Theorem isprm4 12412

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 12410	. 2 |- ( P e. Prime <-> ( P e. ( ZZ>= ` 2 ) /\ A. z e. NN ( z || P -> ( z = 1 \/ z = P ) ) ) )
2		eluz2nn 9686	. . . . . . . 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 9724	. . . . . . . 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 2376	. . . . . . . . . . . . 13 |- ( z =/= 1 <-> -. z = 1 )
12	11	imbi1i 238	. . . . . . . . . . . 12 |- ( ( z =/= 1 -> z = P ) <-> ( -. z = 1 -> z = P ) )
13		nnz 9390	. . . . . . . . . . . . . 14 |- ( z e. NN -> z e. ZZ )
14		1zzd 9398	. . . . . . . . . . . . . 14 |- ( z e. NN -> 1 e. ZZ )
15		zdceq 9447	. . . . . . . . . . . . . 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 2515	. . 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 1372   e. wcel 2175   =/= wne 2375   A.wral 2483   class class class wbr 4043   ` cfv 5270   1c1 7925   NNcn 9035   2c2 9086   ZZcz 9371   ZZ>=cuz 9647   || cdvds 12069   Primecprime 12400

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-coll 4158  ax-sep 4161  ax-nul 4169  ax-pow 4217  ax-pr 4252  ax-un 4479  ax-setind 4584  ax-iinf 4635  ax-cnex 8015  ax-resscn 8016  ax-1cn 8017  ax-1re 8018  ax-icn 8019  ax-addcl 8020  ax-addrcl 8021  ax-mulcl 8022  ax-mulrcl 8023  ax-addcom 8024  ax-mulcom 8025  ax-addass 8026  ax-mulass 8027  ax-distr 8028  ax-i2m1 8029  ax-0lt1 8030  ax-1rid 8031  ax-0id 8032  ax-rnegex 8033  ax-precex 8034  ax-cnre 8035  ax-pre-ltirr 8036  ax-pre-ltwlin 8037  ax-pre-lttrn 8038  ax-pre-apti 8039  ax-pre-ltadd 8040  ax-pre-mulgt0 8041  ax-pre-mulext 8042  ax-arch 8043  ax-caucvg 8044

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-nel 2471  df-ral 2488  df-rex 2489  df-reu 2490  df-rmo 2491  df-rab 2492  df-v 2773  df-sbc 2998  df-csb 3093  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-nul 3460  df-if 3571  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-int 3885  df-iun 3928  df-br 4044  df-opab 4105  df-mpt 4106  df-tr 4142  df-id 4339  df-po 4342  df-iso 4343  df-iord 4412  df-on 4414  df-ilim 4415  df-suc 4417  df-iom 4638  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-res 4686  df-ima 4687  df-iota 5231  df-fun 5272  df-fn 5273  df-f 5274  df-f1 5275  df-fo 5276  df-f1o 5277  df-fv 5278  df-riota 5898  df-ov 5946  df-oprab 5947  df-mpo 5948  df-1st 6225  df-2nd 6226  df-recs 6390  df-frec 6476  df-1o 6501  df-2o 6502  df-er 6619  df-en 6827  df-pnf 8108  df-mnf 8109  df-xr 8110  df-ltxr 8111  df-le 8112  df-sub 8244  df-neg 8245  df-reap 8647  df-ap 8654  df-div 8745  df-inn 9036  df-2 9094  df-3 9095  df-4 9096  df-n0 9295  df-z 9372  df-uz 9648  df-q 9740  df-rp 9775  df-seqfrec 10591  df-exp 10682  df-cj 11124  df-re 11125  df-im 11126  df-rsqrt 11280  df-abs 11281  df-dvds 12070  df-prm 12401

This theorem is referenced by:  nprm  12416  prmuz2  12424  dvdsprm  12430