (/) )
11	5	adantr 481	. . . 4 |- ( ( ( M e. NN0 /\ R e. V /\ F : R --> NN0 ) /\ T =/= (/) ) -> ( M Ramsey F ) = inf ( T , RR* , < ) )
12		xrltso 11974	. . . . . 6 |- < Or RR*
13	12	a1i 11	. . . . 5 |- ( ( ( M e. NN0 /\ R e. V /\ F : R --> NN0 ) /\ T =/= (/) ) -> < Or RR* )
14		ssrab2 3687	. . . . . . . . 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
15	4, 14	eqsstri 3635	. . . . . . . 8 |- T C_ NN0
16		nn0ssre 11296	. . . . . . . 8 |- NN0 C_ RR
17	15, 16	sstri 3612	. . . . . . 7 |- T C_ RR
18		nn0uz 11722	. . . . . . . . . 10 |- NN0 = ( ZZ>= ` 0 )
19	15, 18	sseqtri 3637	. . . . . . . . 9 |- T C_ ( ZZ>= ` 0 )
20	19	a1i 11	. . . . . . . 8 |- ( ( M e. NN0 /\ R e. V /\ F : R --> NN0 ) -> T C_ ( ZZ>= ` 0 ) )
21		infssuzcl 11772	. . . . . . . 8 |- ( ( T C_ ( ZZ>= ` 0 ) /\ T =/= (/) ) -> inf ( T , RR , < ) e. T )
22	20, 21	sylan 488	. . . . . . 7 |- ( ( ( M e. NN0 /\ R e. V /\ F : R --> NN0 ) /\ T =/= (/) ) -> inf ( T , RR , < ) e. T )
23	17, 22	sseldi 3601	. . . . . 6 |- ( ( ( M e. NN0 /\ R e. V /\ F : R --> NN0 ) /\ T =/= (/) ) -> inf ( T , RR , < ) e. RR )
24	23	rexrd 10089	. . . . 5 |- ( ( ( M e. NN0 /\ R e. V /\ F : R --> NN0 ) /\ T =/= (/) ) -> inf ( T , RR , < ) e. RR* )
25		simpr 477	. . . . . . 7 |- ( ( ( ( M e. NN0 /\ R e. V /\ F : R --> NN0 ) /\ T =/= (/) ) /\ z e. T ) -> z e. T )
26		infssuzle 11771	. . . . . . 7 |- ( ( T C_ ( ZZ>= ` 0 ) /\ z e. T ) -> inf ( T , RR , < ) <_ z )
27	19, 25, 26	sylancr 695	. . . . . 6 |- ( ( ( ( M e. NN0 /\ R e. V /\ F : R --> NN0 ) /\ T =/= (/) ) /\ z e. T ) -> inf ( T , RR , < ) <_ z )
28	23	adantr 481	. . . . . . 7 |- ( ( ( ( M e. NN0 /\ R e. V /\ F : R --> NN0 ) /\ T =/= (/) ) /\ z e. T ) -> inf ( T , RR , < ) e. RR )
29	17	a1i 11	. . . . . . . 8 |- ( ( ( M e. NN0 /\ R e. V /\ F : R --> NN0 ) /\ T =/= (/) ) -> T C_ RR )
30	29	sselda 3603	. . . . . . 7 |- ( ( ( ( M e. NN0 /\ R e. V /\ F : R --> NN0 ) /\ T =/= (/) ) /\ z e. T ) -> z e. RR )
31	28, 30	lenltd 10183	. . . . . 6 |- ( ( ( ( M e. NN0 /\ R e. V /\ F : R --> NN0 ) /\ T =/= (/) ) /\ z e. T ) -> (inf ( T , RR , < ) <_ z <-> -. z < inf ( T , RR , < ) ) )
32	27, 31	mpbid 222	. . . . 5 |- ( ( ( ( M e. NN0 /\ R e. V /\ F : R --> NN0 ) /\ T =/= (/) ) /\ z e. T ) -> -. z < inf ( T , RR , < ) )
33	13, 24, 22, 32	infmin 8400	. . . 4 |- ( ( ( M e. NN0 /\ R e. V /\ F : R --> NN0 ) /\ T =/= (/) ) -> inf ( T , RR* , < ) = inf ( T , RR , < ) )
34	11, 33	eqtrd 2656	. . 3 |- ( ( ( M e. NN0 /\ R e. V /\ F : R --> NN0 ) /\ T =/= (/) ) -> ( M Ramsey F ) = inf ( T , RR , < ) )
35	10, 34	sylan2br 493	. 2 |- ( ( ( M e. NN0 /\ R e. V /\ F : R --> NN0 ) /\ -. T = (/) ) -> ( M Ramsey F ) = inf ( T , RR , < ) )
36	1, 2, 9, 35	ifbothda 4123	1 |- ( ( M e. NN0 /\ R e. V /\ F : R --> NN0 ) -> ( M Ramsey F ) = if ( T = (/) , +oo , inf ( T , RR , < ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   /\ w3a 1037   A.wal 1481   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   {crab 2916   _Vcvv 3200   C_ wss 3574   (/)c0 3915   ifcif 4086   ~Pcpw 4158   {csn 4177   class class class wbr 4653   Or wor 5034   `'ccnv 5113   "cima 5117   -->wf 5884   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   ^m cmap 7857  infcinf 8347   RRcr 9935   0cc0 9936   +oocpnf 10071   RR*cxr 10073   < clt 10074   <_ cle 10075   NN0cn0 11292   ZZ>=cuz 11687   #chash 13117   Ramsey cram 15703

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-inf 8349  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-ram 15705

This theorem is referenced by:  ramtcl  15714  ramtcl2  15715  ramtub  15716  ramcl2  15720