Theorem mul4sq 17147

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 17146. (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 17145	. 2 ⊢ (𝐴 ∈ 𝑆 ↔ ∃𝑎 ∈ ℤ[i] ∃𝑏 ∈ ℤ[i] 𝐴 = (((abs‘𝑎)↑2) + ((abs‘𝑏)↑2)))
3	1	4sqlem4 17145	. 2 ⊢ (𝐵 ∈ 𝑆 ↔ ∃𝑐 ∈ ℤ[i] ∃𝑑 ∈ ℤ[i] 𝐵 = (((abs‘𝑐)↑2) + ((abs‘𝑑)↑2)))
4		reeanv 3238	. . 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 3238	. . . . 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 779	. . . . . . . . . . . . 13 ⊢ (((𝑎 ∈ ℤ[i] ∧ 𝑐 ∈ ℤ[i]) ∧ (𝑏 ∈ ℤ[i] ∧ 𝑑 ∈ ℤ[i])) → 𝑎 ∈ ℤ[i])
7		gzabssqcl 17134	. . . . . . . . . . . . 13 ⊢ (𝑎 ∈ ℤ[i] → ((abs‘𝑎)↑2) ∈ ℕ0)
8	6, 7	syl 18	. . . . . . . . . . . 12 ⊢ (((𝑎 ∈ ℤ[i] ∧ 𝑐 ∈ ℤ[i]) ∧ (𝑏 ∈ ℤ[i] ∧ 𝑑 ∈ ℤ[i])) → ((abs‘𝑎)↑2) ∈ ℕ0)
9		simprl 783	. . . . . . . . . . . . 13 ⊢ (((𝑎 ∈ ℤ[i] ∧ 𝑐 ∈ ℤ[i]) ∧ (𝑏 ∈ ℤ[i] ∧ 𝑑 ∈ ℤ[i])) → 𝑏 ∈ ℤ[i])
10		gzabssqcl 17134	. . . . . . . . . . . . 13 ⊢ (𝑏 ∈ ℤ[i] → ((abs‘𝑏)↑2) ∈ ℕ0)
11	9, 10	syl 18	. . . . . . . . . . . 12 ⊢ (((𝑎 ∈ ℤ[i] ∧ 𝑐 ∈ ℤ[i]) ∧ (𝑏 ∈ ℤ[i] ∧ 𝑑 ∈ ℤ[i])) → ((abs‘𝑏)↑2) ∈ ℕ0)
12	8, 11	nn0addcld 12684	. . . . . . . . . . 11 ⊢ (((𝑎 ∈ ℤ[i] ∧ 𝑐 ∈ ℤ[i]) ∧ (𝑏 ∈ ℤ[i] ∧ 𝑑 ∈ ℤ[i])) → (((abs‘𝑎)↑2) + ((abs‘𝑏)↑2)) ∈ ℕ0)
13	12	nn0cnd 12682	. . . . . . . . . 10 ⊢ (((𝑎 ∈ ℤ[i] ∧ 𝑐 ∈ ℤ[i]) ∧ (𝑏 ∈ ℤ[i] ∧ 𝑑 ∈ ℤ[i])) → (((abs‘𝑎)↑2) + ((abs‘𝑏)↑2)) ∈ ℂ)
14	13	div1d 12098	. . . . . . . . 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 781	. . . . . . . . . . . . 13 ⊢ (((𝑎 ∈ ℤ[i] ∧ 𝑐 ∈ ℤ[i]) ∧ (𝑏 ∈ ℤ[i] ∧ 𝑑 ∈ ℤ[i])) → 𝑐 ∈ ℤ[i])
16		gzabssqcl 17134	. . . . . . . . . . . . 13 ⊢ (𝑐 ∈ ℤ[i] → ((abs‘𝑐)↑2) ∈ ℕ0)
17	15, 16	syl 18	. . . . . . . . . . . 12 ⊢ (((𝑎 ∈ ℤ[i] ∧ 𝑐 ∈ ℤ[i]) ∧ (𝑏 ∈ ℤ[i] ∧ 𝑑 ∈ ℤ[i])) → ((abs‘𝑐)↑2) ∈ ℕ0)
18		simprr 785	. . . . . . . . . . . . 13 ⊢ (((𝑎 ∈ ℤ[i] ∧ 𝑐 ∈ ℤ[i]) ∧ (𝑏 ∈ ℤ[i] ∧ 𝑑 ∈ ℤ[i])) → 𝑑 ∈ ℤ[i])
19		gzabssqcl 17134	. . . . . . . . . . . . 13 ⊢ (𝑑 ∈ ℤ[i] → ((abs‘𝑑)↑2) ∈ ℕ0)
20	18, 19	syl 18	. . . . . . . . . . . 12 ⊢ (((𝑎 ∈ ℤ[i] ∧ 𝑐 ∈ ℤ[i]) ∧ (𝑏 ∈ ℤ[i] ∧ 𝑑 ∈ ℤ[i])) → ((abs‘𝑑)↑2) ∈ ℕ0)
21	17, 20	nn0addcld 12684	. . . . . . . . . . 11 ⊢ (((𝑎 ∈ ℤ[i] ∧ 𝑐 ∈ ℤ[i]) ∧ (𝑏 ∈ ℤ[i] ∧ 𝑑 ∈ ℤ[i])) → (((abs‘𝑐)↑2) + ((abs‘𝑑)↑2)) ∈ ℕ0)
22	21	nn0cnd 12682	. . . . . . . . . 10 ⊢ (((𝑎 ∈ ℤ[i] ∧ 𝑐 ∈ ℤ[i]) ∧ (𝑏 ∈ ℤ[i] ∧ 𝑑 ∈ ℤ[i])) → (((abs‘𝑐)↑2) + ((abs‘𝑑)↑2)) ∈ ℂ)
23	22	div1d 12098	. . . . . . . . 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 7448	. . . . . . . 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 2763	. . . . . . . . 9 ⊢ (((abs‘𝑎)↑2) + ((abs‘𝑏)↑2)) = (((abs‘𝑎)↑2) + ((abs‘𝑏)↑2))
26		eqid 2763	. . . . . . . . 9 ⊢ (((abs‘𝑐)↑2) + ((abs‘𝑑)↑2)) = (((abs‘𝑐)↑2) + ((abs‘𝑑)↑2))
27		1nn 12359	. . . . . . . . . 10 ⊢ 1 ∈ ℕ
28	27	a1i 11	. . . . . . . . 9 ⊢ (((𝑎 ∈ ℤ[i] ∧ 𝑐 ∈ ℤ[i]) ∧ (𝑏 ∈ ℤ[i] ∧ 𝑑 ∈ ℤ[i])) → 1 ∈ ℕ)
29		gzsubcl 17133	. . . . . . . . . . . . 13 ⊢ ((𝑎 ∈ ℤ[i] ∧ 𝑐 ∈ ℤ[i]) → (𝑎 − 𝑐) ∈ ℤ[i])
30	29	adantr 486	. . . . . . . . . . . 12 ⊢ (((𝑎 ∈ ℤ[i] ∧ 𝑐 ∈ ℤ[i]) ∧ (𝑏 ∈ ℤ[i] ∧ 𝑑 ∈ ℤ[i])) → (𝑎 − 𝑐) ∈ ℤ[i])
31		gzcn 17125	. . . . . . . . . . . 12 ⊢ ((𝑎 − 𝑐) ∈ ℤ[i] → (𝑎 − 𝑐) ∈ ℂ)
32	30, 31	syl 18	. . . . . . . . . . 11 ⊢ (((𝑎 ∈ ℤ[i] ∧ 𝑐 ∈ ℤ[i]) ∧ (𝑏 ∈ ℤ[i] ∧ 𝑑 ∈ ℤ[i])) → (𝑎 − 𝑐) ∈ ℂ)
33	32	div1d 12098	. . . . . . . . . 10 ⊢ (((𝑎 ∈ ℤ[i] ∧ 𝑐 ∈ ℤ[i]) ∧ (𝑏 ∈ ℤ[i] ∧ 𝑑 ∈ ℤ[i])) → ((𝑎 − 𝑐) / 1) = (𝑎 − 𝑐))
34	33, 30	eqeltrd 2864	. . . . . . . . 9 ⊢ (((𝑎 ∈ ℤ[i] ∧ 𝑐 ∈ ℤ[i]) ∧ (𝑏 ∈ ℤ[i] ∧ 𝑑 ∈ ℤ[i])) → ((𝑎 − 𝑐) / 1) ∈ ℤ[i])
35		gzsubcl 17133	. . . . . . . . . . . . 13 ⊢ ((𝑏 ∈ ℤ[i] ∧ 𝑑 ∈ ℤ[i]) → (𝑏 − 𝑑) ∈ ℤ[i])
36	35	adantl 487	. . . . . . . . . . . 12 ⊢ (((𝑎 ∈ ℤ[i] ∧ 𝑐 ∈ ℤ[i]) ∧ (𝑏 ∈ ℤ[i] ∧ 𝑑 ∈ ℤ[i])) → (𝑏 − 𝑑) ∈ ℤ[i])
37		gzcn 17125	. . . . . . . . . . . 12 ⊢ ((𝑏 − 𝑑) ∈ ℤ[i] → (𝑏 − 𝑑) ∈ ℂ)
38	36, 37	syl 18	. . . . . . . . . . 11 ⊢ (((𝑎 ∈ ℤ[i] ∧ 𝑐 ∈ ℤ[i]) ∧ (𝑏 ∈ ℤ[i] ∧ 𝑑 ∈ ℤ[i])) → (𝑏 − 𝑑) ∈ ℂ)
39	38	div1d 12098	. . . . . . . . . 10 ⊢ (((𝑎 ∈ ℤ[i] ∧ 𝑐 ∈ ℤ[i]) ∧ (𝑏 ∈ ℤ[i] ∧ 𝑑 ∈ ℤ[i])) → ((𝑏 − 𝑑) / 1) = (𝑏 − 𝑑))
40	39, 36	eqeltrd 2864	. . . . . . . . 9 ⊢ (((𝑎 ∈ ℤ[i] ∧ 𝑐 ∈ ℤ[i]) ∧ (𝑏 ∈ ℤ[i] ∧ 𝑑 ∈ ℤ[i])) → ((𝑏 − 𝑑) / 1) ∈ ℤ[i])
41	14, 12	eqeltrd 2864	. . . . . . . . 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 17146	. . . . . . . 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 2865	. . . . . . 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 7439	. . . . . . . 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 2849	. . . . . . 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 250	. . . . . 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 3223	. . . . 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 246	. . . 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 3208	. . 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 238	. 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 610	1 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴 · 𝐵) ∈ 𝑆)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2146  {cab 2741  ∃wrex 3090  ‘cfv 6547  (class class class)co 7430  ℂcc 11211  1c1 11214   + caddc 11216   · cmul 11218   − cmin 11554   / cdiv 11986  ℕcn 12348  2c2 12410  ℕ₀cn0 12619  ℤcz 12706  ↑cexp 14217  abscabs 15416  ℤ[i]cgz 17122

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7761  ax-cnex 11269  ax-resscn 11270  ax-1cn 11271  ax-icn 11272  ax-addcl 11273  ax-addrcl 11274  ax-mulcl 11275  ax-mulrcl 11276  ax-mulcom 11277  ax-addass 11278  ax-mulass 11279  ax-distr 11280  ax-i2m1 11281  ax-1ne0 11282  ax-1rid 11283  ax-rnegex 11284  ax-rrecex 11285  ax-cnre 11286  ax-pre-lttri 11287  ax-pre-lttrn 11288  ax-pre-ltadd 11289  ax-pre-mulgt0 11290  ax-pre-sup 11291

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6311  df-ord 6372  df-on 6373  df-lim 6374  df-suc 6375  df-iota 6501  df-fun 6549  df-fn 6550  df-f 6551  df-f1 6552  df-fo 6553  df-f1o 6554  df-fv 6555  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7888  df-2nd 8012  df-frecs 8313  df-wrecs 8344  df-recs 8393  df-rdg 8432  df-er 8731  df-en 8988  df-dom 8989  df-sdom 8990  df-sup 9447  df-pnf 11358  df-mnf 11359  df-xr 11360  df-ltxr 11361  df-le 11362  df-sub 11556  df-neg 11557  df-div 11987  df-nn 12349  df-2 12418  df-3 12419  df-n0 12620  df-z 12707  df-uz 12979  df-rp 13134  df-seq 14158  df-exp 14218  df-cj 15281  df-re 15282  df-im 15283  df-sqrt 15417  df-abs 15418  df-gz 17123

This theorem is used by:  4sqlem19  17156