4sqlem1 - Metamath Proof Explorer

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Theorem 4sqlem1 16577

Description: Lemma for 4sq 16593. 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 13784	. . . . . . . 8 ⊢ (𝑥 ∈ ℤ → (𝑥↑2) ∈ ℕ₀)
3		zsqcl2 13784	. . . . . . . 8 ⊢ (𝑦 ∈ ℤ → (𝑦↑2) ∈ ℕ₀)
4		nn0addcl 12198	. . . . . . . 8 ⊢ (((𝑥↑2) ∈ ℕ₀ ∧ (𝑦↑2) ∈ ℕ₀) → ((𝑥↑2) + (𝑦↑2)) ∈ ℕ₀)
5	2, 3, 4	syl2an 595	. . . . . . 7 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → ((𝑥↑2) + (𝑦↑2)) ∈ ℕ₀)
6		zsqcl2 13784	. . . . . . . 8 ⊢ (𝑧 ∈ ℤ → (𝑧↑2) ∈ ℕ₀)
7		zsqcl2 13784	. . . . . . . 8 ⊢ (𝑤 ∈ ℤ → (𝑤↑2) ∈ ℕ₀)
8		nn0addcl 12198	. . . . . . . 8 ⊢ (((𝑧↑2) ∈ ℕ₀ ∧ (𝑤↑2) ∈ ℕ₀) → ((𝑧↑2) + (𝑤↑2)) ∈ ℕ₀)
9	6, 7, 8	syl2an 595	. . . . . . 7 ⊢ ((𝑧 ∈ ℤ ∧ 𝑤 ∈ ℤ) → ((𝑧↑2) + (𝑤↑2)) ∈ ℕ₀)
10		nn0addcl 12198	. . . . . . 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 595	. . . . . 6 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑧 ∈ ℤ ∧ 𝑤 ∈ ℤ)) → (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2))) ∈ ℕ₀)
12		eleq1a 2834	. . . . . 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 3222	. . . 4 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → (∃𝑧 ∈ ℤ ∃𝑤 ∈ ℤ 𝑛 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2))) → 𝑛 ∈ ℕ₀))
15	14	rexlimivv 3220	. . 3 ⊢ (∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ ∃𝑧 ∈ ℤ ∃𝑤 ∈ ℤ 𝑛 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2))) → 𝑛 ∈ ℕ₀)
16	15	abssi 3999	. 2 ⊢ {𝑛 ∣ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ ∃𝑧 ∈ ℤ ∃𝑤 ∈ ℤ 𝑛 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))} ⊆ ℕ₀
17	1, 16	eqsstri 3951	1 ⊢ 𝑆 ⊆ ℕ₀

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2108 {cab 2715 ∃wrex 3064 ⊆ wss 3883 (class class class)co 7255 + caddc 10805 2c2 11958 ℕ₀cn0 12163 ℤcz 12249 ↑cexp 13710

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-om 7688 df-2nd 7805 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-er 8456 df-en 8692 df-dom 8693 df-sdom 8694 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-nn 11904 df-2 11966 df-n0 12164 df-z 12250 df-uz 12512 df-seq 13650 df-exp 13711

This theorem is referenced by: 4sqlem19 16592

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