Theorem mul4sq 13030

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 13029. (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 13028	. 2 |- ( A e. S <-> E. a e. ZZ[_i] E. b e. ZZ[_i] A = ( ( ( abs ` a ) ^ 2 ) + ( ( abs ` b ) ^ 2 ) ) )
3	1	4sqlem4 13028	. 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 13017	. . . . . . . . . . . . 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 13017	. . . . . . . . . . . . 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 9504	. . . . . . . . . . 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 9502	. . . . . . . . . 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 9003	. . . . . . . . 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 13017	. . . . . . . . . . . . 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 13017	. . . . . . . . . . . . 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 9504	. . . . . . . . . . 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 9502	. . . . . . . . . 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 9003	. . . . . . . . 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 9197	. . . . . . . . . 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 13016	. . . . . . . . . . . . 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 13008	. . . . . . . . . . . 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 9003	. . . . . . . . . 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 13016	. . . . . . . . . . . . 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 13008	. . . . . . . . . . . 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 9003	. . . . . . . . . 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 13029	. . . . . . . 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 8073   1c1 8076   + caddc 8078   x. cmul 8080   - cmin 8393   / cdiv 8895   NNcn 9186   2c2 9237   NN0cn0 9445   ZZcz 9524   ^cexp 10846   abscabs 11620   ZZ[_i]cgz 13005

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 8166  ax-resscn 8167  ax-1cn 8168  ax-1re 8169  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-mulrcl 8174  ax-addcom 8175  ax-mulcom 8176  ax-addass 8177  ax-mulass 8178  ax-distr 8179  ax-i2m1 8180  ax-0lt1 8181  ax-1rid 8182  ax-0id 8183  ax-rnegex 8184  ax-precex 8185  ax-cnre 8186  ax-pre-ltirr 8187  ax-pre-ltwlin 8188  ax-pre-lttrn 8189  ax-pre-apti 8190  ax-pre-ltadd 8191  ax-pre-mulgt0 8192  ax-pre-mulext 8193  ax-arch 8194  ax-caucvg 8195

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 8259  df-mnf 8260  df-xr 8261  df-ltxr 8262  df-le 8263  df-sub 8395  df-neg 8396  df-reap 8798  df-ap 8805  df-div 8896  df-inn 9187  df-2 9245  df-3 9246  df-4 9247  df-n0 9446  df-z 9525  df-uz 9801  df-rp 9934  df-seqfrec 10756  df-exp 10847  df-cj 11465  df-re 11466  df-im 11467  df-rsqrt 11621  df-abs 11622  df-gz 13006

This theorem is referenced by:  4sqlem19  13045