Theorem ncoprmgcdne1b 12892

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 9372	. . . . . . 7 |- 2 = ( 1 + 1 )
2		2re 9383	. . . . . . . . 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 9947	. . . . . . . . . 10 |- ( i e. ( ZZ>= ` 2 ) -> i e. ZZ )
5	4	ad2antlr 493	. . . . . . . . 9 |- ( ( ( ( A e. NN /\ B e. NN ) /\ i e. ( ZZ>= ` 2 ) ) /\ ( i || A /\ i || B ) ) -> i e. ZZ )
6	5	zred 9779	. . . . . . . 8 |- ( ( ( ( A e. NN /\ B e. NN ) /\ i e. ( ZZ>= ` 2 ) ) /\ ( i || A /\ i || B ) ) -> i e. RR )
7		simplll 539	. . . . . . . . . 10 |- ( ( ( ( A e. NN /\ B e. NN ) /\ i e. ( ZZ>= ` 2 ) ) /\ ( i || A /\ i || B ) ) -> A e. NN )
8		simpllr 540	. . . . . . . . . 10 |- ( ( ( ( A e. NN /\ B e. NN ) /\ i e. ( ZZ>= ` 2 ) ) /\ ( i || A /\ i || B ) ) -> B e. NN )
9		gcdnncl 12769	. . . . . . . . . 10 |- ( ( A e. NN /\ B e. NN ) -> ( A gcd B ) e. NN )
10	7, 8, 9	syl2anc 415	. . . . . . . . 9 |- ( ( ( ( A e. NN /\ B e. NN ) /\ i e. ( ZZ>= ` 2 ) ) /\ ( i || A /\ i || B ) ) -> ( A gcd B ) e. NN )
11	10	nnred 9326	. . . . . . . 8 |- ( ( ( ( A e. NN /\ B e. NN ) /\ i e. ( ZZ>= ` 2 ) ) /\ ( i || A /\ i || B ) ) -> ( A gcd B ) e. RR )
12		eluzle 9950	. . . . . . . . 9 |- ( i e. ( ZZ>= ` 2 ) -> 2 <_ i )
13	12	ad2antlr 493	. . . . . . . 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 9778	. . . . . . . . . . 11 |- ( ( ( ( A e. NN /\ B e. NN ) /\ i e. ( ZZ>= ` 2 ) ) /\ ( i || A /\ i || B ) ) -> A e. ZZ )
16	8	nnzd 9778	. . . . . . . . . . 11 |- ( ( ( ( A e. NN /\ B e. NN ) /\ i e. ( ZZ>= ` 2 ) ) /\ ( i || A /\ i || B ) ) -> B e. ZZ )
17		dvdsgcd 12814	. . . . . . . . . . 11 |- ( ( i e. ZZ /\ A e. ZZ /\ B e. ZZ ) -> ( ( i || A /\ i || B ) -> i || ( A gcd B ) ) )
18	5, 15, 16, 17	syl3anc 1278	. . . . . . . . . 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 12636	. . . . . . . . . 10 |- ( ( i e. ZZ /\ ( A gcd B ) e. NN ) -> ( i || ( A gcd B ) -> i <_ ( A gcd B ) ) )
21	5, 10, 20	syl2anc 415	. . . . . . . . 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 8457	. . . . . . 7 |- ( ( ( ( A e. NN /\ B e. NN ) /\ i e. ( ZZ>= ` 2 ) ) /\ ( i || A /\ i || B ) ) -> 2 <_ ( A gcd B ) )
24	1, 23	eqbrtrrid 4167	. . . . . 6 |- ( ( ( ( A e. NN /\ B e. NN ) /\ i e. ( ZZ>= ` 2 ) ) /\ ( i || A /\ i || B ) ) -> ( 1 + 1 ) <_ ( A gcd B ) )
25		1nn 9324	. . . . . . . 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 9716	. . . . . . 7 |- ( ( 1 e. NN /\ ( A gcd B ) e. NN ) -> ( 1 < ( A gcd B ) <-> ( 1 + 1 ) <_ ( A gcd B ) ) )
28	26, 10, 27	syl2anc 415	. . . . . 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 9348	. . . . . 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 2668	. 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 10020	. . . . 5 |- ( ( A gcd B ) e. ( ZZ>= ` 2 ) <-> ( ( A gcd B ) e. NN /\ ( A gcd B ) =/= 1 ) )
38	35, 36, 37	sylanbrc 421	. . . 4 |- ( ( ( A e. NN /\ B e. NN ) /\ ( A gcd B ) =/= 1 ) -> ( A gcd B ) e. ( ZZ>= ` 2 ) )
39		simpll 531	. . . . . 6 |- ( ( ( A e. NN /\ B e. NN ) /\ ( A gcd B ) =/= 1 ) -> A e. NN )
40	39	nnzd 9778	. . . . 5 |- ( ( ( A e. NN /\ B e. NN ) /\ ( A gcd B ) =/= 1 ) -> A e. ZZ )
41		simplr 533	. . . . . 6 |- ( ( ( A e. NN /\ B e. NN ) /\ ( A gcd B ) =/= 1 ) -> B e. NN )
42	41	nnzd 9778	. . . . 5 |- ( ( ( A e. NN /\ B e. NN ) /\ ( A gcd B ) =/= 1 ) -> B e. ZZ )
43		gcddvds 12765	. . . . 5 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( ( A gcd B ) || A /\ ( A gcd B ) || B ) )
44	40, 42, 43	syl2anc 415	. . . 4 |- ( ( ( A e. NN /\ B e. NN ) /\ ( A gcd B ) =/= 1 ) -> ( ( A gcd B ) || A /\ ( A gcd B ) || B ) )
45		breq1 4134	. . . . . 6 |- ( i = ( A gcd B ) -> ( i || A <-> ( A gcd B ) || A ) )
46		breq1 4134	. . . . . 6 |- ( i = ( A gcd B ) -> ( i || B <-> ( A gcd B ) || B ) )
47	45, 46	anbi12d 477	. . . . 5 |- ( i = ( A gcd B ) -> ( ( i || A /\ i || B ) <-> ( ( A gcd B ) || A /\ ( A gcd B ) || B ) ) )
48	47	rspcev 2929	. . . 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 415	. . 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 ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   =/= wne 2420   E.wrex 2529   class class class wbr 4131   ` cfv 5380  (class class class)co 6088   RRcr 8184   1c1 8186   + caddc 8188   < clt 8366   <_ cle 8367   NNcn 9313   2c2 9364   ZZcz 9655   ZZ>=cuz 9937   || cdvds 12579   gcd cgcd 12755

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-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

This theorem is used by:  ncoprmgcdgt1b  12893