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Theorem gcd0id 12702
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 12688 . . . 4  |-  ( 0  gcd  0 )  =  0
2 oveq2 5866 . . . 4  |-  ( N  =  0  ->  (
0  gcd  N )  =  ( 0  gcd  0 ) )
3 fveq2 5525 . . . . 5  |-  ( N  =  0  ->  ( abs `  N )  =  ( abs `  0
) )
4 abs0 11770 . . . . 5  |-  ( abs `  0 )  =  0
53, 4syl6eq 2331 . . . 4  |-  ( N  =  0  ->  ( abs `  N )  =  0 )
61, 2, 53eqtr4a 2341 . . 3  |-  ( N  =  0  ->  (
0  gcd  N )  =  ( abs `  N
) )
76adantl 452 . 2  |-  ( ( N  e.  ZZ  /\  N  =  0 )  ->  ( 0  gcd 
N )  =  ( abs `  N ) )
8 0z 10035 . . . . . . 7  |-  0  e.  ZZ
9 gcddvds 12694 . . . . . . 7  |-  ( ( 0  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( 0  gcd 
N )  ||  0  /\  ( 0  gcd  N
)  ||  N )
)
108, 9mpan 651 . . . . . 6  |-  ( N  e.  ZZ  ->  (
( 0  gcd  N
)  ||  0  /\  ( 0  gcd  N
)  ||  N )
)
1110simprd 449 . . . . 5  |-  ( N  e.  ZZ  ->  (
0  gcd  N )  ||  N )
1211adantr 451 . . . 4  |-  ( ( N  e.  ZZ  /\  N  =/=  0 )  -> 
( 0  gcd  N
)  ||  N )
13 gcdcl 12696 . . . . . . . 8  |-  ( ( 0  e.  ZZ  /\  N  e.  ZZ )  ->  ( 0  gcd  N
)  e.  NN0 )
148, 13mpan 651 . . . . . . 7  |-  ( N  e.  ZZ  ->  (
0  gcd  N )  e.  NN0 )
1514nn0zd 10115 . . . . . 6  |-  ( N  e.  ZZ  ->  (
0  gcd  N )  e.  ZZ )
16 dvdsleabs 12575 . . . . . 6  |-  ( ( ( 0  gcd  N
)  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  ->  (
( 0  gcd  N
)  ||  N  ->  ( 0  gcd  N )  <_  ( abs `  N
) ) )
1715, 16syl3an1 1215 . . . . 5  |-  ( ( N  e.  ZZ  /\  N  e.  ZZ  /\  N  =/=  0 )  ->  (
( 0  gcd  N
)  ||  N  ->  ( 0  gcd  N )  <_  ( abs `  N
) ) )
18173anidm12 1239 . . . 4  |-  ( ( N  e.  ZZ  /\  N  =/=  0 )  -> 
( ( 0  gcd 
N )  ||  N  ->  ( 0  gcd  N
)  <_  ( abs `  N ) ) )
1912, 18mpd 14 . . 3  |-  ( ( N  e.  ZZ  /\  N  =/=  0 )  -> 
( 0  gcd  N
)  <_  ( abs `  N ) )
20 nn0abscl 11797 . . . . . . . 8  |-  ( N  e.  ZZ  ->  ( abs `  N )  e. 
NN0 )
2120nn0zd 10115 . . . . . . 7  |-  ( N  e.  ZZ  ->  ( abs `  N )  e.  ZZ )
22 dvds0 12544 . . . . . . 7  |-  ( ( abs `  N )  e.  ZZ  ->  ( abs `  N )  ||  0 )
2321, 22syl 15 . . . . . 6  |-  ( N  e.  ZZ  ->  ( abs `  N )  ||  0 )
24 iddvds 12542 . . . . . . 7  |-  ( N  e.  ZZ  ->  N  ||  N )
25 absdvdsb 12547 . . . . . . . 8  |-  ( ( N  e.  ZZ  /\  N  e.  ZZ )  ->  ( N  ||  N  <->  ( abs `  N ) 
||  N ) )
2625anidms 626 . . . . . . 7  |-  ( N  e.  ZZ  ->  ( N  ||  N  <->  ( abs `  N )  ||  N
) )
2724, 26mpbid 201 . . . . . 6  |-  ( N  e.  ZZ  ->  ( abs `  N )  ||  N )
2823, 27jca 518 . . . . 5  |-  ( N  e.  ZZ  ->  (
( abs `  N
)  ||  0  /\  ( abs `  N ) 
||  N ) )
2928adantr 451 . . . 4  |-  ( ( N  e.  ZZ  /\  N  =/=  0 )  -> 
( ( abs `  N
)  ||  0  /\  ( abs `  N ) 
||  N ) )
30 eqid 2283 . . . . . . . 8  |-  0  =  0
3130biantrur 492 . . . . . . 7  |-  ( N  =  0  <->  ( 0  =  0  /\  N  =  0 ) )
3231necon3abii 2476 . . . . . 6  |-  ( N  =/=  0  <->  -.  (
0  =  0  /\  N  =  0 ) )
33 dvdslegcd 12695 . . . . . . . . 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 423 . . . . . . . 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 1265 . . . . . . 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 650 . . . . . 6  |-  ( N  e.  ZZ  ->  ( -.  ( 0  =  0  /\  N  =  0 )  ->  ( (
( abs `  N
)  ||  0  /\  ( abs `  N ) 
||  N )  -> 
( abs `  N
)  <_  ( 0  gcd  N ) ) ) )
3732, 36syl5bi 208 . . . . 5  |-  ( N  e.  ZZ  ->  ( N  =/=  0  ->  (
( ( abs `  N
)  ||  0  /\  ( abs `  N ) 
||  N )  -> 
( abs `  N
)  <_  ( 0  gcd  N ) ) ) )
3837imp 418 . . . 4  |-  ( ( N  e.  ZZ  /\  N  =/=  0 )  -> 
( ( ( abs `  N )  ||  0  /\  ( abs `  N
)  ||  N )  ->  ( abs `  N
)  <_  ( 0  gcd  N ) ) )
3929, 38mpd 14 . . 3  |-  ( ( N  e.  ZZ  /\  N  =/=  0 )  -> 
( abs `  N
)  <_  ( 0  gcd  N ) )
4015zred 10117 . . . . 5  |-  ( N  e.  ZZ  ->  (
0  gcd  N )  e.  RR )
4121zred 10117 . . . . 5  |-  ( N  e.  ZZ  ->  ( abs `  N )  e.  RR )
4240, 41letri3d 8961 . . . 4  |-  ( N  e.  ZZ  ->  (
( 0  gcd  N
)  =  ( abs `  N )  <->  ( (
0  gcd  N )  <_  ( abs `  N
)  /\  ( abs `  N )  <_  (
0  gcd  N )
) ) )
4342adantr 451 . . 3  |-  ( ( N  e.  ZZ  /\  N  =/=  0 )  -> 
( ( 0  gcd 
N )  =  ( abs `  N )  <-> 
( ( 0  gcd 
N )  <_  ( abs `  N )  /\  ( abs `  N )  <_  ( 0  gcd 
N ) ) ) )
4419, 39, 43mpbir2and 888 . 2  |-  ( ( N  e.  ZZ  /\  N  =/=  0 )  -> 
( 0  gcd  N
)  =  ( abs `  N ) )
457, 44pm2.61dane 2524 1  |-  ( N  e.  ZZ  ->  (
0  gcd  N )  =  ( abs `  N
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   0cc0 8737    <_ cle 8868   NN0cn0 9965   ZZcz 10024   abscabs 11719    || cdivides 12531    gcd cgcd 12685
This theorem is referenced by:  gcdid0  12703  nn0gcdsq  12823  dfphi2  12842
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-seq 11047  df-exp 11105  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-dvds 12532  df-gcd 12686
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