Theorem mul4sq 15658

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 15657. (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 15656	. 2 |- ( A e. S <-> E. a e. ZZ[_i] E. b e. ZZ[_i] A = ( ( ( abs ` a ) ^ 2 ) + ( ( abs ` b ) ^ 2 ) ) )
3	1	4sqlem4 15656	. 2 |- ( B e. S <-> E. c e. ZZ[_i] E. d e. ZZ[_i] B = ( ( ( abs ` c ) ^ 2 ) + ( ( abs ` d ) ^ 2 ) ) )
4		reeanv 3107	. . 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 3107	. . . . 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 790	. . . . . . . . . . . . 13 |- ( ( ( a e. ZZ[_i] /\ c e. ZZ[_i] ) /\ ( b e. ZZ[_i] /\ d e. ZZ[_i] ) ) -> a e. ZZ[_i] )
7		gzabssqcl 15645	. . . . . . . . . . . . 13 |- ( a e. ZZ[_i] -> ( ( abs ` a ) ^ 2 ) e. NN0 )
8	6, 7	syl 17	. . . . . . . . . . . 12 |- ( ( ( a e. ZZ[_i] /\ c e. ZZ[_i] ) /\ ( b e. ZZ[_i] /\ d e. ZZ[_i] ) ) -> ( ( abs ` a ) ^ 2 ) e. NN0 )
9		simprl 794	. . . . . . . . . . . . 13 |- ( ( ( a e. ZZ[_i] /\ c e. ZZ[_i] ) /\ ( b e. ZZ[_i] /\ d e. ZZ[_i] ) ) -> b e. ZZ[_i] )
10		gzabssqcl 15645	. . . . . . . . . . . . 13 |- ( b e. ZZ[_i] -> ( ( abs ` b ) ^ 2 ) e. NN0 )
11	9, 10	syl 17	. . . . . . . . . . . 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 11355	. . . . . . . . . . 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 11353	. . . . . . . . . 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 10793	. . . . . . . . 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 792	. . . . . . . . . . . . 13 |- ( ( ( a e. ZZ[_i] /\ c e. ZZ[_i] ) /\ ( b e. ZZ[_i] /\ d e. ZZ[_i] ) ) -> c e. ZZ[_i] )
16		gzabssqcl 15645	. . . . . . . . . . . . 13 |- ( c e. ZZ[_i] -> ( ( abs ` c ) ^ 2 ) e. NN0 )
17	15, 16	syl 17	. . . . . . . . . . . 12 |- ( ( ( a e. ZZ[_i] /\ c e. ZZ[_i] ) /\ ( b e. ZZ[_i] /\ d e. ZZ[_i] ) ) -> ( ( abs ` c ) ^ 2 ) e. NN0 )
18		simprr 796	. . . . . . . . . . . . 13 |- ( ( ( a e. ZZ[_i] /\ c e. ZZ[_i] ) /\ ( b e. ZZ[_i] /\ d e. ZZ[_i] ) ) -> d e. ZZ[_i] )
19		gzabssqcl 15645	. . . . . . . . . . . . 13 |- ( d e. ZZ[_i] -> ( ( abs ` d ) ^ 2 ) e. NN0 )
20	18, 19	syl 17	. . . . . . . . . . . 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 11355	. . . . . . . . . . 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 11353	. . . . . . . . . 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 10793	. . . . . . . . 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 6668	. . . . . . . 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 2622	. . . . . . . . 9 |- ( ( ( abs ` a ) ^ 2 ) + ( ( abs ` b ) ^ 2 ) ) = ( ( ( abs ` a ) ^ 2 ) + ( ( abs ` b ) ^ 2 ) )
26		eqid 2622	. . . . . . . . 9 |- ( ( ( abs ` c ) ^ 2 ) + ( ( abs ` d ) ^ 2 ) ) = ( ( ( abs ` c ) ^ 2 ) + ( ( abs ` d ) ^ 2 ) )
27		1nn 11031	. . . . . . . . . 10 |- 1 e. NN
28	27	a1i 11	. . . . . . . . 9 |- ( ( ( a e. ZZ[_i] /\ c e. ZZ[_i] ) /\ ( b e. ZZ[_i] /\ d e. ZZ[_i] ) ) -> 1 e. NN )
29		gzsubcl 15644	. . . . . . . . . . . . 13 |- ( ( a e. ZZ[_i] /\ c e. ZZ[_i] ) -> ( a - c ) e. ZZ[_i] )
30	29	adantr 481	. . . . . . . . . . . 12 |- ( ( ( a e. ZZ[_i] /\ c e. ZZ[_i] ) /\ ( b e. ZZ[_i] /\ d e. ZZ[_i] ) ) -> ( a - c ) e. ZZ[_i] )
31		gzcn 15636	. . . . . . . . . . . 12 |- ( ( a - c ) e. ZZ[_i] -> ( a - c ) e. CC )
32	30, 31	syl 17	. . . . . . . . . . 11 |- ( ( ( a e. ZZ[_i] /\ c e. ZZ[_i] ) /\ ( b e. ZZ[_i] /\ d e. ZZ[_i] ) ) -> ( a - c ) e. CC )
33	32	div1d 10793	. . . . . . . . . 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 2701	. . . . . . . . 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 15644	. . . . . . . . . . . . 13 |- ( ( b e. ZZ[_i] /\ d e. ZZ[_i] ) -> ( b - d ) e. ZZ[_i] )
36	35	adantl 482	. . . . . . . . . . . 12 |- ( ( ( a e. ZZ[_i] /\ c e. ZZ[_i] ) /\ ( b e. ZZ[_i] /\ d e. ZZ[_i] ) ) -> ( b - d ) e. ZZ[_i] )
37		gzcn 15636	. . . . . . . . . . . 12 |- ( ( b - d ) e. ZZ[_i] -> ( b - d ) e. CC )
38	36, 37	syl 17	. . . . . . . . . . 11 |- ( ( ( a e. ZZ[_i] /\ c e. ZZ[_i] ) /\ ( b e. ZZ[_i] /\ d e. ZZ[_i] ) ) -> ( b - d ) e. CC )
39	38	div1d 10793	. . . . . . . . . 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 2701	. . . . . . . . 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 2701	. . . . . . . . 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 15657	. . . . . . . 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 2702	. . . . . . 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 6659	. . . . . . . 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 2686	. . . . . . 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 237	. . . . . 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 3038	. . . . 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	syl5bir 233	. . . 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 3036	. . 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 225	. 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 496	1 |- ( ( A e. S /\ B e. S ) -> ( A x. B ) e. S )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   {cab 2608   E.wrex 2913   ` cfv 5888  (class class class)co 6650   CCcc 9934   1c1 9937   + caddc 9939   x. cmul 9941   - cmin 10266   / cdiv 10684   NNcn 11020   2c2 11070   NN0cn0 11292   ZZcz 11377   ^cexp 12860   abscabs 13974   ZZ[_i]cgz 15633

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-gz 15634

This theorem is referenced by:  4sqlem19  15667