Theorem mul4sq 12972

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 12971. (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 12970	. 2 ⊢ (𝐴 ∈ 𝑆 ↔ ∃𝑎 ∈ ℤ[i] ∃𝑏 ∈ ℤ[i] 𝐴 = (((abs‘𝑎)↑2) + ((abs‘𝑏)↑2)))
3	1	4sqlem4 12970	. 2 ⊢ (𝐵 ∈ 𝑆 ↔ ∃𝑐 ∈ ℤ[i] ∃𝑑 ∈ ℤ[i] 𝐵 = (((abs‘𝑐)↑2) + ((abs‘𝑑)↑2)))
4		reeanv 2703	. . 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 2703	. . . . 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 12959	. . . . . . . . . . . . 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 531	. . . . . . . . . . . . 13 ⊢ (((𝑎 ∈ ℤ[i] ∧ 𝑐 ∈ ℤ[i]) ∧ (𝑏 ∈ ℤ[i] ∧ 𝑑 ∈ ℤ[i])) → 𝑏 ∈ ℤ[i])
10		gzabssqcl 12959	. . . . . . . . . . . . 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 9459	. . . . . . . . . . 11 ⊢ (((𝑎 ∈ ℤ[i] ∧ 𝑐 ∈ ℤ[i]) ∧ (𝑏 ∈ ℤ[i] ∧ 𝑑 ∈ ℤ[i])) → (((abs‘𝑎)↑2) + ((abs‘𝑏)↑2)) ∈ ℕ0)
13	12	nn0cnd 9457	. . . . . . . . . 10 ⊢ (((𝑎 ∈ ℤ[i] ∧ 𝑐 ∈ ℤ[i]) ∧ (𝑏 ∈ ℤ[i] ∧ 𝑑 ∈ ℤ[i])) → (((abs‘𝑎)↑2) + ((abs‘𝑏)↑2)) ∈ ℂ)
14	13	div1d 8960	. . . . . . . . 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 529	. . . . . . . . . . . . 13 ⊢ (((𝑎 ∈ ℤ[i] ∧ 𝑐 ∈ ℤ[i]) ∧ (𝑏 ∈ ℤ[i] ∧ 𝑑 ∈ ℤ[i])) → 𝑐 ∈ ℤ[i])
16		gzabssqcl 12959	. . . . . . . . . . . . 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 533	. . . . . . . . . . . . 13 ⊢ (((𝑎 ∈ ℤ[i] ∧ 𝑐 ∈ ℤ[i]) ∧ (𝑏 ∈ ℤ[i] ∧ 𝑑 ∈ ℤ[i])) → 𝑑 ∈ ℤ[i])
19		gzabssqcl 12959	. . . . . . . . . . . . 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 9459	. . . . . . . . . . 11 ⊢ (((𝑎 ∈ ℤ[i] ∧ 𝑐 ∈ ℤ[i]) ∧ (𝑏 ∈ ℤ[i] ∧ 𝑑 ∈ ℤ[i])) → (((abs‘𝑐)↑2) + ((abs‘𝑑)↑2)) ∈ ℕ0)
22	21	nn0cnd 9457	. . . . . . . . . 10 ⊢ (((𝑎 ∈ ℤ[i] ∧ 𝑐 ∈ ℤ[i]) ∧ (𝑏 ∈ ℤ[i] ∧ 𝑑 ∈ ℤ[i])) → (((abs‘𝑐)↑2) + ((abs‘𝑑)↑2)) ∈ ℂ)
23	22	div1d 8960	. . . . . . . . 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 6036	. . . . . . . 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 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 9154	. . . . . . . . . 10 ⊢ 1 ∈ ℕ
28	27	a1i 9	. . . . . . . . 9 ⊢ (((𝑎 ∈ ℤ[i] ∧ 𝑐 ∈ ℤ[i]) ∧ (𝑏 ∈ ℤ[i] ∧ 𝑑 ∈ ℤ[i])) → 1 ∈ ℕ)
29		gzsubcl 12958	. . . . . . . . . . . . 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 12950	. . . . . . . . . . . 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 8960	. . . . . . . . . 10 ⊢ (((𝑎 ∈ ℤ[i] ∧ 𝑐 ∈ ℤ[i]) ∧ (𝑏 ∈ ℤ[i] ∧ 𝑑 ∈ ℤ[i])) → ((𝑎 − 𝑐) / 1) = (𝑎 − 𝑐))
34	33, 30	eqeltrd 2308	. . . . . . . . 9 ⊢ (((𝑎 ∈ ℤ[i] ∧ 𝑐 ∈ ℤ[i]) ∧ (𝑏 ∈ ℤ[i] ∧ 𝑑 ∈ ℤ[i])) → ((𝑎 − 𝑐) / 1) ∈ ℤ[i])
35		gzsubcl 12958	. . . . . . . . . . . . 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 12950	. . . . . . . . . . . 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 8960	. . . . . . . . . 10 ⊢ (((𝑎 ∈ ℤ[i] ∧ 𝑐 ∈ ℤ[i]) ∧ (𝑏 ∈ ℤ[i] ∧ 𝑑 ∈ ℤ[i])) → ((𝑏 − 𝑑) / 1) = (𝑏 − 𝑑))
40	39, 36	eqeltrd 2308	. . . . . . . . 9 ⊢ (((𝑎 ∈ ℤ[i] ∧ 𝑐 ∈ ℤ[i]) ∧ (𝑏 ∈ ℤ[i] ∧ 𝑑 ∈ ℤ[i])) → ((𝑏 − 𝑑) / 1) ∈ ℤ[i])
41	14, 12	eqeltrd 2308	. . . . . . . . 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 12971	. . . . . . . 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 2309	. . . . . . 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 6027	. . . . . . . 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 2300	. . . . . . 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 2658	. . . . 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 2656	. . 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 1397   ∈ wcel 2202  {cab 2217  ∃wrex 2511  ‘cfv 5326  (class class class)co 6018  ℂcc 8030  1c1 8033   + caddc 8035   · cmul 8037   − cmin 8350   / cdiv 8852  ℕcn 9143  2c2 9194  ℕ₀cn0 9402  ℤcz 9479  ↑cexp 10801  abscabs 11562  ℤ[i]cgz 12947

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-mulrcl 8131  ax-addcom 8132  ax-mulcom 8133  ax-addass 8134  ax-mulass 8135  ax-distr 8136  ax-i2m1 8137  ax-0lt1 8138  ax-1rid 8139  ax-0id 8140  ax-rnegex 8141  ax-precex 8142  ax-cnre 8143  ax-pre-ltirr 8144  ax-pre-ltwlin 8145  ax-pre-lttrn 8146  ax-pre-apti 8147  ax-pre-ltadd 8148  ax-pre-mulgt0 8149  ax-pre-mulext 8150  ax-arch 8151  ax-caucvg 8152

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-po 4393  df-iso 4394  df-iord 4463  df-on 4465  df-ilim 4466  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1st 6303  df-2nd 6304  df-recs 6471  df-frec 6557  df-pnf 8216  df-mnf 8217  df-xr 8218  df-ltxr 8219  df-le 8220  df-sub 8352  df-neg 8353  df-reap 8755  df-ap 8762  df-div 8853  df-inn 9144  df-2 9202  df-3 9203  df-4 9204  df-n0 9403  df-z 9480  df-uz 9756  df-rp 9889  df-seqfrec 10711  df-exp 10802  df-cj 11407  df-re 11408  df-im 11409  df-rsqrt 11563  df-abs 11564  df-gz 12948

This theorem is referenced by:  4sqlem19  12987