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Theorem mul2sq 20620
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 20618 . 2  |-  ( A  e.  S  <->  E. x  e.  ZZ [ _i ]  A  =  ( ( abs `  x ) ^
2 ) )
312sqlem1 20618 . 2  |-  ( B  e.  S  <->  E. y  e.  ZZ [ _i ]  B  =  ( ( abs `  y ) ^
2 ) )
4 reeanv 2720 . . 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 13001 . . . . . . 7  |-  ( ( x  e.  ZZ [
_i ]  /\  y  e.  ZZ [ _i ]
)  ->  ( x  x.  y )  e.  ZZ [ _i ] )
6 gzcn 12995 . . . . . . . . . 10  |-  ( x  e.  ZZ [ _i ]  ->  x  e.  CC )
7 gzcn 12995 . . . . . . . . . 10  |-  ( y  e.  ZZ [ _i ]  ->  y  e.  CC )
8 absmul 11795 . . . . . . . . . 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 5889 . . . . . . . 8  |-  ( ( x  e.  ZZ [
_i ]  /\  y  e.  ZZ [ _i ]
)  ->  ( ( abs `  ( x  x.  y ) ) ^
2 )  =  ( ( ( abs `  x
)  x.  ( abs `  y ) ) ^
2 ) )
116abscld 11934 . . . . . . . . . 10  |-  ( x  e.  ZZ [ _i ]  ->  ( abs `  x
)  e.  RR )
1211recnd 8877 . . . . . . . . 9  |-  ( x  e.  ZZ [ _i ]  ->  ( abs `  x
)  e.  CC )
137abscld 11934 . . . . . . . . . 10  |-  ( y  e.  ZZ [ _i ]  ->  ( abs `  y
)  e.  RR )
1413recnd 8877 . . . . . . . . 9  |-  ( y  e.  ZZ [ _i ]  ->  ( abs `  y
)  e.  CC )
15 sqmul 11183 . . . . . . . . 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 2329 . . . . . . 7  |-  ( ( x  e.  ZZ [
_i ]  /\  y  e.  ZZ [ _i ]
)  ->  ( (
( abs `  x
) ^ 2 )  x.  ( ( abs `  y ) ^ 2 ) )  =  ( ( abs `  (
x  x.  y ) ) ^ 2 ) )
18 fveq2 5541 . . . . . . . . . 10  |-  ( z  =  ( x  x.  y )  ->  ( abs `  z )  =  ( abs `  (
x  x.  y ) ) )
1918oveq1d 5889 . . . . . . . . 9  |-  ( z  =  ( x  x.  y )  ->  (
( abs `  z
) ^ 2 )  =  ( ( abs `  ( x  x.  y
) ) ^ 2 ) )
2019eqeq2d 2307 . . . . . . . 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 2897 . . . . . . 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 20618 . . . . . 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 5883 . . . . . 6  |-  ( ( A  =  ( ( abs `  x ) ^ 2 )  /\  B  =  ( ( abs `  y ) ^
2 ) )  -> 
( A  x.  B
)  =  ( ( ( abs `  x
) ^ 2 )  x.  ( ( abs `  y ) ^ 2 ) ) )
2625eleq1d 2362 . . . . 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 2685 . . 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 1632    e. wcel 1696   E.wrex 2557    e. cmpt 4093   ran crn 4706   ` cfv 5271  (class class class)co 5874   CCcc 8751    x. cmul 8758   2c2 9811   ^cexp 11120   abscabs 11735   ZZ [ _i ]cgz 12992
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-rp 10371  df-seq 11063  df-exp 11121  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-gz 12993
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