4sqlem1 - Metamath Proof Explorer

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Theorem 4sqlem1 16968

Description: Lemma for 4sq 16984. The set 𝑆 is the set of all numbers that are expressible as a sum of four squares. Our goal is to show that 𝑆 = ℕ₀; here we show one subset direction. (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
4sqlem1	⊢ 𝑆 ⊆ ℕ₀

Distinct variable groups: 𝑤,𝑛,𝑥,𝑦,𝑧 𝑆,𝑛 Allowed substitution hints: 𝑆(𝑥,𝑦,𝑧,𝑤)

**Proof of Theorem 4sqlem1**
Step	Hyp	Ref	Expression
1		4sq.1	. 2 ⊢ 𝑆 = {𝑛 ∣ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ ∃𝑧 ∈ ℤ ∃𝑤 ∈ ℤ 𝑛 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))}
2		zsqcl2 14160	. . . . . . . 8 ⊢ (𝑥 ∈ ℤ → (𝑥↑2) ∈ ℕ₀)
3		zsqcl2 14160	. . . . . . . 8 ⊢ (𝑦 ∈ ℤ → (𝑦↑2) ∈ ℕ₀)
4		nn0addcl 12544	. . . . . . . 8 ⊢ (((𝑥↑2) ∈ ℕ₀ ∧ (𝑦↑2) ∈ ℕ₀) → ((𝑥↑2) + (𝑦↑2)) ∈ ℕ₀)
5	2, 3, 4	syl2an 596	. . . . . . 7 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → ((𝑥↑2) + (𝑦↑2)) ∈ ℕ₀)
6		zsqcl2 14160	. . . . . . . 8 ⊢ (𝑧 ∈ ℤ → (𝑧↑2) ∈ ℕ₀)
7		zsqcl2 14160	. . . . . . . 8 ⊢ (𝑤 ∈ ℤ → (𝑤↑2) ∈ ℕ₀)
8		nn0addcl 12544	. . . . . . . 8 ⊢ (((𝑧↑2) ∈ ℕ₀ ∧ (𝑤↑2) ∈ ℕ₀) → ((𝑧↑2) + (𝑤↑2)) ∈ ℕ₀)
9	6, 7, 8	syl2an 596	. . . . . . 7 ⊢ ((𝑧 ∈ ℤ ∧ 𝑤 ∈ ℤ) → ((𝑧↑2) + (𝑤↑2)) ∈ ℕ₀)
10		nn0addcl 12544	. . . . . . 7 ⊢ ((((𝑥↑2) + (𝑦↑2)) ∈ ℕ₀ ∧ ((𝑧↑2) + (𝑤↑2)) ∈ ℕ₀) → (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2))) ∈ ℕ₀)
11	5, 9, 10	syl2an 596	. . . . . 6 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑧 ∈ ℤ ∧ 𝑤 ∈ ℤ)) → (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2))) ∈ ℕ₀)
12		eleq1a 2828	. . . . . 6 ⊢ ((((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2))) ∈ ℕ₀ → (𝑛 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2))) → 𝑛 ∈ ℕ₀))
13	11, 12	syl 17	. . . . 5 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑧 ∈ ℤ ∧ 𝑤 ∈ ℤ)) → (𝑛 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2))) → 𝑛 ∈ ℕ₀))
14	13	rexlimdvva 3200	. . . 4 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → (∃𝑧 ∈ ℤ ∃𝑤 ∈ ℤ 𝑛 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2))) → 𝑛 ∈ ℕ₀))
15	14	rexlimivv 3188	. . 3 ⊢ (∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ ∃𝑧 ∈ ℤ ∃𝑤 ∈ ℤ 𝑛 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2))) → 𝑛 ∈ ℕ₀)
16	15	abssi 4050	. 2 ⊢ {𝑛 ∣ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ ∃𝑧 ∈ ℤ ∃𝑤 ∈ ℤ 𝑛 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))} ⊆ ℕ₀
17	1, 16	eqsstri 4010	1 ⊢ 𝑆 ⊆ ℕ₀

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2107 {cab 2712 ∃wrex 3059 ⊆ wss 3931 (class class class)co 7413 + caddc 11140 2c2 12303 ℕ₀cn0 12509 ℤcz 12596 ↑cexp 14084

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2706 ax-sep 5276 ax-nul 5286 ax-pow 5345 ax-pr 5412 ax-un 7737 ax-cnex 11193 ax-resscn 11194 ax-1cn 11195 ax-icn 11196 ax-addcl 11197 ax-addrcl 11198 ax-mulcl 11199 ax-mulrcl 11200 ax-mulcom 11201 ax-addass 11202 ax-mulass 11203 ax-distr 11204 ax-i2m1 11205 ax-1ne0 11206 ax-1rid 11207 ax-rnegex 11208 ax-rrecex 11209 ax-cnre 11210 ax-pre-lttri 11211 ax-pre-lttrn 11212 ax-pre-ltadd 11213 ax-pre-mulgt0 11214

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2808 df-nfc 2884 df-ne 2932 df-nel 3036 df-ral 3051 df-rex 3060 df-reu 3364 df-rab 3420 df-v 3465 df-sbc 3771 df-csb 3880 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-pss 3951 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-op 4613 df-uni 4888 df-iun 4973 df-br 5124 df-opab 5186 df-mpt 5206 df-tr 5240 df-id 5558 df-eprel 5564 df-po 5572 df-so 5573 df-fr 5617 df-we 5619 df-xp 5671 df-rel 5672 df-cnv 5673 df-co 5674 df-dm 5675 df-rn 5676 df-res 5677 df-ima 5678 df-pred 6301 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-riota 7370 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7870 df-2nd 7997 df-frecs 8288 df-wrecs 8319 df-recs 8393 df-rdg 8432 df-er 8727 df-en 8968 df-dom 8969 df-sdom 8970 df-pnf 11279 df-mnf 11280 df-xr 11281 df-ltxr 11282 df-le 11283 df-sub 11476 df-neg 11477 df-nn 12249 df-2 12311 df-n0 12510 df-z 12597 df-uz 12861 df-seq 14025 df-exp 14085

This theorem is referenced by: 4sqlem19 16983

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