Theorem gcdmultiplez 10410

Description: Extend gcdmultiple 10409 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 8362	. . . 4 |- 0 e. ZZ
2		zdceq 8423	. . . 4 |- ( ( N e. ZZ /\ 0 e. ZZ ) -> DECID N = 0 )
3	1, 2	mpan2 415	. . 3 |- ( N e. ZZ -> DECID N = 0 )
4		exmiddc 777	. . 3 |- (DECID N = 0 -> ( N = 0 \/ -. N = 0 ) )
5		nncn 8047	. . . . . . . 8 |- ( M e. NN -> M e. CC )
6		mul01 7493	. . . . . . . . 9 |- ( M e. CC -> ( M x. 0 ) = 0 )
7	6	oveq2d 5548	. . . . . . . 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 270	. . . . . 6 |- ( ( M e. NN /\ N e. ZZ ) -> ( M gcd ( M x. 0 ) ) = ( M gcd 0 ) )
10		nnnn0 8295	. . . . . . . 8 |- ( M e. NN -> M e. NN0 )
11		nn0gcdid0 10372	. . . . . . . 8 |- ( M e. NN0 -> ( M gcd 0 ) = M )
12	10, 11	syl 14	. . . . . . 7 |- ( M e. NN -> ( M gcd 0 ) = M )
13	12	adantr 270	. . . . . 6 |- ( ( M e. NN /\ N e. ZZ ) -> ( M gcd 0 ) = M )
14	9, 13	eqtrd 2113	. . . . 5 |- ( ( M e. NN /\ N e. ZZ ) -> ( M gcd ( M x. 0 ) ) = M )
15		oveq2 5540	. . . . . . 7 |- ( N = 0 -> ( M x. N ) = ( M x. 0 ) )
16	15	oveq2d 5548	. . . . . 6 |- ( N = 0 -> ( M gcd ( M x. N ) ) = ( M gcd ( M x. 0 ) ) )
17	16	eqeq1d 2089	. . . . 5 |- ( N = 0 -> ( ( M gcd ( M x. N ) ) = M <-> ( M gcd ( M x. 0 ) ) = M ) )
18	14, 17	syl5ibr 154	. . . 4 |- ( N = 0 -> ( ( M e. NN /\ N e. ZZ ) -> ( M gcd ( M x. N ) ) = M ) )
19		df-ne 2246	. . . . 5 |- ( N =/= 0 <-> -. N = 0 )
20		zcn 8356	. . . . . . . . . . 11 |- ( N e. ZZ -> N e. CC )
21		absmul 9955	. . . . . . . . . . 11 |- ( ( M e. CC /\ N e. CC ) -> ( abs ` ( M x. N ) ) = ( ( abs ` M ) x. ( abs ` N ) ) )
22	5, 20, 21	syl2an 283	. . . . . . . . . 10 |- ( ( M e. NN /\ N e. ZZ ) -> ( abs ` ( M x. N ) ) = ( ( abs ` M ) x. ( abs ` N ) ) )
23		nnre 8046	. . . . . . . . . . . . 13 |- ( M e. NN -> M e. RR )
24	10	nn0ge0d 8344	. . . . . . . . . . . . 13 |- ( M e. NN -> 0 <_ M )
25	23, 24	absidd 10053	. . . . . . . . . . . 12 |- ( M e. NN -> ( abs ` M ) = M )
26	25	oveq1d 5547	. . . . . . . . . . 11 |- ( M e. NN -> ( ( abs ` M ) x. ( abs ` N ) ) = ( M x. ( abs ` N ) ) )
27	26	adantr 270	. . . . . . . . . 10 |- ( ( M e. NN /\ N e. ZZ ) -> ( ( abs ` M ) x. ( abs ` N ) ) = ( M x. ( abs ` N ) ) )
28	22, 27	eqtrd 2113	. . . . . . . . 9 |- ( ( M e. NN /\ N e. ZZ ) -> ( abs ` ( M x. N ) ) = ( M x. ( abs ` N ) ) )
29	28	oveq2d 5548	. . . . . . . 8 |- ( ( M e. NN /\ N e. ZZ ) -> ( M gcd ( abs ` ( M x. N ) ) ) = ( M gcd ( M x. ( abs ` N ) ) ) )
30	29	adantr 270	. . . . . . 7 |- ( ( ( M e. NN /\ N e. ZZ ) /\ N =/= 0 ) -> ( M gcd ( abs ` ( M x. N ) ) ) = ( M gcd ( M x. ( abs ` N ) ) ) )
31		simpll 495	. . . . . . . . 9 |- ( ( ( M e. NN /\ N e. ZZ ) /\ N =/= 0 ) -> M e. NN )
32	31	nnzd 8468	. . . . . . . 8 |- ( ( ( M e. NN /\ N e. ZZ ) /\ N =/= 0 ) -> M e. ZZ )
33		nnz 8370	. . . . . . . . . 10 |- ( M e. NN -> M e. ZZ )
34		zmulcl 8404	. . . . . . . . . 10 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M x. N ) e. ZZ )
35	33, 34	sylan 277	. . . . . . . . 9 |- ( ( M e. NN /\ N e. ZZ ) -> ( M x. N ) e. ZZ )
36	35	adantr 270	. . . . . . . 8 |- ( ( ( M e. NN /\ N e. ZZ ) /\ N =/= 0 ) -> ( M x. N ) e. ZZ )
37		gcdabs2 10381	. . . . . . . 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 403	. . . . . . 7 |- ( ( ( M e. NN /\ N e. ZZ ) /\ N =/= 0 ) -> ( M gcd ( abs ` ( M x. N ) ) ) = ( M gcd ( M x. N ) ) )
39		nnabscl 9986	. . . . . . . . 9 |- ( ( N e. ZZ /\ N =/= 0 ) -> ( abs ` N ) e. NN )
40		gcdmultiple 10409	. . . . . . . . 9 |- ( ( M e. NN /\ ( abs ` N ) e. NN ) -> ( M gcd ( M x. ( abs ` N ) ) ) = M )
41	39, 40	sylan2 280	. . . . . . . 8 |- ( ( M e. NN /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( M gcd ( M x. ( abs ` N ) ) ) = M )
42	41	anassrs 392	. . . . . . 7 |- ( ( ( M e. NN /\ N e. ZZ ) /\ N =/= 0 ) -> ( M gcd ( M x. ( abs ` N ) ) ) = M )
43	30, 38, 42	3eqtr3d 2121	. . . . . 6 |- ( ( ( M e. NN /\ N e. ZZ ) /\ N =/= 0 ) -> ( M gcd ( M x. N ) ) = M )
44	43	expcom 114	. . . . 5 |- ( N =/= 0 -> ( ( M e. NN /\ N e. ZZ ) -> ( M gcd ( M x. N ) ) = M ) )
45	19, 44	sylbir 133	. . . 4 |- ( -. N = 0 -> ( ( M e. NN /\ N e. ZZ ) -> ( M gcd ( M x. N ) ) = M ) )
46	18, 45	jaoi 668	. . 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 545	1 |- ( ( M e. NN /\ N e. ZZ ) -> ( M gcd ( M x. N ) ) = M )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 102   \/ wo 661  DECID wdc 775   = wceq 1284   e. wcel 1433   =/= wne 2245   ` cfv 4922  (class class class)co 5532   CCcc 6979   0cc0 6981   x. cmul 6986   NNcn 8039   NN0cn0 8288   ZZcz 8351   abscabs 9883   gcd cgcd 10338

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-coll 3893  ax-sep 3896  ax-nul 3904  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-iinf 4329  ax-cnex 7067  ax-resscn 7068  ax-1cn 7069  ax-1re 7070  ax-icn 7071  ax-addcl 7072  ax-addrcl 7073  ax-mulcl 7074  ax-mulrcl 7075  ax-addcom 7076  ax-mulcom 7077  ax-addass 7078  ax-mulass 7079  ax-distr 7080  ax-i2m1 7081  ax-0lt1 7082  ax-1rid 7083  ax-0id 7084  ax-rnegex 7085  ax-precex 7086  ax-cnre 7087  ax-pre-ltirr 7088  ax-pre-ltwlin 7089  ax-pre-lttrn 7090  ax-pre-apti 7091  ax-pre-ltadd 7092  ax-pre-mulgt0 7093  ax-pre-mulext 7094  ax-arch 7095  ax-caucvg 7096

This theorem depends on definitions:  df-bi 115  df-dc 776  df-3or 920  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-nel 2340  df-ral 2353  df-rex 2354  df-reu 2355  df-rmo 2356  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-if 3352  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-int 3637  df-iun 3680  df-br 3786  df-opab 3840  df-mpt 3841  df-tr 3876  df-id 4048  df-po 4051  df-iso 4052  df-iord 4121  df-on 4123  df-suc 4126  df-iom 4332  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930  df-riota 5488  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-1st 5787  df-2nd 5788  df-recs 5943  df-frec 6001  df-sup 6397  df-pnf 7155  df-mnf 7156  df-xr 7157  df-ltxr 7158  df-le 7159  df-sub 7281  df-neg 7282  df-reap 7675  df-ap 7682  df-div 7761  df-inn 8040  df-2 8098  df-3 8099  df-4 8100  df-n0 8289  df-z 8352  df-uz 8620  df-q 8705  df-rp 8735  df-fz 9030  df-fzo 9153  df-fl 9274  df-mod 9325  df-iseq 9432  df-iexp 9476  df-cj 9729  df-re 9730  df-im 9731  df-rsqrt 9884  df-abs 9885  df-dvds 10196  df-gcd 10339

This theorem is referenced by: (None)