Theorem prime 9150

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 9073	. . . . . . 7 |- ( x e. NN -> x e. ZZ )
2		1z 9080	. . . . . . . 8 |- 1 e. ZZ
3		zdceq 9126	. . . . . . . 8 |- ( ( x e. ZZ /\ 1 e. ZZ ) -> DECID x = 1 )
4	2, 3	mpan2 421	. . . . . . 7 |- ( x e. ZZ -> DECID x = 1 )
5		dfordc 877	. . . . . . . 8 |- (DECID x = 1 -> ( ( x = 1 \/ x = A ) <-> ( -. x = 1 -> x = A ) ) )
6		df-ne 2309	. . . . . . . . 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 8755	. . . . . . 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 460	. . . . 5 |- ( ( A e. NN /\ x e. NN ) -> ( ( 1 < x /\ ( A / x ) e. NN ) <-> ( x =/= 1 /\ ( A / x ) e. NN ) ) )
19		nnz 9073	. . . . . . . . 9 |- ( ( A / x ) e. NN -> ( A / x ) e. ZZ )
20		nnre 8727	. . . . . . . . . . . . 13 |- ( x e. NN -> x e. RR )
21		gtndiv 9146	. . . . . . . . . . . . . 14 |- ( ( x e. RR /\ A e. NN /\ A < x ) -> -. ( A / x ) e. ZZ )
22	21	3expia 1183	. . . . . . . . . . . . 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 613	. . . . . . . . . . 11 |- ( ( x e. NN /\ A e. NN ) -> ( ( A / x ) e. ZZ -> -. A < x ) )
25		nnre 8727	. . . . . . . . . . . 12 |- ( A e. NN -> A e. RR )
26		lenlt 7840	. . . . . . . . . . . 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 391	. . . . . . 7 |- ( ( A e. NN /\ x e. NN ) -> ( ( A / x ) e. NN <-> ( x <_ A /\ ( A / x ) e. NN ) ) )
32	31	anbi2d 459	. . . . . 6 |- ( ( A e. NN /\ x e. NN ) -> ( ( 1 < x /\ ( A / x ) e. NN ) <-> ( 1 < x /\ ( x <_ A /\ ( A / x ) e. NN ) ) ) )
33		3anass 966	. . . . . 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 2433	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 697  DECID wdc 819   /\ w3a 962   = wceq 1331   e. wcel 1480   =/= wne 2308   A.wral 2416   class class class wbr 3929  (class class class)co 5774   RRcr 7619   1c1 7621   < clt 7800   <_ cle 7801   / cdiv 8432   NNcn 8720   ZZcz 9054

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-cnex 7711  ax-resscn 7712  ax-1cn 7713  ax-1re 7714  ax-icn 7715  ax-addcl 7716  ax-addrcl 7717  ax-mulcl 7718  ax-mulrcl 7719  ax-addcom 7720  ax-mulcom 7721  ax-addass 7722  ax-mulass 7723  ax-distr 7724  ax-i2m1 7725  ax-0lt1 7726  ax-1rid 7727  ax-0id 7728  ax-rnegex 7729  ax-precex 7730  ax-cnre 7731  ax-pre-ltirr 7732  ax-pre-ltwlin 7733  ax-pre-lttrn 7734  ax-pre-apti 7735  ax-pre-ltadd 7736  ax-pre-mulgt0 7737  ax-pre-mulext 7738

This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-br 3930  df-opab 3990  df-id 4215  df-po 4218  df-iso 4219  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-iota 5088  df-fun 5125  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-pnf 7802  df-mnf 7803  df-xr 7804  df-ltxr 7805  df-le 7806  df-sub 7935  df-neg 7936  df-reap 8337  df-ap 8344  df-div 8433  df-inn 8721  df-n0 8978  df-z 9055

This theorem is referenced by: (None)