Theorem gcdmultiple 10616

Description: The GCD of a multiple of a number is the number itself. (Contributed by Scott Fenton, 12-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Assertion

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

Proof of Theorem gcdmultiple

Dummy variables k n are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq2 5571	. . . . . 6 |- ( k = 1 -> ( M x. k ) = ( M x. 1 ) )
2	1	oveq2d 5579	. . . . 5 |- ( k = 1 -> ( M gcd ( M x. k ) ) = ( M gcd ( M x. 1 ) ) )
3	2	eqeq1d 2091	. . . 4 |- ( k = 1 -> ( ( M gcd ( M x. k ) ) = M <-> ( M gcd ( M x. 1 ) ) = M ) )
4	3	imbi2d 228	. . 3 |- ( k = 1 -> ( ( M e. NN -> ( M gcd ( M x. k ) ) = M ) <-> ( M e. NN -> ( M gcd ( M x. 1 ) ) = M ) ) )
5		oveq2 5571	. . . . . 6 |- ( k = n -> ( M x. k ) = ( M x. n ) )
6	5	oveq2d 5579	. . . . 5 |- ( k = n -> ( M gcd ( M x. k ) ) = ( M gcd ( M x. n ) ) )
7	6	eqeq1d 2091	. . . 4 |- ( k = n -> ( ( M gcd ( M x. k ) ) = M <-> ( M gcd ( M x. n ) ) = M ) )
8	7	imbi2d 228	. . 3 |- ( k = n -> ( ( M e. NN -> ( M gcd ( M x. k ) ) = M ) <-> ( M e. NN -> ( M gcd ( M x. n ) ) = M ) ) )
9		oveq2 5571	. . . . . 6 |- ( k = ( n + 1 ) -> ( M x. k ) = ( M x. ( n + 1 ) ) )
10	9	oveq2d 5579	. . . . 5 |- ( k = ( n + 1 ) -> ( M gcd ( M x. k ) ) = ( M gcd ( M x. ( n + 1 ) ) ) )
11	10	eqeq1d 2091	. . . 4 |- ( k = ( n + 1 ) -> ( ( M gcd ( M x. k ) ) = M <-> ( M gcd ( M x. ( n + 1 ) ) ) = M ) )
12	11	imbi2d 228	. . 3 |- ( k = ( n + 1 ) -> ( ( M e. NN -> ( M gcd ( M x. k ) ) = M ) <-> ( M e. NN -> ( M gcd ( M x. ( n + 1 ) ) ) = M ) ) )
13		oveq2 5571	. . . . . 6 |- ( k = N -> ( M x. k ) = ( M x. N ) )
14	13	oveq2d 5579	. . . . 5 |- ( k = N -> ( M gcd ( M x. k ) ) = ( M gcd ( M x. N ) ) )
15	14	eqeq1d 2091	. . . 4 |- ( k = N -> ( ( M gcd ( M x. k ) ) = M <-> ( M gcd ( M x. N ) ) = M ) )
16	15	imbi2d 228	. . 3 |- ( k = N -> ( ( M e. NN -> ( M gcd ( M x. k ) ) = M ) <-> ( M e. NN -> ( M gcd ( M x. N ) ) = M ) ) )
17		nncn 8166	. . . . . 6 |- ( M e. NN -> M e. CC )
18	17	mulid1d 7250	. . . . 5 |- ( M e. NN -> ( M x. 1 ) = M )
19	18	oveq2d 5579	. . . 4 |- ( M e. NN -> ( M gcd ( M x. 1 ) ) = ( M gcd M ) )
20		nnz 8503	. . . . . 6 |- ( M e. NN -> M e. ZZ )
21		gcdid 10584	. . . . . 6 |- ( M e. ZZ -> ( M gcd M ) = ( abs ` M ) )
22	20, 21	syl 14	. . . . 5 |- ( M e. NN -> ( M gcd M ) = ( abs ` M ) )
23		nnre 8165	. . . . . 6 |- ( M e. NN -> M e. RR )
24		nnnn0 8414	. . . . . . 7 |- ( M e. NN -> M e. NN0 )
25	24	nn0ge0d 8463	. . . . . 6 |- ( M e. NN -> 0 <_ M )
26	23, 25	absidd 10254	. . . . 5 |- ( M e. NN -> ( abs ` M ) = M )
27	22, 26	eqtrd 2115	. . . 4 |- ( M e. NN -> ( M gcd M ) = M )
28	19, 27	eqtrd 2115	. . 3 |- ( M e. NN -> ( M gcd ( M x. 1 ) ) = M )
29	20	adantr 270	. . . . . . . . 9 |- ( ( M e. NN /\ n e. NN ) -> M e. ZZ )
30		nnz 8503	. . . . . . . . . 10 |- ( n e. NN -> n e. ZZ )
31		zmulcl 8537	. . . . . . . . . 10 |- ( ( M e. ZZ /\ n e. ZZ ) -> ( M x. n ) e. ZZ )
32	20, 30, 31	syl2an 283	. . . . . . . . 9 |- ( ( M e. NN /\ n e. NN ) -> ( M x. n ) e. ZZ )
33		1z 8510	. . . . . . . . . 10 |- 1 e. ZZ
34		gcdaddm 10582	. . . . . . . . . 10 |- ( ( 1 e. ZZ /\ M e. ZZ /\ ( M x. n ) e. ZZ ) -> ( M gcd ( M x. n ) ) = ( M gcd ( ( M x. n ) + ( 1 x. M ) ) ) )
35	33, 34	mp3an1 1256	. . . . . . . . 9 |- ( ( M e. ZZ /\ ( M x. n ) e. ZZ ) -> ( M gcd ( M x. n ) ) = ( M gcd ( ( M x. n ) + ( 1 x. M ) ) ) )
36	29, 32, 35	syl2anc 403	. . . . . . . 8 |- ( ( M e. NN /\ n e. NN ) -> ( M gcd ( M x. n ) ) = ( M gcd ( ( M x. n ) + ( 1 x. M ) ) ) )
37		nncn 8166	. . . . . . . . . 10 |- ( n e. NN -> n e. CC )
38		ax-1cn 7183	. . . . . . . . . . . 12 |- 1 e. CC
39		adddi 7219	. . . . . . . . . . . 12 |- ( ( M e. CC /\ n e. CC /\ 1 e. CC ) -> ( M x. ( n + 1 ) ) = ( ( M x. n ) + ( M x. 1 ) ) )
40	38, 39	mp3an3 1258	. . . . . . . . . . 11 |- ( ( M e. CC /\ n e. CC ) -> ( M x. ( n + 1 ) ) = ( ( M x. n ) + ( M x. 1 ) ) )
41		mulcom 7216	. . . . . . . . . . . . . 14 |- ( ( M e. CC /\ 1 e. CC ) -> ( M x. 1 ) = ( 1 x. M ) )
42	38, 41	mpan2 416	. . . . . . . . . . . . 13 |- ( M e. CC -> ( M x. 1 ) = ( 1 x. M ) )
43	42	adantr 270	. . . . . . . . . . . 12 |- ( ( M e. CC /\ n e. CC ) -> ( M x. 1 ) = ( 1 x. M ) )
44	43	oveq2d 5579	. . . . . . . . . . 11 |- ( ( M e. CC /\ n e. CC ) -> ( ( M x. n ) + ( M x. 1 ) ) = ( ( M x. n ) + ( 1 x. M ) ) )
45	40, 44	eqtrd 2115	. . . . . . . . . 10 |- ( ( M e. CC /\ n e. CC ) -> ( M x. ( n + 1 ) ) = ( ( M x. n ) + ( 1 x. M ) ) )
46	17, 37, 45	syl2an 283	. . . . . . . . 9 |- ( ( M e. NN /\ n e. NN ) -> ( M x. ( n + 1 ) ) = ( ( M x. n ) + ( 1 x. M ) ) )
47	46	oveq2d 5579	. . . . . . . 8 |- ( ( M e. NN /\ n e. NN ) -> ( M gcd ( M x. ( n + 1 ) ) ) = ( M gcd ( ( M x. n ) + ( 1 x. M ) ) ) )
48	36, 47	eqtr4d 2118	. . . . . . 7 |- ( ( M e. NN /\ n e. NN ) -> ( M gcd ( M x. n ) ) = ( M gcd ( M x. ( n + 1 ) ) ) )
49	48	eqeq1d 2091	. . . . . 6 |- ( ( M e. NN /\ n e. NN ) -> ( ( M gcd ( M x. n ) ) = M <-> ( M gcd ( M x. ( n + 1 ) ) ) = M ) )
50	49	biimpd 142	. . . . 5 |- ( ( M e. NN /\ n e. NN ) -> ( ( M gcd ( M x. n ) ) = M -> ( M gcd ( M x. ( n + 1 ) ) ) = M ) )
51	50	expcom 114	. . . 4 |- ( n e. NN -> ( M e. NN -> ( ( M gcd ( M x. n ) ) = M -> ( M gcd ( M x. ( n + 1 ) ) ) = M ) ) )
52	51	a2d 26	. . 3 |- ( n e. NN -> ( ( M e. NN -> ( M gcd ( M x. n ) ) = M ) -> ( M e. NN -> ( M gcd ( M x. ( n + 1 ) ) ) = M ) ) )
53	4, 8, 12, 16, 28, 52	nnind 8174	. 2 |- ( N e. NN -> ( M e. NN -> ( M gcd ( M x. N ) ) = M ) )
54	53	impcom 123	1 |- ( ( M e. NN /\ N e. NN ) -> ( M gcd ( M x. N ) ) = M )

Syntax hints:   -> wi 4   /\ wa 102   = wceq 1285   e. wcel 1434   ` cfv 4952  (class class class)co 5563   CCcc 7093   1c1 7096   + caddc 7098   x. cmul 7100   NNcn 8158   ZZcz 8484   abscabs 10084   gcd cgcd 10545

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3913  ax-sep 3916  ax-nul 3924  ax-pow 3968  ax-pr 3992  ax-un 4216  ax-setind 4308  ax-iinf 4357  ax-cnex 7181  ax-resscn 7182  ax-1cn 7183  ax-1re 7184  ax-icn 7185  ax-addcl 7186  ax-addrcl 7187  ax-mulcl 7188  ax-mulrcl 7189  ax-addcom 7190  ax-mulcom 7191  ax-addass 7192  ax-mulass 7193  ax-distr 7194  ax-i2m1 7195  ax-0lt1 7196  ax-1rid 7197  ax-0id 7198  ax-rnegex 7199  ax-precex 7200  ax-cnre 7201  ax-pre-ltirr 7202  ax-pre-ltwlin 7203  ax-pre-lttrn 7204  ax-pre-apti 7205  ax-pre-ltadd 7206  ax-pre-mulgt0 7207  ax-pre-mulext 7208  ax-arch 7209  ax-caucvg 7210

This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-nel 2345  df-ral 2358  df-rex 2359  df-reu 2360  df-rmo 2361  df-rab 2362  df-v 2612  df-sbc 2825  df-csb 2918  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-nul 3268  df-if 3369  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-int 3657  df-iun 3700  df-br 3806  df-opab 3860  df-mpt 3861  df-tr 3896  df-id 4076  df-po 4079  df-iso 4080  df-iord 4149  df-on 4151  df-ilim 4152  df-suc 4154  df-iom 4360  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402  df-res 4403  df-ima 4404  df-iota 4917  df-fun 4954  df-fn 4955  df-f 4956  df-f1 4957  df-fo 4958  df-f1o 4959  df-fv 4960  df-riota 5519  df-ov 5566  df-oprab 5567  df-mpt2 5568  df-1st 5818  df-2nd 5819  df-recs 5974  df-frec 6060  df-sup 6491  df-pnf 7269  df-mnf 7270  df-xr 7271  df-ltxr 7272  df-le 7273  df-sub 7400  df-neg 7401  df-reap 7794  df-ap 7801  df-div 7880  df-inn 8159  df-2 8217  df-3 8218  df-4 8219  df-n0 8408  df-z 8485  df-uz 8753  df-q 8838  df-rp 8868  df-fz 9158  df-fzo 9282  df-fl 9404  df-mod 9457  df-iseq 9574  df-iexp 9625  df-cj 9930  df-re 9931  df-im 9932  df-rsqrt 10085  df-abs 10086  df-dvds 10404  df-gcd 10546

This theorem is referenced by:  gcdmultiplez  10617  rpmulgcd  10622