Theorem gcdmultiple 12712

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 6057	. . . . . 6 |- ( k = 1 -> ( M x. k ) = ( M x. 1 ) )
2	1	oveq2d 6065	. . . . 5 |- ( k = 1 -> ( M gcd ( M x. k ) ) = ( M gcd ( M x. 1 ) ) )
3	2	eqeq1d 2241	. . . 4 |- ( k = 1 -> ( ( M gcd ( M x. k ) ) = M <-> ( M gcd ( M x. 1 ) ) = M ) )
4	3	imbi2d 230	. . 3 |- ( k = 1 -> ( ( M e. NN -> ( M gcd ( M x. k ) ) = M ) <-> ( M e. NN -> ( M gcd ( M x. 1 ) ) = M ) ) )
5		oveq2 6057	. . . . . 6 |- ( k = n -> ( M x. k ) = ( M x. n ) )
6	5	oveq2d 6065	. . . . 5 |- ( k = n -> ( M gcd ( M x. k ) ) = ( M gcd ( M x. n ) ) )
7	6	eqeq1d 2241	. . . 4 |- ( k = n -> ( ( M gcd ( M x. k ) ) = M <-> ( M gcd ( M x. n ) ) = M ) )
8	7	imbi2d 230	. . 3 |- ( k = n -> ( ( M e. NN -> ( M gcd ( M x. k ) ) = M ) <-> ( M e. NN -> ( M gcd ( M x. n ) ) = M ) ) )
9		oveq2 6057	. . . . . 6 |- ( k = ( n + 1 ) -> ( M x. k ) = ( M x. ( n + 1 ) ) )
10	9	oveq2d 6065	. . . . 5 |- ( k = ( n + 1 ) -> ( M gcd ( M x. k ) ) = ( M gcd ( M x. ( n + 1 ) ) ) )
11	10	eqeq1d 2241	. . . 4 |- ( k = ( n + 1 ) -> ( ( M gcd ( M x. k ) ) = M <-> ( M gcd ( M x. ( n + 1 ) ) ) = M ) )
12	11	imbi2d 230	. . 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 6057	. . . . . 6 |- ( k = N -> ( M x. k ) = ( M x. N ) )
14	13	oveq2d 6065	. . . . 5 |- ( k = N -> ( M gcd ( M x. k ) ) = ( M gcd ( M x. N ) ) )
15	14	eqeq1d 2241	. . . 4 |- ( k = N -> ( ( M gcd ( M x. k ) ) = M <-> ( M gcd ( M x. N ) ) = M ) )
16	15	imbi2d 230	. . 3 |- ( k = N -> ( ( M e. NN -> ( M gcd ( M x. k ) ) = M ) <-> ( M e. NN -> ( M gcd ( M x. N ) ) = M ) ) )
17		nncn 9244	. . . . . 6 |- ( M e. NN -> M e. CC )
18	17	mulridd 8290	. . . . 5 |- ( M e. NN -> ( M x. 1 ) = M )
19	18	oveq2d 6065	. . . 4 |- ( M e. NN -> ( M gcd ( M x. 1 ) ) = ( M gcd M ) )
20		nnz 9595	. . . . . 6 |- ( M e. NN -> M e. ZZ )
21		gcdid 12678	. . . . . 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 9243	. . . . . 6 |- ( M e. NN -> M e. RR )
24		nnnn0 9502	. . . . . . 7 |- ( M e. NN -> M e. NN0 )
25	24	nn0ge0d 9555	. . . . . 6 |- ( M e. NN -> 0 <_ M )
26	23, 25	absidd 11848	. . . . 5 |- ( M e. NN -> ( abs ` M ) = M )
27	22, 26	eqtrd 2265	. . . 4 |- ( M e. NN -> ( M gcd M ) = M )
28	19, 27	eqtrd 2265	. . 3 |- ( M e. NN -> ( M gcd ( M x. 1 ) ) = M )
29	20	adantr 276	. . . . . . . . 9 |- ( ( M e. NN /\ n e. NN ) -> M e. ZZ )
30		nnz 9595	. . . . . . . . . 10 |- ( n e. NN -> n e. ZZ )
31		zmulcl 9630	. . . . . . . . . 10 |- ( ( M e. ZZ /\ n e. ZZ ) -> ( M x. n ) e. ZZ )
32	20, 30, 31	syl2an 289	. . . . . . . . 9 |- ( ( M e. NN /\ n e. NN ) -> ( M x. n ) e. ZZ )
33		1z 9602	. . . . . . . . . 10 |- 1 e. ZZ
34		gcdaddm 12676	. . . . . . . . . 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 1361	. . . . . . . . 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 411	. . . . . . . 8 |- ( ( M e. NN /\ n e. NN ) -> ( M gcd ( M x. n ) ) = ( M gcd ( ( M x. n ) + ( 1 x. M ) ) ) )
37		nncn 9244	. . . . . . . . . 10 |- ( n e. NN -> n e. CC )
38		ax-1cn 8219	. . . . . . . . . . . 12 |- 1 e. CC
39		adddi 8258	. . . . . . . . . . . 12 |- ( ( M e. CC /\ n e. CC /\ 1 e. CC ) -> ( M x. ( n + 1 ) ) = ( ( M x. n ) + ( M x. 1 ) ) )
40	38, 39	mp3an3 1363	. . . . . . . . . . 11 |- ( ( M e. CC /\ n e. CC ) -> ( M x. ( n + 1 ) ) = ( ( M x. n ) + ( M x. 1 ) ) )
41		mulcom 8255	. . . . . . . . . . . . . 14 |- ( ( M e. CC /\ 1 e. CC ) -> ( M x. 1 ) = ( 1 x. M ) )
42	38, 41	mpan2 425	. . . . . . . . . . . . 13 |- ( M e. CC -> ( M x. 1 ) = ( 1 x. M ) )
43	42	adantr 276	. . . . . . . . . . . 12 |- ( ( M e. CC /\ n e. CC ) -> ( M x. 1 ) = ( 1 x. M ) )
44	43	oveq2d 6065	. . . . . . . . . . 11 |- ( ( M e. CC /\ n e. CC ) -> ( ( M x. n ) + ( M x. 1 ) ) = ( ( M x. n ) + ( 1 x. M ) ) )
45	40, 44	eqtrd 2265	. . . . . . . . . 10 |- ( ( M e. CC /\ n e. CC ) -> ( M x. ( n + 1 ) ) = ( ( M x. n ) + ( 1 x. M ) ) )
46	17, 37, 45	syl2an 289	. . . . . . . . 9 |- ( ( M e. NN /\ n e. NN ) -> ( M x. ( n + 1 ) ) = ( ( M x. n ) + ( 1 x. M ) ) )
47	46	oveq2d 6065	. . . . . . . 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 2268	. . . . . . 7 |- ( ( M e. NN /\ n e. NN ) -> ( M gcd ( M x. n ) ) = ( M gcd ( M x. ( n + 1 ) ) ) )
49	48	eqeq1d 2241	. . . . . 6 |- ( ( M e. NN /\ n e. NN ) -> ( ( M gcd ( M x. n ) ) = M <-> ( M gcd ( M x. ( n + 1 ) ) ) = M ) )
50	49	biimpd 144	. . . . 5 |- ( ( M e. NN /\ n e. NN ) -> ( ( M gcd ( M x. n ) ) = M -> ( M gcd ( M x. ( n + 1 ) ) ) = M ) )
51	50	expcom 116	. . . 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 9252	. 2 |- ( N e. NN -> ( M e. NN -> ( M gcd ( M x. N ) ) = M ) )
54	53	impcom 125	1 |- ( ( M e. NN /\ N e. NN ) -> ( M gcd ( M x. N ) ) = M )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2203   ` cfv 5351  (class class class)co 6049   CCcc 8124   1c1 8127   + caddc 8129   x. cmul 8131   NNcn 9236   ZZcz 9576   abscabs 11678   gcd cgcd 12645

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 2205  ax-14 2206  ax-ext 2214  ax-coll 4224  ax-sep 4227  ax-nul 4235  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-iinf 4709  ax-cnex 8217  ax-resscn 8218  ax-1cn 8219  ax-1re 8220  ax-icn 8221  ax-addcl 8222  ax-addrcl 8223  ax-mulcl 8224  ax-mulrcl 8225  ax-addcom 8226  ax-mulcom 8227  ax-addass 8228  ax-mulass 8229  ax-distr 8230  ax-i2m1 8231  ax-0lt1 8232  ax-1rid 8233  ax-0id 8234  ax-rnegex 8235  ax-precex 8236  ax-cnre 8237  ax-pre-ltirr 8238  ax-pre-ltwlin 8239  ax-pre-lttrn 8240  ax-pre-apti 8241  ax-pre-ltadd 8242  ax-pre-mulgt0 8243  ax-pre-mulext 8244  ax-arch 8245  ax-caucvg 8246

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 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3508  df-if 3620  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-iun 3992  df-br 4109  df-opab 4171  df-mpt 4172  df-tr 4208  df-id 4413  df-po 4416  df-iso 4417  df-iord 4486  df-on 4488  df-ilim 4489  df-suc 4491  df-iom 4712  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-f1 5356  df-fo 5357  df-f1o 5358  df-fv 5359  df-riota 6002  df-ov 6052  df-oprab 6053  df-mpo 6054  df-1st 6333  df-2nd 6334  df-recs 6535  df-frec 6621  df-sup 7274  df-pnf 8309  df-mnf 8310  df-xr 8311  df-ltxr 8312  df-le 8313  df-sub 8445  df-neg 8446  df-reap 8848  df-ap 8855  df-div 8946  df-inn 9237  df-2 9295  df-3 9296  df-4 9297  df-n0 9496  df-z 9577  df-uz 9853  df-q 9951  df-rp 9986  df-fz 10342  df-fzo 10476  df-fl 10629  df-mod 10684  df-seqfrec 10809  df-exp 10900  df-cj 11523  df-re 11524  df-im 11525  df-rsqrt 11679  df-abs 11680  df-dvds 12470  df-gcd 12646

This theorem is referenced by:  gcdmultiplez  12713  rpmulgcd  12718