Theorem prime 9174

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 9097	. . . . . . 7 |- ( x e. NN -> x e. ZZ )
2		1z 9104	. . . . . . . 8 |- 1 e. ZZ
3		zdceq 9150	. . . . . . . 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 2310	. . . . . . . . 9 |- ( x =/= 1 <-> -. x = 1 )
7	6	imbi1i 237	. . . . . . . 8 |- ( ( x =/= 1 -> x = A ) <-> ( -. x = 1 -> x = A ) )
8	5, 7	syl6bbr 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	syl6bbr 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 8779	. . . . . . 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 9097	. . . . . . . . 9 |- ( ( A / x ) e. NN -> ( A / x ) e. ZZ )
20		nnre 8751	. . . . . . . . . . . . 13 |- ( x e. NN -> x e. RR )
21		gtndiv 9170	. . . . . . . . . . . . . 14 |- ( ( x e. RR /\ A e. NN /\ A < x ) -> -. ( A / x ) e. ZZ )
22	21	3expia 1184	. . . . . . . . . . . . 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 8751	. . . . . . . . . . . 12 |- ( A e. NN -> A e. RR )
26		lenlt 7864	. . . . . . . . . . . 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	syl6bbr 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 2434	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 1332   e. wcel 1481   =/= wne 2309   A.wral 2417   class class class wbr 3937  (class class class)co 5782   RRcr 7643   1c1 7645   < clt 7824   <_ cle 7825   / cdiv 8456   NNcn 8744   ZZcz 9078

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460  ax-cnex 7735  ax-resscn 7736  ax-1cn 7737  ax-1re 7738  ax-icn 7739  ax-addcl 7740  ax-addrcl 7741  ax-mulcl 7742  ax-mulrcl 7743  ax-addcom 7744  ax-mulcom 7745  ax-addass 7746  ax-mulass 7747  ax-distr 7748  ax-i2m1 7749  ax-0lt1 7750  ax-1rid 7751  ax-0id 7752  ax-rnegex 7753  ax-precex 7754  ax-cnre 7755  ax-pre-ltirr 7756  ax-pre-ltwlin 7757  ax-pre-lttrn 7758  ax-pre-apti 7759  ax-pre-ltadd 7760  ax-pre-mulgt0 7761  ax-pre-mulext 7762

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-reu 2424  df-rmo 2425  df-rab 2426  df-v 2691  df-sbc 2914  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-int 3780  df-br 3938  df-opab 3998  df-id 4223  df-po 4226  df-iso 4227  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-iota 5096  df-fun 5133  df-fv 5139  df-riota 5738  df-ov 5785  df-oprab 5786  df-mpo 5787  df-pnf 7826  df-mnf 7827  df-xr 7828  df-ltxr 7829  df-le 7830  df-sub 7959  df-neg 7960  df-reap 8361  df-ap 8368  df-div 8457  df-inn 8745  df-n0 9002  df-z 9079

This theorem is referenced by: (None)