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Theorem mul2sq 22847
Description: Fibonacci's identity (actually due to Diophantus). The product of two sums of two squares is also a sum of two squares. We can take advantage of Gaussian integers here to trivialize the proof. (Contributed by Mario Carneiro, 19-Jun-2015.)
Hypothesis
Ref Expression
2sq.1  |-  S  =  ran  ( w  e.  ZZ[_i]  |->  ( ( abs `  w
) ^ 2 ) )
Assertion
Ref Expression
mul2sq  |-  ( ( A  e.  S  /\  B  e.  S )  ->  ( A  x.  B
)  e.  S )

Proof of Theorem mul2sq
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 2sq.1 . . 3  |-  S  =  ran  ( w  e.  ZZ[_i]  |->  ( ( abs `  w
) ^ 2 ) )
212sqlem1 22845 . 2  |-  ( A  e.  S  <->  E. x  e.  ZZ[_i]  A  =  ( ( abs `  x ) ^
2 ) )
312sqlem1 22845 . 2  |-  ( B  e.  S  <->  E. y  e.  ZZ[_i]  B  =  ( ( abs `  y ) ^
2 ) )
4 reeanv 2994 . . 3  |-  ( E. x  e.  ZZ[_i]  E. y  e.  ZZ[_i] 
( A  =  ( ( abs `  x
) ^ 2 )  /\  B  =  ( ( abs `  y
) ^ 2 ) )  <->  ( E. x  e.  ZZ[_i]  A  =  ( ( abs `  x ) ^
2 )  /\  E. y  e.  ZZ[_i]  B  =  ( ( abs `  y
) ^ 2 ) ) )
5 gzmulcl 14121 . . . . . . 7  |-  ( ( x  e.  ZZ[_i]  /\  y  e.  ZZ[_i]
)  ->  ( x  x.  y )  e.  ZZ[_i] )
6 gzcn 14115 . . . . . . . . . 10  |-  ( x  e.  ZZ[_i]  ->  x  e.  CC )
7 gzcn 14115 . . . . . . . . . 10  |-  ( y  e.  ZZ[_i]  ->  y  e.  CC )
8 absmul 12905 . . . . . . . . . 10  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( abs `  (
x  x.  y ) )  =  ( ( abs `  x )  x.  ( abs `  y
) ) )
96, 7, 8syl2an 477 . . . . . . . . 9  |-  ( ( x  e.  ZZ[_i]  /\  y  e.  ZZ[_i]
)  ->  ( abs `  ( x  x.  y
) )  =  ( ( abs `  x
)  x.  ( abs `  y ) ) )
109oveq1d 6218 . . . . . . . 8  |-  ( ( x  e.  ZZ[_i]  /\  y  e.  ZZ[_i]
)  ->  ( ( abs `  ( x  x.  y ) ) ^
2 )  =  ( ( ( abs `  x
)  x.  ( abs `  y ) ) ^
2 ) )
116abscld 13044 . . . . . . . . . 10  |-  ( x  e.  ZZ[_i]  ->  ( abs `  x )  e.  RR )
1211recnd 9527 . . . . . . . . 9  |-  ( x  e.  ZZ[_i]  ->  ( abs `  x )  e.  CC )
137abscld 13044 . . . . . . . . . 10  |-  ( y  e.  ZZ[_i]  ->  ( abs `  y )  e.  RR )
1413recnd 9527 . . . . . . . . 9  |-  ( y  e.  ZZ[_i]  ->  ( abs `  y )  e.  CC )
15 sqmul 12050 . . . . . . . . 9  |-  ( ( ( abs `  x
)  e.  CC  /\  ( abs `  y )  e.  CC )  -> 
( ( ( abs `  x )  x.  ( abs `  y ) ) ^ 2 )  =  ( ( ( abs `  x ) ^ 2 )  x.  ( ( abs `  y ) ^ 2 ) ) )
1612, 14, 15syl2an 477 . . . . . . . 8  |-  ( ( x  e.  ZZ[_i]  /\  y  e.  ZZ[_i]
)  ->  ( (
( abs `  x
)  x.  ( abs `  y ) ) ^
2 )  =  ( ( ( abs `  x
) ^ 2 )  x.  ( ( abs `  y ) ^ 2 ) ) )
1710, 16eqtr2d 2496 . . . . . . 7  |-  ( ( x  e.  ZZ[_i]  /\  y  e.  ZZ[_i]
)  ->  ( (
( abs `  x
) ^ 2 )  x.  ( ( abs `  y ) ^ 2 ) )  =  ( ( abs `  (
x  x.  y ) ) ^ 2 ) )
18 fveq2 5802 . . . . . . . . . 10  |-  ( z  =  ( x  x.  y )  ->  ( abs `  z )  =  ( abs `  (
x  x.  y ) ) )
1918oveq1d 6218 . . . . . . . . 9  |-  ( z  =  ( x  x.  y )  ->  (
( abs `  z
) ^ 2 )  =  ( ( abs `  ( x  x.  y
) ) ^ 2 ) )
2019eqeq2d 2468 . . . . . . . 8  |-  ( z  =  ( x  x.  y )  ->  (
( ( ( abs `  x ) ^ 2 )  x.  ( ( abs `  y ) ^ 2 ) )  =  ( ( abs `  z ) ^ 2 )  <->  ( ( ( abs `  x ) ^ 2 )  x.  ( ( abs `  y
) ^ 2 ) )  =  ( ( abs `  ( x  x.  y ) ) ^ 2 ) ) )
2120rspcev 3179 . . . . . . 7  |-  ( ( ( x  x.  y
)  e.  ZZ[_i]  /\  (
( ( abs `  x
) ^ 2 )  x.  ( ( abs `  y ) ^ 2 ) )  =  ( ( abs `  (
x  x.  y ) ) ^ 2 ) )  ->  E. z  e.  ZZ[_i] 
( ( ( abs `  x ) ^ 2 )  x.  ( ( abs `  y ) ^ 2 ) )  =  ( ( abs `  z ) ^ 2 ) )
225, 17, 21syl2anc 661 . . . . . 6  |-  ( ( x  e.  ZZ[_i]  /\  y  e.  ZZ[_i]
)  ->  E. z  e.  ZZ[_i] 
( ( ( abs `  x ) ^ 2 )  x.  ( ( abs `  y ) ^ 2 ) )  =  ( ( abs `  z ) ^ 2 ) )
2312sqlem1 22845 . . . . . 6  |-  ( ( ( ( abs `  x
) ^ 2 )  x.  ( ( abs `  y ) ^ 2 ) )  e.  S  <->  E. z  e.  ZZ[_i]  ( (
( abs `  x
) ^ 2 )  x.  ( ( abs `  y ) ^ 2 ) )  =  ( ( abs `  z
) ^ 2 ) )
2422, 23sylibr 212 . . . . 5  |-  ( ( x  e.  ZZ[_i]  /\  y  e.  ZZ[_i]
)  ->  ( (
( abs `  x
) ^ 2 )  x.  ( ( abs `  y ) ^ 2 ) )  e.  S
)
25 oveq12 6212 . . . . . 6  |-  ( ( A  =  ( ( abs `  x ) ^ 2 )  /\  B  =  ( ( abs `  y ) ^
2 ) )  -> 
( A  x.  B
)  =  ( ( ( abs `  x
) ^ 2 )  x.  ( ( abs `  y ) ^ 2 ) ) )
2625eleq1d 2523 . . . . 5  |-  ( ( A  =  ( ( abs `  x ) ^ 2 )  /\  B  =  ( ( abs `  y ) ^
2 ) )  -> 
( ( A  x.  B )  e.  S  <->  ( ( ( abs `  x
) ^ 2 )  x.  ( ( abs `  y ) ^ 2 ) )  e.  S
) )
2724, 26syl5ibrcom 222 . . . 4  |-  ( ( x  e.  ZZ[_i]  /\  y  e.  ZZ[_i]
)  ->  ( ( A  =  ( ( abs `  x ) ^
2 )  /\  B  =  ( ( abs `  y ) ^ 2 ) )  ->  ( A  x.  B )  e.  S ) )
2827rexlimivv 2952 . . 3  |-  ( E. x  e.  ZZ[_i]  E. y  e.  ZZ[_i] 
( A  =  ( ( abs `  x
) ^ 2 )  /\  B  =  ( ( abs `  y
) ^ 2 ) )  ->  ( A  x.  B )  e.  S
)
294, 28sylbir 213 . 2  |-  ( ( E. x  e.  ZZ[_i]  A  =  ( ( abs `  x ) ^ 2 )  /\  E. y  e.  ZZ[_i]  B  =  ( ( abs `  y ) ^
2 ) )  -> 
( A  x.  B
)  e.  S )
302, 3, 29syl2anb 479 1  |-  ( ( A  e.  S  /\  B  e.  S )  ->  ( A  x.  B
)  e.  S )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   E.wrex 2800    |-> cmpt 4461   ran crn 4952   ` cfv 5529  (class class class)co 6203   CCcc 9395    x. cmul 9402   2c2 10486   ^cexp 11986   abscabs 12845   ZZ[_i]cgz 14112
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-seq 11928  df-exp 11987  df-cj 12710  df-re 12711  df-im 12712  df-sqr 12846  df-abs 12847  df-gz 14113
This theorem is referenced by: (None)
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