Theorem ramz2 15728

Description: The Ramsey number when F has value zero for some color C. (Contributed by Mario Carneiro, 22-Apr-2015.)

Assertion

Ref	Expression
ramz2	|- ( ( ( M e. NN /\ R e. V /\ F : R --> NN0 ) /\ ( C e. R /\ ( F ` C ) = 0 ) ) -> ( M Ramsey F ) = 0 )

Proof of Theorem ramz2

Dummy variables b f c s x a i are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2622	. . 3 |- ( 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		simpl1 1064	. . . 4 |- ( ( ( M e. NN /\ R e. V /\ F : R --> NN0 ) /\ ( C e. R /\ ( F ` C ) = 0 ) ) -> M e. NN )
3	2	nnnn0d 11351	. . 3 |- ( ( ( M e. NN /\ R e. V /\ F : R --> NN0 ) /\ ( C e. R /\ ( F ` C ) = 0 ) ) -> M e. NN0 )
4		simpl2 1065	. . 3 |- ( ( ( M e. NN /\ R e. V /\ F : R --> NN0 ) /\ ( C e. R /\ ( F ` C ) = 0 ) ) -> R e. V )
5		simpl3 1066	. . 3 |- ( ( ( M e. NN /\ R e. V /\ F : R --> NN0 ) /\ ( C e. R /\ ( F ` C ) = 0 ) ) -> F : R --> NN0 )
6		0nn0 11307	. . . 4 |- 0 e. NN0
7	6	a1i 11	. . 3 |- ( ( ( M e. NN /\ R e. V /\ F : R --> NN0 ) /\ ( C e. R /\ ( F ` C ) = 0 ) ) -> 0 e. NN0 )
8		simplrl 800	. . . 4 |- ( ( ( ( M e. NN /\ R e. V /\ F : R --> NN0 ) /\ ( C e. R /\ ( F ` C ) = 0 ) ) /\ ( 0 <_ ( # ` s ) /\ f : ( s ( a e. _V , i e. NN0 |-> { b e. ~P a | ( # ` b ) = i } ) M ) --> R ) ) -> C e. R )
9		0elpw 4834	. . . . 5 |- (/) e. ~P s
10	9	a1i 11	. . . 4 |- ( ( ( ( M e. NN /\ R e. V /\ F : R --> NN0 ) /\ ( C e. R /\ ( F ` C ) = 0 ) ) /\ ( 0 <_ ( # ` s ) /\ f : ( s ( a e. _V , i e. NN0 |-> { b e. ~P a | ( # ` b ) = i } ) M ) --> R ) ) -> (/) e. ~P s )
11		simplrr 801	. . . . 5 |- ( ( ( ( M e. NN /\ R e. V /\ F : R --> NN0 ) /\ ( C e. R /\ ( F ` C ) = 0 ) ) /\ ( 0 <_ ( # ` s ) /\ f : ( s ( a e. _V , i e. NN0 |-> { b e. ~P a | ( # ` b ) = i } ) M ) --> R ) ) -> ( F ` C ) = 0 )
12		0le0 11110	. . . . 5 |- 0 <_ 0
13	11, 12	syl6eqbr 4692	. . . 4 |- ( ( ( ( M e. NN /\ R e. V /\ F : R --> NN0 ) /\ ( C e. R /\ ( F ` C ) = 0 ) ) /\ ( 0 <_ ( # ` s ) /\ f : ( s ( a e. _V , i e. NN0 |-> { b e. ~P a | ( # ` b ) = i } ) M ) --> R ) ) -> ( F ` C ) <_ 0 )
14		simpll1 1100	. . . . . 6 |- ( ( ( ( M e. NN /\ R e. V /\ F : R --> NN0 ) /\ ( C e. R /\ ( F ` C ) = 0 ) ) /\ ( 0 <_ ( # ` s ) /\ f : ( s ( a e. _V , i e. NN0 |-> { b e. ~P a | ( # ` b ) = i } ) M ) --> R ) ) -> M e. NN )
15	1	0hashbc 15711	. . . . . 6 |- ( M e. NN -> ( (/) ( a e. _V , i e. NN0 |-> { b e. ~P a | ( # ` b ) = i } ) M ) = (/) )
16	14, 15	syl 17	. . . . 5 |- ( ( ( ( M e. NN /\ R e. V /\ F : R --> NN0 ) /\ ( C e. R /\ ( F ` C ) = 0 ) ) /\ ( 0 <_ ( # ` s ) /\ f : ( s ( a e. _V , i e. NN0 |-> { b e. ~P a | ( # ` b ) = i } ) M ) --> R ) ) -> ( (/) ( a e. _V , i e. NN0 |-> { b e. ~P a | ( # ` b ) = i } ) M ) = (/) )
17		0ss 3972	. . . . 5 |- (/) C_ ( `' f " { C } )
18	16, 17	syl6eqss 3655	. . . 4 |- ( ( ( ( M e. NN /\ R e. V /\ F : R --> NN0 ) /\ ( C e. R /\ ( F ` C ) = 0 ) ) /\ ( 0 <_ ( # ` s ) /\ f : ( s ( a e. _V , i e. NN0 |-> { b e. ~P a | ( # ` b ) = i } ) M ) --> R ) ) -> ( (/) ( a e. _V , i e. NN0 |-> { b e. ~P a | ( # ` b ) = i } ) M ) C_ ( `' f " { C } ) )
19		fveq2 6191	. . . . . . 7 |- ( c = C -> ( F ` c ) = ( F ` C ) )
20	19	breq1d 4663	. . . . . 6 |- ( c = C -> ( ( F ` c ) <_ ( # ` x ) <-> ( F ` C ) <_ ( # ` x ) ) )
21		sneq 4187	. . . . . . . 8 |- ( c = C -> { c } = { C } )
22	21	imaeq2d 5466	. . . . . . 7 |- ( c = C -> ( `' f " { c } ) = ( `' f " { C } ) )
23	22	sseq2d 3633	. . . . . 6 |- ( c = C -> ( ( x ( a e. _V , i e. NN0 |-> { b e. ~P a | ( # ` b ) = i } ) M ) C_ ( `' f " { c } ) <-> ( x ( a e. _V , i e. NN0 |-> { b e. ~P a | ( # ` b ) = i } ) M ) C_ ( `' f " { C } ) ) )
24	20, 23	anbi12d 747	. . . . 5 |- ( c = C -> ( ( ( F ` c ) <_ ( # ` x ) /\ ( x ( a e. _V , i e. NN0 |-> { b e. ~P a | ( # ` b ) = i } ) M ) C_ ( `' f " { c } ) ) <-> ( ( F ` C ) <_ ( # ` x ) /\ ( x ( a e. _V , i e. NN0 |-> { b e. ~P a | ( # ` b ) = i } ) M ) C_ ( `' f " { C } ) ) ) )
25		fveq2 6191	. . . . . . . 8 |- ( x = (/) -> ( # ` x ) = ( # ` (/) ) )
26		hash0 13158	. . . . . . . 8 |- ( # ` (/) ) = 0
27	25, 26	syl6eq 2672	. . . . . . 7 |- ( x = (/) -> ( # ` x ) = 0 )
28	27	breq2d 4665	. . . . . 6 |- ( x = (/) -> ( ( F ` C ) <_ ( # ` x ) <-> ( F ` C ) <_ 0 ) )
29		oveq1 6657	. . . . . . 7 |- ( x = (/) -> ( x ( a e. _V , i e. NN0 |-> { b e. ~P a | ( # ` b ) = i } ) M ) = ( (/) ( a e. _V , i e. NN0 |-> { b e. ~P a | ( # ` b ) = i } ) M ) )
30	29	sseq1d 3632	. . . . . 6 |- ( x = (/) -> ( ( x ( a e. _V , i e. NN0 |-> { b e. ~P a | ( # ` b ) = i } ) M ) C_ ( `' f " { C } ) <-> ( (/) ( a e. _V , i e. NN0 |-> { b e. ~P a | ( # ` b ) = i } ) M ) C_ ( `' f " { C } ) ) )
31	28, 30	anbi12d 747	. . . . 5 |- ( x = (/) -> ( ( ( F ` C ) <_ ( # ` x ) /\ ( x ( a e. _V , i e. NN0 |-> { b e. ~P a | ( # ` b ) = i } ) M ) C_ ( `' f " { C } ) ) <-> ( ( F ` C ) <_ 0 /\ ( (/) ( a e. _V , i e. NN0 |-> { b e. ~P a | ( # ` b ) = i } ) M ) C_ ( `' f " { C } ) ) ) )
32	24, 31	rspc2ev 3324	. . . 4 |- ( ( C e. R /\ (/) e. ~P s /\ ( ( F ` C ) <_ 0 /\ ( (/) ( a e. _V , i e. NN0 |-> { b e. ~P a | ( # ` b ) = i } ) M ) C_ ( `' f " { C } ) ) ) -> E. c e. R E. x e. ~P s ( ( F ` c ) <_ ( # ` x ) /\ ( x ( a e. _V , i e. NN0 |-> { b e. ~P a | ( # ` b ) = i } ) M ) C_ ( `' f " { c } ) ) )
33	8, 10, 13, 18, 32	syl112anc 1330	. . 3 |- ( ( ( ( M e. NN /\ R e. V /\ F : R --> NN0 ) /\ ( C e. R /\ ( F ` C ) = 0 ) ) /\ ( 0 <_ ( # ` s ) /\ f : ( s ( a e. _V , i e. NN0 |-> { b e. ~P a | ( # ` b ) = i } ) M ) --> R ) ) -> E. c e. R E. x e. ~P s ( ( F ` c ) <_ ( # ` x ) /\ ( x ( a e. _V , i e. NN0 |-> { b e. ~P a | ( # ` b ) = i } ) M ) C_ ( `' f " { c } ) ) )
34	1, 3, 4, 5, 7, 33	ramub 15717	. 2 |- ( ( ( M e. NN /\ R e. V /\ F : R --> NN0 ) /\ ( C e. R /\ ( F ` C ) = 0 ) ) -> ( M Ramsey F ) <_ 0 )
35		ramubcl 15722	. . . 4 |- ( ( ( M e. NN0 /\ R e. V /\ F : R --> NN0 ) /\ ( 0 e. NN0 /\ ( M Ramsey F ) <_ 0 ) ) -> ( M Ramsey F ) e. NN0 )
36	3, 4, 5, 7, 34, 35	syl32anc 1334	. . 3 |- ( ( ( M e. NN /\ R e. V /\ F : R --> NN0 ) /\ ( C e. R /\ ( F ` C ) = 0 ) ) -> ( M Ramsey F ) e. NN0 )
37		nn0le0eq0 11321	. . 3 |- ( ( M Ramsey F ) e. NN0 -> ( ( M Ramsey F ) <_ 0 <-> ( M Ramsey F ) = 0 ) )
38	36, 37	syl 17	. 2 |- ( ( ( M e. NN /\ R e. V /\ F : R --> NN0 ) /\ ( C e. R /\ ( F ` C ) = 0 ) ) -> ( ( M Ramsey F ) <_ 0 <-> ( M Ramsey F ) = 0 ) )
39	34, 38	mpbid 222	1 |- ( ( ( M e. NN /\ R e. V /\ F : R --> NN0 ) /\ ( C e. R /\ ( F ` C ) = 0 ) ) -> ( M Ramsey F ) = 0 )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   E.wrex 2913   {crab 2916   _Vcvv 3200   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   {csn 4177   class class class wbr 4653   `'ccnv 5113   "cima 5117   -->wf 5884   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   0cc0 9936   <_ cle 10075   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-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-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:  ramz  15729  ramcl  15733