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Theorem gcd0id 13015
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 13001 . . . 4  |-  ( 0  gcd  0 )  =  0
2 oveq2 6081 . . . 4  |-  ( N  =  0  ->  (
0  gcd  N )  =  ( 0  gcd  0 ) )
3 fveq2 5720 . . . . 5  |-  ( N  =  0  ->  ( abs `  N )  =  ( abs `  0
) )
4 abs0 12082 . . . . 5  |-  ( abs `  0 )  =  0
53, 4syl6eq 2483 . . . 4  |-  ( N  =  0  ->  ( abs `  N )  =  0 )
61, 2, 53eqtr4a 2493 . . 3  |-  ( N  =  0  ->  (
0  gcd  N )  =  ( abs `  N
) )
76adantl 453 . 2  |-  ( ( N  e.  ZZ  /\  N  =  0 )  ->  ( 0  gcd 
N )  =  ( abs `  N ) )
8 0z 10285 . . . . . . 7  |-  0  e.  ZZ
9 gcddvds 13007 . . . . . . 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 13009 . . . . . . . 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 10365 . . . . . 6  |-  ( N  e.  ZZ  ->  (
0  gcd  N )  e.  ZZ )
16 dvdsleabs 12888 . . . . . 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 12109 . . . . . . . 8  |-  ( N  e.  ZZ  ->  ( abs `  N )  e. 
NN0 )
2120nn0zd 10365 . . . . . . 7  |-  ( N  e.  ZZ  ->  ( abs `  N )  e.  ZZ )
22 dvds0 12857 . . . . . . 7  |-  ( ( abs `  N )  e.  ZZ  ->  ( abs `  N )  ||  0 )
2321, 22syl 16 . . . . . 6  |-  ( N  e.  ZZ  ->  ( abs `  N )  ||  0 )
24 iddvds 12855 . . . . . . 7  |-  ( N  e.  ZZ  ->  N  ||  N )
25 absdvdsb 12860 . . . . . . . 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 2435 . . . . . . . 8  |-  0  =  0
3130biantrur 493 . . . . . . 7  |-  ( N  =  0  <->  ( 0  =  0  /\  N  =  0 ) )
3231necon3abii 2628 . . . . . 6  |-  ( N  =/=  0  <->  -.  (
0  =  0  /\  N  =  0 ) )
33 dvdslegcd 13008 . . . . . . . . 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 10367 . . . . 5  |-  ( N  e.  ZZ  ->  (
0  gcd  N )  e.  RR )
4121zred 10367 . . . . 5  |-  ( N  e.  ZZ  ->  ( abs `  N )  e.  RR )
4240, 41letri3d 9207 . . . 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 2676 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 1652    e. wcel 1725    =/= wne 2598   class class class wbr 4204   ` cfv 5446  (class class class)co 6073   0cc0 8982    <_ cle 9113   NN0cn0 10213   ZZcz 10274   abscabs 12031    || cdivides 12844    gcd cgcd 12998
This theorem is referenced by:  gcdid0  13016  nn0gcdsq  13136  dfphi2  13155  qqh0  24360
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-sup 7438  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-n0 10214  df-z 10275  df-uz 10481  df-rp 10605  df-seq 11316  df-exp 11375  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033  df-dvds 12845  df-gcd 12999
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