Theorem coprm 12779

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 12746	. . . . . . 7 |- ( P e. Prime -> P e. ZZ )
2		gcddvds 12597	. . . . . . 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 4096	. . . . 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 636	. . 3 |- ( ( P e. Prime /\ N e. ZZ ) -> ( -. P || N -> -. ( P gcd N ) = P ) )
8		0nnn 9212	. . . . . . . . 9 |- -. 0 e. NN
9		prmnn 12745	. . . . . . . . . 10 |- ( P e. Prime -> P e. NN )
10		eleq1 2294	. . . . . . . . . 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 670	. . . . . . . 8 |- ( P e. Prime -> -. P = 0 )
13	12	intnanrd 940	. . . . . . 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 12596	. . . . . . . 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 12752	. . . . . . . 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 4096	. . . . . . . . 9 |- ( z = ( P gcd N ) -> ( z || P <-> ( P gcd N ) || P ) )
23		eqeq1 2238	. . . . . . . . . 10 |- ( z = ( P gcd N ) -> ( z = 1 <-> ( P gcd N ) = 1 ) )
24		eqeq1 2238	. . . . . . . . . 10 |- ( z = ( P gcd N ) -> ( z = P <-> ( P gcd N ) = P ) )
25	23, 24	orbi12d 801	. . . . . . . . 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 2907	. . . . . . 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 752	. . . . 5 |- ( -. ( P gcd N ) = P -> ( ( P gcd N ) = 1 <-> ( ( P gcd N ) = P \/ ( P gcd N ) = 1 ) ) )
32		orcom 736	. . . . 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 12428	. . . . . . 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 12598	. . . . . . . . 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 1332	. . . . . . 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 12767	. . . . . 6 |- ( P e. Prime -> 1 < P )
46	45	adantr 276	. . . . 5 |- ( ( P e. Prime /\ N e. ZZ ) -> 1 < P )
47	1	zred 9646	. . . . . . 7 |- ( P e. Prime -> P e. RR )
48	47	adantr 276	. . . . . 6 |- ( ( P e. Prime /\ N e. ZZ ) -> P e. RR )
49	18	nnred 9198	. . . . . 6 |- ( ( P e. Prime /\ N e. ZZ ) -> ( P gcd N ) e. RR )
50		1re 8221	. . . . . . 7 |- 1 e. RR
51		ltletr 8311	. . . . . . 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 1361	. . . . . 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 8306	. . . . . 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 2461	. 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 716   /\ w3a 1005   = wceq 1398   e. wcel 2202   =/= wne 2403   A.wral 2511   class class class wbr 4093   ` cfv 5333  (class class class)co 6028   RRcr 8074   0cc0 8075   1c1 8076   < clt 8256   <_ cle 8257   NNcn 9185   2c2 9236   ZZcz 9523   ZZ>=cuz 9799   || cdvds 12411   gcd cgcd 12587   Primecprime 12742

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692  ax-cnex 8166  ax-resscn 8167  ax-1cn 8168  ax-1re 8169  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-mulrcl 8174  ax-addcom 8175  ax-mulcom 8176  ax-addass 8177  ax-mulass 8178  ax-distr 8179  ax-i2m1 8180  ax-0lt1 8181  ax-1rid 8182  ax-0id 8183  ax-rnegex 8184  ax-precex 8185  ax-cnre 8186  ax-pre-ltirr 8187  ax-pre-ltwlin 8188  ax-pre-lttrn 8189  ax-pre-apti 8190  ax-pre-ltadd 8191  ax-pre-mulgt0 8192  ax-pre-mulext 8193  ax-arch 8194  ax-caucvg 8195

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-po 4399  df-iso 4400  df-iord 4469  df-on 4471  df-ilim 4472  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-frec 6600  df-1o 6625  df-2o 6626  df-er 6745  df-en 6953  df-sup 7226  df-pnf 8258  df-mnf 8259  df-xr 8260  df-ltxr 8261  df-le 8262  df-sub 8394  df-neg 8395  df-reap 8797  df-ap 8804  df-div 8895  df-inn 9186  df-2 9244  df-3 9245  df-4 9246  df-n0 9445  df-z 9524  df-uz 9800  df-q 9898  df-rp 9933  df-fz 10289  df-fzo 10423  df-fl 10576  df-mod 10631  df-seqfrec 10756  df-exp 10847  df-cj 11465  df-re 11466  df-im 11467  df-rsqrt 11621  df-abs 11622  df-dvds 12412  df-gcd 12588  df-prm 12743

This theorem is referenced by:  prmrp  12780  euclemma  12781  cncongrprm  12792  isoddgcd1  12794  phiprmpw  12857  fermltl  12869  prmdiv  12870  prmdiveq  12871  vfermltl  12887  prmpwdvds  12991  perfect1  15795  perfectlem1  15796  perfectlem2  15797  lgslem1  15802  lgsprme0  15844  gausslemma2dlem0c  15853  lgseisenlem3  15874  lgsquad2lem2  15884