Theorem coprmdvds 12105

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 9271	. . . . . . . . . . 11 |- ( M e. ZZ -> M e. CC )
2		zcn 9271	. . . . . . . . . . 11 |- ( N e. ZZ -> N e. CC )
3		mulcom 7953	. . . . . . . . . . 11 |- ( ( M e. CC /\ N e. CC ) -> ( M x. N ) = ( N x. M ) )
4	1, 2, 3	syl2an 289	. . . . . . . . . 10 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M x. N ) = ( N x. M ) )
5	4	breq2d 4027	. . . . . . . . 9 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( K || ( M x. N ) <-> K || ( N x. M ) ) )
6		dvdsmulgcd 12039	. . . . . . . . . 10 |- ( ( N e. ZZ /\ M e. ZZ ) -> ( K || ( N x. M ) <-> K || ( N x. ( M gcd K ) ) ) )
7	6	ancoms 268	. . . . . . . . 9 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( K || ( N x. M ) <-> K || ( N x. ( M gcd K ) ) ) )
8	5, 7	bitrd 188	. . . . . . . 8 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( K || ( M x. N ) <-> K || ( N x. ( M gcd K ) ) ) )
9	8	3adant1 1016	. . . . . . 7 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( K || ( M x. N ) <-> K || ( N x. ( M gcd K ) ) ) )
10	9	adantr 276	. . . . . 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 11987	. . . . . . . . . . . 12 |- ( ( K e. ZZ /\ M e. ZZ ) -> ( K gcd M ) = ( M gcd K ) )
12	11	3adant3 1018	. . . . . . . . . . 11 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( K gcd M ) = ( M gcd K ) )
13	12	eqeq1d 2196	. . . . . . . . . 10 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( K gcd M ) = 1 <-> ( M gcd K ) = 1 ) )
14		oveq2 5896	. . . . . . . . . 10 |- ( ( M gcd K ) = 1 -> ( N x. ( M gcd K ) ) = ( N x. 1 ) )
15	13, 14	biimtrdi 163	. . . . . . . . 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 124	. . . . . . . 8 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( K gcd M ) = 1 ) -> ( N x. ( M gcd K ) ) = ( N x. 1 ) )
17	2	mulridd 7987	. . . . . . . . . 10 |- ( N e. ZZ -> ( N x. 1 ) = N )
18	17	3ad2ant3 1021	. . . . . . . . 9 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( N x. 1 ) = N )
19	18	adantr 276	. . . . . . . 8 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( K gcd M ) = 1 ) -> ( N x. 1 ) = N )
20	16, 19	eqtrd 2220	. . . . . . 7 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( K gcd M ) = 1 ) -> ( N x. ( M gcd K ) ) = N )
21	20	breq2d 4027	. . . . . 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 188	. . . . 5 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( K gcd M ) = 1 ) -> ( K || ( M x. N ) <-> K || N ) )
23	22	biimpd 144	. . . 4 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( K gcd M ) = 1 ) -> ( K || ( M x. N ) -> K || N ) )
24	23	ex 115	. . 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 254	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 104   <-> wb 105   /\ w3a 979   = wceq 1363   e. wcel 2158   class class class wbr 4015  (class class class)co 5888   CCcc 7822   1c1 7825   x. cmul 7829   ZZcz 9266   || cdvds 11807   gcd cgcd 11956

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 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-coll 4130  ax-sep 4133  ax-nul 4141  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-iinf 4599  ax-cnex 7915  ax-resscn 7916  ax-1cn 7917  ax-1re 7918  ax-icn 7919  ax-addcl 7920  ax-addrcl 7921  ax-mulcl 7922  ax-mulrcl 7923  ax-addcom 7924  ax-mulcom 7925  ax-addass 7926  ax-mulass 7927  ax-distr 7928  ax-i2m1 7929  ax-0lt1 7930  ax-1rid 7931  ax-0id 7932  ax-rnegex 7933  ax-precex 7934  ax-cnre 7935  ax-pre-ltirr 7936  ax-pre-ltwlin 7937  ax-pre-lttrn 7938  ax-pre-apti 7939  ax-pre-ltadd 7940  ax-pre-mulgt0 7941  ax-pre-mulext 7942  ax-arch 7943  ax-caucvg 7944

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 980  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-nel 2453  df-ral 2470  df-rex 2471  df-reu 2472  df-rmo 2473  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-if 3547  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-tr 4114  df-id 4305  df-po 4308  df-iso 4309  df-iord 4378  df-on 4380  df-ilim 4381  df-suc 4383  df-iom 4602  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-riota 5844  df-ov 5891  df-oprab 5892  df-mpo 5893  df-1st 6154  df-2nd 6155  df-recs 6319  df-frec 6405  df-sup 6996  df-pnf 8007  df-mnf 8008  df-xr 8009  df-ltxr 8010  df-le 8011  df-sub 8143  df-neg 8144  df-reap 8545  df-ap 8552  df-div 8643  df-inn 8933  df-2 8991  df-3 8992  df-4 8993  df-n0 9190  df-z 9267  df-uz 9542  df-q 9633  df-rp 9667  df-fz 10022  df-fzo 10156  df-fl 10283  df-mod 10336  df-seqfrec 10459  df-exp 10533  df-cj 10864  df-re 10865  df-im 10866  df-rsqrt 11020  df-abs 11021  df-dvds 11808  df-gcd 11957

This theorem is referenced by:  coprmdvds2  12106  qredeq  12109  cncongr1  12116  euclemma  12159  eulerthlemh  12244  eulerthlemth  12245  prmdiveq  12249  prmpwdvds  12366  lgseisenlem1  14690  lgseisenlem2  14691  2sqlem8  14710