4sqlem13m - Intuitionistic Logic Explorer

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Theorem 4sqlem13m 13101

Description: Lemma for 4sq 13108. (Contributed by Mario Carneiro, 16-Jul-2014.) (Revised by AV, 14-Sep-2020.)

**Hypotheses**
Ref	Expression
4sqlem11.1	⊢ 𝑆 = {𝑛 ∣ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ ∃𝑧 ∈ ℤ ∃𝑤 ∈ ℤ 𝑛 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))}
4sq.2	⊢ (𝜑 → 𝑁 ∈ ℕ)
4sq.3	⊢ (𝜑 → 𝑃 = ((2 · 𝑁) + 1))
4sq.4	⊢ (𝜑 → 𝑃 ∈ ℙ)
4sq.5	⊢ (𝜑 → (0...(2 · 𝑁)) ⊆ 𝑆)
4sq.6	⊢ 𝑇 = {𝑖 ∈ ℕ ∣ (𝑖 · 𝑃) ∈ 𝑆}
4sq.7	⊢ 𝑀 = inf(𝑇, ℝ, < )

**Assertion**
Ref	Expression
4sqlem13m	⊢ (𝜑 → (∃𝑗 𝑗 ∈ 𝑇 ∧ 𝑀 < 𝑃))

Distinct variable groups: 𝑛,𝑁 𝑃,𝑖,𝑛,𝑤,𝑥,𝑦,𝑧 𝑆,𝑖,𝑛 𝑇,𝑗 𝜑,𝑖,𝑛 Allowed substitution hints: 𝜑(𝑥,𝑦,𝑧,𝑤,𝑗) 𝑃(𝑗) 𝑆(𝑥,𝑦,𝑧,𝑤,𝑗) 𝑇(𝑥,𝑦,𝑧,𝑤,𝑖,𝑛) 𝑀(𝑥,𝑦,𝑧,𝑤,𝑖,𝑗,𝑛) 𝑁(𝑥,𝑦,𝑧,𝑤,𝑖,𝑗)

Proof of Theorem 4sqlem13m Dummy variables 𝑘 𝑢 𝑚 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		4sqlem11.1	. . 3 ⊢ 𝑆 = {𝑛 ∣ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ ∃𝑧 ∈ ℤ ∃𝑤 ∈ ℤ 𝑛 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))}
2		4sq.2	. . 3 ⊢ (𝜑 → 𝑁 ∈ ℕ)
3		4sq.3	. . 3 ⊢ (𝜑 → 𝑃 = ((2 · 𝑁) + 1))
4		4sq.4	. . 3 ⊢ (𝜑 → 𝑃 ∈ ℙ)
5		eqid 2232	. . 3 ⊢ {𝑢 ∣ ∃𝑚 ∈ (0...𝑁)𝑢 = ((𝑚↑2) mod 𝑃)} = {𝑢 ∣ ∃𝑚 ∈ (0...𝑁)𝑢 = ((𝑚↑2) mod 𝑃)}
6		eqid 2232	. . 3 ⊢ (𝑣 ∈ {𝑢 ∣ ∃𝑚 ∈ (0...𝑁)𝑢 = ((𝑚↑2) mod 𝑃)} ↦ ((𝑃 − 1) − 𝑣)) = (𝑣 ∈ {𝑢 ∣ ∃𝑚 ∈ (0...𝑁)𝑢 = ((𝑚↑2) mod 𝑃)} ↦ ((𝑃 − 1) − 𝑣))
7	1, 2, 3, 4, 5, 6	4sqlem12 13100	. 2 ⊢ (𝜑 → ∃𝑘 ∈ (1...(𝑃 − 1))∃𝑢 ∈ ℤ[i] (((abs‘𝑢)↑2) + 1) = (𝑘 · 𝑃))
8		simplrl 537	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑘 ∈ (1...(𝑃 − 1)) ∧ 𝑢 ∈ ℤ[i])) ∧ (((abs‘𝑢)↑2) + 1) = (𝑘 · 𝑃)) → 𝑘 ∈ (1...(𝑃 − 1)))
9		elfznn 10388	. . . . . . . 8 ⊢ (𝑘 ∈ (1...(𝑃 − 1)) → 𝑘 ∈ ℕ)
10	8, 9	syl 14	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑘 ∈ (1...(𝑃 − 1)) ∧ 𝑢 ∈ ℤ[i])) ∧ (((abs‘𝑢)↑2) + 1) = (𝑘 · 𝑃)) → 𝑘 ∈ ℕ)
11		simpr 110	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑘 ∈ (1...(𝑃 − 1)) ∧ 𝑢 ∈ ℤ[i])) ∧ (((abs‘𝑢)↑2) + 1) = (𝑘 · 𝑃)) → (((abs‘𝑢)↑2) + 1) = (𝑘 · 𝑃))
12		abs1 11757	. . . . . . . . . . . 12 ⊢ (abs‘1) = 1
13	12	oveq1i 6060	. . . . . . . . . . 11 ⊢ ((abs‘1)↑2) = (1↑2)
14		sq1 10995	. . . . . . . . . . 11 ⊢ (1↑2) = 1
15	13, 14	eqtri 2253	. . . . . . . . . 10 ⊢ ((abs‘1)↑2) = 1
16	15	oveq2i 6061	. . . . . . . . 9 ⊢ (((abs‘𝑢)↑2) + ((abs‘1)↑2)) = (((abs‘𝑢)↑2) + 1)
17		simplrr 538	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑘 ∈ (1...(𝑃 − 1)) ∧ 𝑢 ∈ ℤ[i])) ∧ (((abs‘𝑢)↑2) + 1) = (𝑘 · 𝑃)) → 𝑢 ∈ ℤ[i])
18		1z 9603	. . . . . . . . . . 11 ⊢ 1 ∈ ℤ
19		zgz 13071	. . . . . . . . . . 11 ⊢ (1 ∈ ℤ → 1 ∈ ℤ[i])
20	18, 19	ax-mp 5	. . . . . . . . . 10 ⊢ 1 ∈ ℤ[i]
21	1	4sqlem4a 13089	. . . . . . . . . 10 ⊢ ((𝑢 ∈ ℤ[i] ∧ 1 ∈ ℤ[i]) → (((abs‘𝑢)↑2) + ((abs‘1)↑2)) ∈ 𝑆)
22	17, 20, 21	sylancl 413	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑘 ∈ (1...(𝑃 − 1)) ∧ 𝑢 ∈ ℤ[i])) ∧ (((abs‘𝑢)↑2) + 1) = (𝑘 · 𝑃)) → (((abs‘𝑢)↑2) + ((abs‘1)↑2)) ∈ 𝑆)
23	16, 22	eqeltrrid 2320	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑘 ∈ (1...(𝑃 − 1)) ∧ 𝑢 ∈ ℤ[i])) ∧ (((abs‘𝑢)↑2) + 1) = (𝑘 · 𝑃)) → (((abs‘𝑢)↑2) + 1) ∈ 𝑆)
24	11, 23	eqeltrrd 2310	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑘 ∈ (1...(𝑃 − 1)) ∧ 𝑢 ∈ ℤ[i])) ∧ (((abs‘𝑢)↑2) + 1) = (𝑘 · 𝑃)) → (𝑘 · 𝑃) ∈ 𝑆)
25		oveq1 6057	. . . . . . . . 9 ⊢ (𝑖 = 𝑘 → (𝑖 · 𝑃) = (𝑘 · 𝑃))
26	25	eleq1d 2301	. . . . . . . 8 ⊢ (𝑖 = 𝑘 → ((𝑖 · 𝑃) ∈ 𝑆 ↔ (𝑘 · 𝑃) ∈ 𝑆))
27		4sq.6	. . . . . . . 8 ⊢ 𝑇 = {𝑖 ∈ ℕ ∣ (𝑖 · 𝑃) ∈ 𝑆}
28	26, 27	elrab2 2976	. . . . . . 7 ⊢ (𝑘 ∈ 𝑇 ↔ (𝑘 ∈ ℕ ∧ (𝑘 · 𝑃) ∈ 𝑆))
29	10, 24, 28	sylanbrc 417	. . . . . 6 ⊢ (((𝜑 ∧ (𝑘 ∈ (1...(𝑃 − 1)) ∧ 𝑢 ∈ ℤ[i])) ∧ (((abs‘𝑢)↑2) + 1) = (𝑘 · 𝑃)) → 𝑘 ∈ 𝑇)
30		elex2 2830	. . . . . 6 ⊢ (𝑘 ∈ 𝑇 → ∃𝑗 𝑗 ∈ 𝑇)
31	29, 30	syl 14	. . . . 5 ⊢ (((𝜑 ∧ (𝑘 ∈ (1...(𝑃 − 1)) ∧ 𝑢 ∈ ℤ[i])) ∧ (((abs‘𝑢)↑2) + 1) = (𝑘 · 𝑃)) → ∃𝑗 𝑗 ∈ 𝑇)
32	27	ssrab3 3324	. . . . . . . 8 ⊢ 𝑇 ⊆ ℕ
33		4sq.7	. . . . . . . . 9 ⊢ 𝑀 = inf(𝑇, ℝ, < )
34		1zzd 9604	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑘 ∈ (1...(𝑃 − 1)) ∧ 𝑢 ∈ ℤ[i])) ∧ (((abs‘𝑢)↑2) + 1) = (𝑘 · 𝑃)) → 1 ∈ ℤ)
35		nnuz 9890	. . . . . . . . . . . 12 ⊢ ℕ = (ℤ_≥‘1)
36	35	rabeqi 2806	. . . . . . . . . . 11 ⊢ {𝑖 ∈ ℕ ∣ (𝑖 · 𝑃) ∈ 𝑆} = {𝑖 ∈ (ℤ_≥‘1) ∣ (𝑖 · 𝑃) ∈ 𝑆}
37	27, 36	eqtri 2253	. . . . . . . . . 10 ⊢ 𝑇 = {𝑖 ∈ (ℤ_≥‘1) ∣ (𝑖 · 𝑃) ∈ 𝑆}
38		elfznn 10388	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∈ (1...𝑘) → 𝑖 ∈ ℕ)
39	38	adantl 277	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑘 ∈ (1...(𝑃 − 1)) ∧ 𝑢 ∈ ℤ[i])) ∧ (((abs‘𝑢)↑2) + 1) = (𝑘 · 𝑃)) ∧ 𝑖 ∈ (1...𝑘)) → 𝑖 ∈ ℕ)
40		prmnn 12807	. . . . . . . . . . . . . . 15 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
41	4, 40	syl 14	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑃 ∈ ℕ)
42	41	ad3antrrr 492	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑘 ∈ (1...(𝑃 − 1)) ∧ 𝑢 ∈ ℤ[i])) ∧ (((abs‘𝑢)↑2) + 1) = (𝑘 · 𝑃)) ∧ 𝑖 ∈ (1...𝑘)) → 𝑃 ∈ ℕ)
43	39, 42	nnmulcld 9286	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑘 ∈ (1...(𝑃 − 1)) ∧ 𝑢 ∈ ℤ[i])) ∧ (((abs‘𝑢)↑2) + 1) = (𝑘 · 𝑃)) ∧ 𝑖 ∈ (1...𝑘)) → (𝑖 · 𝑃) ∈ ℕ)
44	43	nnnn0d 9553	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑘 ∈ (1...(𝑃 − 1)) ∧ 𝑢 ∈ ℤ[i])) ∧ (((abs‘𝑢)↑2) + 1) = (𝑘 · 𝑃)) ∧ 𝑖 ∈ (1...𝑘)) → (𝑖 · 𝑃) ∈ ℕ₀)
45	1	4sqlemsdc 13098	. . . . . . . . . . 11 ⊢ ((𝑖 · 𝑃) ∈ ℕ₀ → DECID (𝑖 · 𝑃) ∈ 𝑆)
46	44, 45	syl 14	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑘 ∈ (1...(𝑃 − 1)) ∧ 𝑢 ∈ ℤ[i])) ∧ (((abs‘𝑢)↑2) + 1) = (𝑘 · 𝑃)) ∧ 𝑖 ∈ (1...𝑘)) → DECID (𝑖 · 𝑃) ∈ 𝑆)
47	34, 37, 29, 46	infssuzcldc 10595	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑘 ∈ (1...(𝑃 − 1)) ∧ 𝑢 ∈ ℤ[i])) ∧ (((abs‘𝑢)↑2) + 1) = (𝑘 · 𝑃)) → inf(𝑇, ℝ, < ) ∈ 𝑇)
48	33, 47	eqeltrid 2319	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑘 ∈ (1...(𝑃 − 1)) ∧ 𝑢 ∈ ℤ[i])) ∧ (((abs‘𝑢)↑2) + 1) = (𝑘 · 𝑃)) → 𝑀 ∈ 𝑇)
49	32, 48	sselid 3236	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑘 ∈ (1...(𝑃 − 1)) ∧ 𝑢 ∈ ℤ[i])) ∧ (((abs‘𝑢)↑2) + 1) = (𝑘 · 𝑃)) → 𝑀 ∈ ℕ)
50	49	nnred 9250	. . . . . 6 ⊢ (((𝜑 ∧ (𝑘 ∈ (1...(𝑃 − 1)) ∧ 𝑢 ∈ ℤ[i])) ∧ (((abs‘𝑢)↑2) + 1) = (𝑘 · 𝑃)) → 𝑀 ∈ ℝ)
51	10	nnred 9250	. . . . . 6 ⊢ (((𝜑 ∧ (𝑘 ∈ (1...(𝑃 − 1)) ∧ 𝑢 ∈ ℤ[i])) ∧ (((abs‘𝑢)↑2) + 1) = (𝑘 · 𝑃)) → 𝑘 ∈ ℝ)
52	41	nnred 9250	. . . . . . 7 ⊢ (𝜑 → 𝑃 ∈ ℝ)
53	52	ad2antrr 488	. . . . . 6 ⊢ (((𝜑 ∧ (𝑘 ∈ (1...(𝑃 − 1)) ∧ 𝑢 ∈ ℤ[i])) ∧ (((abs‘𝑢)↑2) + 1) = (𝑘 · 𝑃)) → 𝑃 ∈ ℝ)
54	34, 37, 29, 46	infssuzledc 10594	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑘 ∈ (1...(𝑃 − 1)) ∧ 𝑢 ∈ ℤ[i])) ∧ (((abs‘𝑢)↑2) + 1) = (𝑘 · 𝑃)) → inf(𝑇, ℝ, < ) ≤ 𝑘)
55	33, 54	eqbrtrid 4144	. . . . . 6 ⊢ (((𝜑 ∧ (𝑘 ∈ (1...(𝑃 − 1)) ∧ 𝑢 ∈ ℤ[i])) ∧ (((abs‘𝑢)↑2) + 1) = (𝑘 · 𝑃)) → 𝑀 ≤ 𝑘)
56		prmz 12808	. . . . . . . . . . 11 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℤ)
57	4, 56	syl 14	. . . . . . . . . 10 ⊢ (𝜑 → 𝑃 ∈ ℤ)
58	57	ad2antrr 488	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑘 ∈ (1...(𝑃 − 1)) ∧ 𝑢 ∈ ℤ[i])) ∧ (((abs‘𝑢)↑2) + 1) = (𝑘 · 𝑃)) → 𝑃 ∈ ℤ)
59		elfzm11 10425	. . . . . . . . 9 ⊢ ((1 ∈ ℤ ∧ 𝑃 ∈ ℤ) → (𝑘 ∈ (1...(𝑃 − 1)) ↔ (𝑘 ∈ ℤ ∧ 1 ≤ 𝑘 ∧ 𝑘 < 𝑃)))
60	18, 58, 59	sylancr 414	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑘 ∈ (1...(𝑃 − 1)) ∧ 𝑢 ∈ ℤ[i])) ∧ (((abs‘𝑢)↑2) + 1) = (𝑘 · 𝑃)) → (𝑘 ∈ (1...(𝑃 − 1)) ↔ (𝑘 ∈ ℤ ∧ 1 ≤ 𝑘 ∧ 𝑘 < 𝑃)))
61	8, 60	mpbid 147	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑘 ∈ (1...(𝑃 − 1)) ∧ 𝑢 ∈ ℤ[i])) ∧ (((abs‘𝑢)↑2) + 1) = (𝑘 · 𝑃)) → (𝑘 ∈ ℤ ∧ 1 ≤ 𝑘 ∧ 𝑘 < 𝑃))
62	61	simp3d 1038	. . . . . 6 ⊢ (((𝜑 ∧ (𝑘 ∈ (1...(𝑃 − 1)) ∧ 𝑢 ∈ ℤ[i])) ∧ (((abs‘𝑢)↑2) + 1) = (𝑘 · 𝑃)) → 𝑘 < 𝑃)
63	50, 51, 53, 55, 62	lelttrd 8398	. . . . 5 ⊢ (((𝜑 ∧ (𝑘 ∈ (1...(𝑃 − 1)) ∧ 𝑢 ∈ ℤ[i])) ∧ (((abs‘𝑢)↑2) + 1) = (𝑘 · 𝑃)) → 𝑀 < 𝑃)
64	31, 63	jca 306	. . . 4 ⊢ (((𝜑 ∧ (𝑘 ∈ (1...(𝑃 − 1)) ∧ 𝑢 ∈ ℤ[i])) ∧ (((abs‘𝑢)↑2) + 1) = (𝑘 · 𝑃)) → (∃𝑗 𝑗 ∈ 𝑇 ∧ 𝑀 < 𝑃))
65	64	ex 115	. . 3 ⊢ ((𝜑 ∧ (𝑘 ∈ (1...(𝑃 − 1)) ∧ 𝑢 ∈ ℤ[i])) → ((((abs‘𝑢)↑2) + 1) = (𝑘 · 𝑃) → (∃𝑗 𝑗 ∈ 𝑇 ∧ 𝑀 < 𝑃)))
66	65	rexlimdvva 2668	. 2 ⊢ (𝜑 → (∃𝑘 ∈ (1...(𝑃 − 1))∃𝑢 ∈ ℤ[i] (((abs‘𝑢)↑2) + 1) = (𝑘 · 𝑃) → (∃𝑗 𝑗 ∈ 𝑇 ∧ 𝑀 < 𝑃)))
67	7, 66	mpd 13	1 ⊢ (𝜑 → (∃𝑗 𝑗 ∈ 𝑇 ∧ 𝑀 < 𝑃))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 DECID wdc 842 ∧ w3a 1005 = wceq 1398 ∃wex 1541 ∈ wcel 2203 {cab 2218 ∃wrex 2521 {crab 2524 ⊆ wss 3211 class class class wbr 4109 ↦ cmpt 4171 ‘cfv 5352 (class class class)co 6050 infcinf 7274 ℝcr 8126 0cc0 8127 1c1 8128 + caddc 8130 · cmul 8132 < clt 8308 ≤ cle 8309 − cmin 8444 ℕcn 9237 2c2 9288 ℕ₀cn0 9496 ℤcz 9577 ℤ_≥cuz 9853 ...cfz 10342 mod cmo 10684 ↑cexp 10900 abscabs 11682 ℙcprime 12804 ℤ[i]cgz 13067

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2205 ax-14 2206 ax-ext 2214 ax-coll 4225 ax-sep 4228 ax-nul 4236 ax-pow 4287 ax-pr 4322 ax-un 4554 ax-setind 4659 ax-iinf 4710 ax-cnex 8218 ax-resscn 8219 ax-1cn 8220 ax-1re 8221 ax-icn 8222 ax-addcl 8223 ax-addrcl 8224 ax-mulcl 8225 ax-mulrcl 8226 ax-addcom 8227 ax-mulcom 8228 ax-addass 8229 ax-mulass 8230 ax-distr 8231 ax-i2m1 8232 ax-0lt1 8233 ax-1rid 8234 ax-0id 8235 ax-rnegex 8236 ax-precex 8237 ax-cnre 8238 ax-pre-ltirr 8239 ax-pre-ltwlin 8240 ax-pre-lttrn 8241 ax-pre-apti 8242 ax-pre-ltadd 8243 ax-pre-mulgt0 8244 ax-pre-mulext 8245 ax-arch 8246 ax-caucvg 8247

This theorem depends on definitions: df-bi 117 df-stab 839 df-dc 843 df-3or 1006 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1812 df-eu 2083 df-mo 2084 df-clab 2219 df-cleq 2225 df-clel 2228 df-nfc 2373 df-ne 2413 df-nel 2508 df-ral 2525 df-rex 2526 df-reu 2527 df-rmo 2528 df-rab 2529 df-v 2815 df-sbc 3043 df-csb 3139 df-dif 3213 df-un 3215 df-in 3217 df-ss 3224 df-nul 3509 df-if 3621 df-pw 3671 df-sn 3695 df-pr 3696 df-op 3698 df-uni 3915 df-int 3950 df-iun 3993 df-br 4110 df-opab 4172 df-mpt 4173 df-tr 4209 df-id 4414 df-po 4417 df-iso 4418 df-iord 4487 df-on 4489 df-ilim 4490 df-suc 4492 df-iom 4713 df-xp 4755 df-rel 4756 df-cnv 4757 df-co 4758 df-dm 4759 df-rn 4760 df-res 4761 df-ima 4762 df-iota 5312 df-fun 5354 df-fn 5355 df-f 5356 df-f1 5357 df-fo 5358 df-f1o 5359 df-fv 5360 df-isom 5361 df-riota 6003 df-ov 6053 df-oprab 6054 df-mpo 6055 df-1st 6334 df-2nd 6335 df-recs 6536 df-irdg 6601 df-frec 6622 df-1o 6647 df-2o 6648 df-oadd 6651 df-er 6767 df-en 6976 df-dom 6977 df-fin 6978 df-sup 7275 df-inf 7276 df-pnf 8310 df-mnf 8311 df-xr 8312 df-ltxr 8313 df-le 8314 df-sub 8446 df-neg 8447 df-reap 8849 df-ap 8856 df-div 8947 df-inn 9238 df-2 9296 df-3 9297 df-4 9298 df-n0 9497 df-z 9578 df-uz 9854 df-q 9952 df-rp 9987 df-fz 10343 df-fzo 10477 df-fl 10630 df-mod 10685 df-seqfrec 10810 df-exp 10901 df-ihash 11139 df-cj 11527 df-re 11528 df-im 11529 df-rsqrt 11683 df-abs 11684 df-dvds 12474 df-gcd 12650 df-prm 12805 df-gz 13068

This theorem is referenced by: 4sqlem14 13102 4sqlem17 13105 4sqlem18 13106

W3C validator