Theorem ncoprmgcdne1b 12627

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 9180	. . . . . . 7 |- 2 = ( 1 + 1 )
2		2re 9191	. . . . . . . . 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 9743	. . . . . . . . . 10 |- ( i e. ( ZZ>= ` 2 ) -> i e. ZZ )
5	4	ad2antlr 489	. . . . . . . . 9 |- ( ( ( ( A e. NN /\ B e. NN ) /\ i e. ( ZZ>= ` 2 ) ) /\ ( i || A /\ i || B ) ) -> i e. ZZ )
6	5	zred 9580	. . . . . . . 8 |- ( ( ( ( A e. NN /\ B e. NN ) /\ i e. ( ZZ>= ` 2 ) ) /\ ( i || A /\ i || B ) ) -> i e. RR )
7		simplll 533	. . . . . . . . . 10 |- ( ( ( ( A e. NN /\ B e. NN ) /\ i e. ( ZZ>= ` 2 ) ) /\ ( i || A /\ i || B ) ) -> A e. NN )
8		simpllr 534	. . . . . . . . . 10 |- ( ( ( ( A e. NN /\ B e. NN ) /\ i e. ( ZZ>= ` 2 ) ) /\ ( i || A /\ i || B ) ) -> B e. NN )
9		gcdnncl 12504	. . . . . . . . . 10 |- ( ( A e. NN /\ B e. NN ) -> ( A gcd B ) e. NN )
10	7, 8, 9	syl2anc 411	. . . . . . . . 9 |- ( ( ( ( A e. NN /\ B e. NN ) /\ i e. ( ZZ>= ` 2 ) ) /\ ( i || A /\ i || B ) ) -> ( A gcd B ) e. NN )
11	10	nnred 9134	. . . . . . . 8 |- ( ( ( ( A e. NN /\ B e. NN ) /\ i e. ( ZZ>= ` 2 ) ) /\ ( i || A /\ i || B ) ) -> ( A gcd B ) e. RR )
12		eluzle 9746	. . . . . . . . 9 |- ( i e. ( ZZ>= ` 2 ) -> 2 <_ i )
13	12	ad2antlr 489	. . . . . . . 8 |- ( ( ( ( A e. NN /\ B e. NN ) /\ i e. ( ZZ>= ` 2 ) ) /\ ( i || A /\ i || B ) ) -> 2 <_ i )
14		simpr 110	. . . . . . . . . 10 |- ( ( ( ( A e. NN /\ B e. NN ) /\ i e. ( ZZ>= ` 2 ) ) /\ ( i || A /\ i || B ) ) -> ( i || A /\ i || B ) )
15	7	nnzd 9579	. . . . . . . . . . 11 |- ( ( ( ( A e. NN /\ B e. NN ) /\ i e. ( ZZ>= ` 2 ) ) /\ ( i || A /\ i || B ) ) -> A e. ZZ )
16	8	nnzd 9579	. . . . . . . . . . 11 |- ( ( ( ( A e. NN /\ B e. NN ) /\ i e. ( ZZ>= ` 2 ) ) /\ ( i || A /\ i || B ) ) -> B e. ZZ )
17		dvdsgcd 12549	. . . . . . . . . . 11 |- ( ( i e. ZZ /\ A e. ZZ /\ B e. ZZ ) -> ( ( i || A /\ i || B ) -> i || ( A gcd B ) ) )
18	5, 15, 16, 17	syl3anc 1271	. . . . . . . . . 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 12371	. . . . . . . . . 10 |- ( ( i e. ZZ /\ ( A gcd B ) e. NN ) -> ( i || ( A gcd B ) -> i <_ ( A gcd B ) ) )
21	5, 10, 20	syl2anc 411	. . . . . . . . 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 8281	. . . . . . 7 |- ( ( ( ( A e. NN /\ B e. NN ) /\ i e. ( ZZ>= ` 2 ) ) /\ ( i || A /\ i || B ) ) -> 2 <_ ( A gcd B ) )
24	1, 23	eqbrtrrid 4119	. . . . . 6 |- ( ( ( ( A e. NN /\ B e. NN ) /\ i e. ( ZZ>= ` 2 ) ) /\ ( i || A /\ i || B ) ) -> ( 1 + 1 ) <_ ( A gcd B ) )
25		1nn 9132	. . . . . . . 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 9518	. . . . . . 7 |- ( ( 1 e. NN /\ ( A gcd B ) e. NN ) -> ( 1 < ( A gcd B ) <-> ( 1 + 1 ) <_ ( A gcd B ) ) )
28	26, 10, 27	syl2anc 411	. . . . . 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 167	. . . . 5 |- ( ( ( ( A e. NN /\ B e. NN ) /\ i e. ( ZZ>= ` 2 ) ) /\ ( i || A /\ i || B ) ) -> 1 < ( A gcd B ) )
30		nngt1ne1 9156	. . . . . 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 147	. . . 4 |- ( ( ( ( A e. NN /\ B e. NN ) /\ i e. ( ZZ>= ` 2 ) ) /\ ( i || A /\ i || B ) ) -> ( A gcd B ) =/= 1 )
33	32	ex 115	. . 3 |- ( ( ( A e. NN /\ B e. NN ) /\ i e. ( ZZ>= ` 2 ) ) -> ( ( i || A /\ i || B ) -> ( A gcd B ) =/= 1 ) )
34	33	rexlimdva 2648	. 2 |- ( ( A e. NN /\ B e. NN ) -> ( E. i e. ( ZZ>= ` 2 ) ( i || A /\ i || B ) -> ( A gcd B ) =/= 1 ) )
35	9	adantr 276	. . . . 5 |- ( ( ( A e. NN /\ B e. NN ) /\ ( A gcd B ) =/= 1 ) -> ( A gcd B ) e. NN )
36		simpr 110	. . . . 5 |- ( ( ( A e. NN /\ B e. NN ) /\ ( A gcd B ) =/= 1 ) -> ( A gcd B ) =/= 1 )
37		eluz2b3 9811	. . . . 5 |- ( ( A gcd B ) e. ( ZZ>= ` 2 ) <-> ( ( A gcd B ) e. NN /\ ( A gcd B ) =/= 1 ) )
38	35, 36, 37	sylanbrc 417	. . . 4 |- ( ( ( A e. NN /\ B e. NN ) /\ ( A gcd B ) =/= 1 ) -> ( A gcd B ) e. ( ZZ>= ` 2 ) )
39		simpll 527	. . . . . 6 |- ( ( ( A e. NN /\ B e. NN ) /\ ( A gcd B ) =/= 1 ) -> A e. NN )
40	39	nnzd 9579	. . . . 5 |- ( ( ( A e. NN /\ B e. NN ) /\ ( A gcd B ) =/= 1 ) -> A e. ZZ )
41		simplr 528	. . . . . 6 |- ( ( ( A e. NN /\ B e. NN ) /\ ( A gcd B ) =/= 1 ) -> B e. NN )
42	41	nnzd 9579	. . . . 5 |- ( ( ( A e. NN /\ B e. NN ) /\ ( A gcd B ) =/= 1 ) -> B e. ZZ )
43		gcddvds 12500	. . . . 5 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( ( A gcd B ) || A /\ ( A gcd B ) || B ) )
44	40, 42, 43	syl2anc 411	. . . 4 |- ( ( ( A e. NN /\ B e. NN ) /\ ( A gcd B ) =/= 1 ) -> ( ( A gcd B ) || A /\ ( A gcd B ) || B ) )
45		breq1 4086	. . . . . 6 |- ( i = ( A gcd B ) -> ( i || A <-> ( A gcd B ) || A ) )
46		breq1 4086	. . . . . 6 |- ( i = ( A gcd B ) -> ( i || B <-> ( A gcd B ) || B ) )
47	45, 46	anbi12d 473	. . . . 5 |- ( i = ( A gcd B ) -> ( ( i || A /\ i || B ) <-> ( ( A gcd B ) || A /\ ( A gcd B ) || B ) ) )
48	47	rspcev 2907	. . . 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 411	. . 3 |- ( ( ( A e. NN /\ B e. NN ) /\ ( A gcd B ) =/= 1 ) -> E. i e. ( ZZ>= ` 2 ) ( i || A /\ i || B ) )
50	49	ex 115	. 2 |- ( ( A e. NN /\ B e. NN ) -> ( ( A gcd B ) =/= 1 -> E. i e. ( ZZ>= ` 2 ) ( i || A /\ i || B ) ) )
51	34, 50	impbid 129	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 104   <-> wb 105   = wceq 1395   e. wcel 2200   =/= wne 2400   E.wrex 2509   class class class wbr 4083   ` cfv 5318  (class class class)co 6007   RRcr 8009   1c1 8011   + caddc 8013   < clt 8192   <_ cle 8193   NNcn 9121   2c2 9172   ZZcz 9457   ZZ>=cuz 9733   || cdvds 12314   gcd cgcd 12490

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680  ax-cnex 8101  ax-resscn 8102  ax-1cn 8103  ax-1re 8104  ax-icn 8105  ax-addcl 8106  ax-addrcl 8107  ax-mulcl 8108  ax-mulrcl 8109  ax-addcom 8110  ax-mulcom 8111  ax-addass 8112  ax-mulass 8113  ax-distr 8114  ax-i2m1 8115  ax-0lt1 8116  ax-1rid 8117  ax-0id 8118  ax-rnegex 8119  ax-precex 8120  ax-cnre 8121  ax-pre-ltirr 8122  ax-pre-ltwlin 8123  ax-pre-lttrn 8124  ax-pre-apti 8125  ax-pre-ltadd 8126  ax-pre-mulgt0 8127  ax-pre-mulext 8128  ax-arch 8129  ax-caucvg 8130

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-po 4387  df-iso 4388  df-iord 4457  df-on 4459  df-ilim 4460  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-riota 5960  df-ov 6010  df-oprab 6011  df-mpo 6012  df-1st 6292  df-2nd 6293  df-recs 6457  df-frec 6543  df-sup 7162  df-pnf 8194  df-mnf 8195  df-xr 8196  df-ltxr 8197  df-le 8198  df-sub 8330  df-neg 8331  df-reap 8733  df-ap 8740  df-div 8831  df-inn 9122  df-2 9180  df-3 9181  df-4 9182  df-n0 9381  df-z 9458  df-uz 9734  df-q 9827  df-rp 9862  df-fz 10217  df-fzo 10351  df-fl 10502  df-mod 10557  df-seqfrec 10682  df-exp 10773  df-cj 11369  df-re 11370  df-im 11371  df-rsqrt 11525  df-abs 11526  df-dvds 12315  df-gcd 12491

This theorem is referenced by:  ncoprmgcdgt1b  12628