Theorem vdwmc2 15683

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 15682	. 2 |- ( ph -> ( K MonoAP F <-> E. c E. a e. NN E. d e. NN ( a (AP ` K ) d ) C_ ( `' F " { c } ) ) )
5		vdwapid1 15679	. . . . . . . . . . . 12 |- ( ( K e. NN /\ a e. NN /\ d e. NN ) -> a e. ( a (AP ` K ) d ) )
6		ne0i 3921	. . . . . . . . . . . 12 |- ( a e. ( a (AP ` K ) d ) -> ( a (AP ` K ) d ) =/= (/) )
7	5, 6	syl 17	. . . . . . . . . . 11 |- ( ( K e. NN /\ a e. NN /\ d e. NN ) -> ( a (AP ` K ) d ) =/= (/) )
8	7	3expb 1266	. . . . . . . . . 10 |- ( ( K e. NN /\ ( a e. NN /\ d e. NN ) ) -> ( a (AP ` K ) d ) =/= (/) )
9	8	adantll 750	. . . . . . . . 9 |- ( ( ( ph /\ K e. NN ) /\ ( a e. NN /\ d e. NN ) ) -> ( a (AP ` K ) d ) =/= (/) )
10		ssn0 3976	. . . . . . . . . 10 |- ( ( ( a (AP ` K ) d ) C_ ( `' F " { c } ) /\ ( a (AP ` K ) d ) =/= (/) ) -> ( `' F " { c } ) =/= (/) )
11	10	expcom 451	. . . . . . . . 9 |- ( ( a (AP ` K ) d ) =/= (/) -> ( ( a (AP ` K ) d ) C_ ( `' F " { c } ) -> ( `' F " { c } ) =/= (/) ) )
12	9, 11	syl 17	. . . . . . . 8 |- ( ( ( ph /\ K e. NN ) /\ ( a e. NN /\ d e. NN ) ) -> ( ( a (AP ` K ) d ) C_ ( `' F " { c } ) -> ( `' F " { c } ) =/= (/) ) )
13		disjsn 4246	. . . . . . . . . 10 |- ( ( R i^i { c } ) = (/) <-> -. c e. R )
14	3	adantr 481	. . . . . . . . . . . 12 |- ( ( ph /\ K e. NN ) -> F : X --> R )
15		fimacnvdisj 6083	. . . . . . . . . . . . 13 |- ( ( F : X --> R /\ ( R i^i { c } ) = (/) ) -> ( `' F " { c } ) = (/) )
16	15	ex 450	. . . . . . . . . . . 12 |- ( F : X --> R -> ( ( R i^i { c } ) = (/) -> ( `' F " { c } ) = (/) ) )
17	14, 16	syl 17	. . . . . . . . . . 11 |- ( ( ph /\ K e. NN ) -> ( ( R i^i { c } ) = (/) -> ( `' F " { c } ) = (/) ) )
18	17	adantr 481	. . . . . . . . . 10 |- ( ( ( ph /\ K e. NN ) /\ ( a e. NN /\ d e. NN ) ) -> ( ( R i^i { c } ) = (/) -> ( `' F " { c } ) = (/) ) )
19	13, 18	syl5bir 233	. . . . . . . . 9 |- ( ( ( ph /\ K e. NN ) /\ ( a e. NN /\ d e. NN ) ) -> ( -. c e. R -> ( `' F " { c } ) = (/) ) )
20	19	necon1ad 2811	. . . . . . . 8 |- ( ( ( ph /\ K e. NN ) /\ ( a e. NN /\ d e. NN ) ) -> ( ( `' F " { c } ) =/= (/) -> c e. R ) )
21	12, 20	syld 47	. . . . . . 7 |- ( ( ( ph /\ K e. NN ) /\ ( a e. NN /\ d e. NN ) ) -> ( ( a (AP ` K ) d ) C_ ( `' F " { c } ) -> c e. R ) )
22	21	rexlimdvva 3038	. . . . . 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 667	. . . . 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 1850	. . . 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 2918	. . . 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 278	. . 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 6360	. . . . . . . 8 |- ( ph -> ( F ` A ) e. R )
29		ne0i 3921	. . . . . . . 8 |- ( ( F ` A ) e. R -> R =/= (/) )
30	28, 29	syl 17	. . . . . . 7 |- ( ph -> R =/= (/) )
31	30	adantr 481	. . . . . 6 |- ( ( ph /\ K = 0 ) -> R =/= (/) )
32		1nn 11031	. . . . . . . . 9 |- 1 e. NN
33	32	ne0ii 3923	. . . . . . . 8 |- NN =/= (/)
34		simpllr 799	. . . . . . . . . . . . . . 15 |- ( ( ( ( ph /\ K = 0 ) /\ a e. NN ) /\ d e. NN ) -> K = 0 )
35	34	fveq2d 6195	. . . . . . . . . . . . . 14 |- ( ( ( ( ph /\ K = 0 ) /\ a e. NN ) /\ d e. NN ) -> (AP ` K ) = (AP ` 0 ) )
36	35	oveqd 6667	. . . . . . . . . . . . 13 |- ( ( ( ( ph /\ K = 0 ) /\ a e. NN ) /\ d e. NN ) -> ( a (AP ` K ) d ) = ( a (AP ` 0 ) d ) )
37		vdwap0 15680	. . . . . . . . . . . . . 14 |- ( ( a e. NN /\ d e. NN ) -> ( a (AP ` 0 ) d ) = (/) )
38	37	adantll 750	. . . . . . . . . . . . 13 |- ( ( ( ( ph /\ K = 0 ) /\ a e. NN ) /\ d e. NN ) -> ( a (AP ` 0 ) d ) = (/) )
39	36, 38	eqtrd 2656	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ K = 0 ) /\ a e. NN ) /\ d e. NN ) -> ( a (AP ` K ) d ) = (/) )
40		0ss 3972	. . . . . . . . . . . 12 |- (/) C_ ( `' F " { c } )
41	39, 40	syl6eqss 3655	. . . . . . . . . . 11 |- ( ( ( ( ph /\ K = 0 ) /\ a e. NN ) /\ d e. NN ) -> ( a (AP ` K ) d ) C_ ( `' F " { c } ) )
42	41	ralrimiva 2966	. . . . . . . . . 10 |- ( ( ( ph /\ K = 0 ) /\ a e. NN ) -> A. d e. NN ( a (AP ` K ) d ) C_ ( `' F " { c } ) )
43		r19.2z 4060	. . . . . . . . . 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 695	. . . . . . . . 9 |- ( ( ( ph /\ K = 0 ) /\ a e. NN ) -> E. d e. NN ( a (AP ` K ) d ) C_ ( `' F " { c } ) )
45	44	ralrimiva 2966	. . . . . . . 8 |- ( ( ph /\ K = 0 ) -> A. a e. NN E. d e. NN ( a (AP ` K ) d ) C_ ( `' F " { c } ) )
46		r19.2z 4060	. . . . . . . 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 695	. . . . . . 7 |- ( ( ph /\ K = 0 ) -> E. a e. NN E. d e. NN ( a (AP ` K ) d ) C_ ( `' F " { c } ) )
48	47	ralrimivw 2967	. . . . . 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 4060	. . . . . 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 693	. . . . 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 3002	. . . . 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 17	. . . 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 255	. . 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 11294	. . . 4 |- ( K e. NN0 <-> ( K e. NN \/ K = 0 ) )
55	2, 54	sylib 208	. . 3 |- ( ph -> ( K e. NN \/ K = 0 ) )
56	26, 53, 55	mpjaodan 827	. 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 15677	. . . . . . . . 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 1266	. . . . . . . 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 488	. . . . . . 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 331	. . . . . 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 1849	. . . . 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 3591	. . . . 5 |- ( ( a (AP ` K ) d ) C_ ( `' F " { c } ) <-> A. x ( x e. ( a (AP ` K ) d ) -> x e. ( `' F " { c } ) ) )
63		ralcom4 3224	. . . . . 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 6678	. . . . . . . 8 |- ( a + ( m x. d ) ) e. _V
65		eleq1 2689	. . . . . . . 8 |- ( x = ( a + ( m x. d ) ) -> ( x e. ( `' F " { c } ) <-> ( a + ( m x. d ) ) e. ( `' F " { c } ) ) )
66	64, 65	ceqsalv 3233	. . . . . . 7 |- ( A. x ( x = ( a + ( m x. d ) ) -> x e. ( `' F " { c } ) ) <-> ( a + ( m x. d ) ) e. ( `' F " { c } ) )
67	66	ralbii 2980	. . . . . 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 3023	. . . . . . 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 1747	. . . . . 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 290	. . . . 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 303	. . . 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 3056	. . 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 3052	. 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 294	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 196   \/ wo 383   /\ wa 384   /\ w3a 1037   A.wal 1481   = wceq 1483   E.wex 1704   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   _Vcvv 3200   i^i cin 3573   C_ wss 3574   (/)c0 3915   {csn 4177   class class class wbr 4653   `'ccnv 5113   "cima 5117   -->wf 5884   ` cfv 5888  (class class class)co 6650   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   - cmin 10266   NNcn 11020   NN0cn0 11292   ...cfz 12326  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:  vdw  15698