Theorem vdwmc2 14929

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 14928	. 2 |- ( ph -> ( K MonoAP F <-> E. c E. a e. NN E. d e. NN ( a (AP ` K ) d ) C_ ( `' F " { c } ) ) )
5		vdwapid1 14925	. . . . . . . . . . . 12 |- ( ( K e. NN /\ a e. NN /\ d e. NN ) -> a e. ( a (AP ` K ) d ) )
6		ne0i 3737	. . . . . . . . . . . 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 1209	. . . . . . . . . 10 |- ( ( K e. NN /\ ( a e. NN /\ d e. NN ) ) -> ( a (AP ` K ) d ) =/= (/) )
9	8	adantll 720	. . . . . . . . 9 |- ( ( ( ph /\ K e. NN ) /\ ( a e. NN /\ d e. NN ) ) -> ( a (AP ` K ) d ) =/= (/) )
10		ssn0 3767	. . . . . . . . . 10 |- ( ( ( a (AP ` K ) d ) C_ ( `' F " { c } ) /\ ( a (AP ` K ) d ) =/= (/) ) -> ( `' F " { c } ) =/= (/) )
11	10	expcom 437	. . . . . . . . 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 4032	. . . . . . . . . 10 |- ( ( R i^i { c } ) = (/) <-> -. c e. R )
14	3	adantr 467	. . . . . . . . . . . 12 |- ( ( ph /\ K e. NN ) -> F : X --> R )
15		fimacnvdisj 5761	. . . . . . . . . . . . 13 |- ( ( F : X --> R /\ ( R i^i { c } ) = (/) ) -> ( `' F " { c } ) = (/) )
16	15	ex 436	. . . . . . . . . . . 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 467	. . . . . . . . . 10 |- ( ( ( ph /\ K e. NN ) /\ ( a e. NN /\ d e. NN ) ) -> ( ( R i^i { c } ) = (/) -> ( `' F " { c } ) = (/) ) )
19	13, 18	syl5bir 222	. . . . . . . . 9 |- ( ( ( ph /\ K e. NN ) /\ ( a e. NN /\ d e. NN ) ) -> ( -. c e. R -> ( `' F " { c } ) = (/) ) )
20	19	necon1ad 2641	. . . . . . . 8 |- ( ( ( ph /\ K e. NN ) /\ ( a e. NN /\ d e. NN ) ) -> ( ( `' F " { c } ) =/= (/) -> c e. R ) )
21	12, 20	syld 45	. . . . . . 7 |- ( ( ( ph /\ K e. NN ) /\ ( a e. NN /\ d e. NN ) ) -> ( ( a (AP ` K ) d ) C_ ( `' F " { c } ) -> c e. R ) )
22	21	rexlimdvva 2886	. . . . . 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 641	. . . . 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 1768	. . . 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 2743	. . . 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 267	. . 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 6023	. . . . . . . 8 |- ( ph -> ( F ` A ) e. R )
29		ne0i 3737	. . . . . . . 8 |- ( ( F ` A ) e. R -> R =/= (/) )
30	28, 29	syl 17	. . . . . . 7 |- ( ph -> R =/= (/) )
31	30	adantr 467	. . . . . 6 |- ( ( ph /\ K = 0 ) -> R =/= (/) )
32		1nn 10620	. . . . . . . . 9 |- 1 e. NN
33	32	ne0ii 3738	. . . . . . . 8 |- NN =/= (/)
34		simpllr 769	. . . . . . . . . . . . . . 15 |- ( ( ( ( ph /\ K = 0 ) /\ a e. NN ) /\ d e. NN ) -> K = 0 )
35	34	fveq2d 5869	. . . . . . . . . . . . . 14 |- ( ( ( ( ph /\ K = 0 ) /\ a e. NN ) /\ d e. NN ) -> (AP ` K ) = (AP ` 0 ) )
36	35	oveqd 6307	. . . . . . . . . . . . 13 |- ( ( ( ( ph /\ K = 0 ) /\ a e. NN ) /\ d e. NN ) -> ( a (AP ` K ) d ) = ( a (AP ` 0 ) d ) )
37		vdwap0 14926	. . . . . . . . . . . . . 14 |- ( ( a e. NN /\ d e. NN ) -> ( a (AP ` 0 ) d ) = (/) )
38	37	adantll 720	. . . . . . . . . . . . 13 |- ( ( ( ( ph /\ K = 0 ) /\ a e. NN ) /\ d e. NN ) -> ( a (AP ` 0 ) d ) = (/) )
39	36, 38	eqtrd 2485	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ K = 0 ) /\ a e. NN ) /\ d e. NN ) -> ( a (AP ` K ) d ) = (/) )
40		0ss 3763	. . . . . . . . . . . 12 |- (/) C_ ( `' F " { c } )
41	39, 40	syl6eqss 3482	. . . . . . . . . . 11 |- ( ( ( ( ph /\ K = 0 ) /\ a e. NN ) /\ d e. NN ) -> ( a (AP ` K ) d ) C_ ( `' F " { c } ) )
42	41	ralrimiva 2802	. . . . . . . . . 10 |- ( ( ( ph /\ K = 0 ) /\ a e. NN ) -> A. d e. NN ( a (AP ` K ) d ) C_ ( `' F " { c } ) )
43		r19.2z 3858	. . . . . . . . . 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 669	. . . . . . . . 9 |- ( ( ( ph /\ K = 0 ) /\ a e. NN ) -> E. d e. NN ( a (AP ` K ) d ) C_ ( `' F " { c } ) )
45	44	ralrimiva 2802	. . . . . . . 8 |- ( ( ph /\ K = 0 ) -> A. a e. NN E. d e. NN ( a (AP ` K ) d ) C_ ( `' F " { c } ) )
46		r19.2z 3858	. . . . . . . 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 669	. . . . . . 7 |- ( ( ph /\ K = 0 ) -> E. a e. NN E. d e. NN ( a (AP ` K ) d ) C_ ( `' F " { c } ) )
48	47	ralrimivw 2803	. . . . . 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 3858	. . . . . 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 667	. . . . 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 2844	. . . . 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 244	. . 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 10871	. . . 4 |- ( K e. NN0 <-> ( K e. NN \/ K = 0 ) )
55	2, 54	sylib 200	. . 3 |- ( ph -> ( K e. NN \/ K = 0 ) )
56	26, 53, 55	mpjaodan 795	. 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 14923	. . . . . . . . 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 1209	. . . . . . . 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 474	. . . . . . 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 319	. . . . . 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 1767	. . . . 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 3421	. . . . 5 |- ( ( a (AP ` K ) d ) C_ ( `' F " { c } ) <-> A. x ( x e. ( a (AP ` K ) d ) -> x e. ( `' F " { c } ) ) )
63		ralcom4 3066	. . . . . 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 6318	. . . . . . . 8 |- ( a + ( m x. d ) ) e. _V
65		eleq1 2517	. . . . . . . 8 |- ( x = ( a + ( m x. d ) ) -> ( x e. ( `' F " { c } ) <-> ( a + ( m x. d ) ) e. ( `' F " { c } ) ) )
66	64, 65	ceqsalv 3075	. . . . . . 7 |- ( A. x ( x = ( a + ( m x. d ) ) -> x e. ( `' F " { c } ) ) <-> ( a + ( m x. d ) ) e. ( `' F " { c } ) )
67	66	ralbii 2819	. . . . . 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 2867	. . . . . . 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 1691	. . . . . 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 279	. . . . 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 292	. . . 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 2907	. . 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 2901	. 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 283	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 188   \/ wo 370   /\ wa 371   /\ w3a 985   A.wal 1442   = wceq 1444   E.wex 1663   e. wcel 1887   =/= wne 2622   A.wral 2737   E.wrex 2738   _Vcvv 3045   i^i cin 3403   C_ wss 3404   (/)c0 3731   {csn 3968   class class class wbr 4402   `'ccnv 4833   "cima 4837   -->wf 5578   ` cfv 5582  (class class class)co 6290   0cc0 9539   1c1 9540   + caddc 9542   x. cmul 9544   - cmin 9860   NNcn 10609   NN0cn0 10869   ...cfz 11784  APcvdwa 14915   MonoAP cvdwm 14916

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-1st 6793  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-er 7363  df-en 7570  df-dom 7571  df-sdom 7572  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-nn 10610  df-n0 10870  df-z 10938  df-uz 11160  df-fz 11785  df-vdwap 14918  df-vdwmc 14919

This theorem is referenced by:  vdw  14944