Theorem mul4sq 12532

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 12531. (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 12530	. 2 |- ( A e. S <-> E. a e. ZZ[_i] E. b e. ZZ[_i] A = ( ( ( abs ` a ) ^ 2 ) + ( ( abs ` b ) ^ 2 ) ) )
3	1	4sqlem4 12530	. 2 |- ( B e. S <-> E. c e. ZZ[_i] E. d e. ZZ[_i] B = ( ( ( abs ` c ) ^ 2 ) + ( ( abs ` d ) ^ 2 ) ) )
4		reeanv 2664	. . 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 2664	. . . . 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 12519	. . . . . . . . . . . . 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 529	. . . . . . . . . . . . 13 |- ( ( ( a e. ZZ[_i] /\ c e. ZZ[_i] ) /\ ( b e. ZZ[_i] /\ d e. ZZ[_i] ) ) -> b e. ZZ[_i] )
10		gzabssqcl 12519	. . . . . . . . . . . . 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 9297	. . . . . . . . . . 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 9295	. . . . . . . . . 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 8799	. . . . . . . . 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 528	. . . . . . . . . . . . 13 |- ( ( ( a e. ZZ[_i] /\ c e. ZZ[_i] ) /\ ( b e. ZZ[_i] /\ d e. ZZ[_i] ) ) -> c e. ZZ[_i] )
16		gzabssqcl 12519	. . . . . . . . . . . . 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 531	. . . . . . . . . . . . 13 |- ( ( ( a e. ZZ[_i] /\ c e. ZZ[_i] ) /\ ( b e. ZZ[_i] /\ d e. ZZ[_i] ) ) -> d e. ZZ[_i] )
19		gzabssqcl 12519	. . . . . . . . . . . . 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 9297	. . . . . . . . . . 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 9295	. . . . . . . . . 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 8799	. . . . . . . . 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 5936	. . . . . . . 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 2193	. . . . . . . . 9 |- ( ( ( abs ` a ) ^ 2 ) + ( ( abs ` b ) ^ 2 ) ) = ( ( ( abs ` a ) ^ 2 ) + ( ( abs ` b ) ^ 2 ) )
26		eqid 2193	. . . . . . . . 9 |- ( ( ( abs ` c ) ^ 2 ) + ( ( abs ` d ) ^ 2 ) ) = ( ( ( abs ` c ) ^ 2 ) + ( ( abs ` d ) ^ 2 ) )
27		1nn 8993	. . . . . . . . . 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 12518	. . . . . . . . . . . . 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 12510	. . . . . . . . . . . 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 8799	. . . . . . . . . 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 2270	. . . . . . . . 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 12518	. . . . . . . . . . . . 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 12510	. . . . . . . . . . . 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 8799	. . . . . . . . . 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 2270	. . . . . . . . 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 2270	. . . . . . . . 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 12531	. . . . . . . 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 2271	. . . . . . 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 5927	. . . . . . . 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 2262	. . . . . . 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 2619	. . . . 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 2617	. . 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 1364   e. wcel 2164   {cab 2179   E.wrex 2473   ` cfv 5254  (class class class)co 5918   CCcc 7870   1c1 7873   + caddc 7875   x. cmul 7877   - cmin 8190   / cdiv 8691   NNcn 8982   2c2 9033   NN0cn0 9240   ZZcz 9317   ^cexp 10609   abscabs 11141   ZZ[_i]cgz 12507

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 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-mulrcl 7971  ax-addcom 7972  ax-mulcom 7973  ax-addass 7974  ax-mulass 7975  ax-distr 7976  ax-i2m1 7977  ax-0lt1 7978  ax-1rid 7979  ax-0id 7980  ax-rnegex 7981  ax-precex 7982  ax-cnre 7983  ax-pre-ltirr 7984  ax-pre-ltwlin 7985  ax-pre-lttrn 7986  ax-pre-apti 7987  ax-pre-ltadd 7988  ax-pre-mulgt0 7989  ax-pre-mulext 7990  ax-arch 7991  ax-caucvg 7992

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 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-if 3558  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-id 4324  df-po 4327  df-iso 4328  df-iord 4397  df-on 4399  df-ilim 4400  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-recs 6358  df-frec 6444  df-pnf 8056  df-mnf 8057  df-xr 8058  df-ltxr 8059  df-le 8060  df-sub 8192  df-neg 8193  df-reap 8594  df-ap 8601  df-div 8692  df-inn 8983  df-2 9041  df-3 9042  df-4 9043  df-n0 9241  df-z 9318  df-uz 9593  df-rp 9720  df-seqfrec 10519  df-exp 10610  df-cj 10986  df-re 10987  df-im 10988  df-rsqrt 11142  df-abs 11143  df-gz 12508

This theorem is referenced by:  4sqlem19  12547