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Theorem gcd0id 11678
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
StepHypRef Expression
1 gcd0val 11660 . . . 4  |-  ( 0  gcd  0 )  =  0
2 oveq2 5782 . . . 4  |-  ( N  =  0  ->  (
0  gcd  N )  =  ( 0  gcd  0 ) )
3 fveq2 5421 . . . . 5  |-  ( N  =  0  ->  ( abs `  N )  =  ( abs `  0
) )
4 abs0 10842 . . . . 5  |-  ( abs `  0 )  =  0
53, 4syl6eq 2188 . . . 4  |-  ( N  =  0  ->  ( abs `  N )  =  0 )
61, 2, 53eqtr4a 2198 . . 3  |-  ( N  =  0  ->  (
0  gcd  N )  =  ( abs `  N
) )
76adantl 275 . 2  |-  ( ( N  e.  ZZ  /\  N  =  0 )  ->  ( 0  gcd 
N )  =  ( abs `  N ) )
8 df-ne 2309 . . 3  |-  ( N  =/=  0  <->  -.  N  =  0 )
9 0z 9077 . . . . . . . 8  |-  0  e.  ZZ
10 gcddvds 11663 . . . . . . . 8  |-  ( ( 0  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( 0  gcd 
N )  ||  0  /\  ( 0  gcd  N
)  ||  N )
)
119, 10mpan 420 . . . . . . 7  |-  ( N  e.  ZZ  ->  (
( 0  gcd  N
)  ||  0  /\  ( 0  gcd  N
)  ||  N )
)
1211simprd 113 . . . . . 6  |-  ( N  e.  ZZ  ->  (
0  gcd  N )  ||  N )
1312adantr 274 . . . . 5  |-  ( ( N  e.  ZZ  /\  N  =/=  0 )  -> 
( 0  gcd  N
)  ||  N )
14 gcdcl 11666 . . . . . . . . 9  |-  ( ( 0  e.  ZZ  /\  N  e.  ZZ )  ->  ( 0  gcd  N
)  e.  NN0 )
159, 14mpan 420 . . . . . . . 8  |-  ( N  e.  ZZ  ->  (
0  gcd  N )  e.  NN0 )
1615nn0zd 9183 . . . . . . 7  |-  ( N  e.  ZZ  ->  (
0  gcd  N )  e.  ZZ )
17 dvdsleabs 11554 . . . . . . 7  |-  ( ( ( 0  gcd  N
)  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  ->  (
( 0  gcd  N
)  ||  N  ->  ( 0  gcd  N )  <_  ( abs `  N
) ) )
1816, 17syl3an1 1249 . . . . . 6  |-  ( ( N  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  ->  (
( 0  gcd  N
)  ||  N  ->  ( 0  gcd  N )  <_  ( abs `  N
) ) )
19183anidm12 1273 . . . . 5  |-  ( ( N  e.  ZZ  /\  N  =/=  0 )  -> 
( ( 0  gcd 
N )  ||  N  ->  ( 0  gcd  N
)  <_  ( abs `  N ) ) )
2013, 19mpd 13 . . . 4  |-  ( ( N  e.  ZZ  /\  N  =/=  0 )  -> 
( 0  gcd  N
)  <_  ( abs `  N ) )
21 zabscl 10870 . . . . . . . 8  |-  ( N  e.  ZZ  ->  ( abs `  N )  e.  ZZ )
22 dvds0 11519 . . . . . . . 8  |-  ( ( abs `  N )  e.  ZZ  ->  ( abs `  N )  ||  0 )
2321, 22syl 14 . . . . . . 7  |-  ( N  e.  ZZ  ->  ( abs `  N )  ||  0 )
24 iddvds 11517 . . . . . . . 8  |-  ( N  e.  ZZ  ->  N  ||  N )
25 absdvdsb 11522 . . . . . . . . 9  |-  ( ( N  e.  ZZ  /\  N  e.  ZZ )  ->  ( N  ||  N  <->  ( abs `  N ) 
||  N ) )
2625anidms 394 . . . . . . . 8  |-  ( N  e.  ZZ  ->  ( N  ||  N  <->  ( abs `  N )  ||  N
) )
2724, 26mpbid 146 . . . . . . 7  |-  ( N  e.  ZZ  ->  ( abs `  N )  ||  N )
2823, 27jca 304 . . . . . 6  |-  ( N  e.  ZZ  ->  (
( abs `  N
)  ||  0  /\  ( abs `  N ) 
||  N ) )
2928adantr 274 . . . . 5  |-  ( ( N  e.  ZZ  /\  N  =/=  0 )  -> 
( ( abs `  N
)  ||  0  /\  ( abs `  N ) 
||  N ) )
30 eqid 2139 . . . . . . . . 9  |-  0  =  0
3130biantrur 301 . . . . . . . 8  |-  ( N  =  0  <->  ( 0  =  0  /\  N  =  0 ) )
3231necon3abii 2344 . . . . . . 7  |-  ( N  =/=  0  <->  -.  (
0  =  0  /\  N  =  0 ) )
33 dvdslegcd 11664 . . . . . . . . . 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 ) ) )
3433ex 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 ) ) ) )
359, 34mp3an2 1303 . . . . . . . 8  |-  ( ( ( abs `  N
)  e.  ZZ  /\  N  e.  ZZ )  ->  ( -.  ( 0  =  0  /\  N  =  0 )  -> 
( ( ( abs `  N )  ||  0  /\  ( abs `  N
)  ||  N )  ->  ( abs `  N
)  <_  ( 0  gcd  N ) ) ) )
3621, 35mpancom 418 . . . . . . 7  |-  ( N  e.  ZZ  ->  ( -.  ( 0  =  0  /\  N  =  0 )  ->  ( (
( abs `  N
)  ||  0  /\  ( abs `  N ) 
||  N )  -> 
( abs `  N
)  <_  ( 0  gcd  N ) ) ) )
3732, 36syl5bi 151 . . . . . 6  |-  ( N  e.  ZZ  ->  ( N  =/=  0  ->  (
( ( abs `  N
)  ||  0  /\  ( abs `  N ) 
||  N )  -> 
( abs `  N
)  <_  ( 0  gcd  N ) ) ) )
3837imp 123 . . . . 5  |-  ( ( N  e.  ZZ  /\  N  =/=  0 )  -> 
( ( ( abs `  N )  ||  0  /\  ( abs `  N
)  ||  N )  ->  ( abs `  N
)  <_  ( 0  gcd  N ) ) )
3929, 38mpd 13 . . . 4  |-  ( ( N  e.  ZZ  /\  N  =/=  0 )  -> 
( abs `  N
)  <_  ( 0  gcd  N ) )
4016zred 9185 . . . . . 6  |-  ( N  e.  ZZ  ->  (
0  gcd  N )  e.  RR )
4121zred 9185 . . . . . 6  |-  ( N  e.  ZZ  ->  ( abs `  N )  e.  RR )
4240, 41letri3d 7891 . . . . 5  |-  ( N  e.  ZZ  ->  (
( 0  gcd  N
)  =  ( abs `  N )  <->  ( (
0  gcd  N )  <_  ( abs `  N
)  /\  ( abs `  N )  <_  (
0  gcd  N )
) ) )
4342adantr 274 . . . 4  |-  ( ( N  e.  ZZ  /\  N  =/=  0 )  -> 
( ( 0  gcd 
N )  =  ( abs `  N )  <-> 
( ( 0  gcd 
N )  <_  ( abs `  N )  /\  ( abs `  N )  <_  ( 0  gcd 
N ) ) ) )
4420, 39, 43mpbir2and 928 . . 3  |-  ( ( N  e.  ZZ  /\  N  =/=  0 )  -> 
( 0  gcd  N
)  =  ( abs `  N ) )
458, 44sylan2br 286 . 2  |-  ( ( N  e.  ZZ  /\  -.  N  =  0
)  ->  ( 0  gcd  N )  =  ( abs `  N
) )
46 zdceq 9138 . . . 4  |-  ( ( N  e.  ZZ  /\  0  e.  ZZ )  -> DECID  N  =  0 )
479, 46mpan2 421 . . 3  |-  ( N  e.  ZZ  -> DECID  N  =  0
)
48 exmiddc 821 . . 3  |-  (DECID  N  =  0  ->  ( N  =  0  \/  -.  N  =  0 ) )
4947, 48syl 14 . 2  |-  ( N  e.  ZZ  ->  ( N  =  0  \/  -.  N  =  0
) )
507, 45, 49mpjaodan 787 1  |-  ( N  e.  ZZ  ->  (
0  gcd  N )  =  ( abs `  N
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    <-> wb 104    \/ wo 697  DECID wdc 819    /\ w3a 962    = wceq 1331    e. wcel 1480    =/= wne 2308   class class class wbr 3929   ` cfv 5123  (class class class)co 5774   0cc0 7632    <_ cle 7813   NN0cn0 8989   ZZcz 9066   abscabs 10781    || cdvds 11504    gcd cgcd 11646
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502  ax-cnex 7723  ax-resscn 7724  ax-1cn 7725  ax-1re 7726  ax-icn 7727  ax-addcl 7728  ax-addrcl 7729  ax-mulcl 7730  ax-mulrcl 7731  ax-addcom 7732  ax-mulcom 7733  ax-addass 7734  ax-mulass 7735  ax-distr 7736  ax-i2m1 7737  ax-0lt1 7738  ax-1rid 7739  ax-0id 7740  ax-rnegex 7741  ax-precex 7742  ax-cnre 7743  ax-pre-ltirr 7744  ax-pre-ltwlin 7745  ax-pre-lttrn 7746  ax-pre-apti 7747  ax-pre-ltadd 7748  ax-pre-mulgt0 7749  ax-pre-mulext 7750  ax-arch 7751  ax-caucvg 7752
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-ilim 4291  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-frec 6288  df-sup 6871  df-pnf 7814  df-mnf 7815  df-xr 7816  df-ltxr 7817  df-le 7818  df-sub 7947  df-neg 7948  df-reap 8349  df-ap 8356  df-div 8445  df-inn 8733  df-2 8791  df-3 8792  df-4 8793  df-n0 8990  df-z 9067  df-uz 9339  df-q 9424  df-rp 9454  df-fz 9803  df-fzo 9932  df-fl 10055  df-mod 10108  df-seqfrec 10231  df-exp 10305  df-cj 10626  df-re 10627  df-im 10628  df-rsqrt 10782  df-abs 10783  df-dvds 11505  df-gcd 11647
This theorem is referenced by:  gcdid0  11679  nn0gcdsq  11889  dfphi2  11907
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