Theorem gcd0id 12146

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 12127	. . . 4 |- ( 0 gcd 0 ) = 0
2		oveq2 5930	. . . 4 |- ( N = 0 -> ( 0 gcd N ) = ( 0 gcd 0 ) )
3		fveq2 5558	. . . . 5 |- ( N = 0 -> ( abs ` N ) = ( abs ` 0 ) )
4		abs0 11223	. . . . 5 |- ( abs ` 0 ) = 0
5	3, 4	eqtrdi 2245	. . . 4 |- ( N = 0 -> ( abs ` N ) = 0 )
6	1, 2, 5	3eqtr4a 2255	. . 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 2368	. . 3 |- ( N =/= 0 <-> -. N = 0 )
9		0z 9337	. . . . . . . 8 |- 0 e. ZZ
10		gcddvds 12130	. . . . . . . 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 12133	. . . . . . . . 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 9446	. . . . . . 7 |- ( N e. ZZ -> ( 0 gcd N ) e. ZZ )
17		dvdsleabs 12010	. . . . . . 7 |- ( ( ( 0 gcd N ) e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> ( ( 0 gcd N ) || N -> ( 0 gcd N ) <_ ( abs ` N ) ) )
18	16, 17	syl3an1 1282	. . . . . 6 |- ( ( N e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> ( ( 0 gcd N ) || N -> ( 0 gcd N ) <_ ( abs ` N ) ) )
19	18	3anidm12 1306	. . . . 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 11251	. . . . . . . 8 |- ( N e. ZZ -> ( abs ` N ) e. ZZ )
22		dvds0 11971	. . . . . . . 8 |- ( ( abs ` N ) e. ZZ -> ( abs ` N ) || 0 )
23	21, 22	syl 14	. . . . . . 7 |- ( N e. ZZ -> ( abs ` N ) || 0 )
24		iddvds 11969	. . . . . . . 8 |- ( N e. ZZ -> N || N )
25		absdvdsb 11974	. . . . . . . . 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 2196	. . . . . . . . 9 |- 0 = 0
31	30	biantrur 303	. . . . . . . 8 |- ( N = 0 <-> ( 0 = 0 /\ N = 0 ) )
32	31	necon3abii 2403	. . . . . . 7 |- ( N =/= 0 <-> -. ( 0 = 0 /\ N = 0 ) )
33		dvdslegcd 12131	. . . . . . . . . 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 1336	. . . . . . . 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 9448	. . . . . 6 |- ( N e. ZZ -> ( 0 gcd N ) e. RR )
41	21	zred 9448	. . . . . 6 |- ( N e. ZZ -> ( abs ` N ) e. RR )
42	40, 41	letri3d 8142	. . . . 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 946	. . 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 9401	. . . 4 |- ( ( N e. ZZ /\ 0 e. ZZ ) -> DECID N = 0 )
47	9, 46	mpan2 425	. . 3 |- ( N e. ZZ -> DECID N = 0 )
48		exmiddc 837	. . 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 799	1 |- ( N e. ZZ -> ( 0 gcd N ) = ( abs ` N ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 709  DECID wdc 835   /\ w3a 980   = wceq 1364   e. wcel 2167   =/= wne 2367   class class class wbr 4033   ` cfv 5258  (class class class)co 5922   0cc0 7879   <_ cle 8062   NN0cn0 9249   ZZcz 9326   abscabs 11162   || cdvds 11952   gcd cgcd 12120

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-iinf 4624  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-mulrcl 7978  ax-addcom 7979  ax-mulcom 7980  ax-addass 7981  ax-mulass 7982  ax-distr 7983  ax-i2m1 7984  ax-0lt1 7985  ax-1rid 7986  ax-0id 7987  ax-rnegex 7988  ax-precex 7989  ax-cnre 7990  ax-pre-ltirr 7991  ax-pre-ltwlin 7992  ax-pre-lttrn 7993  ax-pre-apti 7994  ax-pre-ltadd 7995  ax-pre-mulgt0 7996  ax-pre-mulext 7997  ax-arch 7998  ax-caucvg 7999

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-if 3562  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-id 4328  df-po 4331  df-iso 4332  df-iord 4401  df-on 4403  df-ilim 4404  df-suc 4406  df-iom 4627  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-recs 6363  df-frec 6449  df-sup 7050  df-pnf 8063  df-mnf 8064  df-xr 8065  df-ltxr 8066  df-le 8067  df-sub 8199  df-neg 8200  df-reap 8602  df-ap 8609  df-div 8700  df-inn 8991  df-2 9049  df-3 9050  df-4 9051  df-n0 9250  df-z 9327  df-uz 9602  df-q 9694  df-rp 9729  df-fz 10084  df-fzo 10218  df-fl 10360  df-mod 10415  df-seqfrec 10540  df-exp 10631  df-cj 11007  df-re 11008  df-im 11009  df-rsqrt 11163  df-abs 11164  df-dvds 11953  df-gcd 12121

This theorem is referenced by:  gcdid0  12147  nn0gcdsq  12368  dfphi2  12388