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Theorem coprimeprodsq2 13882
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 10656 . . . . . 6  |-  ( A  e.  ZZ  ->  A  e.  CC )
2 nn0cn 10594 . . . . . 6  |-  ( B  e.  NN0  ->  B  e.  CC )
3 mulcom 9373 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  B
)  =  ( B  x.  A ) )
41, 2, 3syl2an 477 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  NN0 )  -> 
( A  x.  B
)  =  ( B  x.  A ) )
543adant3 1008 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  NN0  /\  C  e.  NN0 )  ->  ( A  x.  B )  =  ( B  x.  A ) )
65adantr 465 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  NN0  /\  C  e.  NN0 )  /\  ( ( A  gcd  B )  gcd  C )  =  1 )  -> 
( A  x.  B
)  =  ( B  x.  A ) )
76eqeq2d 2454 . 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 992 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  NN0  /\  C  e.  NN0 )  /\  ( ( A  gcd  B )  gcd  C )  =  1 )  ->  B  e.  NN0 )
9 simpl1 991 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  NN0  /\  C  e.  NN0 )  /\  ( ( A  gcd  B )  gcd  C )  =  1 )  ->  A  e.  ZZ )
10 simpl3 993 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  NN0  /\  C  e.  NN0 )  /\  ( ( A  gcd  B )  gcd  C )  =  1 )  ->  C  e.  NN0 )
11 nn0z 10674 . . . . . 6  |-  ( B  e.  NN0  ->  B  e.  ZZ )
12 gcdcom 13709 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( A  gcd  B
)  =  ( B  gcd  A ) )
1312oveq1d 6111 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( A  gcd  B )  gcd  C )  =  ( ( B  gcd  A )  gcd 
C ) )
1413eqeq1d 2451 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( ( A  gcd  B )  gcd 
C )  =  1  <-> 
( ( B  gcd  A )  gcd  C )  =  1 ) )
1511, 14sylan2 474 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  NN0 )  -> 
( ( ( A  gcd  B )  gcd 
C )  =  1  <-> 
( ( B  gcd  A )  gcd  C )  =  1 ) )
16153adant3 1008 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  NN0  /\  C  e.  NN0 )  ->  (
( ( A  gcd  B )  gcd  C )  =  1  <->  ( ( B  gcd  A )  gcd 
C )  =  1 ) )
1716biimpa 484 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  NN0  /\  C  e.  NN0 )  /\  ( ( A  gcd  B )  gcd  C )  =  1 )  -> 
( ( B  gcd  A )  gcd  C )  =  1 )
18 coprimeprodsq 13881 . . 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 1221 . 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 215 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 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756  (class class class)co 6096   CCcc 9285   1c1 9288    x. cmul 9292   2c2 10376   NN0cn0 10584   ZZcz 10651   ^cexp 11870    gcd cgcd 13695
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364  ax-pre-sup 9365
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-2nd 6583  df-recs 6837  df-rdg 6871  df-er 7106  df-en 7316  df-dom 7317  df-sdom 7318  df-sup 7696  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-div 9999  df-nn 10328  df-2 10385  df-3 10386  df-n0 10585  df-z 10652  df-uz 10867  df-rp 10997  df-fl 11647  df-mod 11714  df-seq 11812  df-exp 11871  df-cj 12593  df-re 12594  df-im 12595  df-sqr 12729  df-abs 12730  df-dvds 13541  df-gcd 13696
This theorem is referenced by:  pythagtriplem7  13894
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