Theorem rpmulgcd 11925

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 11919	. . . . . 6 |- ( ( K e. NN /\ N e. NN ) -> ( K gcd ( K x. N ) ) = K )
2	1	3adant2 1001	. . . . 5 |- ( ( K e. NN /\ M e. NN /\ N e. NN ) -> ( K gcd ( K x. N ) ) = K )
3	2	oveq1d 5841	. . . 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 9191	. . . . . 6 |- ( K e. NN -> K e. ZZ )
5	4	3ad2ant1 1003	. . . . 5 |- ( ( K e. NN /\ M e. NN /\ N e. NN ) -> K e. ZZ )
6		nnz 9191	. . . . . . 7 |- ( N e. NN -> N e. ZZ )
7		zmulcl 9225	. . . . . . 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 1001	. . . . 5 |- ( ( K e. NN /\ M e. NN /\ N e. NN ) -> ( K x. N ) e. ZZ )
10		nnz 9191	. . . . . . 7 |- ( M e. NN -> M e. ZZ )
11		zmulcl 9225	. . . . . . 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 1000	. . . . 5 |- ( ( K e. NN /\ M e. NN /\ N e. NN ) -> ( M x. N ) e. ZZ )
14		gcdass 11914	. . . . 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 1220	. . . 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 2192	. . 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 9102	. . . . . 6 |- ( N e. NN -> N e. NN0 )
19		mulgcdr 11917	. . . . . 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 1262	. . . . 5 |- ( ( K e. NN /\ M e. NN /\ N e. NN ) -> ( ( K x. N ) gcd ( M x. N ) ) = ( ( K gcd M ) x. N ) )
21		oveq1 5833	. . . . 5 |- ( ( K gcd M ) = 1 -> ( ( K gcd M ) x. N ) = ( 1 x. N ) )
22	20, 21	sylan9eq 2210	. . . 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 8846	. . . . . . 7 |- ( N e. NN -> N e. CC )
24	23	3ad2ant3 1005	. . . . . 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 7898	. . . 4 |- ( ( ( K e. NN /\ M e. NN /\ N e. NN ) /\ ( K gcd M ) = 1 ) -> ( 1 x. N ) = N )
27	22, 26	eqtrd 2190	. . 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 5842	. 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 2190	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 963   = wceq 1335   e. wcel 2128  (class class class)co 5826   CCcc 7732   1c1 7735   x. cmul 7739   NNcn 8838   NN0cn0 9095   ZZcz 9172   gcd cgcd 11841

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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4081  ax-sep 4084  ax-nul 4092  ax-pow 4137  ax-pr 4171  ax-un 4395  ax-setind 4498  ax-iinf 4549  ax-cnex 7825  ax-resscn 7826  ax-1cn 7827  ax-1re 7828  ax-icn 7829  ax-addcl 7830  ax-addrcl 7831  ax-mulcl 7832  ax-mulrcl 7833  ax-addcom 7834  ax-mulcom 7835  ax-addass 7836  ax-mulass 7837  ax-distr 7838  ax-i2m1 7839  ax-0lt1 7840  ax-1rid 7841  ax-0id 7842  ax-rnegex 7843  ax-precex 7844  ax-cnre 7845  ax-pre-ltirr 7846  ax-pre-ltwlin 7847  ax-pre-lttrn 7848  ax-pre-apti 7849  ax-pre-ltadd 7850  ax-pre-mulgt0 7851  ax-pre-mulext 7852  ax-arch 7853  ax-caucvg 7854

This theorem depends on definitions:  df-bi 116  df-stab 817  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-reu 2442  df-rmo 2443  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3396  df-if 3507  df-pw 3546  df-sn 3567  df-pr 3568  df-op 3570  df-uni 3775  df-int 3810  df-iun 3853  df-br 3968  df-opab 4028  df-mpt 4029  df-tr 4065  df-id 4255  df-po 4258  df-iso 4259  df-iord 4328  df-on 4330  df-ilim 4331  df-suc 4333  df-iom 4552  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-iota 5137  df-fun 5174  df-fn 5175  df-f 5176  df-f1 5177  df-fo 5178  df-f1o 5179  df-fv 5180  df-riota 5782  df-ov 5829  df-oprab 5830  df-mpo 5831  df-1st 6090  df-2nd 6091  df-recs 6254  df-frec 6340  df-sup 6930  df-pnf 7916  df-mnf 7917  df-xr 7918  df-ltxr 7919  df-le 7920  df-sub 8052  df-neg 8053  df-reap 8454  df-ap 8461  df-div 8550  df-inn 8839  df-2 8897  df-3 8898  df-4 8899  df-n0 9096  df-z 9173  df-uz 9445  df-q 9535  df-rp 9567  df-fz 9919  df-fzo 10051  df-fl 10178  df-mod 10231  df-seqfrec 10354  df-exp 10428  df-cj 10753  df-re 10754  df-im 10755  df-rsqrt 10909  df-abs 10910  df-dvds 11695  df-gcd 11842

This theorem is referenced by:  rplpwr  11926