Theorem vdwmc2 14509

Description: Expand out the definition of an arithmetic progression. (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 )
vdwmc2.4	|- ( ph -> A e. X )

Assertion

Ref	Expression
vdwmc2	|- ( ph -> ( K MonoAP F <-> E. c e. R E. a e. NN E. d e. NN A. m e. ( 0 ... ( K - 1 ) ) ( a + ( m x. d ) ) e. ( `' F " { c } ) ) )

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

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

Proof of Theorem vdwmc2

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		vdwmc.1	. . 3 |- X e. _V
2		vdwmc.2	. . 3 |- ( ph -> K e. NN0 )
3		vdwmc.3	. . 3 |- ( ph -> F : X --> R )
4	1, 2, 3	vdwmc 14508	. 2 |- ( ph -> ( K MonoAP F <-> E. c E. a e. NN E. d e. NN ( a (AP ` K ) d ) C_ ( `' F " { c } ) ) )
5		vdwapid1 14505	. . . . . . . . . . . 12 |- ( ( K e. NN /\ a e. NN /\ d e. NN ) -> a e. ( a (AP ` K ) d ) )
6		ne0i 3799	. . . . . . . . . . . 12 |- ( a e. ( a (AP ` K ) d ) -> ( a (AP ` K ) d ) =/= (/) )
7	5, 6	syl 16	. . . . . . . . . . 11 |- ( ( K e. NN /\ a e. NN /\ d e. NN ) -> ( a (AP ` K ) d ) =/= (/) )
8	7	3expb 1197	. . . . . . . . . 10 |- ( ( K e. NN /\ ( a e. NN /\ d e. NN ) ) -> ( a (AP ` K ) d ) =/= (/) )
9	8	adantll 713	. . . . . . . . 9 |- ( ( ( ph /\ K e. NN ) /\ ( a e. NN /\ d e. NN ) ) -> ( a (AP ` K ) d ) =/= (/) )
10		ssn0 3827	. . . . . . . . . 10 |- ( ( ( a (AP ` K ) d ) C_ ( `' F " { c } ) /\ ( a (AP ` K ) d ) =/= (/) ) -> ( `' F " { c } ) =/= (/) )
11	10	expcom 435	. . . . . . . . 9 |- ( ( a (AP ` K ) d ) =/= (/) -> ( ( a (AP ` K ) d ) C_ ( `' F " { c } ) -> ( `' F " { c } ) =/= (/) ) )
12	9, 11	syl 16	. . . . . . . 8 |- ( ( ( ph /\ K e. NN ) /\ ( a e. NN /\ d e. NN ) ) -> ( ( a (AP ` K ) d ) C_ ( `' F " { c } ) -> ( `' F " { c } ) =/= (/) ) )
13		disjsn 4092	. . . . . . . . . 10 |- ( ( R i^i { c } ) = (/) <-> -. c e. R )
14	3	adantr 465	. . . . . . . . . . . 12 |- ( ( ph /\ K e. NN ) -> F : X --> R )
15		fimacnvdisj 5769	. . . . . . . . . . . . 13 |- ( ( F : X --> R /\ ( R i^i { c } ) = (/) ) -> ( `' F " { c } ) = (/) )
16	15	ex 434	. . . . . . . . . . . 12 |- ( F : X --> R -> ( ( R i^i { c } ) = (/) -> ( `' F " { c } ) = (/) ) )
17	14, 16	syl 16	. . . . . . . . . . 11 |- ( ( ph /\ K e. NN ) -> ( ( R i^i { c } ) = (/) -> ( `' F " { c } ) = (/) ) )
18	17	adantr 465	. . . . . . . . . 10 |- ( ( ( ph /\ K e. NN ) /\ ( a e. NN /\ d e. NN ) ) -> ( ( R i^i { c } ) = (/) -> ( `' F " { c } ) = (/) ) )
19	13, 18	syl5bir 218	. . . . . . . . 9 |- ( ( ( ph /\ K e. NN ) /\ ( a e. NN /\ d e. NN ) ) -> ( -. c e. R -> ( `' F " { c } ) = (/) ) )
20	19	necon1ad 2673	. . . . . . . 8 |- ( ( ( ph /\ K e. NN ) /\ ( a e. NN /\ d e. NN ) ) -> ( ( `' F " { c } ) =/= (/) -> c e. R ) )
21	12, 20	syld 44	. . . . . . 7 |- ( ( ( ph /\ K e. NN ) /\ ( a e. NN /\ d e. NN ) ) -> ( ( a (AP ` K ) d ) C_ ( `' F " { c } ) -> c e. R ) )
22	21	rexlimdvva 2956	. . . . . 6 |- ( ( ph /\ K e. NN ) -> ( E. a e. NN E. d e. NN ( a (AP ` K ) d ) C_ ( `' F " { c } ) -> c e. R ) )
23	22	pm4.71rd 635	. . . . 5 |- ( ( ph /\ K e. NN ) -> ( E. a e. NN E. d e. NN ( a (AP ` K ) d ) C_ ( `' F " { c } ) <-> ( c e. R /\ E. a e. NN E. d e. NN ( a (AP ` K ) d ) C_ ( `' F " { c } ) ) ) )
24	23	exbidv 1715	. . . 4 |- ( ( ph /\ K e. NN ) -> ( E. c E. a e. NN E. d e. NN ( a (AP ` K ) d ) C_ ( `' F " { c } ) <-> E. c ( c e. R /\ E. a e. NN E. d e. NN ( a (AP ` K ) d ) C_ ( `' F " { c } ) ) ) )
25		df-rex 2813	. . . 4 |- ( E. c e. R E. a e. NN E. d e. NN ( a (AP ` K ) d ) C_ ( `' F " { c } ) <-> E. c ( c e. R /\ E. a e. NN E. d e. NN ( a (AP ` K ) d ) C_ ( `' F " { c } ) ) )
26	24, 25	syl6bbr 263	. . 3 |- ( ( ph /\ K e. NN ) -> ( E. c E. a e. NN E. d e. NN ( a (AP ` K ) d ) C_ ( `' F " { c } ) <-> E. c e. R E. a e. NN E. d e. NN ( a (AP ` K ) d ) C_ ( `' F " { c } ) ) )
27		vdwmc2.4	. . . . . . . . 9 |- ( ph -> A e. X )
28	3, 27	ffvelrnd 6033	. . . . . . . 8 |- ( ph -> ( F ` A ) e. R )
29		ne0i 3799	. . . . . . . 8 |- ( ( F ` A ) e. R -> R =/= (/) )
30	28, 29	syl 16	. . . . . . 7 |- ( ph -> R =/= (/) )
31	30	adantr 465	. . . . . 6 |- ( ( ph /\ K = 0 ) -> R =/= (/) )
32		1nn 10567	. . . . . . . . 9 |- 1 e. NN
33	32	ne0ii 3800	. . . . . . . 8 |- NN =/= (/)
34		simpllr 760	. . . . . . . . . . . . . . 15 |- ( ( ( ( ph /\ K = 0 ) /\ a e. NN ) /\ d e. NN ) -> K = 0 )
35	34	fveq2d 5876	. . . . . . . . . . . . . 14 |- ( ( ( ( ph /\ K = 0 ) /\ a e. NN ) /\ d e. NN ) -> (AP ` K ) = (AP ` 0 ) )
36	35	oveqd 6313	. . . . . . . . . . . . 13 |- ( ( ( ( ph /\ K = 0 ) /\ a e. NN ) /\ d e. NN ) -> ( a (AP ` K ) d ) = ( a (AP ` 0 ) d ) )
37		vdwap0 14506	. . . . . . . . . . . . . 14 |- ( ( a e. NN /\ d e. NN ) -> ( a (AP ` 0 ) d ) = (/) )
38	37	adantll 713	. . . . . . . . . . . . 13 |- ( ( ( ( ph /\ K = 0 ) /\ a e. NN ) /\ d e. NN ) -> ( a (AP ` 0 ) d ) = (/) )
39	36, 38	eqtrd 2498	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ K = 0 ) /\ a e. NN ) /\ d e. NN ) -> ( a (AP ` K ) d ) = (/) )
40		0ss 3823	. . . . . . . . . . . 12 |- (/) C_ ( `' F " { c } )
41	39, 40	syl6eqss 3549	. . . . . . . . . . 11 |- ( ( ( ( ph /\ K = 0 ) /\ a e. NN ) /\ d e. NN ) -> ( a (AP ` K ) d ) C_ ( `' F " { c } ) )
42	41	ralrimiva 2871	. . . . . . . . . 10 |- ( ( ( ph /\ K = 0 ) /\ a e. NN ) -> A. d e. NN ( a (AP ` K ) d ) C_ ( `' F " { c } ) )
43		r19.2z 3921	. . . . . . . . . 10 |- ( ( NN =/= (/) /\ A. d e. NN ( a (AP ` K ) d ) C_ ( `' F " { c } ) ) -> E. d e. NN ( a (AP ` K ) d ) C_ ( `' F " { c } ) )
44	33, 42, 43	sylancr 663	. . . . . . . . 9 |- ( ( ( ph /\ K = 0 ) /\ a e. NN ) -> E. d e. NN ( a (AP ` K ) d ) C_ ( `' F " { c } ) )
45	44	ralrimiva 2871	. . . . . . . 8 |- ( ( ph /\ K = 0 ) -> A. a e. NN E. d e. NN ( a (AP ` K ) d ) C_ ( `' F " { c } ) )
46		r19.2z 3921	. . . . . . . 8 |- ( ( NN =/= (/) /\ A. a e. NN E. d e. NN ( a (AP ` K ) d ) C_ ( `' F " { c } ) ) -> E. a e. NN E. d e. NN ( a (AP ` K ) d ) C_ ( `' F " { c } ) )
47	33, 45, 46	sylancr 663	. . . . . . 7 |- ( ( ph /\ K = 0 ) -> E. a e. NN E. d e. NN ( a (AP ` K ) d ) C_ ( `' F " { c } ) )
48	47	ralrimivw 2872	. . . . . 6 |- ( ( ph /\ K = 0 ) -> A. c e. R E. a e. NN E. d e. NN ( a (AP ` K ) d ) C_ ( `' F " { c } ) )
49		r19.2z 3921	. . . . . 6 |- ( ( R =/= (/) /\ A. c e. R E. a e. NN E. d e. NN ( a (AP ` K ) d ) C_ ( `' F " { c } ) ) -> E. c e. R E. a e. NN E. d e. NN ( a (AP ` K ) d ) C_ ( `' F " { c } ) )
50	31, 48, 49	syl2anc 661	. . . . 5 |- ( ( ph /\ K = 0 ) -> E. c e. R E. a e. NN E. d e. NN ( a (AP ` K ) d ) C_ ( `' F " { c } ) )
51		rexex 2914	. . . . 5 |- ( E. c e. R E. a e. NN E. d e. NN ( a (AP ` K ) d ) C_ ( `' F " { c } ) -> E. c E. a e. NN E. d e. NN ( a (AP ` K ) d ) C_ ( `' F " { c } ) )
52	50, 51	syl 16	. . . 4 |- ( ( ph /\ K = 0 ) -> E. c E. a e. NN E. d e. NN ( a (AP ` K ) d ) C_ ( `' F " { c } ) )
53	52, 50	2thd 240	. . 3 |- ( ( ph /\ K = 0 ) -> ( E. c E. a e. NN E. d e. NN ( a (AP ` K ) d ) C_ ( `' F " { c } ) <-> E. c e. R E. a e. NN E. d e. NN ( a (AP ` K ) d ) C_ ( `' F " { c } ) ) )
54		elnn0 10818	. . . 4 |- ( K e. NN0 <-> ( K e. NN \/ K = 0 ) )
55	2, 54	sylib 196	. . 3 |- ( ph -> ( K e. NN \/ K = 0 ) )
56	26, 53, 55	mpjaodan 786	. 2 |- ( ph -> ( E. c E. a e. NN E. d e. NN ( a (AP ` K ) d ) C_ ( `' F " { c } ) <-> E. c e. R E. a e. NN E. d e. NN ( a (AP ` K ) d ) C_ ( `' F " { c } ) ) )
57		vdwapval 14503	. . . . . . . . 9 |- ( ( K e. NN0 /\ a e. NN /\ d e. NN ) -> ( x e. ( a (AP ` K ) d ) <-> E. m e. ( 0 ... ( K - 1 ) ) x = ( a + ( m x. d ) ) ) )
58	57	3expb 1197	. . . . . . . 8 |- ( ( K e. NN0 /\ ( a e. NN /\ d e. NN ) ) -> ( x e. ( a (AP ` K ) d ) <-> E. m e. ( 0 ... ( K - 1 ) ) x = ( a + ( m x. d ) ) ) )
59	2, 58	sylan 471	. . . . . . 7 |- ( ( ph /\ ( a e. NN /\ d e. NN ) ) -> ( x e. ( a (AP ` K ) d ) <-> E. m e. ( 0 ... ( K - 1 ) ) x = ( a + ( m x. d ) ) ) )
60	59	imbi1d 317	. . . . . 6 |- ( ( ph /\ ( a e. NN /\ d e. NN ) ) -> ( ( x e. ( a (AP ` K ) d ) -> x e. ( `' F " { c } ) ) <-> ( E. m e. ( 0 ... ( K - 1 ) ) x = ( a + ( m x. d ) ) -> x e. ( `' F " { c } ) ) ) )
61	60	albidv 1714	. . . . 5 |- ( ( ph /\ ( a e. NN /\ d e. NN ) ) -> ( A. x ( x e. ( a (AP ` K ) d ) -> x e. ( `' F " { c } ) ) <-> A. x ( E. m e. ( 0 ... ( K - 1 ) ) x = ( a + ( m x. d ) ) -> x e. ( `' F " { c } ) ) ) )
62		dfss2 3488	. . . . 5 |- ( ( a (AP ` K ) d ) C_ ( `' F " { c } ) <-> A. x ( x e. ( a (AP ` K ) d ) -> x e. ( `' F " { c } ) ) )
63		ralcom4 3128	. . . . . 6 |- ( A. m e. ( 0 ... ( K - 1 ) ) A. x ( x = ( a + ( m x. d ) ) -> x e. ( `' F " { c } ) ) <-> A. x A. m e. ( 0 ... ( K - 1 ) ) ( x = ( a + ( m x. d ) ) -> x e. ( `' F " { c } ) ) )
64		ovex 6324	. . . . . . . 8 |- ( a + ( m x. d ) ) e. _V
65		eleq1 2529	. . . . . . . 8 |- ( x = ( a + ( m x. d ) ) -> ( x e. ( `' F " { c } ) <-> ( a + ( m x. d ) ) e. ( `' F " { c } ) ) )
66	64, 65	ceqsalv 3137	. . . . . . 7 |- ( A. x ( x = ( a + ( m x. d ) ) -> x e. ( `' F " { c } ) ) <-> ( a + ( m x. d ) ) e. ( `' F " { c } ) )
67	66	ralbii 2888	. . . . . 6 |- ( A. m e. ( 0 ... ( K - 1 ) ) A. x ( x = ( a + ( m x. d ) ) -> x e. ( `' F " { c } ) ) <-> A. m e. ( 0 ... ( K - 1 ) ) ( a + ( m x. d ) ) e. ( `' F " { c } ) )
68		r19.23v 2937	. . . . . . 7 |- ( A. m e. ( 0 ... ( K - 1 ) ) ( x = ( a + ( m x. d ) ) -> x e. ( `' F " { c } ) ) <-> ( E. m e. ( 0 ... ( K - 1 ) ) x = ( a + ( m x. d ) ) -> x e. ( `' F " { c } ) ) )
69	68	albii 1641	. . . . . 6 |- ( A. x A. m e. ( 0 ... ( K - 1 ) ) ( x = ( a + ( m x. d ) ) -> x e. ( `' F " { c } ) ) <-> A. x ( E. m e. ( 0 ... ( K - 1 ) ) x = ( a + ( m x. d ) ) -> x e. ( `' F " { c } ) ) )
70	63, 67, 69	3bitr3i 275	. . . . 5 |- ( A. m e. ( 0 ... ( K - 1 ) ) ( a + ( m x. d ) ) e. ( `' F " { c } ) <-> A. x ( E. m e. ( 0 ... ( K - 1 ) ) x = ( a + ( m x. d ) ) -> x e. ( `' F " { c } ) ) )
71	61, 62, 70	3bitr4g 288	. . . 4 |- ( ( ph /\ ( a e. NN /\ d e. NN ) ) -> ( ( a (AP ` K ) d ) C_ ( `' F " { c } ) <-> A. m e. ( 0 ... ( K - 1 ) ) ( a + ( m x. d ) ) e. ( `' F " { c } ) ) )
72	71	2rexbidva 2974	. . 3 |- ( ph -> ( E. a e. NN E. d e. NN ( a (AP ` K ) d ) C_ ( `' F " { c } ) <-> E. a e. NN E. d e. NN A. m e. ( 0 ... ( K - 1 ) ) ( a + ( m x. d ) ) e. ( `' F " { c } ) ) )
73	72	rexbidv 2968	. 2 |- ( ph -> ( E. c e. R E. a e. NN E. d e. NN ( a (AP ` K ) d ) C_ ( `' F " { c } ) <-> E. c e. R E. a e. NN E. d e. NN A. m e. ( 0 ... ( K - 1 ) ) ( a + ( m x. d ) ) e. ( `' F " { c } ) ) )
74	4, 56, 73	3bitrd 279	1 |- ( ph -> ( K MonoAP F <-> E. c e. R E. a e. NN E. d e. NN A. m e. ( 0 ... ( K - 1 ) ) ( a + ( m x. d ) ) e. ( `' F " { c } ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   \/ wo 368   /\ wa 369   /\ w3a 973   A.wal 1393   = wceq 1395   E.wex 1613   e. wcel 1819   =/= wne 2652   A.wral 2807   E.wrex 2808   _Vcvv 3109   i^i cin 3470   C_ wss 3471   (/)c0 3793   {csn 4032   class class class wbr 4456   `'ccnv 5007   "cima 5011   -->wf 5590   ` cfv 5594  (class class class)co 6296   0cc0 9509   1c1 9510   + caddc 9512   x. cmul 9514   - cmin 9824   NNcn 10556   NN0cn0 10816   ...cfz 11697  APcvdwa 14495   MonoAP cvdwm 14496

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-n0 10817  df-z 10886  df-uz 11107  df-fz 11698  df-vdwap 14498  df-vdwmc 14499

This theorem is referenced by:  vdw  14524