Theorem mul2sq 16235

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.

Step	Hyp	Ref	Expression
1		2sq.1	. . 3 |- S = ran ( w e. ZZ[_i] |-> ( ( abs ` w ) ^ 2 ) )
2	1	2sqlem1 16233	. 2 |- ( A e. S <-> E. x e. ZZ[_i] A = ( ( abs ` x ) ^ 2 ) )
3	1	2sqlem1 16233	. 2 |- ( B e. S <-> E. y e. ZZ[_i] B = ( ( abs ` y ) ^ 2 ) )
4		reeanv 2721	. . 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 13157	. . . . . . 7 |- ( ( x e. ZZ[_i] /\ y e. ZZ[_i] ) -> ( x x. y ) e. ZZ[_i] )
6		gzcn 13151	. . . . . . . . . 10 |- ( x e. ZZ[_i] -> x e. CC )
7		gzcn 13151	. . . . . . . . . 10 |- ( y e. ZZ[_i] -> y e. CC )
8		absmul 11835	. . . . . . . . . 10 |- ( ( x e. CC /\ y e. CC ) -> ( abs ` ( x x. y ) ) = ( ( abs ` x ) x. ( abs ` y ) ) )
9	6, 7, 8	syl2an 289	. . . . . . . . 9 |- ( ( x e. ZZ[_i] /\ y e. ZZ[_i] ) -> ( abs ` ( x x. y ) ) = ( ( abs ` x ) x. ( abs ` y ) ) )
10	9	oveq1d 6100	. . . . . . . 8 |- ( ( x e. ZZ[_i] /\ y e. ZZ[_i] ) -> ( ( abs ` ( x x. y ) ) ^ 2 ) = ( ( ( abs ` x ) x. ( abs ` y ) ) ^ 2 ) )
11	6	abscld 11947	. . . . . . . . . 10 |- ( x e. ZZ[_i] -> ( abs ` x ) e. RR )
12	11	recnd 8354	. . . . . . . . 9 |- ( x e. ZZ[_i] -> ( abs ` x ) e. CC )
13	7	abscld 11947	. . . . . . . . . 10 |- ( y e. ZZ[_i] -> ( abs ` y ) e. RR )
14	13	recnd 8354	. . . . . . . . 9 |- ( y e. ZZ[_i] -> ( abs ` y ) e. CC )
15		sqmul 11038	. . . . . . . . 9 |- ( ( ( abs ` x ) e. CC /\ ( abs ` y ) e. CC ) -> ( ( ( abs ` x ) x. ( abs ` y ) ) ^ 2 ) = ( ( ( abs ` x ) ^ 2 ) x. ( ( abs ` y ) ^ 2 ) ) )
16	12, 14, 15	syl2an 289	. . . . . . . 8 |- ( ( x e. ZZ[_i] /\ y e. ZZ[_i] ) -> ( ( ( abs ` x ) x. ( abs ` y ) ) ^ 2 ) = ( ( ( abs ` x ) ^ 2 ) x. ( ( abs ` y ) ^ 2 ) ) )
17	10, 16	eqtr2d 2272	. . . . . . 7 |- ( ( x e. ZZ[_i] /\ y e. ZZ[_i] ) -> ( ( ( abs ` x ) ^ 2 ) x. ( ( abs ` y ) ^ 2 ) ) = ( ( abs ` ( x x. y ) ) ^ 2 ) )
18		fveq2 5695	. . . . . . . . 9 |- ( z = ( x x. y ) -> ( abs ` z ) = ( abs ` ( x x. y ) ) )
19	18	oveq1d 6100	. . . . . . . 8 |- ( z = ( x x. y ) -> ( ( abs ` z ) ^ 2 ) = ( ( abs ` ( x x. y ) ) ^ 2 ) )
20	19	rspceeqv 2948	. . . . . . 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 ) )
21	5, 17, 20	syl2anc 415	. . . . . 6 |- ( ( x e. ZZ[_i] /\ y e. ZZ[_i] ) -> E. z e. ZZ[_i] ( ( ( abs ` x ) ^ 2 ) x. ( ( abs ` y ) ^ 2 ) ) = ( ( abs ` z ) ^ 2 ) )
22	1	2sqlem1 16233	. . . . . 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 ) )
23	21, 22	sylibr 134	. . . . 5 |- ( ( x e. ZZ[_i] /\ y e. ZZ[_i] ) -> ( ( ( abs ` x ) ^ 2 ) x. ( ( abs ` y ) ^ 2 ) ) e. S )
24		oveq12 6094	. . . . . 6 |- ( ( A = ( ( abs ` x ) ^ 2 ) /\ B = ( ( abs ` y ) ^ 2 ) ) -> ( A x. B ) = ( ( ( abs ` x ) ^ 2 ) x. ( ( abs ` y ) ^ 2 ) ) )
25	24	eleq1d 2307	. . . . 5 |- ( ( A = ( ( abs ` x ) ^ 2 ) /\ B = ( ( abs ` y ) ^ 2 ) ) -> ( ( A x. B ) e. S <-> ( ( ( abs ` x ) ^ 2 ) x. ( ( abs ` y ) ^ 2 ) ) e. S ) )
26	23, 25	syl5ibrcom 157	. . . 4 |- ( ( x e. ZZ[_i] /\ y e. ZZ[_i] ) -> ( ( A = ( ( abs ` x ) ^ 2 ) /\ B = ( ( abs ` y ) ^ 2 ) ) -> ( A x. B ) e. S ) )
27	26	rexlimivv 2674	. . 3 |- ( E. x e. ZZ[_i] E. y e. ZZ[_i] ( A = ( ( abs ` x ) ^ 2 ) /\ B = ( ( abs ` y ) ^ 2 ) ) -> ( A x. B ) e. S )
28	4, 27	sylbir 135	. 2 |- ( ( E. x e. ZZ[_i] A = ( ( abs ` x ) ^ 2 ) /\ E. y e. ZZ[_i] B = ( ( abs ` y ) ^ 2 ) ) -> ( A x. B ) e. S )
29	2, 3, 28	syl2anb 291	1 |- ( ( A e. S /\ B e. S ) -> ( A x. B ) e. S )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   E.wrex 2529   |-> cmpt 4192   ran crn 4775   ` cfv 5377  (class class class)co 6085   CCcc 8177   x. cmul 8184   2c2 9355   ^cexp 10975   abscabs 11763   ZZ[_i]cgz 13148

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295  ax-pre-mulgt0 8296  ax-pre-mulext 8297  ax-arch 8298  ax-caucvg 8299

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-po 4441  df-iso 4442  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-frec 6662  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-reap 8903  df-ap 8910  df-div 9003  df-inn 9305  df-2 9363  df-3 9364  df-4 9365  df-n0 9564  df-z 9645  df-uz 9922  df-rp 10055  df-seqfrec 10885  df-exp 10976  df-cj 11607  df-re 11608  df-im 11609  df-rsqrt 11764  df-abs 11765  df-gz 13149

This theorem is used by: (None)