Theorem vdwapval 15677

Description: Value of the arithmetic progression function. (Contributed by Mario Carneiro, 18-Aug-2014.)

Assertion

Ref	Expression
vdwapval	|- ( ( 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 ) ) ) )

Distinct variable groups:   A, m   D, m   m, K   m, X

Proof of Theorem vdwapval

Dummy variables a d x are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		vdwapfval 15675	. . . . . . 7 |- ( K e. NN0 -> (AP ` K ) = ( a e. NN , d e. NN |-> ran ( m e. ( 0 ... ( K - 1 ) ) |-> ( a + ( m x. d ) ) ) ) )
2	1	3ad2ant1 1082	. . . . . 6 |- ( ( K e. NN0 /\ A e. NN /\ D e. NN ) -> (AP ` K ) = ( a e. NN , d e. NN |-> ran ( m e. ( 0 ... ( K - 1 ) ) |-> ( a + ( m x. d ) ) ) ) )
3	2	oveqd 6667	. . . . 5 |- ( ( K e. NN0 /\ A e. NN /\ D e. NN ) -> ( A (AP ` K ) D ) = ( A ( a e. NN , d e. NN |-> ran ( m e. ( 0 ... ( K - 1 ) ) |-> ( a + ( m x. d ) ) ) ) D ) )
4		oveq2 6658	. . . . . . . . . 10 |- ( d = D -> ( m x. d ) = ( m x. D ) )
5		oveq12 6659	. . . . . . . . . 10 |- ( ( a = A /\ ( m x. d ) = ( m x. D ) ) -> ( a + ( m x. d ) ) = ( A + ( m x. D ) ) )
6	4, 5	sylan2 491	. . . . . . . . 9 |- ( ( a = A /\ d = D ) -> ( a + ( m x. d ) ) = ( A + ( m x. D ) ) )
7	6	mpteq2dv 4745	. . . . . . . 8 |- ( ( a = A /\ d = D ) -> ( m e. ( 0 ... ( K - 1 ) ) |-> ( a + ( m x. d ) ) ) = ( m e. ( 0 ... ( K - 1 ) ) |-> ( A + ( m x. D ) ) ) )
8	7	rneqd 5353	. . . . . . 7 |- ( ( a = A /\ d = D ) -> ran ( m e. ( 0 ... ( K - 1 ) ) |-> ( a + ( m x. d ) ) ) = ran ( m e. ( 0 ... ( K - 1 ) ) |-> ( A + ( m x. D ) ) ) )
9		eqid 2622	. . . . . . 7 |- ( a e. NN , d e. NN |-> ran ( m e. ( 0 ... ( K - 1 ) ) |-> ( a + ( m x. d ) ) ) ) = ( a e. NN , d e. NN |-> ran ( m e. ( 0 ... ( K - 1 ) ) |-> ( a + ( m x. d ) ) ) )
10		ovex 6678	. . . . . . . . 9 |- ( 0 ... ( K - 1 ) ) e. _V
11	10	mptex 6486	. . . . . . . 8 |- ( m e. ( 0 ... ( K - 1 ) ) |-> ( A + ( m x. D ) ) ) e. _V
12	11	rnex 7100	. . . . . . 7 |- ran ( m e. ( 0 ... ( K - 1 ) ) |-> ( A + ( m x. D ) ) ) e. _V
13	8, 9, 12	ovmpt2a 6791	. . . . . 6 |- ( ( A e. NN /\ D e. NN ) -> ( A ( a e. NN , d e. NN |-> ran ( m e. ( 0 ... ( K - 1 ) ) |-> ( a + ( m x. d ) ) ) ) D ) = ran ( m e. ( 0 ... ( K - 1 ) ) |-> ( A + ( m x. D ) ) ) )
14	13	3adant1 1079	. . . . 5 |- ( ( K e. NN0 /\ A e. NN /\ D e. NN ) -> ( A ( a e. NN , d e. NN |-> ran ( m e. ( 0 ... ( K - 1 ) ) |-> ( a + ( m x. d ) ) ) ) D ) = ran ( m e. ( 0 ... ( K - 1 ) ) |-> ( A + ( m x. D ) ) ) )
15	3, 14	eqtrd 2656	. . . 4 |- ( ( K e. NN0 /\ A e. NN /\ D e. NN ) -> ( A (AP ` K ) D ) = ran ( m e. ( 0 ... ( K - 1 ) ) |-> ( A + ( m x. D ) ) ) )
16		eqid 2622	. . . . 5 |- ( m e. ( 0 ... ( K - 1 ) ) |-> ( A + ( m x. D ) ) ) = ( m e. ( 0 ... ( K - 1 ) ) |-> ( A + ( m x. D ) ) )
17	16	rnmpt 5371	. . . 4 |- ran ( m e. ( 0 ... ( K - 1 ) ) |-> ( A + ( m x. D ) ) ) = { x | E. m e. ( 0 ... ( K - 1 ) ) x = ( A + ( m x. D ) ) }
18	15, 17	syl6eq 2672	. . 3 |- ( ( K e. NN0 /\ A e. NN /\ D e. NN ) -> ( A (AP ` K ) D ) = { x | E. m e. ( 0 ... ( K - 1 ) ) x = ( A + ( m x. D ) ) } )
19	18	eleq2d 2687	. 2 |- ( ( K e. NN0 /\ A e. NN /\ D e. NN ) -> ( X e. ( A (AP ` K ) D ) <-> X e. { x | E. m e. ( 0 ... ( K - 1 ) ) x = ( A + ( m x. D ) ) } ) )
20		id 22	. . . . 5 |- ( X = ( A + ( m x. D ) ) -> X = ( A + ( m x. D ) ) )
21		ovex 6678	. . . . 5 |- ( A + ( m x. D ) ) e. _V
22	20, 21	syl6eqel 2709	. . . 4 |- ( X = ( A + ( m x. D ) ) -> X e. _V )
23	22	rexlimivw 3029	. . 3 |- ( E. m e. ( 0 ... ( K - 1 ) ) X = ( A + ( m x. D ) ) -> X e. _V )
24		eqeq1 2626	. . . 4 |- ( x = X -> ( x = ( A + ( m x. D ) ) <-> X = ( A + ( m x. D ) ) ) )
25	24	rexbidv 3052	. . 3 |- ( x = X -> ( E. m e. ( 0 ... ( K - 1 ) ) x = ( A + ( m x. D ) ) <-> E. m e. ( 0 ... ( K - 1 ) ) X = ( A + ( m x. D ) ) ) )
26	23, 25	elab3 3358	. 2 |- ( X e. { x | E. m e. ( 0 ... ( K - 1 ) ) x = ( A + ( m x. D ) ) } <-> E. m e. ( 0 ... ( K - 1 ) ) X = ( A + ( m x. D ) ) )
27	19, 26	syl6bb 276	1 |- ( ( 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 ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   {cab 2608   E.wrex 2913   _Vcvv 3200   |-> cmpt 4729   ran crn 5115   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   - cmin 10266   NNcn 11020   NN0cn0 11292   ...cfz 12326  APcvdwa 15669

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-i2m1 10004  ax-1ne0 10005  ax-rrecex 10008  ax-cnre 10009

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-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-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-nn 11021  df-vdwap 15672

This theorem is referenced by:  vdwapun  15678  vdwap0  15680  vdwmc2  15683  vdwlem1  15685  vdwlem2  15686  vdwlem6  15690  vdwlem8  15692