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Theorem coprimeprodsq2 14772
Description: If three numbers are coprime, and the square of one is the product of the other two, then there is a formula for the other two in terms of  gcd and square. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Assertion
Ref Expression
coprimeprodsq2  |-  ( ( ( A  e.  ZZ  /\  B  e.  NN0  /\  C  e.  NN0 )  /\  ( ( A  gcd  B )  gcd  C )  =  1 )  -> 
( ( C ^
2 )  =  ( A  x.  B )  ->  B  =  ( ( B  gcd  C
) ^ 2 ) ) )

Proof of Theorem coprimeprodsq2
StepHypRef Expression
1 zcn 10949 . . . . . 6  |-  ( A  e.  ZZ  ->  A  e.  CC )
2 nn0cn 10886 . . . . . 6  |-  ( B  e.  NN0  ->  B  e.  CC )
3 mulcom 9630 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  B
)  =  ( B  x.  A ) )
41, 2, 3syl2an 480 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  NN0 )  -> 
( A  x.  B
)  =  ( B  x.  A ) )
543adant3 1029 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  NN0  /\  C  e.  NN0 )  ->  ( A  x.  B )  =  ( B  x.  A ) )
65adantr 467 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  NN0  /\  C  e.  NN0 )  /\  ( ( A  gcd  B )  gcd  C )  =  1 )  -> 
( A  x.  B
)  =  ( B  x.  A ) )
76eqeq2d 2463 . 2  |-  ( ( ( A  e.  ZZ  /\  B  e.  NN0  /\  C  e.  NN0 )  /\  ( ( A  gcd  B )  gcd  C )  =  1 )  -> 
( ( C ^
2 )  =  ( A  x.  B )  <-> 
( C ^ 2 )  =  ( B  x.  A ) ) )
8 simpl2 1013 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  NN0  /\  C  e.  NN0 )  /\  ( ( A  gcd  B )  gcd  C )  =  1 )  ->  B  e.  NN0 )
9 simpl1 1012 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  NN0  /\  C  e.  NN0 )  /\  ( ( A  gcd  B )  gcd  C )  =  1 )  ->  A  e.  ZZ )
10 simpl3 1014 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  NN0  /\  C  e.  NN0 )  /\  ( ( A  gcd  B )  gcd  C )  =  1 )  ->  C  e.  NN0 )
11 nn0z 10967 . . . . . 6  |-  ( B  e.  NN0  ->  B  e.  ZZ )
12 gcdcom 14496 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( A  gcd  B
)  =  ( B  gcd  A ) )
1312oveq1d 6310 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( A  gcd  B )  gcd  C )  =  ( ( B  gcd  A )  gcd 
C ) )
1413eqeq1d 2455 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( ( A  gcd  B )  gcd 
C )  =  1  <-> 
( ( B  gcd  A )  gcd  C )  =  1 ) )
1511, 14sylan2 477 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  NN0 )  -> 
( ( ( A  gcd  B )  gcd 
C )  =  1  <-> 
( ( B  gcd  A )  gcd  C )  =  1 ) )
16153adant3 1029 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  NN0  /\  C  e.  NN0 )  ->  (
( ( A  gcd  B )  gcd  C )  =  1  <->  ( ( B  gcd  A )  gcd 
C )  =  1 ) )
1716biimpa 487 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  NN0  /\  C  e.  NN0 )  /\  ( ( A  gcd  B )  gcd  C )  =  1 )  -> 
( ( B  gcd  A )  gcd  C )  =  1 )
18 coprimeprodsq 14771 . . 3  |-  ( ( ( B  e.  NN0  /\  A  e.  ZZ  /\  C  e.  NN0 )  /\  ( ( B  gcd  A )  gcd  C )  =  1 )  -> 
( ( C ^
2 )  =  ( B  x.  A )  ->  B  =  ( ( B  gcd  C
) ^ 2 ) ) )
198, 9, 10, 17, 18syl31anc 1272 . 2  |-  ( ( ( A  e.  ZZ  /\  B  e.  NN0  /\  C  e.  NN0 )  /\  ( ( A  gcd  B )  gcd  C )  =  1 )  -> 
( ( C ^
2 )  =  ( B  x.  A )  ->  B  =  ( ( B  gcd  C
) ^ 2 ) ) )
207, 19sylbid 219 1  |-  ( ( ( A  e.  ZZ  /\  B  e.  NN0  /\  C  e.  NN0 )  /\  ( ( A  gcd  B )  gcd  C )  =  1 )  -> 
( ( C ^
2 )  =  ( A  x.  B )  ->  B  =  ( ( B  gcd  C
) ^ 2 ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    /\ w3a 986    = wceq 1446    e. wcel 1889  (class class class)co 6295   CCcc 9542   1c1 9545    x. cmul 9549   2c2 10666   NN0cn0 10876   ZZcz 10944   ^cexp 12279    gcd cgcd 14480
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588  ax-cnex 9600  ax-resscn 9601  ax-1cn 9602  ax-icn 9603  ax-addcl 9604  ax-addrcl 9605  ax-mulcl 9606  ax-mulrcl 9607  ax-mulcom 9608  ax-addass 9609  ax-mulass 9610  ax-distr 9611  ax-i2m1 9612  ax-1ne0 9613  ax-1rid 9614  ax-rnegex 9615  ax-rrecex 9616  ax-cnre 9617  ax-pre-lttri 9618  ax-pre-lttrn 9619  ax-pre-ltadd 9620  ax-pre-mulgt0 9621  ax-pre-sup 9622
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-nel 2627  df-ral 2744  df-rex 2745  df-reu 2746  df-rmo 2747  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-pss 3422  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-tp 3975  df-op 3977  df-uni 4202  df-iun 4283  df-br 4406  df-opab 4465  df-mpt 4466  df-tr 4501  df-eprel 4748  df-id 4752  df-po 4758  df-so 4759  df-fr 4796  df-we 4798  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-pred 5383  df-ord 5429  df-on 5430  df-lim 5431  df-suc 5432  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-riota 6257  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6698  df-2nd 6799  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-er 7368  df-en 7575  df-dom 7576  df-sdom 7577  df-sup 7961  df-inf 7962  df-pnf 9682  df-mnf 9683  df-xr 9684  df-ltxr 9685  df-le 9686  df-sub 9867  df-neg 9868  df-div 10277  df-nn 10617  df-2 10675  df-3 10676  df-n0 10877  df-z 10945  df-uz 11167  df-rp 11310  df-fl 12035  df-mod 12104  df-seq 12221  df-exp 12280  df-cj 13174  df-re 13175  df-im 13176  df-sqrt 13310  df-abs 13311  df-dvds 14318  df-gcd 14481
This theorem is referenced by:  pythagtriplem7  14784
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