Theorem mul4sq 12563

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 12562. (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 12561	. 2 ⊢ (𝐴 ∈ 𝑆 ↔ ∃𝑎 ∈ ℤ[i] ∃𝑏 ∈ ℤ[i] 𝐴 = (((abs‘𝑎)↑2) + ((abs‘𝑏)↑2)))
3	1	4sqlem4 12561	. 2 ⊢ (𝐵 ∈ 𝑆 ↔ ∃𝑐 ∈ ℤ[i] ∃𝑑 ∈ ℤ[i] 𝐵 = (((abs‘𝑐)↑2) + ((abs‘𝑑)↑2)))
4		reeanv 2667	. . 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 2667	. . . . 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 12550	. . . . . . . . . . . . 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 12550	. . . . . . . . . . . . 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 9306	. . . . . . . . . . 11 ⊢ (((𝑎 ∈ ℤ[i] ∧ 𝑐 ∈ ℤ[i]) ∧ (𝑏 ∈ ℤ[i] ∧ 𝑑 ∈ ℤ[i])) → (((abs‘𝑎)↑2) + ((abs‘𝑏)↑2)) ∈ ℕ0)
13	12	nn0cnd 9304	. . . . . . . . . 10 ⊢ (((𝑎 ∈ ℤ[i] ∧ 𝑐 ∈ ℤ[i]) ∧ (𝑏 ∈ ℤ[i] ∧ 𝑑 ∈ ℤ[i])) → (((abs‘𝑎)↑2) + ((abs‘𝑏)↑2)) ∈ ℂ)
14	13	div1d 8807	. . . . . . . . 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 12550	. . . . . . . . . . . . 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 12550	. . . . . . . . . . . . 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 9306	. . . . . . . . . . 11 ⊢ (((𝑎 ∈ ℤ[i] ∧ 𝑐 ∈ ℤ[i]) ∧ (𝑏 ∈ ℤ[i] ∧ 𝑑 ∈ ℤ[i])) → (((abs‘𝑐)↑2) + ((abs‘𝑑)↑2)) ∈ ℕ0)
22	21	nn0cnd 9304	. . . . . . . . . 10 ⊢ (((𝑎 ∈ ℤ[i] ∧ 𝑐 ∈ ℤ[i]) ∧ (𝑏 ∈ ℤ[i] ∧ 𝑑 ∈ ℤ[i])) → (((abs‘𝑐)↑2) + ((abs‘𝑑)↑2)) ∈ ℂ)
23	22	div1d 8807	. . . . . . . . 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 5940	. . . . . . . 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 2196	. . . . . . . . 9 ⊢ (((abs‘𝑎)↑2) + ((abs‘𝑏)↑2)) = (((abs‘𝑎)↑2) + ((abs‘𝑏)↑2))
26		eqid 2196	. . . . . . . . 9 ⊢ (((abs‘𝑐)↑2) + ((abs‘𝑑)↑2)) = (((abs‘𝑐)↑2) + ((abs‘𝑑)↑2))
27		1nn 9001	. . . . . . . . . 10 ⊢ 1 ∈ ℕ
28	27	a1i 9	. . . . . . . . 9 ⊢ (((𝑎 ∈ ℤ[i] ∧ 𝑐 ∈ ℤ[i]) ∧ (𝑏 ∈ ℤ[i] ∧ 𝑑 ∈ ℤ[i])) → 1 ∈ ℕ)
29		gzsubcl 12549	. . . . . . . . . . . . 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 12541	. . . . . . . . . . . 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 8807	. . . . . . . . . 10 ⊢ (((𝑎 ∈ ℤ[i] ∧ 𝑐 ∈ ℤ[i]) ∧ (𝑏 ∈ ℤ[i] ∧ 𝑑 ∈ ℤ[i])) → ((𝑎 − 𝑐) / 1) = (𝑎 − 𝑐))
34	33, 30	eqeltrd 2273	. . . . . . . . 9 ⊢ (((𝑎 ∈ ℤ[i] ∧ 𝑐 ∈ ℤ[i]) ∧ (𝑏 ∈ ℤ[i] ∧ 𝑑 ∈ ℤ[i])) → ((𝑎 − 𝑐) / 1) ∈ ℤ[i])
35		gzsubcl 12549	. . . . . . . . . . . . 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 12541	. . . . . . . . . . . 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 8807	. . . . . . . . . 10 ⊢ (((𝑎 ∈ ℤ[i] ∧ 𝑐 ∈ ℤ[i]) ∧ (𝑏 ∈ ℤ[i] ∧ 𝑑 ∈ ℤ[i])) → ((𝑏 − 𝑑) / 1) = (𝑏 − 𝑑))
40	39, 36	eqeltrd 2273	. . . . . . . . 9 ⊢ (((𝑎 ∈ ℤ[i] ∧ 𝑐 ∈ ℤ[i]) ∧ (𝑏 ∈ ℤ[i] ∧ 𝑑 ∈ ℤ[i])) → ((𝑏 − 𝑑) / 1) ∈ ℤ[i])
41	14, 12	eqeltrd 2273	. . . . . . . . 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 12562	. . . . . . . 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 2274	. . . . . . 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 5931	. . . . . . . 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 2265	. . . . . . 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 2622	. . . . 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 2620	. . 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 1364   ∈ wcel 2167  {cab 2182  ∃wrex 2476  ‘cfv 5258  (class class class)co 5922  ℂcc 7877  1c1 7880   + caddc 7882   · cmul 7884   − cmin 8197   / cdiv 8699  ℕcn 8990  2c2 9041  ℕ₀cn0 9249  ℤcz 9326  ↑cexp 10630  abscabs 11162  ℤ[i]cgz 12538

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-iinf 4624  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-mulrcl 7978  ax-addcom 7979  ax-mulcom 7980  ax-addass 7981  ax-mulass 7982  ax-distr 7983  ax-i2m1 7984  ax-0lt1 7985  ax-1rid 7986  ax-0id 7987  ax-rnegex 7988  ax-precex 7989  ax-cnre 7990  ax-pre-ltirr 7991  ax-pre-ltwlin 7992  ax-pre-lttrn 7993  ax-pre-apti 7994  ax-pre-ltadd 7995  ax-pre-mulgt0 7996  ax-pre-mulext 7997  ax-arch 7998  ax-caucvg 7999

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 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-if 3562  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-id 4328  df-po 4331  df-iso 4332  df-iord 4401  df-on 4403  df-ilim 4404  df-suc 4406  df-iom 4627  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-recs 6363  df-frec 6449  df-pnf 8063  df-mnf 8064  df-xr 8065  df-ltxr 8066  df-le 8067  df-sub 8199  df-neg 8200  df-reap 8602  df-ap 8609  df-div 8700  df-inn 8991  df-2 9049  df-3 9050  df-4 9051  df-n0 9250  df-z 9327  df-uz 9602  df-rp 9729  df-seqfrec 10540  df-exp 10631  df-cj 11007  df-re 11008  df-im 11009  df-rsqrt 11163  df-abs 11164  df-gz 12539

This theorem is referenced by:  4sqlem19  12578