Theorem coprmgcdb 12020

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 9210	. . . 4 |- ( A e. NN -> A e. ZZ )
2		nnz 9210	. . . 4 |- ( B e. NN -> B e. ZZ )
3		gcddvds 11896	. . . 4 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( ( A gcd B ) || A /\ ( A gcd B ) || B ) )
4	1, 2, 3	syl2an 287	. . 3 |- ( ( A e. NN /\ B e. NN ) -> ( ( A gcd B ) || A /\ ( A gcd B ) || B ) )
5		simpr 109	. . . 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 11900	. . . . . 6 |- ( ( A e. NN /\ B e. NN ) -> ( A gcd B ) e. NN )
7	6	adantr 274	. . . . 5 |- ( ( ( A e. NN /\ B e. NN ) /\ ( ( A gcd B ) || A /\ ( A gcd B ) || B ) ) -> ( A gcd B ) e. NN )
8		breq1 3985	. . . . . . . 8 |- ( i = ( A gcd B ) -> ( i || A <-> ( A gcd B ) || A ) )
9		breq1 3985	. . . . . . . 8 |- ( i = ( A gcd B ) -> ( i || B <-> ( A gcd B ) || B ) )
10	8, 9	anbi12d 465	. . . . . . 7 |- ( i = ( A gcd B ) -> ( ( i || A /\ i || B ) <-> ( ( A gcd B ) || A /\ ( A gcd B ) || B ) ) )
11		eqeq1 2172	. . . . . . 7 |- ( i = ( A gcd B ) -> ( i = 1 <-> ( A gcd B ) = 1 ) )
12	10, 11	imbi12d 233	. . . . . 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 2826	. . . . 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 418	. 2 |- ( ( A e. NN /\ B e. NN ) -> ( A. i e. NN ( ( i || A /\ i || B ) -> i = 1 ) -> ( A gcd B ) = 1 ) )
17		simpl 108	. . . . . . . . 9 |- ( ( ( A e. NN /\ B e. NN ) /\ ( A gcd B ) = 1 ) -> ( A e. NN /\ B e. NN ) )
18	17	anim1i 338	. . . . . . . 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 265	. . . . . . 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 972	. . . . . . 7 |- ( ( i e. NN /\ A e. NN /\ B e. NN ) <-> ( i e. NN /\ ( A e. NN /\ B e. NN ) ) )
21	19, 20	sylibr 133	. . . . . 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 11898	. . . . . 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 3986	. . . . . . . 8 |- ( ( A gcd B ) = 1 -> ( i <_ ( A gcd B ) <-> i <_ 1 ) )
25	24	adantr 274	. . . . . . 7 |- ( ( ( A gcd B ) = 1 /\ i e. NN ) -> ( i <_ ( A gcd B ) <-> i <_ 1 ) )
26		nnge1 8880	. . . . . . . . 9 |- ( i e. NN -> 1 <_ i )
27		nnre 8864	. . . . . . . . . . 11 |- ( i e. NN -> i e. RR )
28		1red 7914	. . . . . . . . . . 11 |- ( i e. NN -> 1 e. RR )
29	27, 28	letri3d 8014	. . . . . . . . . 10 |- ( i e. NN -> ( i = 1 <-> ( i <_ 1 /\ 1 <_ i ) ) )
30	29	biimprd 157	. . . . . . . . 9 |- ( i e. NN -> ( ( i <_ 1 /\ 1 <_ i ) -> i = 1 ) )
31	26, 30	mpan2d 425	. . . . . . . 8 |- ( i e. NN -> ( i <_ 1 -> i = 1 ) )
32	31	adantl 275	. . . . . . 7 |- ( ( ( A gcd B ) = 1 /\ i e. NN ) -> ( i <_ 1 -> i = 1 ) )
33	25, 32	sylbid 149	. . . . . 6 |- ( ( ( A gcd B ) = 1 /\ i e. NN ) -> ( i <_ ( A gcd B ) -> i = 1 ) )
34	33	adantll 468	. . . . 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 2539	. . 3 |- ( ( ( A e. NN /\ B e. NN ) /\ ( A gcd B ) = 1 ) -> A. i e. NN ( ( i || A /\ i || B ) -> i = 1 ) )
37	36	ex 114	. 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 128	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 103   <-> wb 104   /\ w3a 968   = wceq 1343   e. wcel 2136   A.wral 2444   class class class wbr 3982  (class class class)co 5842   1c1 7754   <_ cle 7934   NNcn 8857   ZZcz 9191   || cdvds 11727   gcd cgcd 11875

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-mulrcl 7852  ax-addcom 7853  ax-mulcom 7854  ax-addass 7855  ax-mulass 7856  ax-distr 7857  ax-i2m1 7858  ax-0lt1 7859  ax-1rid 7860  ax-0id 7861  ax-rnegex 7862  ax-precex 7863  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-ltwlin 7866  ax-pre-lttrn 7867  ax-pre-apti 7868  ax-pre-ltadd 7869  ax-pre-mulgt0 7870  ax-pre-mulext 7871  ax-arch 7872  ax-caucvg 7873

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rmo 2452  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-if 3521  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-id 4271  df-po 4274  df-iso 4275  df-iord 4344  df-on 4346  df-ilim 4347  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-recs 6273  df-frec 6359  df-sup 6949  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-sub 8071  df-neg 8072  df-reap 8473  df-ap 8480  df-div 8569  df-inn 8858  df-2 8916  df-3 8917  df-4 8918  df-n0 9115  df-z 9192  df-uz 9467  df-q 9558  df-rp 9590  df-fz 9945  df-fzo 10078  df-fl 10205  df-mod 10258  df-seqfrec 10381  df-exp 10455  df-cj 10784  df-re 10785  df-im 10786  df-rsqrt 10940  df-abs 10941  df-dvds 11728  df-gcd 11876

This theorem is referenced by:  coprmdvds1  12023