Theorem coprm 11822

Description: A prime number either divides an integer or is coprime to it, but not both. Theorem 1.8 in [ApostolNT] p. 17. (Contributed by Paul Chapman, 22-Jun-2011.)

Assertion

Ref	Expression
coprm	|- ( ( P e. Prime /\ N e. ZZ ) -> ( -. P || N <-> ( P gcd N ) = 1 ) )

Proof of Theorem coprm

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		prmz 11792	. . . . . . 7 |- ( P e. Prime -> P e. ZZ )
2		gcddvds 11652	. . . . . . 7 |- ( ( P e. ZZ /\ N e. ZZ ) -> ( ( P gcd N ) || P /\ ( P gcd N ) || N ) )
3	1, 2	sylan 281	. . . . . 6 |- ( ( P e. Prime /\ N e. ZZ ) -> ( ( P gcd N ) || P /\ ( P gcd N ) || N ) )
4	3	simprd 113	. . . . 5 |- ( ( P e. Prime /\ N e. ZZ ) -> ( P gcd N ) || N )
5		breq1 3932	. . . . 5 |- ( ( P gcd N ) = P -> ( ( P gcd N ) || N <-> P || N ) )
6	4, 5	syl5ibcom 154	. . . 4 |- ( ( P e. Prime /\ N e. ZZ ) -> ( ( P gcd N ) = P -> P || N ) )
7	6	con3d 620	. . 3 |- ( ( P e. Prime /\ N e. ZZ ) -> ( -. P || N -> -. ( P gcd N ) = P ) )
8		0nnn 8747	. . . . . . . . 9 |- -. 0 e. NN
9		prmnn 11791	. . . . . . . . . 10 |- ( P e. Prime -> P e. NN )
10		eleq1 2202	. . . . . . . . . 10 |- ( P = 0 -> ( P e. NN <-> 0 e. NN ) )
11	9, 10	syl5ibcom 154	. . . . . . . . 9 |- ( P e. Prime -> ( P = 0 -> 0 e. NN ) )
12	8, 11	mtoi 653	. . . . . . . 8 |- ( P e. Prime -> -. P = 0 )
13	12	intnanrd 917	. . . . . . 7 |- ( P e. Prime -> -. ( P = 0 /\ N = 0 ) )
14	13	adantr 274	. . . . . 6 |- ( ( P e. Prime /\ N e. ZZ ) -> -. ( P = 0 /\ N = 0 ) )
15		gcdn0cl 11651	. . . . . . . 8 |- ( ( ( P e. ZZ /\ N e. ZZ ) /\ -. ( P = 0 /\ N = 0 ) ) -> ( P gcd N ) e. NN )
16	15	ex 114	. . . . . . 7 |- ( ( P e. ZZ /\ N e. ZZ ) -> ( -. ( P = 0 /\ N = 0 ) -> ( P gcd N ) e. NN ) )
17	1, 16	sylan 281	. . . . . 6 |- ( ( P e. Prime /\ N e. ZZ ) -> ( -. ( P = 0 /\ N = 0 ) -> ( P gcd N ) e. NN ) )
18	14, 17	mpd 13	. . . . 5 |- ( ( P e. Prime /\ N e. ZZ ) -> ( P gcd N ) e. NN )
19	3	simpld 111	. . . . 5 |- ( ( P e. Prime /\ N e. ZZ ) -> ( P gcd N ) || P )
20		isprm2 11798	. . . . . . . 8 |- ( P e. Prime <-> ( P e. ( ZZ>= ` 2 ) /\ A. z e. NN ( z || P -> ( z = 1 \/ z = P ) ) ) )
21	20	simprbi 273	. . . . . . 7 |- ( P e. Prime -> A. z e. NN ( z || P -> ( z = 1 \/ z = P ) ) )
22		breq1 3932	. . . . . . . . 9 |- ( z = ( P gcd N ) -> ( z || P <-> ( P gcd N ) || P ) )
23		eqeq1 2146	. . . . . . . . . 10 |- ( z = ( P gcd N ) -> ( z = 1 <-> ( P gcd N ) = 1 ) )
24		eqeq1 2146	. . . . . . . . . 10 |- ( z = ( P gcd N ) -> ( z = P <-> ( P gcd N ) = P ) )
25	23, 24	orbi12d 782	. . . . . . . . 9 |- ( z = ( P gcd N ) -> ( ( z = 1 \/ z = P ) <-> ( ( P gcd N ) = 1 \/ ( P gcd N ) = P ) ) )
26	22, 25	imbi12d 233	. . . . . . . 8 |- ( z = ( P gcd N ) -> ( ( z || P -> ( z = 1 \/ z = P ) ) <-> ( ( P gcd N ) || P -> ( ( P gcd N ) = 1 \/ ( P gcd N ) = P ) ) ) )
27	26	rspcv 2785	. . . . . . 7 |- ( ( P gcd N ) e. NN -> ( A. z e. NN ( z || P -> ( z = 1 \/ z = P ) ) -> ( ( P gcd N ) || P -> ( ( P gcd N ) = 1 \/ ( P gcd N ) = P ) ) ) )
28	21, 27	syl5com 29	. . . . . 6 |- ( P e. Prime -> ( ( P gcd N ) e. NN -> ( ( P gcd N ) || P -> ( ( P gcd N ) = 1 \/ ( P gcd N ) = P ) ) ) )
29	28	adantr 274	. . . . 5 |- ( ( P e. Prime /\ N e. ZZ ) -> ( ( P gcd N ) e. NN -> ( ( P gcd N ) || P -> ( ( P gcd N ) = 1 \/ ( P gcd N ) = P ) ) ) )
30	18, 19, 29	mp2d 47	. . . 4 |- ( ( P e. Prime /\ N e. ZZ ) -> ( ( P gcd N ) = 1 \/ ( P gcd N ) = P ) )
31		biorf 733	. . . . 5 |- ( -. ( P gcd N ) = P -> ( ( P gcd N ) = 1 <-> ( ( P gcd N ) = P \/ ( P gcd N ) = 1 ) ) )
32		orcom 717	. . . . 5 |- ( ( ( P gcd N ) = P \/ ( P gcd N ) = 1 ) <-> ( ( P gcd N ) = 1 \/ ( P gcd N ) = P ) )
33	31, 32	syl6bb 195	. . . 4 |- ( -. ( P gcd N ) = P -> ( ( P gcd N ) = 1 <-> ( ( P gcd N ) = 1 \/ ( P gcd N ) = P ) ) )
34	30, 33	syl5ibrcom 156	. . 3 |- ( ( P e. Prime /\ N e. ZZ ) -> ( -. ( P gcd N ) = P -> ( P gcd N ) = 1 ) )
35	7, 34	syld 45	. 2 |- ( ( P e. Prime /\ N e. ZZ ) -> ( -. P || N -> ( P gcd N ) = 1 ) )
36		iddvds 11506	. . . . . . 7 |- ( P e. ZZ -> P || P )
37	1, 36	syl 14	. . . . . 6 |- ( P e. Prime -> P || P )
38	37	adantr 274	. . . . 5 |- ( ( P e. Prime /\ N e. ZZ ) -> P || P )
39		dvdslegcd 11653	. . . . . . . . 9 |- ( ( ( P e. ZZ /\ P e. ZZ /\ N e. ZZ ) /\ -. ( P = 0 /\ N = 0 ) ) -> ( ( P || P /\ P || N ) -> P <_ ( P gcd N ) ) )
40	39	ex 114	. . . . . . . 8 |- ( ( P e. ZZ /\ P e. ZZ /\ N e. ZZ ) -> ( -. ( P = 0 /\ N = 0 ) -> ( ( P || P /\ P || N ) -> P <_ ( P gcd N ) ) ) )
41	40	3anidm12 1273	. . . . . . 7 |- ( ( P e. ZZ /\ N e. ZZ ) -> ( -. ( P = 0 /\ N = 0 ) -> ( ( P || P /\ P || N ) -> P <_ ( P gcd N ) ) ) )
42	1, 41	sylan 281	. . . . . 6 |- ( ( P e. Prime /\ N e. ZZ ) -> ( -. ( P = 0 /\ N = 0 ) -> ( ( P || P /\ P || N ) -> P <_ ( P gcd N ) ) ) )
43	14, 42	mpd 13	. . . . 5 |- ( ( P e. Prime /\ N e. ZZ ) -> ( ( P || P /\ P || N ) -> P <_ ( P gcd N ) ) )
44	38, 43	mpand 425	. . . 4 |- ( ( P e. Prime /\ N e. ZZ ) -> ( P || N -> P <_ ( P gcd N ) ) )
45		prmgt1 11812	. . . . . 6 |- ( P e. Prime -> 1 < P )
46	45	adantr 274	. . . . 5 |- ( ( P e. Prime /\ N e. ZZ ) -> 1 < P )
47	1	zred 9173	. . . . . . 7 |- ( P e. Prime -> P e. RR )
48	47	adantr 274	. . . . . 6 |- ( ( P e. Prime /\ N e. ZZ ) -> P e. RR )
49	18	nnred 8733	. . . . . 6 |- ( ( P e. Prime /\ N e. ZZ ) -> ( P gcd N ) e. RR )
50		1re 7765	. . . . . . 7 |- 1 e. RR
51		ltletr 7853	. . . . . . 7 |- ( ( 1 e. RR /\ P e. RR /\ ( P gcd N ) e. RR ) -> ( ( 1 < P /\ P <_ ( P gcd N ) ) -> 1 < ( P gcd N ) ) )
52	50, 51	mp3an1 1302	. . . . . 6 |- ( ( P e. RR /\ ( P gcd N ) e. RR ) -> ( ( 1 < P /\ P <_ ( P gcd N ) ) -> 1 < ( P gcd N ) ) )
53	48, 49, 52	syl2anc 408	. . . . 5 |- ( ( P e. Prime /\ N e. ZZ ) -> ( ( 1 < P /\ P <_ ( P gcd N ) ) -> 1 < ( P gcd N ) ) )
54	46, 53	mpand 425	. . . 4 |- ( ( P e. Prime /\ N e. ZZ ) -> ( P <_ ( P gcd N ) -> 1 < ( P gcd N ) ) )
55		ltne 7849	. . . . . 6 |- ( ( 1 e. RR /\ 1 < ( P gcd N ) ) -> ( P gcd N ) =/= 1 )
56	50, 55	mpan 420	. . . . 5 |- ( 1 < ( P gcd N ) -> ( P gcd N ) =/= 1 )
57	56	a1i 9	. . . 4 |- ( ( P e. Prime /\ N e. ZZ ) -> ( 1 < ( P gcd N ) -> ( P gcd N ) =/= 1 ) )
58	44, 54, 57	3syld 57	. . 3 |- ( ( P e. Prime /\ N e. ZZ ) -> ( P || N -> ( P gcd N ) =/= 1 ) )
59	58	necon2bd 2366	. 2 |- ( ( P e. Prime /\ N e. ZZ ) -> ( ( P gcd N ) = 1 -> -. P || N ) )
60	35, 59	impbid 128	1 |- ( ( P e. Prime /\ N e. ZZ ) -> ( -. P || N <-> ( P gcd N ) = 1 ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 697   /\ w3a 962   = wceq 1331   e. wcel 1480   =/= wne 2308   A.wral 2416   class class class wbr 3929   ` cfv 5123  (class class class)co 5774   RRcr 7619   0cc0 7620   1c1 7621   < clt 7800   <_ cle 7801   NNcn 8720   2c2 8771   ZZcz 9054   ZZ>=cuz 9326   || cdvds 11493   gcd cgcd 11635   Primecprime 11788

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-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502  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  ax-arch 7739  ax-caucvg 7740

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-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-ilim 4291  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-frec 6288  df-1o 6313  df-2o 6314  df-er 6429  df-en 6635  df-sup 6871  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-2 8779  df-3 8780  df-4 8781  df-n0 8978  df-z 9055  df-uz 9327  df-q 9412  df-rp 9442  df-fz 9791  df-fzo 9920  df-fl 10043  df-mod 10096  df-seqfrec 10219  df-exp 10293  df-cj 10614  df-re 10615  df-im 10616  df-rsqrt 10770  df-abs 10771  df-dvds 11494  df-gcd 11636  df-prm 11789

This theorem is referenced by:  prmrp  11823  euclemma  11824  cncongrprm  11835  isoddgcd1  11837  phiprmpw  11898