Theorem gcd0id 12756

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 12737	. . . 4 |- ( 0 gcd 0 ) = 0
2		oveq2 6093	. . . 4 |- ( N = 0 -> ( 0 gcd N ) = ( 0 gcd 0 ) )
3		fveq2 5695	. . . . 5 |- ( N = 0 -> ( abs ` N ) = ( abs ` 0 ) )
4		abs0 11824	. . . . 5 |- ( abs ` 0 ) = 0
5	3, 4	eqtrdi 2287	. . . 4 |- ( N = 0 -> ( abs ` N ) = 0 )
6	1, 2, 5	3eqtr4a 2297	. . 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 2421	. . 3 |- ( N =/= 0 <-> -. N = 0 )
9		0z 9655	. . . . . . . 8 |- 0 e. ZZ
10		gcddvds 12740	. . . . . . . 8 |- ( ( 0 e. ZZ /\ N e. ZZ ) -> ( ( 0 gcd N ) || 0 /\ ( 0 gcd N ) || N ) )
11	9, 10	mpan 428	. . . . . . 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 12743	. . . . . . . . 9 |- ( ( 0 e. ZZ /\ N e. ZZ ) -> ( 0 gcd N ) e. NN0 )
15	9, 14	mpan 428	. . . . . . . 8 |- ( N e. ZZ -> ( 0 gcd N ) e. NN0 )
16	15	nn0zd 9766	. . . . . . 7 |- ( N e. ZZ -> ( 0 gcd N ) e. ZZ )
17		dvdsleabs 12612	. . . . . . 7 |- ( ( ( 0 gcd N ) e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> ( ( 0 gcd N ) || N -> ( 0 gcd N ) <_ ( abs ` N ) ) )
18	16, 17	syl3an1 1311	. . . . . 6 |- ( ( N e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> ( ( 0 gcd N ) || N -> ( 0 gcd N ) <_ ( abs ` N ) ) )
19	18	3anidm12 1336	. . . . 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 11852	. . . . . . . 8 |- ( N e. ZZ -> ( abs ` N ) e. ZZ )
22		dvds0 12573	. . . . . . . 8 |- ( ( abs ` N ) e. ZZ -> ( abs ` N ) || 0 )
23	21, 22	syl 14	. . . . . . 7 |- ( N e. ZZ -> ( abs ` N ) || 0 )
24		iddvds 12571	. . . . . . . 8 |- ( N e. ZZ -> N || N )
25		absdvdsb 12576	. . . . . . . . 9 |- ( ( N e. ZZ /\ N e. ZZ ) -> ( N || N <-> ( abs ` N ) || N ) )
26	25	anidms 401	. . . . . . . 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 2238	. . . . . . . . 9 |- 0 = 0
31	30	biantrur 303	. . . . . . . 8 |- ( N = 0 <-> ( 0 = 0 /\ N = 0 ) )
32	31	necon3abii 2456	. . . . . . 7 |- ( N =/= 0 <-> -. ( 0 = 0 /\ N = 0 ) )
33		dvdslegcd 12741	. . . . . . . . . 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 1366	. . . . . . . 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 426	. . . . . . 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 9768	. . . . . 6 |- ( N e. ZZ -> ( 0 gcd N ) e. RR )
41	21	zred 9768	. . . . . 6 |- ( N e. ZZ -> ( abs ` N ) e. RR )
42	40, 41	letri3d 8441	. . . . 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 957	. . 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 9720	. . . 4 |- ( ( N e. ZZ /\ 0 e. ZZ ) -> DECID N = 0 )
47	9, 46	mpan2 429	. . 3 |- ( N e. ZZ -> DECID N = 0 )
48		exmiddc 848	. . 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 810	1 |- ( N e. ZZ -> ( 0 gcd N ) = ( abs ` N ) )

This proof depends on syntax axioms:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 720  DECID wdc 846   /\ w3a 1009   = wceq 1402   e. wcel 2209   =/= wne 2420   class class class wbr 4130   ` cfv 5377  (class class class)co 6085   0cc0 8179   <_ cle 8361   NN0cn0 9563   ZZcz 9644   abscabs 11763   || cdvds 12554   gcd cgcd 12730

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 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295  ax-pre-mulgt0 8296  ax-pre-mulext 8297  ax-arch 8298  ax-caucvg 8299

This proof depends on definitions:  df-bi 117  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 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-po 4441  df-iso 4442  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-frec 6662  df-sup 7324  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-reap 8903  df-ap 8910  df-div 9003  df-inn 9305  df-2 9363  df-3 9364  df-4 9365  df-n0 9564  df-z 9645  df-uz 9922  df-q 10020  df-rp 10055  df-fz 10412  df-fzo 10550  df-fl 10705  df-mod 10760  df-seqfrec 10885  df-exp 10976  df-cj 11607  df-re 11608  df-im 11609  df-rsqrt 11764  df-abs 11765  df-dvds 12555  df-gcd 12731

This theorem is used by:  gcdid0  12757  nn0gcdsq  12978  dfphi2  12998