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Theorem gcd0id 12950
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 12936 . . . 4  |-  ( 0  gcd  0 )  =  0
2 oveq2 6028 . . . 4  |-  ( N  =  0  ->  (
0  gcd  N )  =  ( 0  gcd  0 ) )
3 fveq2 5668 . . . . 5  |-  ( N  =  0  ->  ( abs `  N )  =  ( abs `  0
) )
4 abs0 12017 . . . . 5  |-  ( abs `  0 )  =  0
53, 4syl6eq 2435 . . . 4  |-  ( N  =  0  ->  ( abs `  N )  =  0 )
61, 2, 53eqtr4a 2445 . . 3  |-  ( N  =  0  ->  (
0  gcd  N )  =  ( abs `  N
) )
76adantl 453 . 2  |-  ( ( N  e.  ZZ  /\  N  =  0 )  ->  ( 0  gcd 
N )  =  ( abs `  N ) )
8 0z 10225 . . . . . . 7  |-  0  e.  ZZ
9 gcddvds 12942 . . . . . . 7  |-  ( ( 0  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( 0  gcd 
N )  ||  0  /\  ( 0  gcd  N
)  ||  N )
)
108, 9mpan 652 . . . . . 6  |-  ( N  e.  ZZ  ->  (
( 0  gcd  N
)  ||  0  /\  ( 0  gcd  N
)  ||  N )
)
1110simprd 450 . . . . 5  |-  ( N  e.  ZZ  ->  (
0  gcd  N )  ||  N )
1211adantr 452 . . . 4  |-  ( ( N  e.  ZZ  /\  N  =/=  0 )  -> 
( 0  gcd  N
)  ||  N )
13 gcdcl 12944 . . . . . . . 8  |-  ( ( 0  e.  ZZ  /\  N  e.  ZZ )  ->  ( 0  gcd  N
)  e.  NN0 )
148, 13mpan 652 . . . . . . 7  |-  ( N  e.  ZZ  ->  (
0  gcd  N )  e.  NN0 )
1514nn0zd 10305 . . . . . 6  |-  ( N  e.  ZZ  ->  (
0  gcd  N )  e.  ZZ )
16 dvdsleabs 12823 . . . . . 6  |-  ( ( ( 0  gcd  N
)  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  ->  (
( 0  gcd  N
)  ||  N  ->  ( 0  gcd  N )  <_  ( abs `  N
) ) )
1715, 16syl3an1 1217 . . . . 5  |-  ( ( N  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  ->  (
( 0  gcd  N
)  ||  N  ->  ( 0  gcd  N )  <_  ( abs `  N
) ) )
18173anidm12 1241 . . . 4  |-  ( ( N  e.  ZZ  /\  N  =/=  0 )  -> 
( ( 0  gcd 
N )  ||  N  ->  ( 0  gcd  N
)  <_  ( abs `  N ) ) )
1912, 18mpd 15 . . 3  |-  ( ( N  e.  ZZ  /\  N  =/=  0 )  -> 
( 0  gcd  N
)  <_  ( abs `  N ) )
20 nn0abscl 12044 . . . . . . . 8  |-  ( N  e.  ZZ  ->  ( abs `  N )  e. 
NN0 )
2120nn0zd 10305 . . . . . . 7  |-  ( N  e.  ZZ  ->  ( abs `  N )  e.  ZZ )
22 dvds0 12792 . . . . . . 7  |-  ( ( abs `  N )  e.  ZZ  ->  ( abs `  N )  ||  0 )
2321, 22syl 16 . . . . . 6  |-  ( N  e.  ZZ  ->  ( abs `  N )  ||  0 )
24 iddvds 12790 . . . . . . 7  |-  ( N  e.  ZZ  ->  N  ||  N )
25 absdvdsb 12795 . . . . . . . 8  |-  ( ( N  e.  ZZ  /\  N  e.  ZZ )  ->  ( N  ||  N  <->  ( abs `  N ) 
||  N ) )
2625anidms 627 . . . . . . 7  |-  ( N  e.  ZZ  ->  ( N  ||  N  <->  ( abs `  N )  ||  N
) )
2724, 26mpbid 202 . . . . . 6  |-  ( N  e.  ZZ  ->  ( abs `  N )  ||  N )
2823, 27jca 519 . . . . 5  |-  ( N  e.  ZZ  ->  (
( abs `  N
)  ||  0  /\  ( abs `  N ) 
||  N ) )
2928adantr 452 . . . 4  |-  ( ( N  e.  ZZ  /\  N  =/=  0 )  -> 
( ( abs `  N
)  ||  0  /\  ( abs `  N ) 
||  N ) )
30 eqid 2387 . . . . . . . 8  |-  0  =  0
3130biantrur 493 . . . . . . 7  |-  ( N  =  0  <->  ( 0  =  0  /\  N  =  0 ) )
3231necon3abii 2580 . . . . . 6  |-  ( N  =/=  0  <->  -.  (
0  =  0  /\  N  =  0 ) )
33 dvdslegcd 12943 . . . . . . . . 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 ) ) )
3433ex 424 . . . . . . . 8  |-  ( ( ( 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 ) ) ) )
358, 34mp3an2 1267 . . . . . . 7  |-  ( ( ( abs `  N
)  e.  ZZ  /\  N  e.  ZZ )  ->  ( -.  ( 0  =  0  /\  N  =  0 )  -> 
( ( ( abs `  N )  ||  0  /\  ( abs `  N
)  ||  N )  ->  ( abs `  N
)  <_  ( 0  gcd  N ) ) ) )
3621, 35mpancom 651 . . . . . 6  |-  ( N  e.  ZZ  ->  ( -.  ( 0  =  0  /\  N  =  0 )  ->  ( (
( abs `  N
)  ||  0  /\  ( abs `  N ) 
||  N )  -> 
( abs `  N
)  <_  ( 0  gcd  N ) ) ) )
3732, 36syl5bi 209 . . . . 5  |-  ( N  e.  ZZ  ->  ( N  =/=  0  ->  (
( ( abs `  N
)  ||  0  /\  ( abs `  N ) 
||  N )  -> 
( abs `  N
)  <_  ( 0  gcd  N ) ) ) )
3837imp 419 . . . 4  |-  ( ( N  e.  ZZ  /\  N  =/=  0 )  -> 
( ( ( abs `  N )  ||  0  /\  ( abs `  N
)  ||  N )  ->  ( abs `  N
)  <_  ( 0  gcd  N ) ) )
3929, 38mpd 15 . . 3  |-  ( ( N  e.  ZZ  /\  N  =/=  0 )  -> 
( abs `  N
)  <_  ( 0  gcd  N ) )
4015zred 10307 . . . . 5  |-  ( N  e.  ZZ  ->  (
0  gcd  N )  e.  RR )
4121zred 10307 . . . . 5  |-  ( N  e.  ZZ  ->  ( abs `  N )  e.  RR )
4240, 41letri3d 9147 . . . 4  |-  ( N  e.  ZZ  ->  (
( 0  gcd  N
)  =  ( abs `  N )  <->  ( (
0  gcd  N )  <_  ( abs `  N
)  /\  ( abs `  N )  <_  (
0  gcd  N )
) ) )
4342adantr 452 . . 3  |-  ( ( N  e.  ZZ  /\  N  =/=  0 )  -> 
( ( 0  gcd 
N )  =  ( abs `  N )  <-> 
( ( 0  gcd 
N )  <_  ( abs `  N )  /\  ( abs `  N )  <_  ( 0  gcd 
N ) ) ) )
4419, 39, 43mpbir2and 889 . 2  |-  ( ( N  e.  ZZ  /\  N  =/=  0 )  -> 
( 0  gcd  N
)  =  ( abs `  N ) )
457, 44pm2.61dane 2628 1  |-  ( N  e.  ZZ  ->  (
0  gcd  N )  =  ( abs `  N
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717    =/= wne 2550   class class class wbr 4153   ` cfv 5394  (class class class)co 6020   0cc0 8923    <_ cle 9054   NN0cn0 10153   ZZcz 10214   abscabs 11966    || cdivides 12779    gcd cgcd 12933
This theorem is referenced by:  gcdid0  12951  nn0gcdsq  13071  dfphi2  13090  qqh0  24167
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-cnex 8979  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-pre-mulgt0 9000  ax-pre-sup 9001
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-2nd 6289  df-riota 6485  df-recs 6569  df-rdg 6604  df-er 6841  df-en 7046  df-dom 7047  df-sdom 7048  df-sup 7381  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059  df-sub 9225  df-neg 9226  df-div 9610  df-nn 9933  df-2 9990  df-3 9991  df-n0 10154  df-z 10215  df-uz 10421  df-rp 10545  df-seq 11251  df-exp 11310  df-cj 11831  df-re 11832  df-im 11833  df-sqr 11967  df-abs 11968  df-dvds 12780  df-gcd 12934
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