Theorem vdwmc 15682

Description: The predicate " The <. R , N >.-coloring F contains a monochromatic AP of length K". (Contributed by Mario Carneiro, 18-Aug-2014.)

Hypotheses

Ref	Expression
vdwmc.1	|- X e. _V
vdwmc.2	|- ( ph -> K e. NN0 )
vdwmc.3	|- ( ph -> F : X --> R )

Assertion

Ref	Expression
vdwmc	|- ( ph -> ( K MonoAP F <-> E. c E. a e. NN E. d e. NN ( a (AP ` K ) d ) C_ ( `' F " { c } ) ) )

Distinct variable groups:   a, c, d, F   K, a, c, d   ph, c

Allowed substitution hints:   ph( a, d)   R( a, c, d)   X( a, c, d)

Proof of Theorem vdwmc

Dummy variables f k w z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		vdwmc.2	. . 3 |- ( ph -> K e. NN0 )
2		vdwmc.3	. . . 4 |- ( ph -> F : X --> R )
3		vdwmc.1	. . . 4 |- X e. _V
4		fex 6490	. . . 4 |- ( ( F : X --> R /\ X e. _V ) -> F e. _V )
5	2, 3, 4	sylancl 694	. . 3 |- ( ph -> F e. _V )
6		fveq2 6191	. . . . . . . 8 |- ( k = K -> (AP ` k ) = (AP ` K ) )
7	6	rneqd 5353	. . . . . . 7 |- ( k = K -> ran (AP ` k ) = ran (AP ` K ) )
8		cnveq 5296	. . . . . . . . 9 |- ( f = F -> `' f = `' F )
9	8	imaeq1d 5465	. . . . . . . 8 |- ( f = F -> ( `' f " { c } ) = ( `' F " { c } ) )
10	9	pweqd 4163	. . . . . . 7 |- ( f = F -> ~P ( `' f " { c } ) = ~P ( `' F " { c } ) )
11	7, 10	ineqan12d 3816	. . . . . 6 |- ( ( k = K /\ f = F ) -> ( ran (AP ` k ) i^i ~P ( `' f " { c } ) ) = ( ran (AP ` K ) i^i ~P ( `' F " { c } ) ) )
12	11	neeq1d 2853	. . . . 5 |- ( ( k = K /\ f = F ) -> ( ( ran (AP ` k ) i^i ~P ( `' f " { c } ) ) =/= (/) <-> ( ran (AP ` K ) i^i ~P ( `' F " { c } ) ) =/= (/) ) )
13	12	exbidv 1850	. . . 4 |- ( ( k = K /\ f = F ) -> ( E. c ( ran (AP ` k ) i^i ~P ( `' f " { c } ) ) =/= (/) <-> E. c ( ran (AP ` K ) i^i ~P ( `' F " { c } ) ) =/= (/) ) )
14		df-vdwmc 15673	. . . 4 |- MonoAP = { <. k , f >. | E. c ( ran (AP ` k ) i^i ~P ( `' f " { c } ) ) =/= (/) }
15	13, 14	brabga 4989	. . 3 |- ( ( K e. NN0 /\ F e. _V ) -> ( K MonoAP F <-> E. c ( ran (AP ` K ) i^i ~P ( `' F " { c } ) ) =/= (/) ) )
16	1, 5, 15	syl2anc 693	. 2 |- ( ph -> ( K MonoAP F <-> E. c ( ran (AP ` K ) i^i ~P ( `' F " { c } ) ) =/= (/) ) )
17		vdwapf 15676	. . . . 5 |- ( K e. NN0 -> (AP ` K ) : ( NN X. NN ) --> ~P NN )
18		ffn 6045	. . . . 5 |- ( (AP ` K ) : ( NN X. NN ) --> ~P NN -> (AP ` K ) Fn ( NN X. NN ) )
19		selpw 4165	. . . . . . 7 |- ( z e. ~P ( `' F " { c } ) <-> z C_ ( `' F " { c } ) )
20		sseq1 3626	. . . . . . 7 |- ( z = ( (AP ` K ) ` w ) -> ( z C_ ( `' F " { c } ) <-> ( (AP ` K ) ` w ) C_ ( `' F " { c } ) ) )
21	19, 20	syl5bb 272	. . . . . 6 |- ( z = ( (AP ` K ) ` w ) -> ( z e. ~P ( `' F " { c } ) <-> ( (AP ` K ) ` w ) C_ ( `' F " { c } ) ) )
22	21	rexrn 6361	. . . . 5 |- ( (AP ` K ) Fn ( NN X. NN ) -> ( E. z e. ran (AP ` K ) z e. ~P ( `' F " { c } ) <-> E. w e. ( NN X. NN ) ( (AP ` K ) ` w ) C_ ( `' F " { c } ) ) )
23	1, 17, 18, 22	4syl 19	. . . 4 |- ( ph -> ( E. z e. ran (AP ` K ) z e. ~P ( `' F " { c } ) <-> E. w e. ( NN X. NN ) ( (AP ` K ) ` w ) C_ ( `' F " { c } ) ) )
24		elin 3796	. . . . . 6 |- ( z e. ( ran (AP ` K ) i^i ~P ( `' F " { c } ) ) <-> ( z e. ran (AP ` K ) /\ z e. ~P ( `' F " { c } ) ) )
25	24	exbii 1774	. . . . 5 |- ( E. z z e. ( ran (AP ` K ) i^i ~P ( `' F " { c } ) ) <-> E. z ( z e. ran (AP ` K ) /\ z e. ~P ( `' F " { c } ) ) )
26		n0 3931	. . . . 5 |- ( ( ran (AP ` K ) i^i ~P ( `' F " { c } ) ) =/= (/) <-> E. z z e. ( ran (AP ` K ) i^i ~P ( `' F " { c } ) ) )
27		df-rex 2918	. . . . 5 |- ( E. z e. ran (AP ` K ) z e. ~P ( `' F " { c } ) <-> E. z ( z e. ran (AP ` K ) /\ z e. ~P ( `' F " { c } ) ) )
28	25, 26, 27	3bitr4ri 293	. . . 4 |- ( E. z e. ran (AP ` K ) z e. ~P ( `' F " { c } ) <-> ( ran (AP ` K ) i^i ~P ( `' F " { c } ) ) =/= (/) )
29		fveq2 6191	. . . . . . 7 |- ( w = <. a , d >. -> ( (AP ` K ) ` w ) = ( (AP ` K ) ` <. a , d >. ) )
30		df-ov 6653	. . . . . . 7 |- ( a (AP ` K ) d ) = ( (AP ` K ) ` <. a , d >. )
31	29, 30	syl6eqr 2674	. . . . . 6 |- ( w = <. a , d >. -> ( (AP ` K ) ` w ) = ( a (AP ` K ) d ) )
32	31	sseq1d 3632	. . . . 5 |- ( w = <. a , d >. -> ( ( (AP ` K ) ` w ) C_ ( `' F " { c } ) <-> ( a (AP ` K ) d ) C_ ( `' F " { c } ) ) )
33	32	rexxp 5264	. . . 4 |- ( E. w e. ( NN X. NN ) ( (AP ` K ) ` w ) C_ ( `' F " { c } ) <-> E. a e. NN E. d e. NN ( a (AP ` K ) d ) C_ ( `' F " { c } ) )
34	23, 28, 33	3bitr3g 302	. . 3 |- ( ph -> ( ( ran (AP ` K ) i^i ~P ( `' F " { c } ) ) =/= (/) <-> E. a e. NN E. d e. NN ( a (AP ` K ) d ) C_ ( `' F " { c } ) ) )
35	34	exbidv 1850	. 2 |- ( ph -> ( E. c ( ran (AP ` K ) i^i ~P ( `' F " { c } ) ) =/= (/) <-> E. c E. a e. NN E. d e. NN ( a (AP ` K ) d ) C_ ( `' F " { c } ) ) )
36	16, 35	bitrd 268	1 |- ( ph -> ( K MonoAP F <-> E. c E. a e. NN E. d e. NN ( a (AP ` K ) d ) C_ ( `' F " { c } ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   =/= wne 2794   E.wrex 2913   _Vcvv 3200   i^i cin 3573   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   {csn 4177   <.cop 4183   class class class wbr 4653   X. cxp 5112   `'ccnv 5113   ran crn 5115   "cima 5117   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   NNcn 11020   NN0cn0 11292  APcvdwa 15669   MonoAP cvdwm 15670

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-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-en 7956  df-dom 7957  df-sdom 7958  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-vdwap 15672  df-vdwmc 15673

This theorem is referenced by:  vdwmc2  15683  vdwlem1  15685  vdwlem2  15686  vdwlem9  15693  vdwlem10  15694  vdwlem12  15696  vdwlem13  15697