Theorem mul2sq 15273

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 15271	. 2 |- ( A e. S <-> E. x e. ZZ[_i] A = ( ( abs ` x ) ^ 2 ) )
3	1	2sqlem1 15271	. 2 |- ( B e. S <-> E. y e. ZZ[_i] B = ( ( abs ` y ) ^ 2 ) )
4		reeanv 2664	. . 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 12519	. . . . . . 7 |- ( ( x e. ZZ[_i] /\ y e. ZZ[_i] ) -> ( x x. y ) e. ZZ[_i] )
6		gzcn 12513	. . . . . . . . . 10 |- ( x e. ZZ[_i] -> x e. CC )
7		gzcn 12513	. . . . . . . . . 10 |- ( y e. ZZ[_i] -> y e. CC )
8		absmul 11216	. . . . . . . . . 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 5934	. . . . . . . 8 |- ( ( x e. ZZ[_i] /\ y e. ZZ[_i] ) -> ( ( abs ` ( x x. y ) ) ^ 2 ) = ( ( ( abs ` x ) x. ( abs ` y ) ) ^ 2 ) )
11	6	abscld 11328	. . . . . . . . . 10 |- ( x e. ZZ[_i] -> ( abs ` x ) e. RR )
12	11	recnd 8050	. . . . . . . . 9 |- ( x e. ZZ[_i] -> ( abs ` x ) e. CC )
13	7	abscld 11328	. . . . . . . . . 10 |- ( y e. ZZ[_i] -> ( abs ` y ) e. RR )
14	13	recnd 8050	. . . . . . . . 9 |- ( y e. ZZ[_i] -> ( abs ` y ) e. CC )
15		sqmul 10675	. . . . . . . . 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 2227	. . . . . . 7 |- ( ( x e. ZZ[_i] /\ y e. ZZ[_i] ) -> ( ( ( abs ` x ) ^ 2 ) x. ( ( abs ` y ) ^ 2 ) ) = ( ( abs ` ( x x. y ) ) ^ 2 ) )
18		fveq2 5555	. . . . . . . . 9 |- ( z = ( x x. y ) -> ( abs ` z ) = ( abs ` ( x x. y ) ) )
19	18	oveq1d 5934	. . . . . . . 8 |- ( z = ( x x. y ) -> ( ( abs ` z ) ^ 2 ) = ( ( abs ` ( x x. y ) ) ^ 2 ) )
20	19	rspceeqv 2883	. . . . . . 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 15271	. . . . . 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 5928	. . . . . 6 |- ( ( A = ( ( abs ` x ) ^ 2 ) /\ B = ( ( abs ` y ) ^ 2 ) ) -> ( A x. B ) = ( ( ( abs ` x ) ^ 2 ) x. ( ( abs ` y ) ^ 2 ) ) )
25	24	eleq1d 2262	. . . . 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 2617	. . 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 1364   e. wcel 2164   E.wrex 2473   |-> cmpt 4091   ran crn 4661   ` cfv 5255  (class class class)co 5919   CCcc 7872   x. cmul 7879   2c2 9035   ^cexp 10612   abscabs 11144   ZZ[_i]cgz 12510

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4145  ax-sep 4148  ax-nul 4156  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-iinf 4621  ax-cnex 7965  ax-resscn 7966  ax-1cn 7967  ax-1re 7968  ax-icn 7969  ax-addcl 7970  ax-addrcl 7971  ax-mulcl 7972  ax-mulrcl 7973  ax-addcom 7974  ax-mulcom 7975  ax-addass 7976  ax-mulass 7977  ax-distr 7978  ax-i2m1 7979  ax-0lt1 7980  ax-1rid 7981  ax-0id 7982  ax-rnegex 7983  ax-precex 7984  ax-cnre 7985  ax-pre-ltirr 7986  ax-pre-ltwlin 7987  ax-pre-lttrn 7988  ax-pre-apti 7989  ax-pre-ltadd 7990  ax-pre-mulgt0 7991  ax-pre-mulext 7992  ax-arch 7993  ax-caucvg 7994

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-if 3559  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-tr 4129  df-id 4325  df-po 4328  df-iso 4329  df-iord 4398  df-on 4400  df-ilim 4401  df-suc 4403  df-iom 4624  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-riota 5874  df-ov 5922  df-oprab 5923  df-mpo 5924  df-1st 6195  df-2nd 6196  df-recs 6360  df-frec 6446  df-pnf 8058  df-mnf 8059  df-xr 8060  df-ltxr 8061  df-le 8062  df-sub 8194  df-neg 8195  df-reap 8596  df-ap 8603  df-div 8694  df-inn 8985  df-2 9043  df-3 9044  df-4 9045  df-n0 9244  df-z 9321  df-uz 9596  df-rp 9723  df-seqfrec 10522  df-exp 10613  df-cj 10989  df-re 10990  df-im 10991  df-rsqrt 11145  df-abs 11146  df-gz 12511

This theorem is referenced by: (None)