Theorem coprmgcdb 12042

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 9231	. . . 4 |- ( A e. NN -> A e. ZZ )
2		nnz 9231	. . . 4 |- ( B e. NN -> B e. ZZ )
3		gcddvds 11918	. . . 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 11922	. . . . . 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 3992	. . . . . . . 8 |- ( i = ( A gcd B ) -> ( i || A <-> ( A gcd B ) || A ) )
9		breq1 3992	. . . . . . . 8 |- ( i = ( A gcd B ) -> ( i || B <-> ( A gcd B ) || B ) )
10	8, 9	anbi12d 470	. . . . . . 7 |- ( i = ( A gcd B ) -> ( ( i || A /\ i || B ) <-> ( ( A gcd B ) || A /\ ( A gcd B ) || B ) ) )
11		eqeq1 2177	. . . . . . 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 2830	. . . . 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 419	. 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 977	. . . . . . 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 11920	. . . . . 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 3993	. . . . . . . 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 8901	. . . . . . . . 9 |- ( i e. NN -> 1 <_ i )
27		nnre 8885	. . . . . . . . . . 11 |- ( i e. NN -> i e. RR )
28		1red 7935	. . . . . . . . . . 11 |- ( i e. NN -> 1 e. RR )
29	27, 28	letri3d 8035	. . . . . . . . . 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 426	. . . . . . . 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 473	. . . . 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 2543	. . 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 973   = wceq 1348   e. wcel 2141   A.wral 2448   class class class wbr 3989  (class class class)co 5853   1c1 7775   <_ cle 7955   NNcn 8878   ZZcz 9212   || cdvds 11749   gcd cgcd 11897

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891  ax-pre-mulext 7892  ax-arch 7893  ax-caucvg 7894

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-ilim 4354  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-frec 6370  df-sup 6961  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501  df-div 8590  df-inn 8879  df-2 8937  df-3 8938  df-4 8939  df-n0 9136  df-z 9213  df-uz 9488  df-q 9579  df-rp 9611  df-fz 9966  df-fzo 10099  df-fl 10226  df-mod 10279  df-seqfrec 10402  df-exp 10476  df-cj 10806  df-re 10807  df-im 10808  df-rsqrt 10962  df-abs 10963  df-dvds 11750  df-gcd 11898

This theorem is referenced by:  coprmdvds1  12045