MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nn0gcdsq Structured version   Unicode version

Theorem nn0gcdsq 13144
Description: Squaring commutes with GCD, in particular two coprime numbers have coprime squares. (Contributed by Stefan O'Rear, 15-Sep-2014.)
Assertion
Ref Expression
nn0gcdsq  |-  ( ( A  e.  NN0  /\  B  e.  NN0 )  -> 
( ( A  gcd  B ) ^ 2 )  =  ( ( A ^ 2 )  gcd  ( B ^ 2 ) ) )

Proof of Theorem nn0gcdsq
StepHypRef Expression
1 elnn0 10223 . 2  |-  ( A  e.  NN0  <->  ( A  e.  NN  \/  A  =  0 ) )
2 elnn0 10223 . 2  |-  ( B  e.  NN0  <->  ( B  e.  NN  \/  B  =  0 ) )
3 sqgcd 13058 . . 3  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( ( A  gcd  B ) ^ 2 )  =  ( ( A ^ 2 )  gcd  ( B ^ 2 ) ) )
4 nncn 10008 . . . . . . 7  |-  ( B  e.  NN  ->  B  e.  CC )
5 abssq 12111 . . . . . . 7  |-  ( B  e.  CC  ->  (
( abs `  B
) ^ 2 )  =  ( abs `  ( B ^ 2 ) ) )
64, 5syl 16 . . . . . 6  |-  ( B  e.  NN  ->  (
( abs `  B
) ^ 2 )  =  ( abs `  ( B ^ 2 ) ) )
7 nnz 10303 . . . . . . . 8  |-  ( B  e.  NN  ->  B  e.  ZZ )
8 gcd0id 13023 . . . . . . . 8  |-  ( B  e.  ZZ  ->  (
0  gcd  B )  =  ( abs `  B
) )
97, 8syl 16 . . . . . . 7  |-  ( B  e.  NN  ->  (
0  gcd  B )  =  ( abs `  B
) )
109oveq1d 6096 . . . . . 6  |-  ( B  e.  NN  ->  (
( 0  gcd  B
) ^ 2 )  =  ( ( abs `  B ) ^ 2 ) )
11 sq0 11473 . . . . . . . . 9  |-  ( 0 ^ 2 )  =  0
1211a1i 11 . . . . . . . 8  |-  ( B  e.  NN  ->  (
0 ^ 2 )  =  0 )
1312oveq1d 6096 . . . . . . 7  |-  ( B  e.  NN  ->  (
( 0 ^ 2 )  gcd  ( B ^ 2 ) )  =  ( 0  gcd  ( B ^ 2 ) ) )
14 zsqcl 11452 . . . . . . . 8  |-  ( B  e.  ZZ  ->  ( B ^ 2 )  e.  ZZ )
15 gcd0id 13023 . . . . . . . 8  |-  ( ( B ^ 2 )  e.  ZZ  ->  (
0  gcd  ( B ^ 2 ) )  =  ( abs `  ( B ^ 2 ) ) )
167, 14, 153syl 19 . . . . . . 7  |-  ( B  e.  NN  ->  (
0  gcd  ( B ^ 2 ) )  =  ( abs `  ( B ^ 2 ) ) )
1713, 16eqtrd 2468 . . . . . 6  |-  ( B  e.  NN  ->  (
( 0 ^ 2 )  gcd  ( B ^ 2 ) )  =  ( abs `  ( B ^ 2 ) ) )
186, 10, 173eqtr4d 2478 . . . . 5  |-  ( B  e.  NN  ->  (
( 0  gcd  B
) ^ 2 )  =  ( ( 0 ^ 2 )  gcd  ( B ^ 2 ) ) )
1918adantl 453 . . . 4  |-  ( ( A  =  0  /\  B  e.  NN )  ->  ( ( 0  gcd  B ) ^
2 )  =  ( ( 0 ^ 2 )  gcd  ( B ^ 2 ) ) )
20 oveq1 6088 . . . . . . 7  |-  ( A  =  0  ->  ( A  gcd  B )  =  ( 0  gcd  B
) )
2120oveq1d 6096 . . . . . 6  |-  ( A  =  0  ->  (
( A  gcd  B
) ^ 2 )  =  ( ( 0  gcd  B ) ^
2 ) )
22 oveq1 6088 . . . . . . 7  |-  ( A  =  0  ->  ( A ^ 2 )  =  ( 0 ^ 2 ) )
2322oveq1d 6096 . . . . . 6  |-  ( A  =  0  ->  (
( A ^ 2 )  gcd  ( B ^ 2 ) )  =  ( ( 0 ^ 2 )  gcd  ( B ^ 2 ) ) )
2421, 23eqeq12d 2450 . . . . 5  |-  ( A  =  0  ->  (
( ( A  gcd  B ) ^ 2 )  =  ( ( A ^ 2 )  gcd  ( B ^ 2 ) )  <->  ( (
0  gcd  B ) ^ 2 )  =  ( ( 0 ^ 2 )  gcd  ( B ^ 2 ) ) ) )
2524adantr 452 . . . 4  |-  ( ( A  =  0  /\  B  e.  NN )  ->  ( ( ( A  gcd  B ) ^ 2 )  =  ( ( A ^
2 )  gcd  ( B ^ 2 ) )  <-> 
( ( 0  gcd 
B ) ^ 2 )  =  ( ( 0 ^ 2 )  gcd  ( B ^
2 ) ) ) )
2619, 25mpbird 224 . . 3  |-  ( ( A  =  0  /\  B  e.  NN )  ->  ( ( A  gcd  B ) ^
2 )  =  ( ( A ^ 2 )  gcd  ( B ^ 2 ) ) )
27 nncn 10008 . . . . . . 7  |-  ( A  e.  NN  ->  A  e.  CC )
28 abssq 12111 . . . . . . 7  |-  ( A  e.  CC  ->  (
( abs `  A
) ^ 2 )  =  ( abs `  ( A ^ 2 ) ) )
2927, 28syl 16 . . . . . 6  |-  ( A  e.  NN  ->  (
( abs `  A
) ^ 2 )  =  ( abs `  ( A ^ 2 ) ) )
30 nnz 10303 . . . . . . . 8  |-  ( A  e.  NN  ->  A  e.  ZZ )
31 gcdid0 13024 . . . . . . . 8  |-  ( A  e.  ZZ  ->  ( A  gcd  0 )  =  ( abs `  A
) )
3230, 31syl 16 . . . . . . 7  |-  ( A  e.  NN  ->  ( A  gcd  0 )  =  ( abs `  A
) )
3332oveq1d 6096 . . . . . 6  |-  ( A  e.  NN  ->  (
( A  gcd  0
) ^ 2 )  =  ( ( abs `  A ) ^ 2 ) )
3411a1i 11 . . . . . . . 8  |-  ( A  e.  NN  ->  (
0 ^ 2 )  =  0 )
3534oveq2d 6097 . . . . . . 7  |-  ( A  e.  NN  ->  (
( A ^ 2 )  gcd  ( 0 ^ 2 ) )  =  ( ( A ^ 2 )  gcd  0 ) )
36 zsqcl 11452 . . . . . . . 8  |-  ( A  e.  ZZ  ->  ( A ^ 2 )  e.  ZZ )
37 gcdid0 13024 . . . . . . . 8  |-  ( ( A ^ 2 )  e.  ZZ  ->  (
( A ^ 2 )  gcd  0 )  =  ( abs `  ( A ^ 2 ) ) )
3830, 36, 373syl 19 . . . . . . 7  |-  ( A  e.  NN  ->  (
( A ^ 2 )  gcd  0 )  =  ( abs `  ( A ^ 2 ) ) )
3935, 38eqtrd 2468 . . . . . 6  |-  ( A  e.  NN  ->  (
( A ^ 2 )  gcd  ( 0 ^ 2 ) )  =  ( abs `  ( A ^ 2 ) ) )
4029, 33, 393eqtr4d 2478 . . . . 5  |-  ( A  e.  NN  ->  (
( A  gcd  0
) ^ 2 )  =  ( ( A ^ 2 )  gcd  ( 0 ^ 2 ) ) )
4140adantr 452 . . . 4  |-  ( ( A  e.  NN  /\  B  =  0 )  ->  ( ( A  gcd  0 ) ^
2 )  =  ( ( A ^ 2 )  gcd  ( 0 ^ 2 ) ) )
42 oveq2 6089 . . . . . . 7  |-  ( B  =  0  ->  ( A  gcd  B )  =  ( A  gcd  0
) )
4342oveq1d 6096 . . . . . 6  |-  ( B  =  0  ->  (
( A  gcd  B
) ^ 2 )  =  ( ( A  gcd  0 ) ^
2 ) )
44 oveq1 6088 . . . . . . 7  |-  ( B  =  0  ->  ( B ^ 2 )  =  ( 0 ^ 2 ) )
4544oveq2d 6097 . . . . . 6  |-  ( B  =  0  ->  (
( A ^ 2 )  gcd  ( B ^ 2 ) )  =  ( ( A ^ 2 )  gcd  ( 0 ^ 2 ) ) )
4643, 45eqeq12d 2450 . . . . 5  |-  ( B  =  0  ->  (
( ( A  gcd  B ) ^ 2 )  =  ( ( A ^ 2 )  gcd  ( B ^ 2 ) )  <->  ( ( A  gcd  0 ) ^
2 )  =  ( ( A ^ 2 )  gcd  ( 0 ^ 2 ) ) ) )
4746adantl 453 . . . 4  |-  ( ( A  e.  NN  /\  B  =  0 )  ->  ( ( ( A  gcd  B ) ^ 2 )  =  ( ( A ^
2 )  gcd  ( B ^ 2 ) )  <-> 
( ( A  gcd  0 ) ^ 2 )  =  ( ( A ^ 2 )  gcd  ( 0 ^ 2 ) ) ) )
4841, 47mpbird 224 . . 3  |-  ( ( A  e.  NN  /\  B  =  0 )  ->  ( ( A  gcd  B ) ^
2 )  =  ( ( A ^ 2 )  gcd  ( B ^ 2 ) ) )
49 gcd0val 13009 . . . . . 6  |-  ( 0  gcd  0 )  =  0
5049oveq1i 6091 . . . . 5  |-  ( ( 0  gcd  0 ) ^ 2 )  =  ( 0 ^ 2 )
5111, 11oveq12i 6093 . . . . . 6  |-  ( ( 0 ^ 2 )  gcd  ( 0 ^ 2 ) )  =  ( 0  gcd  0
)
5251, 49eqtri 2456 . . . . 5  |-  ( ( 0 ^ 2 )  gcd  ( 0 ^ 2 ) )  =  0
5311, 50, 523eqtr4i 2466 . . . 4  |-  ( ( 0  gcd  0 ) ^ 2 )  =  ( ( 0 ^ 2 )  gcd  (
0 ^ 2 ) )
54 oveq12 6090 . . . . 5  |-  ( ( A  =  0  /\  B  =  0 )  ->  ( A  gcd  B )  =  ( 0  gcd  0 ) )
5554oveq1d 6096 . . . 4  |-  ( ( A  =  0  /\  B  =  0 )  ->  ( ( A  gcd  B ) ^
2 )  =  ( ( 0  gcd  0
) ^ 2 ) )
5622, 44oveqan12d 6100 . . . 4  |-  ( ( A  =  0  /\  B  =  0 )  ->  ( ( A ^ 2 )  gcd  ( B ^ 2 ) )  =  ( ( 0 ^ 2 )  gcd  ( 0 ^ 2 ) ) )
5753, 55, 563eqtr4a 2494 . . 3  |-  ( ( A  =  0  /\  B  =  0 )  ->  ( ( A  gcd  B ) ^
2 )  =  ( ( A ^ 2 )  gcd  ( B ^ 2 ) ) )
583, 26, 48, 57ccase 913 . 2  |-  ( ( ( A  e.  NN  \/  A  =  0
)  /\  ( B  e.  NN  \/  B  =  0 ) )  -> 
( ( A  gcd  B ) ^ 2 )  =  ( ( A ^ 2 )  gcd  ( B ^ 2 ) ) )
591, 2, 58syl2anb 466 1  |-  ( ( A  e.  NN0  /\  B  e.  NN0 )  -> 
( ( A  gcd  B ) ^ 2 )  =  ( ( A ^ 2 )  gcd  ( B ^ 2 ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    = wceq 1652    e. wcel 1725   ` cfv 5454  (class class class)co 6081   CCcc 8988   0cc0 8990   NNcn 10000   2c2 10049   NN0cn0 10221   ZZcz 10282   ^cexp 11382   abscabs 12039    gcd cgcd 13006
This theorem is referenced by:  zgcdsq  13145
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 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-sup 7446  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-2 10058  df-3 10059  df-n0 10222  df-z 10283  df-uz 10489  df-rp 10613  df-fl 11202  df-mod 11251  df-seq 11324  df-exp 11383  df-cj 11904  df-re 11905  df-im 11906  df-sqr 12040  df-abs 12041  df-dvds 12853  df-gcd 13007
  Copyright terms: Public domain W3C validator