Theorem ncoprmgcdne1b 11759

Description: Two positive integers are not coprime, i.e. there is an integer greater than 1 which divides both integers, iff their greatest common divisor is not 1. (Contributed by AV, 9-Aug-2020.)

Assertion

Ref	Expression
ncoprmgcdne1b	|- ( ( A e. NN /\ B e. NN ) -> ( E. i e. ( ZZ>= ` 2 ) ( i || A /\ i || B ) <-> ( A gcd B ) =/= 1 ) )

Distinct variable groups:   A, i   B, i

Proof of Theorem ncoprmgcdne1b

Step	Hyp	Ref	Expression
1		df-2 8772	. . . . . . 7 |- 2 = ( 1 + 1 )
2		2re 8783	. . . . . . . . 9 |- 2 e. RR
3	2	a1i 9	. . . . . . . 8 |- ( ( ( ( A e. NN /\ B e. NN ) /\ i e. ( ZZ>= ` 2 ) ) /\ ( i || A /\ i || B ) ) -> 2 e. RR )
4		eluzelz 9328	. . . . . . . . . 10 |- ( i e. ( ZZ>= ` 2 ) -> i e. ZZ )
5	4	ad2antlr 480	. . . . . . . . 9 |- ( ( ( ( A e. NN /\ B e. NN ) /\ i e. ( ZZ>= ` 2 ) ) /\ ( i || A /\ i || B ) ) -> i e. ZZ )
6	5	zred 9166	. . . . . . . 8 |- ( ( ( ( A e. NN /\ B e. NN ) /\ i e. ( ZZ>= ` 2 ) ) /\ ( i || A /\ i || B ) ) -> i e. RR )
7		simplll 522	. . . . . . . . . 10 |- ( ( ( ( A e. NN /\ B e. NN ) /\ i e. ( ZZ>= ` 2 ) ) /\ ( i || A /\ i || B ) ) -> A e. NN )
8		simpllr 523	. . . . . . . . . 10 |- ( ( ( ( A e. NN /\ B e. NN ) /\ i e. ( ZZ>= ` 2 ) ) /\ ( i || A /\ i || B ) ) -> B e. NN )
9		gcdnncl 11645	. . . . . . . . . 10 |- ( ( A e. NN /\ B e. NN ) -> ( A gcd B ) e. NN )
10	7, 8, 9	syl2anc 408	. . . . . . . . 9 |- ( ( ( ( A e. NN /\ B e. NN ) /\ i e. ( ZZ>= ` 2 ) ) /\ ( i || A /\ i || B ) ) -> ( A gcd B ) e. NN )
11	10	nnred 8726	. . . . . . . 8 |- ( ( ( ( A e. NN /\ B e. NN ) /\ i e. ( ZZ>= ` 2 ) ) /\ ( i || A /\ i || B ) ) -> ( A gcd B ) e. RR )
12		eluzle 9331	. . . . . . . . 9 |- ( i e. ( ZZ>= ` 2 ) -> 2 <_ i )
13	12	ad2antlr 480	. . . . . . . 8 |- ( ( ( ( A e. NN /\ B e. NN ) /\ i e. ( ZZ>= ` 2 ) ) /\ ( i || A /\ i || B ) ) -> 2 <_ i )
14		simpr 109	. . . . . . . . . 10 |- ( ( ( ( A e. NN /\ B e. NN ) /\ i e. ( ZZ>= ` 2 ) ) /\ ( i || A /\ i || B ) ) -> ( i || A /\ i || B ) )
15	7	nnzd 9165	. . . . . . . . . . 11 |- ( ( ( ( A e. NN /\ B e. NN ) /\ i e. ( ZZ>= ` 2 ) ) /\ ( i || A /\ i || B ) ) -> A e. ZZ )
16	8	nnzd 9165	. . . . . . . . . . 11 |- ( ( ( ( A e. NN /\ B e. NN ) /\ i e. ( ZZ>= ` 2 ) ) /\ ( i || A /\ i || B ) ) -> B e. ZZ )
17		dvdsgcd 11689	. . . . . . . . . . 11 |- ( ( i e. ZZ /\ A e. ZZ /\ B e. ZZ ) -> ( ( i || A /\ i || B ) -> i || ( A gcd B ) ) )
18	5, 15, 16, 17	syl3anc 1216	. . . . . . . . . 10 |- ( ( ( ( A e. NN /\ B e. NN ) /\ i e. ( ZZ>= ` 2 ) ) /\ ( i || A /\ i || B ) ) -> ( ( i || A /\ i || B ) -> i || ( A gcd B ) ) )
19	14, 18	mpd 13	. . . . . . . . 9 |- ( ( ( ( A e. NN /\ B e. NN ) /\ i e. ( ZZ>= ` 2 ) ) /\ ( i || A /\ i || B ) ) -> i || ( A gcd B ) )
20		dvdsle 11531	. . . . . . . . . 10 |- ( ( i e. ZZ /\ ( A gcd B ) e. NN ) -> ( i || ( A gcd B ) -> i <_ ( A gcd B ) ) )
21	5, 10, 20	syl2anc 408	. . . . . . . . 9 |- ( ( ( ( A e. NN /\ B e. NN ) /\ i e. ( ZZ>= ` 2 ) ) /\ ( i || A /\ i || B ) ) -> ( i || ( A gcd B ) -> i <_ ( A gcd B ) ) )
22	19, 21	mpd 13	. . . . . . . 8 |- ( ( ( ( A e. NN /\ B e. NN ) /\ i e. ( ZZ>= ` 2 ) ) /\ ( i || A /\ i || B ) ) -> i <_ ( A gcd B ) )
23	3, 6, 11, 13, 22	letrd 7879	. . . . . . 7 |- ( ( ( ( A e. NN /\ B e. NN ) /\ i e. ( ZZ>= ` 2 ) ) /\ ( i || A /\ i || B ) ) -> 2 <_ ( A gcd B ) )
24	1, 23	eqbrtrrid 3959	. . . . . 6 |- ( ( ( ( A e. NN /\ B e. NN ) /\ i e. ( ZZ>= ` 2 ) ) /\ ( i || A /\ i || B ) ) -> ( 1 + 1 ) <_ ( A gcd B ) )
25		1nn 8724	. . . . . . . 8 |- 1 e. NN
26	25	a1i 9	. . . . . . 7 |- ( ( ( ( A e. NN /\ B e. NN ) /\ i e. ( ZZ>= ` 2 ) ) /\ ( i || A /\ i || B ) ) -> 1 e. NN )
27		nnltp1le 9107	. . . . . . 7 |- ( ( 1 e. NN /\ ( A gcd B ) e. NN ) -> ( 1 < ( A gcd B ) <-> ( 1 + 1 ) <_ ( A gcd B ) ) )
28	26, 10, 27	syl2anc 408	. . . . . 6 |- ( ( ( ( A e. NN /\ B e. NN ) /\ i e. ( ZZ>= ` 2 ) ) /\ ( i || A /\ i || B ) ) -> ( 1 < ( A gcd B ) <-> ( 1 + 1 ) <_ ( A gcd B ) ) )
29	24, 28	mpbird 166	. . . . 5 |- ( ( ( ( A e. NN /\ B e. NN ) /\ i e. ( ZZ>= ` 2 ) ) /\ ( i || A /\ i || B ) ) -> 1 < ( A gcd B ) )
30		nngt1ne1 8748	. . . . . 6 |- ( ( A gcd B ) e. NN -> ( 1 < ( A gcd B ) <-> ( A gcd B ) =/= 1 ) )
31	10, 30	syl 14	. . . . 5 |- ( ( ( ( A e. NN /\ B e. NN ) /\ i e. ( ZZ>= ` 2 ) ) /\ ( i || A /\ i || B ) ) -> ( 1 < ( A gcd B ) <-> ( A gcd B ) =/= 1 ) )
32	29, 31	mpbid 146	. . . 4 |- ( ( ( ( A e. NN /\ B e. NN ) /\ i e. ( ZZ>= ` 2 ) ) /\ ( i || A /\ i || B ) ) -> ( A gcd B ) =/= 1 )
33	32	ex 114	. . 3 |- ( ( ( A e. NN /\ B e. NN ) /\ i e. ( ZZ>= ` 2 ) ) -> ( ( i || A /\ i || B ) -> ( A gcd B ) =/= 1 ) )
34	33	rexlimdva 2547	. 2 |- ( ( A e. NN /\ B e. NN ) -> ( E. i e. ( ZZ>= ` 2 ) ( i || A /\ i || B ) -> ( A gcd B ) =/= 1 ) )
35	9	adantr 274	. . . . 5 |- ( ( ( A e. NN /\ B e. NN ) /\ ( A gcd B ) =/= 1 ) -> ( A gcd B ) e. NN )
36		simpr 109	. . . . 5 |- ( ( ( A e. NN /\ B e. NN ) /\ ( A gcd B ) =/= 1 ) -> ( A gcd B ) =/= 1 )
37		eluz2b3 9391	. . . . 5 |- ( ( A gcd B ) e. ( ZZ>= ` 2 ) <-> ( ( A gcd B ) e. NN /\ ( A gcd B ) =/= 1 ) )
38	35, 36, 37	sylanbrc 413	. . . 4 |- ( ( ( A e. NN /\ B e. NN ) /\ ( A gcd B ) =/= 1 ) -> ( A gcd B ) e. ( ZZ>= ` 2 ) )
39		simpll 518	. . . . . 6 |- ( ( ( A e. NN /\ B e. NN ) /\ ( A gcd B ) =/= 1 ) -> A e. NN )
40	39	nnzd 9165	. . . . 5 |- ( ( ( A e. NN /\ B e. NN ) /\ ( A gcd B ) =/= 1 ) -> A e. ZZ )
41		simplr 519	. . . . . 6 |- ( ( ( A e. NN /\ B e. NN ) /\ ( A gcd B ) =/= 1 ) -> B e. NN )
42	41	nnzd 9165	. . . . 5 |- ( ( ( A e. NN /\ B e. NN ) /\ ( A gcd B ) =/= 1 ) -> B e. ZZ )
43		gcddvds 11641	. . . . 5 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( ( A gcd B ) || A /\ ( A gcd B ) || B ) )
44	40, 42, 43	syl2anc 408	. . . 4 |- ( ( ( A e. NN /\ B e. NN ) /\ ( A gcd B ) =/= 1 ) -> ( ( A gcd B ) || A /\ ( A gcd B ) || B ) )
45		breq1 3927	. . . . . 6 |- ( i = ( A gcd B ) -> ( i || A <-> ( A gcd B ) || A ) )
46		breq1 3927	. . . . . 6 |- ( i = ( A gcd B ) -> ( i || B <-> ( A gcd B ) || B ) )
47	45, 46	anbi12d 464	. . . . 5 |- ( i = ( A gcd B ) -> ( ( i || A /\ i || B ) <-> ( ( A gcd B ) || A /\ ( A gcd B ) || B ) ) )
48	47	rspcev 2784	. . . 4 |- ( ( ( A gcd B ) e. ( ZZ>= ` 2 ) /\ ( ( A gcd B ) || A /\ ( A gcd B ) || B ) ) -> E. i e. ( ZZ>= ` 2 ) ( i || A /\ i || B ) )
49	38, 44, 48	syl2anc 408	. . 3 |- ( ( ( A e. NN /\ B e. NN ) /\ ( A gcd B ) =/= 1 ) -> E. i e. ( ZZ>= ` 2 ) ( i || A /\ i || B ) )
50	49	ex 114	. 2 |- ( ( A e. NN /\ B e. NN ) -> ( ( A gcd B ) =/= 1 -> E. i e. ( ZZ>= ` 2 ) ( i || A /\ i || B ) ) )
51	34, 50	impbid 128	1 |- ( ( A e. NN /\ B e. NN ) -> ( E. i e. ( ZZ>= ` 2 ) ( i || A /\ i || B ) <-> ( A gcd B ) =/= 1 ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1331   e. wcel 1480   =/= wne 2306   E.wrex 2415   class class class wbr 3924   ` cfv 5118  (class class class)co 5767   RRcr 7612   1c1 7614   + caddc 7616   < clt 7793   <_ cle 7794   NNcn 8713   2c2 8764   ZZcz 9047   ZZ>=cuz 9319   || cdvds 11482   gcd cgcd 11624

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-coll 4038  ax-sep 4041  ax-nul 4049  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-iinf 4497  ax-cnex 7704  ax-resscn 7705  ax-1cn 7706  ax-1re 7707  ax-icn 7708  ax-addcl 7709  ax-addrcl 7710  ax-mulcl 7711  ax-mulrcl 7712  ax-addcom 7713  ax-mulcom 7714  ax-addass 7715  ax-mulass 7716  ax-distr 7717  ax-i2m1 7718  ax-0lt1 7719  ax-1rid 7720  ax-0id 7721  ax-rnegex 7722  ax-precex 7723  ax-cnre 7724  ax-pre-ltirr 7725  ax-pre-ltwlin 7726  ax-pre-lttrn 7727  ax-pre-apti 7728  ax-pre-ltadd 7729  ax-pre-mulgt0 7730  ax-pre-mulext 7731  ax-arch 7732  ax-caucvg 7733

This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-nel 2402  df-ral 2419  df-rex 2420  df-reu 2421  df-rmo 2422  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-if 3470  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-tr 4022  df-id 4210  df-po 4213  df-iso 4214  df-iord 4283  df-on 4285  df-ilim 4286  df-suc 4288  df-iom 4500  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-riota 5723  df-ov 5770  df-oprab 5771  df-mpo 5772  df-1st 6031  df-2nd 6032  df-recs 6195  df-frec 6281  df-sup 6864  df-pnf 7795  df-mnf 7796  df-xr 7797  df-ltxr 7798  df-le 7799  df-sub 7928  df-neg 7929  df-reap 8330  df-ap 8337  df-div 8426  df-inn 8714  df-2 8772  df-3 8773  df-4 8774  df-n0 8971  df-z 9048  df-uz 9320  df-q 9405  df-rp 9435  df-fz 9784  df-fzo 9913  df-fl 10036  df-mod 10089  df-seqfrec 10212  df-exp 10286  df-cj 10607  df-re 10608  df-im 10609  df-rsqrt 10763  df-abs 10764  df-dvds 11483  df-gcd 11625

This theorem is referenced by:  ncoprmgcdgt1b  11760