Theorem gcdmultiple 12822

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 6096	. . . . . 6 |- ( k = 1 -> ( M x. k ) = ( M x. 1 ) )
2	1	oveq2d 6104	. . . . 5 |- ( k = 1 -> ( M gcd ( M x. k ) ) = ( M gcd ( M x. 1 ) ) )
3	2	eqeq1d 2247	. . . 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 6096	. . . . . 6 |- ( k = n -> ( M x. k ) = ( M x. n ) )
6	5	oveq2d 6104	. . . . 5 |- ( k = n -> ( M gcd ( M x. k ) ) = ( M gcd ( M x. n ) ) )
7	6	eqeq1d 2247	. . . 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 6096	. . . . . 6 |- ( k = ( n + 1 ) -> ( M x. k ) = ( M x. ( n + 1 ) ) )
10	9	oveq2d 6104	. . . . 5 |- ( k = ( n + 1 ) -> ( M gcd ( M x. k ) ) = ( M gcd ( M x. ( n + 1 ) ) ) )
11	10	eqeq1d 2247	. . . 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 6096	. . . . . 6 |- ( k = N -> ( M x. k ) = ( M x. N ) )
14	13	oveq2d 6104	. . . . 5 |- ( k = N -> ( M gcd ( M x. k ) ) = ( M gcd ( M x. N ) ) )
15	14	eqeq1d 2247	. . . 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 9321	. . . . . 6 |- ( M e. NN -> M e. CC )
18	17	mulridd 8349	. . . . 5 |- ( M e. NN -> ( M x. 1 ) = M )
19	18	oveq2d 6104	. . . 4 |- ( M e. NN -> ( M gcd ( M x. 1 ) ) = ( M gcd M ) )
20		nnz 9674	. . . . . 6 |- ( M e. NN -> M e. ZZ )
21		gcdid 12788	. . . . . 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 9320	. . . . . 6 |- ( M e. NN -> M e. RR )
24		nnnn0 9581	. . . . . . 7 |- ( M e. NN -> M e. NN0 )
25	24	nn0ge0d 9634	. . . . . 6 |- ( M e. NN -> 0 <_ M )
26	23, 25	absidd 11956	. . . . 5 |- ( M e. NN -> ( abs ` M ) = M )
27	22, 26	eqtrd 2271	. . . 4 |- ( M e. NN -> ( M gcd M ) = M )
28	19, 27	eqtrd 2271	. . 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 9674	. . . . . . . . . 10 |- ( n e. NN -> n e. ZZ )
31		zmulcl 9709	. . . . . . . . . 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 9681	. . . . . . . . . 10 |- 1 e. ZZ
34		gcdaddm 12786	. . . . . . . . . 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 1365	. . . . . . . . 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 415	. . . . . . . 8 |- ( ( M e. NN /\ n e. NN ) -> ( M gcd ( M x. n ) ) = ( M gcd ( ( M x. n ) + ( 1 x. M ) ) ) )
37		nncn 9321	. . . . . . . . . 10 |- ( n e. NN -> n e. CC )
38		ax-1cn 8278	. . . . . . . . . . . 12 |- 1 e. CC
39		adddi 8317	. . . . . . . . . . . 12 |- ( ( M e. CC /\ n e. CC /\ 1 e. CC ) -> ( M x. ( n + 1 ) ) = ( ( M x. n ) + ( M x. 1 ) ) )
40	38, 39	mp3an3 1367	. . . . . . . . . . 11 |- ( ( M e. CC /\ n e. CC ) -> ( M x. ( n + 1 ) ) = ( ( M x. n ) + ( M x. 1 ) ) )
41		mulcom 8314	. . . . . . . . . . . . . 14 |- ( ( M e. CC /\ 1 e. CC ) -> ( M x. 1 ) = ( 1 x. M ) )
42	38, 41	mpan2 429	. . . . . . . . . . . . 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 6104	. . . . . . . . . . 11 |- ( ( M e. CC /\ n e. CC ) -> ( ( M x. n ) + ( M x. 1 ) ) = ( ( M x. n ) + ( 1 x. M ) ) )
45	40, 44	eqtrd 2271	. . . . . . . . . 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 6104	. . . . . . . 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 2274	. . . . . . 7 |- ( ( M e. NN /\ n e. NN ) -> ( M gcd ( M x. n ) ) = ( M gcd ( M x. ( n + 1 ) ) ) )
49	48	eqeq1d 2247	. . . . . 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 9329	. 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 )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   ` cfv 5380  (class class class)co 6088   CCcc 8183   1c1 8186   + caddc 8188   x. cmul 8190   NNcn 9313   ZZcz 9655   abscabs 11785   gcd cgcd 12755

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-iinf 4737  ax-cnex 8276  ax-resscn 8277  ax-1cn 8278  ax-1re 8279  ax-icn 8280  ax-addcl 8281  ax-addrcl 8282  ax-mulcl 8283  ax-mulrcl 8284  ax-addcom 8285  ax-mulcom 8286  ax-addass 8287  ax-mulass 8288  ax-distr 8289  ax-i2m1 8290  ax-0lt1 8291  ax-1rid 8292  ax-0id 8293  ax-rnegex 8294  ax-precex 8295  ax-cnre 8296  ax-pre-ltirr 8297  ax-pre-ltwlin 8298  ax-pre-lttrn 8299  ax-pre-apti 8300  ax-pre-ltadd 8301  ax-pre-mulgt0 8302  ax-pre-mulext 8303  ax-arch 8304  ax-caucvg 8305

This proof depends on definitions:  df-bi 117  df-stab 843  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-id 4440  df-po 4443  df-iso 4444  df-iord 4513  df-on 4515  df-ilim 4516  df-suc 4518  df-iom 4740  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-recs 6580  df-frec 6666  df-sup 7330  df-pnf 8368  df-mnf 8369  df-xr 8370  df-ltxr 8371  df-le 8372  df-sub 8506  df-neg 8507  df-reap 8911  df-ap 8918  df-div 9011  df-inn 9314  df-2 9372  df-3 9373  df-4 9374  df-n0 9575  df-z 9656  df-uz 9938  df-q 10036  df-rp 10072  df-fz 10429  df-fzo 10567  df-fl 10722  df-mod 10781  df-seqfrec 10906  df-exp 10997  df-cj 11629  df-re 11630  df-im 11631  df-rsqrt 11786  df-abs 11787  df-dvds 12580  df-gcd 12756

This theorem is used by:  gcdmultiplez  12823  rpmulgcd  12828