Theorem coprmgcdb 12610

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 9465	. . . 4 |- ( A e. NN -> A e. ZZ )
2		nnz 9465	. . . 4 |- ( B e. NN -> B e. ZZ )
3		gcddvds 12484	. . . 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 12488	. . . . . 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 4086	. . . . . . . 8 |- ( i = ( A gcd B ) -> ( i || A <-> ( A gcd B ) || A ) )
9		breq1 4086	. . . . . . . 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 2236	. . . . . . 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 2903	. . . . 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 1006	. . . . . . 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 12486	. . . . . 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 4087	. . . . . . . 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 9133	. . . . . . . . 9 |- ( i e. NN -> 1 <_ i )
27		nnre 9117	. . . . . . . . . . 11 |- ( i e. NN -> i e. RR )
28		1red 8161	. . . . . . . . . . 11 |- ( i e. NN -> 1 e. RR )
29	27, 28	letri3d 8262	. . . . . . . . . 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 2603	. . 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 1002   = wceq 1395   e. wcel 2200   A.wral 2508   class class class wbr 4083  (class class class)co 6001   1c1 8000   <_ cle 8182   NNcn 9110   ZZcz 9446   || cdvds 12298   gcd cgcd 12474

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680  ax-cnex 8090  ax-resscn 8091  ax-1cn 8092  ax-1re 8093  ax-icn 8094  ax-addcl 8095  ax-addrcl 8096  ax-mulcl 8097  ax-mulrcl 8098  ax-addcom 8099  ax-mulcom 8100  ax-addass 8101  ax-mulass 8102  ax-distr 8103  ax-i2m1 8104  ax-0lt1 8105  ax-1rid 8106  ax-0id 8107  ax-rnegex 8108  ax-precex 8109  ax-cnre 8110  ax-pre-ltirr 8111  ax-pre-ltwlin 8112  ax-pre-lttrn 8113  ax-pre-apti 8114  ax-pre-ltadd 8115  ax-pre-mulgt0 8116  ax-pre-mulext 8117  ax-arch 8118  ax-caucvg 8119

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-po 4387  df-iso 4388  df-iord 4457  df-on 4459  df-ilim 4460  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-riota 5954  df-ov 6004  df-oprab 6005  df-mpo 6006  df-1st 6286  df-2nd 6287  df-recs 6451  df-frec 6537  df-sup 7151  df-pnf 8183  df-mnf 8184  df-xr 8185  df-ltxr 8186  df-le 8187  df-sub 8319  df-neg 8320  df-reap 8722  df-ap 8729  df-div 8820  df-inn 9111  df-2 9169  df-3 9170  df-4 9171  df-n0 9370  df-z 9447  df-uz 9723  df-q 9815  df-rp 9850  df-fz 10205  df-fzo 10339  df-fl 10490  df-mod 10545  df-seqfrec 10670  df-exp 10761  df-cj 11353  df-re 11354  df-im 11355  df-rsqrt 11509  df-abs 11510  df-dvds 12299  df-gcd 12475

This theorem is referenced by:  coprmdvds1  12613