Theorem gcdmultiple 11953

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 5850	. . . . . 6 |- ( k = 1 -> ( M x. k ) = ( M x. 1 ) )
2	1	oveq2d 5858	. . . . 5 |- ( k = 1 -> ( M gcd ( M x. k ) ) = ( M gcd ( M x. 1 ) ) )
3	2	eqeq1d 2174	. . . 4 |- ( k = 1 -> ( ( M gcd ( M x. k ) ) = M <-> ( M gcd ( M x. 1 ) ) = M ) )
4	3	imbi2d 229	. . 3 |- ( k = 1 -> ( ( M e. NN -> ( M gcd ( M x. k ) ) = M ) <-> ( M e. NN -> ( M gcd ( M x. 1 ) ) = M ) ) )
5		oveq2 5850	. . . . . 6 |- ( k = n -> ( M x. k ) = ( M x. n ) )
6	5	oveq2d 5858	. . . . 5 |- ( k = n -> ( M gcd ( M x. k ) ) = ( M gcd ( M x. n ) ) )
7	6	eqeq1d 2174	. . . 4 |- ( k = n -> ( ( M gcd ( M x. k ) ) = M <-> ( M gcd ( M x. n ) ) = M ) )
8	7	imbi2d 229	. . 3 |- ( k = n -> ( ( M e. NN -> ( M gcd ( M x. k ) ) = M ) <-> ( M e. NN -> ( M gcd ( M x. n ) ) = M ) ) )
9		oveq2 5850	. . . . . 6 |- ( k = ( n + 1 ) -> ( M x. k ) = ( M x. ( n + 1 ) ) )
10	9	oveq2d 5858	. . . . 5 |- ( k = ( n + 1 ) -> ( M gcd ( M x. k ) ) = ( M gcd ( M x. ( n + 1 ) ) ) )
11	10	eqeq1d 2174	. . . 4 |- ( k = ( n + 1 ) -> ( ( M gcd ( M x. k ) ) = M <-> ( M gcd ( M x. ( n + 1 ) ) ) = M ) )
12	11	imbi2d 229	. . 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 5850	. . . . . 6 |- ( k = N -> ( M x. k ) = ( M x. N ) )
14	13	oveq2d 5858	. . . . 5 |- ( k = N -> ( M gcd ( M x. k ) ) = ( M gcd ( M x. N ) ) )
15	14	eqeq1d 2174	. . . 4 |- ( k = N -> ( ( M gcd ( M x. k ) ) = M <-> ( M gcd ( M x. N ) ) = M ) )
16	15	imbi2d 229	. . 3 |- ( k = N -> ( ( M e. NN -> ( M gcd ( M x. k ) ) = M ) <-> ( M e. NN -> ( M gcd ( M x. N ) ) = M ) ) )
17		nncn 8865	. . . . . 6 |- ( M e. NN -> M e. CC )
18	17	mulid1d 7916	. . . . 5 |- ( M e. NN -> ( M x. 1 ) = M )
19	18	oveq2d 5858	. . . 4 |- ( M e. NN -> ( M gcd ( M x. 1 ) ) = ( M gcd M ) )
20		nnz 9210	. . . . . 6 |- ( M e. NN -> M e. ZZ )
21		gcdid 11919	. . . . . 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 8864	. . . . . 6 |- ( M e. NN -> M e. RR )
24		nnnn0 9121	. . . . . . 7 |- ( M e. NN -> M e. NN0 )
25	24	nn0ge0d 9170	. . . . . 6 |- ( M e. NN -> 0 <_ M )
26	23, 25	absidd 11109	. . . . 5 |- ( M e. NN -> ( abs ` M ) = M )
27	22, 26	eqtrd 2198	. . . 4 |- ( M e. NN -> ( M gcd M ) = M )
28	19, 27	eqtrd 2198	. . 3 |- ( M e. NN -> ( M gcd ( M x. 1 ) ) = M )
29	20	adantr 274	. . . . . . . . 9 |- ( ( M e. NN /\ n e. NN ) -> M e. ZZ )
30		nnz 9210	. . . . . . . . . 10 |- ( n e. NN -> n e. ZZ )
31		zmulcl 9244	. . . . . . . . . 10 |- ( ( M e. ZZ /\ n e. ZZ ) -> ( M x. n ) e. ZZ )
32	20, 30, 31	syl2an 287	. . . . . . . . 9 |- ( ( M e. NN /\ n e. NN ) -> ( M x. n ) e. ZZ )
33		1z 9217	. . . . . . . . . 10 |- 1 e. ZZ
34		gcdaddm 11917	. . . . . . . . . 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 1314	. . . . . . . . 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 409	. . . . . . . 8 |- ( ( M e. NN /\ n e. NN ) -> ( M gcd ( M x. n ) ) = ( M gcd ( ( M x. n ) + ( 1 x. M ) ) ) )
37		nncn 8865	. . . . . . . . . 10 |- ( n e. NN -> n e. CC )
38		ax-1cn 7846	. . . . . . . . . . . 12 |- 1 e. CC
39		adddi 7885	. . . . . . . . . . . 12 |- ( ( M e. CC /\ n e. CC /\ 1 e. CC ) -> ( M x. ( n + 1 ) ) = ( ( M x. n ) + ( M x. 1 ) ) )
40	38, 39	mp3an3 1316	. . . . . . . . . . 11 |- ( ( M e. CC /\ n e. CC ) -> ( M x. ( n + 1 ) ) = ( ( M x. n ) + ( M x. 1 ) ) )
41		mulcom 7882	. . . . . . . . . . . . . 14 |- ( ( M e. CC /\ 1 e. CC ) -> ( M x. 1 ) = ( 1 x. M ) )
42	38, 41	mpan2 422	. . . . . . . . . . . . 13 |- ( M e. CC -> ( M x. 1 ) = ( 1 x. M ) )
43	42	adantr 274	. . . . . . . . . . . 12 |- ( ( M e. CC /\ n e. CC ) -> ( M x. 1 ) = ( 1 x. M ) )
44	43	oveq2d 5858	. . . . . . . . . . 11 |- ( ( M e. CC /\ n e. CC ) -> ( ( M x. n ) + ( M x. 1 ) ) = ( ( M x. n ) + ( 1 x. M ) ) )
45	40, 44	eqtrd 2198	. . . . . . . . . 10 |- ( ( M e. CC /\ n e. CC ) -> ( M x. ( n + 1 ) ) = ( ( M x. n ) + ( 1 x. M ) ) )
46	17, 37, 45	syl2an 287	. . . . . . . . 9 |- ( ( M e. NN /\ n e. NN ) -> ( M x. ( n + 1 ) ) = ( ( M x. n ) + ( 1 x. M ) ) )
47	46	oveq2d 5858	. . . . . . . 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 2201	. . . . . . 7 |- ( ( M e. NN /\ n e. NN ) -> ( M gcd ( M x. n ) ) = ( M gcd ( M x. ( n + 1 ) ) ) )
49	48	eqeq1d 2174	. . . . . 6 |- ( ( M e. NN /\ n e. NN ) -> ( ( M gcd ( M x. n ) ) = M <-> ( M gcd ( M x. ( n + 1 ) ) ) = M ) )
50	49	biimpd 143	. . . . 5 |- ( ( M e. NN /\ n e. NN ) -> ( ( M gcd ( M x. n ) ) = M -> ( M gcd ( M x. ( n + 1 ) ) ) = M ) )
51	50	expcom 115	. . . 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 8873	. 2 |- ( N e. NN -> ( M e. NN -> ( M gcd ( M x. N ) ) = M ) )
54	53	impcom 124	1 |- ( ( M e. NN /\ N e. NN ) -> ( M gcd ( M x. N ) ) = M )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1343   e. wcel 2136   ` cfv 5188  (class class class)co 5842   CCcc 7751   1c1 7754   + caddc 7756   x. cmul 7758   NNcn 8857   ZZcz 9191   abscabs 10939   gcd cgcd 11875

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-mulrcl 7852  ax-addcom 7853  ax-mulcom 7854  ax-addass 7855  ax-mulass 7856  ax-distr 7857  ax-i2m1 7858  ax-0lt1 7859  ax-1rid 7860  ax-0id 7861  ax-rnegex 7862  ax-precex 7863  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-ltwlin 7866  ax-pre-lttrn 7867  ax-pre-apti 7868  ax-pre-ltadd 7869  ax-pre-mulgt0 7870  ax-pre-mulext 7871  ax-arch 7872  ax-caucvg 7873

This theorem depends on definitions:  df-bi 116  df-stab 821  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rmo 2452  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-if 3521  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-id 4271  df-po 4274  df-iso 4275  df-iord 4344  df-on 4346  df-ilim 4347  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-recs 6273  df-frec 6359  df-sup 6949  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-sub 8071  df-neg 8072  df-reap 8473  df-ap 8480  df-div 8569  df-inn 8858  df-2 8916  df-3 8917  df-4 8918  df-n0 9115  df-z 9192  df-uz 9467  df-q 9558  df-rp 9590  df-fz 9945  df-fzo 10078  df-fl 10205  df-mod 10258  df-seqfrec 10381  df-exp 10455  df-cj 10784  df-re 10785  df-im 10786  df-rsqrt 10940  df-abs 10941  df-dvds 11728  df-gcd 11876

This theorem is referenced by:  gcdmultiplez  11954  rpmulgcd  11959