Theorem ncoprmgcdne1b 12021

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 8916	. . . . . . 7 |- 2 = ( 1 + 1 )
2		2re 8927	. . . . . . . . 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 9475	. . . . . . . . . 10 |- ( i e. ( ZZ>= ` 2 ) -> i e. ZZ )
5	4	ad2antlr 481	. . . . . . . . 9 |- ( ( ( ( A e. NN /\ B e. NN ) /\ i e. ( ZZ>= ` 2 ) ) /\ ( i || A /\ i || B ) ) -> i e. ZZ )
6	5	zred 9313	. . . . . . . 8 |- ( ( ( ( A e. NN /\ B e. NN ) /\ i e. ( ZZ>= ` 2 ) ) /\ ( i || A /\ i || B ) ) -> i e. RR )
7		simplll 523	. . . . . . . . . 10 |- ( ( ( ( A e. NN /\ B e. NN ) /\ i e. ( ZZ>= ` 2 ) ) /\ ( i || A /\ i || B ) ) -> A e. NN )
8		simpllr 524	. . . . . . . . . 10 |- ( ( ( ( A e. NN /\ B e. NN ) /\ i e. ( ZZ>= ` 2 ) ) /\ ( i || A /\ i || B ) ) -> B e. NN )
9		gcdnncl 11900	. . . . . . . . . 10 |- ( ( A e. NN /\ B e. NN ) -> ( A gcd B ) e. NN )
10	7, 8, 9	syl2anc 409	. . . . . . . . 9 |- ( ( ( ( A e. NN /\ B e. NN ) /\ i e. ( ZZ>= ` 2 ) ) /\ ( i || A /\ i || B ) ) -> ( A gcd B ) e. NN )
11	10	nnred 8870	. . . . . . . 8 |- ( ( ( ( A e. NN /\ B e. NN ) /\ i e. ( ZZ>= ` 2 ) ) /\ ( i || A /\ i || B ) ) -> ( A gcd B ) e. RR )
12		eluzle 9478	. . . . . . . . 9 |- ( i e. ( ZZ>= ` 2 ) -> 2 <_ i )
13	12	ad2antlr 481	. . . . . . . 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 9312	. . . . . . . . . . 11 |- ( ( ( ( A e. NN /\ B e. NN ) /\ i e. ( ZZ>= ` 2 ) ) /\ ( i || A /\ i || B ) ) -> A e. ZZ )
16	8	nnzd 9312	. . . . . . . . . . 11 |- ( ( ( ( A e. NN /\ B e. NN ) /\ i e. ( ZZ>= ` 2 ) ) /\ ( i || A /\ i || B ) ) -> B e. ZZ )
17		dvdsgcd 11945	. . . . . . . . . . 11 |- ( ( i e. ZZ /\ A e. ZZ /\ B e. ZZ ) -> ( ( i || A /\ i || B ) -> i || ( A gcd B ) ) )
18	5, 15, 16, 17	syl3anc 1228	. . . . . . . . . 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 11782	. . . . . . . . . 10 |- ( ( i e. ZZ /\ ( A gcd B ) e. NN ) -> ( i || ( A gcd B ) -> i <_ ( A gcd B ) ) )
21	5, 10, 20	syl2anc 409	. . . . . . . . 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 8022	. . . . . . 7 |- ( ( ( ( A e. NN /\ B e. NN ) /\ i e. ( ZZ>= ` 2 ) ) /\ ( i || A /\ i || B ) ) -> 2 <_ ( A gcd B ) )
24	1, 23	eqbrtrrid 4018	. . . . . 6 |- ( ( ( ( A e. NN /\ B e. NN ) /\ i e. ( ZZ>= ` 2 ) ) /\ ( i || A /\ i || B ) ) -> ( 1 + 1 ) <_ ( A gcd B ) )
25		1nn 8868	. . . . . . . 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 9251	. . . . . . 7 |- ( ( 1 e. NN /\ ( A gcd B ) e. NN ) -> ( 1 < ( A gcd B ) <-> ( 1 + 1 ) <_ ( A gcd B ) ) )
28	26, 10, 27	syl2anc 409	. . . . . 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 8892	. . . . . 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 2583	. 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 9542	. . . . 5 |- ( ( A gcd B ) e. ( ZZ>= ` 2 ) <-> ( ( A gcd B ) e. NN /\ ( A gcd B ) =/= 1 ) )
38	35, 36, 37	sylanbrc 414	. . . 4 |- ( ( ( A e. NN /\ B e. NN ) /\ ( A gcd B ) =/= 1 ) -> ( A gcd B ) e. ( ZZ>= ` 2 ) )
39		simpll 519	. . . . . 6 |- ( ( ( A e. NN /\ B e. NN ) /\ ( A gcd B ) =/= 1 ) -> A e. NN )
40	39	nnzd 9312	. . . . 5 |- ( ( ( A e. NN /\ B e. NN ) /\ ( A gcd B ) =/= 1 ) -> A e. ZZ )
41		simplr 520	. . . . . 6 |- ( ( ( A e. NN /\ B e. NN ) /\ ( A gcd B ) =/= 1 ) -> B e. NN )
42	41	nnzd 9312	. . . . 5 |- ( ( ( A e. NN /\ B e. NN ) /\ ( A gcd B ) =/= 1 ) -> B e. ZZ )
43		gcddvds 11896	. . . . 5 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( ( A gcd B ) || A /\ ( A gcd B ) || B ) )
44	40, 42, 43	syl2anc 409	. . . 4 |- ( ( ( A e. NN /\ B e. NN ) /\ ( A gcd B ) =/= 1 ) -> ( ( A gcd B ) || A /\ ( A gcd B ) || B ) )
45		breq1 3985	. . . . . 6 |- ( i = ( A gcd B ) -> ( i || A <-> ( A gcd B ) || A ) )
46		breq1 3985	. . . . . 6 |- ( i = ( A gcd B ) -> ( i || B <-> ( A gcd B ) || B ) )
47	45, 46	anbi12d 465	. . . . 5 |- ( i = ( A gcd B ) -> ( ( i || A /\ i || B ) <-> ( ( A gcd B ) || A /\ ( A gcd B ) || B ) ) )
48	47	rspcev 2830	. . . 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 409	. . 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 1343   e. wcel 2136   =/= wne 2336   E.wrex 2445   class class class wbr 3982   ` cfv 5188  (class class class)co 5842   RRcr 7752   1c1 7754   + caddc 7756   < clt 7933   <_ cle 7934   NNcn 8857   2c2 8908   ZZcz 9191   ZZ>=cuz 9466   || cdvds 11727   gcd cgcd 11875

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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-mulrcl 7852  ax-addcom 7853  ax-mulcom 7854  ax-addass 7855  ax-mulass 7856  ax-distr 7857  ax-i2m1 7858  ax-0lt1 7859  ax-1rid 7860  ax-0id 7861  ax-rnegex 7862  ax-precex 7863  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-ltwlin 7866  ax-pre-lttrn 7867  ax-pre-apti 7868  ax-pre-ltadd 7869  ax-pre-mulgt0 7870  ax-pre-mulext 7871  ax-arch 7872  ax-caucvg 7873

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rmo 2452  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-if 3521  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-id 4271  df-po 4274  df-iso 4275  df-iord 4344  df-on 4346  df-ilim 4347  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-recs 6273  df-frec 6359  df-sup 6949  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-sub 8071  df-neg 8072  df-reap 8473  df-ap 8480  df-div 8569  df-inn 8858  df-2 8916  df-3 8917  df-4 8918  df-n0 9115  df-z 9192  df-uz 9467  df-q 9558  df-rp 9590  df-fz 9945  df-fzo 10078  df-fl 10205  df-mod 10258  df-seqfrec 10381  df-exp 10455  df-cj 10784  df-re 10785  df-im 10786  df-rsqrt 10940  df-abs 10941  df-dvds 11728  df-gcd 11876

This theorem is referenced by:  ncoprmgcdgt1b  12022