Theorem coprmdvds 12230

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 9322	. . . . . . . . . . 11 |- ( M e. ZZ -> M e. CC )
2		zcn 9322	. . . . . . . . . . 11 |- ( N e. ZZ -> N e. CC )
3		mulcom 8001	. . . . . . . . . . 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 4041	. . . . . . . . 9 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( K || ( M x. N ) <-> K || ( N x. M ) ) )
6		dvdsmulgcd 12162	. . . . . . . . . 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 1017	. . . . . . 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 12110	. . . . . . . . . . . 12 |- ( ( K e. ZZ /\ M e. ZZ ) -> ( K gcd M ) = ( M gcd K ) )
12	11	3adant3 1019	. . . . . . . . . . 11 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( K gcd M ) = ( M gcd K ) )
13	12	eqeq1d 2202	. . . . . . . . . 10 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( K gcd M ) = 1 <-> ( M gcd K ) = 1 ) )
14		oveq2 5926	. . . . . . . . . 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 8036	. . . . . . . . . 10 |- ( N e. ZZ -> ( N x. 1 ) = N )
18	17	3ad2ant3 1022	. . . . . . . . 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 2226	. . . . . . 7 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( K gcd M ) = 1 ) -> ( N x. ( M gcd K ) ) = N )
21	20	breq2d 4041	. . . . . 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 980   = wceq 1364   e. wcel 2164   class class class wbr 4029  (class class class)co 5918   CCcc 7870   1c1 7873   x. cmul 7877   ZZcz 9317   || cdvds 11930   gcd cgcd 12079

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-mulrcl 7971  ax-addcom 7972  ax-mulcom 7973  ax-addass 7974  ax-mulass 7975  ax-distr 7976  ax-i2m1 7977  ax-0lt1 7978  ax-1rid 7979  ax-0id 7980  ax-rnegex 7981  ax-precex 7982  ax-cnre 7983  ax-pre-ltirr 7984  ax-pre-ltwlin 7985  ax-pre-lttrn 7986  ax-pre-apti 7987  ax-pre-ltadd 7988  ax-pre-mulgt0 7989  ax-pre-mulext 7990  ax-arch 7991  ax-caucvg 7992

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-if 3558  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-id 4324  df-po 4327  df-iso 4328  df-iord 4397  df-on 4399  df-ilim 4400  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-recs 6358  df-frec 6444  df-sup 7043  df-pnf 8056  df-mnf 8057  df-xr 8058  df-ltxr 8059  df-le 8060  df-sub 8192  df-neg 8193  df-reap 8594  df-ap 8601  df-div 8692  df-inn 8983  df-2 9041  df-3 9042  df-4 9043  df-n0 9241  df-z 9318  df-uz 9593  df-q 9685  df-rp 9720  df-fz 10075  df-fzo 10209  df-fl 10339  df-mod 10394  df-seqfrec 10519  df-exp 10610  df-cj 10986  df-re 10987  df-im 10988  df-rsqrt 11142  df-abs 11143  df-dvds 11931  df-gcd 12080

This theorem is referenced by:  coprmdvds2  12231  qredeq  12234  cncongr1  12241  euclemma  12284  eulerthlemh  12369  eulerthlemth  12370  prmdiveq  12374  prmpwdvds  12493  lgseisenlem1  15186  lgseisenlem2  15187  2sqlem8  15210