Theorem gcd0id 12700

Description: The gcd of 0 and an integer is the integer's absolute value. (Contributed by Paul Chapman, 21-Mar-2011.)

Assertion

Ref	Expression
gcd0id	|- ( N e. ZZ -> ( 0 gcd N ) = ( abs ` N ) )

Proof of Theorem gcd0id

Step	Hyp	Ref	Expression
1		gcd0val 12681	. . . 4 |- ( 0 gcd 0 ) = 0
2		oveq2 6066	. . . 4 |- ( N = 0 -> ( 0 gcd N ) = ( 0 gcd 0 ) )
3		fveq2 5675	. . . . 5 |- ( N = 0 -> ( abs ` N ) = ( abs ` 0 ) )
4		abs0 11768	. . . . 5 |- ( abs ` 0 ) = 0
5	3, 4	eqtrdi 2283	. . . 4 |- ( N = 0 -> ( abs ` N ) = 0 )
6	1, 2, 5	3eqtr4a 2293	. . 3 |- ( N = 0 -> ( 0 gcd N ) = ( abs ` N ) )
7	6	adantl 277	. 2 |- ( ( N e. ZZ /\ N = 0 ) -> ( 0 gcd N ) = ( abs ` N ) )
8		df-ne 2415	. . 3 |- ( N =/= 0 <-> -. N = 0 )
9		0z 9605	. . . . . . . 8 |- 0 e. ZZ
10		gcddvds 12684	. . . . . . . 8 |- ( ( 0 e. ZZ /\ N e. ZZ ) -> ( ( 0 gcd N ) || 0 /\ ( 0 gcd N ) || N ) )
11	9, 10	mpan 424	. . . . . . 7 |- ( N e. ZZ -> ( ( 0 gcd N ) || 0 /\ ( 0 gcd N ) || N ) )
12	11	simprd 114	. . . . . 6 |- ( N e. ZZ -> ( 0 gcd N ) || N )
13	12	adantr 276	. . . . 5 |- ( ( N e. ZZ /\ N =/= 0 ) -> ( 0 gcd N ) || N )
14		gcdcl 12687	. . . . . . . . 9 |- ( ( 0 e. ZZ /\ N e. ZZ ) -> ( 0 gcd N ) e. NN0 )
15	9, 14	mpan 424	. . . . . . . 8 |- ( N e. ZZ -> ( 0 gcd N ) e. NN0 )
16	15	nn0zd 9716	. . . . . . 7 |- ( N e. ZZ -> ( 0 gcd N ) e. ZZ )
17		dvdsleabs 12556	. . . . . . 7 |- ( ( ( 0 gcd N ) e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> ( ( 0 gcd N ) || N -> ( 0 gcd N ) <_ ( abs ` N ) ) )
18	16, 17	syl3an1 1307	. . . . . 6 |- ( ( N e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> ( ( 0 gcd N ) || N -> ( 0 gcd N ) <_ ( abs ` N ) ) )
19	18	3anidm12 1332	. . . . 5 |- ( ( N e. ZZ /\ N =/= 0 ) -> ( ( 0 gcd N ) || N -> ( 0 gcd N ) <_ ( abs ` N ) ) )
20	13, 19	mpd 13	. . . 4 |- ( ( N e. ZZ /\ N =/= 0 ) -> ( 0 gcd N ) <_ ( abs ` N ) )
21		zabscl 11796	. . . . . . . 8 |- ( N e. ZZ -> ( abs ` N ) e. ZZ )
22		dvds0 12517	. . . . . . . 8 |- ( ( abs ` N ) e. ZZ -> ( abs ` N ) || 0 )
23	21, 22	syl 14	. . . . . . 7 |- ( N e. ZZ -> ( abs ` N ) || 0 )
24		iddvds 12515	. . . . . . . 8 |- ( N e. ZZ -> N || N )
25		absdvdsb 12520	. . . . . . . . 9 |- ( ( N e. ZZ /\ N e. ZZ ) -> ( N || N <-> ( abs ` N ) || N ) )
26	25	anidms 397	. . . . . . . 8 |- ( N e. ZZ -> ( N || N <-> ( abs ` N ) || N ) )
27	24, 26	mpbid 147	. . . . . . 7 |- ( N e. ZZ -> ( abs ` N ) || N )
28	23, 27	jca 306	. . . . . 6 |- ( N e. ZZ -> ( ( abs ` N ) || 0 /\ ( abs ` N ) || N ) )
29	28	adantr 276	. . . . 5 |- ( ( N e. ZZ /\ N =/= 0 ) -> ( ( abs ` N ) || 0 /\ ( abs ` N ) || N ) )
30		eqid 2234	. . . . . . . . 9 |- 0 = 0
31	30	biantrur 303	. . . . . . . 8 |- ( N = 0 <-> ( 0 = 0 /\ N = 0 ) )
32	31	necon3abii 2450	. . . . . . 7 |- ( N =/= 0 <-> -. ( 0 = 0 /\ N = 0 ) )
33		dvdslegcd 12685	. . . . . . . . . 10 |- ( ( ( ( abs ` N ) e. ZZ /\ 0 e. ZZ /\ N e. ZZ ) /\ -. ( 0 = 0 /\ N = 0 ) ) -> ( ( ( abs ` N ) || 0 /\ ( abs ` N ) || N ) -> ( abs ` N ) <_ ( 0 gcd N ) ) )
34	33	ex 115	. . . . . . . . 9 |- ( ( ( abs ` N ) e. ZZ /\ 0 e. ZZ /\ N e. ZZ ) -> ( -. ( 0 = 0 /\ N = 0 ) -> ( ( ( abs ` N ) || 0 /\ ( abs ` N ) || N ) -> ( abs ` N ) <_ ( 0 gcd N ) ) ) )
35	9, 34	mp3an2 1362	. . . . . . . 8 |- ( ( ( abs ` N ) e. ZZ /\ N e. ZZ ) -> ( -. ( 0 = 0 /\ N = 0 ) -> ( ( ( abs ` N ) || 0 /\ ( abs ` N ) || N ) -> ( abs ` N ) <_ ( 0 gcd N ) ) ) )
36	21, 35	mpancom 422	. . . . . . 7 |- ( N e. ZZ -> ( -. ( 0 = 0 /\ N = 0 ) -> ( ( ( abs ` N ) || 0 /\ ( abs ` N ) || N ) -> ( abs ` N ) <_ ( 0 gcd N ) ) ) )
37	32, 36	biimtrid 152	. . . . . 6 |- ( N e. ZZ -> ( N =/= 0 -> ( ( ( abs ` N ) || 0 /\ ( abs ` N ) || N ) -> ( abs ` N ) <_ ( 0 gcd N ) ) ) )
38	37	imp 124	. . . . 5 |- ( ( N e. ZZ /\ N =/= 0 ) -> ( ( ( abs ` N ) || 0 /\ ( abs ` N ) || N ) -> ( abs ` N ) <_ ( 0 gcd N ) ) )
39	29, 38	mpd 13	. . . 4 |- ( ( N e. ZZ /\ N =/= 0 ) -> ( abs ` N ) <_ ( 0 gcd N ) )
40	16	zred 9718	. . . . . 6 |- ( N e. ZZ -> ( 0 gcd N ) e. RR )
41	21	zred 9718	. . . . . 6 |- ( N e. ZZ -> ( abs ` N ) e. RR )
42	40, 41	letri3d 8405	. . . . 5 |- ( N e. ZZ -> ( ( 0 gcd N ) = ( abs ` N ) <-> ( ( 0 gcd N ) <_ ( abs ` N ) /\ ( abs ` N ) <_ ( 0 gcd N ) ) ) )
43	42	adantr 276	. . . 4 |- ( ( N e. ZZ /\ N =/= 0 ) -> ( ( 0 gcd N ) = ( abs ` N ) <-> ( ( 0 gcd N ) <_ ( abs ` N ) /\ ( abs ` N ) <_ ( 0 gcd N ) ) ) )
44	20, 39, 43	mpbir2and 953	. . 3 |- ( ( N e. ZZ /\ N =/= 0 ) -> ( 0 gcd N ) = ( abs ` N ) )
45	8, 44	sylan2br 288	. 2 |- ( ( N e. ZZ /\ -. N = 0 ) -> ( 0 gcd N ) = ( abs ` N ) )
46		zdceq 9670	. . . 4 |- ( ( N e. ZZ /\ 0 e. ZZ ) -> DECID N = 0 )
47	9, 46	mpan2 425	. . 3 |- ( N e. ZZ -> DECID N = 0 )
48		exmiddc 844	. . 3 |- (DECID N = 0 -> ( N = 0 \/ -. N = 0 ) )
49	47, 48	syl 14	. 2 |- ( N e. ZZ -> ( N = 0 \/ -. N = 0 ) )
50	7, 45, 49	mpjaodan 806	1 |- ( N e. ZZ -> ( 0 gcd N ) = ( abs ` N ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 716  DECID wdc 842   /\ w3a 1005   = wceq 1398   e. wcel 2205   =/= wne 2414   class class class wbr 4114   ` cfv 5357  (class class class)co 6058   0cc0 8143   <_ cle 8325   NN0cn0 9513   ZZcz 9594   abscabs 11707   || cdvds 12498   gcd cgcd 12674

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 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-mulrcl 8242  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-precex 8253  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259  ax-pre-mulgt0 8260  ax-pre-mulext 8261  ax-arch 8262  ax-caucvg 8263

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-po 4422  df-iso 4423  df-iord 4492  df-on 4494  df-ilim 4495  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-frec 6635  df-sup 7288  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-reap 8866  df-ap 8873  df-div 8964  df-inn 9255  df-2 9313  df-3 9314  df-4 9315  df-n0 9514  df-z 9595  df-uz 9872  df-q 9970  df-rp 10005  df-fz 10362  df-fzo 10499  df-fl 10654  df-mod 10709  df-seqfrec 10834  df-exp 10925  df-cj 11552  df-re 11553  df-im 11554  df-rsqrt 11708  df-abs 11709  df-dvds 12499  df-gcd 12675

This theorem is referenced by:  gcdid0  12701  nn0gcdsq  12922  dfphi2  12942