Theorem 4sqlem2 16821

Description: Lemma for 4sq 16836. Change bound variables in 𝑆. (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
4sqlem2	⊢ (𝐴 ∈ 𝑆 ↔ ∃𝑎 ∈ ℤ ∃𝑏 ∈ ℤ ∃𝑐 ∈ ℤ ∃𝑑 ∈ ℤ 𝐴 = (((𝑎↑2) + (𝑏↑2)) + ((𝑐↑2) + (𝑑↑2))))

Distinct variable groups:   𝑎,𝑏,𝑐,𝑑,𝑛,𝑤,𝑥,𝑦,𝑧   𝐴,𝑎,𝑏,𝑐,𝑑,𝑛   𝑆,𝑎,𝑏,𝑐,𝑑,𝑛

Allowed substitution hints:   𝐴(𝑥,𝑦,𝑧,𝑤)   𝑆(𝑥,𝑦,𝑧,𝑤)

Proof of Theorem 4sqlem2

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	eleq2i 2829	. 2 ⊢ (𝐴 ∈ 𝑆 ↔ 𝐴 ∈ {𝑛 ∣ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ ∃𝑧 ∈ ℤ ∃𝑤 ∈ ℤ 𝑛 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))})
3		id 22	. . . . . . 7 ⊢ (𝐴 = (((𝑎↑2) + (𝑏↑2)) + ((𝑐↑2) + (𝑑↑2))) → 𝐴 = (((𝑎↑2) + (𝑏↑2)) + ((𝑐↑2) + (𝑑↑2))))
4		ovex 7390	. . . . . . 7 ⊢ (((𝑎↑2) + (𝑏↑2)) + ((𝑐↑2) + (𝑑↑2))) ∈ V
5	3, 4	eqeltrdi 2846	. . . . . 6 ⊢ (𝐴 = (((𝑎↑2) + (𝑏↑2)) + ((𝑐↑2) + (𝑑↑2))) → 𝐴 ∈ V)
6	5	a1i 11	. . . . 5 ⊢ (((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) ∧ (𝑐 ∈ ℤ ∧ 𝑑 ∈ ℤ)) → (𝐴 = (((𝑎↑2) + (𝑏↑2)) + ((𝑐↑2) + (𝑑↑2))) → 𝐴 ∈ V))
7	6	rexlimdvva 3205	. . . 4 ⊢ ((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) → (∃𝑐 ∈ ℤ ∃𝑑 ∈ ℤ 𝐴 = (((𝑎↑2) + (𝑏↑2)) + ((𝑐↑2) + (𝑑↑2))) → 𝐴 ∈ V))
8	7	rexlimivv 3196	. . 3 ⊢ (∃𝑎 ∈ ℤ ∃𝑏 ∈ ℤ ∃𝑐 ∈ ℤ ∃𝑑 ∈ ℤ 𝐴 = (((𝑎↑2) + (𝑏↑2)) + ((𝑐↑2) + (𝑑↑2))) → 𝐴 ∈ V)
9		oveq1 7364	. . . . . . . . 9 ⊢ (𝑥 = 𝑎 → (𝑥↑2) = (𝑎↑2))
10	9	oveq1d 7372	. . . . . . . 8 ⊢ (𝑥 = 𝑎 → ((𝑥↑2) + (𝑦↑2)) = ((𝑎↑2) + (𝑦↑2)))
11	10	oveq1d 7372	. . . . . . 7 ⊢ (𝑥 = 𝑎 → (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2))) = (((𝑎↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2))))
12	11	eqeq2d 2747	. . . . . 6 ⊢ (𝑥 = 𝑎 → (𝑛 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2))) ↔ 𝑛 = (((𝑎↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))))
13	12	2rexbidv 3213	. . . . 5 ⊢ (𝑥 = 𝑎 → (∃𝑧 ∈ ℤ ∃𝑤 ∈ ℤ 𝑛 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2))) ↔ ∃𝑧 ∈ ℤ ∃𝑤 ∈ ℤ 𝑛 = (((𝑎↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))))
14		oveq1 7364	. . . . . . . . 9 ⊢ (𝑦 = 𝑏 → (𝑦↑2) = (𝑏↑2))
15	14	oveq2d 7373	. . . . . . . 8 ⊢ (𝑦 = 𝑏 → ((𝑎↑2) + (𝑦↑2)) = ((𝑎↑2) + (𝑏↑2)))
16	15	oveq1d 7372	. . . . . . 7 ⊢ (𝑦 = 𝑏 → (((𝑎↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2))) = (((𝑎↑2) + (𝑏↑2)) + ((𝑧↑2) + (𝑤↑2))))
17	16	eqeq2d 2747	. . . . . 6 ⊢ (𝑦 = 𝑏 → (𝑛 = (((𝑎↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2))) ↔ 𝑛 = (((𝑎↑2) + (𝑏↑2)) + ((𝑧↑2) + (𝑤↑2)))))
18	17	2rexbidv 3213	. . . . 5 ⊢ (𝑦 = 𝑏 → (∃𝑧 ∈ ℤ ∃𝑤 ∈ ℤ 𝑛 = (((𝑎↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2))) ↔ ∃𝑧 ∈ ℤ ∃𝑤 ∈ ℤ 𝑛 = (((𝑎↑2) + (𝑏↑2)) + ((𝑧↑2) + (𝑤↑2)))))
19	13, 18	cbvrex2vw 3228	. . . 4 ⊢ (∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ ∃𝑧 ∈ ℤ ∃𝑤 ∈ ℤ 𝑛 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2))) ↔ ∃𝑎 ∈ ℤ ∃𝑏 ∈ ℤ ∃𝑧 ∈ ℤ ∃𝑤 ∈ ℤ 𝑛 = (((𝑎↑2) + (𝑏↑2)) + ((𝑧↑2) + (𝑤↑2))))
20		oveq1 7364	. . . . . . . . . 10 ⊢ (𝑧 = 𝑐 → (𝑧↑2) = (𝑐↑2))
21	20	oveq1d 7372	. . . . . . . . 9 ⊢ (𝑧 = 𝑐 → ((𝑧↑2) + (𝑤↑2)) = ((𝑐↑2) + (𝑤↑2)))
22	21	oveq2d 7373	. . . . . . . 8 ⊢ (𝑧 = 𝑐 → (((𝑎↑2) + (𝑏↑2)) + ((𝑧↑2) + (𝑤↑2))) = (((𝑎↑2) + (𝑏↑2)) + ((𝑐↑2) + (𝑤↑2))))
23	22	eqeq2d 2747	. . . . . . 7 ⊢ (𝑧 = 𝑐 → (𝑛 = (((𝑎↑2) + (𝑏↑2)) + ((𝑧↑2) + (𝑤↑2))) ↔ 𝑛 = (((𝑎↑2) + (𝑏↑2)) + ((𝑐↑2) + (𝑤↑2)))))
24		oveq1 7364	. . . . . . . . . 10 ⊢ (𝑤 = 𝑑 → (𝑤↑2) = (𝑑↑2))
25	24	oveq2d 7373	. . . . . . . . 9 ⊢ (𝑤 = 𝑑 → ((𝑐↑2) + (𝑤↑2)) = ((𝑐↑2) + (𝑑↑2)))
26	25	oveq2d 7373	. . . . . . . 8 ⊢ (𝑤 = 𝑑 → (((𝑎↑2) + (𝑏↑2)) + ((𝑐↑2) + (𝑤↑2))) = (((𝑎↑2) + (𝑏↑2)) + ((𝑐↑2) + (𝑑↑2))))
27	26	eqeq2d 2747	. . . . . . 7 ⊢ (𝑤 = 𝑑 → (𝑛 = (((𝑎↑2) + (𝑏↑2)) + ((𝑐↑2) + (𝑤↑2))) ↔ 𝑛 = (((𝑎↑2) + (𝑏↑2)) + ((𝑐↑2) + (𝑑↑2)))))
28	23, 27	cbvrex2vw 3228	. . . . . 6 ⊢ (∃𝑧 ∈ ℤ ∃𝑤 ∈ ℤ 𝑛 = (((𝑎↑2) + (𝑏↑2)) + ((𝑧↑2) + (𝑤↑2))) ↔ ∃𝑐 ∈ ℤ ∃𝑑 ∈ ℤ 𝑛 = (((𝑎↑2) + (𝑏↑2)) + ((𝑐↑2) + (𝑑↑2))))
29		eqeq1 2740	. . . . . . 7 ⊢ (𝑛 = 𝐴 → (𝑛 = (((𝑎↑2) + (𝑏↑2)) + ((𝑐↑2) + (𝑑↑2))) ↔ 𝐴 = (((𝑎↑2) + (𝑏↑2)) + ((𝑐↑2) + (𝑑↑2)))))
30	29	2rexbidv 3213	. . . . . 6 ⊢ (𝑛 = 𝐴 → (∃𝑐 ∈ ℤ ∃𝑑 ∈ ℤ 𝑛 = (((𝑎↑2) + (𝑏↑2)) + ((𝑐↑2) + (𝑑↑2))) ↔ ∃𝑐 ∈ ℤ ∃𝑑 ∈ ℤ 𝐴 = (((𝑎↑2) + (𝑏↑2)) + ((𝑐↑2) + (𝑑↑2)))))
31	28, 30	bitrid 282	. . . . 5 ⊢ (𝑛 = 𝐴 → (∃𝑧 ∈ ℤ ∃𝑤 ∈ ℤ 𝑛 = (((𝑎↑2) + (𝑏↑2)) + ((𝑧↑2) + (𝑤↑2))) ↔ ∃𝑐 ∈ ℤ ∃𝑑 ∈ ℤ 𝐴 = (((𝑎↑2) + (𝑏↑2)) + ((𝑐↑2) + (𝑑↑2)))))
32	31	2rexbidv 3213	. . . 4 ⊢ (𝑛 = 𝐴 → (∃𝑎 ∈ ℤ ∃𝑏 ∈ ℤ ∃𝑧 ∈ ℤ ∃𝑤 ∈ ℤ 𝑛 = (((𝑎↑2) + (𝑏↑2)) + ((𝑧↑2) + (𝑤↑2))) ↔ ∃𝑎 ∈ ℤ ∃𝑏 ∈ ℤ ∃𝑐 ∈ ℤ ∃𝑑 ∈ ℤ 𝐴 = (((𝑎↑2) + (𝑏↑2)) + ((𝑐↑2) + (𝑑↑2)))))
33	19, 32	bitrid 282	. . 3 ⊢ (𝑛 = 𝐴 → (∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ ∃𝑧 ∈ ℤ ∃𝑤 ∈ ℤ 𝑛 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2))) ↔ ∃𝑎 ∈ ℤ ∃𝑏 ∈ ℤ ∃𝑐 ∈ ℤ ∃𝑑 ∈ ℤ 𝐴 = (((𝑎↑2) + (𝑏↑2)) + ((𝑐↑2) + (𝑑↑2)))))
34	8, 33	elab3 3638	. 2 ⊢ (𝐴 ∈ {𝑛 ∣ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ ∃𝑧 ∈ ℤ ∃𝑤 ∈ ℤ 𝑛 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))} ↔ ∃𝑎 ∈ ℤ ∃𝑏 ∈ ℤ ∃𝑐 ∈ ℤ ∃𝑑 ∈ ℤ 𝐴 = (((𝑎↑2) + (𝑏↑2)) + ((𝑐↑2) + (𝑑↑2))))
35	2, 34	bitri 274	1 ⊢ (𝐴 ∈ 𝑆 ↔ ∃𝑎 ∈ ℤ ∃𝑏 ∈ ℤ ∃𝑐 ∈ ℤ ∃𝑑 ∈ ℤ 𝐴 = (((𝑎↑2) + (𝑏↑2)) + ((𝑐↑2) + (𝑑↑2))))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  {cab 2713  ∃wrex 3073  Vcvv 3445  (class class class)co 7357   + caddc 11054  2c2 12208  ℤcz 12499  ↑cexp 13967

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2707  ax-nul 5263

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-ne 2944  df-rex 3074  df-rab 3408  df-v 3447  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-iota 6448  df-fv 6504  df-ov 7360

This theorem is referenced by:  4sqlem3  16822  4sqlem4  16824  4sqlem18  16834  4sq  16836