Theorem mul2sq 15593

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 15591	. 2 |- ( A e. S <-> E. x e. ZZ[_i] A = ( ( abs ` x ) ^ 2 ) )
3	1	2sqlem1 15591	. 2 |- ( B e. S <-> E. y e. ZZ[_i] B = ( ( abs ` y ) ^ 2 ) )
4		reeanv 2676	. . 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 12701	. . . . . . 7 |- ( ( x e. ZZ[_i] /\ y e. ZZ[_i] ) -> ( x x. y ) e. ZZ[_i] )
6		gzcn 12695	. . . . . . . . . 10 |- ( x e. ZZ[_i] -> x e. CC )
7		gzcn 12695	. . . . . . . . . 10 |- ( y e. ZZ[_i] -> y e. CC )
8		absmul 11380	. . . . . . . . . 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 5959	. . . . . . . 8 |- ( ( x e. ZZ[_i] /\ y e. ZZ[_i] ) -> ( ( abs ` ( x x. y ) ) ^ 2 ) = ( ( ( abs ` x ) x. ( abs ` y ) ) ^ 2 ) )
11	6	abscld 11492	. . . . . . . . . 10 |- ( x e. ZZ[_i] -> ( abs ` x ) e. RR )
12	11	recnd 8101	. . . . . . . . 9 |- ( x e. ZZ[_i] -> ( abs ` x ) e. CC )
13	7	abscld 11492	. . . . . . . . . 10 |- ( y e. ZZ[_i] -> ( abs ` y ) e. RR )
14	13	recnd 8101	. . . . . . . . 9 |- ( y e. ZZ[_i] -> ( abs ` y ) e. CC )
15		sqmul 10746	. . . . . . . . 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 2239	. . . . . . 7 |- ( ( x e. ZZ[_i] /\ y e. ZZ[_i] ) -> ( ( ( abs ` x ) ^ 2 ) x. ( ( abs ` y ) ^ 2 ) ) = ( ( abs ` ( x x. y ) ) ^ 2 ) )
18		fveq2 5576	. . . . . . . . 9 |- ( z = ( x x. y ) -> ( abs ` z ) = ( abs ` ( x x. y ) ) )
19	18	oveq1d 5959	. . . . . . . 8 |- ( z = ( x x. y ) -> ( ( abs ` z ) ^ 2 ) = ( ( abs ` ( x x. y ) ) ^ 2 ) )
20	19	rspceeqv 2895	. . . . . . 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 15591	. . . . . 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 5953	. . . . . 6 |- ( ( A = ( ( abs ` x ) ^ 2 ) /\ B = ( ( abs ` y ) ^ 2 ) ) -> ( A x. B ) = ( ( ( abs ` x ) ^ 2 ) x. ( ( abs ` y ) ^ 2 ) ) )
25	24	eleq1d 2274	. . . . 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 2629	. . 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 2176   E.wrex 2485   |-> cmpt 4105   ran crn 4676   ` cfv 5271  (class class class)co 5944   CCcc 7923   x. cmul 7930   2c2 9087   ^cexp 10683   abscabs 11308   ZZ[_i]cgz 12692

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-coll 4159  ax-sep 4162  ax-nul 4170  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-iinf 4636  ax-cnex 8016  ax-resscn 8017  ax-1cn 8018  ax-1re 8019  ax-icn 8020  ax-addcl 8021  ax-addrcl 8022  ax-mulcl 8023  ax-mulrcl 8024  ax-addcom 8025  ax-mulcom 8026  ax-addass 8027  ax-mulass 8028  ax-distr 8029  ax-i2m1 8030  ax-0lt1 8031  ax-1rid 8032  ax-0id 8033  ax-rnegex 8034  ax-precex 8035  ax-cnre 8036  ax-pre-ltirr 8037  ax-pre-ltwlin 8038  ax-pre-lttrn 8039  ax-pre-apti 8040  ax-pre-ltadd 8041  ax-pre-mulgt0 8042  ax-pre-mulext 8043  ax-arch 8044  ax-caucvg 8045

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 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-reu 2491  df-rmo 2492  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-if 3572  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  df-br 4045  df-opab 4106  df-mpt 4107  df-tr 4143  df-id 4340  df-po 4343  df-iso 4344  df-iord 4413  df-on 4415  df-ilim 4416  df-suc 4418  df-iom 4639  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-riota 5899  df-ov 5947  df-oprab 5948  df-mpo 5949  df-1st 6226  df-2nd 6227  df-recs 6391  df-frec 6477  df-pnf 8109  df-mnf 8110  df-xr 8111  df-ltxr 8112  df-le 8113  df-sub 8245  df-neg 8246  df-reap 8648  df-ap 8655  df-div 8746  df-inn 9037  df-2 9095  df-3 9096  df-4 9097  df-n0 9296  df-z 9373  df-uz 9649  df-rp 9776  df-seqfrec 10593  df-exp 10684  df-cj 11153  df-re 11154  df-im 11155  df-rsqrt 11309  df-abs 11310  df-gz 12693

This theorem is referenced by: (None)