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

Theorem dvdssq 12739
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 4026 . . 3  |-  ( M  =  0  ->  ( M  ||  N  <->  0  ||  N ) )
2 sq0i 11196 . . . 4  |-  ( M  =  0  ->  ( M ^ 2 )  =  0 )
32breq1d 4033 . . 3  |-  ( M  =  0  ->  (
( M ^ 2 )  ||  ( N ^ 2 )  <->  0  ||  ( N ^ 2 ) ) )
41, 3bibi12d 312 . 2  |-  ( M  =  0  ->  (
( M  ||  N  <->  ( M ^ 2 ) 
||  ( N ^
2 ) )  <->  ( 0 
||  N  <->  0  ||  ( N ^ 2 ) ) ) )
5 nnabscl 11809 . . . . 5  |-  ( ( M  e.  ZZ  /\  M  =/=  0 )  -> 
( abs `  M
)  e.  NN )
6 breq2 4027 . . . . . . 7  |-  ( N  =  0  ->  (
( abs `  M
)  ||  N  <->  ( abs `  M )  ||  0
) )
7 sq0i 11196 . . . . . . . 8  |-  ( N  =  0  ->  ( N ^ 2 )  =  0 )
87breq2d 4035 . . . . . . 7  |-  ( N  =  0  ->  (
( ( abs `  M
) ^ 2 ) 
||  ( N ^
2 )  <->  ( ( abs `  M ) ^
2 )  ||  0
) )
96, 8bibi12d 312 . . . . . 6  |-  ( N  =  0  ->  (
( ( abs `  M
)  ||  N  <->  ( ( abs `  M ) ^
2 )  ||  ( N ^ 2 ) )  <-> 
( ( abs `  M
)  ||  0  <->  ( ( abs `  M ) ^
2 )  ||  0
) ) )
10 nnabscl 11809 . . . . . . . . 9  |-  ( ( N  e.  ZZ  /\  N  =/=  0 )  -> 
( abs `  N
)  e.  NN )
11 dvdssqlem 12738 . . . . . . . . 9  |-  ( ( ( abs `  M
)  e.  NN  /\  ( abs `  N )  e.  NN )  -> 
( ( abs `  M
)  ||  ( abs `  N )  <->  ( ( abs `  M ) ^
2 )  ||  (
( abs `  N
) ^ 2 ) ) )
1210, 11sylan2 460 . . . . . . . 8  |-  ( ( ( abs `  M
)  e.  NN  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  ->  ( ( abs `  M )  ||  ( abs `  N )  <-> 
( ( abs `  M
) ^ 2 ) 
||  ( ( abs `  N ) ^ 2 ) ) )
13 nnz 10045 . . . . . . . . 9  |-  ( ( abs `  M )  e.  NN  ->  ( abs `  M )  e.  ZZ )
14 simpl 443 . . . . . . . . 9  |-  ( ( N  e.  ZZ  /\  N  =/=  0 )  ->  N  e.  ZZ )
15 dvdsabsb 12548 . . . . . . . . 9  |-  ( ( ( abs `  M
)  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( abs `  M
)  ||  N  <->  ( abs `  M )  ||  ( abs `  N ) ) )
1613, 14, 15syl2an 463 . . . . . . . 8  |-  ( ( ( abs `  M
)  e.  NN  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  ->  ( ( abs `  M )  ||  N 
<->  ( abs `  M
)  ||  ( abs `  N ) ) )
17 nnsqcl 11173 . . . . . . . . . . 11  |-  ( ( abs `  M )  e.  NN  ->  (
( abs `  M
) ^ 2 )  e.  NN )
1817nnzd 10116 . . . . . . . . . 10  |-  ( ( abs `  M )  e.  NN  ->  (
( abs `  M
) ^ 2 )  e.  ZZ )
19 zsqcl 11174 . . . . . . . . . . 11  |-  ( N  e.  ZZ  ->  ( N ^ 2 )  e.  ZZ )
2019adantr 451 . . . . . . . . . 10  |-  ( ( N  e.  ZZ  /\  N  =/=  0 )  -> 
( N ^ 2 )  e.  ZZ )
21 dvdsabsb 12548 . . . . . . . . . 10  |-  ( ( ( ( abs `  M
) ^ 2 )  e.  ZZ  /\  ( N ^ 2 )  e.  ZZ )  ->  (
( ( abs `  M
) ^ 2 ) 
||  ( N ^
2 )  <->  ( ( abs `  M ) ^
2 )  ||  ( abs `  ( N ^
2 ) ) ) )
2218, 20, 21syl2an 463 . . . . . . . . 9  |-  ( ( ( abs `  M
)  e.  NN  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  ->  ( (
( abs `  M
) ^ 2 ) 
||  ( N ^
2 )  <->  ( ( abs `  M ) ^
2 )  ||  ( abs `  ( N ^
2 ) ) ) )
23 zcn 10029 . . . . . . . . . . . . 13  |-  ( N  e.  ZZ  ->  N  e.  CC )
2423adantr 451 . . . . . . . . . . . 12  |-  ( ( N  e.  ZZ  /\  N  =/=  0 )  ->  N  e.  CC )
25 abssq 11791 . . . . . . . . . . . 12  |-  ( N  e.  CC  ->  (
( abs `  N
) ^ 2 )  =  ( abs `  ( N ^ 2 ) ) )
2624, 25syl 15 . . . . . . . . . . 11  |-  ( ( N  e.  ZZ  /\  N  =/=  0 )  -> 
( ( abs `  N
) ^ 2 )  =  ( abs `  ( N ^ 2 ) ) )
2726breq2d 4035 . . . . . . . . . 10  |-  ( ( N  e.  ZZ  /\  N  =/=  0 )  -> 
( ( ( abs `  M ) ^ 2 )  ||  ( ( abs `  N ) ^ 2 )  <->  ( ( abs `  M ) ^
2 )  ||  ( abs `  ( N ^
2 ) ) ) )
2827adantl 452 . . . . . . . . 9  |-  ( ( ( abs `  M
)  e.  NN  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  ->  ( (
( abs `  M
) ^ 2 ) 
||  ( ( abs `  N ) ^ 2 )  <->  ( ( abs `  M ) ^ 2 )  ||  ( abs `  ( N ^ 2 ) ) ) )
2922, 28bitr4d 247 . . . . . . . 8  |-  ( ( ( abs `  M
)  e.  NN  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  ->  ( (
( abs `  M
) ^ 2 ) 
||  ( N ^
2 )  <->  ( ( abs `  M ) ^
2 )  ||  (
( abs `  N
) ^ 2 ) ) )
3012, 16, 293bitr4d 276 . . . . . . 7  |-  ( ( ( abs `  M
)  e.  NN  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  ->  ( ( abs `  M )  ||  N 
<->  ( ( abs `  M
) ^ 2 ) 
||  ( N ^
2 ) ) )
3130anassrs 629 . . . . . 6  |-  ( ( ( ( abs `  M
)  e.  NN  /\  N  e.  ZZ )  /\  N  =/=  0
)  ->  ( ( abs `  M )  ||  N 
<->  ( ( abs `  M
) ^ 2 ) 
||  ( N ^
2 ) ) )
32 dvds0 12544 . . . . . . . . 9  |-  ( ( abs `  M )  e.  ZZ  ->  ( abs `  M )  ||  0 )
33 zsqcl 11174 . . . . . . . . . 10  |-  ( ( abs `  M )  e.  ZZ  ->  (
( abs `  M
) ^ 2 )  e.  ZZ )
34 dvds0 12544 . . . . . . . . . 10  |-  ( ( ( abs `  M
) ^ 2 )  e.  ZZ  ->  (
( abs `  M
) ^ 2 ) 
||  0 )
3533, 34syl 15 . . . . . . . . 9  |-  ( ( abs `  M )  e.  ZZ  ->  (
( abs `  M
) ^ 2 ) 
||  0 )
3632, 352thd 231 . . . . . . . 8  |-  ( ( abs `  M )  e.  ZZ  ->  (
( abs `  M
)  ||  0  <->  ( ( abs `  M ) ^
2 )  ||  0
) )
3713, 36syl 15 . . . . . . 7  |-  ( ( abs `  M )  e.  NN  ->  (
( abs `  M
)  ||  0  <->  ( ( abs `  M ) ^
2 )  ||  0
) )
3837adantr 451 . . . . . 6  |-  ( ( ( abs `  M
)  e.  NN  /\  N  e.  ZZ )  ->  ( ( abs `  M
)  ||  0  <->  ( ( abs `  M ) ^
2 )  ||  0
) )
399, 31, 38pm2.61ne 2521 . . . . 5  |-  ( ( ( abs `  M
)  e.  NN  /\  N  e.  ZZ )  ->  ( ( abs `  M
)  ||  N  <->  ( ( abs `  M ) ^
2 )  ||  ( N ^ 2 ) ) )
405, 39sylan 457 . . . 4  |-  ( ( ( M  e.  ZZ  /\  M  =/=  0 )  /\  N  e.  ZZ )  ->  ( ( abs `  M )  ||  N  <->  ( ( abs `  M
) ^ 2 ) 
||  ( N ^
2 ) ) )
41 absdvdsb 12547 . . . . 5  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  ||  N  <->  ( abs `  M ) 
||  N ) )
4241adantlr 695 . . . 4  |-  ( ( ( M  e.  ZZ  /\  M  =/=  0 )  /\  N  e.  ZZ )  ->  ( M  ||  N 
<->  ( abs `  M
)  ||  N )
)
43 zsqcl 11174 . . . . . . 7  |-  ( M  e.  ZZ  ->  ( M ^ 2 )  e.  ZZ )
4443adantr 451 . . . . . 6  |-  ( ( M  e.  ZZ  /\  M  =/=  0 )  -> 
( M ^ 2 )  e.  ZZ )
45 absdvdsb 12547 . . . . . 6  |-  ( ( ( M ^ 2 )  e.  ZZ  /\  ( N ^ 2 )  e.  ZZ )  -> 
( ( M ^
2 )  ||  ( N ^ 2 )  <->  ( abs `  ( M ^ 2 ) )  ||  ( N ^ 2 ) ) )
4644, 19, 45syl2an 463 . . . . 5  |-  ( ( ( M  e.  ZZ  /\  M  =/=  0 )  /\  N  e.  ZZ )  ->  ( ( M ^ 2 )  ||  ( N ^ 2 )  <-> 
( abs `  ( M ^ 2 ) ) 
||  ( N ^
2 ) ) )
47 zcn 10029 . . . . . . . . . 10  |-  ( M  e.  ZZ  ->  M  e.  CC )
48 abssq 11791 . . . . . . . . . 10  |-  ( M  e.  CC  ->  (
( abs `  M
) ^ 2 )  =  ( abs `  ( M ^ 2 ) ) )
4947, 48syl 15 . . . . . . . . 9  |-  ( M  e.  ZZ  ->  (
( abs `  M
) ^ 2 )  =  ( abs `  ( M ^ 2 ) ) )
5049eqcomd 2288 . . . . . . . 8  |-  ( M  e.  ZZ  ->  ( abs `  ( M ^
2 ) )  =  ( ( abs `  M
) ^ 2 ) )
5150adantr 451 . . . . . . 7  |-  ( ( M  e.  ZZ  /\  M  =/=  0 )  -> 
( abs `  ( M ^ 2 ) )  =  ( ( abs `  M ) ^ 2 ) )
5251breq1d 4033 . . . . . 6  |-  ( ( M  e.  ZZ  /\  M  =/=  0 )  -> 
( ( abs `  ( M ^ 2 ) ) 
||  ( N ^
2 )  <->  ( ( abs `  M ) ^
2 )  ||  ( N ^ 2 ) ) )
5352adantr 451 . . . . 5  |-  ( ( ( M  e.  ZZ  /\  M  =/=  0 )  /\  N  e.  ZZ )  ->  ( ( abs `  ( M ^ 2 ) )  ||  ( N ^ 2 )  <->  ( ( abs `  M ) ^
2 )  ||  ( N ^ 2 ) ) )
5446, 53bitrd 244 . . . 4  |-  ( ( ( M  e.  ZZ  /\  M  =/=  0 )  /\  N  e.  ZZ )  ->  ( ( M ^ 2 )  ||  ( N ^ 2 )  <-> 
( ( abs `  M
) ^ 2 ) 
||  ( N ^
2 ) ) )
5540, 42, 543bitr4d 276 . . 3  |-  ( ( ( M  e.  ZZ  /\  M  =/=  0 )  /\  N  e.  ZZ )  ->  ( M  ||  N 
<->  ( M ^ 2 )  ||  ( N ^ 2 ) ) )
5655an32s 779 . 2  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  M  =/=  0
)  ->  ( M  ||  N  <->  ( M ^
2 )  ||  ( N ^ 2 ) ) )
57 0dvds 12549 . . . . 5  |-  ( N  e.  ZZ  ->  (
0  ||  N  <->  N  = 
0 ) )
58 sqeq0 11168 . . . . . 6  |-  ( N  e.  CC  ->  (
( N ^ 2 )  =  0  <->  N  =  0 ) )
5923, 58syl 15 . . . . 5  |-  ( N  e.  ZZ  ->  (
( N ^ 2 )  =  0  <->  N  =  0 ) )
6057, 59bitr4d 247 . . . 4  |-  ( N  e.  ZZ  ->  (
0  ||  N  <->  ( N ^ 2 )  =  0 ) )
61 0dvds 12549 . . . . 5  |-  ( ( N ^ 2 )  e.  ZZ  ->  (
0  ||  ( N ^ 2 )  <->  ( N ^ 2 )  =  0 ) )
6219, 61syl 15 . . . 4  |-  ( N  e.  ZZ  ->  (
0  ||  ( N ^ 2 )  <->  ( N ^ 2 )  =  0 ) )
6360, 62bitr4d 247 . . 3  |-  ( N  e.  ZZ  ->  (
0  ||  N  <->  0  ||  ( N ^ 2 ) ) )
6463adantl 452 . 2  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( 0  ||  N  <->  0 
||  ( N ^
2 ) ) )
654, 56, 64pm2.61ne 2521 1  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  ||  N  <->  ( M ^ 2 ) 
||  ( N ^
2 ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   CCcc 8735   0cc0 8737   NNcn 9746   2c2 9795   ZZcz 10024   ^cexp 11104   abscabs 11719    || cdivides 12531
This theorem is referenced by:  pythagtriplem19  12886  4sqlem9  12993  4sqlem10  12994  lgsdir  20569  2sqlem8a  20610
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-fl 10925  df-mod 10974  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
  Copyright terms: Public domain W3C validator