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Mirrors > Home > MPE Home > Th. List > 4sqlem1 | Structured version Visualization version GIF version |
Description: Lemma for 4sq 16162. 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.) |
Ref | Expression |
---|---|
4sq.1 | ⊢ 𝑆 = {𝑛 ∣ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ ∃𝑧 ∈ ℤ ∃𝑤 ∈ ℤ 𝑛 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))} |
Ref | Expression |
---|---|
4sqlem1 | ⊢ 𝑆 ⊆ ℕ0 |
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 13323 | . . . . . . . 8 ⊢ (𝑥 ∈ ℤ → (𝑥↑2) ∈ ℕ0) | |
3 | zsqcl2 13323 | . . . . . . . 8 ⊢ (𝑦 ∈ ℤ → (𝑦↑2) ∈ ℕ0) | |
4 | nn0addcl 11750 | . . . . . . . 8 ⊢ (((𝑥↑2) ∈ ℕ0 ∧ (𝑦↑2) ∈ ℕ0) → ((𝑥↑2) + (𝑦↑2)) ∈ ℕ0) | |
5 | 2, 3, 4 | syl2an 587 | . . . . . . 7 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → ((𝑥↑2) + (𝑦↑2)) ∈ ℕ0) |
6 | zsqcl2 13323 | . . . . . . . 8 ⊢ (𝑧 ∈ ℤ → (𝑧↑2) ∈ ℕ0) | |
7 | zsqcl2 13323 | . . . . . . . 8 ⊢ (𝑤 ∈ ℤ → (𝑤↑2) ∈ ℕ0) | |
8 | nn0addcl 11750 | . . . . . . . 8 ⊢ (((𝑧↑2) ∈ ℕ0 ∧ (𝑤↑2) ∈ ℕ0) → ((𝑧↑2) + (𝑤↑2)) ∈ ℕ0) | |
9 | 6, 7, 8 | syl2an 587 | . . . . . . 7 ⊢ ((𝑧 ∈ ℤ ∧ 𝑤 ∈ ℤ) → ((𝑧↑2) + (𝑤↑2)) ∈ ℕ0) |
10 | nn0addcl 11750 | . . . . . . 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 587 | . . . . . 6 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑧 ∈ ℤ ∧ 𝑤 ∈ ℤ)) → (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2))) ∈ ℕ0) |
12 | eleq1a 2863 | . . . . . 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 3241 | . . . 4 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → (∃𝑧 ∈ ℤ ∃𝑤 ∈ ℤ 𝑛 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2))) → 𝑛 ∈ ℕ0)) |
15 | 14 | rexlimivv 3239 | . . 3 ⊢ (∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ ∃𝑧 ∈ ℤ ∃𝑤 ∈ ℤ 𝑛 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2))) → 𝑛 ∈ ℕ0) |
16 | 15 | abssi 3938 | . 2 ⊢ {𝑛 ∣ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ ∃𝑧 ∈ ℤ ∃𝑤 ∈ ℤ 𝑛 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))} ⊆ ℕ0 |
17 | 1, 16 | eqsstri 3893 | 1 ⊢ 𝑆 ⊆ ℕ0 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 = wceq 1508 ∈ wcel 2051 {cab 2760 ∃wrex 3091 ⊆ wss 3831 (class class class)co 6982 + caddc 10344 2c2 11501 ℕ0cn0 11713 ℤcz 11799 ↑cexp 13250 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2752 ax-sep 5064 ax-nul 5071 ax-pow 5123 ax-pr 5190 ax-un 7285 ax-cnex 10397 ax-resscn 10398 ax-1cn 10399 ax-icn 10400 ax-addcl 10401 ax-addrcl 10402 ax-mulcl 10403 ax-mulrcl 10404 ax-mulcom 10405 ax-addass 10406 ax-mulass 10407 ax-distr 10408 ax-i2m1 10409 ax-1ne0 10410 ax-1rid 10411 ax-rnegex 10412 ax-rrecex 10413 ax-cnre 10414 ax-pre-lttri 10415 ax-pre-lttrn 10416 ax-pre-ltadd 10417 ax-pre-mulgt0 10418 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3or 1070 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2551 df-eu 2589 df-clab 2761 df-cleq 2773 df-clel 2848 df-nfc 2920 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rab 3099 df-v 3419 df-sbc 3684 df-csb 3789 df-dif 3834 df-un 3836 df-in 3838 df-ss 3845 df-pss 3847 df-nul 4182 df-if 4354 df-pw 4427 df-sn 4445 df-pr 4447 df-tp 4449 df-op 4451 df-uni 4718 df-iun 4799 df-br 4935 df-opab 4997 df-mpt 5014 df-tr 5036 df-id 5316 df-eprel 5321 df-po 5330 df-so 5331 df-fr 5370 df-we 5372 df-xp 5417 df-rel 5418 df-cnv 5419 df-co 5420 df-dm 5421 df-rn 5422 df-res 5423 df-ima 5424 df-pred 5991 df-ord 6037 df-on 6038 df-lim 6039 df-suc 6040 df-iota 6157 df-fun 6195 df-fn 6196 df-f 6197 df-f1 6198 df-fo 6199 df-f1o 6200 df-fv 6201 df-riota 6943 df-ov 6985 df-oprab 6986 df-mpo 6987 df-om 7403 df-2nd 7508 df-wrecs 7756 df-recs 7818 df-rdg 7856 df-er 8095 df-en 8313 df-dom 8314 df-sdom 8315 df-pnf 10482 df-mnf 10483 df-xr 10484 df-ltxr 10485 df-le 10486 df-sub 10678 df-neg 10679 df-nn 11446 df-2 11509 df-n0 11714 df-z 11800 df-uz 12065 df-seq 13191 df-exp 13251 |
This theorem is referenced by: 4sqlem19 16161 |
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