Theorem coprmgcdb 10470

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 8370	. . . 4 |- ( A e. NN -> A e. ZZ )
2		nnz 8370	. . . 4 |- ( B e. NN -> B e. ZZ )
3		gcddvds 10355	. . . 4 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( ( A gcd B ) || A /\ ( A gcd B ) || B ) )
4	1, 2, 3	syl2an 283	. . 3 |- ( ( A e. NN /\ B e. NN ) -> ( ( A gcd B ) || A /\ ( A gcd B ) || B ) )
5		simpr 108	. . . 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 10359	. . . . . 6 |- ( ( A e. NN /\ B e. NN ) -> ( A gcd B ) e. NN )
7	6	adantr 270	. . . . 5 |- ( ( ( A e. NN /\ B e. NN ) /\ ( ( A gcd B ) || A /\ ( A gcd B ) || B ) ) -> ( A gcd B ) e. NN )
8		breq1 3788	. . . . . . . 8 |- ( i = ( A gcd B ) -> ( i || A <-> ( A gcd B ) || A ) )
9		breq1 3788	. . . . . . . 8 |- ( i = ( A gcd B ) -> ( i || B <-> ( A gcd B ) || B ) )
10	8, 9	anbi12d 456	. . . . . . 7 |- ( i = ( A gcd B ) -> ( ( i || A /\ i || B ) <-> ( ( A gcd B ) || A /\ ( A gcd B ) || B ) ) )
11		eqeq1 2087	. . . . . . 7 |- ( i = ( A gcd B ) -> ( i = 1 <-> ( A gcd B ) = 1 ) )
12	10, 11	imbi12d 232	. . . . . 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 2697	. . . . 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 41	. . 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 412	. 2 |- ( ( A e. NN /\ B e. NN ) -> ( A. i e. NN ( ( i || A /\ i || B ) -> i = 1 ) -> ( A gcd B ) = 1 ) )
17		simpl 107	. . . . . . . . 9 |- ( ( ( A e. NN /\ B e. NN ) /\ ( A gcd B ) = 1 ) -> ( A e. NN /\ B e. NN ) )
18	17	anim1i 333	. . . . . . . 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 263	. . . . . . 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 923	. . . . . . 7 |- ( ( i e. NN /\ A e. NN /\ B e. NN ) <-> ( i e. NN /\ ( A e. NN /\ B e. NN ) ) )
21	19, 20	sylibr 132	. . . . . 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 10357	. . . . . 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 3789	. . . . . . . 8 |- ( ( A gcd B ) = 1 -> ( i <_ ( A gcd B ) <-> i <_ 1 ) )
25	24	adantr 270	. . . . . . 7 |- ( ( ( A gcd B ) = 1 /\ i e. NN ) -> ( i <_ ( A gcd B ) <-> i <_ 1 ) )
26		nnge1 8062	. . . . . . . . 9 |- ( i e. NN -> 1 <_ i )
27		nnre 8046	. . . . . . . . . . 11 |- ( i e. NN -> i e. RR )
28		1red 7134	. . . . . . . . . . 11 |- ( i e. NN -> 1 e. RR )
29	27, 28	letri3d 7226	. . . . . . . . . 10 |- ( i e. NN -> ( i = 1 <-> ( i <_ 1 /\ 1 <_ i ) ) )
30	29	biimprd 156	. . . . . . . . 9 |- ( i e. NN -> ( ( i <_ 1 /\ 1 <_ i ) -> i = 1 ) )
31	26, 30	mpan2d 418	. . . . . . . 8 |- ( i e. NN -> ( i <_ 1 -> i = 1 ) )
32	31	adantl 271	. . . . . . 7 |- ( ( ( A gcd B ) = 1 /\ i e. NN ) -> ( i <_ 1 -> i = 1 ) )
33	25, 32	sylbid 148	. . . . . 6 |- ( ( ( A gcd B ) = 1 /\ i e. NN ) -> ( i <_ ( A gcd B ) -> i = 1 ) )
34	33	adantll 459	. . . . 5 |- ( ( ( ( A e. NN /\ B e. NN ) /\ ( A gcd B ) = 1 ) /\ i e. NN ) -> ( i <_ ( A gcd B ) -> i = 1 ) )
35	23, 34	syld 44	. . . 4 |- ( ( ( ( A e. NN /\ B e. NN ) /\ ( A gcd B ) = 1 ) /\ i e. NN ) -> ( ( i || A /\ i || B ) -> i = 1 ) )
36	35	ralrimiva 2434	. . 3 |- ( ( ( A e. NN /\ B e. NN ) /\ ( A gcd B ) = 1 ) -> A. i e. NN ( ( i || A /\ i || B ) -> i = 1 ) )
37	36	ex 113	. 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 127	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 102   <-> wb 103   /\ w3a 919   = wceq 1284   e. wcel 1433   A.wral 2348   class class class wbr 3785  (class class class)co 5532   1c1 6982   <_ cle 7154   NNcn 8039   ZZcz 8351   || cdvds 10195   gcd cgcd 10338

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-coll 3893  ax-sep 3896  ax-nul 3904  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-iinf 4329  ax-cnex 7067  ax-resscn 7068  ax-1cn 7069  ax-1re 7070  ax-icn 7071  ax-addcl 7072  ax-addrcl 7073  ax-mulcl 7074  ax-mulrcl 7075  ax-addcom 7076  ax-mulcom 7077  ax-addass 7078  ax-mulass 7079  ax-distr 7080  ax-i2m1 7081  ax-0lt1 7082  ax-1rid 7083  ax-0id 7084  ax-rnegex 7085  ax-precex 7086  ax-cnre 7087  ax-pre-ltirr 7088  ax-pre-ltwlin 7089  ax-pre-lttrn 7090  ax-pre-apti 7091  ax-pre-ltadd 7092  ax-pre-mulgt0 7093  ax-pre-mulext 7094  ax-arch 7095  ax-caucvg 7096

This theorem depends on definitions:  df-bi 115  df-dc 776  df-3or 920  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-nel 2340  df-ral 2353  df-rex 2354  df-reu 2355  df-rmo 2356  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-if 3352  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-int 3637  df-iun 3680  df-br 3786  df-opab 3840  df-mpt 3841  df-tr 3876  df-id 4048  df-po 4051  df-iso 4052  df-iord 4121  df-on 4123  df-suc 4126  df-iom 4332  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930  df-riota 5488  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-1st 5787  df-2nd 5788  df-recs 5943  df-frec 6001  df-sup 6397  df-pnf 7155  df-mnf 7156  df-xr 7157  df-ltxr 7158  df-le 7159  df-sub 7281  df-neg 7282  df-reap 7675  df-ap 7682  df-div 7761  df-inn 8040  df-2 8098  df-3 8099  df-4 8100  df-n0 8289  df-z 8352  df-uz 8620  df-q 8705  df-rp 8735  df-fz 9030  df-fzo 9153  df-fl 9274  df-mod 9325  df-iseq 9432  df-iexp 9476  df-cj 9729  df-re 9730  df-im 9731  df-rsqrt 9884  df-abs 9885  df-dvds 10196  df-gcd 10339

This theorem is referenced by:  coprmdvds1  10473