Theorem gcdmultiple 12596

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 6026	. . . . . 6 |- ( k = 1 -> ( M x. k ) = ( M x. 1 ) )
2	1	oveq2d 6034	. . . . 5 |- ( k = 1 -> ( M gcd ( M x. k ) ) = ( M gcd ( M x. 1 ) ) )
3	2	eqeq1d 2240	. . . 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 6026	. . . . . 6 |- ( k = n -> ( M x. k ) = ( M x. n ) )
6	5	oveq2d 6034	. . . . 5 |- ( k = n -> ( M gcd ( M x. k ) ) = ( M gcd ( M x. n ) ) )
7	6	eqeq1d 2240	. . . 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 6026	. . . . . 6 |- ( k = ( n + 1 ) -> ( M x. k ) = ( M x. ( n + 1 ) ) )
10	9	oveq2d 6034	. . . . 5 |- ( k = ( n + 1 ) -> ( M gcd ( M x. k ) ) = ( M gcd ( M x. ( n + 1 ) ) ) )
11	10	eqeq1d 2240	. . . 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 6026	. . . . . 6 |- ( k = N -> ( M x. k ) = ( M x. N ) )
14	13	oveq2d 6034	. . . . 5 |- ( k = N -> ( M gcd ( M x. k ) ) = ( M gcd ( M x. N ) ) )
15	14	eqeq1d 2240	. . . 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 9151	. . . . . 6 |- ( M e. NN -> M e. CC )
18	17	mulridd 8196	. . . . 5 |- ( M e. NN -> ( M x. 1 ) = M )
19	18	oveq2d 6034	. . . 4 |- ( M e. NN -> ( M gcd ( M x. 1 ) ) = ( M gcd M ) )
20		nnz 9498	. . . . . 6 |- ( M e. NN -> M e. ZZ )
21		gcdid 12562	. . . . . 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 9150	. . . . . 6 |- ( M e. NN -> M e. RR )
24		nnnn0 9409	. . . . . . 7 |- ( M e. NN -> M e. NN0 )
25	24	nn0ge0d 9458	. . . . . 6 |- ( M e. NN -> 0 <_ M )
26	23, 25	absidd 11732	. . . . 5 |- ( M e. NN -> ( abs ` M ) = M )
27	22, 26	eqtrd 2264	. . . 4 |- ( M e. NN -> ( M gcd M ) = M )
28	19, 27	eqtrd 2264	. . 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 9498	. . . . . . . . . 10 |- ( n e. NN -> n e. ZZ )
31		zmulcl 9533	. . . . . . . . . 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 9505	. . . . . . . . . 10 |- 1 e. ZZ
34		gcdaddm 12560	. . . . . . . . . 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 1360	. . . . . . . . 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 9151	. . . . . . . . . 10 |- ( n e. NN -> n e. CC )
38		ax-1cn 8125	. . . . . . . . . . . 12 |- 1 e. CC
39		adddi 8164	. . . . . . . . . . . 12 |- ( ( M e. CC /\ n e. CC /\ 1 e. CC ) -> ( M x. ( n + 1 ) ) = ( ( M x. n ) + ( M x. 1 ) ) )
40	38, 39	mp3an3 1362	. . . . . . . . . . 11 |- ( ( M e. CC /\ n e. CC ) -> ( M x. ( n + 1 ) ) = ( ( M x. n ) + ( M x. 1 ) ) )
41		mulcom 8161	. . . . . . . . . . . . . 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 6034	. . . . . . . . . . 11 |- ( ( M e. CC /\ n e. CC ) -> ( ( M x. n ) + ( M x. 1 ) ) = ( ( M x. n ) + ( 1 x. M ) ) )
45	40, 44	eqtrd 2264	. . . . . . . . . 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 6034	. . . . . . . 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 2267	. . . . . . 7 |- ( ( M e. NN /\ n e. NN ) -> ( M gcd ( M x. n ) ) = ( M gcd ( M x. ( n + 1 ) ) ) )
49	48	eqeq1d 2240	. . . . . 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 9159	. 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 1397   e. wcel 2202   ` cfv 5326  (class class class)co 6018   CCcc 8030   1c1 8033   + caddc 8035   x. cmul 8037   NNcn 9143   ZZcz 9479   abscabs 11562   gcd cgcd 12529

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-mulrcl 8131  ax-addcom 8132  ax-mulcom 8133  ax-addass 8134  ax-mulass 8135  ax-distr 8136  ax-i2m1 8137  ax-0lt1 8138  ax-1rid 8139  ax-0id 8140  ax-rnegex 8141  ax-precex 8142  ax-cnre 8143  ax-pre-ltirr 8144  ax-pre-ltwlin 8145  ax-pre-lttrn 8146  ax-pre-apti 8147  ax-pre-ltadd 8148  ax-pre-mulgt0 8149  ax-pre-mulext 8150  ax-arch 8151  ax-caucvg 8152

This theorem depends on definitions:  df-bi 117  df-stab 838  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-po 4393  df-iso 4394  df-iord 4463  df-on 4465  df-ilim 4466  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1st 6303  df-2nd 6304  df-recs 6471  df-frec 6557  df-sup 7183  df-pnf 8216  df-mnf 8217  df-xr 8218  df-ltxr 8219  df-le 8220  df-sub 8352  df-neg 8353  df-reap 8755  df-ap 8762  df-div 8853  df-inn 9144  df-2 9202  df-3 9203  df-4 9204  df-n0 9403  df-z 9480  df-uz 9756  df-q 9854  df-rp 9889  df-fz 10244  df-fzo 10378  df-fl 10531  df-mod 10586  df-seqfrec 10711  df-exp 10802  df-cj 11407  df-re 11408  df-im 11409  df-rsqrt 11563  df-abs 11564  df-dvds 12354  df-gcd 12530

This theorem is referenced by:  gcdmultiplez  12597  rpmulgcd  12602