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Theorem dvdssq 14074
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 4456 . . 3  |-  ( M  =  0  ->  ( M  ||  N  <->  0  ||  N ) )
2 sq0i 12240 . . . 4  |-  ( M  =  0  ->  ( M ^ 2 )  =  0 )
32breq1d 4463 . . 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 13138 . . . . 5  |-  ( ( M  e.  ZZ  /\  M  =/=  0 )  -> 
( abs `  M
)  e.  NN )
6 breq2 4457 . . . . . . 7  |-  ( N  =  0  ->  (
( abs `  M
)  ||  N  <->  ( abs `  M )  ||  0
) )
7 sq0i 12240 . . . . . . . 8  |-  ( N  =  0  ->  ( N ^ 2 )  =  0 )
87breq2d 4465 . . . . . . 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 13138 . . . . . . . . 9  |-  ( ( N  e.  ZZ  /\  N  =/=  0 )  -> 
( abs `  N
)  e.  NN )
11 dvdssqlem 14073 . . . . . . . . 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 10898 . . . . . . . . 9  |-  ( ( abs `  M )  e.  NN  ->  ( abs `  M )  e.  ZZ )
14 simpl 457 . . . . . . . . 9  |-  ( ( N  e.  ZZ  /\  N  =/=  0 )  ->  N  e.  ZZ )
15 dvdsabsb 13881 . . . . . . . . 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 12217 . . . . . . . . . . 11  |-  ( ( abs `  M )  e.  NN  ->  (
( abs `  M
) ^ 2 )  e.  NN )
1817nnzd 10977 . . . . . . . . . 10  |-  ( ( abs `  M )  e.  NN  ->  (
( abs `  M
) ^ 2 )  e.  ZZ )
19 zsqcl 12218 . . . . . . . . . . 11  |-  ( N  e.  ZZ  ->  ( N ^ 2 )  e.  ZZ )
2019adantr 465 . . . . . . . . . 10  |-  ( ( N  e.  ZZ  /\  N  =/=  0 )  -> 
( N ^ 2 )  e.  ZZ )
21 dvdsabsb 13881 . . . . . . . . . 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 10881 . . . . . . . . . . . . 13  |-  ( N  e.  ZZ  ->  N  e.  CC )
2423adantr 465 . . . . . . . . . . . 12  |-  ( ( N  e.  ZZ  /\  N  =/=  0 )  ->  N  e.  CC )
25 abssq 13119 . . . . . . . . . . . 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 4465 . . . . . . . . . 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 13877 . . . . . . . . 9  |-  ( ( abs `  M )  e.  ZZ  ->  ( abs `  M )  ||  0 )
33 zsqcl 12218 . . . . . . . . . 10  |-  ( ( abs `  M )  e.  ZZ  ->  (
( abs `  M
) ^ 2 )  e.  ZZ )
34 dvds0 13877 . . . . . . . . . 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 2782 . . . . 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 13880 . . . . 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 12218 . . . . . . 7  |-  ( M  e.  ZZ  ->  ( M ^ 2 )  e.  ZZ )
4443adantr 465 . . . . . 6  |-  ( ( M  e.  ZZ  /\  M  =/=  0 )  -> 
( M ^ 2 )  e.  ZZ )
45 absdvdsb 13880 . . . . . 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 10881 . . . . . . . . . 10  |-  ( M  e.  ZZ  ->  M  e.  CC )
48 abssq 13119 . . . . . . . . . 10  |-  ( M  e.  CC  ->  (
( abs `  M
) ^ 2 )  =  ( abs `  ( M ^ 2 ) ) )
4947, 48syl 16 . . . . . . . . 9  |-  ( M  e.  ZZ  ->  (
( abs `  M
) ^ 2 )  =  ( abs `  ( M ^ 2 ) ) )
5049eqcomd 2475 . . . . . . . 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 4463 . . . . . 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 13882 . . . . 5  |-  ( N  e.  ZZ  ->  (
0  ||  N  <->  N  = 
0 ) )
58 sqeq0 12212 . . . . . 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 13882 . . . . 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 2782 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 1379    e. wcel 1767    =/= wne 2662   class class class wbr 4453   ` cfv 5594  (class class class)co 6295   CCcc 9502   0cc0 9504   NNcn 10548   2c2 10597   ZZcz 10876   ^cexp 12146   abscabs 13047    || cdivides 13864
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-pre-sup 9582
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-2nd 6796  df-recs 7054  df-rdg 7088  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-sup 7913  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-div 10219  df-nn 10549  df-2 10606  df-3 10607  df-n0 10808  df-z 10877  df-uz 11095  df-rp 11233  df-fl 11909  df-mod 11977  df-seq 12088  df-exp 12147  df-cj 12912  df-re 12913  df-im 12914  df-sqrt 13048  df-abs 13049  df-dvds 13865  df-gcd 14021
This theorem is referenced by:  pythagtriplem19  14233  4sqlem9  14340  4sqlem10  14341  lgsdir  23471  2sqlem8a  23512
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