Theorem mul2sq 15708

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 15706	. 2 |- ( A e. S <-> E. x e. ZZ[_i] A = ( ( abs ` x ) ^ 2 ) )
3	1	2sqlem1 15706	. 2 |- ( B e. S <-> E. y e. ZZ[_i] B = ( ( abs ` y ) ^ 2 ) )
4		reeanv 2678	. . 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 12816	. . . . . . 7 |- ( ( x e. ZZ[_i] /\ y e. ZZ[_i] ) -> ( x x. y ) e. ZZ[_i] )
6		gzcn 12810	. . . . . . . . . 10 |- ( x e. ZZ[_i] -> x e. CC )
7		gzcn 12810	. . . . . . . . . 10 |- ( y e. ZZ[_i] -> y e. CC )
8		absmul 11495	. . . . . . . . . 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 5982	. . . . . . . 8 |- ( ( x e. ZZ[_i] /\ y e. ZZ[_i] ) -> ( ( abs ` ( x x. y ) ) ^ 2 ) = ( ( ( abs ` x ) x. ( abs ` y ) ) ^ 2 ) )
11	6	abscld 11607	. . . . . . . . . 10 |- ( x e. ZZ[_i] -> ( abs ` x ) e. RR )
12	11	recnd 8136	. . . . . . . . 9 |- ( x e. ZZ[_i] -> ( abs ` x ) e. CC )
13	7	abscld 11607	. . . . . . . . . 10 |- ( y e. ZZ[_i] -> ( abs ` y ) e. RR )
14	13	recnd 8136	. . . . . . . . 9 |- ( y e. ZZ[_i] -> ( abs ` y ) e. CC )
15		sqmul 10783	. . . . . . . . 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 2241	. . . . . . 7 |- ( ( x e. ZZ[_i] /\ y e. ZZ[_i] ) -> ( ( ( abs ` x ) ^ 2 ) x. ( ( abs ` y ) ^ 2 ) ) = ( ( abs ` ( x x. y ) ) ^ 2 ) )
18		fveq2 5599	. . . . . . . . 9 |- ( z = ( x x. y ) -> ( abs ` z ) = ( abs ` ( x x. y ) ) )
19	18	oveq1d 5982	. . . . . . . 8 |- ( z = ( x x. y ) -> ( ( abs ` z ) ^ 2 ) = ( ( abs ` ( x x. y ) ) ^ 2 ) )
20	19	rspceeqv 2902	. . . . . . 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 411	. . . . . 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 15706	. . . . . 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 5976	. . . . . 6 |- ( ( A = ( ( abs ` x ) ^ 2 ) /\ B = ( ( abs ` y ) ^ 2 ) ) -> ( A x. B ) = ( ( ( abs ` x ) ^ 2 ) x. ( ( abs ` y ) ^ 2 ) ) )
25	24	eleq1d 2276	. . . . 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 2631	. . 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 )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   e. wcel 2178   E.wrex 2487   |-> cmpt 4121   ran crn 4694   ` cfv 5290  (class class class)co 5967   CCcc 7958   x. cmul 7965   2c2 9122   ^cexp 10720   abscabs 11423   ZZ[_i]cgz 12807

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-nul 4186  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-iinf 4654  ax-cnex 8051  ax-resscn 8052  ax-1cn 8053  ax-1re 8054  ax-icn 8055  ax-addcl 8056  ax-addrcl 8057  ax-mulcl 8058  ax-mulrcl 8059  ax-addcom 8060  ax-mulcom 8061  ax-addass 8062  ax-mulass 8063  ax-distr 8064  ax-i2m1 8065  ax-0lt1 8066  ax-1rid 8067  ax-0id 8068  ax-rnegex 8069  ax-precex 8070  ax-cnre 8071  ax-pre-ltirr 8072  ax-pre-ltwlin 8073  ax-pre-lttrn 8074  ax-pre-apti 8075  ax-pre-ltadd 8076  ax-pre-mulgt0 8077  ax-pre-mulext 8078  ax-arch 8079  ax-caucvg 8080

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-reu 2493  df-rmo 2494  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-if 3580  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-tr 4159  df-id 4358  df-po 4361  df-iso 4362  df-iord 4431  df-on 4433  df-ilim 4434  df-suc 4436  df-iom 4657  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-riota 5922  df-ov 5970  df-oprab 5971  df-mpo 5972  df-1st 6249  df-2nd 6250  df-recs 6414  df-frec 6500  df-pnf 8144  df-mnf 8145  df-xr 8146  df-ltxr 8147  df-le 8148  df-sub 8280  df-neg 8281  df-reap 8683  df-ap 8690  df-div 8781  df-inn 9072  df-2 9130  df-3 9131  df-4 9132  df-n0 9331  df-z 9408  df-uz 9684  df-rp 9811  df-seqfrec 10630  df-exp 10721  df-cj 11268  df-re 11269  df-im 11270  df-rsqrt 11424  df-abs 11425  df-gz 12808

This theorem is referenced by: (None)