Theorem mul4sq 12429

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 12428. (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 12427	. 2 ⊢ (𝐴 ∈ 𝑆 ↔ ∃𝑎 ∈ ℤ[i] ∃𝑏 ∈ ℤ[i] 𝐴 = (((abs‘𝑎)↑2) + ((abs‘𝑏)↑2)))
3	1	4sqlem4 12427	. 2 ⊢ (𝐵 ∈ 𝑆 ↔ ∃𝑐 ∈ ℤ[i] ∃𝑑 ∈ ℤ[i] 𝐵 = (((abs‘𝑐)↑2) + ((abs‘𝑑)↑2)))
4		reeanv 2660	. . 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 2660	. . . . 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 12416	. . . . . . . . . . . . 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 12416	. . . . . . . . . . . . 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 9264	. . . . . . . . . . 11 ⊢ (((𝑎 ∈ ℤ[i] ∧ 𝑐 ∈ ℤ[i]) ∧ (𝑏 ∈ ℤ[i] ∧ 𝑑 ∈ ℤ[i])) → (((abs‘𝑎)↑2) + ((abs‘𝑏)↑2)) ∈ ℕ0)
13	12	nn0cnd 9262	. . . . . . . . . 10 ⊢ (((𝑎 ∈ ℤ[i] ∧ 𝑐 ∈ ℤ[i]) ∧ (𝑏 ∈ ℤ[i] ∧ 𝑑 ∈ ℤ[i])) → (((abs‘𝑎)↑2) + ((abs‘𝑏)↑2)) ∈ ℂ)
14	13	div1d 8768	. . . . . . . . 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 12416	. . . . . . . . . . . . 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 12416	. . . . . . . . . . . . 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 9264	. . . . . . . . . . 11 ⊢ (((𝑎 ∈ ℤ[i] ∧ 𝑐 ∈ ℤ[i]) ∧ (𝑏 ∈ ℤ[i] ∧ 𝑑 ∈ ℤ[i])) → (((abs‘𝑐)↑2) + ((abs‘𝑑)↑2)) ∈ ℕ0)
22	21	nn0cnd 9262	. . . . . . . . . 10 ⊢ (((𝑎 ∈ ℤ[i] ∧ 𝑐 ∈ ℤ[i]) ∧ (𝑏 ∈ ℤ[i] ∧ 𝑑 ∈ ℤ[i])) → (((abs‘𝑐)↑2) + ((abs‘𝑑)↑2)) ∈ ℂ)
23	22	div1d 8768	. . . . . . . . 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 5915	. . . . . . . 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 2189	. . . . . . . . 9 ⊢ (((abs‘𝑎)↑2) + ((abs‘𝑏)↑2)) = (((abs‘𝑎)↑2) + ((abs‘𝑏)↑2))
26		eqid 2189	. . . . . . . . 9 ⊢ (((abs‘𝑐)↑2) + ((abs‘𝑑)↑2)) = (((abs‘𝑐)↑2) + ((abs‘𝑑)↑2))
27		1nn 8961	. . . . . . . . . 10 ⊢ 1 ∈ ℕ
28	27	a1i 9	. . . . . . . . 9 ⊢ (((𝑎 ∈ ℤ[i] ∧ 𝑐 ∈ ℤ[i]) ∧ (𝑏 ∈ ℤ[i] ∧ 𝑑 ∈ ℤ[i])) → 1 ∈ ℕ)
29		gzsubcl 12415	. . . . . . . . . . . . 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 12407	. . . . . . . . . . . 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 8768	. . . . . . . . . 10 ⊢ (((𝑎 ∈ ℤ[i] ∧ 𝑐 ∈ ℤ[i]) ∧ (𝑏 ∈ ℤ[i] ∧ 𝑑 ∈ ℤ[i])) → ((𝑎 − 𝑐) / 1) = (𝑎 − 𝑐))
34	33, 30	eqeltrd 2266	. . . . . . . . 9 ⊢ (((𝑎 ∈ ℤ[i] ∧ 𝑐 ∈ ℤ[i]) ∧ (𝑏 ∈ ℤ[i] ∧ 𝑑 ∈ ℤ[i])) → ((𝑎 − 𝑐) / 1) ∈ ℤ[i])
35		gzsubcl 12415	. . . . . . . . . . . . 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 12407	. . . . . . . . . . . 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 8768	. . . . . . . . . 10 ⊢ (((𝑎 ∈ ℤ[i] ∧ 𝑐 ∈ ℤ[i]) ∧ (𝑏 ∈ ℤ[i] ∧ 𝑑 ∈ ℤ[i])) → ((𝑏 − 𝑑) / 1) = (𝑏 − 𝑑))
40	39, 36	eqeltrd 2266	. . . . . . . . 9 ⊢ (((𝑎 ∈ ℤ[i] ∧ 𝑐 ∈ ℤ[i]) ∧ (𝑏 ∈ ℤ[i] ∧ 𝑑 ∈ ℤ[i])) → ((𝑏 − 𝑑) / 1) ∈ ℤ[i])
41	14, 12	eqeltrd 2266	. . . . . . . . 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 12428	. . . . . . . 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 2267	. . . . . . 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 5906	. . . . . . . 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 2258	. . . . . . 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 2615	. . . . 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 2613	. . 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 2160  {cab 2175  ∃wrex 2469  ‘cfv 5235  (class class class)co 5897  ℂcc 7840  1c1 7843   + caddc 7845   · cmul 7847   − cmin 8159   / cdiv 8660  ℕcn 8950  2c2 9001  ℕ₀cn0 9207  ℤcz 9284  ↑cexp 10553  abscabs 11041  ℤ[i]cgz 12404

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-iinf 4605  ax-cnex 7933  ax-resscn 7934  ax-1cn 7935  ax-1re 7936  ax-icn 7937  ax-addcl 7938  ax-addrcl 7939  ax-mulcl 7940  ax-mulrcl 7941  ax-addcom 7942  ax-mulcom 7943  ax-addass 7944  ax-mulass 7945  ax-distr 7946  ax-i2m1 7947  ax-0lt1 7948  ax-1rid 7949  ax-0id 7950  ax-rnegex 7951  ax-precex 7952  ax-cnre 7953  ax-pre-ltirr 7954  ax-pre-ltwlin 7955  ax-pre-lttrn 7956  ax-pre-apti 7957  ax-pre-ltadd 7958  ax-pre-mulgt0 7959  ax-pre-mulext 7960  ax-arch 7961  ax-caucvg 7962

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 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-if 3550  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4311  df-po 4314  df-iso 4315  df-iord 4384  df-on 4386  df-ilim 4387  df-suc 4389  df-iom 4608  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-riota 5852  df-ov 5900  df-oprab 5901  df-mpo 5902  df-1st 6166  df-2nd 6167  df-recs 6331  df-frec 6417  df-pnf 8025  df-mnf 8026  df-xr 8027  df-ltxr 8028  df-le 8029  df-sub 8161  df-neg 8162  df-reap 8563  df-ap 8570  df-div 8661  df-inn 8951  df-2 9009  df-3 9010  df-4 9011  df-n0 9208  df-z 9285  df-uz 9560  df-rp 9686  df-seqfrec 10479  df-exp 10554  df-cj 10886  df-re 10887  df-im 10888  df-rsqrt 11042  df-abs 11043  df-gz 12405

This theorem is referenced by:  4sqlem19  12444