Theorem coprmdvds 12744

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 9545	. . . . . . . . . . 11 |- ( M e. ZZ -> M e. CC )
2		zcn 9545	. . . . . . . . . . 11 |- ( N e. ZZ -> N e. CC )
3		mulcom 8221	. . . . . . . . . . 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 4105	. . . . . . . . 9 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( K || ( M x. N ) <-> K || ( N x. M ) ) )
6		dvdsmulgcd 12676	. . . . . . . . . 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 1042	. . . . . . 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 12624	. . . . . . . . . . . 12 |- ( ( K e. ZZ /\ M e. ZZ ) -> ( K gcd M ) = ( M gcd K ) )
12	11	3adant3 1044	. . . . . . . . . . 11 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( K gcd M ) = ( M gcd K ) )
13	12	eqeq1d 2240	. . . . . . . . . 10 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( K gcd M ) = 1 <-> ( M gcd K ) = 1 ) )
14		oveq2 6036	. . . . . . . . . 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 8256	. . . . . . . . . 10 |- ( N e. ZZ -> ( N x. 1 ) = N )
18	17	3ad2ant3 1047	. . . . . . . . 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 2264	. . . . . . 7 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( K gcd M ) = 1 ) -> ( N x. ( M gcd K ) ) = N )
21	20	breq2d 4105	. . . . . 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 1005   = wceq 1398   e. wcel 2202   class class class wbr 4093  (class class class)co 6028   CCcc 8090   1c1 8093   x. cmul 8097   ZZcz 9540   || cdvds 12428   gcd cgcd 12604

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692  ax-cnex 8183  ax-resscn 8184  ax-1cn 8185  ax-1re 8186  ax-icn 8187  ax-addcl 8188  ax-addrcl 8189  ax-mulcl 8190  ax-mulrcl 8191  ax-addcom 8192  ax-mulcom 8193  ax-addass 8194  ax-mulass 8195  ax-distr 8196  ax-i2m1 8197  ax-0lt1 8198  ax-1rid 8199  ax-0id 8200  ax-rnegex 8201  ax-precex 8202  ax-cnre 8203  ax-pre-ltirr 8204  ax-pre-ltwlin 8205  ax-pre-lttrn 8206  ax-pre-apti 8207  ax-pre-ltadd 8208  ax-pre-mulgt0 8209  ax-pre-mulext 8210  ax-arch 8211  ax-caucvg 8212

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-po 4399  df-iso 4400  df-iord 4469  df-on 4471  df-ilim 4472  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-frec 6600  df-sup 7243  df-pnf 8275  df-mnf 8276  df-xr 8277  df-ltxr 8278  df-le 8279  df-sub 8411  df-neg 8412  df-reap 8814  df-ap 8821  df-div 8912  df-inn 9203  df-2 9261  df-3 9262  df-4 9263  df-n0 9462  df-z 9541  df-uz 9817  df-q 9915  df-rp 9950  df-fz 10306  df-fzo 10440  df-fl 10593  df-mod 10648  df-seqfrec 10773  df-exp 10864  df-cj 11482  df-re 11483  df-im 11484  df-rsqrt 11638  df-abs 11639  df-dvds 12429  df-gcd 12605

This theorem is referenced by:  coprmdvds2  12745  qredeq  12748  cncongr1  12755  euclemma  12798  eulerthlemh  12883  eulerthlemth  12884  prmdiveq  12888  prmpwdvds  13008  mpodvdsmulf1o  15804  perfectlem1  15813  lgseisenlem1  15889  lgseisenlem2  15890  lgsquadlem2  15897  lgsquadlem3  15898  2sqlem8  15942