4sqlem13m - Intuitionistic Logic Explorer

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Theorem 4sqlem13m 12438

Description: Lemma for 4sq 12445. (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 2189	. . 3 ⊢ {𝑢 ∣ ∃𝑚 ∈ (0...𝑁)𝑢 = ((𝑚↑2) mod 𝑃)} = {𝑢 ∣ ∃𝑚 ∈ (0...𝑁)𝑢 = ((𝑚↑2) mod 𝑃)}
6		eqid 2189	. . 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 12437	. 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 10086	. . . . . . . 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 11116	. . . . . . . . . . . 12 ⊢ (abs‘1) = 1
13	12	oveq1i 5907	. . . . . . . . . . 11 ⊢ ((abs‘1)↑2) = (1↑2)
14		sq1 10648	. . . . . . . . . . 11 ⊢ (1↑2) = 1
15	13, 14	eqtri 2210	. . . . . . . . . 10 ⊢ ((abs‘1)↑2) = 1
16	15	oveq2i 5908	. . . . . . . . 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 9310	. . . . . . . . . . 11 ⊢ 1 ∈ ℤ
19		zgz 12408	. . . . . . . . . . 11 ⊢ (1 ∈ ℤ → 1 ∈ ℤ[i])
20	18, 19	ax-mp 5	. . . . . . . . . 10 ⊢ 1 ∈ ℤ[i]
21	1	4sqlem4a 12426	. . . . . . . . . 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 2277	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑘 ∈ (1...(𝑃 − 1)) ∧ 𝑢 ∈ ℤ[i])) ∧ (((abs‘𝑢)↑2) + 1) = (𝑘 · 𝑃)) → (((abs‘𝑢)↑2) + 1) ∈ 𝑆)
24	11, 23	eqeltrrd 2267	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑘 ∈ (1...(𝑃 − 1)) ∧ 𝑢 ∈ ℤ[i])) ∧ (((abs‘𝑢)↑2) + 1) = (𝑘 · 𝑃)) → (𝑘 · 𝑃) ∈ 𝑆)
25		oveq1 5904	. . . . . . . . 9 ⊢ (𝑖 = 𝑘 → (𝑖 · 𝑃) = (𝑘 · 𝑃))
26	25	eleq1d 2258	. . . . . . . 8 ⊢ (𝑖 = 𝑘 → ((𝑖 · 𝑃) ∈ 𝑆 ↔ (𝑘 · 𝑃) ∈ 𝑆))
27		4sq.6	. . . . . . . 8 ⊢ 𝑇 = {𝑖 ∈ ℕ ∣ (𝑖 · 𝑃) ∈ 𝑆}
28	26, 27	elrab2 2911	. . . . . . 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 2768	. . . . . 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 3256	. . . . . . . 8 ⊢ 𝑇 ⊆ ℕ
33		4sq.7	. . . . . . . . 9 ⊢ 𝑀 = inf(𝑇, ℝ, < )
34		1zzd 9311	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑘 ∈ (1...(𝑃 − 1)) ∧ 𝑢 ∈ ℤ[i])) ∧ (((abs‘𝑢)↑2) + 1) = (𝑘 · 𝑃)) → 1 ∈ ℤ)
35		nnuz 9595	. . . . . . . . . . . 12 ⊢ ℕ = (ℤ_≥‘1)
36	35	rabeqi 2745	. . . . . . . . . . 11 ⊢ {𝑖 ∈ ℕ ∣ (𝑖 · 𝑃) ∈ 𝑆} = {𝑖 ∈ (ℤ_≥‘1) ∣ (𝑖 · 𝑃) ∈ 𝑆}
37	27, 36	eqtri 2210	. . . . . . . . . 10 ⊢ 𝑇 = {𝑖 ∈ (ℤ_≥‘1) ∣ (𝑖 · 𝑃) ∈ 𝑆}
38		elfznn 10086	. . . . . . . . . . . . . 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 12145	. . . . . . . . . . . . . . 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 8999	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑘 ∈ (1...(𝑃 − 1)) ∧ 𝑢 ∈ ℤ[i])) ∧ (((abs‘𝑢)↑2) + 1) = (𝑘 · 𝑃)) ∧ 𝑖 ∈ (1...𝑘)) → (𝑖 · 𝑃) ∈ ℕ)
44	43	nnnn0d 9260	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑘 ∈ (1...(𝑃 − 1)) ∧ 𝑢 ∈ ℤ[i])) ∧ (((abs‘𝑢)↑2) + 1) = (𝑘 · 𝑃)) ∧ 𝑖 ∈ (1...𝑘)) → (𝑖 · 𝑃) ∈ ℕ₀)
45	1	4sqlemsdc 12435	. . . . . . . . . . 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 11987	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑘 ∈ (1...(𝑃 − 1)) ∧ 𝑢 ∈ ℤ[i])) ∧ (((abs‘𝑢)↑2) + 1) = (𝑘 · 𝑃)) → inf(𝑇, ℝ, < ) ∈ 𝑇)
48	33, 47	eqeltrid 2276	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑘 ∈ (1...(𝑃 − 1)) ∧ 𝑢 ∈ ℤ[i])) ∧ (((abs‘𝑢)↑2) + 1) = (𝑘 · 𝑃)) → 𝑀 ∈ 𝑇)
49	32, 48	sselid 3168	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑘 ∈ (1...(𝑃 − 1)) ∧ 𝑢 ∈ ℤ[i])) ∧ (((abs‘𝑢)↑2) + 1) = (𝑘 · 𝑃)) → 𝑀 ∈ ℕ)
50	49	nnred 8963	. . . . . 6 ⊢ (((𝜑 ∧ (𝑘 ∈ (1...(𝑃 − 1)) ∧ 𝑢 ∈ ℤ[i])) ∧ (((abs‘𝑢)↑2) + 1) = (𝑘 · 𝑃)) → 𝑀 ∈ ℝ)
51	10	nnred 8963	. . . . . 6 ⊢ (((𝜑 ∧ (𝑘 ∈ (1...(𝑃 − 1)) ∧ 𝑢 ∈ ℤ[i])) ∧ (((abs‘𝑢)↑2) + 1) = (𝑘 · 𝑃)) → 𝑘 ∈ ℝ)
52	41	nnred 8963	. . . . . . 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 11986	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑘 ∈ (1...(𝑃 − 1)) ∧ 𝑢 ∈ ℤ[i])) ∧ (((abs‘𝑢)↑2) + 1) = (𝑘 · 𝑃)) → inf(𝑇, ℝ, < ) ≤ 𝑘)
55	33, 54	eqbrtrid 4053	. . . . . 6 ⊢ (((𝜑 ∧ (𝑘 ∈ (1...(𝑃 − 1)) ∧ 𝑢 ∈ ℤ[i])) ∧ (((abs‘𝑢)↑2) + 1) = (𝑘 · 𝑃)) → 𝑀 ≤ 𝑘)
56		prmz 12146	. . . . . . . . . . 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 10123	. . . . . . . . 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 8113	. . . . 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 2615	. 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 1503 ∈ wcel 2160 {cab 2175 ∃wrex 2469 {crab 2472 ⊆ wss 3144 class class class wbr 4018 ↦ cmpt 4079 ‘cfv 5235 (class class class)co 5897 infcinf 7013 ℝcr 7841 0cc0 7842 1c1 7843 + caddc 7845 · cmul 7847 < clt 8023 ≤ cle 8024 − cmin 8159 ℕcn 8950 2c2 9001 ℕ₀cn0 9207 ℤcz 9284 ℤ_≥cuz 9559 ...cfz 10040 mod cmo 10355 ↑cexp 10553 abscabs 11041 ℙcprime 12142 ℤ[i]cgz 12404

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 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-coll 4133 ax-sep 4136 ax-nul 4144 ax-pow 4192 ax-pr 4227 ax-un 4451 ax-setind 4554 ax-iinf 4605 ax-cnex 7933 ax-resscn 7934 ax-1cn 7935 ax-1re 7936 ax-icn 7937 ax-addcl 7938 ax-addrcl 7939 ax-mulcl 7940 ax-mulrcl 7941 ax-addcom 7942 ax-mulcom 7943 ax-addass 7944 ax-mulass 7945 ax-distr 7946 ax-i2m1 7947 ax-0lt1 7948 ax-1rid 7949 ax-0id 7950 ax-rnegex 7951 ax-precex 7952 ax-cnre 7953 ax-pre-ltirr 7954 ax-pre-ltwlin 7955 ax-pre-lttrn 7956 ax-pre-apti 7957 ax-pre-ltadd 7958 ax-pre-mulgt0 7959 ax-pre-mulext 7960 ax-arch 7961 ax-caucvg 7962

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 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rmo 2476 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-if 3550 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-iun 3903 df-br 4019 df-opab 4080 df-mpt 4081 df-tr 4117 df-id 4311 df-po 4314 df-iso 4315 df-iord 4384 df-on 4386 df-ilim 4387 df-suc 4389 df-iom 4608 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-rn 4655 df-res 4656 df-ima 4657 df-iota 5196 df-fun 5237 df-fn 5238 df-f 5239 df-f1 5240 df-fo 5241 df-f1o 5242 df-fv 5243 df-isom 5244 df-riota 5852 df-ov 5900 df-oprab 5901 df-mpo 5902 df-1st 6166 df-2nd 6167 df-recs 6331 df-irdg 6396 df-frec 6417 df-1o 6442 df-2o 6443 df-oadd 6446 df-er 6560 df-en 6768 df-dom 6769 df-fin 6770 df-sup 7014 df-inf 7015 df-pnf 8025 df-mnf 8026 df-xr 8027 df-ltxr 8028 df-le 8029 df-sub 8161 df-neg 8162 df-reap 8563 df-ap 8570 df-div 8661 df-inn 8951 df-2 9009 df-3 9010 df-4 9011 df-n0 9208 df-z 9285 df-uz 9560 df-q 9652 df-rp 9686 df-fz 10041 df-fzo 10175 df-fl 10303 df-mod 10356 df-seqfrec 10479 df-exp 10554 df-ihash 10791 df-cj 10886 df-re 10887 df-im 10888 df-rsqrt 11042 df-abs 11043 df-dvds 11830 df-gcd 11979 df-prm 12143 df-gz 12405

This theorem is referenced by: 4sqlem14 12439 4sqlem17 12442 4sqlem18 12443

W3C validator