Theorem mul4sq 12932

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 12931. (For the curious, the explicit formula that is used is ( ∣ 𝑎 ∣ ↑2 + ∣ 𝑏 ∣ ↑2)( ∣ 𝑐 ∣ ↑2 + ∣ 𝑑 ∣ ↑2) = ∣ 𝑎∗ · 𝑐 + 𝑏 · 𝑑∗ ∣ ↑2 + ∣ 𝑎∗ · 𝑑 − 𝑏 · 𝑐∗ ∣ ↑2.) (Contributed by Mario Carneiro, 14-Jul-2014.)

Hypothesis

Ref	Expression
4sq.1	⊢ 𝑆 = {𝑛 ∣ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ ∃𝑧 ∈ ℤ ∃𝑤 ∈ ℤ 𝑛 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))}

Assertion

Ref	Expression
mul4sq	⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴 · 𝐵) ∈ 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 ∣ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ ∃𝑧 ∈ ℤ ∃𝑤 ∈ ℤ 𝑛 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))}
2	1	4sqlem4 12930	. 2 ⊢ (𝐴 ∈ 𝑆 ↔ ∃𝑎 ∈ ℤ[i] ∃𝑏 ∈ ℤ[i] 𝐴 = (((abs‘𝑎)↑2) + ((abs‘𝑏)↑2)))
3	1	4sqlem4 12930	. 2 ⊢ (𝐵 ∈ 𝑆 ↔ ∃𝑐 ∈ ℤ[i] ∃𝑑 ∈ ℤ[i] 𝐵 = (((abs‘𝑐)↑2) + ((abs‘𝑑)↑2)))
4		reeanv 2701	. . 3 ⊢ (∃𝑎 ∈ ℤ[i] ∃𝑐 ∈ ℤ[i] (∃𝑏 ∈ ℤ[i] 𝐴 = (((abs‘𝑎)↑2) + ((abs‘𝑏)↑2)) ∧ ∃𝑑 ∈ ℤ[i] 𝐵 = (((abs‘𝑐)↑2) + ((abs‘𝑑)↑2))) ↔ (∃𝑎 ∈ ℤ[i] ∃𝑏 ∈ ℤ[i] 𝐴 = (((abs‘𝑎)↑2) + ((abs‘𝑏)↑2)) ∧ ∃𝑐 ∈ ℤ[i] ∃𝑑 ∈ ℤ[i] 𝐵 = (((abs‘𝑐)↑2) + ((abs‘𝑑)↑2))))
5		reeanv 2701	. . . . 5 ⊢ (∃𝑏 ∈ ℤ[i] ∃𝑑 ∈ ℤ[i] (𝐴 = (((abs‘𝑎)↑2) + ((abs‘𝑏)↑2)) ∧ 𝐵 = (((abs‘𝑐)↑2) + ((abs‘𝑑)↑2))) ↔ (∃𝑏 ∈ ℤ[i] 𝐴 = (((abs‘𝑎)↑2) + ((abs‘𝑏)↑2)) ∧ ∃𝑑 ∈ ℤ[i] 𝐵 = (((abs‘𝑐)↑2) + ((abs‘𝑑)↑2))))
6		simpll 527	. . . . . . . . . . . . 13 ⊢ (((𝑎 ∈ ℤ[i] ∧ 𝑐 ∈ ℤ[i]) ∧ (𝑏 ∈ ℤ[i] ∧ 𝑑 ∈ ℤ[i])) → 𝑎 ∈ ℤ[i])
7		gzabssqcl 12919	. . . . . . . . . . . . 13 ⊢ (𝑎 ∈ ℤ[i] → ((abs‘𝑎)↑2) ∈ ℕ0)
8	6, 7	syl 14	. . . . . . . . . . . 12 ⊢ (((𝑎 ∈ ℤ[i] ∧ 𝑐 ∈ ℤ[i]) ∧ (𝑏 ∈ ℤ[i] ∧ 𝑑 ∈ ℤ[i])) → ((abs‘𝑎)↑2) ∈ ℕ0)
9		simprl 529	. . . . . . . . . . . . 13 ⊢ (((𝑎 ∈ ℤ[i] ∧ 𝑐 ∈ ℤ[i]) ∧ (𝑏 ∈ ℤ[i] ∧ 𝑑 ∈ ℤ[i])) → 𝑏 ∈ ℤ[i])
10		gzabssqcl 12919	. . . . . . . . . . . . 13 ⊢ (𝑏 ∈ ℤ[i] → ((abs‘𝑏)↑2) ∈ ℕ0)
11	9, 10	syl 14	. . . . . . . . . . . 12 ⊢ (((𝑎 ∈ ℤ[i] ∧ 𝑐 ∈ ℤ[i]) ∧ (𝑏 ∈ ℤ[i] ∧ 𝑑 ∈ ℤ[i])) → ((abs‘𝑏)↑2) ∈ ℕ0)
12	8, 11	nn0addcld 9437	. . . . . . . . . . 11 ⊢ (((𝑎 ∈ ℤ[i] ∧ 𝑐 ∈ ℤ[i]) ∧ (𝑏 ∈ ℤ[i] ∧ 𝑑 ∈ ℤ[i])) → (((abs‘𝑎)↑2) + ((abs‘𝑏)↑2)) ∈ ℕ0)
13	12	nn0cnd 9435	. . . . . . . . . 10 ⊢ (((𝑎 ∈ ℤ[i] ∧ 𝑐 ∈ ℤ[i]) ∧ (𝑏 ∈ ℤ[i] ∧ 𝑑 ∈ ℤ[i])) → (((abs‘𝑎)↑2) + ((abs‘𝑏)↑2)) ∈ ℂ)
14	13	div1d 8938	. . . . . . . . 9 ⊢ (((𝑎 ∈ ℤ[i] ∧ 𝑐 ∈ ℤ[i]) ∧ (𝑏 ∈ ℤ[i] ∧ 𝑑 ∈ ℤ[i])) → ((((abs‘𝑎)↑2) + ((abs‘𝑏)↑2)) / 1) = (((abs‘𝑎)↑2) + ((abs‘𝑏)↑2)))
15		simplr 528	. . . . . . . . . . . . 13 ⊢ (((𝑎 ∈ ℤ[i] ∧ 𝑐 ∈ ℤ[i]) ∧ (𝑏 ∈ ℤ[i] ∧ 𝑑 ∈ ℤ[i])) → 𝑐 ∈ ℤ[i])
16		gzabssqcl 12919	. . . . . . . . . . . . 13 ⊢ (𝑐 ∈ ℤ[i] → ((abs‘𝑐)↑2) ∈ ℕ0)
17	15, 16	syl 14	. . . . . . . . . . . 12 ⊢ (((𝑎 ∈ ℤ[i] ∧ 𝑐 ∈ ℤ[i]) ∧ (𝑏 ∈ ℤ[i] ∧ 𝑑 ∈ ℤ[i])) → ((abs‘𝑐)↑2) ∈ ℕ0)
18		simprr 531	. . . . . . . . . . . . 13 ⊢ (((𝑎 ∈ ℤ[i] ∧ 𝑐 ∈ ℤ[i]) ∧ (𝑏 ∈ ℤ[i] ∧ 𝑑 ∈ ℤ[i])) → 𝑑 ∈ ℤ[i])
19		gzabssqcl 12919	. . . . . . . . . . . . 13 ⊢ (𝑑 ∈ ℤ[i] → ((abs‘𝑑)↑2) ∈ ℕ0)
20	18, 19	syl 14	. . . . . . . . . . . 12 ⊢ (((𝑎 ∈ ℤ[i] ∧ 𝑐 ∈ ℤ[i]) ∧ (𝑏 ∈ ℤ[i] ∧ 𝑑 ∈ ℤ[i])) → ((abs‘𝑑)↑2) ∈ ℕ0)
21	17, 20	nn0addcld 9437	. . . . . . . . . . 11 ⊢ (((𝑎 ∈ ℤ[i] ∧ 𝑐 ∈ ℤ[i]) ∧ (𝑏 ∈ ℤ[i] ∧ 𝑑 ∈ ℤ[i])) → (((abs‘𝑐)↑2) + ((abs‘𝑑)↑2)) ∈ ℕ0)
22	21	nn0cnd 9435	. . . . . . . . . 10 ⊢ (((𝑎 ∈ ℤ[i] ∧ 𝑐 ∈ ℤ[i]) ∧ (𝑏 ∈ ℤ[i] ∧ 𝑑 ∈ ℤ[i])) → (((abs‘𝑐)↑2) + ((abs‘𝑑)↑2)) ∈ ℂ)
23	22	div1d 8938	. . . . . . . . 9 ⊢ (((𝑎 ∈ ℤ[i] ∧ 𝑐 ∈ ℤ[i]) ∧ (𝑏 ∈ ℤ[i] ∧ 𝑑 ∈ ℤ[i])) → ((((abs‘𝑐)↑2) + ((abs‘𝑑)↑2)) / 1) = (((abs‘𝑐)↑2) + ((abs‘𝑑)↑2)))
24	14, 23	oveq12d 6025	. . . . . . . 8 ⊢ (((𝑎 ∈ ℤ[i] ∧ 𝑐 ∈ ℤ[i]) ∧ (𝑏 ∈ ℤ[i] ∧ 𝑑 ∈ ℤ[i])) → (((((abs‘𝑎)↑2) + ((abs‘𝑏)↑2)) / 1) · ((((abs‘𝑐)↑2) + ((abs‘𝑑)↑2)) / 1)) = ((((abs‘𝑎)↑2) + ((abs‘𝑏)↑2)) · (((abs‘𝑐)↑2) + ((abs‘𝑑)↑2))))
25		eqid 2229	. . . . . . . . 9 ⊢ (((abs‘𝑎)↑2) + ((abs‘𝑏)↑2)) = (((abs‘𝑎)↑2) + ((abs‘𝑏)↑2))
26		eqid 2229	. . . . . . . . 9 ⊢ (((abs‘𝑐)↑2) + ((abs‘𝑑)↑2)) = (((abs‘𝑐)↑2) + ((abs‘𝑑)↑2))
27		1nn 9132	. . . . . . . . . 10 ⊢ 1 ∈ ℕ
28	27	a1i 9	. . . . . . . . 9 ⊢ (((𝑎 ∈ ℤ[i] ∧ 𝑐 ∈ ℤ[i]) ∧ (𝑏 ∈ ℤ[i] ∧ 𝑑 ∈ ℤ[i])) → 1 ∈ ℕ)
29		gzsubcl 12918	. . . . . . . . . . . . 13 ⊢ ((𝑎 ∈ ℤ[i] ∧ 𝑐 ∈ ℤ[i]) → (𝑎 − 𝑐) ∈ ℤ[i])
30	29	adantr 276	. . . . . . . . . . . 12 ⊢ (((𝑎 ∈ ℤ[i] ∧ 𝑐 ∈ ℤ[i]) ∧ (𝑏 ∈ ℤ[i] ∧ 𝑑 ∈ ℤ[i])) → (𝑎 − 𝑐) ∈ ℤ[i])
31		gzcn 12910	. . . . . . . . . . . 12 ⊢ ((𝑎 − 𝑐) ∈ ℤ[i] → (𝑎 − 𝑐) ∈ ℂ)
32	30, 31	syl 14	. . . . . . . . . . 11 ⊢ (((𝑎 ∈ ℤ[i] ∧ 𝑐 ∈ ℤ[i]) ∧ (𝑏 ∈ ℤ[i] ∧ 𝑑 ∈ ℤ[i])) → (𝑎 − 𝑐) ∈ ℂ)
33	32	div1d 8938	. . . . . . . . . 10 ⊢ (((𝑎 ∈ ℤ[i] ∧ 𝑐 ∈ ℤ[i]) ∧ (𝑏 ∈ ℤ[i] ∧ 𝑑 ∈ ℤ[i])) → ((𝑎 − 𝑐) / 1) = (𝑎 − 𝑐))
34	33, 30	eqeltrd 2306	. . . . . . . . 9 ⊢ (((𝑎 ∈ ℤ[i] ∧ 𝑐 ∈ ℤ[i]) ∧ (𝑏 ∈ ℤ[i] ∧ 𝑑 ∈ ℤ[i])) → ((𝑎 − 𝑐) / 1) ∈ ℤ[i])
35		gzsubcl 12918	. . . . . . . . . . . . 13 ⊢ ((𝑏 ∈ ℤ[i] ∧ 𝑑 ∈ ℤ[i]) → (𝑏 − 𝑑) ∈ ℤ[i])
36	35	adantl 277	. . . . . . . . . . . 12 ⊢ (((𝑎 ∈ ℤ[i] ∧ 𝑐 ∈ ℤ[i]) ∧ (𝑏 ∈ ℤ[i] ∧ 𝑑 ∈ ℤ[i])) → (𝑏 − 𝑑) ∈ ℤ[i])
37		gzcn 12910	. . . . . . . . . . . 12 ⊢ ((𝑏 − 𝑑) ∈ ℤ[i] → (𝑏 − 𝑑) ∈ ℂ)
38	36, 37	syl 14	. . . . . . . . . . 11 ⊢ (((𝑎 ∈ ℤ[i] ∧ 𝑐 ∈ ℤ[i]) ∧ (𝑏 ∈ ℤ[i] ∧ 𝑑 ∈ ℤ[i])) → (𝑏 − 𝑑) ∈ ℂ)
39	38	div1d 8938	. . . . . . . . . 10 ⊢ (((𝑎 ∈ ℤ[i] ∧ 𝑐 ∈ ℤ[i]) ∧ (𝑏 ∈ ℤ[i] ∧ 𝑑 ∈ ℤ[i])) → ((𝑏 − 𝑑) / 1) = (𝑏 − 𝑑))
40	39, 36	eqeltrd 2306	. . . . . . . . 9 ⊢ (((𝑎 ∈ ℤ[i] ∧ 𝑐 ∈ ℤ[i]) ∧ (𝑏 ∈ ℤ[i] ∧ 𝑑 ∈ ℤ[i])) → ((𝑏 − 𝑑) / 1) ∈ ℤ[i])
41	14, 12	eqeltrd 2306	. . . . . . . . 9 ⊢ (((𝑎 ∈ ℤ[i] ∧ 𝑐 ∈ ℤ[i]) ∧ (𝑏 ∈ ℤ[i] ∧ 𝑑 ∈ ℤ[i])) → ((((abs‘𝑎)↑2) + ((abs‘𝑏)↑2)) / 1) ∈ ℕ0)
42	1, 6, 9, 15, 18, 25, 26, 28, 34, 40, 41	mul4sqlem 12931	. . . . . . . 8 ⊢ (((𝑎 ∈ ℤ[i] ∧ 𝑐 ∈ ℤ[i]) ∧ (𝑏 ∈ ℤ[i] ∧ 𝑑 ∈ ℤ[i])) → (((((abs‘𝑎)↑2) + ((abs‘𝑏)↑2)) / 1) · ((((abs‘𝑐)↑2) + ((abs‘𝑑)↑2)) / 1)) ∈ 𝑆)
43	24, 42	eqeltrrd 2307	. . . . . . 7 ⊢ (((𝑎 ∈ ℤ[i] ∧ 𝑐 ∈ ℤ[i]) ∧ (𝑏 ∈ ℤ[i] ∧ 𝑑 ∈ ℤ[i])) → ((((abs‘𝑎)↑2) + ((abs‘𝑏)↑2)) · (((abs‘𝑐)↑2) + ((abs‘𝑑)↑2))) ∈ 𝑆)
44		oveq12 6016	. . . . . . . 8 ⊢ ((𝐴 = (((abs‘𝑎)↑2) + ((abs‘𝑏)↑2)) ∧ 𝐵 = (((abs‘𝑐)↑2) + ((abs‘𝑑)↑2))) → (𝐴 · 𝐵) = ((((abs‘𝑎)↑2) + ((abs‘𝑏)↑2)) · (((abs‘𝑐)↑2) + ((abs‘𝑑)↑2))))
45	44	eleq1d 2298	. . . . . . 7 ⊢ ((𝐴 = (((abs‘𝑎)↑2) + ((abs‘𝑏)↑2)) ∧ 𝐵 = (((abs‘𝑐)↑2) + ((abs‘𝑑)↑2))) → ((𝐴 · 𝐵) ∈ 𝑆 ↔ ((((abs‘𝑎)↑2) + ((abs‘𝑏)↑2)) · (((abs‘𝑐)↑2) + ((abs‘𝑑)↑2))) ∈ 𝑆))
46	43, 45	syl5ibrcom 157	. . . . . 6 ⊢ (((𝑎 ∈ ℤ[i] ∧ 𝑐 ∈ ℤ[i]) ∧ (𝑏 ∈ ℤ[i] ∧ 𝑑 ∈ ℤ[i])) → ((𝐴 = (((abs‘𝑎)↑2) + ((abs‘𝑏)↑2)) ∧ 𝐵 = (((abs‘𝑐)↑2) + ((abs‘𝑑)↑2))) → (𝐴 · 𝐵) ∈ 𝑆))
47	46	rexlimdvva 2656	. . . . 5 ⊢ ((𝑎 ∈ ℤ[i] ∧ 𝑐 ∈ ℤ[i]) → (∃𝑏 ∈ ℤ[i] ∃𝑑 ∈ ℤ[i] (𝐴 = (((abs‘𝑎)↑2) + ((abs‘𝑏)↑2)) ∧ 𝐵 = (((abs‘𝑐)↑2) + ((abs‘𝑑)↑2))) → (𝐴 · 𝐵) ∈ 𝑆))
48	5, 47	biimtrrid 153	. . . 4 ⊢ ((𝑎 ∈ ℤ[i] ∧ 𝑐 ∈ ℤ[i]) → ((∃𝑏 ∈ ℤ[i] 𝐴 = (((abs‘𝑎)↑2) + ((abs‘𝑏)↑2)) ∧ ∃𝑑 ∈ ℤ[i] 𝐵 = (((abs‘𝑐)↑2) + ((abs‘𝑑)↑2))) → (𝐴 · 𝐵) ∈ 𝑆))
49	48	rexlimivv 2654	. . 3 ⊢ (∃𝑎 ∈ ℤ[i] ∃𝑐 ∈ ℤ[i] (∃𝑏 ∈ ℤ[i] 𝐴 = (((abs‘𝑎)↑2) + ((abs‘𝑏)↑2)) ∧ ∃𝑑 ∈ ℤ[i] 𝐵 = (((abs‘𝑐)↑2) + ((abs‘𝑑)↑2))) → (𝐴 · 𝐵) ∈ 𝑆)
50	4, 49	sylbir 135	. 2 ⊢ ((∃𝑎 ∈ ℤ[i] ∃𝑏 ∈ ℤ[i] 𝐴 = (((abs‘𝑎)↑2) + ((abs‘𝑏)↑2)) ∧ ∃𝑐 ∈ ℤ[i] ∃𝑑 ∈ ℤ[i] 𝐵 = (((abs‘𝑐)↑2) + ((abs‘𝑑)↑2))) → (𝐴 · 𝐵) ∈ 𝑆)
51	2, 3, 50	syl2anb 291	1 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴 · 𝐵) ∈ 𝑆)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1395   ∈ wcel 2200  {cab 2215  ∃wrex 2509  ‘cfv 5318  (class class class)co 6007  ℂcc 8008  1c1 8011   + caddc 8013   · cmul 8015   − cmin 8328   / cdiv 8830  ℕcn 9121  2c2 9172  ℕ₀cn0 9380  ℤcz 9457  ↑cexp 10772  abscabs 11523  ℤ[i]cgz 12907

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680  ax-cnex 8101  ax-resscn 8102  ax-1cn 8103  ax-1re 8104  ax-icn 8105  ax-addcl 8106  ax-addrcl 8107  ax-mulcl 8108  ax-mulrcl 8109  ax-addcom 8110  ax-mulcom 8111  ax-addass 8112  ax-mulass 8113  ax-distr 8114  ax-i2m1 8115  ax-0lt1 8116  ax-1rid 8117  ax-0id 8118  ax-rnegex 8119  ax-precex 8120  ax-cnre 8121  ax-pre-ltirr 8122  ax-pre-ltwlin 8123  ax-pre-lttrn 8124  ax-pre-apti 8125  ax-pre-ltadd 8126  ax-pre-mulgt0 8127  ax-pre-mulext 8128  ax-arch 8129  ax-caucvg 8130

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-po 4387  df-iso 4388  df-iord 4457  df-on 4459  df-ilim 4460  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-riota 5960  df-ov 6010  df-oprab 6011  df-mpo 6012  df-1st 6292  df-2nd 6293  df-recs 6457  df-frec 6543  df-pnf 8194  df-mnf 8195  df-xr 8196  df-ltxr 8197  df-le 8198  df-sub 8330  df-neg 8331  df-reap 8733  df-ap 8740  df-div 8831  df-inn 9122  df-2 9180  df-3 9181  df-4 9182  df-n0 9381  df-z 9458  df-uz 9734  df-rp 9862  df-seqfrec 10682  df-exp 10773  df-cj 11368  df-re 11369  df-im 11370  df-rsqrt 11524  df-abs 11525  df-gz 12908

This theorem is referenced by:  4sqlem19  12947