Theorem vdwpc 15684

Description: The predicate " The coloring F contains a polychromatic M-tuple of AP's of length K". A polychromatic M-tuple of AP's is a set of AP's with the same base point but different step lengths, such that each individual AP is monochromatic, but the AP's all have mutually distinct colors. (The common basepoint is not required to have the same color as any of the AP's.) (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 )
vdwpc.4	|- ( ph -> M e. NN )
vdwpc.5	|- J = ( 1 ... M )

Assertion

Ref	Expression
vdwpc	|- ( ph -> ( <. M , K >. PolyAP F <-> E. a e. NN E. d e. ( NN ^m J ) ( A. i e. J ( ( a + ( d ` i ) ) (AP ` K ) ( d ` i ) ) C_ ( `' F " { ( F ` ( a + ( d ` i ) ) ) } ) /\ ( # ` ran ( i e. J |-> ( F ` ( a + ( d ` i ) ) ) ) ) = M ) ) )

Distinct variable groups:   a, d, i, F   K, a, d, i   J, d, i   M, a, d, i

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

Proof of Theorem vdwpc

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

Step	Hyp	Ref	Expression
1		vdwpc.4	. 2 |- ( ph -> M e. NN )
2		vdwmc.2	. 2 |- ( ph -> K e. NN0 )
3		vdwmc.3	. . 3 |- ( ph -> F : X --> R )
4		vdwmc.1	. . 3 |- X e. _V
5		fex 6490	. . 3 |- ( ( F : X --> R /\ X e. _V ) -> F e. _V )
6	3, 4, 5	sylancl 694	. 2 |- ( ph -> F e. _V )
7		df-br 4654	. . . 4 |- ( <. M , K >. PolyAP F <-> <. <. M , K >. , F >. e. PolyAP )
8		df-vdwpc 15674	. . . . 5 |- PolyAP = { <. <. m , k >. , f >. | E. a e. NN E. d e. ( NN ^m ( 1 ... m ) ) ( A. i e. ( 1 ... m ) ( ( a + ( d ` i ) ) (AP ` k ) ( d ` i ) ) C_ ( `' f " { ( f ` ( a + ( d ` i ) ) ) } ) /\ ( # ` ran ( i e. ( 1 ... m ) |-> ( f ` ( a + ( d ` i ) ) ) ) ) = m ) }
9	8	eleq2i 2693	. . . 4 |- ( <. <. M , K >. , F >. e. PolyAP <-> <. <. M , K >. , F >. e. { <. <. m , k >. , f >. | E. a e. NN E. d e. ( NN ^m ( 1 ... m ) ) ( A. i e. ( 1 ... m ) ( ( a + ( d ` i ) ) (AP ` k ) ( d ` i ) ) C_ ( `' f " { ( f ` ( a + ( d ` i ) ) ) } ) /\ ( # ` ran ( i e. ( 1 ... m ) |-> ( f ` ( a + ( d ` i ) ) ) ) ) = m ) } )
10	7, 9	bitri 264	. . 3 |- ( <. M , K >. PolyAP F <-> <. <. M , K >. , F >. e. { <. <. m , k >. , f >. | E. a e. NN E. d e. ( NN ^m ( 1 ... m ) ) ( A. i e. ( 1 ... m ) ( ( a + ( d ` i ) ) (AP ` k ) ( d ` i ) ) C_ ( `' f " { ( f ` ( a + ( d ` i ) ) ) } ) /\ ( # ` ran ( i e. ( 1 ... m ) |-> ( f ` ( a + ( d ` i ) ) ) ) ) = m ) } )
11		simp1 1061	. . . . . . . . 9 |- ( ( m = M /\ k = K /\ f = F ) -> m = M )
12	11	oveq2d 6666	. . . . . . . 8 |- ( ( m = M /\ k = K /\ f = F ) -> ( 1 ... m ) = ( 1 ... M ) )
13		vdwpc.5	. . . . . . . 8 |- J = ( 1 ... M )
14	12, 13	syl6eqr 2674	. . . . . . 7 |- ( ( m = M /\ k = K /\ f = F ) -> ( 1 ... m ) = J )
15	14	oveq2d 6666	. . . . . 6 |- ( ( m = M /\ k = K /\ f = F ) -> ( NN ^m ( 1 ... m ) ) = ( NN ^m J ) )
16		simp2 1062	. . . . . . . . . . 11 |- ( ( m = M /\ k = K /\ f = F ) -> k = K )
17	16	fveq2d 6195	. . . . . . . . . 10 |- ( ( m = M /\ k = K /\ f = F ) -> (AP ` k ) = (AP ` K ) )
18	17	oveqd 6667	. . . . . . . . 9 |- ( ( m = M /\ k = K /\ f = F ) -> ( ( a + ( d ` i ) ) (AP ` k ) ( d ` i ) ) = ( ( a + ( d ` i ) ) (AP ` K ) ( d ` i ) ) )
19		simp3 1063	. . . . . . . . . . 11 |- ( ( m = M /\ k = K /\ f = F ) -> f = F )
20	19	cnveqd 5298	. . . . . . . . . 10 |- ( ( m = M /\ k = K /\ f = F ) -> `' f = `' F )
21	19	fveq1d 6193	. . . . . . . . . . 11 |- ( ( m = M /\ k = K /\ f = F ) -> ( f ` ( a + ( d ` i ) ) ) = ( F ` ( a + ( d ` i ) ) ) )
22	21	sneqd 4189	. . . . . . . . . 10 |- ( ( m = M /\ k = K /\ f = F ) -> { ( f ` ( a + ( d ` i ) ) ) } = { ( F ` ( a + ( d ` i ) ) ) } )
23	20, 22	imaeq12d 5467	. . . . . . . . 9 |- ( ( m = M /\ k = K /\ f = F ) -> ( `' f " { ( f ` ( a + ( d ` i ) ) ) } ) = ( `' F " { ( F ` ( a + ( d ` i ) ) ) } ) )
24	18, 23	sseq12d 3634	. . . . . . . 8 |- ( ( m = M /\ k = K /\ f = F ) -> ( ( ( a + ( d ` i ) ) (AP ` k ) ( d ` i ) ) C_ ( `' f " { ( f ` ( a + ( d ` i ) ) ) } ) <-> ( ( a + ( d ` i ) ) (AP ` K ) ( d ` i ) ) C_ ( `' F " { ( F ` ( a + ( d ` i ) ) ) } ) ) )
25	14, 24	raleqbidv 3152	. . . . . . 7 |- ( ( m = M /\ k = K /\ f = F ) -> ( A. i e. ( 1 ... m ) ( ( a + ( d ` i ) ) (AP ` k ) ( d ` i ) ) C_ ( `' f " { ( f ` ( a + ( d ` i ) ) ) } ) <-> A. i e. J ( ( a + ( d ` i ) ) (AP ` K ) ( d ` i ) ) C_ ( `' F " { ( F ` ( a + ( d ` i ) ) ) } ) ) )
26	14, 21	mpteq12dv 4733	. . . . . . . . . 10 |- ( ( m = M /\ k = K /\ f = F ) -> ( i e. ( 1 ... m ) |-> ( f ` ( a + ( d ` i ) ) ) ) = ( i e. J |-> ( F ` ( a + ( d ` i ) ) ) ) )
27	26	rneqd 5353	. . . . . . . . 9 |- ( ( m = M /\ k = K /\ f = F ) -> ran ( i e. ( 1 ... m ) |-> ( f ` ( a + ( d ` i ) ) ) ) = ran ( i e. J |-> ( F ` ( a + ( d ` i ) ) ) ) )
28	27	fveq2d 6195	. . . . . . . 8 |- ( ( m = M /\ k = K /\ f = F ) -> ( # ` ran ( i e. ( 1 ... m ) |-> ( f ` ( a + ( d ` i ) ) ) ) ) = ( # ` ran ( i e. J |-> ( F ` ( a + ( d ` i ) ) ) ) ) )
29	28, 11	eqeq12d 2637	. . . . . . 7 |- ( ( m = M /\ k = K /\ f = F ) -> ( ( # ` ran ( i e. ( 1 ... m ) |-> ( f ` ( a + ( d ` i ) ) ) ) ) = m <-> ( # ` ran ( i e. J |-> ( F ` ( a + ( d ` i ) ) ) ) ) = M ) )
30	25, 29	anbi12d 747	. . . . . 6 |- ( ( m = M /\ k = K /\ f = F ) -> ( ( A. i e. ( 1 ... m ) ( ( a + ( d ` i ) ) (AP ` k ) ( d ` i ) ) C_ ( `' f " { ( f ` ( a + ( d ` i ) ) ) } ) /\ ( # ` ran ( i e. ( 1 ... m ) |-> ( f ` ( a + ( d ` i ) ) ) ) ) = m ) <-> ( A. i e. J ( ( a + ( d ` i ) ) (AP ` K ) ( d ` i ) ) C_ ( `' F " { ( F ` ( a + ( d ` i ) ) ) } ) /\ ( # ` ran ( i e. J |-> ( F ` ( a + ( d ` i ) ) ) ) ) = M ) ) )
31	15, 30	rexeqbidv 3153	. . . . 5 |- ( ( m = M /\ k = K /\ f = F ) -> ( E. d e. ( NN ^m ( 1 ... m ) ) ( A. i e. ( 1 ... m ) ( ( a + ( d ` i ) ) (AP ` k ) ( d ` i ) ) C_ ( `' f " { ( f ` ( a + ( d ` i ) ) ) } ) /\ ( # ` ran ( i e. ( 1 ... m ) |-> ( f ` ( a + ( d ` i ) ) ) ) ) = m ) <-> E. d e. ( NN ^m J ) ( A. i e. J ( ( a + ( d ` i ) ) (AP ` K ) ( d ` i ) ) C_ ( `' F " { ( F ` ( a + ( d ` i ) ) ) } ) /\ ( # ` ran ( i e. J |-> ( F ` ( a + ( d ` i ) ) ) ) ) = M ) ) )
32	31	rexbidv 3052	. . . 4 |- ( ( m = M /\ k = K /\ f = F ) -> ( E. a e. NN E. d e. ( NN ^m ( 1 ... m ) ) ( A. i e. ( 1 ... m ) ( ( a + ( d ` i ) ) (AP ` k ) ( d ` i ) ) C_ ( `' f " { ( f ` ( a + ( d ` i ) ) ) } ) /\ ( # ` ran ( i e. ( 1 ... m ) |-> ( f ` ( a + ( d ` i ) ) ) ) ) = m ) <-> E. a e. NN E. d e. ( NN ^m J ) ( A. i e. J ( ( a + ( d ` i ) ) (AP ` K ) ( d ` i ) ) C_ ( `' F " { ( F ` ( a + ( d ` i ) ) ) } ) /\ ( # ` ran ( i e. J |-> ( F ` ( a + ( d ` i ) ) ) ) ) = M ) ) )
33	32	eloprabga 6747	. . 3 |- ( ( M e. NN /\ K e. NN0 /\ F e. _V ) -> ( <. <. M , K >. , F >. e. { <. <. m , k >. , f >. | E. a e. NN E. d e. ( NN ^m ( 1 ... m ) ) ( A. i e. ( 1 ... m ) ( ( a + ( d ` i ) ) (AP ` k ) ( d ` i ) ) C_ ( `' f " { ( f ` ( a + ( d ` i ) ) ) } ) /\ ( # ` ran ( i e. ( 1 ... m ) |-> ( f ` ( a + ( d ` i ) ) ) ) ) = m ) } <-> E. a e. NN E. d e. ( NN ^m J ) ( A. i e. J ( ( a + ( d ` i ) ) (AP ` K ) ( d ` i ) ) C_ ( `' F " { ( F ` ( a + ( d ` i ) ) ) } ) /\ ( # ` ran ( i e. J |-> ( F ` ( a + ( d ` i ) ) ) ) ) = M ) ) )
34	10, 33	syl5bb 272	. 2 |- ( ( M e. NN /\ K e. NN0 /\ F e. _V ) -> ( <. M , K >. PolyAP F <-> E. a e. NN E. d e. ( NN ^m J ) ( A. i e. J ( ( a + ( d ` i ) ) (AP ` K ) ( d ` i ) ) C_ ( `' F " { ( F ` ( a + ( d ` i ) ) ) } ) /\ ( # ` ran ( i e. J |-> ( F ` ( a + ( d ` i ) ) ) ) ) = M ) ) )
35	1, 2, 6, 34	syl3anc 1326	1 |- ( ph -> ( <. M , K >. PolyAP F <-> E. a e. NN E. d e. ( NN ^m J ) ( A. i e. J ( ( a + ( d ` i ) ) (AP ` K ) ( d ` i ) ) C_ ( `' F " { ( F ` ( a + ( d ` i ) ) ) } ) /\ ( # ` ran ( i e. J |-> ( F ` ( a + ( d ` i ) ) ) ) ) = M ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   _Vcvv 3200   C_ wss 3574   {csn 4177   <.cop 4183   class class class wbr 4653   |-> cmpt 4729   `'ccnv 5113   ran crn 5115   "cima 5117   -->wf 5884   ` cfv 5888  (class class class)co 6650   {coprab 6651   ^m cmap 7857   1c1 9937   + caddc 9939   NNcn 11020   NN0cn0 11292   ...cfz 12326   #chash 13117  APcvdwa 15669   PolyAP cvdwp 15671

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-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-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  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-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-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  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-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-vdwpc 15674

This theorem is referenced by:  vdwlem6  15690  vdwlem7  15691  vdwlem8  15692  vdwlem11  15695