Theorem gcd0id 11703

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 11685	. . . 4 |- ( 0 gcd 0 ) = 0
2		oveq2 5790	. . . 4 |- ( N = 0 -> ( 0 gcd N ) = ( 0 gcd 0 ) )
3		fveq2 5429	. . . . 5 |- ( N = 0 -> ( abs ` N ) = ( abs ` 0 ) )
4		abs0 10862	. . . . 5 |- ( abs ` 0 ) = 0
5	3, 4	eqtrdi 2189	. . . 4 |- ( N = 0 -> ( abs ` N ) = 0 )
6	1, 2, 5	3eqtr4a 2199	. . 3 |- ( N = 0 -> ( 0 gcd N ) = ( abs ` N ) )
7	6	adantl 275	. 2 |- ( ( N e. ZZ /\ N = 0 ) -> ( 0 gcd N ) = ( abs ` N ) )
8		df-ne 2310	. . 3 |- ( N =/= 0 <-> -. N = 0 )
9		0z 9089	. . . . . . . 8 |- 0 e. ZZ
10		gcddvds 11688	. . . . . . . 8 |- ( ( 0 e. ZZ /\ N e. ZZ ) -> ( ( 0 gcd N ) || 0 /\ ( 0 gcd N ) || N ) )
11	9, 10	mpan 421	. . . . . . 7 |- ( N e. ZZ -> ( ( 0 gcd N ) || 0 /\ ( 0 gcd N ) || N ) )
12	11	simprd 113	. . . . . 6 |- ( N e. ZZ -> ( 0 gcd N ) || N )
13	12	adantr 274	. . . . 5 |- ( ( N e. ZZ /\ N =/= 0 ) -> ( 0 gcd N ) || N )
14		gcdcl 11691	. . . . . . . . 9 |- ( ( 0 e. ZZ /\ N e. ZZ ) -> ( 0 gcd N ) e. NN0 )
15	9, 14	mpan 421	. . . . . . . 8 |- ( N e. ZZ -> ( 0 gcd N ) e. NN0 )
16	15	nn0zd 9195	. . . . . . 7 |- ( N e. ZZ -> ( 0 gcd N ) e. ZZ )
17		dvdsleabs 11579	. . . . . . 7 |- ( ( ( 0 gcd N ) e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> ( ( 0 gcd N ) || N -> ( 0 gcd N ) <_ ( abs ` N ) ) )
18	16, 17	syl3an1 1250	. . . . . 6 |- ( ( N e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> ( ( 0 gcd N ) || N -> ( 0 gcd N ) <_ ( abs ` N ) ) )
19	18	3anidm12 1274	. . . . 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 10890	. . . . . . . 8 |- ( N e. ZZ -> ( abs ` N ) e. ZZ )
22		dvds0 11544	. . . . . . . 8 |- ( ( abs ` N ) e. ZZ -> ( abs ` N ) || 0 )
23	21, 22	syl 14	. . . . . . 7 |- ( N e. ZZ -> ( abs ` N ) || 0 )
24		iddvds 11542	. . . . . . . 8 |- ( N e. ZZ -> N || N )
25		absdvdsb 11547	. . . . . . . . 9 |- ( ( N e. ZZ /\ N e. ZZ ) -> ( N || N <-> ( abs ` N ) || N ) )
26	25	anidms 395	. . . . . . . 8 |- ( N e. ZZ -> ( N || N <-> ( abs ` N ) || N ) )
27	24, 26	mpbid 146	. . . . . . 7 |- ( N e. ZZ -> ( abs ` N ) || N )
28	23, 27	jca 304	. . . . . 6 |- ( N e. ZZ -> ( ( abs ` N ) || 0 /\ ( abs ` N ) || N ) )
29	28	adantr 274	. . . . 5 |- ( ( N e. ZZ /\ N =/= 0 ) -> ( ( abs ` N ) || 0 /\ ( abs ` N ) || N ) )
30		eqid 2140	. . . . . . . . 9 |- 0 = 0
31	30	biantrur 301	. . . . . . . 8 |- ( N = 0 <-> ( 0 = 0 /\ N = 0 ) )
32	31	necon3abii 2345	. . . . . . 7 |- ( N =/= 0 <-> -. ( 0 = 0 /\ N = 0 ) )
33		dvdslegcd 11689	. . . . . . . . . 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 114	. . . . . . . . 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 1304	. . . . . . . 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 419	. . . . . . 7 |- ( N e. ZZ -> ( -. ( 0 = 0 /\ N = 0 ) -> ( ( ( abs ` N ) || 0 /\ ( abs ` N ) || N ) -> ( abs ` N ) <_ ( 0 gcd N ) ) ) )
37	32, 36	syl5bi 151	. . . . . 6 |- ( N e. ZZ -> ( N =/= 0 -> ( ( ( abs ` N ) || 0 /\ ( abs ` N ) || N ) -> ( abs ` N ) <_ ( 0 gcd N ) ) ) )
38	37	imp 123	. . . . 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 9197	. . . . . 6 |- ( N e. ZZ -> ( 0 gcd N ) e. RR )
41	21	zred 9197	. . . . . 6 |- ( N e. ZZ -> ( abs ` N ) e. RR )
42	40, 41	letri3d 7903	. . . . 5 |- ( N e. ZZ -> ( ( 0 gcd N ) = ( abs ` N ) <-> ( ( 0 gcd N ) <_ ( abs ` N ) /\ ( abs ` N ) <_ ( 0 gcd N ) ) ) )
43	42	adantr 274	. . . 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 929	. . 3 |- ( ( N e. ZZ /\ N =/= 0 ) -> ( 0 gcd N ) = ( abs ` N ) )
45	8, 44	sylan2br 286	. 2 |- ( ( N e. ZZ /\ -. N = 0 ) -> ( 0 gcd N ) = ( abs ` N ) )
46		zdceq 9150	. . . 4 |- ( ( N e. ZZ /\ 0 e. ZZ ) -> DECID N = 0 )
47	9, 46	mpan2 422	. . 3 |- ( N e. ZZ -> DECID N = 0 )
48		exmiddc 822	. . 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 788	1 |- ( N e. ZZ -> ( 0 gcd N ) = ( abs ` N ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 698  DECID wdc 820   /\ w3a 963   = wceq 1332   e. wcel 1481   =/= wne 2309   class class class wbr 3937   ` cfv 5131  (class class class)co 5782   0cc0 7644   <_ cle 7825   NN0cn0 9001   ZZcz 9078   abscabs 10801   || cdvds 11529   gcd cgcd 11671

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4051  ax-sep 4054  ax-nul 4062  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460  ax-iinf 4510  ax-cnex 7735  ax-resscn 7736  ax-1cn 7737  ax-1re 7738  ax-icn 7739  ax-addcl 7740  ax-addrcl 7741  ax-mulcl 7742  ax-mulrcl 7743  ax-addcom 7744  ax-mulcom 7745  ax-addass 7746  ax-mulass 7747  ax-distr 7748  ax-i2m1 7749  ax-0lt1 7750  ax-1rid 7751  ax-0id 7752  ax-rnegex 7753  ax-precex 7754  ax-cnre 7755  ax-pre-ltirr 7756  ax-pre-ltwlin 7757  ax-pre-lttrn 7758  ax-pre-apti 7759  ax-pre-ltadd 7760  ax-pre-mulgt0 7761  ax-pre-mulext 7762  ax-arch 7763  ax-caucvg 7764

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-reu 2424  df-rmo 2425  df-rab 2426  df-v 2691  df-sbc 2914  df-csb 3008  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-if 3480  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-int 3780  df-iun 3823  df-br 3938  df-opab 3998  df-mpt 3999  df-tr 4035  df-id 4223  df-po 4226  df-iso 4227  df-iord 4296  df-on 4298  df-ilim 4299  df-suc 4301  df-iom 4513  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-fv 5139  df-riota 5738  df-ov 5785  df-oprab 5786  df-mpo 5787  df-1st 6046  df-2nd 6047  df-recs 6210  df-frec 6296  df-sup 6879  df-pnf 7826  df-mnf 7827  df-xr 7828  df-ltxr 7829  df-le 7830  df-sub 7959  df-neg 7960  df-reap 8361  df-ap 8368  df-div 8457  df-inn 8745  df-2 8803  df-3 8804  df-4 8805  df-n0 9002  df-z 9079  df-uz 9351  df-q 9439  df-rp 9471  df-fz 9822  df-fzo 9951  df-fl 10074  df-mod 10127  df-seqfrec 10250  df-exp 10324  df-cj 10646  df-re 10647  df-im 10648  df-rsqrt 10802  df-abs 10803  df-dvds 11530  df-gcd 11672

This theorem is referenced by:  gcdid0  11704  nn0gcdsq  11914  dfphi2  11932