Theorem 0ramcl 15727

Description: Lemma for ramcl 15733: Existence of the Ramsey number when M = 0. (Contributed by Mario Carneiro, 23-Apr-2015.)

Assertion

Ref	Expression
0ramcl	|- ( ( R e. Fin /\ F : R --> NN0 ) -> ( 0 Ramsey F ) e. NN0 )

Proof of Theorem 0ramcl

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ffn 6045	. . . . . . . 8 |- ( F : R --> NN0 -> F Fn R )
2		dffn4 6121	. . . . . . . 8 |- ( F Fn R <-> F : R -onto-> ran F )
3	1, 2	sylib 208	. . . . . . 7 |- ( F : R --> NN0 -> F : R -onto-> ran F )
4	3	ad2antlr 763	. . . . . 6 |- ( ( ( R e. Fin /\ F : R --> NN0 ) /\ R = (/) ) -> F : R -onto-> ran F )
5		foeq2 6112	. . . . . . 7 |- ( R = (/) -> ( F : R -onto-> ran F <-> F : (/) -onto-> ran F ) )
6	5	adantl 482	. . . . . 6 |- ( ( ( R e. Fin /\ F : R --> NN0 ) /\ R = (/) ) -> ( F : R -onto-> ran F <-> F : (/) -onto-> ran F ) )
7	4, 6	mpbid 222	. . . . 5 |- ( ( ( R e. Fin /\ F : R --> NN0 ) /\ R = (/) ) -> F : (/) -onto-> ran F )
8		fo00 6172	. . . . . 6 |- ( F : (/) -onto-> ran F <-> ( F = (/) /\ ran F = (/) ) )
9	8	simplbi 476	. . . . 5 |- ( F : (/) -onto-> ran F -> F = (/) )
10	7, 9	syl 17	. . . 4 |- ( ( ( R e. Fin /\ F : R --> NN0 ) /\ R = (/) ) -> F = (/) )
11	10	oveq2d 6666	. . 3 |- ( ( ( R e. Fin /\ F : R --> NN0 ) /\ R = (/) ) -> ( 0 Ramsey F ) = ( 0 Ramsey (/) ) )
12		0nn0 11307	. . . . 5 |- 0 e. NN0
13		ram0 15726	. . . . 5 |- ( 0 e. NN0 -> ( 0 Ramsey (/) ) = 0 )
14	12, 13	ax-mp 5	. . . 4 |- ( 0 Ramsey (/) ) = 0
15	14, 12	eqeltri 2697	. . 3 |- ( 0 Ramsey (/) ) e. NN0
16	11, 15	syl6eqel 2709	. 2 |- ( ( ( R e. Fin /\ F : R --> NN0 ) /\ R = (/) ) -> ( 0 Ramsey F ) e. NN0 )
17		0ram2 15725	. . . . 5 |- ( ( R e. Fin /\ R =/= (/) /\ F : R --> NN0 ) -> ( 0 Ramsey F ) = sup ( ran F , RR , < ) )
18		frn 6053	. . . . . . 7 |- ( F : R --> NN0 -> ran F C_ NN0 )
19	18	3ad2ant3 1084	. . . . . 6 |- ( ( R e. Fin /\ R =/= (/) /\ F : R --> NN0 ) -> ran F C_ NN0 )
20		nn0ssz 11398	. . . . . . . 8 |- NN0 C_ ZZ
21	19, 20	syl6ss 3615	. . . . . . 7 |- ( ( R e. Fin /\ R =/= (/) /\ F : R --> NN0 ) -> ran F C_ ZZ )
22		fdm 6051	. . . . . . . . . 10 |- ( F : R --> NN0 -> dom F = R )
23	22	3ad2ant3 1084	. . . . . . . . 9 |- ( ( R e. Fin /\ R =/= (/) /\ F : R --> NN0 ) -> dom F = R )
24		simp2 1062	. . . . . . . . 9 |- ( ( R e. Fin /\ R =/= (/) /\ F : R --> NN0 ) -> R =/= (/) )
25	23, 24	eqnetrd 2861	. . . . . . . 8 |- ( ( R e. Fin /\ R =/= (/) /\ F : R --> NN0 ) -> dom F =/= (/) )
26		dm0rn0 5342	. . . . . . . . 9 |- ( dom F = (/) <-> ran F = (/) )
27	26	necon3bii 2846	. . . . . . . 8 |- ( dom F =/= (/) <-> ran F =/= (/) )
28	25, 27	sylib 208	. . . . . . 7 |- ( ( R e. Fin /\ R =/= (/) /\ F : R --> NN0 ) -> ran F =/= (/) )
29		nn0ssre 11296	. . . . . . . . . 10 |- NN0 C_ RR
30	19, 29	syl6ss 3615	. . . . . . . . 9 |- ( ( R e. Fin /\ R =/= (/) /\ F : R --> NN0 ) -> ran F C_ RR )
31		simp1 1061	. . . . . . . . . 10 |- ( ( R e. Fin /\ R =/= (/) /\ F : R --> NN0 ) -> R e. Fin )
32	3	3ad2ant3 1084	. . . . . . . . . 10 |- ( ( R e. Fin /\ R =/= (/) /\ F : R --> NN0 ) -> F : R -onto-> ran F )
33		fofi 8252	. . . . . . . . . 10 |- ( ( R e. Fin /\ F : R -onto-> ran F ) -> ran F e. Fin )
34	31, 32, 33	syl2anc 693	. . . . . . . . 9 |- ( ( R e. Fin /\ R =/= (/) /\ F : R --> NN0 ) -> ran F e. Fin )
35		fimaxre 10968	. . . . . . . . 9 |- ( ( ran F C_ RR /\ ran F e. Fin /\ ran F =/= (/) ) -> E. x e. ran F A. y e. ran F y <_ x )
36	30, 34, 28, 35	syl3anc 1326	. . . . . . . 8 |- ( ( R e. Fin /\ R =/= (/) /\ F : R --> NN0 ) -> E. x e. ran F A. y e. ran F y <_ x )
37		ssrexv 3667	. . . . . . . 8 |- ( ran F C_ ZZ -> ( E. x e. ran F A. y e. ran F y <_ x -> E. x e. ZZ A. y e. ran F y <_ x ) )
38	21, 36, 37	sylc 65	. . . . . . 7 |- ( ( R e. Fin /\ R =/= (/) /\ F : R --> NN0 ) -> E. x e. ZZ A. y e. ran F y <_ x )
39		suprzcl2 11778	. . . . . . 7 |- ( ( ran F C_ ZZ /\ ran F =/= (/) /\ E. x e. ZZ A. y e. ran F y <_ x ) -> sup ( ran F , RR , < ) e. ran F )
40	21, 28, 38, 39	syl3anc 1326	. . . . . 6 |- ( ( R e. Fin /\ R =/= (/) /\ F : R --> NN0 ) -> sup ( ran F , RR , < ) e. ran F )
41	19, 40	sseldd 3604	. . . . 5 |- ( ( R e. Fin /\ R =/= (/) /\ F : R --> NN0 ) -> sup ( ran F , RR , < ) e. NN0 )
42	17, 41	eqeltrd 2701	. . . 4 |- ( ( R e. Fin /\ R =/= (/) /\ F : R --> NN0 ) -> ( 0 Ramsey F ) e. NN0 )
43	42	3expa 1265	. . 3 |- ( ( ( R e. Fin /\ R =/= (/) ) /\ F : R --> NN0 ) -> ( 0 Ramsey F ) e. NN0 )
44	43	an32s 846	. 2 |- ( ( ( R e. Fin /\ F : R --> NN0 ) /\ R =/= (/) ) -> ( 0 Ramsey F ) e. NN0 )
45	16, 44	pm2.61dane 2881	1 |- ( ( R e. Fin /\ F : R --> NN0 ) -> ( 0 Ramsey F ) e. NN0 )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   C_ wss 3574   (/)c0 3915   class class class wbr 4653   dom cdm 5114   ran crn 5115   Fn wfn 5883   -->wf 5884   -onto->wfo 5886  (class class class)co 6650   Fincfn 7955   supcsup 8346   RRcr 9935   0cc0 9936   < clt 10074   <_ cle 10075   NN0cn0 11292   ZZcz 11377   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-2o 7561  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-div 10685  df-nn 11021  df-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-seq 12802  df-fac 13061  df-bc 13090  df-hash 13118  df-ram 15705

This theorem is referenced by:  ramcl  15733