Theorem coprmdvds 12046

Description: Euclid's Lemma (see ProofWiki "Euclid's Lemma", 10-Jul-2021, https://proofwiki.org/wiki/Euclid's_Lemma): If an integer divides the product of two integers and is coprime to one of them, then it divides the other. See also theorem 1.5 in [ApostolNT] p. 16. (Contributed by Paul Chapman, 22-Jun-2011.) (Proof shortened by AV, 10-Jul-2021.)

Assertion

Ref	Expression
coprmdvds	|- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( K || ( M x. N ) /\ ( K gcd M ) = 1 ) -> K || N ) )

Proof of Theorem coprmdvds

Step	Hyp	Ref	Expression
1		zcn 9217	. . . . . . . . . . 11 |- ( M e. ZZ -> M e. CC )
2		zcn 9217	. . . . . . . . . . 11 |- ( N e. ZZ -> N e. CC )
3		mulcom 7903	. . . . . . . . . . 11 |- ( ( M e. CC /\ N e. CC ) -> ( M x. N ) = ( N x. M ) )
4	1, 2, 3	syl2an 287	. . . . . . . . . 10 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M x. N ) = ( N x. M ) )
5	4	breq2d 4001	. . . . . . . . 9 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( K || ( M x. N ) <-> K || ( N x. M ) ) )
6		dvdsmulgcd 11980	. . . . . . . . . 10 |- ( ( N e. ZZ /\ M e. ZZ ) -> ( K || ( N x. M ) <-> K || ( N x. ( M gcd K ) ) ) )
7	6	ancoms 266	. . . . . . . . 9 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( K || ( N x. M ) <-> K || ( N x. ( M gcd K ) ) ) )
8	5, 7	bitrd 187	. . . . . . . 8 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( K || ( M x. N ) <-> K || ( N x. ( M gcd K ) ) ) )
9	8	3adant1 1010	. . . . . . 7 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( K || ( M x. N ) <-> K || ( N x. ( M gcd K ) ) ) )
10	9	adantr 274	. . . . . 6 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( K gcd M ) = 1 ) -> ( K || ( M x. N ) <-> K || ( N x. ( M gcd K ) ) ) )
11		gcdcom 11928	. . . . . . . . . . . 12 |- ( ( K e. ZZ /\ M e. ZZ ) -> ( K gcd M ) = ( M gcd K ) )
12	11	3adant3 1012	. . . . . . . . . . 11 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( K gcd M ) = ( M gcd K ) )
13	12	eqeq1d 2179	. . . . . . . . . 10 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( K gcd M ) = 1 <-> ( M gcd K ) = 1 ) )
14		oveq2 5861	. . . . . . . . . 10 |- ( ( M gcd K ) = 1 -> ( N x. ( M gcd K ) ) = ( N x. 1 ) )
15	13, 14	syl6bi 162	. . . . . . . . 9 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( K gcd M ) = 1 -> ( N x. ( M gcd K ) ) = ( N x. 1 ) ) )
16	15	imp 123	. . . . . . . 8 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( K gcd M ) = 1 ) -> ( N x. ( M gcd K ) ) = ( N x. 1 ) )
17	2	mulid1d 7937	. . . . . . . . . 10 |- ( N e. ZZ -> ( N x. 1 ) = N )
18	17	3ad2ant3 1015	. . . . . . . . 9 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( N x. 1 ) = N )
19	18	adantr 274	. . . . . . . 8 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( K gcd M ) = 1 ) -> ( N x. 1 ) = N )
20	16, 19	eqtrd 2203	. . . . . . 7 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( K gcd M ) = 1 ) -> ( N x. ( M gcd K ) ) = N )
21	20	breq2d 4001	. . . . . 6 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( K gcd M ) = 1 ) -> ( K || ( N x. ( M gcd K ) ) <-> K || N ) )
22	10, 21	bitrd 187	. . . . 5 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( K gcd M ) = 1 ) -> ( K || ( M x. N ) <-> K || N ) )
23	22	biimpd 143	. . . 4 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( K gcd M ) = 1 ) -> ( K || ( M x. N ) -> K || N ) )
24	23	ex 114	. . 3 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( K gcd M ) = 1 -> ( K || ( M x. N ) -> K || N ) ) )
25	24	com23 78	. 2 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( K || ( M x. N ) -> ( ( K gcd M ) = 1 -> K || N ) ) )
26	25	impd 252	1 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( K || ( M x. N ) /\ ( K gcd M ) = 1 ) -> K || N ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 973   = wceq 1348   e. wcel 2141   class class class wbr 3989  (class class class)co 5853   CCcc 7772   1c1 7775   x. cmul 7779   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:  coprmdvds2  12047  qredeq  12050  cncongr1  12057  euclemma  12100  eulerthlemh  12185  eulerthlemth  12186  prmdiveq  12190  prmpwdvds  12307  2sqlem8  13753