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Theorem gcd0id 11412
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 11394 . . . 4  |-  ( 0  gcd  0 )  =  0
2 oveq2 5698 . . . 4  |-  ( N  =  0  ->  (
0  gcd  N )  =  ( 0  gcd  0 ) )
3 fveq2 5340 . . . . 5  |-  ( N  =  0  ->  ( abs `  N )  =  ( abs `  0
) )
4 abs0 10622 . . . . 5  |-  ( abs `  0 )  =  0
53, 4syl6eq 2143 . . . 4  |-  ( N  =  0  ->  ( abs `  N )  =  0 )
61, 2, 53eqtr4a 2153 . . 3  |-  ( N  =  0  ->  (
0  gcd  N )  =  ( abs `  N
) )
76adantl 272 . 2  |-  ( ( N  e.  ZZ  /\  N  =  0 )  ->  ( 0  gcd 
N )  =  ( abs `  N ) )
8 df-ne 2263 . . 3  |-  ( N  =/=  0  <->  -.  N  =  0 )
9 0z 8859 . . . . . . . 8  |-  0  e.  ZZ
10 gcddvds 11397 . . . . . . . 8  |-  ( ( 0  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( 0  gcd 
N )  ||  0  /\  ( 0  gcd  N
)  ||  N )
)
119, 10mpan 416 . . . . . . 7  |-  ( N  e.  ZZ  ->  (
( 0  gcd  N
)  ||  0  /\  ( 0  gcd  N
)  ||  N )
)
1211simprd 113 . . . . . 6  |-  ( N  e.  ZZ  ->  (
0  gcd  N )  ||  N )
1312adantr 271 . . . . 5  |-  ( ( N  e.  ZZ  /\  N  =/=  0 )  -> 
( 0  gcd  N
)  ||  N )
14 gcdcl 11400 . . . . . . . . 9  |-  ( ( 0  e.  ZZ  /\  N  e.  ZZ )  ->  ( 0  gcd  N
)  e.  NN0 )
159, 14mpan 416 . . . . . . . 8  |-  ( N  e.  ZZ  ->  (
0  gcd  N )  e.  NN0 )
1615nn0zd 8965 . . . . . . 7  |-  ( N  e.  ZZ  ->  (
0  gcd  N )  e.  ZZ )
17 dvdsleabs 11288 . . . . . . 7  |-  ( ( ( 0  gcd  N
)  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  ->  (
( 0  gcd  N
)  ||  N  ->  ( 0  gcd  N )  <_  ( abs `  N
) ) )
1816, 17syl3an1 1214 . . . . . 6  |-  ( ( N  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  ->  (
( 0  gcd  N
)  ||  N  ->  ( 0  gcd  N )  <_  ( abs `  N
) ) )
19183anidm12 1238 . . . . 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 10650 . . . . . . . 8  |-  ( N  e.  ZZ  ->  ( abs `  N )  e.  ZZ )
22 dvds0 11253 . . . . . . . 8  |-  ( ( abs `  N )  e.  ZZ  ->  ( abs `  N )  ||  0 )
2321, 22syl 14 . . . . . . 7  |-  ( N  e.  ZZ  ->  ( abs `  N )  ||  0 )
24 iddvds 11251 . . . . . . . 8  |-  ( N  e.  ZZ  ->  N  ||  N )
25 absdvdsb 11256 . . . . . . . . 9  |-  ( ( N  e.  ZZ  /\  N  e.  ZZ )  ->  ( N  ||  N  <->  ( abs `  N ) 
||  N ) )
2625anidms 390 . . . . . . . 8  |-  ( N  e.  ZZ  ->  ( N  ||  N  <->  ( abs `  N )  ||  N
) )
2724, 26mpbid 146 . . . . . . 7  |-  ( N  e.  ZZ  ->  ( abs `  N )  ||  N )
2823, 27jca 301 . . . . . 6  |-  ( N  e.  ZZ  ->  (
( abs `  N
)  ||  0  /\  ( abs `  N ) 
||  N ) )
2928adantr 271 . . . . 5  |-  ( ( N  e.  ZZ  /\  N  =/=  0 )  -> 
( ( abs `  N
)  ||  0  /\  ( abs `  N ) 
||  N ) )
30 eqid 2095 . . . . . . . . 9  |-  0  =  0
3130biantrur 298 . . . . . . . 8  |-  ( N  =  0  <->  ( 0  =  0  /\  N  =  0 ) )
3231necon3abii 2298 . . . . . . 7  |-  ( N  =/=  0  <->  -.  (
0  =  0  /\  N  =  0 ) )
33 dvdslegcd 11398 . . . . . . . . . 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 1268 . . . . . . . 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 414 . . . . . . 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 8967 . . . . . 6  |-  ( N  e.  ZZ  ->  (
0  gcd  N )  e.  RR )
4121zred 8967 . . . . . 6  |-  ( N  e.  ZZ  ->  ( abs `  N )  e.  RR )
4240, 41letri3d 7697 . . . . 5  |-  ( N  e.  ZZ  ->  (
( 0  gcd  N
)  =  ( abs `  N )  <->  ( (
0  gcd  N )  <_  ( abs `  N
)  /\  ( abs `  N )  <_  (
0  gcd  N )
) ) )
4342adantr 271 . . . 4  |-  ( ( N  e.  ZZ  /\  N  =/=  0 )  -> 
( ( 0  gcd 
N )  =  ( abs `  N )  <-> 
( ( 0  gcd 
N )  <_  ( abs `  N )  /\  ( abs `  N )  <_  ( 0  gcd 
N ) ) ) )
4420, 39, 43mpbir2and 893 . . 3  |-  ( ( N  e.  ZZ  /\  N  =/=  0 )  -> 
( 0  gcd  N
)  =  ( abs `  N ) )
458, 44sylan2br 283 . 2  |-  ( ( N  e.  ZZ  /\  -.  N  =  0
)  ->  ( 0  gcd  N )  =  ( abs `  N
) )
46 zdceq 8920 . . . 4  |-  ( ( N  e.  ZZ  /\  0  e.  ZZ )  -> DECID  N  =  0 )
479, 46mpan2 417 . . 3  |-  ( N  e.  ZZ  -> DECID  N  =  0
)
48 exmiddc 785 . . 3  |-  (DECID  N  =  0  ->  ( N  =  0  \/  -.  N  =  0 ) )
4947, 48syl 14 . 2  |-  ( N  e.  ZZ  ->  ( N  =  0  \/  -.  N  =  0
) )
507, 45, 49mpjaodan 750 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 667  DECID wdc 783    /\ w3a 927    = wceq 1296    e. wcel 1445    =/= wne 2262   class class class wbr 3867   ` cfv 5049  (class class class)co 5690   0cc0 7447    <_ cle 7620   NN0cn0 8771   ZZcz 8848   abscabs 10561    || cdvds 11238    gcd cgcd 11380
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-13 1456  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-coll 3975  ax-sep 3978  ax-nul 3986  ax-pow 4030  ax-pr 4060  ax-un 4284  ax-setind 4381  ax-iinf 4431  ax-cnex 7533  ax-resscn 7534  ax-1cn 7535  ax-1re 7536  ax-icn 7537  ax-addcl 7538  ax-addrcl 7539  ax-mulcl 7540  ax-mulrcl 7541  ax-addcom 7542  ax-mulcom 7543  ax-addass 7544  ax-mulass 7545  ax-distr 7546  ax-i2m1 7547  ax-0lt1 7548  ax-1rid 7549  ax-0id 7550  ax-rnegex 7551  ax-precex 7552  ax-cnre 7553  ax-pre-ltirr 7554  ax-pre-ltwlin 7555  ax-pre-lttrn 7556  ax-pre-apti 7557  ax-pre-ltadd 7558  ax-pre-mulgt0 7559  ax-pre-mulext 7560  ax-arch 7561  ax-caucvg 7562
This theorem depends on definitions:  df-bi 116  df-dc 784  df-3or 928  df-3an 929  df-tru 1299  df-fal 1302  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ne 2263  df-nel 2358  df-ral 2375  df-rex 2376  df-reu 2377  df-rmo 2378  df-rab 2379  df-v 2635  df-sbc 2855  df-csb 2948  df-dif 3015  df-un 3017  df-in 3019  df-ss 3026  df-nul 3303  df-if 3414  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-int 3711  df-iun 3754  df-br 3868  df-opab 3922  df-mpt 3923  df-tr 3959  df-id 4144  df-po 4147  df-iso 4148  df-iord 4217  df-on 4219  df-ilim 4220  df-suc 4222  df-iom 4434  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-rn 4478  df-res 4479  df-ima 4480  df-iota 5014  df-fun 5051  df-fn 5052  df-f 5053  df-f1 5054  df-fo 5055  df-f1o 5056  df-fv 5057  df-riota 5646  df-ov 5693  df-oprab 5694  df-mpt2 5695  df-1st 5949  df-2nd 5950  df-recs 6108  df-frec 6194  df-sup 6759  df-pnf 7621  df-mnf 7622  df-xr 7623  df-ltxr 7624  df-le 7625  df-sub 7752  df-neg 7753  df-reap 8149  df-ap 8156  df-div 8237  df-inn 8521  df-2 8579  df-3 8580  df-4 8581  df-n0 8772  df-z 8849  df-uz 9119  df-q 9204  df-rp 9234  df-fz 9574  df-fzo 9703  df-fl 9826  df-mod 9879  df-iseq 10002  df-seq3 10003  df-exp 10086  df-cj 10407  df-re 10408  df-im 10409  df-rsqrt 10562  df-abs 10563  df-dvds 11239  df-gcd 11381
This theorem is referenced by:  gcdid0  11413  nn0gcdsq  11620  dfphi2  11638
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