Theorem isprm4 11789

Description: The predicate "is a prime number". A prime number is an integer greater than or equal to 2 whose only divisor greater than or equal to 2 is itself. (Contributed by Paul Chapman, 26-Oct-2012.)

Assertion

Ref	Expression
isprm4	|- ( P e. Prime <-> ( P e. ( ZZ>= ` 2 ) /\ A. z e. ( ZZ>= ` 2 ) ( z || P -> z = P ) ) )

Distinct variable group:   z, P

Proof of Theorem isprm4

Step	Hyp	Ref	Expression
1		isprm2 11787	. 2 |- ( P e. Prime <-> ( P e. ( ZZ>= ` 2 ) /\ A. z e. NN ( z || P -> ( z = 1 \/ z = P ) ) ) )
2		eluz2nn 9357	. . . . . . . 8 |- ( z e. ( ZZ>= ` 2 ) -> z e. NN )
3	2	pm4.71ri 389	. . . . . . 7 |- ( z e. ( ZZ>= ` 2 ) <-> ( z e. NN /\ z e. ( ZZ>= ` 2 ) ) )
4	3	imbi1i 237	. . . . . 6 |- ( ( z e. ( ZZ>= ` 2 ) -> ( z || P -> z = P ) ) <-> ( ( z e. NN /\ z e. ( ZZ>= ` 2 ) ) -> ( z || P -> z = P ) ) )
5		impexp 261	. . . . . 6 |- ( ( ( z e. NN /\ z e. ( ZZ>= ` 2 ) ) -> ( z || P -> z = P ) ) <-> ( z e. NN -> ( z e. ( ZZ>= ` 2 ) -> ( z || P -> z = P ) ) ) )
6	4, 5	bitri 183	. . . . 5 |- ( ( z e. ( ZZ>= ` 2 ) -> ( z || P -> z = P ) ) <-> ( z e. NN -> ( z e. ( ZZ>= ` 2 ) -> ( z || P -> z = P ) ) ) )
7		eluz2b3 9391	. . . . . . . 8 |- ( z e. ( ZZ>= ` 2 ) <-> ( z e. NN /\ z =/= 1 ) )
8	7	imbi1i 237	. . . . . . 7 |- ( ( z e. ( ZZ>= ` 2 ) -> ( z || P -> z = P ) ) <-> ( ( z e. NN /\ z =/= 1 ) -> ( z || P -> z = P ) ) )
9		impexp 261	. . . . . . . 8 |- ( ( ( z e. NN /\ z =/= 1 ) -> ( z || P -> z = P ) ) <-> ( z e. NN -> ( z =/= 1 -> ( z || P -> z = P ) ) ) )
10		bi2.04 247	. . . . . . . . . 10 |- ( ( z =/= 1 -> ( z || P -> z = P ) ) <-> ( z || P -> ( z =/= 1 -> z = P ) ) )
11		nnz 9066	. . . . . . . . . . . . . 14 |- ( z e. NN -> z e. ZZ )
12		1zzd 9074	. . . . . . . . . . . . . 14 |- ( z e. NN -> 1 e. ZZ )
13		zdceq 9119	. . . . . . . . . . . . . 14 |- ( ( z e. ZZ /\ 1 e. ZZ ) -> DECID z = 1 )
14	11, 12, 13	syl2anc 408	. . . . . . . . . . . . 13 |- ( z e. NN -> DECID z = 1 )
15		dfordc 877	. . . . . . . . . . . . 13 |- (DECID z = 1 -> ( ( z = 1 \/ z = P ) <-> ( -. z = 1 -> z = P ) ) )
16	14, 15	syl 14	. . . . . . . . . . . 12 |- ( z e. NN -> ( ( z = 1 \/ z = P ) <-> ( -. z = 1 -> z = P ) ) )
17		df-ne 2307	. . . . . . . . . . . . 13 |- ( z =/= 1 <-> -. z = 1 )
18	17	imbi1i 237	. . . . . . . . . . . 12 |- ( ( z =/= 1 -> z = P ) <-> ( -. z = 1 -> z = P ) )
19	16, 18	syl6rbbr 198	. . . . . . . . . . 11 |- ( z e. NN -> ( ( z =/= 1 -> z = P ) <-> ( z = 1 \/ z = P ) ) )
20	19	imbi2d 229	. . . . . . . . . 10 |- ( z e. NN -> ( ( z || P -> ( z =/= 1 -> z = P ) ) <-> ( z || P -> ( z = 1 \/ z = P ) ) ) )
21	10, 20	syl5bb 191	. . . . . . . . 9 |- ( z e. NN -> ( ( z =/= 1 -> ( z || P -> z = P ) ) <-> ( z || P -> ( z = 1 \/ z = P ) ) ) )
22	21	imbi2d 229	. . . . . . . 8 |- ( z e. NN -> ( ( z e. NN -> ( z =/= 1 -> ( z || P -> z = P ) ) ) <-> ( z e. NN -> ( z || P -> ( z = 1 \/ z = P ) ) ) ) )
23	9, 22	syl5bb 191	. . . . . . 7 |- ( z e. NN -> ( ( ( z e. NN /\ z =/= 1 ) -> ( z || P -> z = P ) ) <-> ( z e. NN -> ( z || P -> ( z = 1 \/ z = P ) ) ) ) )
24	8, 23	syl5bb 191	. . . . . 6 |- ( z e. NN -> ( ( z e. ( ZZ>= ` 2 ) -> ( z || P -> z = P ) ) <-> ( z e. NN -> ( z || P -> ( z = 1 \/ z = P ) ) ) ) )
25	24	pm5.74i 179	. . . . 5 |- ( ( z e. NN -> ( z e. ( ZZ>= ` 2 ) -> ( z || P -> z = P ) ) ) <-> ( z e. NN -> ( z e. NN -> ( z || P -> ( z = 1 \/ z = P ) ) ) ) )
26		pm5.4 248	. . . . 5 |- ( ( z e. NN -> ( z e. NN -> ( z || P -> ( z = 1 \/ z = P ) ) ) ) <-> ( z e. NN -> ( z || P -> ( z = 1 \/ z = P ) ) ) )
27	6, 25, 26	3bitri 205	. . . 4 |- ( ( z e. ( ZZ>= ` 2 ) -> ( z || P -> z = P ) ) <-> ( z e. NN -> ( z || P -> ( z = 1 \/ z = P ) ) ) )
28	27	ralbii2 2443	. . 3 |- ( A. z e. ( ZZ>= ` 2 ) ( z || P -> z = P ) <-> A. z e. NN ( z || P -> ( z = 1 \/ z = P ) ) )
29	28	anbi2i 452	. 2 |- ( ( P e. ( ZZ>= ` 2 ) /\ A. z e. ( ZZ>= ` 2 ) ( z || P -> z = P ) ) <-> ( P e. ( ZZ>= ` 2 ) /\ A. z e. NN ( z || P -> ( z = 1 \/ z = P ) ) ) )
30	1, 29	bitr4i 186	1 |- ( P e. Prime <-> ( P e. ( ZZ>= ` 2 ) /\ A. z e. ( ZZ>= ` 2 ) ( z || P -> z = P ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 697  DECID wdc 819   = wceq 1331   e. wcel 1480   =/= wne 2306   A.wral 2414   class class class wbr 3924   ` cfv 5118   1c1 7614   NNcn 8713   2c2 8764   ZZcz 9047   ZZ>=cuz 9319   || cdvds 11482   Primecprime 11777

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-coll 4038  ax-sep 4041  ax-nul 4049  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-iinf 4497  ax-cnex 7704  ax-resscn 7705  ax-1cn 7706  ax-1re 7707  ax-icn 7708  ax-addcl 7709  ax-addrcl 7710  ax-mulcl 7711  ax-mulrcl 7712  ax-addcom 7713  ax-mulcom 7714  ax-addass 7715  ax-mulass 7716  ax-distr 7717  ax-i2m1 7718  ax-0lt1 7719  ax-1rid 7720  ax-0id 7721  ax-rnegex 7722  ax-precex 7723  ax-cnre 7724  ax-pre-ltirr 7725  ax-pre-ltwlin 7726  ax-pre-lttrn 7727  ax-pre-apti 7728  ax-pre-ltadd 7729  ax-pre-mulgt0 7730  ax-pre-mulext 7731  ax-arch 7732  ax-caucvg 7733

This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-nel 2402  df-ral 2419  df-rex 2420  df-reu 2421  df-rmo 2422  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-if 3470  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-tr 4022  df-id 4210  df-po 4213  df-iso 4214  df-iord 4283  df-on 4285  df-ilim 4286  df-suc 4288  df-iom 4500  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-riota 5723  df-ov 5770  df-oprab 5771  df-mpo 5772  df-1st 6031  df-2nd 6032  df-recs 6195  df-frec 6281  df-1o 6306  df-2o 6307  df-er 6422  df-en 6628  df-pnf 7795  df-mnf 7796  df-xr 7797  df-ltxr 7798  df-le 7799  df-sub 7928  df-neg 7929  df-reap 8330  df-ap 8337  df-div 8426  df-inn 8714  df-2 8772  df-3 8773  df-4 8774  df-n0 8971  df-z 9048  df-uz 9320  df-q 9405  df-rp 9435  df-seqfrec 10212  df-exp 10286  df-cj 10607  df-re 10608  df-im 10609  df-rsqrt 10763  df-abs 10764  df-dvds 11483  df-prm 11778

This theorem is referenced by:  nprm  11793  prmuz2  11800  dvdsprm  11806