Theorem isprm4 12287

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 12285	. 2 |- ( P e. Prime <-> ( P e. ( ZZ>= ` 2 ) /\ A. z e. NN ( z || P -> ( z = 1 \/ z = P ) ) ) )
2		eluz2nn 9640	. . . . . . . 8 |- ( z e. ( ZZ>= ` 2 ) -> z e. NN )
3	2	pm4.71ri 392	. . . . . . 7 |- ( z e. ( ZZ>= ` 2 ) <-> ( z e. NN /\ z e. ( ZZ>= ` 2 ) ) )
4	3	imbi1i 238	. . . . . 6 |- ( ( z e. ( ZZ>= ` 2 ) -> ( z || P -> z = P ) ) <-> ( ( z e. NN /\ z e. ( ZZ>= ` 2 ) ) -> ( z || P -> z = P ) ) )
5		impexp 263	. . . . . 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 184	. . . . 5 |- ( ( z e. ( ZZ>= ` 2 ) -> ( z || P -> z = P ) ) <-> ( z e. NN -> ( z e. ( ZZ>= ` 2 ) -> ( z || P -> z = P ) ) ) )
7		eluz2b3 9678	. . . . . . . 8 |- ( z e. ( ZZ>= ` 2 ) <-> ( z e. NN /\ z =/= 1 ) )
8	7	imbi1i 238	. . . . . . 7 |- ( ( z e. ( ZZ>= ` 2 ) -> ( z || P -> z = P ) ) <-> ( ( z e. NN /\ z =/= 1 ) -> ( z || P -> z = P ) ) )
9		impexp 263	. . . . . . . 8 |- ( ( ( z e. NN /\ z =/= 1 ) -> ( z || P -> z = P ) ) <-> ( z e. NN -> ( z =/= 1 -> ( z || P -> z = P ) ) ) )
10		bi2.04 248	. . . . . . . . . 10 |- ( ( z =/= 1 -> ( z || P -> z = P ) ) <-> ( z || P -> ( z =/= 1 -> z = P ) ) )
11		df-ne 2368	. . . . . . . . . . . . 13 |- ( z =/= 1 <-> -. z = 1 )
12	11	imbi1i 238	. . . . . . . . . . . 12 |- ( ( z =/= 1 -> z = P ) <-> ( -. z = 1 -> z = P ) )
13		nnz 9345	. . . . . . . . . . . . . 14 |- ( z e. NN -> z e. ZZ )
14		1zzd 9353	. . . . . . . . . . . . . 14 |- ( z e. NN -> 1 e. ZZ )
15		zdceq 9401	. . . . . . . . . . . . . 14 |- ( ( z e. ZZ /\ 1 e. ZZ ) -> DECID z = 1 )
16	13, 14, 15	syl2anc 411	. . . . . . . . . . . . 13 |- ( z e. NN -> DECID z = 1 )
17		dfordc 893	. . . . . . . . . . . . 13 |- (DECID z = 1 -> ( ( z = 1 \/ z = P ) <-> ( -. z = 1 -> z = P ) ) )
18	16, 17	syl 14	. . . . . . . . . . . 12 |- ( z e. NN -> ( ( z = 1 \/ z = P ) <-> ( -. z = 1 -> z = P ) ) )
19	12, 18	bitr4id 199	. . . . . . . . . . 11 |- ( z e. NN -> ( ( z =/= 1 -> z = P ) <-> ( z = 1 \/ z = P ) ) )
20	19	imbi2d 230	. . . . . . . . . 10 |- ( z e. NN -> ( ( z || P -> ( z =/= 1 -> z = P ) ) <-> ( z || P -> ( z = 1 \/ z = P ) ) ) )
21	10, 20	bitrid 192	. . . . . . . . 9 |- ( z e. NN -> ( ( z =/= 1 -> ( z || P -> z = P ) ) <-> ( z || P -> ( z = 1 \/ z = P ) ) ) )
22	21	imbi2d 230	. . . . . . . 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	bitrid 192	. . . . . . 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	bitrid 192	. . . . . 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 180	. . . . 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		pm5.4 249	. . . . 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 206	. . . 4 |- ( ( z e. ( ZZ>= ` 2 ) -> ( z || P -> z = P ) ) <-> ( z e. NN -> ( z || P -> ( z = 1 \/ z = P ) ) ) )
28	27	ralbii2 2507	. . 3 |- ( A. z e. ( ZZ>= ` 2 ) ( z || P -> z = P ) <-> A. z e. NN ( z || P -> ( z = 1 \/ z = P ) ) )
29	28	anbi2i 457	. 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 187	1 |- ( P e. Prime <-> ( P e. ( ZZ>= ` 2 ) /\ A. z e. ( ZZ>= ` 2 ) ( z || P -> z = P ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 709  DECID wdc 835   = wceq 1364   e. wcel 2167   =/= wne 2367   A.wral 2475   class class class wbr 4033   ` cfv 5258   1c1 7880   NNcn 8990   2c2 9041   ZZcz 9326   ZZ>=cuz 9601   || cdvds 11952   Primecprime 12275

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-iinf 4624  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-mulrcl 7978  ax-addcom 7979  ax-mulcom 7980  ax-addass 7981  ax-mulass 7982  ax-distr 7983  ax-i2m1 7984  ax-0lt1 7985  ax-1rid 7986  ax-0id 7987  ax-rnegex 7988  ax-precex 7989  ax-cnre 7990  ax-pre-ltirr 7991  ax-pre-ltwlin 7992  ax-pre-lttrn 7993  ax-pre-apti 7994  ax-pre-ltadd 7995  ax-pre-mulgt0 7996  ax-pre-mulext 7997  ax-arch 7998  ax-caucvg 7999

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-if 3562  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-id 4328  df-po 4331  df-iso 4332  df-iord 4401  df-on 4403  df-ilim 4404  df-suc 4406  df-iom 4627  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-recs 6363  df-frec 6449  df-1o 6474  df-2o 6475  df-er 6592  df-en 6800  df-pnf 8063  df-mnf 8064  df-xr 8065  df-ltxr 8066  df-le 8067  df-sub 8199  df-neg 8200  df-reap 8602  df-ap 8609  df-div 8700  df-inn 8991  df-2 9049  df-3 9050  df-4 9051  df-n0 9250  df-z 9327  df-uz 9602  df-q 9694  df-rp 9729  df-seqfrec 10540  df-exp 10631  df-cj 11007  df-re 11008  df-im 11009  df-rsqrt 11163  df-abs 11164  df-dvds 11953  df-prm 12276

This theorem is referenced by:  nprm  12291  prmuz2  12299  dvdsprm  12305