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Theorem dvdssq 13866
Description: Two numbers are divisible iff their squares are. (Contributed by Scott Fenton, 18-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Assertion
Ref Expression
dvdssq  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  ||  N  <->  ( M ^ 2 ) 
||  ( N ^
2 ) ) )

Proof of Theorem dvdssq
StepHypRef Expression
1 breq1 4406 . . 3  |-  ( M  =  0  ->  ( M  ||  N  <->  0  ||  N ) )
2 sq0i 12079 . . . 4  |-  ( M  =  0  ->  ( M ^ 2 )  =  0 )
32breq1d 4413 . . 3  |-  ( M  =  0  ->  (
( M ^ 2 )  ||  ( N ^ 2 )  <->  0  ||  ( N ^ 2 ) ) )
41, 3bibi12d 321 . 2  |-  ( M  =  0  ->  (
( M  ||  N  <->  ( M ^ 2 ) 
||  ( N ^
2 ) )  <->  ( 0 
||  N  <->  0  ||  ( N ^ 2 ) ) ) )
5 nnabscl 12935 . . . . 5  |-  ( ( M  e.  ZZ  /\  M  =/=  0 )  -> 
( abs `  M
)  e.  NN )
6 breq2 4407 . . . . . . 7  |-  ( N  =  0  ->  (
( abs `  M
)  ||  N  <->  ( abs `  M )  ||  0
) )
7 sq0i 12079 . . . . . . . 8  |-  ( N  =  0  ->  ( N ^ 2 )  =  0 )
87breq2d 4415 . . . . . . 7  |-  ( N  =  0  ->  (
( ( abs `  M
) ^ 2 ) 
||  ( N ^
2 )  <->  ( ( abs `  M ) ^
2 )  ||  0
) )
96, 8bibi12d 321 . . . . . 6  |-  ( N  =  0  ->  (
( ( abs `  M
)  ||  N  <->  ( ( abs `  M ) ^
2 )  ||  ( N ^ 2 ) )  <-> 
( ( abs `  M
)  ||  0  <->  ( ( abs `  M ) ^
2 )  ||  0
) ) )
10 nnabscl 12935 . . . . . . . . 9  |-  ( ( N  e.  ZZ  /\  N  =/=  0 )  -> 
( abs `  N
)  e.  NN )
11 dvdssqlem 13865 . . . . . . . . 9  |-  ( ( ( abs `  M
)  e.  NN  /\  ( abs `  N )  e.  NN )  -> 
( ( abs `  M
)  ||  ( abs `  N )  <->  ( ( abs `  M ) ^
2 )  ||  (
( abs `  N
) ^ 2 ) ) )
1210, 11sylan2 474 . . . . . . . 8  |-  ( ( ( abs `  M
)  e.  NN  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  ->  ( ( abs `  M )  ||  ( abs `  N )  <-> 
( ( abs `  M
) ^ 2 ) 
||  ( ( abs `  N ) ^ 2 ) ) )
13 nnz 10783 . . . . . . . . 9  |-  ( ( abs `  M )  e.  NN  ->  ( abs `  M )  e.  ZZ )
14 simpl 457 . . . . . . . . 9  |-  ( ( N  e.  ZZ  /\  N  =/=  0 )  ->  N  e.  ZZ )
15 dvdsabsb 13674 . . . . . . . . 9  |-  ( ( ( abs `  M
)  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( abs `  M
)  ||  N  <->  ( abs `  M )  ||  ( abs `  N ) ) )
1613, 14, 15syl2an 477 . . . . . . . 8  |-  ( ( ( abs `  M
)  e.  NN  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  ->  ( ( abs `  M )  ||  N 
<->  ( abs `  M
)  ||  ( abs `  N ) ) )
17 nnsqcl 12056 . . . . . . . . . . 11  |-  ( ( abs `  M )  e.  NN  ->  (
( abs `  M
) ^ 2 )  e.  NN )
1817nnzd 10861 . . . . . . . . . 10  |-  ( ( abs `  M )  e.  NN  ->  (
( abs `  M
) ^ 2 )  e.  ZZ )
19 zsqcl 12057 . . . . . . . . . . 11  |-  ( N  e.  ZZ  ->  ( N ^ 2 )  e.  ZZ )
2019adantr 465 . . . . . . . . . 10  |-  ( ( N  e.  ZZ  /\  N  =/=  0 )  -> 
( N ^ 2 )  e.  ZZ )
21 dvdsabsb 13674 . . . . . . . . . 10  |-  ( ( ( ( abs `  M
) ^ 2 )  e.  ZZ  /\  ( N ^ 2 )  e.  ZZ )  ->  (
( ( abs `  M
) ^ 2 ) 
||  ( N ^
2 )  <->  ( ( abs `  M ) ^
2 )  ||  ( abs `  ( N ^
2 ) ) ) )
2218, 20, 21syl2an 477 . . . . . . . . 9  |-  ( ( ( abs `  M
)  e.  NN  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  ->  ( (
( abs `  M
) ^ 2 ) 
||  ( N ^
2 )  <->  ( ( abs `  M ) ^
2 )  ||  ( abs `  ( N ^
2 ) ) ) )
23 zcn 10766 . . . . . . . . . . . . 13  |-  ( N  e.  ZZ  ->  N  e.  CC )
2423adantr 465 . . . . . . . . . . . 12  |-  ( ( N  e.  ZZ  /\  N  =/=  0 )  ->  N  e.  CC )
25 abssq 12917 . . . . . . . . . . . 12  |-  ( N  e.  CC  ->  (
( abs `  N
) ^ 2 )  =  ( abs `  ( N ^ 2 ) ) )
2624, 25syl 16 . . . . . . . . . . 11  |-  ( ( N  e.  ZZ  /\  N  =/=  0 )  -> 
( ( abs `  N
) ^ 2 )  =  ( abs `  ( N ^ 2 ) ) )
2726breq2d 4415 . . . . . . . . . 10  |-  ( ( N  e.  ZZ  /\  N  =/=  0 )  -> 
( ( ( abs `  M ) ^ 2 )  ||  ( ( abs `  N ) ^ 2 )  <->  ( ( abs `  M ) ^
2 )  ||  ( abs `  ( N ^
2 ) ) ) )
2827adantl 466 . . . . . . . . 9  |-  ( ( ( abs `  M
)  e.  NN  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  ->  ( (
( abs `  M
) ^ 2 ) 
||  ( ( abs `  N ) ^ 2 )  <->  ( ( abs `  M ) ^ 2 )  ||  ( abs `  ( N ^ 2 ) ) ) )
2922, 28bitr4d 256 . . . . . . . 8  |-  ( ( ( abs `  M
)  e.  NN  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  ->  ( (
( abs `  M
) ^ 2 ) 
||  ( N ^
2 )  <->  ( ( abs `  M ) ^
2 )  ||  (
( abs `  N
) ^ 2 ) ) )
3012, 16, 293bitr4d 285 . . . . . . 7  |-  ( ( ( abs `  M
)  e.  NN  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  ->  ( ( abs `  M )  ||  N 
<->  ( ( abs `  M
) ^ 2 ) 
||  ( N ^
2 ) ) )
3130anassrs 648 . . . . . 6  |-  ( ( ( ( abs `  M
)  e.  NN  /\  N  e.  ZZ )  /\  N  =/=  0
)  ->  ( ( abs `  M )  ||  N 
<->  ( ( abs `  M
) ^ 2 ) 
||  ( N ^
2 ) ) )
32 dvds0 13670 . . . . . . . . 9  |-  ( ( abs `  M )  e.  ZZ  ->  ( abs `  M )  ||  0 )
33 zsqcl 12057 . . . . . . . . . 10  |-  ( ( abs `  M )  e.  ZZ  ->  (
( abs `  M
) ^ 2 )  e.  ZZ )
34 dvds0 13670 . . . . . . . . . 10  |-  ( ( ( abs `  M
) ^ 2 )  e.  ZZ  ->  (
( abs `  M
) ^ 2 ) 
||  0 )
3533, 34syl 16 . . . . . . . . 9  |-  ( ( abs `  M )  e.  ZZ  ->  (
( abs `  M
) ^ 2 ) 
||  0 )
3632, 352thd 240 . . . . . . . 8  |-  ( ( abs `  M )  e.  ZZ  ->  (
( abs `  M
)  ||  0  <->  ( ( abs `  M ) ^
2 )  ||  0
) )
3713, 36syl 16 . . . . . . 7  |-  ( ( abs `  M )  e.  NN  ->  (
( abs `  M
)  ||  0  <->  ( ( abs `  M ) ^
2 )  ||  0
) )
3837adantr 465 . . . . . 6  |-  ( ( ( abs `  M
)  e.  NN  /\  N  e.  ZZ )  ->  ( ( abs `  M
)  ||  0  <->  ( ( abs `  M ) ^
2 )  ||  0
) )
399, 31, 38pm2.61ne 2767 . . . . 5  |-  ( ( ( abs `  M
)  e.  NN  /\  N  e.  ZZ )  ->  ( ( abs `  M
)  ||  N  <->  ( ( abs `  M ) ^
2 )  ||  ( N ^ 2 ) ) )
405, 39sylan 471 . . . 4  |-  ( ( ( M  e.  ZZ  /\  M  =/=  0 )  /\  N  e.  ZZ )  ->  ( ( abs `  M )  ||  N  <->  ( ( abs `  M
) ^ 2 ) 
||  ( N ^
2 ) ) )
41 absdvdsb 13673 . . . . 5  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  ||  N  <->  ( abs `  M ) 
||  N ) )
4241adantlr 714 . . . 4  |-  ( ( ( M  e.  ZZ  /\  M  =/=  0 )  /\  N  e.  ZZ )  ->  ( M  ||  N 
<->  ( abs `  M
)  ||  N )
)
43 zsqcl 12057 . . . . . . 7  |-  ( M  e.  ZZ  ->  ( M ^ 2 )  e.  ZZ )
4443adantr 465 . . . . . 6  |-  ( ( M  e.  ZZ  /\  M  =/=  0 )  -> 
( M ^ 2 )  e.  ZZ )
45 absdvdsb 13673 . . . . . 6  |-  ( ( ( M ^ 2 )  e.  ZZ  /\  ( N ^ 2 )  e.  ZZ )  -> 
( ( M ^
2 )  ||  ( N ^ 2 )  <->  ( abs `  ( M ^ 2 ) )  ||  ( N ^ 2 ) ) )
4644, 19, 45syl2an 477 . . . . 5  |-  ( ( ( M  e.  ZZ  /\  M  =/=  0 )  /\  N  e.  ZZ )  ->  ( ( M ^ 2 )  ||  ( N ^ 2 )  <-> 
( abs `  ( M ^ 2 ) ) 
||  ( N ^
2 ) ) )
47 zcn 10766 . . . . . . . . . 10  |-  ( M  e.  ZZ  ->  M  e.  CC )
48 abssq 12917 . . . . . . . . . 10  |-  ( M  e.  CC  ->  (
( abs `  M
) ^ 2 )  =  ( abs `  ( M ^ 2 ) ) )
4947, 48syl 16 . . . . . . . . 9  |-  ( M  e.  ZZ  ->  (
( abs `  M
) ^ 2 )  =  ( abs `  ( M ^ 2 ) ) )
5049eqcomd 2462 . . . . . . . 8  |-  ( M  e.  ZZ  ->  ( abs `  ( M ^
2 ) )  =  ( ( abs `  M
) ^ 2 ) )
5150adantr 465 . . . . . . 7  |-  ( ( M  e.  ZZ  /\  M  =/=  0 )  -> 
( abs `  ( M ^ 2 ) )  =  ( ( abs `  M ) ^ 2 ) )
5251breq1d 4413 . . . . . 6  |-  ( ( M  e.  ZZ  /\  M  =/=  0 )  -> 
( ( abs `  ( M ^ 2 ) ) 
||  ( N ^
2 )  <->  ( ( abs `  M ) ^
2 )  ||  ( N ^ 2 ) ) )
5352adantr 465 . . . . 5  |-  ( ( ( M  e.  ZZ  /\  M  =/=  0 )  /\  N  e.  ZZ )  ->  ( ( abs `  ( M ^ 2 ) )  ||  ( N ^ 2 )  <->  ( ( abs `  M ) ^
2 )  ||  ( N ^ 2 ) ) )
5446, 53bitrd 253 . . . 4  |-  ( ( ( M  e.  ZZ  /\  M  =/=  0 )  /\  N  e.  ZZ )  ->  ( ( M ^ 2 )  ||  ( N ^ 2 )  <-> 
( ( abs `  M
) ^ 2 ) 
||  ( N ^
2 ) ) )
5540, 42, 543bitr4d 285 . . 3  |-  ( ( ( M  e.  ZZ  /\  M  =/=  0 )  /\  N  e.  ZZ )  ->  ( M  ||  N 
<->  ( M ^ 2 )  ||  ( N ^ 2 ) ) )
5655an32s 802 . 2  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  M  =/=  0
)  ->  ( M  ||  N  <->  ( M ^
2 )  ||  ( N ^ 2 ) ) )
57 0dvds 13675 . . . . 5  |-  ( N  e.  ZZ  ->  (
0  ||  N  <->  N  = 
0 ) )
58 sqeq0 12051 . . . . . 6  |-  ( N  e.  CC  ->  (
( N ^ 2 )  =  0  <->  N  =  0 ) )
5923, 58syl 16 . . . . 5  |-  ( N  e.  ZZ  ->  (
( N ^ 2 )  =  0  <->  N  =  0 ) )
6057, 59bitr4d 256 . . . 4  |-  ( N  e.  ZZ  ->  (
0  ||  N  <->  ( N ^ 2 )  =  0 ) )
61 0dvds 13675 . . . . 5  |-  ( ( N ^ 2 )  e.  ZZ  ->  (
0  ||  ( N ^ 2 )  <->  ( N ^ 2 )  =  0 ) )
6219, 61syl 16 . . . 4  |-  ( N  e.  ZZ  ->  (
0  ||  ( N ^ 2 )  <->  ( N ^ 2 )  =  0 ) )
6360, 62bitr4d 256 . . 3  |-  ( N  e.  ZZ  ->  (
0  ||  N  <->  0  ||  ( N ^ 2 ) ) )
6463adantl 466 . 2  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( 0  ||  N  <->  0 
||  ( N ^
2 ) ) )
654, 56, 64pm2.61ne 2767 1  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  ||  N  <->  ( M ^ 2 ) 
||  ( N ^
2 ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758    =/= wne 2648   class class class wbr 4403   ` cfv 5529  (class class class)co 6203   CCcc 9395   0cc0 9397   NNcn 10437   2c2 10486   ZZcz 10761   ^cexp 11986   abscabs 12845    || cdivides 13657
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9453  ax-resscn 9454  ax-1cn 9455  ax-icn 9456  ax-addcl 9457  ax-addrcl 9458  ax-mulcl 9459  ax-mulrcl 9460  ax-mulcom 9461  ax-addass 9462  ax-mulass 9463  ax-distr 9464  ax-i2m1 9465  ax-1ne0 9466  ax-1rid 9467  ax-rnegex 9468  ax-rrecex 9469  ax-cnre 9470  ax-pre-lttri 9471  ax-pre-lttrn 9472  ax-pre-ltadd 9473  ax-pre-mulgt0 9474  ax-pre-sup 9475
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-2nd 6691  df-recs 6945  df-rdg 6979  df-er 7214  df-en 7424  df-dom 7425  df-sdom 7426  df-sup 7806  df-pnf 9535  df-mnf 9536  df-xr 9537  df-ltxr 9538  df-le 9539  df-sub 9712  df-neg 9713  df-div 10109  df-nn 10438  df-2 10495  df-3 10496  df-n0 10695  df-z 10762  df-uz 10977  df-rp 11107  df-fl 11763  df-mod 11830  df-seq 11928  df-exp 11987  df-cj 12710  df-re 12711  df-im 12712  df-sqr 12846  df-abs 12847  df-dvds 13658  df-gcd 13813
This theorem is referenced by:  pythagtriplem19  14022  4sqlem9  14129  4sqlem10  14130  lgsdir  22812  2sqlem8a  22853
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