Theorem rpmulgcd 11714

Description: If K and M are relatively prime, then the GCD of K and M x. N is the GCD of K and N. (Contributed by Scott Fenton, 12-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Assertion

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

Proof of Theorem rpmulgcd

Step	Hyp	Ref	Expression
1		gcdmultiple 11708	. . . . . 6 |- ( ( K e. NN /\ N e. NN ) -> ( K gcd ( K x. N ) ) = K )
2	1	3adant2 1000	. . . . 5 |- ( ( K e. NN /\ M e. NN /\ N e. NN ) -> ( K gcd ( K x. N ) ) = K )
3	2	oveq1d 5789	. . . 4 |- ( ( K e. NN /\ M e. NN /\ N e. NN ) -> ( ( K gcd ( K x. N ) ) gcd ( M x. N ) ) = ( K gcd ( M x. N ) ) )
4		nnz 9073	. . . . . 6 |- ( K e. NN -> K e. ZZ )
5	4	3ad2ant1 1002	. . . . 5 |- ( ( K e. NN /\ M e. NN /\ N e. NN ) -> K e. ZZ )
6		nnz 9073	. . . . . . 7 |- ( N e. NN -> N e. ZZ )
7		zmulcl 9107	. . . . . . 7 |- ( ( K e. ZZ /\ N e. ZZ ) -> ( K x. N ) e. ZZ )
8	4, 6, 7	syl2an 287	. . . . . 6 |- ( ( K e. NN /\ N e. NN ) -> ( K x. N ) e. ZZ )
9	8	3adant2 1000	. . . . 5 |- ( ( K e. NN /\ M e. NN /\ N e. NN ) -> ( K x. N ) e. ZZ )
10		nnz 9073	. . . . . . 7 |- ( M e. NN -> M e. ZZ )
11		zmulcl 9107	. . . . . . 7 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M x. N ) e. ZZ )
12	10, 6, 11	syl2an 287	. . . . . 6 |- ( ( M e. NN /\ N e. NN ) -> ( M x. N ) e. ZZ )
13	12	3adant1 999	. . . . 5 |- ( ( K e. NN /\ M e. NN /\ N e. NN ) -> ( M x. N ) e. ZZ )
14		gcdass 11703	. . . . 5 |- ( ( K e. ZZ /\ ( K x. N ) e. ZZ /\ ( M x. N ) e. ZZ ) -> ( ( K gcd ( K x. N ) ) gcd ( M x. N ) ) = ( K gcd ( ( K x. N ) gcd ( M x. N ) ) ) )
15	5, 9, 13, 14	syl3anc 1216	. . . 4 |- ( ( K e. NN /\ M e. NN /\ N e. NN ) -> ( ( K gcd ( K x. N ) ) gcd ( M x. N ) ) = ( K gcd ( ( K x. N ) gcd ( M x. N ) ) ) )
16	3, 15	eqtr3d 2174	. . 3 |- ( ( K e. NN /\ M e. NN /\ N e. NN ) -> ( K gcd ( M x. N ) ) = ( K gcd ( ( K x. N ) gcd ( M x. N ) ) ) )
17	16	adantr 274	. 2 |- ( ( ( K e. NN /\ M e. NN /\ N e. NN ) /\ ( K gcd M ) = 1 ) -> ( K gcd ( M x. N ) ) = ( K gcd ( ( K x. N ) gcd ( M x. N ) ) ) )
18		nnnn0 8984	. . . . . 6 |- ( N e. NN -> N e. NN0 )
19		mulgcdr 11706	. . . . . 6 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. NN0 ) -> ( ( K x. N ) gcd ( M x. N ) ) = ( ( K gcd M ) x. N ) )
20	4, 10, 18, 19	syl3an 1258	. . . . 5 |- ( ( K e. NN /\ M e. NN /\ N e. NN ) -> ( ( K x. N ) gcd ( M x. N ) ) = ( ( K gcd M ) x. N ) )
21		oveq1 5781	. . . . 5 |- ( ( K gcd M ) = 1 -> ( ( K gcd M ) x. N ) = ( 1 x. N ) )
22	20, 21	sylan9eq 2192	. . . 4 |- ( ( ( K e. NN /\ M e. NN /\ N e. NN ) /\ ( K gcd M ) = 1 ) -> ( ( K x. N ) gcd ( M x. N ) ) = ( 1 x. N ) )
23		nncn 8728	. . . . . . 7 |- ( N e. NN -> N e. CC )
24	23	3ad2ant3 1004	. . . . . 6 |- ( ( K e. NN /\ M e. NN /\ N e. NN ) -> N e. CC )
25	24	adantr 274	. . . . 5 |- ( ( ( K e. NN /\ M e. NN /\ N e. NN ) /\ ( K gcd M ) = 1 ) -> N e. CC )
26	25	mulid2d 7784	. . . 4 |- ( ( ( K e. NN /\ M e. NN /\ N e. NN ) /\ ( K gcd M ) = 1 ) -> ( 1 x. N ) = N )
27	22, 26	eqtrd 2172	. . 3 |- ( ( ( K e. NN /\ M e. NN /\ N e. NN ) /\ ( K gcd M ) = 1 ) -> ( ( K x. N ) gcd ( M x. N ) ) = N )
28	27	oveq2d 5790	. 2 |- ( ( ( K e. NN /\ M e. NN /\ N e. NN ) /\ ( K gcd M ) = 1 ) -> ( K gcd ( ( K x. N ) gcd ( M x. N ) ) ) = ( K gcd N ) )
29	17, 28	eqtrd 2172	1 |- ( ( ( K e. NN /\ M e. NN /\ N e. NN ) /\ ( K gcd M ) = 1 ) -> ( K gcd ( M x. N ) ) = ( K gcd N ) )

Syntax hints:   -> wi 4   /\ wa 103   /\ w3a 962   = wceq 1331   e. wcel 1480  (class class class)co 5774   CCcc 7618   1c1 7621   x. cmul 7625   NNcn 8720   NN0cn0 8977   ZZcz 9054   gcd cgcd 11635

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502  ax-cnex 7711  ax-resscn 7712  ax-1cn 7713  ax-1re 7714  ax-icn 7715  ax-addcl 7716  ax-addrcl 7717  ax-mulcl 7718  ax-mulrcl 7719  ax-addcom 7720  ax-mulcom 7721  ax-addass 7722  ax-mulass 7723  ax-distr 7724  ax-i2m1 7725  ax-0lt1 7726  ax-1rid 7727  ax-0id 7728  ax-rnegex 7729  ax-precex 7730  ax-cnre 7731  ax-pre-ltirr 7732  ax-pre-ltwlin 7733  ax-pre-lttrn 7734  ax-pre-apti 7735  ax-pre-ltadd 7736  ax-pre-mulgt0 7737  ax-pre-mulext 7738  ax-arch 7739  ax-caucvg 7740

This theorem depends on definitions:  df-bi 116  df-stab 816  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-ilim 4291  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-frec 6288  df-sup 6871  df-pnf 7802  df-mnf 7803  df-xr 7804  df-ltxr 7805  df-le 7806  df-sub 7935  df-neg 7936  df-reap 8337  df-ap 8344  df-div 8433  df-inn 8721  df-2 8779  df-3 8780  df-4 8781  df-n0 8978  df-z 9055  df-uz 9327  df-q 9412  df-rp 9442  df-fz 9791  df-fzo 9920  df-fl 10043  df-mod 10096  df-seqfrec 10219  df-exp 10293  df-cj 10614  df-re 10615  df-im 10616  df-rsqrt 10770  df-abs 10771  df-dvds 11494  df-gcd 11636

This theorem is referenced by:  rplpwr  11715