Theorem vdwmc2 14159

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 14158	. 2 |- ( ph -> ( K MonoAP F <-> E. c E. a e. NN E. d e. NN ( a (AP ` K ) d ) C_ ( `' F " { c } ) ) )
5		vdwapid1 14155	. . . . . . . . . . . 12 |- ( ( K e. NN /\ a e. NN /\ d e. NN ) -> a e. ( a (AP ` K ) d ) )
6		ne0i 3752	. . . . . . . . . . . 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 1189	. . . . . . . . . 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 3779	. . . . . . . . . 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 4045	. . . . . . . . . 10 |- ( ( R i^i { c } ) = (/) <-> -. c e. R )
14	3	adantr 465	. . . . . . . . . . . 12 |- ( ( ph /\ K e. NN ) -> F : X --> R )
15		fimacnvdisj 5698	. . . . . . . . . . . . 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 2668	. . . . . . . 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 2954	. . . . . 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 1681	. . . 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 2805	. . . 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 5954	. . . . . . . 8 |- ( ph -> ( F ` A ) e. R )
29		ne0i 3752	. . . . . . . 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 10445	. . . . . . . . 9 |- 1 e. NN
33		ne0i 3752	. . . . . . . . 9 |- ( 1 e. NN -> NN =/= (/) )
34	32, 33	ax-mp 5	. . . . . . . 8 |- NN =/= (/)
35		simpllr 758	. . . . . . . . . . . . . . 15 |- ( ( ( ( ph /\ K = 0 ) /\ a e. NN ) /\ d e. NN ) -> K = 0 )
36	35	fveq2d 5804	. . . . . . . . . . . . . 14 |- ( ( ( ( ph /\ K = 0 ) /\ a e. NN ) /\ d e. NN ) -> (AP ` K ) = (AP ` 0 ) )
37	36	oveqd 6218	. . . . . . . . . . . . 13 |- ( ( ( ( ph /\ K = 0 ) /\ a e. NN ) /\ d e. NN ) -> ( a (AP ` K ) d ) = ( a (AP ` 0 ) d ) )
38		vdwap0 14156	. . . . . . . . . . . . . 14 |- ( ( a e. NN /\ d e. NN ) -> ( a (AP ` 0 ) d ) = (/) )
39	38	adantll 713	. . . . . . . . . . . . 13 |- ( ( ( ( ph /\ K = 0 ) /\ a e. NN ) /\ d e. NN ) -> ( a (AP ` 0 ) d ) = (/) )
40	37, 39	eqtrd 2495	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ K = 0 ) /\ a e. NN ) /\ d e. NN ) -> ( a (AP ` K ) d ) = (/) )
41		0ss 3775	. . . . . . . . . . . 12 |- (/) C_ ( `' F " { c } )
42	40, 41	syl6eqss 3515	. . . . . . . . . . 11 |- ( ( ( ( ph /\ K = 0 ) /\ a e. NN ) /\ d e. NN ) -> ( a (AP ` K ) d ) C_ ( `' F " { c } ) )
43	42	ralrimiva 2830	. . . . . . . . . 10 |- ( ( ( ph /\ K = 0 ) /\ a e. NN ) -> A. d e. NN ( a (AP ` K ) d ) C_ ( `' F " { c } ) )
44		r19.2z 3878	. . . . . . . . . 10 |- ( ( NN =/= (/) /\ A. d e. NN ( a (AP ` K ) d ) C_ ( `' F " { c } ) ) -> E. d e. NN ( a (AP ` K ) d ) C_ ( `' F " { c } ) )
45	34, 43, 44	sylancr 663	. . . . . . . . 9 |- ( ( ( ph /\ K = 0 ) /\ a e. NN ) -> E. d e. NN ( a (AP ` K ) d ) C_ ( `' F " { c } ) )
46	45	ralrimiva 2830	. . . . . . . 8 |- ( ( ph /\ K = 0 ) -> A. a e. NN E. d e. NN ( a (AP ` K ) d ) C_ ( `' F " { c } ) )
47		r19.2z 3878	. . . . . . . 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 } ) )
48	34, 46, 47	sylancr 663	. . . . . . 7 |- ( ( ph /\ K = 0 ) -> E. a e. NN E. d e. NN ( a (AP ` K ) d ) C_ ( `' F " { c } ) )
49	48	ralrimivw 2831	. . . . . 6 |- ( ( ph /\ K = 0 ) -> A. c e. R E. a e. NN E. d e. NN ( a (AP ` K ) d ) C_ ( `' F " { c } ) )
50		r19.2z 3878	. . . . . 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 } ) )
51	31, 49, 50	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 } ) )
52		rexex 2893	. . . . 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 } ) )
53	51, 52	syl 16	. . . 4 |- ( ( ph /\ K = 0 ) -> E. c E. a e. NN E. d e. NN ( a (AP ` K ) d ) C_ ( `' F " { c } ) )
54	53, 51	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 } ) ) )
55		elnn0 10693	. . . 4 |- ( K e. NN0 <-> ( K e. NN \/ K = 0 ) )
56	2, 55	sylib 196	. . 3 |- ( ph -> ( K e. NN \/ K = 0 ) )
57	26, 54, 56	mpjaodan 784	. 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 } ) ) )
58		vdwapval 14153	. . . . . . . . 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 ) ) ) )
59	58	3expb 1189	. . . . . . . 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 ) ) ) )
60	2, 59	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 ) ) ) )
61	60	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 } ) ) ) )
62	61	albidv 1680	. . . . 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 } ) ) ) )
63		dfss2 3454	. . . . 5 |- ( ( a (AP ` K ) d ) C_ ( `' F " { c } ) <-> A. x ( x e. ( a (AP ` K ) d ) -> x e. ( `' F " { c } ) ) )
64		ralcom4 3097	. . . . . 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 } ) ) )
65		ovex 6226	. . . . . . . 8 |- ( a + ( m x. d ) ) e. _V
66		eleq1 2526	. . . . . . . 8 |- ( x = ( a + ( m x. d ) ) -> ( x e. ( `' F " { c } ) <-> ( a + ( m x. d ) ) e. ( `' F " { c } ) ) )
67	65, 66	ceqsalv 3106	. . . . . . 7 |- ( A. x ( x = ( a + ( m x. d ) ) -> x e. ( `' F " { c } ) ) <-> ( a + ( m x. d ) ) e. ( `' F " { c } ) )
68	67	ralbii 2839	. . . . . 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 } ) )
69		r19.23v 2939	. . . . . . 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 } ) ) )
70	69	albii 1611	. . . . . 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 } ) ) )
71	64, 68, 70	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 } ) ) )
72	62, 63, 71	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 } ) ) )
73	72	2rexbidva 2874	. . 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 } ) ) )
74	73	rexbidv 2868	. 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 } ) ) )
75	4, 57, 74	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 965   A.wal 1368   = wceq 1370   E.wex 1587   e. wcel 1758   =/= wne 2648   A.wral 2799   E.wrex 2800   _Vcvv 3078   i^i cin 3436   C_ wss 3437   (/)c0 3746   {csn 3986   class class class wbr 4401   `'ccnv 4948   "cima 4952   -->wf 5523   ` cfv 5527  (class class class)co 6201   0cc0 9394   1c1 9395   + caddc 9397   x. cmul 9399   - cmin 9707   NNcn 10434   NN0cn0 10691   ...cfz 11555  APcvdwa 14145   MonoAP cvdwm 14146

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483  ax-cnex 9450  ax-resscn 9451  ax-1cn 9452  ax-icn 9453  ax-addcl 9454  ax-addrcl 9455  ax-mulcl 9456  ax-mulrcl 9457  ax-mulcom 9458  ax-addass 9459  ax-mulass 9460  ax-distr 9461  ax-i2m1 9462  ax-1ne0 9463  ax-1rid 9464  ax-rnegex 9465  ax-rrecex 9466  ax-cnre 9467  ax-pre-lttri 9468  ax-pre-lttrn 9469  ax-pre-ltadd 9470  ax-pre-mulgt0 9471

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-tp 3991  df-op 3993  df-uni 4201  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-tr 4495  df-eprel 4741  df-id 4745  df-po 4750  df-so 4751  df-fr 4788  df-we 4790  df-ord 4831  df-on 4832  df-lim 4833  df-suc 4834  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-riota 6162  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-om 6588  df-1st 6688  df-2nd 6689  df-recs 6943  df-rdg 6977  df-er 7212  df-en 7422  df-dom 7423  df-sdom 7424  df-pnf 9532  df-mnf 9533  df-xr 9534  df-ltxr 9535  df-le 9536  df-sub 9709  df-neg 9710  df-nn 10435  df-n0 10692  df-z 10759  df-uz 10974  df-fz 11556  df-vdwap 14148  df-vdwmc 14149

This theorem is referenced by:  vdw  14174