Theorem prime 9352

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 9272	. . . . . . 7 |- ( x e. NN -> x e. ZZ )
2		1z 9279	. . . . . . . 8 |- 1 e. ZZ
3		zdceq 9328	. . . . . . . 8 |- ( ( x e. ZZ /\ 1 e. ZZ ) -> DECID x = 1 )
4	2, 3	mpan2 425	. . . . . . 7 |- ( x e. ZZ -> DECID x = 1 )
5		dfordc 892	. . . . . . . 8 |- (DECID x = 1 -> ( ( x = 1 \/ x = A ) <-> ( -. x = 1 -> x = A ) ) )
6		df-ne 2348	. . . . . . . . 9 |- ( x =/= 1 <-> -. x = 1 )
7	6	imbi1i 238	. . . . . . . 8 |- ( ( x =/= 1 -> x = A ) <-> ( -. x = 1 -> x = A ) )
8	5, 7	bitr4di 198	. . . . . . 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 230	. . . . 5 |- ( x e. NN -> ( ( ( A / x ) e. NN -> ( x = 1 \/ x = A ) ) <-> ( ( A / x ) e. NN -> ( x =/= 1 -> x = A ) ) ) )
11		impexp 263	. . . . . 6 |- ( ( ( x =/= 1 /\ ( A / x ) e. NN ) -> x = A ) <-> ( x =/= 1 -> ( ( A / x ) e. NN -> x = A ) ) )
12		bi2.04 248	. . . . . 6 |- ( ( x =/= 1 -> ( ( A / x ) e. NN -> x = A ) ) <-> ( ( A / x ) e. NN -> ( x =/= 1 -> x = A ) ) )
13	11, 12	bitri 184	. . . . 5 |- ( ( ( x =/= 1 /\ ( A / x ) e. NN ) -> x = A ) <-> ( ( A / x ) e. NN -> ( x =/= 1 -> x = A ) ) )
14	10, 13	bitr4di 198	. . . 4 |- ( x e. NN -> ( ( ( A / x ) e. NN -> ( x = 1 \/ x = A ) ) <-> ( ( x =/= 1 /\ ( A / x ) e. NN ) -> x = A ) ) )
15	14	adantl 277	. . 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 8954	. . . . . . 7 |- ( x e. NN -> ( 1 < x <-> x =/= 1 ) )
17	16	adantl 277	. . . . . 6 |- ( ( A e. NN /\ x e. NN ) -> ( 1 < x <-> x =/= 1 ) )
18	17	anbi1d 465	. . . . 5 |- ( ( A e. NN /\ x e. NN ) -> ( ( 1 < x /\ ( A / x ) e. NN ) <-> ( x =/= 1 /\ ( A / x ) e. NN ) ) )
19		nnz 9272	. . . . . . . . 9 |- ( ( A / x ) e. NN -> ( A / x ) e. ZZ )
20		nnre 8926	. . . . . . . . . . . . 13 |- ( x e. NN -> x e. RR )
21		gtndiv 9348	. . . . . . . . . . . . . 14 |- ( ( x e. RR /\ A e. NN /\ A < x ) -> -. ( A / x ) e. ZZ )
22	21	3expia 1205	. . . . . . . . . . . . 13 |- ( ( x e. RR /\ A e. NN ) -> ( A < x -> -. ( A / x ) e. ZZ ) )
23	20, 22	sylan 283	. . . . . . . . . . . 12 |- ( ( x e. NN /\ A e. NN ) -> ( A < x -> -. ( A / x ) e. ZZ ) )
24	23	con2d 624	. . . . . . . . . . 11 |- ( ( x e. NN /\ A e. NN ) -> ( ( A / x ) e. ZZ -> -. A < x ) )
25		nnre 8926	. . . . . . . . . . . 12 |- ( A e. NN -> A e. RR )
26		lenlt 8033	. . . . . . . . . . . 12 |- ( ( x e. RR /\ A e. RR ) -> ( x <_ A <-> -. A < x ) )
27	20, 25, 26	syl2an 289	. . . . . . . . . . 11 |- ( ( x e. NN /\ A e. NN ) -> ( x <_ A <-> -. A < x ) )
28	24, 27	sylibrd 169	. . . . . . . . . 10 |- ( ( x e. NN /\ A e. NN ) -> ( ( A / x ) e. ZZ -> x <_ A ) )
29	28	ancoms 268	. . . . . . . . 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 394	. . . . . . 7 |- ( ( A e. NN /\ x e. NN ) -> ( ( A / x ) e. NN <-> ( x <_ A /\ ( A / x ) e. NN ) ) )
32	31	anbi2d 464	. . . . . 6 |- ( ( A e. NN /\ x e. NN ) -> ( ( 1 < x /\ ( A / x ) e. NN ) <-> ( 1 < x /\ ( x <_ A /\ ( A / x ) e. NN ) ) ) )
33		3anass 982	. . . . . 6 |- ( ( 1 < x /\ x <_ A /\ ( A / x ) e. NN ) <-> ( 1 < x /\ ( x <_ A /\ ( A / x ) e. NN ) ) )
34	32, 33	bitr4di 198	. . . . 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 190	. . . 4 |- ( ( A e. NN /\ x e. NN ) -> ( ( x =/= 1 /\ ( A / x ) e. NN ) <-> ( 1 < x /\ x <_ A /\ ( A / x ) e. NN ) ) )
36	35	imbi1d 231	. . 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 188	. 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 2473	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 104   <-> wb 105   \/ wo 708  DECID wdc 834   /\ w3a 978   = wceq 1353   e. wcel 2148   =/= wne 2347   A.wral 2455   class class class wbr 4004  (class class class)co 5875   RRcr 7810   1c1 7812   < clt 7992   <_ cle 7993   / cdiv 8629   NNcn 8919   ZZcz 9253

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-cnex 7902  ax-resscn 7903  ax-1cn 7904  ax-1re 7905  ax-icn 7906  ax-addcl 7907  ax-addrcl 7908  ax-mulcl 7909  ax-mulrcl 7910  ax-addcom 7911  ax-mulcom 7912  ax-addass 7913  ax-mulass 7914  ax-distr 7915  ax-i2m1 7916  ax-0lt1 7917  ax-1rid 7918  ax-0id 7919  ax-rnegex 7920  ax-precex 7921  ax-cnre 7922  ax-pre-ltirr 7923  ax-pre-ltwlin 7924  ax-pre-lttrn 7925  ax-pre-apti 7926  ax-pre-ltadd 7927  ax-pre-mulgt0 7928  ax-pre-mulext 7929

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2740  df-sbc 2964  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-br 4005  df-opab 4066  df-id 4294  df-po 4297  df-iso 4298  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-iota 5179  df-fun 5219  df-fv 5225  df-riota 5831  df-ov 5878  df-oprab 5879  df-mpo 5880  df-pnf 7994  df-mnf 7995  df-xr 7996  df-ltxr 7997  df-le 7998  df-sub 8130  df-neg 8131  df-reap 8532  df-ap 8539  df-div 8630  df-inn 8920  df-n0 9177  df-z 9254

This theorem is referenced by:  infpnlem1  12357