Theorem isprm4 12441

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 12439	. 2 |- ( P e. Prime <-> ( P e. ( ZZ>= ` 2 ) /\ A. z e. NN ( z || P -> ( z = 1 \/ z = P ) ) ) )
2		eluz2nn 9687	. . . . . . . 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 9725	. . . . . . . 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 2377	. . . . . . . . . . . . 13 |- ( z =/= 1 <-> -. z = 1 )
12	11	imbi1i 238	. . . . . . . . . . . 12 |- ( ( z =/= 1 -> z = P ) <-> ( -. z = 1 -> z = P ) )
13		nnz 9391	. . . . . . . . . . . . . 14 |- ( z e. NN -> z e. ZZ )
14		1zzd 9399	. . . . . . . . . . . . . 14 |- ( z e. NN -> 1 e. ZZ )
15		zdceq 9448	. . . . . . . . . . . . . 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 894	. . . . . . . . . . . . 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 2516	. . 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 710  DECID wdc 836   = wceq 1373   e. wcel 2176   =/= wne 2376   A.wral 2484   class class class wbr 4044   ` cfv 5271   1c1 7926   NNcn 9036   2c2 9087   ZZcz 9372   ZZ>=cuz 9648   || cdvds 12098   Primecprime 12429

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-coll 4159  ax-sep 4162  ax-nul 4170  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-iinf 4636  ax-cnex 8016  ax-resscn 8017  ax-1cn 8018  ax-1re 8019  ax-icn 8020  ax-addcl 8021  ax-addrcl 8022  ax-mulcl 8023  ax-mulrcl 8024  ax-addcom 8025  ax-mulcom 8026  ax-addass 8027  ax-mulass 8028  ax-distr 8029  ax-i2m1 8030  ax-0lt1 8031  ax-1rid 8032  ax-0id 8033  ax-rnegex 8034  ax-precex 8035  ax-cnre 8036  ax-pre-ltirr 8037  ax-pre-ltwlin 8038  ax-pre-lttrn 8039  ax-pre-apti 8040  ax-pre-ltadd 8041  ax-pre-mulgt0 8042  ax-pre-mulext 8043  ax-arch 8044  ax-caucvg 8045

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-reu 2491  df-rmo 2492  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-if 3572  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  df-br 4045  df-opab 4106  df-mpt 4107  df-tr 4143  df-id 4340  df-po 4343  df-iso 4344  df-iord 4413  df-on 4415  df-ilim 4416  df-suc 4418  df-iom 4639  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-riota 5899  df-ov 5947  df-oprab 5948  df-mpo 5949  df-1st 6226  df-2nd 6227  df-recs 6391  df-frec 6477  df-1o 6502  df-2o 6503  df-er 6620  df-en 6828  df-pnf 8109  df-mnf 8110  df-xr 8111  df-ltxr 8112  df-le 8113  df-sub 8245  df-neg 8246  df-reap 8648  df-ap 8655  df-div 8746  df-inn 9037  df-2 9095  df-3 9096  df-4 9097  df-n0 9296  df-z 9373  df-uz 9649  df-q 9741  df-rp 9776  df-seqfrec 10593  df-exp 10684  df-cj 11153  df-re 11154  df-im 11155  df-rsqrt 11309  df-abs 11310  df-dvds 12099  df-prm 12430

This theorem is referenced by:  nprm  12445  prmuz2  12453  dvdsprm  12459