Theorem coprmgcdb 12229

Description: Two positive integers are coprime, i.e. the only positive integer that divides both of them is 1, iff their greatest common divisor is 1. (Contributed by AV, 9-Aug-2020.)

Assertion

Ref	Expression
coprmgcdb	|- ( ( A e. NN /\ B e. NN ) -> ( A. i e. NN ( ( i || A /\ i || B ) -> i = 1 ) <-> ( A gcd B ) = 1 ) )

Distinct variable groups:   A, i   B, i

Proof of Theorem coprmgcdb

Step	Hyp	Ref	Expression
1		nnz 9339	. . . 4 |- ( A e. NN -> A e. ZZ )
2		nnz 9339	. . . 4 |- ( B e. NN -> B e. ZZ )
3		gcddvds 12103	. . . 4 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( ( A gcd B ) || A /\ ( A gcd B ) || B ) )
4	1, 2, 3	syl2an 289	. . 3 |- ( ( A e. NN /\ B e. NN ) -> ( ( A gcd B ) || A /\ ( A gcd B ) || B ) )
5		simpr 110	. . . 4 |- ( ( ( A e. NN /\ B e. NN ) /\ ( ( A gcd B ) || A /\ ( A gcd B ) || B ) ) -> ( ( A gcd B ) || A /\ ( A gcd B ) || B ) )
6		gcdnncl 12107	. . . . . 6 |- ( ( A e. NN /\ B e. NN ) -> ( A gcd B ) e. NN )
7	6	adantr 276	. . . . 5 |- ( ( ( A e. NN /\ B e. NN ) /\ ( ( A gcd B ) || A /\ ( A gcd B ) || B ) ) -> ( A gcd B ) e. NN )
8		breq1 4033	. . . . . . . 8 |- ( i = ( A gcd B ) -> ( i || A <-> ( A gcd B ) || A ) )
9		breq1 4033	. . . . . . . 8 |- ( i = ( A gcd B ) -> ( i || B <-> ( A gcd B ) || B ) )
10	8, 9	anbi12d 473	. . . . . . 7 |- ( i = ( A gcd B ) -> ( ( i || A /\ i || B ) <-> ( ( A gcd B ) || A /\ ( A gcd B ) || B ) ) )
11		eqeq1 2200	. . . . . . 7 |- ( i = ( A gcd B ) -> ( i = 1 <-> ( A gcd B ) = 1 ) )
12	10, 11	imbi12d 234	. . . . . 6 |- ( i = ( A gcd B ) -> ( ( ( i || A /\ i || B ) -> i = 1 ) <-> ( ( ( A gcd B ) || A /\ ( A gcd B ) || B ) -> ( A gcd B ) = 1 ) ) )
13	12	rspcv 2861	. . . . 5 |- ( ( A gcd B ) e. NN -> ( A. i e. NN ( ( i || A /\ i || B ) -> i = 1 ) -> ( ( ( A gcd B ) || A /\ ( A gcd B ) || B ) -> ( A gcd B ) = 1 ) ) )
14	7, 13	syl 14	. . . 4 |- ( ( ( A e. NN /\ B e. NN ) /\ ( ( A gcd B ) || A /\ ( A gcd B ) || B ) ) -> ( A. i e. NN ( ( i || A /\ i || B ) -> i = 1 ) -> ( ( ( A gcd B ) || A /\ ( A gcd B ) || B ) -> ( A gcd B ) = 1 ) ) )
15	5, 14	mpid 42	. . 3 |- ( ( ( A e. NN /\ B e. NN ) /\ ( ( A gcd B ) || A /\ ( A gcd B ) || B ) ) -> ( A. i e. NN ( ( i || A /\ i || B ) -> i = 1 ) -> ( A gcd B ) = 1 ) )
16	4, 15	mpdan 421	. 2 |- ( ( A e. NN /\ B e. NN ) -> ( A. i e. NN ( ( i || A /\ i || B ) -> i = 1 ) -> ( A gcd B ) = 1 ) )
17		simpl 109	. . . . . . . . 9 |- ( ( ( A e. NN /\ B e. NN ) /\ ( A gcd B ) = 1 ) -> ( A e. NN /\ B e. NN ) )
18	17	anim1i 340	. . . . . . . 8 |- ( ( ( ( A e. NN /\ B e. NN ) /\ ( A gcd B ) = 1 ) /\ i e. NN ) -> ( ( A e. NN /\ B e. NN ) /\ i e. NN ) )
19	18	ancomd 267	. . . . . . 7 |- ( ( ( ( A e. NN /\ B e. NN ) /\ ( A gcd B ) = 1 ) /\ i e. NN ) -> ( i e. NN /\ ( A e. NN /\ B e. NN ) ) )
20		3anass 984	. . . . . . 7 |- ( ( i e. NN /\ A e. NN /\ B e. NN ) <-> ( i e. NN /\ ( A e. NN /\ B e. NN ) ) )
21	19, 20	sylibr 134	. . . . . 6 |- ( ( ( ( A e. NN /\ B e. NN ) /\ ( A gcd B ) = 1 ) /\ i e. NN ) -> ( i e. NN /\ A e. NN /\ B e. NN ) )
22		nndvdslegcd 12105	. . . . . 6 |- ( ( i e. NN /\ A e. NN /\ B e. NN ) -> ( ( i || A /\ i || B ) -> i <_ ( A gcd B ) ) )
23	21, 22	syl 14	. . . . 5 |- ( ( ( ( A e. NN /\ B e. NN ) /\ ( A gcd B ) = 1 ) /\ i e. NN ) -> ( ( i || A /\ i || B ) -> i <_ ( A gcd B ) ) )
24		breq2 4034	. . . . . . . 8 |- ( ( A gcd B ) = 1 -> ( i <_ ( A gcd B ) <-> i <_ 1 ) )
25	24	adantr 276	. . . . . . 7 |- ( ( ( A gcd B ) = 1 /\ i e. NN ) -> ( i <_ ( A gcd B ) <-> i <_ 1 ) )
26		nnge1 9007	. . . . . . . . 9 |- ( i e. NN -> 1 <_ i )
27		nnre 8991	. . . . . . . . . . 11 |- ( i e. NN -> i e. RR )
28		1red 8036	. . . . . . . . . . 11 |- ( i e. NN -> 1 e. RR )
29	27, 28	letri3d 8137	. . . . . . . . . 10 |- ( i e. NN -> ( i = 1 <-> ( i <_ 1 /\ 1 <_ i ) ) )
30	29	biimprd 158	. . . . . . . . 9 |- ( i e. NN -> ( ( i <_ 1 /\ 1 <_ i ) -> i = 1 ) )
31	26, 30	mpan2d 428	. . . . . . . 8 |- ( i e. NN -> ( i <_ 1 -> i = 1 ) )
32	31	adantl 277	. . . . . . 7 |- ( ( ( A gcd B ) = 1 /\ i e. NN ) -> ( i <_ 1 -> i = 1 ) )
33	25, 32	sylbid 150	. . . . . 6 |- ( ( ( A gcd B ) = 1 /\ i e. NN ) -> ( i <_ ( A gcd B ) -> i = 1 ) )
34	33	adantll 476	. . . . 5 |- ( ( ( ( A e. NN /\ B e. NN ) /\ ( A gcd B ) = 1 ) /\ i e. NN ) -> ( i <_ ( A gcd B ) -> i = 1 ) )
35	23, 34	syld 45	. . . 4 |- ( ( ( ( A e. NN /\ B e. NN ) /\ ( A gcd B ) = 1 ) /\ i e. NN ) -> ( ( i || A /\ i || B ) -> i = 1 ) )
36	35	ralrimiva 2567	. . 3 |- ( ( ( A e. NN /\ B e. NN ) /\ ( A gcd B ) = 1 ) -> A. i e. NN ( ( i || A /\ i || B ) -> i = 1 ) )
37	36	ex 115	. 2 |- ( ( A e. NN /\ B e. NN ) -> ( ( A gcd B ) = 1 -> A. i e. NN ( ( i || A /\ i || B ) -> i = 1 ) ) )
38	16, 37	impbid 129	1 |- ( ( A e. NN /\ B e. NN ) -> ( A. i e. NN ( ( i || A /\ i || B ) -> i = 1 ) <-> ( A gcd B ) = 1 ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 980   = wceq 1364   e. wcel 2164   A.wral 2472   class class class wbr 4030  (class class class)co 5919   1c1 7875   <_ cle 8057   NNcn 8984   ZZcz 9320   || cdvds 11933   gcd cgcd 12082

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4145  ax-sep 4148  ax-nul 4156  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-iinf 4621  ax-cnex 7965  ax-resscn 7966  ax-1cn 7967  ax-1re 7968  ax-icn 7969  ax-addcl 7970  ax-addrcl 7971  ax-mulcl 7972  ax-mulrcl 7973  ax-addcom 7974  ax-mulcom 7975  ax-addass 7976  ax-mulass 7977  ax-distr 7978  ax-i2m1 7979  ax-0lt1 7980  ax-1rid 7981  ax-0id 7982  ax-rnegex 7983  ax-precex 7984  ax-cnre 7985  ax-pre-ltirr 7986  ax-pre-ltwlin 7987  ax-pre-lttrn 7988  ax-pre-apti 7989  ax-pre-ltadd 7990  ax-pre-mulgt0 7991  ax-pre-mulext 7992  ax-arch 7993  ax-caucvg 7994

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-if 3559  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-tr 4129  df-id 4325  df-po 4328  df-iso 4329  df-iord 4398  df-on 4400  df-ilim 4401  df-suc 4403  df-iom 4624  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-riota 5874  df-ov 5922  df-oprab 5923  df-mpo 5924  df-1st 6195  df-2nd 6196  df-recs 6360  df-frec 6446  df-sup 7045  df-pnf 8058  df-mnf 8059  df-xr 8060  df-ltxr 8061  df-le 8062  df-sub 8194  df-neg 8195  df-reap 8596  df-ap 8603  df-div 8694  df-inn 8985  df-2 9043  df-3 9044  df-4 9045  df-n0 9244  df-z 9321  df-uz 9596  df-q 9688  df-rp 9723  df-fz 10078  df-fzo 10212  df-fl 10342  df-mod 10397  df-seqfrec 10522  df-exp 10613  df-cj 10989  df-re 10990  df-im 10991  df-rsqrt 11145  df-abs 11146  df-dvds 11934  df-gcd 12083

This theorem is referenced by:  coprmdvds1  12232