Theorem coprm 12146

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 12113	. . . . . . 7 |- ( P e. Prime -> P e. ZZ )
2		gcddvds 11966	. . . . . . 7 |- ( ( P e. ZZ /\ N e. ZZ ) -> ( ( P gcd N ) || P /\ ( P gcd N ) || N ) )
3	1, 2	sylan 283	. . . . . 6 |- ( ( P e. Prime /\ N e. ZZ ) -> ( ( P gcd N ) || P /\ ( P gcd N ) || N ) )
4	3	simprd 114	. . . . 5 |- ( ( P e. Prime /\ N e. ZZ ) -> ( P gcd N ) || N )
5		breq1 4008	. . . . 5 |- ( ( P gcd N ) = P -> ( ( P gcd N ) || N <-> P || N ) )
6	4, 5	syl5ibcom 155	. . . 4 |- ( ( P e. Prime /\ N e. ZZ ) -> ( ( P gcd N ) = P -> P || N ) )
7	6	con3d 631	. . 3 |- ( ( P e. Prime /\ N e. ZZ ) -> ( -. P || N -> -. ( P gcd N ) = P ) )
8		0nnn 8948	. . . . . . . . 9 |- -. 0 e. NN
9		prmnn 12112	. . . . . . . . . 10 |- ( P e. Prime -> P e. NN )
10		eleq1 2240	. . . . . . . . . 10 |- ( P = 0 -> ( P e. NN <-> 0 e. NN ) )
11	9, 10	syl5ibcom 155	. . . . . . . . 9 |- ( P e. Prime -> ( P = 0 -> 0 e. NN ) )
12	8, 11	mtoi 664	. . . . . . . 8 |- ( P e. Prime -> -. P = 0 )
13	12	intnanrd 932	. . . . . . 7 |- ( P e. Prime -> -. ( P = 0 /\ N = 0 ) )
14	13	adantr 276	. . . . . 6 |- ( ( P e. Prime /\ N e. ZZ ) -> -. ( P = 0 /\ N = 0 ) )
15		gcdn0cl 11965	. . . . . . . 8 |- ( ( ( P e. ZZ /\ N e. ZZ ) /\ -. ( P = 0 /\ N = 0 ) ) -> ( P gcd N ) e. NN )
16	15	ex 115	. . . . . . 7 |- ( ( P e. ZZ /\ N e. ZZ ) -> ( -. ( P = 0 /\ N = 0 ) -> ( P gcd N ) e. NN ) )
17	1, 16	sylan 283	. . . . . 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 112	. . . . 5 |- ( ( P e. Prime /\ N e. ZZ ) -> ( P gcd N ) || P )
20		isprm2 12119	. . . . . . . 8 |- ( P e. Prime <-> ( P e. ( ZZ>= ` 2 ) /\ A. z e. NN ( z || P -> ( z = 1 \/ z = P ) ) ) )
21	20	simprbi 275	. . . . . . 7 |- ( P e. Prime -> A. z e. NN ( z || P -> ( z = 1 \/ z = P ) ) )
22		breq1 4008	. . . . . . . . 9 |- ( z = ( P gcd N ) -> ( z || P <-> ( P gcd N ) || P ) )
23		eqeq1 2184	. . . . . . . . . 10 |- ( z = ( P gcd N ) -> ( z = 1 <-> ( P gcd N ) = 1 ) )
24		eqeq1 2184	. . . . . . . . . 10 |- ( z = ( P gcd N ) -> ( z = P <-> ( P gcd N ) = P ) )
25	23, 24	orbi12d 793	. . . . . . . . 9 |- ( z = ( P gcd N ) -> ( ( z = 1 \/ z = P ) <-> ( ( P gcd N ) = 1 \/ ( P gcd N ) = P ) ) )
26	22, 25	imbi12d 234	. . . . . . . 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 2839	. . . . . . 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 276	. . . . 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 744	. . . . 5 |- ( -. ( P gcd N ) = P -> ( ( P gcd N ) = 1 <-> ( ( P gcd N ) = P \/ ( P gcd N ) = 1 ) ) )
32		orcom 728	. . . . 5 |- ( ( ( P gcd N ) = P \/ ( P gcd N ) = 1 ) <-> ( ( P gcd N ) = 1 \/ ( P gcd N ) = P ) )
33	31, 32	bitrdi 196	. . . 4 |- ( -. ( P gcd N ) = P -> ( ( P gcd N ) = 1 <-> ( ( P gcd N ) = 1 \/ ( P gcd N ) = P ) ) )
34	30, 33	syl5ibrcom 157	. . 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 11813	. . . . . . 7 |- ( P e. ZZ -> P || P )
37	1, 36	syl 14	. . . . . 6 |- ( P e. Prime -> P || P )
38	37	adantr 276	. . . . 5 |- ( ( P e. Prime /\ N e. ZZ ) -> P || P )
39		dvdslegcd 11967	. . . . . . . . 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 115	. . . . . . . 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 1295	. . . . . . 7 |- ( ( P e. ZZ /\ N e. ZZ ) -> ( -. ( P = 0 /\ N = 0 ) -> ( ( P || P /\ P || N ) -> P <_ ( P gcd N ) ) ) )
42	1, 41	sylan 283	. . . . . 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 429	. . . 4 |- ( ( P e. Prime /\ N e. ZZ ) -> ( P || N -> P <_ ( P gcd N ) ) )
45		prmgt1 12134	. . . . . 6 |- ( P e. Prime -> 1 < P )
46	45	adantr 276	. . . . 5 |- ( ( P e. Prime /\ N e. ZZ ) -> 1 < P )
47	1	zred 9377	. . . . . . 7 |- ( P e. Prime -> P e. RR )
48	47	adantr 276	. . . . . 6 |- ( ( P e. Prime /\ N e. ZZ ) -> P e. RR )
49	18	nnred 8934	. . . . . 6 |- ( ( P e. Prime /\ N e. ZZ ) -> ( P gcd N ) e. RR )
50		1re 7958	. . . . . . 7 |- 1 e. RR
51		ltletr 8049	. . . . . . 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 1324	. . . . . 6 |- ( ( P e. RR /\ ( P gcd N ) e. RR ) -> ( ( 1 < P /\ P <_ ( P gcd N ) ) -> 1 < ( P gcd N ) ) )
53	48, 49, 52	syl2anc 411	. . . . 5 |- ( ( P e. Prime /\ N e. ZZ ) -> ( ( 1 < P /\ P <_ ( P gcd N ) ) -> 1 < ( P gcd N ) ) )
54	46, 53	mpand 429	. . . 4 |- ( ( P e. Prime /\ N e. ZZ ) -> ( P <_ ( P gcd N ) -> 1 < ( P gcd N ) ) )
55		ltne 8044	. . . . . 6 |- ( ( 1 e. RR /\ 1 < ( P gcd N ) ) -> ( P gcd N ) =/= 1 )
56	50, 55	mpan 424	. . . . 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 2405	. 2 |- ( ( P e. Prime /\ N e. ZZ ) -> ( ( P gcd N ) = 1 -> -. P || N ) )
60	35, 59	impbid 129	1 |- ( ( P e. Prime /\ N e. ZZ ) -> ( -. P || N <-> ( P gcd N ) = 1 ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 708   /\ w3a 978   = wceq 1353   e. wcel 2148   =/= wne 2347   A.wral 2455   class class class wbr 4005   ` cfv 5218  (class class class)co 5877   RRcr 7812   0cc0 7813   1c1 7814   < clt 7994   <_ cle 7995   NNcn 8921   2c2 8972   ZZcz 9255   ZZ>=cuz 9530   || cdvds 11796   gcd cgcd 11945   Primecprime 12109

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-coll 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589  ax-cnex 7904  ax-resscn 7905  ax-1cn 7906  ax-1re 7907  ax-icn 7908  ax-addcl 7909  ax-addrcl 7910  ax-mulcl 7911  ax-mulrcl 7912  ax-addcom 7913  ax-mulcom 7914  ax-addass 7915  ax-mulass 7916  ax-distr 7917  ax-i2m1 7918  ax-0lt1 7919  ax-1rid 7920  ax-0id 7921  ax-rnegex 7922  ax-precex 7923  ax-cnre 7924  ax-pre-ltirr 7925  ax-pre-ltwlin 7926  ax-pre-lttrn 7927  ax-pre-apti 7928  ax-pre-ltadd 7929  ax-pre-mulgt0 7930  ax-pre-mulext 7931  ax-arch 7932  ax-caucvg 7933

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 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-if 3537  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-id 4295  df-po 4298  df-iso 4299  df-iord 4368  df-on 4370  df-ilim 4371  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-riota 5833  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-recs 6308  df-frec 6394  df-1o 6419  df-2o 6420  df-er 6537  df-en 6743  df-sup 6985  df-pnf 7996  df-mnf 7997  df-xr 7998  df-ltxr 7999  df-le 8000  df-sub 8132  df-neg 8133  df-reap 8534  df-ap 8541  df-div 8632  df-inn 8922  df-2 8980  df-3 8981  df-4 8982  df-n0 9179  df-z 9256  df-uz 9531  df-q 9622  df-rp 9656  df-fz 10011  df-fzo 10145  df-fl 10272  df-mod 10325  df-seqfrec 10448  df-exp 10522  df-cj 10853  df-re 10854  df-im 10855  df-rsqrt 11009  df-abs 11010  df-dvds 11797  df-gcd 11946  df-prm 12110

This theorem is referenced by:  prmrp  12147  euclemma  12148  cncongrprm  12159  isoddgcd1  12161  phiprmpw  12224  fermltl  12236  prmdiv  12237  prmdiveq  12238  vfermltl  12253  prmpwdvds  12355  lgslem1  14486  lgsprme0  14528