Theorem vdwapf 15676

Description: The arithmetic progression function is a function. (Contributed by Mario Carneiro, 18-Aug-2014.)

Assertion

Ref	Expression
vdwapf	|- ( K e. NN0 -> (AP ` K ) : ( NN X. NN ) --> ~P NN )

Proof of Theorem vdwapf

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

Step	Hyp	Ref	Expression
1		simpll 790	. . . . . . . 8 |- ( ( ( a e. NN /\ d e. NN ) /\ m e. ( 0 ... ( K - 1 ) ) ) -> a e. NN )
2		elfznn0 12433	. . . . . . . . . 10 |- ( m e. ( 0 ... ( K - 1 ) ) -> m e. NN0 )
3	2	adantl 482	. . . . . . . . 9 |- ( ( ( a e. NN /\ d e. NN ) /\ m e. ( 0 ... ( K - 1 ) ) ) -> m e. NN0 )
4		nnnn0 11299	. . . . . . . . . 10 |- ( d e. NN -> d e. NN0 )
5	4	ad2antlr 763	. . . . . . . . 9 |- ( ( ( a e. NN /\ d e. NN ) /\ m e. ( 0 ... ( K - 1 ) ) ) -> d e. NN0 )
6	3, 5	nn0mulcld 11356	. . . . . . . 8 |- ( ( ( a e. NN /\ d e. NN ) /\ m e. ( 0 ... ( K - 1 ) ) ) -> ( m x. d ) e. NN0 )
7		nnnn0addcl 11323	. . . . . . . 8 |- ( ( a e. NN /\ ( m x. d ) e. NN0 ) -> ( a + ( m x. d ) ) e. NN )
8	1, 6, 7	syl2anc 693	. . . . . . 7 |- ( ( ( a e. NN /\ d e. NN ) /\ m e. ( 0 ... ( K - 1 ) ) ) -> ( a + ( m x. d ) ) e. NN )
9		eqid 2622	. . . . . . 7 |- ( m e. ( 0 ... ( K - 1 ) ) |-> ( a + ( m x. d ) ) ) = ( m e. ( 0 ... ( K - 1 ) ) |-> ( a + ( m x. d ) ) )
10	8, 9	fmptd 6385	. . . . . 6 |- ( ( a e. NN /\ d e. NN ) -> ( m e. ( 0 ... ( K - 1 ) ) |-> ( a + ( m x. d ) ) ) : ( 0 ... ( K - 1 ) ) --> NN )
11		frn 6053	. . . . . 6 |- ( ( m e. ( 0 ... ( K - 1 ) ) |-> ( a + ( m x. d ) ) ) : ( 0 ... ( K - 1 ) ) --> NN -> ran ( m e. ( 0 ... ( K - 1 ) ) |-> ( a + ( m x. d ) ) ) C_ NN )
12	10, 11	syl 17	. . . . 5 |- ( ( a e. NN /\ d e. NN ) -> ran ( m e. ( 0 ... ( K - 1 ) ) |-> ( a + ( m x. d ) ) ) C_ NN )
13		nnex 11026	. . . . . 6 |- NN e. _V
14	13	elpw2 4828	. . . . 5 |- ( ran ( m e. ( 0 ... ( K - 1 ) ) |-> ( a + ( m x. d ) ) ) e. ~P NN <-> ran ( m e. ( 0 ... ( K - 1 ) ) |-> ( a + ( m x. d ) ) ) C_ NN )
15	12, 14	sylibr 224	. . . 4 |- ( ( a e. NN /\ d e. NN ) -> ran ( m e. ( 0 ... ( K - 1 ) ) |-> ( a + ( m x. d ) ) ) e. ~P NN )
16	15	rgen2a 2977	. . 3 |- A. a e. NN A. d e. NN ran ( m e. ( 0 ... ( K - 1 ) ) |-> ( a + ( m x. d ) ) ) e. ~P NN
17		eqid 2622	. . . 4 |- ( 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 ) ) ) )
18	17	fmpt2 7237	. . 3 |- ( A. a e. NN A. d e. NN ran ( m e. ( 0 ... ( K - 1 ) ) |-> ( a + ( m x. d ) ) ) e. ~P NN <-> ( a e. NN , d e. NN |-> ran ( m e. ( 0 ... ( K - 1 ) ) |-> ( a + ( m x. d ) ) ) ) : ( NN X. NN ) --> ~P NN )
19	16, 18	mpbi 220	. 2 |- ( a e. NN , d e. NN |-> ran ( m e. ( 0 ... ( K - 1 ) ) |-> ( a + ( m x. d ) ) ) ) : ( NN X. NN ) --> ~P NN
20		vdwapfval 15675	. . 3 |- ( K e. NN0 -> (AP ` K ) = ( a e. NN , d e. NN |-> ran ( m e. ( 0 ... ( K - 1 ) ) |-> ( a + ( m x. d ) ) ) ) )
21	20	feq1d 6030	. 2 |- ( K e. NN0 -> ( (AP ` K ) : ( NN X. NN ) --> ~P NN <-> ( a e. NN , d e. NN |-> ran ( m e. ( 0 ... ( K - 1 ) ) |-> ( a + ( m x. d ) ) ) ) : ( NN X. NN ) --> ~P NN ) )
22	19, 21	mpbiri 248	1 |- ( K e. NN0 -> (AP ` K ) : ( NN X. NN ) --> ~P NN )

Syntax hints:   -> wi 4   /\ wa 384   e. wcel 1990   A.wral 2912   C_ wss 3574   ~Pcpw 4158   |-> cmpt 4729   X. cxp 5112   ran crn 5115   -->wf 5884   ` 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-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

This theorem is referenced by:  vdwmc  15682