Theorem coprm 13782

Description: A prime number either divides an integer or is coprime to it, but not both. (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

Step	Hyp	Ref	Expression
1		breq1 3341	. . . . 5 |- ((P gcd N) = P -> ((P gcd N)||N <-> P||N))
2		gcddvds 13722	. . . . . . 7 |- ((P e. ZZ /\ N e. ZZ) -> ((P gcd N)||P /\ (P gcd N)||N))
3		isprm2 13775	. . . . . . . 8 |- (P e. Prime <-> (P e. ZZ /\ 1 < P /\ A.z e. NN (z||P -> (z = 1 \/ z = P))))
4	3	simp1bi 891	. . . . . . 7 |- (P e. Prime -> P e. ZZ)
5	2, 4	sylan 497	. . . . . 6 |- ((P e. Prime /\ N e. ZZ) -> ((P gcd N)||P /\ (P gcd N)||N))
6	5	simprd 352	. . . . 5 |- ((P e. Prime /\ N e. ZZ) -> (P gcd N)||N)
7	1, 6	syl5cbi 226	. . . 4 |- ((P e. Prime /\ N e. ZZ) -> ((P gcd N) = P -> P||N))
8	7	con3d 111	. . 3 |- ((P e. Prime /\ N e. ZZ) -> (-. P||N -> -. (P gcd N) = P))
9		biorf 807	. . . . 5 |- (-. (P gcd N) = P -> ((P gcd N) = 1 <-> ((P gcd N) = P \/ (P gcd N) = 1)))
10		orcom 266	. . . . 5 |- (((P gcd N) = P \/ (P gcd N) = 1) <-> ((P gcd N) = 1 \/ (P gcd N) = P))
11	9, 10	syl6bb 595	. . . 4 |- (-. (P gcd N) = P -> ((P gcd N) = 1 <-> ((P gcd N) = 1 \/ (P gcd N) = P)))
12	5	simplld 348	. . . . 5 |- ((P e. Prime /\ N e. ZZ) -> (P gcd N)||P)
13		0nnn 7131	. . . . . . . . . 10 |- -. 0 e. NN
14		prmnn 13778	. . . . . . . . . . 11 |- (P e. Prime -> P e. NN)
15		eleq1 1957	. . . . . . . . . . . 12 |- (P = 0 -> (P e. NN <-> 0 e. NN))
16	15	biimpcd 172	. . . . . . . . . . 11 |- (P e. NN -> (P = 0 -> 0 e. NN))
17	14, 16	syl 12	. . . . . . . . . 10 |- (P e. Prime -> (P = 0 -> 0 e. NN))
18	13, 17	mtoi 122	. . . . . . . . 9 |- (P e. Prime -> -. P = 0)
19	18	intnanrd 758	. . . . . . . 8 |- (P e. Prime -> -. (P = 0 /\ N = 0))
20	19	adantr 425	. . . . . . 7 |- ((P e. Prime /\ N e. ZZ) -> -. (P = 0 /\ N = 0))
21		gcdn0cl 13721	. . . . . . . . 9 |- (((P e. ZZ /\ N e. ZZ) /\ -. (P = 0 /\ N = 0)) -> (P gcd N) e. NN)
22	21	ex 402	. . . . . . . 8 |- ((P e. ZZ /\ N e. ZZ) -> (-. (P = 0 /\ N = 0) -> (P gcd N) e. NN))
23	22, 4	sylan 497	. . . . . . 7 |- ((P e. Prime /\ N e. ZZ) -> (-. (P = 0 /\ N = 0) -> (P gcd N) e. NN))
24	20, 23	mpd 29	. . . . . 6 |- ((P e. Prime /\ N e. ZZ) -> (P gcd N) e. NN)
25		breq1 3341	. . . . . . . . . . 11 |- (z = (P gcd N) -> (z||P <-> (P gcd N)||P))
26		eqeq1 1890	. . . . . . . . . . . 12 |- (z = (P gcd N) -> (z = 1 <-> (P gcd N) = 1))
27		eqeq1 1890	. . . . . . . . . . . 12 |- (z = (P gcd N) -> (z = P <-> (P gcd N) = P))
28	26, 27	orbi12d 689	. . . . . . . . . . 11 |- (z = (P gcd N) -> ((z = 1 \/ z = P) <-> ((P gcd N) = 1 \/ (P gcd N) = P)))
29	25, 28	imbi12d 688	. . . . . . . . . 10 |- (z = (P gcd N) -> ((z||P -> (z = 1 \/ z = P)) <-> ((P gcd N)||P -> ((P gcd N) = 1 \/ (P gcd N) = P))))
30	29	rcla4v 2376	. . . . . . . . 9 |- ((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))))
31	3	simp3bi 893	. . . . . . . . 9 |- (P e. Prime -> A.z e. NN (z||P -> (z = 1 \/ z = P)))
32	30, 31	syl5 20	. . . . . . . 8 |- ((P gcd N) e. NN -> (P e. Prime -> ((P gcd N)||P -> ((P gcd N) = 1 \/ (P gcd N) = P))))
33	32	com12 14	. . . . . . 7 |- (P e. Prime -> ((P gcd N) e. NN -> ((P gcd N)||P -> ((P gcd N) = 1 \/ (P gcd N) = P))))
34	33	adantr 425	. . . . . 6 |- ((P e. Prime /\ N e. ZZ) -> ((P gcd N) e. NN -> ((P gcd N)||P -> ((P gcd N) = 1 \/ (P gcd N) = P))))
35	24, 34	mpd 29	. . . . 5 |- ((P e. Prime /\ N e. ZZ) -> ((P gcd N)||P -> ((P gcd N) = 1 \/ (P gcd N) = P)))
36	12, 35	mpd 29	. . . 4 |- ((P e. Prime /\ N e. ZZ) -> ((P gcd N) = 1 \/ (P gcd N) = P))
37	11, 36	syl5cbir 228	. . 3 |- ((P e. Prime /\ N e. ZZ) -> (-. (P gcd N) = P -> (P gcd N) = 1))
38	8, 37	syld 30	. 2 |- ((P e. Prime /\ N e. ZZ) -> (-. P||N -> (P gcd N) = 1))
39		iddvds 13668	. . . . . . . 8 |- (P e. ZZ -> P||P)
40	4, 39	syl 12	. . . . . . 7 |- (P e. Prime -> P||P)
41	40	adantr 425	. . . . . 6 |- ((P e. Prime /\ N e. ZZ) -> P||P)
42		dvdslegcd 13723	. . . . . . . . . 10 |- (((P e. ZZ /\ P e. ZZ /\ N e. ZZ) /\ -. (P = 0 /\ N = 0)) -> ((P||P /\ P||N) -> P <_ (P gcd N)))
43	42	ex 402	. . . . . . . . 9 |- ((P e. ZZ /\ P e. ZZ /\ N e. ZZ) -> (-. (P = 0 /\ N = 0) -> ((P||P /\ P||N) -> P <_ (P gcd N))))
44	43	3anidm12 1154	. . . . . . . 8 |- ((P e. ZZ /\ N e. ZZ) -> (-. (P = 0 /\ N = 0) -> ((P||P /\ P||N) -> P <_ (P gcd N))))
45	44, 4	sylan 497	. . . . . . 7 |- ((P e. Prime /\ N e. ZZ) -> (-. (P = 0 /\ N = 0) -> ((P||P /\ P||N) -> P <_ (P gcd N))))
46	20, 45	mpd 29	. . . . . 6 |- ((P e. Prime /\ N e. ZZ) -> ((P||P /\ P||N) -> P <_ (P gcd N)))
47	41, 46	mpand 765	. . . . 5 |- ((P e. Prime /\ N e. ZZ) -> (P||N -> P <_ (P gcd N)))
48	3	simp2bi 892	. . . . . . 7 |- (P e. Prime -> 1 < P)
49	48	adantr 425	. . . . . 6 |- ((P e. Prime /\ N e. ZZ) -> 1 < P)
50		zre 7348	. . . . . . . . 9 |- (P e. ZZ -> P e. RR)
51	4, 50	syl 12	. . . . . . . 8 |- (P e. Prime -> P e. RR)
52	51	adantr 425	. . . . . . 7 |- ((P e. Prime /\ N e. ZZ) -> P e. RR)
53		nnre 7112	. . . . . . . 8 |- ((P gcd N) e. NN -> (P gcd N) e. RR)
54	24, 53	syl 12	. . . . . . 7 |- ((P e. Prime /\ N e. ZZ) -> (P gcd N) e. RR)
55		1re 6598	. . . . . . . 8 |- 1 e. RR
56		ltletr 6694	. . . . . . . 8 |- ((1 e. RR /\ P e. RR /\ (P gcd N) e. RR) -> ((1 < P /\ P <_ (P gcd N)) -> 1 < (P gcd N)))
57	55, 56	mp3an1 1178	. . . . . . 7 |- ((P e. RR /\ (P gcd N) e. RR) -> ((1 < P /\ P <_ (P gcd N)) -> 1 < (P gcd N)))
58	52, 54, 57	syl11anc 524	. . . . . 6 |- ((P e. Prime /\ N e. ZZ) -> ((1 < P /\ P <_ (P gcd N)) -> 1 < (P gcd N)))
59	49, 58	mpand 765	. . . . 5 |- ((P e. Prime /\ N e. ZZ) -> (P <_ (P gcd N) -> 1 < (P gcd N)))
60		ltne 6686	. . . . . . . 8 |- ((1 e. RR /\ (P gcd N) e. RR /\ 1 < (P gcd N)) -> (P gcd N) =/= 1)
61	60	3expia 1069	. . . . . . 7 |- ((1 e. RR /\ (P gcd N) e. RR) -> (1 < (P gcd N) -> (P gcd N) =/= 1))
62	55, 61	mpan 759	. . . . . 6 |- ((P gcd N) e. RR -> (1 < (P gcd N) -> (P gcd N) =/= 1))
63	54, 62	syl 12	. . . . 5 |- ((P e. Prime /\ N e. ZZ) -> (1 < (P gcd N) -> (P gcd N) =/= 1))
64	47, 59, 63	3syld 31	. . . 4 |- ((P e. Prime /\ N e. ZZ) -> (P||N -> (P gcd N) =/= 1))
65		df-ne 2019	. . . 4 |- ((P gcd N) =/= 1 <-> -. (P gcd N) = 1)
66	64, 65	syl6ib 229	. . 3 |- ((P e. Prime /\ N e. ZZ) -> (P||N -> -. (P gcd N) = 1))
67	66	con2d 107	. 2 |- ((P e. Prime /\ N e. ZZ) -> ((P gcd N) = 1 -> -. P||N))
68	38, 67	impbid 574	1 |- ((P e. Prime /\ N e. ZZ) -> (-. P||N <-> (P gcd N) = 1))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   \/ wo 239   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300   =/= wne 2017  A.wral 2105   class class class wbr 3338  (class class class)co 4884  RRcr 6385  0cc0 6386  1c1 6387   <_ cle 6448  NNcn 6449  ZZcz 6451   < clt 6653  ||cdivides 13662   gcd cgcd 13713  Primecprime 13766

This theorem is referenced by:  prmdvdsmul 13784

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-inf2 5731

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-nel 2020  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-oprab 4887  df-mpt 5006  df-1st 5020  df-2nd 5021  df-iota 5089  df-rdg 5140  df-1o 5177  df-2o 5178  df-oadd 5179  df-omul 5180  df-er 5318  df-ec 5320  df-qs 5323  df-en 5427  df-dom 5428  df-sdom 5429  df-fin 5430  df-undef 5556  df-riota 5560  df-sup 5664  df-ni 6152  df-pli 6153  df-mi 6154  df-lti 6155  df-plpq 6187  df-mpq 6188  df-enq 6189  df-nq 6190  df-plq 6191  df-mq 6192  df-rq 6193  df-ltq 6194  df-1q 6195  df-np 6238  df-1p 6239  df-plp 6240  df-mp 6241  df-ltp 6242  df-plpr 6316  df-mpr 6317  df-enr 6318  df-nr 6319  df-plr 6320  df-mr 6321  df-ltr 6322  df-0r 6323  df-1r 6324  df-m1r 6325  df-c 6392  df-0 6393  df-1 6394  df-i 6395  df-r 6396  df-plus 6397  df-mul 6398  df-lt 6399  df-sub 6511  df-neg 6513  df-pnf 6654  df-mnf 6655  df-xr 6656  df-ltxr 6657  df-le 6658  df-div 6892  df-n 7108  df-2 7154  df-n0 7309  df-z 7345  df-uz 7587  df-seq1 7721  df-exp 7812  df-sqr 7920  df-re 8001  df-im 8002  df-cj 8003  df-abs 8004  df-divides 13663  df-gcd 13714  df-prime 13767

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