Theorem prime 9472

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 9391	. . . . . . 7 |- ( x e. NN -> x e. ZZ )
2		1z 9398	. . . . . . . 8 |- 1 e. ZZ
3		zdceq 9448	. . . . . . . 8 |- ( ( x e. ZZ /\ 1 e. ZZ ) -> DECID x = 1 )
4	2, 3	mpan2 425	. . . . . . 7 |- ( x e. ZZ -> DECID x = 1 )
5		dfordc 894	. . . . . . . 8 |- (DECID x = 1 -> ( ( x = 1 \/ x = A ) <-> ( -. x = 1 -> x = A ) ) )
6		df-ne 2377	. . . . . . . . 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 9071	. . . . . . 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 9391	. . . . . . . . 9 |- ( ( A / x ) e. NN -> ( A / x ) e. ZZ )
20		nnre 9043	. . . . . . . . . . . . 13 |- ( x e. NN -> x e. RR )
21		gtndiv 9468	. . . . . . . . . . . . . 14 |- ( ( x e. RR /\ A e. NN /\ A < x ) -> -. ( A / x ) e. ZZ )
22	21	3expia 1208	. . . . . . . . . . . . 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 625	. . . . . . . . . . 11 |- ( ( x e. NN /\ A e. NN ) -> ( ( A / x ) e. ZZ -> -. A < x ) )
25		nnre 9043	. . . . . . . . . . . 12 |- ( A e. NN -> A e. RR )
26		lenlt 8148	. . . . . . . . . . . 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 985	. . . . . 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 2502	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 710  DECID wdc 836   /\ w3a 981   = wceq 1373   e. wcel 2176   =/= wne 2376   A.wral 2484   class class class wbr 4044  (class class class)co 5944   RRcr 7924   1c1 7926   < clt 8107   <_ cle 8108   / cdiv 8745   NNcn 9036   ZZcz 9372

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-sep 4162  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585  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

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-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-br 4045  df-opab 4106  df-id 4340  df-po 4343  df-iso 4344  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-iota 5232  df-fun 5273  df-fv 5279  df-riota 5899  df-ov 5947  df-oprab 5948  df-mpo 5949  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-n0 9296  df-z 9373

This theorem is referenced by:  infpnlem1  12682