Theorem mul2sq 13746

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 13744	. 2 |- ( A e. S <-> E. x e. ZZ[_i] A = ( ( abs ` x ) ^ 2 ) )
3	1	2sqlem1 13744	. 2 |- ( B e. S <-> E. y e. ZZ[_i] B = ( ( abs ` y ) ^ 2 ) )
4		reeanv 2639	. . 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 12330	. . . . . . 7 |- ( ( x e. ZZ[_i] /\ y e. ZZ[_i] ) -> ( x x. y ) e. ZZ[_i] )
6		gzcn 12324	. . . . . . . . . 10 |- ( x e. ZZ[_i] -> x e. CC )
7		gzcn 12324	. . . . . . . . . 10 |- ( y e. ZZ[_i] -> y e. CC )
8		absmul 11033	. . . . . . . . . 10 |- ( ( x e. CC /\ y e. CC ) -> ( abs ` ( x x. y ) ) = ( ( abs ` x ) x. ( abs ` y ) ) )
9	6, 7, 8	syl2an 287	. . . . . . . . 9 |- ( ( x e. ZZ[_i] /\ y e. ZZ[_i] ) -> ( abs ` ( x x. y ) ) = ( ( abs ` x ) x. ( abs ` y ) ) )
10	9	oveq1d 5868	. . . . . . . 8 |- ( ( x e. ZZ[_i] /\ y e. ZZ[_i] ) -> ( ( abs ` ( x x. y ) ) ^ 2 ) = ( ( ( abs ` x ) x. ( abs ` y ) ) ^ 2 ) )
11	6	abscld 11145	. . . . . . . . . 10 |- ( x e. ZZ[_i] -> ( abs ` x ) e. RR )
12	11	recnd 7948	. . . . . . . . 9 |- ( x e. ZZ[_i] -> ( abs ` x ) e. CC )
13	7	abscld 11145	. . . . . . . . . 10 |- ( y e. ZZ[_i] -> ( abs ` y ) e. RR )
14	13	recnd 7948	. . . . . . . . 9 |- ( y e. ZZ[_i] -> ( abs ` y ) e. CC )
15		sqmul 10538	. . . . . . . . 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 287	. . . . . . . 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 2204	. . . . . . 7 |- ( ( x e. ZZ[_i] /\ y e. ZZ[_i] ) -> ( ( ( abs ` x ) ^ 2 ) x. ( ( abs ` y ) ^ 2 ) ) = ( ( abs ` ( x x. y ) ) ^ 2 ) )
18		fveq2 5496	. . . . . . . . 9 |- ( z = ( x x. y ) -> ( abs ` z ) = ( abs ` ( x x. y ) ) )
19	18	oveq1d 5868	. . . . . . . 8 |- ( z = ( x x. y ) -> ( ( abs ` z ) ^ 2 ) = ( ( abs ` ( x x. y ) ) ^ 2 ) )
20	19	rspceeqv 2852	. . . . . . 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 409	. . . . . 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 13744	. . . . . 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 133	. . . . 5 |- ( ( x e. ZZ[_i] /\ y e. ZZ[_i] ) -> ( ( ( abs ` x ) ^ 2 ) x. ( ( abs ` y ) ^ 2 ) ) e. S )
24		oveq12 5862	. . . . . 6 |- ( ( A = ( ( abs ` x ) ^ 2 ) /\ B = ( ( abs ` y ) ^ 2 ) ) -> ( A x. B ) = ( ( ( abs ` x ) ^ 2 ) x. ( ( abs ` y ) ^ 2 ) ) )
25	24	eleq1d 2239	. . . . 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 156	. . . 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 2593	. . 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 134	. 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 289	1 |- ( ( A e. S /\ B e. S ) -> ( A x. B ) e. S )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1348   e. wcel 2141   E.wrex 2449   |-> cmpt 4050   ran crn 4612   ` cfv 5198  (class class class)co 5853   CCcc 7772   x. cmul 7779   2c2 8929   ^cexp 10475   abscabs 10961   ZZ[_i]cgz 12321

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891  ax-pre-mulext 7892  ax-arch 7893  ax-caucvg 7894

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-ilim 4354  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-frec 6370  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501  df-div 8590  df-inn 8879  df-2 8937  df-3 8938  df-4 8939  df-n0 9136  df-z 9213  df-uz 9488  df-rp 9611  df-seqfrec 10402  df-exp 10476  df-cj 10806  df-re 10807  df-im 10808  df-rsqrt 10962  df-abs 10963  df-gz 12322

This theorem is referenced by: (None)