Theorem gcdmultiplez 12823

Description: Extend gcdmultiple 12822 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 9666	. . . 4 |- 0 e. ZZ
2		zdceq 9731	. . . 4 |- ( ( N e. ZZ /\ 0 e. ZZ ) -> DECID N = 0 )
3	1, 2	mpan2 429	. . 3 |- ( N e. ZZ -> DECID N = 0 )
4		exmiddc 848	. . 3 |- (DECID N = 0 -> ( N = 0 \/ -. N = 0 ) )
5		nncn 9321	. . . . . . . 8 |- ( M e. NN -> M e. CC )
6		mul01 8723	. . . . . . . . 9 |- ( M e. CC -> ( M x. 0 ) = 0 )
7	6	oveq2d 6104	. . . . . . . 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 9581	. . . . . . . 8 |- ( M e. NN -> M e. NN0 )
11		nn0gcdid0 12783	. . . . . . . 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 2271	. . . . 5 |- ( ( M e. NN /\ N e. ZZ ) -> ( M gcd ( M x. 0 ) ) = M )
15		oveq2 6096	. . . . . . 7 |- ( N = 0 -> ( M x. N ) = ( M x. 0 ) )
16	15	oveq2d 6104	. . . . . 6 |- ( N = 0 -> ( M gcd ( M x. N ) ) = ( M gcd ( M x. 0 ) ) )
17	16	eqeq1d 2247	. . . . 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 2421	. . . . 5 |- ( N =/= 0 <-> -. N = 0 )
20		zcn 9660	. . . . . . . . . . 11 |- ( N e. ZZ -> N e. CC )
21		absmul 11857	. . . . . . . . . . 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 9320	. . . . . . . . . . . . 13 |- ( M e. NN -> M e. RR )
24	10	nn0ge0d 9634	. . . . . . . . . . . . 13 |- ( M e. NN -> 0 <_ M )
25	23, 24	absidd 11956	. . . . . . . . . . . 12 |- ( M e. NN -> ( abs ` M ) = M )
26	25	oveq1d 6103	. . . . . . . . . . 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 2271	. . . . . . . . 9 |- ( ( M e. NN /\ N e. ZZ ) -> ( abs ` ( M x. N ) ) = ( M x. ( abs ` N ) ) )
29	28	oveq2d 6104	. . . . . . . 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 531	. . . . . . . . 9 |- ( ( ( M e. NN /\ N e. ZZ ) /\ N =/= 0 ) -> M e. NN )
32	31	nnzd 9778	. . . . . . . 8 |- ( ( ( M e. NN /\ N e. ZZ ) /\ N =/= 0 ) -> M e. ZZ )
33		nnz 9674	. . . . . . . . . 10 |- ( M e. NN -> M e. ZZ )
34		zmulcl 9709	. . . . . . . . . 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 12792	. . . . . . . 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 415	. . . . . . 7 |- ( ( ( M e. NN /\ N e. ZZ ) /\ N =/= 0 ) -> ( M gcd ( abs ` ( M x. N ) ) ) = ( M gcd ( M x. N ) ) )
39		nnabscl 11889	. . . . . . . . 9 |- ( ( N e. ZZ /\ N =/= 0 ) -> ( abs ` N ) e. NN )
40		gcdmultiple 12822	. . . . . . . . 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 404	. . . . . . 7 |- ( ( ( M e. NN /\ N e. ZZ ) /\ N =/= 0 ) -> ( M gcd ( M x. ( abs ` N ) ) ) = M )
43	30, 38, 42	3eqtr3d 2279	. . . . . 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 728	. . 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 587	1 |- ( ( M e. NN /\ N e. ZZ ) -> ( M gcd ( M x. N ) ) = M )

This proof depends on syntax axioms:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 720  DECID wdc 846   = wceq 1402   e. wcel 2209   =/= wne 2420   ` cfv 5380  (class class class)co 6088   CCcc 8183   0cc0 8185   x. cmul 8190   NNcn 9313   NN0cn0 9574   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: (None)