Theorem prime 9246

Description: Two ways to express " A is a prime number (or 1)." (Contributed by NM, 4-May-2005.)

Assertion

Ref	Expression
prime	|- ( A e. NN -> ( A. x e. NN ( ( A / x ) e. NN -> ( x = 1 \/ x = A ) ) <-> A. x e. NN ( ( 1 < x /\ x <_ A /\ ( A / x ) e. NN ) -> x = A ) ) )

Distinct variable group:   x, A

Proof of Theorem prime

Step	Hyp	Ref	Expression
1		nnz 9169	. . . . . . 7 |- ( x e. NN -> x e. ZZ )
2		1z 9176	. . . . . . . 8 |- 1 e. ZZ
3		zdceq 9222	. . . . . . . 8 |- ( ( x e. ZZ /\ 1 e. ZZ ) -> DECID x = 1 )
4	2, 3	mpan2 422	. . . . . . 7 |- ( x e. ZZ -> DECID x = 1 )
5		dfordc 878	. . . . . . . 8 |- (DECID x = 1 -> ( ( x = 1 \/ x = A ) <-> ( -. x = 1 -> x = A ) ) )
6		df-ne 2328	. . . . . . . . 9 |- ( x =/= 1 <-> -. x = 1 )
7	6	imbi1i 237	. . . . . . . 8 |- ( ( x =/= 1 -> x = A ) <-> ( -. x = 1 -> x = A ) )
8	5, 7	bitr4di 197	. . . . . . 7 |- (DECID x = 1 -> ( ( x = 1 \/ x = A ) <-> ( x =/= 1 -> x = A ) ) )
9	1, 4, 8	3syl 17	. . . . . 6 |- ( x e. NN -> ( ( x = 1 \/ x = A ) <-> ( x =/= 1 -> x = A ) ) )
10	9	imbi2d 229	. . . . 5 |- ( x e. NN -> ( ( ( A / x ) e. NN -> ( x = 1 \/ x = A ) ) <-> ( ( A / x ) e. NN -> ( x =/= 1 -> x = A ) ) ) )
11		impexp 261	. . . . . 6 |- ( ( ( x =/= 1 /\ ( A / x ) e. NN ) -> x = A ) <-> ( x =/= 1 -> ( ( A / x ) e. NN -> x = A ) ) )
12		bi2.04 247	. . . . . 6 |- ( ( x =/= 1 -> ( ( A / x ) e. NN -> x = A ) ) <-> ( ( A / x ) e. NN -> ( x =/= 1 -> x = A ) ) )
13	11, 12	bitri 183	. . . . 5 |- ( ( ( x =/= 1 /\ ( A / x ) e. NN ) -> x = A ) <-> ( ( A / x ) e. NN -> ( x =/= 1 -> x = A ) ) )
14	10, 13	bitr4di 197	. . . 4 |- ( x e. NN -> ( ( ( A / x ) e. NN -> ( x = 1 \/ x = A ) ) <-> ( ( x =/= 1 /\ ( A / x ) e. NN ) -> x = A ) ) )
15	14	adantl 275	. . 3 |- ( ( A e. NN /\ x e. NN ) -> ( ( ( A / x ) e. NN -> ( x = 1 \/ x = A ) ) <-> ( ( x =/= 1 /\ ( A / x ) e. NN ) -> x = A ) ) )
16		nngt1ne1 8851	. . . . . . 7 |- ( x e. NN -> ( 1 < x <-> x =/= 1 ) )
17	16	adantl 275	. . . . . 6 |- ( ( A e. NN /\ x e. NN ) -> ( 1 < x <-> x =/= 1 ) )
18	17	anbi1d 461	. . . . 5 |- ( ( A e. NN /\ x e. NN ) -> ( ( 1 < x /\ ( A / x ) e. NN ) <-> ( x =/= 1 /\ ( A / x ) e. NN ) ) )
19		nnz 9169	. . . . . . . . 9 |- ( ( A / x ) e. NN -> ( A / x ) e. ZZ )
20		nnre 8823	. . . . . . . . . . . . 13 |- ( x e. NN -> x e. RR )
21		gtndiv 9242	. . . . . . . . . . . . . 14 |- ( ( x e. RR /\ A e. NN /\ A < x ) -> -. ( A / x ) e. ZZ )
22	21	3expia 1187	. . . . . . . . . . . . 13 |- ( ( x e. RR /\ A e. NN ) -> ( A < x -> -. ( A / x ) e. ZZ ) )
23	20, 22	sylan 281	. . . . . . . . . . . 12 |- ( ( x e. NN /\ A e. NN ) -> ( A < x -> -. ( A / x ) e. ZZ ) )
24	23	con2d 614	. . . . . . . . . . 11 |- ( ( x e. NN /\ A e. NN ) -> ( ( A / x ) e. ZZ -> -. A < x ) )
25		nnre 8823	. . . . . . . . . . . 12 |- ( A e. NN -> A e. RR )
26		lenlt 7936	. . . . . . . . . . . 12 |- ( ( x e. RR /\ A e. RR ) -> ( x <_ A <-> -. A < x ) )
27	20, 25, 26	syl2an 287	. . . . . . . . . . 11 |- ( ( x e. NN /\ A e. NN ) -> ( x <_ A <-> -. A < x ) )
28	24, 27	sylibrd 168	. . . . . . . . . 10 |- ( ( x e. NN /\ A e. NN ) -> ( ( A / x ) e. ZZ -> x <_ A ) )
29	28	ancoms 266	. . . . . . . . 9 |- ( ( A e. NN /\ x e. NN ) -> ( ( A / x ) e. ZZ -> x <_ A ) )
30	19, 29	syl5 32	. . . . . . . 8 |- ( ( A e. NN /\ x e. NN ) -> ( ( A / x ) e. NN -> x <_ A ) )
31	30	pm4.71rd 392	. . . . . . 7 |- ( ( A e. NN /\ x e. NN ) -> ( ( A / x ) e. NN <-> ( x <_ A /\ ( A / x ) e. NN ) ) )
32	31	anbi2d 460	. . . . . 6 |- ( ( A e. NN /\ x e. NN ) -> ( ( 1 < x /\ ( A / x ) e. NN ) <-> ( 1 < x /\ ( x <_ A /\ ( A / x ) e. NN ) ) ) )
33		3anass 967	. . . . . 6 |- ( ( 1 < x /\ x <_ A /\ ( A / x ) e. NN ) <-> ( 1 < x /\ ( x <_ A /\ ( A / x ) e. NN ) ) )
34	32, 33	bitr4di 197	. . . . 5 |- ( ( A e. NN /\ x e. NN ) -> ( ( 1 < x /\ ( A / x ) e. NN ) <-> ( 1 < x /\ x <_ A /\ ( A / x ) e. NN ) ) )
35	18, 34	bitr3d 189	. . . 4 |- ( ( A e. NN /\ x e. NN ) -> ( ( x =/= 1 /\ ( A / x ) e. NN ) <-> ( 1 < x /\ x <_ A /\ ( A / x ) e. NN ) ) )
36	35	imbi1d 230	. . 3 |- ( ( A e. NN /\ x e. NN ) -> ( ( ( x =/= 1 /\ ( A / x ) e. NN ) -> x = A ) <-> ( ( 1 < x /\ x <_ A /\ ( A / x ) e. NN ) -> x = A ) ) )
37	15, 36	bitrd 187	. 2 |- ( ( A e. NN /\ x e. NN ) -> ( ( ( A / x ) e. NN -> ( x = 1 \/ x = A ) ) <-> ( ( 1 < x /\ x <_ A /\ ( A / x ) e. NN ) -> x = A ) ) )
38	37	ralbidva 2453	1 |- ( A e. NN -> ( A. x e. NN ( ( A / x ) e. NN -> ( x = 1 \/ x = A ) ) <-> A. x e. NN ( ( 1 < x /\ x <_ A /\ ( A / x ) e. NN ) -> x = A ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 698  DECID wdc 820   /\ w3a 963   = wceq 1335   e. wcel 2128   =/= wne 2327   A.wral 2435   class class class wbr 3965  (class class class)co 5818   RRcr 7714   1c1 7716   < clt 7895   <_ cle 7896   / cdiv 8528   NNcn 8816   ZZcz 9150

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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4134  ax-pr 4168  ax-un 4392  ax-setind 4494  ax-cnex 7806  ax-resscn 7807  ax-1cn 7808  ax-1re 7809  ax-icn 7810  ax-addcl 7811  ax-addrcl 7812  ax-mulcl 7813  ax-mulrcl 7814  ax-addcom 7815  ax-mulcom 7816  ax-addass 7817  ax-mulass 7818  ax-distr 7819  ax-i2m1 7820  ax-0lt1 7821  ax-1rid 7822  ax-0id 7823  ax-rnegex 7824  ax-precex 7825  ax-cnre 7826  ax-pre-ltirr 7827  ax-pre-ltwlin 7828  ax-pre-lttrn 7829  ax-pre-apti 7830  ax-pre-ltadd 7831  ax-pre-mulgt0 7832  ax-pre-mulext 7833

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-reu 2442  df-rmo 2443  df-rab 2444  df-v 2714  df-sbc 2938  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-int 3808  df-br 3966  df-opab 4026  df-id 4252  df-po 4255  df-iso 4256  df-xp 4589  df-rel 4590  df-cnv 4591  df-co 4592  df-dm 4593  df-iota 5132  df-fun 5169  df-fv 5175  df-riota 5774  df-ov 5821  df-oprab 5822  df-mpo 5823  df-pnf 7897  df-mnf 7898  df-xr 7899  df-ltxr 7900  df-le 7901  df-sub 8031  df-neg 8032  df-reap 8433  df-ap 8440  df-div 8529  df-inn 8817  df-n0 9074  df-z 9151

This theorem is referenced by: (None)