Theorem gcdmultiplez 12713

Description: Extend gcdmultiple 12712 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 9587	. . . 4 |- 0 e. ZZ
2		zdceq 9652	. . . 4 |- ( ( N e. ZZ /\ 0 e. ZZ ) -> DECID N = 0 )
3	1, 2	mpan2 425	. . 3 |- ( N e. ZZ -> DECID N = 0 )
4		exmiddc 844	. . 3 |- (DECID N = 0 -> ( N = 0 \/ -. N = 0 ) )
5		nncn 9244	. . . . . . . 8 |- ( M e. NN -> M e. CC )
6		mul01 8661	. . . . . . . . 9 |- ( M e. CC -> ( M x. 0 ) = 0 )
7	6	oveq2d 6065	. . . . . . . 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 9502	. . . . . . . 8 |- ( M e. NN -> M e. NN0 )
11		nn0gcdid0 12673	. . . . . . . 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 2265	. . . . 5 |- ( ( M e. NN /\ N e. ZZ ) -> ( M gcd ( M x. 0 ) ) = M )
15		oveq2 6057	. . . . . . 7 |- ( N = 0 -> ( M x. N ) = ( M x. 0 ) )
16	15	oveq2d 6065	. . . . . 6 |- ( N = 0 -> ( M gcd ( M x. N ) ) = ( M gcd ( M x. 0 ) ) )
17	16	eqeq1d 2241	. . . . 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 2413	. . . . 5 |- ( N =/= 0 <-> -. N = 0 )
20		zcn 9581	. . . . . . . . . . 11 |- ( N e. ZZ -> N e. CC )
21		absmul 11750	. . . . . . . . . . 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 9243	. . . . . . . . . . . . 13 |- ( M e. NN -> M e. RR )
24	10	nn0ge0d 9555	. . . . . . . . . . . . 13 |- ( M e. NN -> 0 <_ M )
25	23, 24	absidd 11848	. . . . . . . . . . . 12 |- ( M e. NN -> ( abs ` M ) = M )
26	25	oveq1d 6064	. . . . . . . . . . 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 2265	. . . . . . . . 9 |- ( ( M e. NN /\ N e. ZZ ) -> ( abs ` ( M x. N ) ) = ( M x. ( abs ` N ) ) )
29	28	oveq2d 6065	. . . . . . . 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 9698	. . . . . . . 8 |- ( ( ( M e. NN /\ N e. ZZ ) /\ N =/= 0 ) -> M e. ZZ )
33		nnz 9595	. . . . . . . . . 10 |- ( M e. NN -> M e. ZZ )
34		zmulcl 9630	. . . . . . . . . 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 12682	. . . . . . . 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 11781	. . . . . . . . 9 |- ( ( N e. ZZ /\ N =/= 0 ) -> ( abs ` N ) e. NN )
40		gcdmultiple 12712	. . . . . . . . 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 2273	. . . . . 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 724	. . 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 583	1 |- ( ( M e. NN /\ N e. ZZ ) -> ( M gcd ( M x. N ) ) = M )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 716  DECID wdc 842   = wceq 1398   e. wcel 2203   =/= wne 2412   ` cfv 5351  (class class class)co 6049   CCcc 8124   0cc0 8126   x. cmul 8131   NNcn 9236   NN0cn0 9495   ZZcz 9576   abscabs 11678   gcd cgcd 12645

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4224  ax-sep 4227  ax-nul 4235  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-iinf 4709  ax-cnex 8217  ax-resscn 8218  ax-1cn 8219  ax-1re 8220  ax-icn 8221  ax-addcl 8222  ax-addrcl 8223  ax-mulcl 8224  ax-mulrcl 8225  ax-addcom 8226  ax-mulcom 8227  ax-addass 8228  ax-mulass 8229  ax-distr 8230  ax-i2m1 8231  ax-0lt1 8232  ax-1rid 8233  ax-0id 8234  ax-rnegex 8235  ax-precex 8236  ax-cnre 8237  ax-pre-ltirr 8238  ax-pre-ltwlin 8239  ax-pre-lttrn 8240  ax-pre-apti 8241  ax-pre-ltadd 8242  ax-pre-mulgt0 8243  ax-pre-mulext 8244  ax-arch 8245  ax-caucvg 8246

This theorem depends on definitions:  df-bi 117  df-stab 839  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3508  df-if 3620  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-iun 3992  df-br 4109  df-opab 4171  df-mpt 4172  df-tr 4208  df-id 4413  df-po 4416  df-iso 4417  df-iord 4486  df-on 4488  df-ilim 4489  df-suc 4491  df-iom 4712  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-f1 5356  df-fo 5357  df-f1o 5358  df-fv 5359  df-riota 6002  df-ov 6052  df-oprab 6053  df-mpo 6054  df-1st 6333  df-2nd 6334  df-recs 6535  df-frec 6621  df-sup 7274  df-pnf 8309  df-mnf 8310  df-xr 8311  df-ltxr 8312  df-le 8313  df-sub 8445  df-neg 8446  df-reap 8848  df-ap 8855  df-div 8946  df-inn 9237  df-2 9295  df-3 9296  df-4 9297  df-n0 9496  df-z 9577  df-uz 9853  df-q 9951  df-rp 9986  df-fz 10342  df-fzo 10476  df-fl 10629  df-mod 10684  df-seqfrec 10809  df-exp 10900  df-cj 11523  df-re 11524  df-im 11525  df-rsqrt 11679  df-abs 11680  df-dvds 12470  df-gcd 12646

This theorem is referenced by: (None)