Theorem coprm 12158

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 12125	. . . . . . 7 |- ( P e. Prime -> P e. ZZ )
2		gcddvds 11978	. . . . . . 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 4018	. . . . 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 632	. . 3 |- ( ( P e. Prime /\ N e. ZZ ) -> ( -. P || N -> -. ( P gcd N ) = P ) )
8		0nnn 8960	. . . . . . . . 9 |- -. 0 e. NN
9		prmnn 12124	. . . . . . . . . 10 |- ( P e. Prime -> P e. NN )
10		eleq1 2250	. . . . . . . . . 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 665	. . . . . . . 8 |- ( P e. Prime -> -. P = 0 )
13	12	intnanrd 933	. . . . . . 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 11977	. . . . . . . 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 12131	. . . . . . . 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 4018	. . . . . . . . 9 |- ( z = ( P gcd N ) -> ( z || P <-> ( P gcd N ) || P ) )
23		eqeq1 2194	. . . . . . . . . 10 |- ( z = ( P gcd N ) -> ( z = 1 <-> ( P gcd N ) = 1 ) )
24		eqeq1 2194	. . . . . . . . . 10 |- ( z = ( P gcd N ) -> ( z = P <-> ( P gcd N ) = P ) )
25	23, 24	orbi12d 794	. . . . . . . . 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 2849	. . . . . . 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 745	. . . . 5 |- ( -. ( P gcd N ) = P -> ( ( P gcd N ) = 1 <-> ( ( P gcd N ) = P \/ ( P gcd N ) = 1 ) ) )
32		orcom 729	. . . . 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 11825	. . . . . . 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 11979	. . . . . . . . 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 1305	. . . . . . 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 12146	. . . . . 6 |- ( P e. Prime -> 1 < P )
46	45	adantr 276	. . . . 5 |- ( ( P e. Prime /\ N e. ZZ ) -> 1 < P )
47	1	zred 9389	. . . . . . 7 |- ( P e. Prime -> P e. RR )
48	47	adantr 276	. . . . . 6 |- ( ( P e. Prime /\ N e. ZZ ) -> P e. RR )
49	18	nnred 8946	. . . . . 6 |- ( ( P e. Prime /\ N e. ZZ ) -> ( P gcd N ) e. RR )
50		1re 7970	. . . . . . 7 |- 1 e. RR
51		ltletr 8061	. . . . . . 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 1334	. . . . . 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 8056	. . . . . 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 2415	. 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 709   /\ w3a 979   = wceq 1363   e. wcel 2158   =/= wne 2357   A.wral 2465   class class class wbr 4015   ` cfv 5228  (class class class)co 5888   RRcr 7824   0cc0 7825   1c1 7826   < clt 8006   <_ cle 8007   NNcn 8933   2c2 8984   ZZcz 9267   ZZ>=cuz 9542   || cdvds 11808   gcd cgcd 11957   Primecprime 12121

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 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-coll 4130  ax-sep 4133  ax-nul 4141  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-iinf 4599  ax-cnex 7916  ax-resscn 7917  ax-1cn 7918  ax-1re 7919  ax-icn 7920  ax-addcl 7921  ax-addrcl 7922  ax-mulcl 7923  ax-mulrcl 7924  ax-addcom 7925  ax-mulcom 7926  ax-addass 7927  ax-mulass 7928  ax-distr 7929  ax-i2m1 7930  ax-0lt1 7931  ax-1rid 7932  ax-0id 7933  ax-rnegex 7934  ax-precex 7935  ax-cnre 7936  ax-pre-ltirr 7937  ax-pre-ltwlin 7938  ax-pre-lttrn 7939  ax-pre-apti 7940  ax-pre-ltadd 7941  ax-pre-mulgt0 7942  ax-pre-mulext 7943  ax-arch 7944  ax-caucvg 7945

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 980  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-nel 2453  df-ral 2470  df-rex 2471  df-reu 2472  df-rmo 2473  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-if 3547  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-tr 4114  df-id 4305  df-po 4308  df-iso 4309  df-iord 4378  df-on 4380  df-ilim 4381  df-suc 4383  df-iom 4602  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-riota 5844  df-ov 5891  df-oprab 5892  df-mpo 5893  df-1st 6155  df-2nd 6156  df-recs 6320  df-frec 6406  df-1o 6431  df-2o 6432  df-er 6549  df-en 6755  df-sup 6997  df-pnf 8008  df-mnf 8009  df-xr 8010  df-ltxr 8011  df-le 8012  df-sub 8144  df-neg 8145  df-reap 8546  df-ap 8553  df-div 8644  df-inn 8934  df-2 8992  df-3 8993  df-4 8994  df-n0 9191  df-z 9268  df-uz 9543  df-q 9634  df-rp 9668  df-fz 10023  df-fzo 10157  df-fl 10284  df-mod 10337  df-seqfrec 10460  df-exp 10534  df-cj 10865  df-re 10866  df-im 10867  df-rsqrt 11021  df-abs 11022  df-dvds 11809  df-gcd 11958  df-prm 12122

This theorem is referenced by:  prmrp  12159  euclemma  12160  cncongrprm  12171  isoddgcd1  12173  phiprmpw  12236  fermltl  12248  prmdiv  12249  prmdiveq  12250  vfermltl  12265  prmpwdvds  12367  lgslem1  14697  lgsprme0  14739