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Theorem mul2sq 20604
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 20602 . 2  |-  ( A  e.  S  <->  E. x  e.  ZZ [ _i ]  A  =  ( ( abs `  x ) ^
2 ) )
312sqlem1 20602 . 2  |-  ( B  e.  S  <->  E. y  e.  ZZ [ _i ]  B  =  ( ( abs `  y ) ^
2 ) )
4 reeanv 2707 . . 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 12985 . . . . . . 7  |-  ( ( x  e.  ZZ [
_i ]  /\  y  e.  ZZ [ _i ]
)  ->  ( x  x.  y )  e.  ZZ [ _i ] )
6 gzcn 12979 . . . . . . . . . 10  |-  ( x  e.  ZZ [ _i ]  ->  x  e.  CC )
7 gzcn 12979 . . . . . . . . . 10  |-  ( y  e.  ZZ [ _i ]  ->  y  e.  CC )
8 absmul 11779 . . . . . . . . . 10  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( abs `  (
x  x.  y ) )  =  ( ( abs `  x )  x.  ( abs `  y
) ) )
96, 7, 8syl2an 463 . . . . . . . . 9  |-  ( ( x  e.  ZZ [
_i ]  /\  y  e.  ZZ [ _i ]
)  ->  ( abs `  ( x  x.  y
) )  =  ( ( abs `  x
)  x.  ( abs `  y ) ) )
109oveq1d 5873 . . . . . . . 8  |-  ( ( x  e.  ZZ [
_i ]  /\  y  e.  ZZ [ _i ]
)  ->  ( ( abs `  ( x  x.  y ) ) ^
2 )  =  ( ( ( abs `  x
)  x.  ( abs `  y ) ) ^
2 ) )
116abscld 11918 . . . . . . . . . 10  |-  ( x  e.  ZZ [ _i ]  ->  ( abs `  x
)  e.  RR )
1211recnd 8861 . . . . . . . . 9  |-  ( x  e.  ZZ [ _i ]  ->  ( abs `  x
)  e.  CC )
137abscld 11918 . . . . . . . . . 10  |-  ( y  e.  ZZ [ _i ]  ->  ( abs `  y
)  e.  RR )
1413recnd 8861 . . . . . . . . 9  |-  ( y  e.  ZZ [ _i ]  ->  ( abs `  y
)  e.  CC )
15 sqmul 11167 . . . . . . . . 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 463 . . . . . . . 8  |-  ( ( x  e.  ZZ [
_i ]  /\  y  e.  ZZ [ _i ]
)  ->  ( (
( abs `  x
)  x.  ( abs `  y ) ) ^
2 )  =  ( ( ( abs `  x
) ^ 2 )  x.  ( ( abs `  y ) ^ 2 ) ) )
1710, 16eqtr2d 2316 . . . . . . 7  |-  ( ( x  e.  ZZ [
_i ]  /\  y  e.  ZZ [ _i ]
)  ->  ( (
( abs `  x
) ^ 2 )  x.  ( ( abs `  y ) ^ 2 ) )  =  ( ( abs `  (
x  x.  y ) ) ^ 2 ) )
18 fveq2 5525 . . . . . . . . . 10  |-  ( z  =  ( x  x.  y )  ->  ( abs `  z )  =  ( abs `  (
x  x.  y ) ) )
1918oveq1d 5873 . . . . . . . . 9  |-  ( z  =  ( x  x.  y )  ->  (
( abs `  z
) ^ 2 )  =  ( ( abs `  ( x  x.  y
) ) ^ 2 ) )
2019eqeq2d 2294 . . . . . . . 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 2884 . . . . . . 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 642 . . . . . 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 20602 . . . . . 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 203 . . . . 5  |-  ( ( x  e.  ZZ [
_i ]  /\  y  e.  ZZ [ _i ]
)  ->  ( (
( abs `  x
) ^ 2 )  x.  ( ( abs `  y ) ^ 2 ) )  e.  S
)
25 oveq12 5867 . . . . . 6  |-  ( ( A  =  ( ( abs `  x ) ^ 2 )  /\  B  =  ( ( abs `  y ) ^
2 ) )  -> 
( A  x.  B
)  =  ( ( ( abs `  x
) ^ 2 )  x.  ( ( abs `  y ) ^ 2 ) ) )
2625eleq1d 2349 . . . . 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 213 . . . 4  |-  ( ( x  e.  ZZ [
_i ]  /\  y  e.  ZZ [ _i ]
)  ->  ( ( A  =  ( ( abs `  x ) ^
2 )  /\  B  =  ( ( abs `  y ) ^ 2 ) )  ->  ( A  x.  B )  e.  S ) )
2827rexlimivv 2672 . . 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 204 . 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 465 1  |-  ( ( A  e.  S  /\  B  e.  S )  ->  ( A  x.  B
)  e.  S )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   E.wrex 2544    e. cmpt 4077   ran crn 4690   ` cfv 5255  (class class class)co 5858   CCcc 8735    x. cmul 8742   2c2 9795   ^cexp 11104   abscabs 11719   ZZ [ _i ]cgz 12976
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-seq 11047  df-exp 11105  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-gz 12977
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