Theorem isprm2 15395

Description: The predicate "is a prime number". A prime number is an integer greater than or equal to 2 whose only positive divisors are 1 and itself. Definition in [ApostolNT] p. 16. (Contributed by Paul Chapman, 26-Oct-2012.)

Assertion

Ref	Expression
isprm2	|- ( P e. Prime <-> ( P e. ( ZZ>= ` 2 ) /\ A. z e. NN ( z || P -> ( z = 1 \/ z = P ) ) ) )

Distinct variable group:   z, P

Proof of Theorem isprm2

Dummy variable n is distinct from all other variables.

Step	Hyp	Ref	Expression
1		1nprm 15392	. . . . 5 |- -. 1 e. Prime
2		eleq1 2689	. . . . . 6 |- ( P = 1 -> ( P e. Prime <-> 1 e. Prime ) )
3	2	biimpcd 239	. . . . 5 |- ( P e. Prime -> ( P = 1 -> 1 e. Prime ) )
4	1, 3	mtoi 190	. . . 4 |- ( P e. Prime -> -. P = 1 )
5	4	neqned 2801	. . 3 |- ( P e. Prime -> P =/= 1 )
6	5	pm4.71i 664	. 2 |- ( P e. Prime <-> ( P e. Prime /\ P =/= 1 ) )
7		isprm 15387	. . . 4 |- ( P e. Prime <-> ( P e. NN /\ { n e. NN | n || P } ~~ 2o ) )
8		isprm2lem 15394	. . . . . . 7 |- ( ( P e. NN /\ P =/= 1 ) -> ( { n e. NN | n || P } ~~ 2o <-> { n e. NN | n || P } = { 1 , P } ) )
9		eqss 3618	. . . . . . . . . . 11 |- ( { n e. NN | n || P } = { 1 , P } <-> ( { n e. NN | n || P } C_ { 1 , P } /\ { 1 , P } C_ { n e. NN | n || P } ) )
10	9	imbi2i 326	. . . . . . . . . 10 |- ( ( P e. NN -> { n e. NN | n || P } = { 1 , P } ) <-> ( P e. NN -> ( { n e. NN | n || P } C_ { 1 , P } /\ { 1 , P } C_ { n e. NN | n || P } ) ) )
11		1idssfct 15393	. . . . . . . . . . 11 |- ( P e. NN -> { 1 , P } C_ { n e. NN | n || P } )
12		jcab 907	. . . . . . . . . . 11 |- ( ( P e. NN -> ( { n e. NN | n || P } C_ { 1 , P } /\ { 1 , P } C_ { n e. NN | n || P } ) ) <-> ( ( P e. NN -> { n e. NN | n || P } C_ { 1 , P } ) /\ ( P e. NN -> { 1 , P } C_ { n e. NN | n || P } ) ) )
13	11, 12	mpbiran2 954	. . . . . . . . . 10 |- ( ( P e. NN -> ( { n e. NN | n || P } C_ { 1 , P } /\ { 1 , P } C_ { n e. NN | n || P } ) ) <-> ( P e. NN -> { n e. NN | n || P } C_ { 1 , P } ) )
14	10, 13	bitri 264	. . . . . . . . 9 |- ( ( P e. NN -> { n e. NN | n || P } = { 1 , P } ) <-> ( P e. NN -> { n e. NN | n || P } C_ { 1 , P } ) )
15	14	pm5.74ri 261	. . . . . . . 8 |- ( P e. NN -> ( { n e. NN | n || P } = { 1 , P } <-> { n e. NN | n || P } C_ { 1 , P } ) )
16	15	adantr 481	. . . . . . 7 |- ( ( P e. NN /\ P =/= 1 ) -> ( { n e. NN | n || P } = { 1 , P } <-> { n e. NN | n || P } C_ { 1 , P } ) )
17	8, 16	bitrd 268	. . . . . 6 |- ( ( P e. NN /\ P =/= 1 ) -> ( { n e. NN | n || P } ~~ 2o <-> { n e. NN | n || P } C_ { 1 , P } ) )
18	17	expcom 451	. . . . 5 |- ( P =/= 1 -> ( P e. NN -> ( { n e. NN | n || P } ~~ 2o <-> { n e. NN | n || P } C_ { 1 , P } ) ) )
19	18	pm5.32d 671	. . . 4 |- ( P =/= 1 -> ( ( P e. NN /\ { n e. NN | n || P } ~~ 2o ) <-> ( P e. NN /\ { n e. NN | n || P } C_ { 1 , P } ) ) )
20	7, 19	syl5bb 272	. . 3 |- ( P =/= 1 -> ( P e. Prime <-> ( P e. NN /\ { n e. NN | n || P } C_ { 1 , P } ) ) )
21	20	pm5.32ri 670	. 2 |- ( ( P e. Prime /\ P =/= 1 ) <-> ( ( P e. NN /\ { n e. NN | n || P } C_ { 1 , P } ) /\ P =/= 1 ) )
22		ancom 466	. . . 4 |- ( ( ( P e. NN /\ { n e. NN | n || P } C_ { 1 , P } ) /\ P =/= 1 ) <-> ( P =/= 1 /\ ( P e. NN /\ { n e. NN | n || P } C_ { 1 , P } ) ) )
23		anass 681	. . . 4 |- ( ( ( P =/= 1 /\ P e. NN ) /\ { n e. NN | n || P } C_ { 1 , P } ) <-> ( P =/= 1 /\ ( P e. NN /\ { n e. NN | n || P } C_ { 1 , P } ) ) )
24	22, 23	bitr4i 267	. . 3 |- ( ( ( P e. NN /\ { n e. NN | n || P } C_ { 1 , P } ) /\ P =/= 1 ) <-> ( ( P =/= 1 /\ P e. NN ) /\ { n e. NN | n || P } C_ { 1 , P } ) )
25		ancom 466	. . . . 5 |- ( ( P =/= 1 /\ P e. NN ) <-> ( P e. NN /\ P =/= 1 ) )
26		eluz2b3 11762	. . . . 5 |- ( P e. ( ZZ>= ` 2 ) <-> ( P e. NN /\ P =/= 1 ) )
27	25, 26	bitr4i 267	. . . 4 |- ( ( P =/= 1 /\ P e. NN ) <-> P e. ( ZZ>= ` 2 ) )
28	27	anbi1i 731	. . 3 |- ( ( ( P =/= 1 /\ P e. NN ) /\ { n e. NN | n || P } C_ { 1 , P } ) <-> ( P e. ( ZZ>= ` 2 ) /\ { n e. NN | n || P } C_ { 1 , P } ) )
29		dfss2 3591	. . . . 5 |- ( { n e. NN | n || P } C_ { 1 , P } <-> A. z ( z e. { n e. NN | n || P } -> z e. { 1 , P } ) )
30		breq1 4656	. . . . . . . . . 10 |- ( n = z -> ( n || P <-> z || P ) )
31	30	elrab 3363	. . . . . . . . 9 |- ( z e. { n e. NN | n || P } <-> ( z e. NN /\ z || P ) )
32		vex 3203	. . . . . . . . . 10 |- z e. _V
33	32	elpr 4198	. . . . . . . . 9 |- ( z e. { 1 , P } <-> ( z = 1 \/ z = P ) )
34	31, 33	imbi12i 340	. . . . . . . 8 |- ( ( z e. { n e. NN | n || P } -> z e. { 1 , P } ) <-> ( ( z e. NN /\ z || P ) -> ( z = 1 \/ z = P ) ) )
35		impexp 462	. . . . . . . 8 |- ( ( ( z e. NN /\ z || P ) -> ( z = 1 \/ z = P ) ) <-> ( z e. NN -> ( z || P -> ( z = 1 \/ z = P ) ) ) )
36	34, 35	bitri 264	. . . . . . 7 |- ( ( z e. { n e. NN | n || P } -> z e. { 1 , P } ) <-> ( z e. NN -> ( z || P -> ( z = 1 \/ z = P ) ) ) )
37	36	albii 1747	. . . . . 6 |- ( A. z ( z e. { n e. NN | n || P } -> z e. { 1 , P } ) <-> A. z ( z e. NN -> ( z || P -> ( z = 1 \/ z = P ) ) ) )
38		df-ral 2917	. . . . . 6 |- ( A. z e. NN ( z || P -> ( z = 1 \/ z = P ) ) <-> A. z ( z e. NN -> ( z || P -> ( z = 1 \/ z = P ) ) ) )
39	37, 38	bitr4i 267	. . . . 5 |- ( A. z ( z e. { n e. NN | n || P } -> z e. { 1 , P } ) <-> A. z e. NN ( z || P -> ( z = 1 \/ z = P ) ) )
40	29, 39	bitri 264	. . . 4 |- ( { n e. NN | n || P } C_ { 1 , P } <-> A. z e. NN ( z || P -> ( z = 1 \/ z = P ) ) )
41	40	anbi2i 730	. . 3 |- ( ( P e. ( ZZ>= ` 2 ) /\ { n e. NN | n || P } C_ { 1 , P } ) <-> ( P e. ( ZZ>= ` 2 ) /\ A. z e. NN ( z || P -> ( z = 1 \/ z = P ) ) ) )
42	24, 28, 41	3bitri 286	. 2 |- ( ( ( P e. NN /\ { n e. NN | n || P } C_ { 1 , P } ) /\ P =/= 1 ) <-> ( P e. ( ZZ>= ` 2 ) /\ A. z e. NN ( z || P -> ( z = 1 \/ z = P ) ) ) )
43	6, 21, 42	3bitri 286	1 |- ( P e. Prime <-> ( P e. ( ZZ>= ` 2 ) /\ A. z e. NN ( z || P -> ( z = 1 \/ z = P ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   A.wal 1481   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   {crab 2916   C_ wss 3574   {cpr 4179   class class class wbr 4653   ` cfv 5888   2oc2o 7554   ~~ cen 7952   1c1 9937   NNcn 11020   2c2 11070   ZZ>=cuz 11687   || cdvds 14983   Primecprime 15385

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-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  ax-pre-sup 10014

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-rmo 2920  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-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-dvds 14984  df-prm 15386

This theorem is referenced by:  isprm3  15396  isprm4  15397  dvdsprime  15400  coprm  15423  isprm6  15426  prmirredlem  19841  znidomb  19910  perfectlem2  24955  perfectALTVlem2  41631