Theorem coprm 12948

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 12914	. . . . . . 7 |- ( P e. Prime -> P e. ZZ )
2		gcddvds 12765	. . . . . . 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 4134	. . . . 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 640	. . 3 |- ( ( P e. Prime /\ N e. ZZ ) -> ( -. P || N -> -. ( P gcd N ) = P ) )
8		0nnn 9340	. . . . . . . . 9 |- -. 0 e. NN
9		prmnn 12913	. . . . . . . . . 10 |- ( P e. Prime -> P e. NN )
10		eleq1 2301	. . . . . . . . . 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 674	. . . . . . . 8 |- ( P e. Prime -> -. P = 0 )
13	12	intnanrd 944	. . . . . . 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 12764	. . . . . . . 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 12920	. . . . . . . 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 4134	. . . . . . . . 9 |- ( z = ( P gcd N ) -> ( z || P <-> ( P gcd N ) || P ) )
23		eqeq1 2245	. . . . . . . . . 10 |- ( z = ( P gcd N ) -> ( z = 1 <-> ( P gcd N ) = 1 ) )
24		eqeq1 2245	. . . . . . . . . 10 |- ( z = ( P gcd N ) -> ( z = P <-> ( P gcd N ) = P ) )
25	23, 24	orbi12d 805	. . . . . . . . 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 2925	. . . . . . 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 756	. . . . 5 |- ( -. ( P gcd N ) = P -> ( ( P gcd N ) = 1 <-> ( ( P gcd N ) = P \/ ( P gcd N ) = 1 ) ) )
32		orcom 740	. . . . 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 12596	. . . . . . 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 12766	. . . . . . . . 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 1336	. . . . . . 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 433	. . . 4 |- ( ( P e. Prime /\ N e. ZZ ) -> ( P || N -> P <_ ( P gcd N ) ) )
45		prmgt1 12936	. . . . . 6 |- ( P e. Prime -> 1 < P )
46	45	adantr 276	. . . . 5 |- ( ( P e. Prime /\ N e. ZZ ) -> 1 < P )
47	1	zred 9779	. . . . . . 7 |- ( P e. Prime -> P e. RR )
48	47	adantr 276	. . . . . 6 |- ( ( P e. Prime /\ N e. ZZ ) -> P e. RR )
49	18	nnred 9326	. . . . . 6 |- ( ( P e. Prime /\ N e. ZZ ) -> ( P gcd N ) e. RR )
50		1re 8331	. . . . . . 7 |- 1 e. RR
51		ltletr 8421	. . . . . . 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 1365	. . . . . 6 |- ( ( P e. RR /\ ( P gcd N ) e. RR ) -> ( ( 1 < P /\ P <_ ( P gcd N ) ) -> 1 < ( P gcd N ) ) )
53	48, 49, 52	syl2anc 415	. . . . 5 |- ( ( P e. Prime /\ N e. ZZ ) -> ( ( 1 < P /\ P <_ ( P gcd N ) ) -> 1 < ( P gcd N ) ) )
54	46, 53	mpand 433	. . . 4 |- ( ( P e. Prime /\ N e. ZZ ) -> ( P <_ ( P gcd N ) -> 1 < ( P gcd N ) ) )
55		ltne 8416	. . . . . 6 |- ( ( 1 e. RR /\ 1 < ( P gcd N ) ) -> ( P gcd N ) =/= 1 )
56	50, 55	mpan 428	. . . . 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 2478	. 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 ) )

This proof depends on syntax axioms:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 720   /\ w3a 1009   = wceq 1402   e. wcel 2209   =/= wne 2420   A.wral 2528   class class class wbr 4131   ` cfv 5380  (class class class)co 6088   RRcr 8184   0cc0 8185   1c1 8186   < clt 8366   <_ cle 8367   NNcn 9313   2c2 9364   ZZcz 9655   ZZ>=cuz 9937   || cdvds 12579   gcd cgcd 12755   Primecprime 12910

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-iinf 4737  ax-cnex 8276  ax-resscn 8277  ax-1cn 8278  ax-1re 8279  ax-icn 8280  ax-addcl 8281  ax-addrcl 8282  ax-mulcl 8283  ax-mulrcl 8284  ax-addcom 8285  ax-mulcom 8286  ax-addass 8287  ax-mulass 8288  ax-distr 8289  ax-i2m1 8290  ax-0lt1 8291  ax-1rid 8292  ax-0id 8293  ax-rnegex 8294  ax-precex 8295  ax-cnre 8296  ax-pre-ltirr 8297  ax-pre-ltwlin 8298  ax-pre-lttrn 8299  ax-pre-apti 8300  ax-pre-ltadd 8301  ax-pre-mulgt0 8302  ax-pre-mulext 8303  ax-arch 8304  ax-caucvg 8305

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-id 4440  df-po 4443  df-iso 4444  df-iord 4513  df-on 4515  df-ilim 4516  df-suc 4518  df-iom 4740  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-recs 6580  df-frec 6666  df-1o 6691  df-2o 6692  df-er 6811  df-en 7027  df-sup 7330  df-pnf 8368  df-mnf 8369  df-xr 8370  df-ltxr 8371  df-le 8372  df-sub 8506  df-neg 8507  df-reap 8911  df-ap 8918  df-div 9011  df-inn 9314  df-2 9372  df-3 9373  df-4 9374  df-n0 9575  df-z 9656  df-uz 9938  df-q 10036  df-rp 10072  df-fz 10429  df-fzo 10567  df-fl 10722  df-mod 10781  df-seqfrec 10906  df-exp 10997  df-cj 11629  df-re 11630  df-im 11631  df-rsqrt 11786  df-abs 11787  df-dvds 12580  df-gcd 12756  df-prm 12911

This theorem is used by:  prmrp  12949  euclemma  12950  cncongrprm  12961  isoddgcd1  12963  phiprmpw  13029  fermltl  13041  prmdiv  13042  prmdiveq  13043  vfermltl  13059  prmpwdvds  13163  1259lem5  13275  zprmlogbaplem2  16216  perfect1  16298  perfectlem1  16299  perfectlem2  16300  lgslem1  16324  lgsprme0  16366  gausslemma2dlem0c  16375  lgseisenlem3  16396  lgsquad2lem2  16406