Theorem gcd0id 11934

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 11915	. . . 4 |- ( 0 gcd 0 ) = 0
2		oveq2 5861	. . . 4 |- ( N = 0 -> ( 0 gcd N ) = ( 0 gcd 0 ) )
3		fveq2 5496	. . . . 5 |- ( N = 0 -> ( abs ` N ) = ( abs ` 0 ) )
4		abs0 11022	. . . . 5 |- ( abs ` 0 ) = 0
5	3, 4	eqtrdi 2219	. . . 4 |- ( N = 0 -> ( abs ` N ) = 0 )
6	1, 2, 5	3eqtr4a 2229	. . 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 2341	. . 3 |- ( N =/= 0 <-> -. N = 0 )
9		0z 9223	. . . . . . . 8 |- 0 e. ZZ
10		gcddvds 11918	. . . . . . . 8 |- ( ( 0 e. ZZ /\ N e. ZZ ) -> ( ( 0 gcd N ) || 0 /\ ( 0 gcd N ) || N ) )
11	9, 10	mpan 422	. . . . . . 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 11921	. . . . . . . . 9 |- ( ( 0 e. ZZ /\ N e. ZZ ) -> ( 0 gcd N ) e. NN0 )
15	9, 14	mpan 422	. . . . . . . 8 |- ( N e. ZZ -> ( 0 gcd N ) e. NN0 )
16	15	nn0zd 9332	. . . . . . 7 |- ( N e. ZZ -> ( 0 gcd N ) e. ZZ )
17		dvdsleabs 11805	. . . . . . 7 |- ( ( ( 0 gcd N ) e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> ( ( 0 gcd N ) || N -> ( 0 gcd N ) <_ ( abs ` N ) ) )
18	16, 17	syl3an1 1266	. . . . . 6 |- ( ( N e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> ( ( 0 gcd N ) || N -> ( 0 gcd N ) <_ ( abs ` N ) ) )
19	18	3anidm12 1290	. . . . 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 11050	. . . . . . . 8 |- ( N e. ZZ -> ( abs ` N ) e. ZZ )
22		dvds0 11768	. . . . . . . 8 |- ( ( abs ` N ) e. ZZ -> ( abs ` N ) || 0 )
23	21, 22	syl 14	. . . . . . 7 |- ( N e. ZZ -> ( abs ` N ) || 0 )
24		iddvds 11766	. . . . . . . 8 |- ( N e. ZZ -> N || N )
25		absdvdsb 11771	. . . . . . . . 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 2170	. . . . . . . . 9 |- 0 = 0
31	30	biantrur 301	. . . . . . . 8 |- ( N = 0 <-> ( 0 = 0 /\ N = 0 ) )
32	31	necon3abii 2376	. . . . . . 7 |- ( N =/= 0 <-> -. ( 0 = 0 /\ N = 0 ) )
33		dvdslegcd 11919	. . . . . . . . . 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 1320	. . . . . . . 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 420	. . . . . . 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 9334	. . . . . 6 |- ( N e. ZZ -> ( 0 gcd N ) e. RR )
41	21	zred 9334	. . . . . 6 |- ( N e. ZZ -> ( abs ` N ) e. RR )
42	40, 41	letri3d 8035	. . . . 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 939	. . 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 9287	. . . 4 |- ( ( N e. ZZ /\ 0 e. ZZ ) -> DECID N = 0 )
47	9, 46	mpan2 423	. . 3 |- ( N e. ZZ -> DECID N = 0 )
48		exmiddc 831	. . 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 793	1 |- ( N e. ZZ -> ( 0 gcd N ) = ( abs ` N ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 703  DECID wdc 829   /\ w3a 973   = wceq 1348   e. wcel 2141   =/= wne 2340   class class class wbr 3989   ` cfv 5198  (class class class)co 5853   0cc0 7774   <_ cle 7955   NN0cn0 9135   ZZcz 9212   abscabs 10961   || cdvds 11749   gcd cgcd 11897

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891  ax-pre-mulext 7892  ax-arch 7893  ax-caucvg 7894

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-ilim 4354  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-frec 6370  df-sup 6961  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501  df-div 8590  df-inn 8879  df-2 8937  df-3 8938  df-4 8939  df-n0 9136  df-z 9213  df-uz 9488  df-q 9579  df-rp 9611  df-fz 9966  df-fzo 10099  df-fl 10226  df-mod 10279  df-seqfrec 10402  df-exp 10476  df-cj 10806  df-re 10807  df-im 10808  df-rsqrt 10962  df-abs 10963  df-dvds 11750  df-gcd 11898

This theorem is referenced by:  gcdid0  11935  nn0gcdsq  12154  dfphi2  12174