4sqlem1 - Metamath Proof Explorer

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Theorem 4sqlem1 16914

Description: Lemma for 4sq 16930. 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 14095	. . . . . . . 8 ⊢ (𝑥 ∈ ℤ → (𝑥↑2) ∈ ℕ₀)
3		zsqcl2 14095	. . . . . . . 8 ⊢ (𝑦 ∈ ℤ → (𝑦↑2) ∈ ℕ₀)
4		nn0addcl 12467	. . . . . . . 8 ⊢ (((𝑥↑2) ∈ ℕ₀ ∧ (𝑦↑2) ∈ ℕ₀) → ((𝑥↑2) + (𝑦↑2)) ∈ ℕ₀)
5	2, 3, 4	syl2an 597	. . . . . . 7 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → ((𝑥↑2) + (𝑦↑2)) ∈ ℕ₀)
6		zsqcl2 14095	. . . . . . . 8 ⊢ (𝑧 ∈ ℤ → (𝑧↑2) ∈ ℕ₀)
7		zsqcl2 14095	. . . . . . . 8 ⊢ (𝑤 ∈ ℤ → (𝑤↑2) ∈ ℕ₀)
8		nn0addcl 12467	. . . . . . . 8 ⊢ (((𝑧↑2) ∈ ℕ₀ ∧ (𝑤↑2) ∈ ℕ₀) → ((𝑧↑2) + (𝑤↑2)) ∈ ℕ₀)
9	6, 7, 8	syl2an 597	. . . . . . 7 ⊢ ((𝑧 ∈ ℤ ∧ 𝑤 ∈ ℤ) → ((𝑧↑2) + (𝑤↑2)) ∈ ℕ₀)
10		nn0addcl 12467	. . . . . . 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 597	. . . . . 6 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑧 ∈ ℤ ∧ 𝑤 ∈ ℤ)) → (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2))) ∈ ℕ₀)
12		eleq1a 2832	. . . . . 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 3195	. . . 4 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → (∃𝑧 ∈ ℤ ∃𝑤 ∈ ℤ 𝑛 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2))) → 𝑛 ∈ ℕ₀))
15	14	rexlimivv 3180	. . 3 ⊢ (∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ ∃𝑧 ∈ ℤ ∃𝑤 ∈ ℤ 𝑛 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2))) → 𝑛 ∈ ℕ₀)
16	15	abssi 4009	. 2 ⊢ {𝑛 ∣ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ ∃𝑧 ∈ ℤ ∃𝑤 ∈ ℤ 𝑛 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))} ⊆ ℕ₀
17	1, 16	eqsstri 3969	1 ⊢ 𝑆 ⊆ ℕ₀

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1542 ∈ wcel 2114 {cab 2715 ∃wrex 3062 ⊆ wss 3890 (class class class)co 7362 + caddc 11036 2c2 12231 ℕ₀cn0 12432 ℤcz 12519 ↑cexp 14018

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-sep 5232 ax-nul 5242 ax-pow 5304 ax-pr 5372 ax-un 7684 ax-cnex 11089 ax-resscn 11090 ax-1cn 11091 ax-icn 11092 ax-addcl 11093 ax-addrcl 11094 ax-mulcl 11095 ax-mulrcl 11096 ax-mulcom 11097 ax-addass 11098 ax-mulass 11099 ax-distr 11100 ax-i2m1 11101 ax-1ne0 11102 ax-1rid 11103 ax-rnegex 11104 ax-rrecex 11105 ax-cnre 11106 ax-pre-lttri 11107 ax-pre-lttrn 11108 ax-pre-ltadd 11109 ax-pre-mulgt0 11110

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5521 df-eprel 5526 df-po 5534 df-so 5535 df-fr 5579 df-we 5581 df-xp 5632 df-rel 5633 df-cnv 5634 df-co 5635 df-dm 5636 df-rn 5637 df-res 5638 df-ima 5639 df-pred 6261 df-ord 6322 df-on 6323 df-lim 6324 df-suc 6325 df-iota 6450 df-fun 6496 df-fn 6497 df-f 6498 df-f1 6499 df-fo 6500 df-f1o 6501 df-fv 6502 df-riota 7319 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7813 df-2nd 7938 df-frecs 8226 df-wrecs 8257 df-recs 8306 df-rdg 8344 df-er 8638 df-en 8889 df-dom 8890 df-sdom 8891 df-pnf 11176 df-mnf 11177 df-xr 11178 df-ltxr 11179 df-le 11180 df-sub 11374 df-neg 11375 df-nn 12170 df-2 12239 df-n0 12433 df-z 12520 df-uz 12784 df-seq 13959 df-exp 14019

This theorem is referenced by: 4sqlem19 16929

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