Theorem rpmulgcd 12747

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 12741	. . . . . 6 |- ( ( K e. NN /\ N e. NN ) -> ( K gcd ( K x. N ) ) = K )
2	1	3adant2 1043	. . . . 5 |- ( ( K e. NN /\ M e. NN /\ N e. NN ) -> ( K gcd ( K x. N ) ) = K )
3	2	oveq1d 6073	. . . 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 9613	. . . . . 6 |- ( K e. NN -> K e. ZZ )
5	4	3ad2ant1 1045	. . . . 5 |- ( ( K e. NN /\ M e. NN /\ N e. NN ) -> K e. ZZ )
6		nnz 9613	. . . . . . 7 |- ( N e. NN -> N e. ZZ )
7		zmulcl 9648	. . . . . . 7 |- ( ( K e. ZZ /\ N e. ZZ ) -> ( K x. N ) e. ZZ )
8	4, 6, 7	syl2an 289	. . . . . 6 |- ( ( K e. NN /\ N e. NN ) -> ( K x. N ) e. ZZ )
9	8	3adant2 1043	. . . . 5 |- ( ( K e. NN /\ M e. NN /\ N e. NN ) -> ( K x. N ) e. ZZ )
10		nnz 9613	. . . . . . 7 |- ( M e. NN -> M e. ZZ )
11		zmulcl 9648	. . . . . . 7 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M x. N ) e. ZZ )
12	10, 6, 11	syl2an 289	. . . . . 6 |- ( ( M e. NN /\ N e. NN ) -> ( M x. N ) e. ZZ )
13	12	3adant1 1042	. . . . 5 |- ( ( K e. NN /\ M e. NN /\ N e. NN ) -> ( M x. N ) e. ZZ )
14		gcdass 12736	. . . . 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 1274	. . . 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 2269	. . 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 276	. 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 9520	. . . . . 6 |- ( N e. NN -> N e. NN0 )
19		mulgcdr 12739	. . . . . 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 1316	. . . . 5 |- ( ( K e. NN /\ M e. NN /\ N e. NN ) -> ( ( K x. N ) gcd ( M x. N ) ) = ( ( K gcd M ) x. N ) )
21		oveq1 6065	. . . . 5 |- ( ( K gcd M ) = 1 -> ( ( K gcd M ) x. N ) = ( 1 x. N ) )
22	20, 21	sylan9eq 2287	. . . 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 9262	. . . . . . 7 |- ( N e. NN -> N e. CC )
24	23	3ad2ant3 1047	. . . . . 6 |- ( ( K e. NN /\ M e. NN /\ N e. NN ) -> N e. CC )
25	24	adantr 276	. . . . 5 |- ( ( ( K e. NN /\ M e. NN /\ N e. NN ) /\ ( K gcd M ) = 1 ) -> N e. CC )
26	25	mullidd 8308	. . . 4 |- ( ( ( K e. NN /\ M e. NN /\ N e. NN ) /\ ( K gcd M ) = 1 ) -> ( 1 x. N ) = N )
27	22, 26	eqtrd 2267	. . 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 6074	. 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 2267	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 104   /\ w3a 1005   = wceq 1398   e. wcel 2205  (class class class)co 6058   CCcc 8141   1c1 8144   x. cmul 8148   NNcn 9254   NN0cn0 9513   ZZcz 9594   gcd cgcd 12674

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 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-mulrcl 8242  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-precex 8253  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259  ax-pre-mulgt0 8260  ax-pre-mulext 8261  ax-arch 8262  ax-caucvg 8263

This theorem depends on definitions:  df-bi 117  df-stab 839  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-po 4422  df-iso 4423  df-iord 4492  df-on 4494  df-ilim 4495  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-frec 6635  df-sup 7288  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-reap 8866  df-ap 8873  df-div 8964  df-inn 9255  df-2 9313  df-3 9314  df-4 9315  df-n0 9514  df-z 9595  df-uz 9872  df-q 9970  df-rp 10005  df-fz 10362  df-fzo 10499  df-fl 10654  df-mod 10709  df-seqfrec 10834  df-exp 10925  df-cj 11552  df-re 11553  df-im 11554  df-rsqrt 11708  df-abs 11709  df-dvds 12499  df-gcd 12675

This theorem is referenced by:  rplpwr  12748  lgsquad2lem2  16081