Theorem mul4sq 13047

Description: Euler's four-square identity: The product of two sums of four squares is also a sum of four squares. This is usually quoted as an explicit formula involving eight real variables; we save some time by working with complex numbers (gaussian integers) instead, so that we only have to work with four variables, and also hiding the actual formula for the product in the proof of mul4sqlem 13046. (For the curious, the explicit formula that is used is ( | a | ^ 2 + | b | ^ 2 ) ( | c | ^ 2 + | d | ^ 2 ) = | a * x. c + b x. d * | ^ 2 + | a * x. d - b x. c * | ^ 2.) (Contributed by Mario Carneiro, 14-Jul-2014.)

Hypothesis

Ref	Expression
4sq.1	|- S = { n | E. x e. ZZ E. y e. ZZ E. z e. ZZ E. w e. ZZ n = ( ( ( x ^ 2 ) + ( y ^ 2 ) ) + ( ( z ^ 2 ) + ( w ^ 2 ) ) ) }

Assertion

Ref	Expression
mul4sq	|- ( ( A e. S /\ B e. S ) -> ( A x. B ) e. S )

Distinct variable groups:   w, n, x, y, z   B, n   A, n   S, n

Allowed substitution hints:   A( x, y, z, w)   B( x, y, z, w)   S( x, y, z, w)

Proof of Theorem mul4sq

Dummy variables a b c d are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		4sq.1	. . 3 |- S = { n | E. x e. ZZ E. y e. ZZ E. z e. ZZ E. w e. ZZ n = ( ( ( x ^ 2 ) + ( y ^ 2 ) ) + ( ( z ^ 2 ) + ( w ^ 2 ) ) ) }
2	1	4sqlem4 13045	. 2 |- ( A e. S <-> E. a e. ZZ[_i] E. b e. ZZ[_i] A = ( ( ( abs ` a ) ^ 2 ) + ( ( abs ` b ) ^ 2 ) ) )
3	1	4sqlem4 13045	. 2 |- ( B e. S <-> E. c e. ZZ[_i] E. d e. ZZ[_i] B = ( ( ( abs ` c ) ^ 2 ) + ( ( abs ` d ) ^ 2 ) ) )
4		reeanv 2704	. . 3 |- ( E. a e. ZZ[_i] E. c e. ZZ[_i] ( E. b e. ZZ[_i] A = ( ( ( abs ` a ) ^ 2 ) + ( ( abs ` b ) ^ 2 ) ) /\ E. d e. ZZ[_i] B = ( ( ( abs ` c ) ^ 2 ) + ( ( abs ` d ) ^ 2 ) ) ) <-> ( E. a e. ZZ[_i] E. b e. ZZ[_i] A = ( ( ( abs ` a ) ^ 2 ) + ( ( abs ` b ) ^ 2 ) ) /\ E. c e. ZZ[_i] E. d e. ZZ[_i] B = ( ( ( abs ` c ) ^ 2 ) + ( ( abs ` d ) ^ 2 ) ) ) )
5		reeanv 2704	. . . . 5 |- ( E. b e. ZZ[_i] E. d e. ZZ[_i] ( A = ( ( ( abs ` a ) ^ 2 ) + ( ( abs ` b ) ^ 2 ) ) /\ B = ( ( ( abs ` c ) ^ 2 ) + ( ( abs ` d ) ^ 2 ) ) ) <-> ( E. b e. ZZ[_i] A = ( ( ( abs ` a ) ^ 2 ) + ( ( abs ` b ) ^ 2 ) ) /\ E. d e. ZZ[_i] B = ( ( ( abs ` c ) ^ 2 ) + ( ( abs ` d ) ^ 2 ) ) ) )
6		simpll 527	. . . . . . . . . . . . 13 |- ( ( ( a e. ZZ[_i] /\ c e. ZZ[_i] ) /\ ( b e. ZZ[_i] /\ d e. ZZ[_i] ) ) -> a e. ZZ[_i] )
7		gzabssqcl 13034	. . . . . . . . . . . . 13 |- ( a e. ZZ[_i] -> ( ( abs ` a ) ^ 2 ) e. NN0 )
8	6, 7	syl 14	. . . . . . . . . . . 12 |- ( ( ( a e. ZZ[_i] /\ c e. ZZ[_i] ) /\ ( b e. ZZ[_i] /\ d e. ZZ[_i] ) ) -> ( ( abs ` a ) ^ 2 ) e. NN0 )
9		simprl 531	. . . . . . . . . . . . 13 |- ( ( ( a e. ZZ[_i] /\ c e. ZZ[_i] ) /\ ( b e. ZZ[_i] /\ d e. ZZ[_i] ) ) -> b e. ZZ[_i] )
10		gzabssqcl 13034	. . . . . . . . . . . . 13 |- ( b e. ZZ[_i] -> ( ( abs ` b ) ^ 2 ) e. NN0 )
11	9, 10	syl 14	. . . . . . . . . . . 12 |- ( ( ( a e. ZZ[_i] /\ c e. ZZ[_i] ) /\ ( b e. ZZ[_i] /\ d e. ZZ[_i] ) ) -> ( ( abs ` b ) ^ 2 ) e. NN0 )
12	8, 11	nn0addcld 9520	. . . . . . . . . . 11 |- ( ( ( a e. ZZ[_i] /\ c e. ZZ[_i] ) /\ ( b e. ZZ[_i] /\ d e. ZZ[_i] ) ) -> ( ( ( abs ` a ) ^ 2 ) + ( ( abs ` b ) ^ 2 ) ) e. NN0 )
13	12	nn0cnd 9518	. . . . . . . . . 10 |- ( ( ( a e. ZZ[_i] /\ c e. ZZ[_i] ) /\ ( b e. ZZ[_i] /\ d e. ZZ[_i] ) ) -> ( ( ( abs ` a ) ^ 2 ) + ( ( abs ` b ) ^ 2 ) ) e. CC )
14	13	div1d 9019	. . . . . . . . 9 |- ( ( ( a e. ZZ[_i] /\ c e. ZZ[_i] ) /\ ( b e. ZZ[_i] /\ d e. ZZ[_i] ) ) -> ( ( ( ( abs ` a ) ^ 2 ) + ( ( abs ` b ) ^ 2 ) ) / 1 ) = ( ( ( abs ` a ) ^ 2 ) + ( ( abs ` b ) ^ 2 ) ) )
15		simplr 529	. . . . . . . . . . . . 13 |- ( ( ( a e. ZZ[_i] /\ c e. ZZ[_i] ) /\ ( b e. ZZ[_i] /\ d e. ZZ[_i] ) ) -> c e. ZZ[_i] )
16		gzabssqcl 13034	. . . . . . . . . . . . 13 |- ( c e. ZZ[_i] -> ( ( abs ` c ) ^ 2 ) e. NN0 )
17	15, 16	syl 14	. . . . . . . . . . . 12 |- ( ( ( a e. ZZ[_i] /\ c e. ZZ[_i] ) /\ ( b e. ZZ[_i] /\ d e. ZZ[_i] ) ) -> ( ( abs ` c ) ^ 2 ) e. NN0 )
18		simprr 533	. . . . . . . . . . . . 13 |- ( ( ( a e. ZZ[_i] /\ c e. ZZ[_i] ) /\ ( b e. ZZ[_i] /\ d e. ZZ[_i] ) ) -> d e. ZZ[_i] )
19		gzabssqcl 13034	. . . . . . . . . . . . 13 |- ( d e. ZZ[_i] -> ( ( abs ` d ) ^ 2 ) e. NN0 )
20	18, 19	syl 14	. . . . . . . . . . . 12 |- ( ( ( a e. ZZ[_i] /\ c e. ZZ[_i] ) /\ ( b e. ZZ[_i] /\ d e. ZZ[_i] ) ) -> ( ( abs ` d ) ^ 2 ) e. NN0 )
21	17, 20	nn0addcld 9520	. . . . . . . . . . 11 |- ( ( ( a e. ZZ[_i] /\ c e. ZZ[_i] ) /\ ( b e. ZZ[_i] /\ d e. ZZ[_i] ) ) -> ( ( ( abs ` c ) ^ 2 ) + ( ( abs ` d ) ^ 2 ) ) e. NN0 )
22	21	nn0cnd 9518	. . . . . . . . . 10 |- ( ( ( a e. ZZ[_i] /\ c e. ZZ[_i] ) /\ ( b e. ZZ[_i] /\ d e. ZZ[_i] ) ) -> ( ( ( abs ` c ) ^ 2 ) + ( ( abs ` d ) ^ 2 ) ) e. CC )
23	22	div1d 9019	. . . . . . . . 9 |- ( ( ( a e. ZZ[_i] /\ c e. ZZ[_i] ) /\ ( b e. ZZ[_i] /\ d e. ZZ[_i] ) ) -> ( ( ( ( abs ` c ) ^ 2 ) + ( ( abs ` d ) ^ 2 ) ) / 1 ) = ( ( ( abs ` c ) ^ 2 ) + ( ( abs ` d ) ^ 2 ) ) )
24	14, 23	oveq12d 6046	. . . . . . . 8 |- ( ( ( a e. ZZ[_i] /\ c e. ZZ[_i] ) /\ ( b e. ZZ[_i] /\ d e. ZZ[_i] ) ) -> ( ( ( ( ( abs ` a ) ^ 2 ) + ( ( abs ` b ) ^ 2 ) ) / 1 ) x. ( ( ( ( abs ` c ) ^ 2 ) + ( ( abs ` d ) ^ 2 ) ) / 1 ) ) = ( ( ( ( abs ` a ) ^ 2 ) + ( ( abs ` b ) ^ 2 ) ) x. ( ( ( abs ` c ) ^ 2 ) + ( ( abs ` d ) ^ 2 ) ) ) )
25		eqid 2231	. . . . . . . . 9 |- ( ( ( abs ` a ) ^ 2 ) + ( ( abs ` b ) ^ 2 ) ) = ( ( ( abs ` a ) ^ 2 ) + ( ( abs ` b ) ^ 2 ) )
26		eqid 2231	. . . . . . . . 9 |- ( ( ( abs ` c ) ^ 2 ) + ( ( abs ` d ) ^ 2 ) ) = ( ( ( abs ` c ) ^ 2 ) + ( ( abs ` d ) ^ 2 ) )
27		1nn 9213	. . . . . . . . . 10 |- 1 e. NN
28	27	a1i 9	. . . . . . . . 9 |- ( ( ( a e. ZZ[_i] /\ c e. ZZ[_i] ) /\ ( b e. ZZ[_i] /\ d e. ZZ[_i] ) ) -> 1 e. NN )
29		gzsubcl 13033	. . . . . . . . . . . . 13 |- ( ( a e. ZZ[_i] /\ c e. ZZ[_i] ) -> ( a - c ) e. ZZ[_i] )
30	29	adantr 276	. . . . . . . . . . . 12 |- ( ( ( a e. ZZ[_i] /\ c e. ZZ[_i] ) /\ ( b e. ZZ[_i] /\ d e. ZZ[_i] ) ) -> ( a - c ) e. ZZ[_i] )
31		gzcn 13025	. . . . . . . . . . . 12 |- ( ( a - c ) e. ZZ[_i] -> ( a - c ) e. CC )
32	30, 31	syl 14	. . . . . . . . . . 11 |- ( ( ( a e. ZZ[_i] /\ c e. ZZ[_i] ) /\ ( b e. ZZ[_i] /\ d e. ZZ[_i] ) ) -> ( a - c ) e. CC )
33	32	div1d 9019	. . . . . . . . . 10 |- ( ( ( a e. ZZ[_i] /\ c e. ZZ[_i] ) /\ ( b e. ZZ[_i] /\ d e. ZZ[_i] ) ) -> ( ( a - c ) / 1 ) = ( a - c ) )
34	33, 30	eqeltrd 2308	. . . . . . . . 9 |- ( ( ( a e. ZZ[_i] /\ c e. ZZ[_i] ) /\ ( b e. ZZ[_i] /\ d e. ZZ[_i] ) ) -> ( ( a - c ) / 1 ) e. ZZ[_i] )
35		gzsubcl 13033	. . . . . . . . . . . . 13 |- ( ( b e. ZZ[_i] /\ d e. ZZ[_i] ) -> ( b - d ) e. ZZ[_i] )
36	35	adantl 277	. . . . . . . . . . . 12 |- ( ( ( a e. ZZ[_i] /\ c e. ZZ[_i] ) /\ ( b e. ZZ[_i] /\ d e. ZZ[_i] ) ) -> ( b - d ) e. ZZ[_i] )
37		gzcn 13025	. . . . . . . . . . . 12 |- ( ( b - d ) e. ZZ[_i] -> ( b - d ) e. CC )
38	36, 37	syl 14	. . . . . . . . . . 11 |- ( ( ( a e. ZZ[_i] /\ c e. ZZ[_i] ) /\ ( b e. ZZ[_i] /\ d e. ZZ[_i] ) ) -> ( b - d ) e. CC )
39	38	div1d 9019	. . . . . . . . . 10 |- ( ( ( a e. ZZ[_i] /\ c e. ZZ[_i] ) /\ ( b e. ZZ[_i] /\ d e. ZZ[_i] ) ) -> ( ( b - d ) / 1 ) = ( b - d ) )
40	39, 36	eqeltrd 2308	. . . . . . . . 9 |- ( ( ( a e. ZZ[_i] /\ c e. ZZ[_i] ) /\ ( b e. ZZ[_i] /\ d e. ZZ[_i] ) ) -> ( ( b - d ) / 1 ) e. ZZ[_i] )
41	14, 12	eqeltrd 2308	. . . . . . . . 9 |- ( ( ( a e. ZZ[_i] /\ c e. ZZ[_i] ) /\ ( b e. ZZ[_i] /\ d e. ZZ[_i] ) ) -> ( ( ( ( abs ` a ) ^ 2 ) + ( ( abs ` b ) ^ 2 ) ) / 1 ) e. NN0 )
42	1, 6, 9, 15, 18, 25, 26, 28, 34, 40, 41	mul4sqlem 13046	. . . . . . . 8 |- ( ( ( a e. ZZ[_i] /\ c e. ZZ[_i] ) /\ ( b e. ZZ[_i] /\ d e. ZZ[_i] ) ) -> ( ( ( ( ( abs ` a ) ^ 2 ) + ( ( abs ` b ) ^ 2 ) ) / 1 ) x. ( ( ( ( abs ` c ) ^ 2 ) + ( ( abs ` d ) ^ 2 ) ) / 1 ) ) e. S )
43	24, 42	eqeltrrd 2309	. . . . . . 7 |- ( ( ( a e. ZZ[_i] /\ c e. ZZ[_i] ) /\ ( b e. ZZ[_i] /\ d e. ZZ[_i] ) ) -> ( ( ( ( abs ` a ) ^ 2 ) + ( ( abs ` b ) ^ 2 ) ) x. ( ( ( abs ` c ) ^ 2 ) + ( ( abs ` d ) ^ 2 ) ) ) e. S )
44		oveq12 6037	. . . . . . . 8 |- ( ( A = ( ( ( abs ` a ) ^ 2 ) + ( ( abs ` b ) ^ 2 ) ) /\ B = ( ( ( abs ` c ) ^ 2 ) + ( ( abs ` d ) ^ 2 ) ) ) -> ( A x. B ) = ( ( ( ( abs ` a ) ^ 2 ) + ( ( abs ` b ) ^ 2 ) ) x. ( ( ( abs ` c ) ^ 2 ) + ( ( abs ` d ) ^ 2 ) ) ) )
45	44	eleq1d 2300	. . . . . . 7 |- ( ( A = ( ( ( abs ` a ) ^ 2 ) + ( ( abs ` b ) ^ 2 ) ) /\ B = ( ( ( abs ` c ) ^ 2 ) + ( ( abs ` d ) ^ 2 ) ) ) -> ( ( A x. B ) e. S <-> ( ( ( ( abs ` a ) ^ 2 ) + ( ( abs ` b ) ^ 2 ) ) x. ( ( ( abs ` c ) ^ 2 ) + ( ( abs ` d ) ^ 2 ) ) ) e. S ) )
46	43, 45	syl5ibrcom 157	. . . . . 6 |- ( ( ( a e. ZZ[_i] /\ c e. ZZ[_i] ) /\ ( b e. ZZ[_i] /\ d e. ZZ[_i] ) ) -> ( ( A = ( ( ( abs ` a ) ^ 2 ) + ( ( abs ` b ) ^ 2 ) ) /\ B = ( ( ( abs ` c ) ^ 2 ) + ( ( abs ` d ) ^ 2 ) ) ) -> ( A x. B ) e. S ) )
47	46	rexlimdvva 2659	. . . . 5 |- ( ( a e. ZZ[_i] /\ c e. ZZ[_i] ) -> ( E. b e. ZZ[_i] E. d e. ZZ[_i] ( A = ( ( ( abs ` a ) ^ 2 ) + ( ( abs ` b ) ^ 2 ) ) /\ B = ( ( ( abs ` c ) ^ 2 ) + ( ( abs ` d ) ^ 2 ) ) ) -> ( A x. B ) e. S ) )
48	5, 47	biimtrrid 153	. . . 4 |- ( ( a e. ZZ[_i] /\ c e. ZZ[_i] ) -> ( ( E. b e. ZZ[_i] A = ( ( ( abs ` a ) ^ 2 ) + ( ( abs ` b ) ^ 2 ) ) /\ E. d e. ZZ[_i] B = ( ( ( abs ` c ) ^ 2 ) + ( ( abs ` d ) ^ 2 ) ) ) -> ( A x. B ) e. S ) )
49	48	rexlimivv 2657	. . 3 |- ( E. a e. ZZ[_i] E. c e. ZZ[_i] ( E. b e. ZZ[_i] A = ( ( ( abs ` a ) ^ 2 ) + ( ( abs ` b ) ^ 2 ) ) /\ E. d e. ZZ[_i] B = ( ( ( abs ` c ) ^ 2 ) + ( ( abs ` d ) ^ 2 ) ) ) -> ( A x. B ) e. S )
50	4, 49	sylbir 135	. 2 |- ( ( E. a e. ZZ[_i] E. b e. ZZ[_i] A = ( ( ( abs ` a ) ^ 2 ) + ( ( abs ` b ) ^ 2 ) ) /\ E. c e. ZZ[_i] E. d e. ZZ[_i] B = ( ( ( abs ` c ) ^ 2 ) + ( ( abs ` d ) ^ 2 ) ) ) -> ( A x. B ) e. S )
51	2, 3, 50	syl2anb 291	1 |- ( ( A e. S /\ B e. S ) -> ( A x. B ) e. S )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2202   {cab 2217   E.wrex 2512   ` cfv 5333  (class class class)co 6028   CCcc 8090   1c1 8093   + caddc 8095   x. cmul 8097   - cmin 8409   / cdiv 8911   NNcn 9202   2c2 9253   NN0cn0 9461   ZZcz 9540   ^cexp 10863   abscabs 11637   ZZ[_i]cgz 13022

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692  ax-cnex 8183  ax-resscn 8184  ax-1cn 8185  ax-1re 8186  ax-icn 8187  ax-addcl 8188  ax-addrcl 8189  ax-mulcl 8190  ax-mulrcl 8191  ax-addcom 8192  ax-mulcom 8193  ax-addass 8194  ax-mulass 8195  ax-distr 8196  ax-i2m1 8197  ax-0lt1 8198  ax-1rid 8199  ax-0id 8200  ax-rnegex 8201  ax-precex 8202  ax-cnre 8203  ax-pre-ltirr 8204  ax-pre-ltwlin 8205  ax-pre-lttrn 8206  ax-pre-apti 8207  ax-pre-ltadd 8208  ax-pre-mulgt0 8209  ax-pre-mulext 8210  ax-arch 8211  ax-caucvg 8212

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-po 4399  df-iso 4400  df-iord 4469  df-on 4471  df-ilim 4472  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-frec 6600  df-pnf 8275  df-mnf 8276  df-xr 8277  df-ltxr 8278  df-le 8279  df-sub 8411  df-neg 8412  df-reap 8814  df-ap 8821  df-div 8912  df-inn 9203  df-2 9261  df-3 9262  df-4 9263  df-n0 9462  df-z 9541  df-uz 9817  df-rp 9950  df-seqfrec 10773  df-exp 10864  df-cj 11482  df-re 11483  df-im 11484  df-rsqrt 11638  df-abs 11639  df-gz 13023

This theorem is referenced by:  4sqlem19  13062