4sqlem13m - Intuitionistic Logic Explorer

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Theorem 4sqlem13m 12572

Description: Lemma for 4sq 12579. (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 2196	. . 3 ⊢ {𝑢 ∣ ∃𝑚 ∈ (0...𝑁)𝑢 = ((𝑚↑2) mod 𝑃)} = {𝑢 ∣ ∃𝑚 ∈ (0...𝑁)𝑢 = ((𝑚↑2) mod 𝑃)}
6		eqid 2196	. . 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 12571	. 2 ⊢ (𝜑 → ∃𝑘 ∈ (1...(𝑃 − 1))∃𝑢 ∈ ℤ[i] (((abs‘𝑢)↑2) + 1) = (𝑘 · 𝑃))
8		simplrl 535	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑘 ∈ (1...(𝑃 − 1)) ∧ 𝑢 ∈ ℤ[i])) ∧ (((abs‘𝑢)↑2) + 1) = (𝑘 · 𝑃)) → 𝑘 ∈ (1...(𝑃 − 1)))
9		elfznn 10129	. . . . . . . 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 11237	. . . . . . . . . . . 12 ⊢ (abs‘1) = 1
13	12	oveq1i 5932	. . . . . . . . . . 11 ⊢ ((abs‘1)↑2) = (1↑2)
14		sq1 10725	. . . . . . . . . . 11 ⊢ (1↑2) = 1
15	13, 14	eqtri 2217	. . . . . . . . . 10 ⊢ ((abs‘1)↑2) = 1
16	15	oveq2i 5933	. . . . . . . . 9 ⊢ (((abs‘𝑢)↑2) + ((abs‘1)↑2)) = (((abs‘𝑢)↑2) + 1)
17		simplrr 536	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑘 ∈ (1...(𝑃 − 1)) ∧ 𝑢 ∈ ℤ[i])) ∧ (((abs‘𝑢)↑2) + 1) = (𝑘 · 𝑃)) → 𝑢 ∈ ℤ[i])
18		1z 9352	. . . . . . . . . . 11 ⊢ 1 ∈ ℤ
19		zgz 12542	. . . . . . . . . . 11 ⊢ (1 ∈ ℤ → 1 ∈ ℤ[i])
20	18, 19	ax-mp 5	. . . . . . . . . 10 ⊢ 1 ∈ ℤ[i]
21	1	4sqlem4a 12560	. . . . . . . . . 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 2284	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑘 ∈ (1...(𝑃 − 1)) ∧ 𝑢 ∈ ℤ[i])) ∧ (((abs‘𝑢)↑2) + 1) = (𝑘 · 𝑃)) → (((abs‘𝑢)↑2) + 1) ∈ 𝑆)
24	11, 23	eqeltrrd 2274	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑘 ∈ (1...(𝑃 − 1)) ∧ 𝑢 ∈ ℤ[i])) ∧ (((abs‘𝑢)↑2) + 1) = (𝑘 · 𝑃)) → (𝑘 · 𝑃) ∈ 𝑆)
25		oveq1 5929	. . . . . . . . 9 ⊢ (𝑖 = 𝑘 → (𝑖 · 𝑃) = (𝑘 · 𝑃))
26	25	eleq1d 2265	. . . . . . . 8 ⊢ (𝑖 = 𝑘 → ((𝑖 · 𝑃) ∈ 𝑆 ↔ (𝑘 · 𝑃) ∈ 𝑆))
27		4sq.6	. . . . . . . 8 ⊢ 𝑇 = {𝑖 ∈ ℕ ∣ (𝑖 · 𝑃) ∈ 𝑆}
28	26, 27	elrab2 2923	. . . . . . 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 2779	. . . . . 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 3269	. . . . . . . 8 ⊢ 𝑇 ⊆ ℕ
33		4sq.7	. . . . . . . . 9 ⊢ 𝑀 = inf(𝑇, ℝ, < )
34		1zzd 9353	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑘 ∈ (1...(𝑃 − 1)) ∧ 𝑢 ∈ ℤ[i])) ∧ (((abs‘𝑢)↑2) + 1) = (𝑘 · 𝑃)) → 1 ∈ ℤ)
35		nnuz 9637	. . . . . . . . . . . 12 ⊢ ℕ = (ℤ_≥‘1)
36	35	rabeqi 2756	. . . . . . . . . . 11 ⊢ {𝑖 ∈ ℕ ∣ (𝑖 · 𝑃) ∈ 𝑆} = {𝑖 ∈ (ℤ_≥‘1) ∣ (𝑖 · 𝑃) ∈ 𝑆}
37	27, 36	eqtri 2217	. . . . . . . . . 10 ⊢ 𝑇 = {𝑖 ∈ (ℤ_≥‘1) ∣ (𝑖 · 𝑃) ∈ 𝑆}
38		elfznn 10129	. . . . . . . . . . . . . 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 12278	. . . . . . . . . . . . . . 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 9039	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑘 ∈ (1...(𝑃 − 1)) ∧ 𝑢 ∈ ℤ[i])) ∧ (((abs‘𝑢)↑2) + 1) = (𝑘 · 𝑃)) ∧ 𝑖 ∈ (1...𝑘)) → (𝑖 · 𝑃) ∈ ℕ)
44	43	nnnn0d 9302	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑘 ∈ (1...(𝑃 − 1)) ∧ 𝑢 ∈ ℤ[i])) ∧ (((abs‘𝑢)↑2) + 1) = (𝑘 · 𝑃)) ∧ 𝑖 ∈ (1...𝑘)) → (𝑖 · 𝑃) ∈ ℕ₀)
45	1	4sqlemsdc 12569	. . . . . . . . . . 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 10325	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑘 ∈ (1...(𝑃 − 1)) ∧ 𝑢 ∈ ℤ[i])) ∧ (((abs‘𝑢)↑2) + 1) = (𝑘 · 𝑃)) → inf(𝑇, ℝ, < ) ∈ 𝑇)
48	33, 47	eqeltrid 2283	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑘 ∈ (1...(𝑃 − 1)) ∧ 𝑢 ∈ ℤ[i])) ∧ (((abs‘𝑢)↑2) + 1) = (𝑘 · 𝑃)) → 𝑀 ∈ 𝑇)
49	32, 48	sselid 3181	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑘 ∈ (1...(𝑃 − 1)) ∧ 𝑢 ∈ ℤ[i])) ∧ (((abs‘𝑢)↑2) + 1) = (𝑘 · 𝑃)) → 𝑀 ∈ ℕ)
50	49	nnred 9003	. . . . . 6 ⊢ (((𝜑 ∧ (𝑘 ∈ (1...(𝑃 − 1)) ∧ 𝑢 ∈ ℤ[i])) ∧ (((abs‘𝑢)↑2) + 1) = (𝑘 · 𝑃)) → 𝑀 ∈ ℝ)
51	10	nnred 9003	. . . . . 6 ⊢ (((𝜑 ∧ (𝑘 ∈ (1...(𝑃 − 1)) ∧ 𝑢 ∈ ℤ[i])) ∧ (((abs‘𝑢)↑2) + 1) = (𝑘 · 𝑃)) → 𝑘 ∈ ℝ)
52	41	nnred 9003	. . . . . . 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 10324	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑘 ∈ (1...(𝑃 − 1)) ∧ 𝑢 ∈ ℤ[i])) ∧ (((abs‘𝑢)↑2) + 1) = (𝑘 · 𝑃)) → inf(𝑇, ℝ, < ) ≤ 𝑘)
55	33, 54	eqbrtrid 4068	. . . . . 6 ⊢ (((𝜑 ∧ (𝑘 ∈ (1...(𝑃 − 1)) ∧ 𝑢 ∈ ℤ[i])) ∧ (((abs‘𝑢)↑2) + 1) = (𝑘 · 𝑃)) → 𝑀 ≤ 𝑘)
56		prmz 12279	. . . . . . . . . . 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 10166	. . . . . . . . 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 1013	. . . . . 6 ⊢ (((𝜑 ∧ (𝑘 ∈ (1...(𝑃 − 1)) ∧ 𝑢 ∈ ℤ[i])) ∧ (((abs‘𝑢)↑2) + 1) = (𝑘 · 𝑃)) → 𝑘 < 𝑃)
63	50, 51, 53, 55, 62	lelttrd 8151	. . . . 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 2622	. 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 835 ∧ w3a 980 = wceq 1364 ∃wex 1506 ∈ wcel 2167 {cab 2182 ∃wrex 2476 {crab 2479 ⊆ wss 3157 class class class wbr 4033 ↦ cmpt 4094 ‘cfv 5258 (class class class)co 5922 infcinf 7049 ℝcr 7878 0cc0 7879 1c1 7880 + caddc 7882 · cmul 7884 < clt 8061 ≤ cle 8062 − cmin 8197 ℕcn 8990 2c2 9041 ℕ₀cn0 9249 ℤcz 9326 ℤ_≥cuz 9601 ...cfz 10083 mod cmo 10414 ↑cexp 10630 abscabs 11162 ℙcprime 12275 ℤ[i]cgz 12538

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 615 ax-in2 616 ax-io 710 ax-5 1461 ax-7 1462 ax-gen 1463 ax-ie1 1507 ax-ie2 1508 ax-8 1518 ax-10 1519 ax-11 1520 ax-i12 1521 ax-bndl 1523 ax-4 1524 ax-17 1540 ax-i9 1544 ax-ial 1548 ax-i5r 1549 ax-13 2169 ax-14 2170 ax-ext 2178 ax-coll 4148 ax-sep 4151 ax-nul 4159 ax-pow 4207 ax-pr 4242 ax-un 4468 ax-setind 4573 ax-iinf 4624 ax-cnex 7970 ax-resscn 7971 ax-1cn 7972 ax-1re 7973 ax-icn 7974 ax-addcl 7975 ax-addrcl 7976 ax-mulcl 7977 ax-mulrcl 7978 ax-addcom 7979 ax-mulcom 7980 ax-addass 7981 ax-mulass 7982 ax-distr 7983 ax-i2m1 7984 ax-0lt1 7985 ax-1rid 7986 ax-0id 7987 ax-rnegex 7988 ax-precex 7989 ax-cnre 7990 ax-pre-ltirr 7991 ax-pre-ltwlin 7992 ax-pre-lttrn 7993 ax-pre-apti 7994 ax-pre-ltadd 7995 ax-pre-mulgt0 7996 ax-pre-mulext 7997 ax-arch 7998 ax-caucvg 7999

This theorem depends on definitions: df-bi 117 df-stab 832 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1475 df-sb 1777 df-eu 2048 df-mo 2049 df-clab 2183 df-cleq 2189 df-clel 2192 df-nfc 2328 df-ne 2368 df-nel 2463 df-ral 2480 df-rex 2481 df-reu 2482 df-rmo 2483 df-rab 2484 df-v 2765 df-sbc 2990 df-csb 3085 df-dif 3159 df-un 3161 df-in 3163 df-ss 3170 df-nul 3451 df-if 3562 df-pw 3607 df-sn 3628 df-pr 3629 df-op 3631 df-uni 3840 df-int 3875 df-iun 3918 df-br 4034 df-opab 4095 df-mpt 4096 df-tr 4132 df-id 4328 df-po 4331 df-iso 4332 df-iord 4401 df-on 4403 df-ilim 4404 df-suc 4406 df-iom 4627 df-xp 4669 df-rel 4670 df-cnv 4671 df-co 4672 df-dm 4673 df-rn 4674 df-res 4675 df-ima 4676 df-iota 5219 df-fun 5260 df-fn 5261 df-f 5262 df-f1 5263 df-fo 5264 df-f1o 5265 df-fv 5266 df-isom 5267 df-riota 5877 df-ov 5925 df-oprab 5926 df-mpo 5927 df-1st 6198 df-2nd 6199 df-recs 6363 df-irdg 6428 df-frec 6449 df-1o 6474 df-2o 6475 df-oadd 6478 df-er 6592 df-en 6800 df-dom 6801 df-fin 6802 df-sup 7050 df-inf 7051 df-pnf 8063 df-mnf 8064 df-xr 8065 df-ltxr 8066 df-le 8067 df-sub 8199 df-neg 8200 df-reap 8602 df-ap 8609 df-div 8700 df-inn 8991 df-2 9049 df-3 9050 df-4 9051 df-n0 9250 df-z 9327 df-uz 9602 df-q 9694 df-rp 9729 df-fz 10084 df-fzo 10218 df-fl 10360 df-mod 10415 df-seqfrec 10540 df-exp 10631 df-ihash 10868 df-cj 11007 df-re 11008 df-im 11009 df-rsqrt 11163 df-abs 11164 df-dvds 11953 df-gcd 12121 df-prm 12276 df-gz 12539

This theorem is referenced by: 4sqlem14 12573 4sqlem17 12576 4sqlem18 12577

W3C validator