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Theorem 4sqlem19 16291
Description: Lemma for 4sq 16292. The proof is by strong induction - we show that if all the integers less than 𝑘 are in 𝑆, then 𝑘 is as well. In this part of the proof we do the induction argument and dispense with all the cases except the odd prime case, which is sent to 4sqlem18 16290. If 𝑘 is 0, 1, 2, we show 𝑘𝑆 directly; otherwise if 𝑘 is composite, 𝑘 is the product of two numbers less than it (and hence in 𝑆 by assumption), so by mul4sq 16282 𝑘𝑆. (Contributed by Mario Carneiro, 14-Jul-2014.) (Revised by Mario Carneiro, 20-Jun-2015.)
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
4sqlem19 0 = 𝑆
Distinct variable groups:   𝑤,𝑛,𝑥,𝑦,𝑧   𝑆,𝑛
Allowed substitution hints:   𝑆(𝑥,𝑦,𝑧,𝑤)

Proof of Theorem 4sqlem19
Dummy variables 𝑗 𝑘 𝑖 𝑚 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elnn0 11891 . . . 4 (𝑘 ∈ ℕ0 ↔ (𝑘 ∈ ℕ ∨ 𝑘 = 0))
2 eleq1 2904 . . . . . 6 (𝑗 = 1 → (𝑗𝑆 ↔ 1 ∈ 𝑆))
3 eleq1 2904 . . . . . 6 (𝑗 = 𝑚 → (𝑗𝑆𝑚𝑆))
4 eleq1 2904 . . . . . 6 (𝑗 = 𝑖 → (𝑗𝑆𝑖𝑆))
5 eleq1 2904 . . . . . 6 (𝑗 = (𝑚 · 𝑖) → (𝑗𝑆 ↔ (𝑚 · 𝑖) ∈ 𝑆))
6 eleq1 2904 . . . . . 6 (𝑗 = 𝑘 → (𝑗𝑆𝑘𝑆))
7 abs1 14650 . . . . . . . . . . 11 (abs‘1) = 1
87oveq1i 7161 . . . . . . . . . 10 ((abs‘1)↑2) = (1↑2)
9 sq1 13551 . . . . . . . . . 10 (1↑2) = 1
108, 9eqtri 2848 . . . . . . . . 9 ((abs‘1)↑2) = 1
11 abs0 14638 . . . . . . . . . . 11 (abs‘0) = 0
1211oveq1i 7161 . . . . . . . . . 10 ((abs‘0)↑2) = (0↑2)
13 sq0 13548 . . . . . . . . . 10 (0↑2) = 0
1412, 13eqtri 2848 . . . . . . . . 9 ((abs‘0)↑2) = 0
1510, 14oveq12i 7163 . . . . . . . 8 (((abs‘1)↑2) + ((abs‘0)↑2)) = (1 + 0)
16 1p0e1 11753 . . . . . . . 8 (1 + 0) = 1
1715, 16eqtri 2848 . . . . . . 7 (((abs‘1)↑2) + ((abs‘0)↑2)) = 1
18 1z 12004 . . . . . . . . 9 1 ∈ ℤ
19 zgz 16261 . . . . . . . . 9 (1 ∈ ℤ → 1 ∈ ℤ[i])
2018, 19ax-mp 5 . . . . . . . 8 1 ∈ ℤ[i]
21 0z 11984 . . . . . . . . 9 0 ∈ ℤ
22 zgz 16261 . . . . . . . . 9 (0 ∈ ℤ → 0 ∈ ℤ[i])
2321, 22ax-mp 5 . . . . . . . 8 0 ∈ ℤ[i]
24 4sq.1 . . . . . . . . 9 𝑆 = {𝑛 ∣ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ ∃𝑧 ∈ ℤ ∃𝑤 ∈ ℤ 𝑛 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))}
25244sqlem4a 16279 . . . . . . . 8 ((1 ∈ ℤ[i] ∧ 0 ∈ ℤ[i]) → (((abs‘1)↑2) + ((abs‘0)↑2)) ∈ 𝑆)
2620, 23, 25mp2an 688 . . . . . . 7 (((abs‘1)↑2) + ((abs‘0)↑2)) ∈ 𝑆
2717, 26eqeltrri 2914 . . . . . 6 1 ∈ 𝑆
28 eleq1 2904 . . . . . . 7 (𝑗 = 2 → (𝑗𝑆 ↔ 2 ∈ 𝑆))
29 eldifsn 4717 . . . . . . . . 9 (𝑗 ∈ (ℙ ∖ {2}) ↔ (𝑗 ∈ ℙ ∧ 𝑗 ≠ 2))
30 oddprm 16139 . . . . . . . . . . 11 (𝑗 ∈ (ℙ ∖ {2}) → ((𝑗 − 1) / 2) ∈ ℕ)
3130adantr 481 . . . . . . . . . 10 ((𝑗 ∈ (ℙ ∖ {2}) ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚𝑆) → ((𝑗 − 1) / 2) ∈ ℕ)
32 eldifi 4106 . . . . . . . . . . . . . . . 16 (𝑗 ∈ (ℙ ∖ {2}) → 𝑗 ∈ ℙ)
3332adantr 481 . . . . . . . . . . . . . . 15 ((𝑗 ∈ (ℙ ∖ {2}) ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚𝑆) → 𝑗 ∈ ℙ)
34 prmnn 16010 . . . . . . . . . . . . . . 15 (𝑗 ∈ ℙ → 𝑗 ∈ ℕ)
35 nncn 11638 . . . . . . . . . . . . . . 15 (𝑗 ∈ ℕ → 𝑗 ∈ ℂ)
3633, 34, 353syl 18 . . . . . . . . . . . . . 14 ((𝑗 ∈ (ℙ ∖ {2}) ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚𝑆) → 𝑗 ∈ ℂ)
37 ax-1cn 10587 . . . . . . . . . . . . . 14 1 ∈ ℂ
38 subcl 10877 . . . . . . . . . . . . . 14 ((𝑗 ∈ ℂ ∧ 1 ∈ ℂ) → (𝑗 − 1) ∈ ℂ)
3936, 37, 38sylancl 586 . . . . . . . . . . . . 13 ((𝑗 ∈ (ℙ ∖ {2}) ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚𝑆) → (𝑗 − 1) ∈ ℂ)
40 2cnd 11707 . . . . . . . . . . . . 13 ((𝑗 ∈ (ℙ ∖ {2}) ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚𝑆) → 2 ∈ ℂ)
41 2ne0 11733 . . . . . . . . . . . . . 14 2 ≠ 0
4241a1i 11 . . . . . . . . . . . . 13 ((𝑗 ∈ (ℙ ∖ {2}) ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚𝑆) → 2 ≠ 0)
4339, 40, 42divcan2d 11410 . . . . . . . . . . . 12 ((𝑗 ∈ (ℙ ∖ {2}) ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚𝑆) → (2 · ((𝑗 − 1) / 2)) = (𝑗 − 1))
4443oveq1d 7166 . . . . . . . . . . 11 ((𝑗 ∈ (ℙ ∖ {2}) ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚𝑆) → ((2 · ((𝑗 − 1) / 2)) + 1) = ((𝑗 − 1) + 1))
45 npcan 10887 . . . . . . . . . . . 12 ((𝑗 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑗 − 1) + 1) = 𝑗)
4636, 37, 45sylancl 586 . . . . . . . . . . 11 ((𝑗 ∈ (ℙ ∖ {2}) ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚𝑆) → ((𝑗 − 1) + 1) = 𝑗)
4744, 46eqtr2d 2861 . . . . . . . . . 10 ((𝑗 ∈ (ℙ ∖ {2}) ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚𝑆) → 𝑗 = ((2 · ((𝑗 − 1) / 2)) + 1))
4843oveq2d 7167 . . . . . . . . . . . 12 ((𝑗 ∈ (ℙ ∖ {2}) ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚𝑆) → (0...(2 · ((𝑗 − 1) / 2))) = (0...(𝑗 − 1)))
49 nnm1nn0 11930 . . . . . . . . . . . . . . 15 (𝑗 ∈ ℕ → (𝑗 − 1) ∈ ℕ0)
5033, 34, 493syl 18 . . . . . . . . . . . . . 14 ((𝑗 ∈ (ℙ ∖ {2}) ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚𝑆) → (𝑗 − 1) ∈ ℕ0)
51 elnn0uz 12275 . . . . . . . . . . . . . 14 ((𝑗 − 1) ∈ ℕ0 ↔ (𝑗 − 1) ∈ (ℤ‘0))
5250, 51sylib 219 . . . . . . . . . . . . 13 ((𝑗 ∈ (ℙ ∖ {2}) ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚𝑆) → (𝑗 − 1) ∈ (ℤ‘0))
53 eluzfz1 12907 . . . . . . . . . . . . 13 ((𝑗 − 1) ∈ (ℤ‘0) → 0 ∈ (0...(𝑗 − 1)))
54 fzsplit 12926 . . . . . . . . . . . . 13 (0 ∈ (0...(𝑗 − 1)) → (0...(𝑗 − 1)) = ((0...0) ∪ ((0 + 1)...(𝑗 − 1))))
5552, 53, 543syl 18 . . . . . . . . . . . 12 ((𝑗 ∈ (ℙ ∖ {2}) ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚𝑆) → (0...(𝑗 − 1)) = ((0...0) ∪ ((0 + 1)...(𝑗 − 1))))
5648, 55eqtrd 2860 . . . . . . . . . . 11 ((𝑗 ∈ (ℙ ∖ {2}) ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚𝑆) → (0...(2 · ((𝑗 − 1) / 2))) = ((0...0) ∪ ((0 + 1)...(𝑗 − 1))))
57 fz0sn 13000 . . . . . . . . . . . . . 14 (0...0) = {0}
5814, 14oveq12i 7163 . . . . . . . . . . . . . . . . 17 (((abs‘0)↑2) + ((abs‘0)↑2)) = (0 + 0)
59 00id 10807 . . . . . . . . . . . . . . . . 17 (0 + 0) = 0
6058, 59eqtri 2848 . . . . . . . . . . . . . . . 16 (((abs‘0)↑2) + ((abs‘0)↑2)) = 0
61244sqlem4a 16279 . . . . . . . . . . . . . . . . 17 ((0 ∈ ℤ[i] ∧ 0 ∈ ℤ[i]) → (((abs‘0)↑2) + ((abs‘0)↑2)) ∈ 𝑆)
6223, 23, 61mp2an 688 . . . . . . . . . . . . . . . 16 (((abs‘0)↑2) + ((abs‘0)↑2)) ∈ 𝑆
6360, 62eqeltrri 2914 . . . . . . . . . . . . . . 15 0 ∈ 𝑆
64 snssi 4739 . . . . . . . . . . . . . . 15 (0 ∈ 𝑆 → {0} ⊆ 𝑆)
6563, 64ax-mp 5 . . . . . . . . . . . . . 14 {0} ⊆ 𝑆
6657, 65eqsstri 4004 . . . . . . . . . . . . 13 (0...0) ⊆ 𝑆
6766a1i 11 . . . . . . . . . . . 12 ((𝑗 ∈ (ℙ ∖ {2}) ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚𝑆) → (0...0) ⊆ 𝑆)
68 0p1e1 11751 . . . . . . . . . . . . . 14 (0 + 1) = 1
6968oveq1i 7161 . . . . . . . . . . . . 13 ((0 + 1)...(𝑗 − 1)) = (1...(𝑗 − 1))
70 simpr 485 . . . . . . . . . . . . . 14 ((𝑗 ∈ (ℙ ∖ {2}) ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚𝑆) → ∀𝑚 ∈ (1...(𝑗 − 1))𝑚𝑆)
71 dfss3 3959 . . . . . . . . . . . . . 14 ((1...(𝑗 − 1)) ⊆ 𝑆 ↔ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚𝑆)
7270, 71sylibr 235 . . . . . . . . . . . . 13 ((𝑗 ∈ (ℙ ∖ {2}) ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚𝑆) → (1...(𝑗 − 1)) ⊆ 𝑆)
7369, 72eqsstrid 4018 . . . . . . . . . . . 12 ((𝑗 ∈ (ℙ ∖ {2}) ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚𝑆) → ((0 + 1)...(𝑗 − 1)) ⊆ 𝑆)
7467, 73unssd 4165 . . . . . . . . . . 11 ((𝑗 ∈ (ℙ ∖ {2}) ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚𝑆) → ((0...0) ∪ ((0 + 1)...(𝑗 − 1))) ⊆ 𝑆)
7556, 74eqsstrd 4008 . . . . . . . . . 10 ((𝑗 ∈ (ℙ ∖ {2}) ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚𝑆) → (0...(2 · ((𝑗 − 1) / 2))) ⊆ 𝑆)
76 oveq1 7158 . . . . . . . . . . . 12 (𝑘 = 𝑖 → (𝑘 · 𝑗) = (𝑖 · 𝑗))
7776eleq1d 2901 . . . . . . . . . . 11 (𝑘 = 𝑖 → ((𝑘 · 𝑗) ∈ 𝑆 ↔ (𝑖 · 𝑗) ∈ 𝑆))
7877cbvrabv 3496 . . . . . . . . . 10 {𝑘 ∈ ℕ ∣ (𝑘 · 𝑗) ∈ 𝑆} = {𝑖 ∈ ℕ ∣ (𝑖 · 𝑗) ∈ 𝑆}
79 eqid 2824 . . . . . . . . . 10 inf({𝑘 ∈ ℕ ∣ (𝑘 · 𝑗) ∈ 𝑆}, ℝ, < ) = inf({𝑘 ∈ ℕ ∣ (𝑘 · 𝑗) ∈ 𝑆}, ℝ, < )
8024, 31, 47, 33, 75, 78, 794sqlem18 16290 . . . . . . . . 9 ((𝑗 ∈ (ℙ ∖ {2}) ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚𝑆) → 𝑗𝑆)
8129, 80sylanbr 582 . . . . . . . 8 (((𝑗 ∈ ℙ ∧ 𝑗 ≠ 2) ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚𝑆) → 𝑗𝑆)
8281an32s 648 . . . . . . 7 (((𝑗 ∈ ℙ ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚𝑆) ∧ 𝑗 ≠ 2) → 𝑗𝑆)
8310, 10oveq12i 7163 . . . . . . . . . 10 (((abs‘1)↑2) + ((abs‘1)↑2)) = (1 + 1)
84 df-2 11692 . . . . . . . . . 10 2 = (1 + 1)
8583, 84eqtr4i 2851 . . . . . . . . 9 (((abs‘1)↑2) + ((abs‘1)↑2)) = 2
86244sqlem4a 16279 . . . . . . . . . 10 ((1 ∈ ℤ[i] ∧ 1 ∈ ℤ[i]) → (((abs‘1)↑2) + ((abs‘1)↑2)) ∈ 𝑆)
8720, 20, 86mp2an 688 . . . . . . . . 9 (((abs‘1)↑2) + ((abs‘1)↑2)) ∈ 𝑆
8885, 87eqeltrri 2914 . . . . . . . 8 2 ∈ 𝑆
8988a1i 11 . . . . . . 7 ((𝑗 ∈ ℙ ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚𝑆) → 2 ∈ 𝑆)
9028, 82, 89pm2.61ne 3106 . . . . . 6 ((𝑗 ∈ ℙ ∧ ∀𝑚 ∈ (1...(𝑗 − 1))𝑚𝑆) → 𝑗𝑆)
9124mul4sq 16282 . . . . . . 7 ((𝑚𝑆𝑖𝑆) → (𝑚 · 𝑖) ∈ 𝑆)
9291a1i 11 . . . . . 6 ((𝑚 ∈ (ℤ‘2) ∧ 𝑖 ∈ (ℤ‘2)) → ((𝑚𝑆𝑖𝑆) → (𝑚 · 𝑖) ∈ 𝑆))
932, 3, 4, 5, 6, 27, 90, 92prmind2 16021 . . . . 5 (𝑘 ∈ ℕ → 𝑘𝑆)
94 id 22 . . . . . 6 (𝑘 = 0 → 𝑘 = 0)
9594, 63syl6eqel 2925 . . . . 5 (𝑘 = 0 → 𝑘𝑆)
9693, 95jaoi 853 . . . 4 ((𝑘 ∈ ℕ ∨ 𝑘 = 0) → 𝑘𝑆)
971, 96sylbi 218 . . 3 (𝑘 ∈ ℕ0𝑘𝑆)
9897ssriv 3974 . 2 0𝑆
99244sqlem1 16276 . 2 𝑆 ⊆ ℕ0
10098, 99eqssi 3986 1 0 = 𝑆
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wo 843   = wceq 1530  wcel 2106  {cab 2802  wne 3020  wral 3142  wrex 3143  {crab 3146  cdif 3936  cun 3937  wss 3939  {csn 4563  cfv 6351  (class class class)co 7151  infcinf 8897  cc 10527  cr 10528  0cc0 10529  1c1 10530   + caddc 10532   · cmul 10534   < clt 10667  cmin 10862   / cdiv 11289  cn 11630  2c2 11684  0cn0 11889  cz 11973  cuz 12235  ...cfz 12885  cexp 13422  abscabs 14586  cprime 16007  ℤ[i]cgz 16257
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2152  ax-12 2167  ax-ext 2796  ax-rep 5186  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606  ax-pre-sup 10607
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2615  df-eu 2649  df-clab 2803  df-cleq 2817  df-clel 2897  df-nfc 2967  df-ne 3021  df-nel 3128  df-ral 3147  df-rex 3148  df-reu 3149  df-rmo 3150  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-tp 4568  df-op 4570  df-uni 4837  df-int 4874  df-iun 4918  df-br 5063  df-opab 5125  df-mpt 5143  df-tr 5169  df-id 5458  df-eprel 5463  df-po 5472  df-so 5473  df-fr 5512  df-we 5514  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-pred 6145  df-ord 6191  df-on 6192  df-lim 6193  df-suc 6194  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-riota 7109  df-ov 7154  df-oprab 7155  df-mpo 7156  df-om 7572  df-1st 7683  df-2nd 7684  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-1o 8096  df-2o 8097  df-oadd 8100  df-er 8282  df-en 8502  df-dom 8503  df-sdom 8504  df-fin 8505  df-sup 8898  df-inf 8899  df-dju 9322  df-card 9360  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-div 11290  df-nn 11631  df-2 11692  df-3 11693  df-4 11694  df-n0 11890  df-xnn0 11960  df-z 11974  df-uz 12236  df-rp 12383  df-fz 12886  df-fl 13155  df-mod 13231  df-seq 13363  df-exp 13423  df-hash 13684  df-cj 14451  df-re 14452  df-im 14453  df-sqrt 14587  df-abs 14588  df-dvds 15600  df-gcd 15836  df-prm 16008  df-gz 16258
This theorem is referenced by:  4sq  16292
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