Theorem vdwmc2 14345

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 14344	. 2 |- ( ph -> ( K MonoAP F <-> E. c E. a e. NN E. d e. NN ( a (AP ` K ) d ) C_ ( `' F " { c } ) ) )
5		vdwapid1 14341	. . . . . . . . . . . 12 |- ( ( K e. NN /\ a e. NN /\ d e. NN ) -> a e. ( a (AP ` K ) d ) )
6		ne0i 3784	. . . . . . . . . . . 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 1192	. . . . . . . . . 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 3811	. . . . . . . . . 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 4081	. . . . . . . . . 10 |- ( ( R i^i { c } ) = (/) <-> -. c e. R )
14	3	adantr 465	. . . . . . . . . . . 12 |- ( ( ph /\ K e. NN ) -> F : X --> R )
15		fimacnvdisj 5754	. . . . . . . . . . . . 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 2676	. . . . . . . 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 2955	. . . . . 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 1685	. . . 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 6013	. . . . . . . 8 |- ( ph -> ( F ` A ) e. R )
29		ne0i 3784	. . . . . . . 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 10536	. . . . . . . . 9 |- 1 e. NN
33		ne0i 3784	. . . . . . . . 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 5861	. . . . . . . . . . . . . 14 |- ( ( ( ( ph /\ K = 0 ) /\ a e. NN ) /\ d e. NN ) -> (AP ` K ) = (AP ` 0 ) )
37	36	oveqd 6292	. . . . . . . . . . . . 13 |- ( ( ( ( ph /\ K = 0 ) /\ a e. NN ) /\ d e. NN ) -> ( a (AP ` K ) d ) = ( a (AP ` 0 ) d ) )
38		vdwap0 14342	. . . . . . . . . . . . . 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 2501	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ K = 0 ) /\ a e. NN ) /\ d e. NN ) -> ( a (AP ` K ) d ) = (/) )
41		0ss 3807	. . . . . . . . . . . 12 |- (/) C_ ( `' F " { c } )
42	40, 41	syl6eqss 3547	. . . . . . . . . . 11 |- ( ( ( ( ph /\ K = 0 ) /\ a e. NN ) /\ d e. NN ) -> ( a (AP ` K ) d ) C_ ( `' F " { c } ) )
43	42	ralrimiva 2871	. . . . . . . . . 10 |- ( ( ( ph /\ K = 0 ) /\ a e. NN ) -> A. d e. NN ( a (AP ` K ) d ) C_ ( `' F " { c } ) )
44		r19.2z 3910	. . . . . . . . . 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 2871	. . . . . . . 8 |- ( ( ph /\ K = 0 ) -> A. a e. NN E. d e. NN ( a (AP ` K ) d ) C_ ( `' F " { c } ) )
47		r19.2z 3910	. . . . . . . 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 2872	. . . . . 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 3910	. . . . . 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 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 } ) )
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 10786	. . . 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 14339	. . . . . . . . 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 1192	. . . . . . . 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 1684	. . . . 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 3486	. . . . 5 |- ( ( a (AP ` K ) d ) C_ ( `' F " { c } ) <-> A. x ( x e. ( a (AP ` K ) d ) -> x e. ( `' F " { c } ) ) )
64		ralcom4 3125	. . . . . 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 6300	. . . . . . . 8 |- ( a + ( m x. d ) ) e. _V
66		eleq1 2532	. . . . . . . 8 |- ( x = ( a + ( m x. d ) ) -> ( x e. ( `' F " { c } ) <-> ( a + ( m x. d ) ) e. ( `' F " { c } ) ) )
67	65, 66	ceqsalv 3134	. . . . . . 7 |- ( A. x ( x = ( a + ( m x. d ) ) -> x e. ( `' F " { c } ) ) <-> ( a + ( m x. d ) ) e. ( `' F " { c } ) )
68	67	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 } ) )
69		r19.23v 2936	. . . . . . 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 1615	. . . . . 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 2972	. . 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 2966	. 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 968   A.wal 1372   = wceq 1374   E.wex 1591   e. wcel 1762   =/= wne 2655   A.wral 2807   E.wrex 2808   _Vcvv 3106   i^i cin 3468   C_ wss 3469   (/)c0 3778   {csn 4020   class class class wbr 4440   `'ccnv 4991   "cima 4995   -->wf 5575   ` cfv 5579  (class class class)co 6275   0cc0 9481   1c1 9482   + caddc 9484   x. cmul 9486   - cmin 9794   NNcn 10525   NN0cn0 10784   ...cfz 11661  APcvdwa 14331   MonoAP cvdwm 14332

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-1st 6774  df-2nd 6775  df-recs 7032  df-rdg 7066  df-er 7301  df-en 7507  df-dom 7508  df-sdom 7509  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-nn 10526  df-n0 10785  df-z 10854  df-uz 11072  df-fz 11662  df-vdwap 14334  df-vdwmc 14335

This theorem is referenced by:  vdw  14360