Theorem ramub1 15732

Description: Inductive step for Ramsey's theorem, in the form of an explicit upper bound. (Contributed by Mario Carneiro, 23-Apr-2015.)

Hypotheses

Ref	Expression
ramub1.m	|- ( ph -> M e. NN )
ramub1.r	|- ( ph -> R e. Fin )
ramub1.f	|- ( ph -> F : R --> NN )
ramub1.g	|- G = ( x e. R |-> ( M Ramsey ( y e. R |-> if ( y = x , ( ( F ` x ) - 1 ) , ( F ` y ) ) ) ) )
ramub1.1	|- ( ph -> G : R --> NN0 )
ramub1.2	|- ( ph -> ( ( M - 1 ) Ramsey G ) e. NN0 )

Assertion

Ref	Expression
ramub1	|- ( ph -> ( M Ramsey F ) <_ ( ( ( M - 1 ) Ramsey G ) + 1 ) )

Distinct variable groups:   x, y, F   x, M, y   x, G, y   x, R, y   ph, x, y

Proof of Theorem ramub1

Dummy variables u c f s v w z a b i are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2622	. 2 |- ( a e. _V , i e. NN0 |-> { b e. ~P a | ( # ` b ) = i } ) = ( a e. _V , i e. NN0 |-> { b e. ~P a | ( # ` b ) = i } )
2		ramub1.m	. . 3 |- ( ph -> M e. NN )
3	2	nnnn0d 11351	. 2 |- ( ph -> M e. NN0 )
4		ramub1.r	. 2 |- ( ph -> R e. Fin )
5		ramub1.f	. . 3 |- ( ph -> F : R --> NN )
6		nnssnn0 11295	. . 3 |- NN C_ NN0
7		fss 6056	. . 3 |- ( ( F : R --> NN /\ NN C_ NN0 ) -> F : R --> NN0 )
8	5, 6, 7	sylancl 694	. 2 |- ( ph -> F : R --> NN0 )
9		ramub1.2	. . 3 |- ( ph -> ( ( M - 1 ) Ramsey G ) e. NN0 )
10		peano2nn0 11333	. . 3 |- ( ( ( M - 1 ) Ramsey G ) e. NN0 -> ( ( ( M - 1 ) Ramsey G ) + 1 ) e. NN0 )
11	9, 10	syl 17	. 2 |- ( ph -> ( ( ( M - 1 ) Ramsey G ) + 1 ) e. NN0 )
12		simprl 794	. . . . . 6 |- ( ( ph /\ ( ( # ` s ) = ( ( ( M - 1 ) Ramsey G ) + 1 ) /\ f : ( s ( a e. _V , i e. NN0 |-> { b e. ~P a | ( # ` b ) = i } ) M ) --> R ) ) -> ( # ` s ) = ( ( ( M - 1 ) Ramsey G ) + 1 ) )
13	9	adantr 481	. . . . . . 7 |- ( ( ph /\ ( ( # ` s ) = ( ( ( M - 1 ) Ramsey G ) + 1 ) /\ f : ( s ( a e. _V , i e. NN0 |-> { b e. ~P a | ( # ` b ) = i } ) M ) --> R ) ) -> ( ( M - 1 ) Ramsey G ) e. NN0 )
14		nn0p1nn 11332	. . . . . . 7 |- ( ( ( M - 1 ) Ramsey G ) e. NN0 -> ( ( ( M - 1 ) Ramsey G ) + 1 ) e. NN )
15	13, 14	syl 17	. . . . . 6 |- ( ( ph /\ ( ( # ` s ) = ( ( ( M - 1 ) Ramsey G ) + 1 ) /\ f : ( s ( a e. _V , i e. NN0 |-> { b e. ~P a | ( # ` b ) = i } ) M ) --> R ) ) -> ( ( ( M - 1 ) Ramsey G ) + 1 ) e. NN )
16	12, 15	eqeltrd 2701	. . . . 5 |- ( ( ph /\ ( ( # ` s ) = ( ( ( M - 1 ) Ramsey G ) + 1 ) /\ f : ( s ( a e. _V , i e. NN0 |-> { b e. ~P a | ( # ` b ) = i } ) M ) --> R ) ) -> ( # ` s ) e. NN )
17	16	nnnn0d 11351	. . . . . . 7 |- ( ( ph /\ ( ( # ` s ) = ( ( ( M - 1 ) Ramsey G ) + 1 ) /\ f : ( s ( a e. _V , i e. NN0 |-> { b e. ~P a | ( # ` b ) = i } ) M ) --> R ) ) -> ( # ` s ) e. NN0 )
18		vex 3203	. . . . . . . 8 |- s e. _V
19		hashclb 13149	. . . . . . . 8 |- ( s e. _V -> ( s e. Fin <-> ( # ` s ) e. NN0 ) )
20	18, 19	ax-mp 5	. . . . . . 7 |- ( s e. Fin <-> ( # ` s ) e. NN0 )
21	17, 20	sylibr 224	. . . . . 6 |- ( ( ph /\ ( ( # ` s ) = ( ( ( M - 1 ) Ramsey G ) + 1 ) /\ f : ( s ( a e. _V , i e. NN0 |-> { b e. ~P a | ( # ` b ) = i } ) M ) --> R ) ) -> s e. Fin )
22		hashnncl 13157	. . . . . 6 |- ( s e. Fin -> ( ( # ` s ) e. NN <-> s =/= (/) ) )
23	21, 22	syl 17	. . . . 5 |- ( ( ph /\ ( ( # ` s ) = ( ( ( M - 1 ) Ramsey G ) + 1 ) /\ f : ( s ( a e. _V , i e. NN0 |-> { b e. ~P a | ( # ` b ) = i } ) M ) --> R ) ) -> ( ( # ` s ) e. NN <-> s =/= (/) ) )
24	16, 23	mpbid 222	. . . 4 |- ( ( ph /\ ( ( # ` s ) = ( ( ( M - 1 ) Ramsey G ) + 1 ) /\ f : ( s ( a e. _V , i e. NN0 |-> { b e. ~P a | ( # ` b ) = i } ) M ) --> R ) ) -> s =/= (/) )
25		n0 3931	. . . 4 |- ( s =/= (/) <-> E. w w e. s )
26	24, 25	sylib 208	. . 3 |- ( ( ph /\ ( ( # ` s ) = ( ( ( M - 1 ) Ramsey G ) + 1 ) /\ f : ( s ( a e. _V , i e. NN0 |-> { b e. ~P a | ( # ` b ) = i } ) M ) --> R ) ) -> E. w w e. s )
27	2	adantr 481	. . . . . 6 |- ( ( ph /\ ( ( ( # ` s ) = ( ( ( M - 1 ) Ramsey G ) + 1 ) /\ f : ( s ( a e. _V , i e. NN0 |-> { b e. ~P a | ( # ` b ) = i } ) M ) --> R ) /\ w e. s ) ) -> M e. NN )
28	4	adantr 481	. . . . . 6 |- ( ( ph /\ ( ( ( # ` s ) = ( ( ( M - 1 ) Ramsey G ) + 1 ) /\ f : ( s ( a e. _V , i e. NN0 |-> { b e. ~P a | ( # ` b ) = i } ) M ) --> R ) /\ w e. s ) ) -> R e. Fin )
29	5	adantr 481	. . . . . 6 |- ( ( ph /\ ( ( ( # ` s ) = ( ( ( M - 1 ) Ramsey G ) + 1 ) /\ f : ( s ( a e. _V , i e. NN0 |-> { b e. ~P a | ( # ` b ) = i } ) M ) --> R ) /\ w e. s ) ) -> F : R --> NN )
30		ramub1.g	. . . . . 6 |- G = ( x e. R |-> ( M Ramsey ( y e. R |-> if ( y = x , ( ( F ` x ) - 1 ) , ( F ` y ) ) ) ) )
31		ramub1.1	. . . . . . 7 |- ( ph -> G : R --> NN0 )
32	31	adantr 481	. . . . . 6 |- ( ( ph /\ ( ( ( # ` s ) = ( ( ( M - 1 ) Ramsey G ) + 1 ) /\ f : ( s ( a e. _V , i e. NN0 |-> { b e. ~P a | ( # ` b ) = i } ) M ) --> R ) /\ w e. s ) ) -> G : R --> NN0 )
33	9	adantr 481	. . . . . 6 |- ( ( ph /\ ( ( ( # ` s ) = ( ( ( M - 1 ) Ramsey G ) + 1 ) /\ f : ( s ( a e. _V , i e. NN0 |-> { b e. ~P a | ( # ` b ) = i } ) M ) --> R ) /\ w e. s ) ) -> ( ( M - 1 ) Ramsey G ) e. NN0 )
34	21	adantrr 753	. . . . . 6 |- ( ( ph /\ ( ( ( # ` s ) = ( ( ( M - 1 ) Ramsey G ) + 1 ) /\ f : ( s ( a e. _V , i e. NN0 |-> { b e. ~P a | ( # ` b ) = i } ) M ) --> R ) /\ w e. s ) ) -> s e. Fin )
35		simprll 802	. . . . . 6 |- ( ( ph /\ ( ( ( # ` s ) = ( ( ( M - 1 ) Ramsey G ) + 1 ) /\ f : ( s ( a e. _V , i e. NN0 |-> { b e. ~P a | ( # ` b ) = i } ) M ) --> R ) /\ w e. s ) ) -> ( # ` s ) = ( ( ( M - 1 ) Ramsey G ) + 1 ) )
36		simprlr 803	. . . . . 6 |- ( ( ph /\ ( ( ( # ` s ) = ( ( ( M - 1 ) Ramsey G ) + 1 ) /\ f : ( s ( a e. _V , i e. NN0 |-> { b e. ~P a | ( # ` b ) = i } ) M ) --> R ) /\ w e. s ) ) -> f : ( s ( a e. _V , i e. NN0 |-> { b e. ~P a | ( # ` b ) = i } ) M ) --> R )
37		simprr 796	. . . . . 6 |- ( ( ph /\ ( ( ( # ` s ) = ( ( ( M - 1 ) Ramsey G ) + 1 ) /\ f : ( s ( a e. _V , i e. NN0 |-> { b e. ~P a | ( # ` b ) = i } ) M ) --> R ) /\ w e. s ) ) -> w e. s )
38		uneq1 3760	. . . . . . . 8 |- ( v = u -> ( v u. { w } ) = ( u u. { w } ) )
39	38	fveq2d 6195	. . . . . . 7 |- ( v = u -> ( f ` ( v u. { w } ) ) = ( f ` ( u u. { w } ) ) )
40	39	cbvmptv 4750	. . . . . 6 |- ( v e. ( ( s \ { w } ) ( a e. _V , i e. NN0 |-> { b e. ~P a | ( # ` b ) = i } ) ( M - 1 ) ) |-> ( f ` ( v u. { w } ) ) ) = ( u e. ( ( s \ { w } ) ( a e. _V , i e. NN0 |-> { b e. ~P a | ( # ` b ) = i } ) ( M - 1 ) ) |-> ( f ` ( u u. { w } ) ) )
41	27, 28, 29, 30, 32, 33, 1, 34, 35, 36, 37, 40	ramub1lem2 15731	. . . . 5 |- ( ( ph /\ ( ( ( # ` s ) = ( ( ( M - 1 ) Ramsey G ) + 1 ) /\ f : ( s ( a e. _V , i e. NN0 |-> { b e. ~P a | ( # ` b ) = i } ) M ) --> R ) /\ w e. s ) ) -> E. c e. R E. z e. ~P s ( ( F ` c ) <_ ( # ` z ) /\ ( z ( a e. _V , i e. NN0 |-> { b e. ~P a | ( # ` b ) = i } ) M ) C_ ( `' f " { c } ) ) )
42	41	expr 643	. . . 4 |- ( ( ph /\ ( ( # ` s ) = ( ( ( M - 1 ) Ramsey G ) + 1 ) /\ f : ( s ( a e. _V , i e. NN0 |-> { b e. ~P a | ( # ` b ) = i } ) M ) --> R ) ) -> ( w e. s -> E. c e. R E. z e. ~P s ( ( F ` c ) <_ ( # ` z ) /\ ( z ( a e. _V , i e. NN0 |-> { b e. ~P a | ( # ` b ) = i } ) M ) C_ ( `' f " { c } ) ) ) )
43	42	exlimdv 1861	. . 3 |- ( ( ph /\ ( ( # ` s ) = ( ( ( M - 1 ) Ramsey G ) + 1 ) /\ f : ( s ( a e. _V , i e. NN0 |-> { b e. ~P a | ( # ` b ) = i } ) M ) --> R ) ) -> ( E. w w e. s -> E. c e. R E. z e. ~P s ( ( F ` c ) <_ ( # ` z ) /\ ( z ( a e. _V , i e. NN0 |-> { b e. ~P a | ( # ` b ) = i } ) M ) C_ ( `' f " { c } ) ) ) )
44	26, 43	mpd 15	. 2 |- ( ( ph /\ ( ( # ` s ) = ( ( ( M - 1 ) Ramsey G ) + 1 ) /\ f : ( s ( a e. _V , i e. NN0 |-> { b e. ~P a | ( # ` b ) = i } ) M ) --> R ) ) -> E. c e. R E. z e. ~P s ( ( F ` c ) <_ ( # ` z ) /\ ( z ( a e. _V , i e. NN0 |-> { b e. ~P a | ( # ` b ) = i } ) M ) C_ ( `' f " { c } ) ) )
45	1, 3, 4, 8, 11, 44	ramub2 15718	1 |- ( ph -> ( M Ramsey F ) <_ ( ( ( M - 1 ) Ramsey G ) + 1 ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   =/= wne 2794   E.wrex 2913   {crab 2916   _Vcvv 3200   \ cdif 3571   u. cun 3572   C_ wss 3574   (/)c0 3915   ifcif 4086   ~Pcpw 4158   {csn 4177   class class class wbr 4653   |-> cmpt 4729   `'ccnv 5113   "cima 5117   -->wf 5884   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   Fincfn 7955   1c1 9937   + caddc 9939   <_ cle 10075   - cmin 10266   NNcn 11020   NN0cn0 11292   #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-int 4476  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-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-card 8765  df-cda 8990  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-xnn0 11364  df-z 11378  df-uz 11688  df-fz 12327  df-hash 13118  df-ram 15705

This theorem is referenced by:  ramcl  15733