Theorem gcdmultiplez 12563

Description: Extend gcdmultiple 12562 so N can be an integer. (Contributed by Scott Fenton, 18-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Assertion

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

Proof of Theorem gcdmultiplez

Step	Hyp	Ref	Expression
1		0z 9473	. . . 4 |- 0 e. ZZ
2		zdceq 9538	. . . 4 |- ( ( N e. ZZ /\ 0 e. ZZ ) -> DECID N = 0 )
3	1, 2	mpan2 425	. . 3 |- ( N e. ZZ -> DECID N = 0 )
4		exmiddc 841	. . 3 |- (DECID N = 0 -> ( N = 0 \/ -. N = 0 ) )
5		nncn 9134	. . . . . . . 8 |- ( M e. NN -> M e. CC )
6		mul01 8551	. . . . . . . . 9 |- ( M e. CC -> ( M x. 0 ) = 0 )
7	6	oveq2d 6026	. . . . . . . 8 |- ( M e. CC -> ( M gcd ( M x. 0 ) ) = ( M gcd 0 ) )
8	5, 7	syl 14	. . . . . . 7 |- ( M e. NN -> ( M gcd ( M x. 0 ) ) = ( M gcd 0 ) )
9	8	adantr 276	. . . . . 6 |- ( ( M e. NN /\ N e. ZZ ) -> ( M gcd ( M x. 0 ) ) = ( M gcd 0 ) )
10		nnnn0 9392	. . . . . . . 8 |- ( M e. NN -> M e. NN0 )
11		nn0gcdid0 12523	. . . . . . . 8 |- ( M e. NN0 -> ( M gcd 0 ) = M )
12	10, 11	syl 14	. . . . . . 7 |- ( M e. NN -> ( M gcd 0 ) = M )
13	12	adantr 276	. . . . . 6 |- ( ( M e. NN /\ N e. ZZ ) -> ( M gcd 0 ) = M )
14	9, 13	eqtrd 2262	. . . . 5 |- ( ( M e. NN /\ N e. ZZ ) -> ( M gcd ( M x. 0 ) ) = M )
15		oveq2 6018	. . . . . . 7 |- ( N = 0 -> ( M x. N ) = ( M x. 0 ) )
16	15	oveq2d 6026	. . . . . 6 |- ( N = 0 -> ( M gcd ( M x. N ) ) = ( M gcd ( M x. 0 ) ) )
17	16	eqeq1d 2238	. . . . 5 |- ( N = 0 -> ( ( M gcd ( M x. N ) ) = M <-> ( M gcd ( M x. 0 ) ) = M ) )
18	14, 17	imbitrrid 156	. . . 4 |- ( N = 0 -> ( ( M e. NN /\ N e. ZZ ) -> ( M gcd ( M x. N ) ) = M ) )
19		df-ne 2401	. . . . 5 |- ( N =/= 0 <-> -. N = 0 )
20		zcn 9467	. . . . . . . . . . 11 |- ( N e. ZZ -> N e. CC )
21		absmul 11601	. . . . . . . . . . 11 |- ( ( M e. CC /\ N e. CC ) -> ( abs ` ( M x. N ) ) = ( ( abs ` M ) x. ( abs ` N ) ) )
22	5, 20, 21	syl2an 289	. . . . . . . . . 10 |- ( ( M e. NN /\ N e. ZZ ) -> ( abs ` ( M x. N ) ) = ( ( abs ` M ) x. ( abs ` N ) ) )
23		nnre 9133	. . . . . . . . . . . . 13 |- ( M e. NN -> M e. RR )
24	10	nn0ge0d 9441	. . . . . . . . . . . . 13 |- ( M e. NN -> 0 <_ M )
25	23, 24	absidd 11699	. . . . . . . . . . . 12 |- ( M e. NN -> ( abs ` M ) = M )
26	25	oveq1d 6025	. . . . . . . . . . 11 |- ( M e. NN -> ( ( abs ` M ) x. ( abs ` N ) ) = ( M x. ( abs ` N ) ) )
27	26	adantr 276	. . . . . . . . . 10 |- ( ( M e. NN /\ N e. ZZ ) -> ( ( abs ` M ) x. ( abs ` N ) ) = ( M x. ( abs ` N ) ) )
28	22, 27	eqtrd 2262	. . . . . . . . 9 |- ( ( M e. NN /\ N e. ZZ ) -> ( abs ` ( M x. N ) ) = ( M x. ( abs ` N ) ) )
29	28	oveq2d 6026	. . . . . . . 8 |- ( ( M e. NN /\ N e. ZZ ) -> ( M gcd ( abs ` ( M x. N ) ) ) = ( M gcd ( M x. ( abs ` N ) ) ) )
30	29	adantr 276	. . . . . . 7 |- ( ( ( M e. NN /\ N e. ZZ ) /\ N =/= 0 ) -> ( M gcd ( abs ` ( M x. N ) ) ) = ( M gcd ( M x. ( abs ` N ) ) ) )
31		simpll 527	. . . . . . . . 9 |- ( ( ( M e. NN /\ N e. ZZ ) /\ N =/= 0 ) -> M e. NN )
32	31	nnzd 9584	. . . . . . . 8 |- ( ( ( M e. NN /\ N e. ZZ ) /\ N =/= 0 ) -> M e. ZZ )
33		nnz 9481	. . . . . . . . . 10 |- ( M e. NN -> M e. ZZ )
34		zmulcl 9516	. . . . . . . . . 10 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M x. N ) e. ZZ )
35	33, 34	sylan 283	. . . . . . . . 9 |- ( ( M e. NN /\ N e. ZZ ) -> ( M x. N ) e. ZZ )
36	35	adantr 276	. . . . . . . 8 |- ( ( ( M e. NN /\ N e. ZZ ) /\ N =/= 0 ) -> ( M x. N ) e. ZZ )
37		gcdabs2 12532	. . . . . . . 8 |- ( ( M e. ZZ /\ ( M x. N ) e. ZZ ) -> ( M gcd ( abs ` ( M x. N ) ) ) = ( M gcd ( M x. N ) ) )
38	32, 36, 37	syl2anc 411	. . . . . . 7 |- ( ( ( M e. NN /\ N e. ZZ ) /\ N =/= 0 ) -> ( M gcd ( abs ` ( M x. N ) ) ) = ( M gcd ( M x. N ) ) )
39		nnabscl 11632	. . . . . . . . 9 |- ( ( N e. ZZ /\ N =/= 0 ) -> ( abs ` N ) e. NN )
40		gcdmultiple 12562	. . . . . . . . 9 |- ( ( M e. NN /\ ( abs ` N ) e. NN ) -> ( M gcd ( M x. ( abs ` N ) ) ) = M )
41	39, 40	sylan2 286	. . . . . . . 8 |- ( ( M e. NN /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( M gcd ( M x. ( abs ` N ) ) ) = M )
42	41	anassrs 400	. . . . . . 7 |- ( ( ( M e. NN /\ N e. ZZ ) /\ N =/= 0 ) -> ( M gcd ( M x. ( abs ` N ) ) ) = M )
43	30, 38, 42	3eqtr3d 2270	. . . . . 6 |- ( ( ( M e. NN /\ N e. ZZ ) /\ N =/= 0 ) -> ( M gcd ( M x. N ) ) = M )
44	43	expcom 116	. . . . 5 |- ( N =/= 0 -> ( ( M e. NN /\ N e. ZZ ) -> ( M gcd ( M x. N ) ) = M ) )
45	19, 44	sylbir 135	. . . 4 |- ( -. N = 0 -> ( ( M e. NN /\ N e. ZZ ) -> ( M gcd ( M x. N ) ) = M ) )
46	18, 45	jaoi 721	. . 3 |- ( ( N = 0 \/ -. N = 0 ) -> ( ( M e. NN /\ N e. ZZ ) -> ( M gcd ( M x. N ) ) = M ) )
47	3, 4, 46	3syl 17	. 2 |- ( N e. ZZ -> ( ( M e. NN /\ N e. ZZ ) -> ( M gcd ( M x. N ) ) = M ) )
48	47	anabsi7 581	1 |- ( ( M e. NN /\ N e. ZZ ) -> ( M gcd ( M x. N ) ) = M )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 713  DECID wdc 839   = wceq 1395   e. wcel 2200   =/= wne 2400   ` cfv 5321  (class class class)co 6010   CCcc 8013   0cc0 8015   x. cmul 8020   NNcn 9126   NN0cn0 9385   ZZcz 9462   abscabs 11529   gcd cgcd 12495

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4259  ax-pr 4294  ax-un 4525  ax-setind 4630  ax-iinf 4681  ax-cnex 8106  ax-resscn 8107  ax-1cn 8108  ax-1re 8109  ax-icn 8110  ax-addcl 8111  ax-addrcl 8112  ax-mulcl 8113  ax-mulrcl 8114  ax-addcom 8115  ax-mulcom 8116  ax-addass 8117  ax-mulass 8118  ax-distr 8119  ax-i2m1 8120  ax-0lt1 8121  ax-1rid 8122  ax-0id 8123  ax-rnegex 8124  ax-precex 8125  ax-cnre 8126  ax-pre-ltirr 8127  ax-pre-ltwlin 8128  ax-pre-lttrn 8129  ax-pre-apti 8130  ax-pre-ltadd 8131  ax-pre-mulgt0 8132  ax-pre-mulext 8133  ax-arch 8134  ax-caucvg 8135

This theorem depends on definitions:  df-bi 117  df-stab 836  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4385  df-po 4388  df-iso 4389  df-iord 4458  df-on 4460  df-ilim 4461  df-suc 4463  df-iom 4684  df-xp 4726  df-rel 4727  df-cnv 4728  df-co 4729  df-dm 4730  df-rn 4731  df-res 4732  df-ima 4733  df-iota 5281  df-fun 5323  df-fn 5324  df-f 5325  df-f1 5326  df-fo 5327  df-f1o 5328  df-fv 5329  df-riota 5963  df-ov 6013  df-oprab 6014  df-mpo 6015  df-1st 6295  df-2nd 6296  df-recs 6462  df-frec 6548  df-sup 7167  df-pnf 8199  df-mnf 8200  df-xr 8201  df-ltxr 8202  df-le 8203  df-sub 8335  df-neg 8336  df-reap 8738  df-ap 8745  df-div 8836  df-inn 9127  df-2 9185  df-3 9186  df-4 9187  df-n0 9386  df-z 9463  df-uz 9739  df-q 9832  df-rp 9867  df-fz 10222  df-fzo 10356  df-fl 10507  df-mod 10562  df-seqfrec 10687  df-exp 10778  df-cj 11374  df-re 11375  df-im 11376  df-rsqrt 11530  df-abs 11531  df-dvds 12320  df-gcd 12496

This theorem is referenced by: (None)