Theorem ramlb 15723

Description: Establish a lower bound on a Ramsey number. (Contributed by Mario Carneiro, 22-Apr-2015.)

Hypotheses

Ref	Expression
ramlb.c	|- C = ( a e. _V , i e. NN0 |-> { b e. ~P a | ( # ` b ) = i } )
ramlb.m	|- ( ph -> M e. NN0 )
ramlb.r	|- ( ph -> R e. V )
ramlb.f	|- ( ph -> F : R --> NN0 )
ramlb.s	|- ( ph -> N e. NN0 )
ramlb.g	|- ( ph -> G : ( ( 1 ... N ) C M ) --> R )
ramlb.i	|- ( ( ph /\ ( c e. R /\ x C_ ( 1 ... N ) ) ) -> ( ( x C M ) C_ ( `' G " { c } ) -> ( # ` x ) < ( F ` c ) ) )

Assertion

Ref	Expression
ramlb	|- ( ph -> N < ( M Ramsey F ) )

Distinct variable groups:   x, c, C   F, c, x   G, c, x   a, b, c, i, x, M   ph, c, x   N, c, x   R, c, x   V, c, x

Allowed substitution hints:   ph( i, a, b)   C( i, a, b)   R( i, a, b)   F( i, a, b)   G( i, a, b)   N( i, a, b)   V( i, a, b)

Proof of Theorem ramlb

Step	Hyp	Ref	Expression
1		ramlb.c	. . . . 5 |- C = ( a e. _V , i e. NN0 |-> { b e. ~P a | ( # ` b ) = i } )
2		ramlb.m	. . . . . 6 |- ( ph -> M e. NN0 )
3	2	adantr 481	. . . . 5 |- ( ( ph /\ ( M Ramsey F ) <_ N ) -> M e. NN0 )
4		ramlb.r	. . . . . 6 |- ( ph -> R e. V )
5	4	adantr 481	. . . . 5 |- ( ( ph /\ ( M Ramsey F ) <_ N ) -> R e. V )
6		ramlb.f	. . . . . 6 |- ( ph -> F : R --> NN0 )
7	6	adantr 481	. . . . 5 |- ( ( ph /\ ( M Ramsey F ) <_ N ) -> F : R --> NN0 )
8		ramlb.s	. . . . . . 7 |- ( ph -> N e. NN0 )
9	8	adantr 481	. . . . . 6 |- ( ( ph /\ ( M Ramsey F ) <_ N ) -> N e. NN0 )
10		simpr 477	. . . . . 6 |- ( ( ph /\ ( M Ramsey F ) <_ N ) -> ( M Ramsey F ) <_ N )
11		ramubcl 15722	. . . . . 6 |- ( ( ( M e. NN0 /\ R e. V /\ F : R --> NN0 ) /\ ( N e. NN0 /\ ( M Ramsey F ) <_ N ) ) -> ( M Ramsey F ) e. NN0 )
12	3, 5, 7, 9, 10, 11	syl32anc 1334	. . . . 5 |- ( ( ph /\ ( M Ramsey F ) <_ N ) -> ( M Ramsey F ) e. NN0 )
13		fzfid 12772	. . . . 5 |- ( ( ph /\ ( M Ramsey F ) <_ N ) -> ( 1 ... N ) e. Fin )
14		hashfz1 13134	. . . . . . . 8 |- ( N e. NN0 -> ( # ` ( 1 ... N ) ) = N )
15	8, 14	syl 17	. . . . . . 7 |- ( ph -> ( # ` ( 1 ... N ) ) = N )
16	15	breq2d 4665	. . . . . 6 |- ( ph -> ( ( M Ramsey F ) <_ ( # ` ( 1 ... N ) ) <-> ( M Ramsey F ) <_ N ) )
17	16	biimpar 502	. . . . 5 |- ( ( ph /\ ( M Ramsey F ) <_ N ) -> ( M Ramsey F ) <_ ( # ` ( 1 ... N ) ) )
18		ramlb.g	. . . . . 6 |- ( ph -> G : ( ( 1 ... N ) C M ) --> R )
19	18	adantr 481	. . . . 5 |- ( ( ph /\ ( M Ramsey F ) <_ N ) -> G : ( ( 1 ... N ) C M ) --> R )
20	1, 3, 5, 7, 12, 13, 17, 19	rami 15719	. . . 4 |- ( ( ph /\ ( M Ramsey F ) <_ N ) -> E. c e. R E. x e. ~P ( 1 ... N ) ( ( F ` c ) <_ ( # ` x ) /\ ( x C M ) C_ ( `' G " { c } ) ) )
21		elpwi 4168	. . . . . . . . 9 |- ( x e. ~P ( 1 ... N ) -> x C_ ( 1 ... N ) )
22		ramlb.i	. . . . . . . . . . 11 |- ( ( ph /\ ( c e. R /\ x C_ ( 1 ... N ) ) ) -> ( ( x C M ) C_ ( `' G " { c } ) -> ( # ` x ) < ( F ` c ) ) )
23	22	adantlr 751	. . . . . . . . . 10 |- ( ( ( ph /\ ( M Ramsey F ) <_ N ) /\ ( c e. R /\ x C_ ( 1 ... N ) ) ) -> ( ( x C M ) C_ ( `' G " { c } ) -> ( # ` x ) < ( F ` c ) ) )
24		fzfid 12772	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ ( M Ramsey F ) <_ N ) /\ ( c e. R /\ x C_ ( 1 ... N ) ) ) -> ( 1 ... N ) e. Fin )
25		simprr 796	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ ( M Ramsey F ) <_ N ) /\ ( c e. R /\ x C_ ( 1 ... N ) ) ) -> x C_ ( 1 ... N ) )
26		ssfi 8180	. . . . . . . . . . . . . 14 |- ( ( ( 1 ... N ) e. Fin /\ x C_ ( 1 ... N ) ) -> x e. Fin )
27	24, 25, 26	syl2anc 693	. . . . . . . . . . . . 13 |- ( ( ( ph /\ ( M Ramsey F ) <_ N ) /\ ( c e. R /\ x C_ ( 1 ... N ) ) ) -> x e. Fin )
28		hashcl 13147	. . . . . . . . . . . . 13 |- ( x e. Fin -> ( # ` x ) e. NN0 )
29	27, 28	syl 17	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( M Ramsey F ) <_ N ) /\ ( c e. R /\ x C_ ( 1 ... N ) ) ) -> ( # ` x ) e. NN0 )
30	29	nn0red 11352	. . . . . . . . . . 11 |- ( ( ( ph /\ ( M Ramsey F ) <_ N ) /\ ( c e. R /\ x C_ ( 1 ... N ) ) ) -> ( # ` x ) e. RR )
31		simpl 473	. . . . . . . . . . . . 13 |- ( ( c e. R /\ x C_ ( 1 ... N ) ) -> c e. R )
32		ffvelrn 6357	. . . . . . . . . . . . 13 |- ( ( F : R --> NN0 /\ c e. R ) -> ( F ` c ) e. NN0 )
33	7, 31, 32	syl2an 494	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( M Ramsey F ) <_ N ) /\ ( c e. R /\ x C_ ( 1 ... N ) ) ) -> ( F ` c ) e. NN0 )
34	33	nn0red 11352	. . . . . . . . . . 11 |- ( ( ( ph /\ ( M Ramsey F ) <_ N ) /\ ( c e. R /\ x C_ ( 1 ... N ) ) ) -> ( F ` c ) e. RR )
35	30, 34	ltnled 10184	. . . . . . . . . 10 |- ( ( ( ph /\ ( M Ramsey F ) <_ N ) /\ ( c e. R /\ x C_ ( 1 ... N ) ) ) -> ( ( # ` x ) < ( F ` c ) <-> -. ( F ` c ) <_ ( # ` x ) ) )
36	23, 35	sylibd 229	. . . . . . . . 9 |- ( ( ( ph /\ ( M Ramsey F ) <_ N ) /\ ( c e. R /\ x C_ ( 1 ... N ) ) ) -> ( ( x C M ) C_ ( `' G " { c } ) -> -. ( F ` c ) <_ ( # ` x ) ) )
37	21, 36	sylanr2 685	. . . . . . . 8 |- ( ( ( ph /\ ( M Ramsey F ) <_ N ) /\ ( c e. R /\ x e. ~P ( 1 ... N ) ) ) -> ( ( x C M ) C_ ( `' G " { c } ) -> -. ( F ` c ) <_ ( # ` x ) ) )
38	37	con2d 129	. . . . . . 7 |- ( ( ( ph /\ ( M Ramsey F ) <_ N ) /\ ( c e. R /\ x e. ~P ( 1 ... N ) ) ) -> ( ( F ` c ) <_ ( # ` x ) -> -. ( x C M ) C_ ( `' G " { c } ) ) )
39		imnan 438	. . . . . . 7 |- ( ( ( F ` c ) <_ ( # ` x ) -> -. ( x C M ) C_ ( `' G " { c } ) ) <-> -. ( ( F ` c ) <_ ( # ` x ) /\ ( x C M ) C_ ( `' G " { c } ) ) )
40	38, 39	sylib 208	. . . . . 6 |- ( ( ( ph /\ ( M Ramsey F ) <_ N ) /\ ( c e. R /\ x e. ~P ( 1 ... N ) ) ) -> -. ( ( F ` c ) <_ ( # ` x ) /\ ( x C M ) C_ ( `' G " { c } ) ) )
41	40	pm2.21d 118	. . . . 5 |- ( ( ( ph /\ ( M Ramsey F ) <_ N ) /\ ( c e. R /\ x e. ~P ( 1 ... N ) ) ) -> ( ( ( F ` c ) <_ ( # ` x ) /\ ( x C M ) C_ ( `' G " { c } ) ) -> -. ( M Ramsey F ) <_ N ) )
42	41	rexlimdvva 3038	. . . 4 |- ( ( ph /\ ( M Ramsey F ) <_ N ) -> ( E. c e. R E. x e. ~P ( 1 ... N ) ( ( F ` c ) <_ ( # ` x ) /\ ( x C M ) C_ ( `' G " { c } ) ) -> -. ( M Ramsey F ) <_ N ) )
43	20, 42	mpd 15	. . 3 |- ( ( ph /\ ( M Ramsey F ) <_ N ) -> -. ( M Ramsey F ) <_ N )
44	43	pm2.01da 458	. 2 |- ( ph -> -. ( M Ramsey F ) <_ N )
45	8	nn0red 11352	. . . 4 |- ( ph -> N e. RR )
46	45	rexrd 10089	. . 3 |- ( ph -> N e. RR* )
47		ramxrcl 15721	. . . 4 |- ( ( M e. NN0 /\ R e. V /\ F : R --> NN0 ) -> ( M Ramsey F ) e. RR* )
48	2, 4, 6, 47	syl3anc 1326	. . 3 |- ( ph -> ( M Ramsey F ) e. RR* )
49		xrltnle 10105	. . 3 |- ( ( N e. RR* /\ ( M Ramsey F ) e. RR* ) -> ( N < ( M Ramsey F ) <-> -. ( M Ramsey F ) <_ N ) )
50	46, 48, 49	syl2anc 693	. 2 |- ( ph -> ( N < ( M Ramsey F ) <-> -. ( M Ramsey F ) <_ N ) )
51	44, 50	mpbird 247	1 |- ( ph -> N < ( M Ramsey F ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   E.wrex 2913   {crab 2916   _Vcvv 3200   C_ wss 3574   ~Pcpw 4158   {csn 4177   class class class wbr 4653   `'ccnv 5113   "cima 5117   -->wf 5884   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   Fincfn 7955   1c1 9937   RR*cxr 10073   < clt 10074   <_ cle 10075   NN0cn0 11292   ...cfz 12326   #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-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-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-fz 12327  df-hash 13118  df-ram 15705

This theorem is referenced by:  0ram  15724  ram0  15726