Theorem ncoprmgcdne1b 12411

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 9095	. . . . . . 7 |- 2 = ( 1 + 1 )
2		2re 9106	. . . . . . . . 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 9657	. . . . . . . . . 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 9495	. . . . . . . 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 12288	. . . . . . . . . 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 9049	. . . . . . . 8 |- ( ( ( ( A e. NN /\ B e. NN ) /\ i e. ( ZZ>= ` 2 ) ) /\ ( i || A /\ i || B ) ) -> ( A gcd B ) e. RR )
12		eluzle 9660	. . . . . . . . 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 9494	. . . . . . . . . . 11 |- ( ( ( ( A e. NN /\ B e. NN ) /\ i e. ( ZZ>= ` 2 ) ) /\ ( i || A /\ i || B ) ) -> A e. ZZ )
16	8	nnzd 9494	. . . . . . . . . . 11 |- ( ( ( ( A e. NN /\ B e. NN ) /\ i e. ( ZZ>= ` 2 ) ) /\ ( i || A /\ i || B ) ) -> B e. ZZ )
17		dvdsgcd 12333	. . . . . . . . . . 11 |- ( ( i e. ZZ /\ A e. ZZ /\ B e. ZZ ) -> ( ( i || A /\ i || B ) -> i || ( A gcd B ) ) )
18	5, 15, 16, 17	syl3anc 1250	. . . . . . . . . 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 12155	. . . . . . . . . 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 8196	. . . . . . 7 |- ( ( ( ( A e. NN /\ B e. NN ) /\ i e. ( ZZ>= ` 2 ) ) /\ ( i || A /\ i || B ) ) -> 2 <_ ( A gcd B ) )
24	1, 23	eqbrtrrid 4080	. . . . . 6 |- ( ( ( ( A e. NN /\ B e. NN ) /\ i e. ( ZZ>= ` 2 ) ) /\ ( i || A /\ i || B ) ) -> ( 1 + 1 ) <_ ( A gcd B ) )
25		1nn 9047	. . . . . . . 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 9433	. . . . . . 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 9071	. . . . . 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 2623	. 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 9725	. . . . 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 9494	. . . . 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 9494	. . . . 5 |- ( ( ( A e. NN /\ B e. NN ) /\ ( A gcd B ) =/= 1 ) -> B e. ZZ )
43		gcddvds 12284	. . . . 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 4047	. . . . . 6 |- ( i = ( A gcd B ) -> ( i || A <-> ( A gcd B ) || A ) )
46		breq1 4047	. . . . . 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 2877	. . . 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 1373   e. wcel 2176   =/= wne 2376   E.wrex 2485   class class class wbr 4044   ` cfv 5271  (class class class)co 5944   RRcr 7924   1c1 7926   + caddc 7928   < clt 8107   <_ cle 8108   NNcn 9036   2c2 9087   ZZcz 9372   ZZ>=cuz 9648   || cdvds 12098   gcd cgcd 12274

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 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-coll 4159  ax-sep 4162  ax-nul 4170  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-iinf 4636  ax-cnex 8016  ax-resscn 8017  ax-1cn 8018  ax-1re 8019  ax-icn 8020  ax-addcl 8021  ax-addrcl 8022  ax-mulcl 8023  ax-mulrcl 8024  ax-addcom 8025  ax-mulcom 8026  ax-addass 8027  ax-mulass 8028  ax-distr 8029  ax-i2m1 8030  ax-0lt1 8031  ax-1rid 8032  ax-0id 8033  ax-rnegex 8034  ax-precex 8035  ax-cnre 8036  ax-pre-ltirr 8037  ax-pre-ltwlin 8038  ax-pre-lttrn 8039  ax-pre-apti 8040  ax-pre-ltadd 8041  ax-pre-mulgt0 8042  ax-pre-mulext 8043  ax-arch 8044  ax-caucvg 8045

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-reu 2491  df-rmo 2492  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-if 3572  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  df-br 4045  df-opab 4106  df-mpt 4107  df-tr 4143  df-id 4340  df-po 4343  df-iso 4344  df-iord 4413  df-on 4415  df-ilim 4416  df-suc 4418  df-iom 4639  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-riota 5899  df-ov 5947  df-oprab 5948  df-mpo 5949  df-1st 6226  df-2nd 6227  df-recs 6391  df-frec 6477  df-sup 7086  df-pnf 8109  df-mnf 8110  df-xr 8111  df-ltxr 8112  df-le 8113  df-sub 8245  df-neg 8246  df-reap 8648  df-ap 8655  df-div 8746  df-inn 9037  df-2 9095  df-3 9096  df-4 9097  df-n0 9296  df-z 9373  df-uz 9649  df-q 9741  df-rp 9776  df-fz 10131  df-fzo 10265  df-fl 10413  df-mod 10468  df-seqfrec 10593  df-exp 10684  df-cj 11153  df-re 11154  df-im 11155  df-rsqrt 11309  df-abs 11310  df-dvds 12099  df-gcd 12275

This theorem is referenced by:  ncoprmgcdgt1b  12412