4sqlem12 - Metamath Proof Explorer

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Theorem 4sqlem12 17146

Description: Lemma for 4sq 17154. For any odd prime 𝑃, there is a 𝑘 < 𝑃 such that 𝑘𝑃 − 1 is a sum of two squares. (Contributed by Mario Carneiro, 15-Jul-2014.)

**Hypotheses**
Ref	Expression
4sq.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)
4sqlem11.5	⊢ 𝐴 = {𝑢 ∣ ∃𝑚 ∈ (0...𝑁)𝑢 = ((𝑚↑2) mod 𝑃)}
4sqlem11.6	⊢ 𝐹 = (𝑣 ∈ 𝐴 ↦ ((𝑃 − 1) − 𝑣))

**Assertion**
Ref	Expression
4sqlem12	⊢ (𝜑 → ∃𝑘 ∈ (1...(𝑃 − 1))∃𝑢 ∈ ℤ[i] (((abs‘𝑢)↑2) + 1) = (𝑘 · 𝑃))

Distinct variable groups: 𝑤,𝑛,𝑥,𝑦,𝑧 𝑘,𝑛,𝑣,𝐴 𝑛,𝐹 𝑢,𝑘,𝑛,𝑚,𝑁,𝑣 𝑃,𝑘,𝑚,𝑛,𝑢,𝑣 𝜑,𝑘,𝑚,𝑛,𝑢,𝑣 𝑆,𝑘,𝑚,𝑛,𝑢,𝑣 Allowed substitution hints: 𝜑(𝑥, 𝑦, 𝑧, 𝑤) 𝐴(𝑥, 𝑦, 𝑧, 𝑤, 𝑢, 𝑚) 𝑃(𝑥, 𝑦, 𝑧, 𝑤) 𝑆(𝑥, 𝑦, 𝑧, 𝑤) 𝐹(𝑥, 𝑦, 𝑧, 𝑤, 𝑣, 𝑢, 𝑘, 𝑚) 𝑁(𝑥, 𝑦, 𝑧, 𝑤)

Proof of Theorem 4sqlem12 Dummy variable 𝑗 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		4sq.1	. . . 4 ⊢ 𝑆 = {𝑛 ∣ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ ∃𝑧 ∈ ℤ ∃𝑤 ∈ ℤ 𝑛 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))}
2		4sq.2	. . . 4 ⊢ (𝜑 → 𝑁 ∈ ℕ)
3		4sq.3	. . . 4 ⊢ (𝜑 → 𝑃 = ((2 · 𝑁) + 1))
4		4sq.4	. . . 4 ⊢ (𝜑 → 𝑃 ∈ ℙ)
5		4sqlem11.5	. . . 4 ⊢ 𝐴 = {𝑢 ∣ ∃𝑚 ∈ (0...𝑁)𝑢 = ((𝑚↑2) mod 𝑃)}
6		4sqlem11.6	. . . 4 ⊢ 𝐹 = (𝑣 ∈ 𝐴 ↦ ((𝑃 − 1) − 𝑣))
7	1, 2, 3, 4, 5, 6	4sqlem11 17145	. . 3 ⊢ (𝜑 → (𝐴 ∩ ran 𝐹) ≠ ∅)
8		n0 4302	. . 3 ⊢ ((𝐴 ∩ ran 𝐹) ≠ ∅ ↔ ∃𝑗 𝑗 ∈ (𝐴 ∩ ran 𝐹))
9	7, 8	sylib 221	. 2 ⊢ (𝜑 → ∃𝑗 𝑗 ∈ (𝐴 ∩ ran 𝐹))
10		vex 3457	. . . . . . 7 ⊢ 𝑗 ∈ V
11		eqeq1 2767	. . . . . . . 8 ⊢ (𝑢 = 𝑗 → (𝑢 = ((𝑚↑2) mod 𝑃) ↔ 𝑗 = ((𝑚↑2) mod 𝑃)))
12	11	rexbidv 3189	. . . . . . 7 ⊢ (𝑢 = 𝑗 → (∃𝑚 ∈ (0...𝑁)𝑢 = ((𝑚↑2) mod 𝑃) ↔ ∃𝑚 ∈ (0...𝑁)𝑗 = ((𝑚↑2) mod 𝑃)))
13	10, 12, 5	elab2 3638	. . . . . 6 ⊢ (𝑗 ∈ 𝐴 ↔ ∃𝑚 ∈ (0...𝑁)𝑗 = ((𝑚↑2) mod 𝑃))
14		abid 2745	. . . . . . . . 9 ⊢ (𝑗 ∈ {𝑗 ∣ ∃𝑣 ∈ 𝐴 𝑗 = ((𝑃 − 1) − 𝑣)} ↔ ∃𝑣 ∈ 𝐴 𝑗 = ((𝑃 − 1) − 𝑣))
15	5	rexeqi 3321	. . . . . . . . 9 ⊢ (∃𝑣 ∈ 𝐴 𝑗 = ((𝑃 − 1) − 𝑣) ↔ ∃𝑣 ∈ {𝑢 ∣ ∃𝑚 ∈ (0...𝑁)𝑢 = ((𝑚↑2) mod 𝑃)}𝑗 = ((𝑃 − 1) − 𝑣))
16		oveq1 7434	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 𝑛 → (𝑚↑2) = (𝑛↑2))
17	16	oveq1d 7442	. . . . . . . . . . . . 13 ⊢ (𝑚 = 𝑛 → ((𝑚↑2) mod 𝑃) = ((𝑛↑2) mod 𝑃))
18	17	eqeq2d 2774	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑛 → (𝑢 = ((𝑚↑2) mod 𝑃) ↔ 𝑢 = ((𝑛↑2) mod 𝑃)))
19	18	cbvrexvw 3244	. . . . . . . . . . 11 ⊢ (∃𝑚 ∈ (0...𝑁)𝑢 = ((𝑚↑2) mod 𝑃) ↔ ∃𝑛 ∈ (0...𝑁)𝑢 = ((𝑛↑2) mod 𝑃))
20		eqeq1 2767	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝑣 → (𝑢 = ((𝑛↑2) mod 𝑃) ↔ 𝑣 = ((𝑛↑2) mod 𝑃)))
21	20	rexbidv 3189	. . . . . . . . . . 11 ⊢ (𝑢 = 𝑣 → (∃𝑛 ∈ (0...𝑁)𝑢 = ((𝑛↑2) mod 𝑃) ↔ ∃𝑛 ∈ (0...𝑁)𝑣 = ((𝑛↑2) mod 𝑃)))
22	19, 21	bitrid 286	. . . . . . . . . 10 ⊢ (𝑢 = 𝑣 → (∃𝑚 ∈ (0...𝑁)𝑢 = ((𝑚↑2) mod 𝑃) ↔ ∃𝑛 ∈ (0...𝑁)𝑣 = ((𝑛↑2) mod 𝑃)))
23	22	rexab 3655	. . . . . . . . 9 ⊢ (∃𝑣 ∈ {𝑢 ∣ ∃𝑚 ∈ (0...𝑁)𝑢 = ((𝑚↑2) mod 𝑃)}𝑗 = ((𝑃 − 1) − 𝑣) ↔ ∃𝑣(∃𝑛 ∈ (0...𝑁)𝑣 = ((𝑛↑2) mod 𝑃) ∧ 𝑗 = ((𝑃 − 1) − 𝑣)))
24	14, 15, 23	3bitri 300	. . . . . . . 8 ⊢ (𝑗 ∈ {𝑗 ∣ ∃𝑣 ∈ 𝐴 𝑗 = ((𝑃 − 1) − 𝑣)} ↔ ∃𝑣(∃𝑛 ∈ (0...𝑁)𝑣 = ((𝑛↑2) mod 𝑃) ∧ 𝑗 = ((𝑃 − 1) − 𝑣)))
25	6	rnmpt 5943	. . . . . . . . 9 ⊢ ran 𝐹 = {𝑗 ∣ ∃𝑣 ∈ 𝐴 𝑗 = ((𝑃 − 1) − 𝑣)}
26	25	eleq2i 2855	. . . . . . . 8 ⊢ (𝑗 ∈ ran 𝐹 ↔ 𝑗 ∈ {𝑗 ∣ ∃𝑣 ∈ 𝐴 𝑗 = ((𝑃 − 1) − 𝑣)})
27		rexcom4 3292	. . . . . . . . 9 ⊢ (∃𝑛 ∈ (0...𝑁)∃𝑣(𝑣 = ((𝑛↑2) mod 𝑃) ∧ 𝑗 = ((𝑃 − 1) − 𝑣)) ↔ ∃𝑣∃𝑛 ∈ (0...𝑁)(𝑣 = ((𝑛↑2) mod 𝑃) ∧ 𝑗 = ((𝑃 − 1) − 𝑣)))
28		r19.41v 3195	. . . . . . . . . 10 ⊢ (∃𝑛 ∈ (0...𝑁)(𝑣 = ((𝑛↑2) mod 𝑃) ∧ 𝑗 = ((𝑃 − 1) − 𝑣)) ↔ (∃𝑛 ∈ (0...𝑁)𝑣 = ((𝑛↑2) mod 𝑃) ∧ 𝑗 = ((𝑃 − 1) − 𝑣)))
29	28	exbii 1880	. . . . . . . . 9 ⊢ (∃𝑣∃𝑛 ∈ (0...𝑁)(𝑣 = ((𝑛↑2) mod 𝑃) ∧ 𝑗 = ((𝑃 − 1) − 𝑣)) ↔ ∃𝑣(∃𝑛 ∈ (0...𝑁)𝑣 = ((𝑛↑2) mod 𝑃) ∧ 𝑗 = ((𝑃 − 1) − 𝑣)))
30	27, 29	bitri 278	. . . . . . . 8 ⊢ (∃𝑛 ∈ (0...𝑁)∃𝑣(𝑣 = ((𝑛↑2) mod 𝑃) ∧ 𝑗 = ((𝑃 − 1) − 𝑣)) ↔ ∃𝑣(∃𝑛 ∈ (0...𝑁)𝑣 = ((𝑛↑2) mod 𝑃) ∧ 𝑗 = ((𝑃 − 1) − 𝑣)))
31	24, 26, 30	3bitr4i 306	. . . . . . 7 ⊢ (𝑗 ∈ ran 𝐹 ↔ ∃𝑛 ∈ (0...𝑁)∃𝑣(𝑣 = ((𝑛↑2) mod 𝑃) ∧ 𝑗 = ((𝑃 − 1) − 𝑣)))
32		ovex 7460	. . . . . . . . 9 ⊢ ((𝑛↑2) mod 𝑃) ∈ V
33		oveq2 7435	. . . . . . . . . 10 ⊢ (𝑣 = ((𝑛↑2) mod 𝑃) → ((𝑃 − 1) − 𝑣) = ((𝑃 − 1) − ((𝑛↑2) mod 𝑃)))
34	33	eqeq2d 2774	. . . . . . . . 9 ⊢ (𝑣 = ((𝑛↑2) mod 𝑃) → (𝑗 = ((𝑃 − 1) − 𝑣) ↔ 𝑗 = ((𝑃 − 1) − ((𝑛↑2) mod 𝑃))))
35	32, 34	ceqsexv 3501	. . . . . . . 8 ⊢ (∃𝑣(𝑣 = ((𝑛↑2) mod 𝑃) ∧ 𝑗 = ((𝑃 − 1) − 𝑣)) ↔ 𝑗 = ((𝑃 − 1) − ((𝑛↑2) mod 𝑃)))
36	35	rexbii 3112	. . . . . . 7 ⊢ (∃𝑛 ∈ (0...𝑁)∃𝑣(𝑣 = ((𝑛↑2) mod 𝑃) ∧ 𝑗 = ((𝑃 − 1) − 𝑣)) ↔ ∃𝑛 ∈ (0...𝑁)𝑗 = ((𝑃 − 1) − ((𝑛↑2) mod 𝑃)))
37	31, 36	bitri 278	. . . . . 6 ⊢ (𝑗 ∈ ran 𝐹 ↔ ∃𝑛 ∈ (0...𝑁)𝑗 = ((𝑃 − 1) − ((𝑛↑2) mod 𝑃)))
38	13, 37	anbi12i 640	. . . . 5 ⊢ ((𝑗 ∈ 𝐴 ∧ 𝑗 ∈ ran 𝐹) ↔ (∃𝑚 ∈ (0...𝑁)𝑗 = ((𝑚↑2) mod 𝑃) ∧ ∃𝑛 ∈ (0...𝑁)𝑗 = ((𝑃 − 1) − ((𝑛↑2) mod 𝑃))))
39		elin 4095	. . . . 5 ⊢ (𝑗 ∈ (𝐴 ∩ ran 𝐹) ↔ (𝑗 ∈ 𝐴 ∧ 𝑗 ∈ ran 𝐹))
40		reeanv 3237	. . . . 5 ⊢ (∃𝑚 ∈ (0...𝑁)∃𝑛 ∈ (0...𝑁)(𝑗 = ((𝑚↑2) mod 𝑃) ∧ 𝑗 = ((𝑃 − 1) − ((𝑛↑2) mod 𝑃))) ↔ (∃𝑚 ∈ (0...𝑁)𝑗 = ((𝑚↑2) mod 𝑃) ∧ ∃𝑛 ∈ (0...𝑁)𝑗 = ((𝑃 − 1) − ((𝑛↑2) mod 𝑃))))
41	38, 39, 40	3bitr4i 306	. . . 4 ⊢ (𝑗 ∈ (𝐴 ∩ ran 𝐹) ↔ ∃𝑚 ∈ (0...𝑁)∃𝑛 ∈ (0...𝑁)(𝑗 = ((𝑚↑2) mod 𝑃) ∧ 𝑗 = ((𝑃 − 1) − ((𝑛↑2) mod 𝑃))))
42		eqtr2 2784	. . . . . 6 ⊢ ((𝑗 = ((𝑚↑2) mod 𝑃) ∧ 𝑗 = ((𝑃 − 1) − ((𝑛↑2) mod 𝑃))) → ((𝑚↑2) mod 𝑃) = ((𝑃 − 1) − ((𝑛↑2) mod 𝑃)))
43	4	3ad2ant1 1151	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑛 ∈ (0...𝑁)) ∧ ((𝑚↑2) mod 𝑃) = ((𝑃 − 1) − ((𝑛↑2) mod 𝑃))) → 𝑃 ∈ ℙ)
44		prmnn 16861	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
45	43, 44	syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑛 ∈ (0...𝑁)) ∧ ((𝑚↑2) mod 𝑃) = ((𝑃 − 1) − ((𝑛↑2) mod 𝑃))) → 𝑃 ∈ ℕ)
46		nnm1nn0 12657	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑃 ∈ ℕ → (𝑃 − 1) ∈ ℕ₀)
47	45, 46	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑛 ∈ (0...𝑁)) ∧ ((𝑚↑2) mod 𝑃) = ((𝑃 − 1) − ((𝑛↑2) mod 𝑃))) → (𝑃 − 1) ∈ ℕ₀)
48	47	nn0red 12678	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑛 ∈ (0...𝑁)) ∧ ((𝑚↑2) mod 𝑃) = ((𝑃 − 1) − ((𝑛↑2) mod 𝑃))) → (𝑃 − 1) ∈ ℝ)
49	45	nnrpd 13172	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑛 ∈ (0...𝑁)) ∧ ((𝑚↑2) mod 𝑃) = ((𝑃 − 1) − ((𝑛↑2) mod 𝑃))) → 𝑃 ∈ ℝ⁺)
50	47	nn0ge0d 12680	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑛 ∈ (0...𝑁)) ∧ ((𝑚↑2) mod 𝑃) = ((𝑃 − 1) − ((𝑛↑2) mod 𝑃))) → 0 ≤ (𝑃 − 1))
51	45	nnred 12360	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑛 ∈ (0...𝑁)) ∧ ((𝑚↑2) mod 𝑃) = ((𝑃 − 1) − ((𝑛↑2) mod 𝑃))) → 𝑃 ∈ ℝ)
52	51	ltm1d 12259	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑛 ∈ (0...𝑁)) ∧ ((𝑚↑2) mod 𝑃) = ((𝑃 − 1) − ((𝑛↑2) mod 𝑃))) → (𝑃 − 1) < 𝑃)
53		modid 14046	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑃 − 1) ∈ ℝ ∧ 𝑃 ∈ ℝ⁺) ∧ (0 ≤ (𝑃 − 1) ∧ (𝑃 − 1) < 𝑃)) → ((𝑃 − 1) mod 𝑃) = (𝑃 − 1))
54	48, 49, 50, 52, 53	syl22anc 852	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑛 ∈ (0...𝑁)) ∧ ((𝑚↑2) mod 𝑃) = ((𝑃 − 1) − ((𝑛↑2) mod 𝑃))) → ((𝑃 − 1) mod 𝑃) = (𝑃 − 1))
55	54	oveq1d 7442	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑛 ∈ (0...𝑁)) ∧ ((𝑚↑2) mod 𝑃) = ((𝑃 − 1) − ((𝑛↑2) mod 𝑃))) → (((𝑃 − 1) mod 𝑃) − ((𝑛↑2) mod 𝑃)) = ((𝑃 − 1) − ((𝑛↑2) mod 𝑃)))
56		simp2r 1219	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑛 ∈ (0...𝑁)) ∧ ((𝑚↑2) mod 𝑃) = ((𝑃 − 1) − ((𝑛↑2) mod 𝑃))) → 𝑛 ∈ (0...𝑁))
57	56	elfzelzd 13667	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑛 ∈ (0...𝑁)) ∧ ((𝑚↑2) mod 𝑃) = ((𝑃 − 1) − ((𝑛↑2) mod 𝑃))) → 𝑛 ∈ ℤ)
58		zsqcl2 14291	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 ∈ ℤ → (𝑛↑2) ∈ ℕ₀)
59	57, 58	syl 18	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑛 ∈ (0...𝑁)) ∧ ((𝑚↑2) mod 𝑃) = ((𝑃 − 1) − ((𝑛↑2) mod 𝑃))) → (𝑛↑2) ∈ ℕ₀)
60	59	nn0red 12678	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑛 ∈ (0...𝑁)) ∧ ((𝑚↑2) mod 𝑃) = ((𝑃 − 1) − ((𝑛↑2) mod 𝑃))) → (𝑛↑2) ∈ ℝ)
61		modlt 14030	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑛↑2) ∈ ℝ ∧ 𝑃 ∈ ℝ⁺) → ((𝑛↑2) mod 𝑃) < 𝑃)
62	60, 49, 61	syl2anc 596	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑛 ∈ (0...𝑁)) ∧ ((𝑚↑2) mod 𝑃) = ((𝑃 − 1) − ((𝑛↑2) mod 𝑃))) → ((𝑛↑2) mod 𝑃) < 𝑃)
63		zsqcl 14282	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 ∈ ℤ → (𝑛↑2) ∈ ℤ)
64	57, 63	syl 18	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑛 ∈ (0...𝑁)) ∧ ((𝑚↑2) mod 𝑃) = ((𝑃 − 1) − ((𝑛↑2) mod 𝑃))) → (𝑛↑2) ∈ ℤ)
65	64, 45	zmodcld 14042	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑛 ∈ (0...𝑁)) ∧ ((𝑚↑2) mod 𝑃) = ((𝑃 − 1) − ((𝑛↑2) mod 𝑃))) → ((𝑛↑2) mod 𝑃) ∈ ℕ₀)
66	65	nn0zd 12728	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑛 ∈ (0...𝑁)) ∧ ((𝑚↑2) mod 𝑃) = ((𝑃 − 1) − ((𝑛↑2) mod 𝑃))) → ((𝑛↑2) mod 𝑃) ∈ ℤ)
67		prmz 16862	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℤ)
68	43, 67	syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑛 ∈ (0...𝑁)) ∧ ((𝑚↑2) mod 𝑃) = ((𝑃 − 1) − ((𝑛↑2) mod 𝑃))) → 𝑃 ∈ ℤ)
69		zltlem1 12759	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑛↑2) mod 𝑃) ∈ ℤ ∧ 𝑃 ∈ ℤ) → (((𝑛↑2) mod 𝑃) < 𝑃 ↔ ((𝑛↑2) mod 𝑃) ≤ (𝑃 − 1)))
70	66, 68, 69	syl2anc 596	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑛 ∈ (0...𝑁)) ∧ ((𝑚↑2) mod 𝑃) = ((𝑃 − 1) − ((𝑛↑2) mod 𝑃))) → (((𝑛↑2) mod 𝑃) < 𝑃 ↔ ((𝑛↑2) mod 𝑃) ≤ (𝑃 − 1)))
71	62, 70	mpbid 235	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑛 ∈ (0...𝑁)) ∧ ((𝑚↑2) mod 𝑃) = ((𝑃 − 1) − ((𝑛↑2) mod 𝑃))) → ((𝑛↑2) mod 𝑃) ≤ (𝑃 − 1))
72	71, 54	breqtrrd 5135	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑛 ∈ (0...𝑁)) ∧ ((𝑚↑2) mod 𝑃) = ((𝑃 − 1) − ((𝑛↑2) mod 𝑃))) → ((𝑛↑2) mod 𝑃) ≤ ((𝑃 − 1) mod 𝑃))
73		modsubdir 14093	. . . . . . . . . . . . . . . 16 ⊢ (((𝑃 − 1) ∈ ℝ ∧ (𝑛↑2) ∈ ℝ ∧ 𝑃 ∈ ℝ⁺) → (((𝑛↑2) mod 𝑃) ≤ ((𝑃 − 1) mod 𝑃) ↔ (((𝑃 − 1) − (𝑛↑2)) mod 𝑃) = (((𝑃 − 1) mod 𝑃) − ((𝑛↑2) mod 𝑃))))
74	48, 60, 49, 73	syl3anc 1398	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑛 ∈ (0...𝑁)) ∧ ((𝑚↑2) mod 𝑃) = ((𝑃 − 1) − ((𝑛↑2) mod 𝑃))) → (((𝑛↑2) mod 𝑃) ≤ ((𝑃 − 1) mod 𝑃) ↔ (((𝑃 − 1) − (𝑛↑2)) mod 𝑃) = (((𝑃 − 1) mod 𝑃) − ((𝑛↑2) mod 𝑃))))
75	72, 74	mpbid 235	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑛 ∈ (0...𝑁)) ∧ ((𝑚↑2) mod 𝑃) = ((𝑃 − 1) − ((𝑛↑2) mod 𝑃))) → (((𝑃 − 1) − (𝑛↑2)) mod 𝑃) = (((𝑃 − 1) mod 𝑃) − ((𝑛↑2) mod 𝑃)))
76		simp3 1156	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑛 ∈ (0...𝑁)) ∧ ((𝑚↑2) mod 𝑃) = ((𝑃 − 1) − ((𝑛↑2) mod 𝑃))) → ((𝑚↑2) mod 𝑃) = ((𝑃 − 1) − ((𝑛↑2) mod 𝑃)))
77	55, 75, 76	3eqtr4rd 2809	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑛 ∈ (0...𝑁)) ∧ ((𝑚↑2) mod 𝑃) = ((𝑃 − 1) − ((𝑛↑2) mod 𝑃))) → ((𝑚↑2) mod 𝑃) = (((𝑃 − 1) − (𝑛↑2)) mod 𝑃))
78		simp2l 1218	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑛 ∈ (0...𝑁)) ∧ ((𝑚↑2) mod 𝑃) = ((𝑃 − 1) − ((𝑛↑2) mod 𝑃))) → 𝑚 ∈ (0...𝑁))
79	78	elfzelzd 13667	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑛 ∈ (0...𝑁)) ∧ ((𝑚↑2) mod 𝑃) = ((𝑃 − 1) − ((𝑛↑2) mod 𝑃))) → 𝑚 ∈ ℤ)
80		zsqcl 14282	. . . . . . . . . . . . . . 15 ⊢ (𝑚 ∈ ℤ → (𝑚↑2) ∈ ℤ)
81	79, 80	syl 18	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑛 ∈ (0...𝑁)) ∧ ((𝑚↑2) mod 𝑃) = ((𝑃 − 1) − ((𝑛↑2) mod 𝑃))) → (𝑚↑2) ∈ ℤ)
82	47	nn0zd 12728	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑛 ∈ (0...𝑁)) ∧ ((𝑚↑2) mod 𝑃) = ((𝑃 − 1) − ((𝑛↑2) mod 𝑃))) → (𝑃 − 1) ∈ ℤ)
83	82, 64	zsubcld 12818	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑛 ∈ (0...𝑁)) ∧ ((𝑚↑2) mod 𝑃) = ((𝑃 − 1) − ((𝑛↑2) mod 𝑃))) → ((𝑃 − 1) − (𝑛↑2)) ∈ ℤ)
84		moddvds 16445	. . . . . . . . . . . . . 14 ⊢ ((𝑃 ∈ ℕ ∧ (𝑚↑2) ∈ ℤ ∧ ((𝑃 − 1) − (𝑛↑2)) ∈ ℤ) → (((𝑚↑2) mod 𝑃) = (((𝑃 − 1) − (𝑛↑2)) mod 𝑃) ↔ 𝑃 ∥ ((𝑚↑2) − ((𝑃 − 1) − (𝑛↑2)))))
85	45, 81, 83, 84	syl3anc 1398	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑛 ∈ (0...𝑁)) ∧ ((𝑚↑2) mod 𝑃) = ((𝑃 − 1) − ((𝑛↑2) mod 𝑃))) → (((𝑚↑2) mod 𝑃) = (((𝑃 − 1) − (𝑛↑2)) mod 𝑃) ↔ 𝑃 ∥ ((𝑚↑2) − ((𝑃 − 1) − (𝑛↑2)))))
86	77, 85	mpbid 235	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑛 ∈ (0...𝑁)) ∧ ((𝑚↑2) mod 𝑃) = ((𝑃 − 1) − ((𝑛↑2) mod 𝑃))) → 𝑃 ∥ ((𝑚↑2) − ((𝑃 − 1) − (𝑛↑2))))
87		zsqcl2 14291	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 ∈ ℤ → (𝑚↑2) ∈ ℕ₀)
88	79, 87	syl 18	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑛 ∈ (0...𝑁)) ∧ ((𝑚↑2) mod 𝑃) = ((𝑃 − 1) − ((𝑛↑2) mod 𝑃))) → (𝑚↑2) ∈ ℕ₀)
89	88	nn0cnd 12679	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑛 ∈ (0...𝑁)) ∧ ((𝑚↑2) mod 𝑃) = ((𝑃 − 1) − ((𝑛↑2) mod 𝑃))) → (𝑚↑2) ∈ ℂ)
90	47	nn0cnd 12679	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑛 ∈ (0...𝑁)) ∧ ((𝑚↑2) mod 𝑃) = ((𝑃 − 1) − ((𝑛↑2) mod 𝑃))) → (𝑃 − 1) ∈ ℂ)
91	59	nn0cnd 12679	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑛 ∈ (0...𝑁)) ∧ ((𝑚↑2) mod 𝑃) = ((𝑃 − 1) − ((𝑛↑2) mod 𝑃))) → (𝑛↑2) ∈ ℂ)
92	89, 90, 91	subsub3d 11709	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑛 ∈ (0...𝑁)) ∧ ((𝑚↑2) mod 𝑃) = ((𝑃 − 1) − ((𝑛↑2) mod 𝑃))) → ((𝑚↑2) − ((𝑃 − 1) − (𝑛↑2))) = (((𝑚↑2) + (𝑛↑2)) − (𝑃 − 1)))
93	88, 59	nn0addcld 12681	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑛 ∈ (0...𝑁)) ∧ ((𝑚↑2) mod 𝑃) = ((𝑃 − 1) − ((𝑛↑2) mod 𝑃))) → ((𝑚↑2) + (𝑛↑2)) ∈ ℕ₀)
94	93	nn0cnd 12679	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑛 ∈ (0...𝑁)) ∧ ((𝑚↑2) mod 𝑃) = ((𝑃 − 1) − ((𝑛↑2) mod 𝑃))) → ((𝑚↑2) + (𝑛↑2)) ∈ ℂ)
95	45	nncnd 12361	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑛 ∈ (0...𝑁)) ∧ ((𝑚↑2) mod 𝑃) = ((𝑃 − 1) − ((𝑛↑2) mod 𝑃))) → 𝑃 ∈ ℂ)
96		1cnd 11312	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑛 ∈ (0...𝑁)) ∧ ((𝑚↑2) mod 𝑃) = ((𝑃 − 1) − ((𝑛↑2) mod 𝑃))) → 1 ∈ ℂ)
97	94, 95, 96	subsub3d 11709	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑛 ∈ (0...𝑁)) ∧ ((𝑚↑2) mod 𝑃) = ((𝑃 − 1) − ((𝑛↑2) mod 𝑃))) → (((𝑚↑2) + (𝑛↑2)) − (𝑃 − 1)) = ((((𝑚↑2) + (𝑛↑2)) + 1) − 𝑃))
98	92, 97	eqtrd 2798	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑛 ∈ (0...𝑁)) ∧ ((𝑚↑2) mod 𝑃) = ((𝑃 − 1) − ((𝑛↑2) mod 𝑃))) → ((𝑚↑2) − ((𝑃 − 1) − (𝑛↑2))) = ((((𝑚↑2) + (𝑛↑2)) + 1) − 𝑃))
99	86, 98	breqtrd 5133	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑛 ∈ (0...𝑁)) ∧ ((𝑚↑2) mod 𝑃) = ((𝑃 − 1) − ((𝑛↑2) mod 𝑃))) → 𝑃 ∥ ((((𝑚↑2) + (𝑛↑2)) + 1) − 𝑃))
100		nn0p1nn 12655	. . . . . . . . . . . . . 14 ⊢ (((𝑚↑2) + (𝑛↑2)) ∈ ℕ₀ → (((𝑚↑2) + (𝑛↑2)) + 1) ∈ ℕ)
101	93, 100	syl 18	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑛 ∈ (0...𝑁)) ∧ ((𝑚↑2) mod 𝑃) = ((𝑃 − 1) − ((𝑛↑2) mod 𝑃))) → (((𝑚↑2) + (𝑛↑2)) + 1) ∈ ℕ)
102	101	nnzd 12729	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑛 ∈ (0...𝑁)) ∧ ((𝑚↑2) mod 𝑃) = ((𝑃 − 1) − ((𝑛↑2) mod 𝑃))) → (((𝑚↑2) + (𝑛↑2)) + 1) ∈ ℤ)
103		dvdssubr 16487	. . . . . . . . . . . 12 ⊢ ((𝑃 ∈ ℤ ∧ (((𝑚↑2) + (𝑛↑2)) + 1) ∈ ℤ) → (𝑃 ∥ (((𝑚↑2) + (𝑛↑2)) + 1) ↔ 𝑃 ∥ ((((𝑚↑2) + (𝑛↑2)) + 1) − 𝑃)))
104	68, 102, 103	syl2anc 596	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑛 ∈ (0...𝑁)) ∧ ((𝑚↑2) mod 𝑃) = ((𝑃 − 1) − ((𝑛↑2) mod 𝑃))) → (𝑃 ∥ (((𝑚↑2) + (𝑛↑2)) + 1) ↔ 𝑃 ∥ ((((𝑚↑2) + (𝑛↑2)) + 1) − 𝑃)))
105	99, 104	mpbird 260	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑛 ∈ (0...𝑁)) ∧ ((𝑚↑2) mod 𝑃) = ((𝑃 − 1) − ((𝑛↑2) mod 𝑃))) → 𝑃 ∥ (((𝑚↑2) + (𝑛↑2)) + 1))
106	45	nnne0d 12398	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑛 ∈ (0...𝑁)) ∧ ((𝑚↑2) mod 𝑃) = ((𝑃 − 1) − ((𝑛↑2) mod 𝑃))) → 𝑃 ≠ 0)
107		dvdsval2 16437	. . . . . . . . . . 11 ⊢ ((𝑃 ∈ ℤ ∧ 𝑃 ≠ 0 ∧ (((𝑚↑2) + (𝑛↑2)) + 1) ∈ ℤ) → (𝑃 ∥ (((𝑚↑2) + (𝑛↑2)) + 1) ↔ ((((𝑚↑2) + (𝑛↑2)) + 1) / 𝑃) ∈ ℤ))
108	68, 106, 102, 107	syl3anc 1398	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑛 ∈ (0...𝑁)) ∧ ((𝑚↑2) mod 𝑃) = ((𝑃 − 1) − ((𝑛↑2) mod 𝑃))) → (𝑃 ∥ (((𝑚↑2) + (𝑛↑2)) + 1) ↔ ((((𝑚↑2) + (𝑛↑2)) + 1) / 𝑃) ∈ ℤ))
109	105, 108	mpbid 235	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑛 ∈ (0...𝑁)) ∧ ((𝑚↑2) mod 𝑃) = ((𝑃 − 1) − ((𝑛↑2) mod 𝑃))) → ((((𝑚↑2) + (𝑛↑2)) + 1) / 𝑃) ∈ ℤ)
110		nnrp 13142	. . . . . . . . . . . . . 14 ⊢ ((((𝑚↑2) + (𝑛↑2)) + 1) ∈ ℕ → (((𝑚↑2) + (𝑛↑2)) + 1) ∈ ℝ⁺)
111		nnrp 13142	. . . . . . . . . . . . . 14 ⊢ (𝑃 ∈ ℕ → 𝑃 ∈ ℝ⁺)
112		rpdivcl 13157	. . . . . . . . . . . . . 14 ⊢ (((((𝑚↑2) + (𝑛↑2)) + 1) ∈ ℝ⁺ ∧ 𝑃 ∈ ℝ⁺) → ((((𝑚↑2) + (𝑛↑2)) + 1) / 𝑃) ∈ ℝ⁺)
113	110, 111, 112	syl2an 608	. . . . . . . . . . . . 13 ⊢ (((((𝑚↑2) + (𝑛↑2)) + 1) ∈ ℕ ∧ 𝑃 ∈ ℕ) → ((((𝑚↑2) + (𝑛↑2)) + 1) / 𝑃) ∈ ℝ⁺)
114	101, 45, 113	syl2anc 596	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑛 ∈ (0...𝑁)) ∧ ((𝑚↑2) mod 𝑃) = ((𝑃 − 1) − ((𝑛↑2) mod 𝑃))) → ((((𝑚↑2) + (𝑛↑2)) + 1) / 𝑃) ∈ ℝ⁺)
115	114	rpgt0d 13177	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑛 ∈ (0...𝑁)) ∧ ((𝑚↑2) mod 𝑃) = ((𝑃 − 1) − ((𝑛↑2) mod 𝑃))) → 0 < ((((𝑚↑2) + (𝑛↑2)) + 1) / 𝑃))
116		elnnz 12713	. . . . . . . . . . 11 ⊢ (((((𝑚↑2) + (𝑛↑2)) + 1) / 𝑃) ∈ ℕ ↔ (((((𝑚↑2) + (𝑛↑2)) + 1) / 𝑃) ∈ ℤ ∧ 0 < ((((𝑚↑2) + (𝑛↑2)) + 1) / 𝑃)))
117	109, 115, 116	sylanbrc 595	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑛 ∈ (0...𝑁)) ∧ ((𝑚↑2) mod 𝑃) = ((𝑃 − 1) − ((𝑛↑2) mod 𝑃))) → ((((𝑚↑2) + (𝑛↑2)) + 1) / 𝑃) ∈ ℕ)
118	117	nnge1d 12396	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑛 ∈ (0...𝑁)) ∧ ((𝑚↑2) mod 𝑃) = ((𝑃 − 1) − ((𝑛↑2) mod 𝑃))) → 1 ≤ ((((𝑚↑2) + (𝑛↑2)) + 1) / 𝑃))
119	93	nn0red 12678	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑛 ∈ (0...𝑁)) ∧ ((𝑚↑2) mod 𝑃) = ((𝑃 − 1) − ((𝑛↑2) mod 𝑃))) → ((𝑚↑2) + (𝑛↑2)) ∈ ℝ)
120		2nn 12426	. . . . . . . . . . . . . . . 16 ⊢ 2 ∈ ℕ
121	2	3ad2ant1 1151	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑛 ∈ (0...𝑁)) ∧ ((𝑚↑2) mod 𝑃) = ((𝑃 − 1) − ((𝑛↑2) mod 𝑃))) → 𝑁 ∈ ℕ)
122		nnmulcl 12369	. . . . . . . . . . . . . . . 16 ⊢ ((2 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (2 · 𝑁) ∈ ℕ)
123	120, 121, 122	sylancr 599	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑛 ∈ (0...𝑁)) ∧ ((𝑚↑2) mod 𝑃) = ((𝑃 − 1) − ((𝑛↑2) mod 𝑃))) → (2 · 𝑁) ∈ ℕ)
124	123	nnred 12360	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑛 ∈ (0...𝑁)) ∧ ((𝑚↑2) mod 𝑃) = ((𝑃 − 1) − ((𝑛↑2) mod 𝑃))) → (2 · 𝑁) ∈ ℝ)
125	124	resqcld 14278	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑛 ∈ (0...𝑁)) ∧ ((𝑚↑2) mod 𝑃) = ((𝑃 − 1) − ((𝑛↑2) mod 𝑃))) → ((2 · 𝑁)↑2) ∈ ℝ)
126		nnmulcl 12369	. . . . . . . . . . . . . . 15 ⊢ ((2 ∈ ℕ ∧ (2 · 𝑁) ∈ ℕ) → (2 · (2 · 𝑁)) ∈ ℕ)
127	120, 123, 126	sylancr 599	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑛 ∈ (0...𝑁)) ∧ ((𝑚↑2) mod 𝑃) = ((𝑃 − 1) − ((𝑛↑2) mod 𝑃))) → (2 · (2 · 𝑁)) ∈ ℕ)
128	127	nnred 12360	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑛 ∈ (0...𝑁)) ∧ ((𝑚↑2) mod 𝑃) = ((𝑃 − 1) − ((𝑛↑2) mod 𝑃))) → (2 · (2 · 𝑁)) ∈ ℝ)
129	125, 128	readdcld 11348	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑛 ∈ (0...𝑁)) ∧ ((𝑚↑2) mod 𝑃) = ((𝑃 − 1) − ((𝑛↑2) mod 𝑃))) → (((2 · 𝑁)↑2) + (2 · (2 · 𝑁))) ∈ ℝ)
130		1red 11319	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑛 ∈ (0...𝑁)) ∧ ((𝑚↑2) mod 𝑃) = ((𝑃 − 1) − ((𝑛↑2) mod 𝑃))) → 1 ∈ ℝ)
131	121	nnsqcld 14398	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑛 ∈ (0...𝑁)) ∧ ((𝑚↑2) mod 𝑃) = ((𝑃 − 1) − ((𝑛↑2) mod 𝑃))) → (𝑁↑2) ∈ ℕ)
132		nnmulcl 12369	. . . . . . . . . . . . . . . 16 ⊢ ((2 ∈ ℕ ∧ (𝑁↑2) ∈ ℕ) → (2 · (𝑁↑2)) ∈ ℕ)
133	120, 131, 132	sylancr 599	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑛 ∈ (0...𝑁)) ∧ ((𝑚↑2) mod 𝑃) = ((𝑃 − 1) − ((𝑛↑2) mod 𝑃))) → (2 · (𝑁↑2)) ∈ ℕ)
134	133	nnred 12360	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑛 ∈ (0...𝑁)) ∧ ((𝑚↑2) mod 𝑃) = ((𝑃 − 1) − ((𝑛↑2) mod 𝑃))) → (2 · (𝑁↑2)) ∈ ℝ)
135	88	nn0red 12678	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑛 ∈ (0...𝑁)) ∧ ((𝑚↑2) mod 𝑃) = ((𝑃 − 1) − ((𝑛↑2) mod 𝑃))) → (𝑚↑2) ∈ ℝ)
136	131	nnred 12360	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑛 ∈ (0...𝑁)) ∧ ((𝑚↑2) mod 𝑃) = ((𝑃 − 1) − ((𝑛↑2) mod 𝑃))) → (𝑁↑2) ∈ ℝ)
137	79	zred 12813	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑛 ∈ (0...𝑁)) ∧ ((𝑚↑2) mod 𝑃) = ((𝑃 − 1) − ((𝑛↑2) mod 𝑃))) → 𝑚 ∈ ℝ)
138		elfzle1 13670	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑚 ∈ (0...𝑁) → 0 ≤ 𝑚)
139	78, 138	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑛 ∈ (0...𝑁)) ∧ ((𝑚↑2) mod 𝑃) = ((𝑃 − 1) − ((𝑛↑2) mod 𝑃))) → 0 ≤ 𝑚)
140	121	nnred 12360	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑛 ∈ (0...𝑁)) ∧ ((𝑚↑2) mod 𝑃) = ((𝑃 − 1) − ((𝑛↑2) mod 𝑃))) → 𝑁 ∈ ℝ)
141		elfzle2 13671	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑚 ∈ (0...𝑁) → 𝑚 ≤ 𝑁)
142	78, 141	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑛 ∈ (0...𝑁)) ∧ ((𝑚↑2) mod 𝑃) = ((𝑃 − 1) − ((𝑛↑2) mod 𝑃))) → 𝑚 ≤ 𝑁)
143		le2sq2 14288	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑚 ∈ ℝ ∧ 0 ≤ 𝑚) ∧ (𝑁 ∈ ℝ ∧ 𝑚 ≤ 𝑁)) → (𝑚↑2) ≤ (𝑁↑2))
144	137, 139, 140, 142, 143	syl22anc 852	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑛 ∈ (0...𝑁)) ∧ ((𝑚↑2) mod 𝑃) = ((𝑃 − 1) − ((𝑛↑2) mod 𝑃))) → (𝑚↑2) ≤ (𝑁↑2))
145	57	zred 12813	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑛 ∈ (0...𝑁)) ∧ ((𝑚↑2) mod 𝑃) = ((𝑃 − 1) − ((𝑛↑2) mod 𝑃))) → 𝑛 ∈ ℝ)
146		elfzle1 13670	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ (0...𝑁) → 0 ≤ 𝑛)
147	56, 146	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑛 ∈ (0...𝑁)) ∧ ((𝑚↑2) mod 𝑃) = ((𝑃 − 1) − ((𝑛↑2) mod 𝑃))) → 0 ≤ 𝑛)
148		elfzle2 13671	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ (0...𝑁) → 𝑛 ≤ 𝑁)
149	56, 148	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑛 ∈ (0...𝑁)) ∧ ((𝑚↑2) mod 𝑃) = ((𝑃 − 1) − ((𝑛↑2) mod 𝑃))) → 𝑛 ≤ 𝑁)
150		le2sq2 14288	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑛 ∈ ℝ ∧ 0 ≤ 𝑛) ∧ (𝑁 ∈ ℝ ∧ 𝑛 ≤ 𝑁)) → (𝑛↑2) ≤ (𝑁↑2))
151	145, 147, 140, 149, 150	syl22anc 852	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑛 ∈ (0...𝑁)) ∧ ((𝑚↑2) mod 𝑃) = ((𝑃 − 1) − ((𝑛↑2) mod 𝑃))) → (𝑛↑2) ≤ (𝑁↑2))
152	135, 60, 136, 136, 144, 151	le2addd 11945	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑛 ∈ (0...𝑁)) ∧ ((𝑚↑2) mod 𝑃) = ((𝑃 − 1) − ((𝑛↑2) mod 𝑃))) → ((𝑚↑2) + (𝑛↑2)) ≤ ((𝑁↑2) + (𝑁↑2)))
153	131	nncnd 12361	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑛 ∈ (0...𝑁)) ∧ ((𝑚↑2) mod 𝑃) = ((𝑃 − 1) − ((𝑛↑2) mod 𝑃))) → (𝑁↑2) ∈ ℂ)
154	153	2timesd 12599	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑛 ∈ (0...𝑁)) ∧ ((𝑚↑2) mod 𝑃) = ((𝑃 − 1) − ((𝑛↑2) mod 𝑃))) → (2 · (𝑁↑2)) = ((𝑁↑2) + (𝑁↑2)))
155	152, 154	breqtrrd 5135	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑛 ∈ (0...𝑁)) ∧ ((𝑚↑2) mod 𝑃) = ((𝑃 − 1) − ((𝑛↑2) mod 𝑃))) → ((𝑚↑2) + (𝑛↑2)) ≤ (2 · (𝑁↑2)))
156		2lt4 12530	. . . . . . . . . . . . . . . 16 ⊢ 2 < 4
157		2re 12427	. . . . . . . . . . . . . . . . . 18 ⊢ 2 ∈ ℝ
158	157	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑛 ∈ (0...𝑁)) ∧ ((𝑚↑2) mod 𝑃) = ((𝑃 − 1) − ((𝑛↑2) mod 𝑃))) → 2 ∈ ℝ)
159		4re 12437	. . . . . . . . . . . . . . . . . 18 ⊢ 4 ∈ ℝ
160	159	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑛 ∈ (0...𝑁)) ∧ ((𝑚↑2) mod 𝑃) = ((𝑃 − 1) − ((𝑛↑2) mod 𝑃))) → 4 ∈ ℝ)
161	131	nngt0d 12397	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑛 ∈ (0...𝑁)) ∧ ((𝑚↑2) mod 𝑃) = ((𝑃 − 1) − ((𝑛↑2) mod 𝑃))) → 0 < (𝑁↑2))
162		ltmul1 12177	. . . . . . . . . . . . . . . . 17 ⊢ ((2 ∈ ℝ ∧ 4 ∈ ℝ ∧ ((𝑁↑2) ∈ ℝ ∧ 0 < (𝑁↑2))) → (2 < 4 ↔ (2 · (𝑁↑2)) < (4 · (𝑁↑2))))
163	158, 160, 136, 161, 162	syl112anc 1401	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑛 ∈ (0...𝑁)) ∧ ((𝑚↑2) mod 𝑃) = ((𝑃 − 1) − ((𝑛↑2) mod 𝑃))) → (2 < 4 ↔ (2 · (𝑁↑2)) < (4 · (𝑁↑2))))
164	156, 163	mpbii 236	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑛 ∈ (0...𝑁)) ∧ ((𝑚↑2) mod 𝑃) = ((𝑃 − 1) − ((𝑛↑2) mod 𝑃))) → (2 · (𝑁↑2)) < (4 · (𝑁↑2)))
165		2cn 12428	. . . . . . . . . . . . . . . . 17 ⊢ 2 ∈ ℂ
166	121	nncnd 12361	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑛 ∈ (0...𝑁)) ∧ ((𝑚↑2) mod 𝑃) = ((𝑃 − 1) − ((𝑛↑2) mod 𝑃))) → 𝑁 ∈ ℂ)
167		sqmul 14272	. . . . . . . . . . . . . . . . 17 ⊢ ((2 ∈ ℂ ∧ 𝑁 ∈ ℂ) → ((2 · 𝑁)↑2) = ((2↑2) · (𝑁↑2)))
168	165, 166, 167	sylancr 599	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑛 ∈ (0...𝑁)) ∧ ((𝑚↑2) mod 𝑃) = ((𝑃 − 1) − ((𝑛↑2) mod 𝑃))) → ((2 · 𝑁)↑2) = ((2↑2) · (𝑁↑2)))
169		sq2 14350	. . . . . . . . . . . . . . . . 17 ⊢ (2↑2) = 4
170	169	oveq1i 7437	. . . . . . . . . . . . . . . 16 ⊢ ((2↑2) · (𝑁↑2)) = (4 · (𝑁↑2))
171	168, 170	eqtrdi 2814	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑛 ∈ (0...𝑁)) ∧ ((𝑚↑2) mod 𝑃) = ((𝑃 − 1) − ((𝑛↑2) mod 𝑃))) → ((2 · 𝑁)↑2) = (4 · (𝑁↑2)))
172	164, 171	breqtrrd 5135	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑛 ∈ (0...𝑁)) ∧ ((𝑚↑2) mod 𝑃) = ((𝑃 − 1) − ((𝑛↑2) mod 𝑃))) → (2 · (𝑁↑2)) < ((2 · 𝑁)↑2))
173	119, 134, 125, 155, 172	lelttrd 11478	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑛 ∈ (0...𝑁)) ∧ ((𝑚↑2) mod 𝑃) = ((𝑃 − 1) − ((𝑛↑2) mod 𝑃))) → ((𝑚↑2) + (𝑛↑2)) < ((2 · 𝑁)↑2))
174	127	nnrpd 13172	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑛 ∈ (0...𝑁)) ∧ ((𝑚↑2) mod 𝑃) = ((𝑃 − 1) − ((𝑛↑2) mod 𝑃))) → (2 · (2 · 𝑁)) ∈ ℝ⁺)
175	125, 174	ltaddrpd 13207	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑛 ∈ (0...𝑁)) ∧ ((𝑚↑2) mod 𝑃) = ((𝑃 − 1) − ((𝑛↑2) mod 𝑃))) → ((2 · 𝑁)↑2) < (((2 · 𝑁)↑2) + (2 · (2 · 𝑁))))
176	119, 125, 129, 173, 175	lttrd 11481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑛 ∈ (0...𝑁)) ∧ ((𝑚↑2) mod 𝑃) = ((𝑃 − 1) − ((𝑛↑2) mod 𝑃))) → ((𝑚↑2) + (𝑛↑2)) < (((2 · 𝑁)↑2) + (2 · (2 · 𝑁))))
177	119, 129, 130, 176	ltadd1dd 11937	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑛 ∈ (0...𝑁)) ∧ ((𝑚↑2) mod 𝑃) = ((𝑃 − 1) − ((𝑛↑2) mod 𝑃))) → (((𝑚↑2) + (𝑛↑2)) + 1) < ((((2 · 𝑁)↑2) + (2 · (2 · 𝑁))) + 1))
178	3	3ad2ant1 1151	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑛 ∈ (0...𝑁)) ∧ ((𝑚↑2) mod 𝑃) = ((𝑃 − 1) − ((𝑛↑2) mod 𝑃))) → 𝑃 = ((2 · 𝑁) + 1))
179	178	oveq1d 7442	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑛 ∈ (0...𝑁)) ∧ ((𝑚↑2) mod 𝑃) = ((𝑃 − 1) − ((𝑛↑2) mod 𝑃))) → (𝑃↑2) = (((2 · 𝑁) + 1)↑2))
180	95	sqvald 14296	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑛 ∈ (0...𝑁)) ∧ ((𝑚↑2) mod 𝑃) = ((𝑃 − 1) − ((𝑛↑2) mod 𝑃))) → (𝑃↑2) = (𝑃 · 𝑃))
181	123	nncnd 12361	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑛 ∈ (0...𝑁)) ∧ ((𝑚↑2) mod 𝑃) = ((𝑃 − 1) − ((𝑛↑2) mod 𝑃))) → (2 · 𝑁) ∈ ℂ)
182		binom21 14373	. . . . . . . . . . . . 13 ⊢ ((2 · 𝑁) ∈ ℂ → (((2 · 𝑁) + 1)↑2) = ((((2 · 𝑁)↑2) + (2 · (2 · 𝑁))) + 1))
183	181, 182	syl 18	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑛 ∈ (0...𝑁)) ∧ ((𝑚↑2) mod 𝑃) = ((𝑃 − 1) − ((𝑛↑2) mod 𝑃))) → (((2 · 𝑁) + 1)↑2) = ((((2 · 𝑁)↑2) + (2 · (2 · 𝑁))) + 1))
184	179, 180, 183	3eqtr3d 2806	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑛 ∈ (0...𝑁)) ∧ ((𝑚↑2) mod 𝑃) = ((𝑃 − 1) − ((𝑛↑2) mod 𝑃))) → (𝑃 · 𝑃) = ((((2 · 𝑁)↑2) + (2 · (2 · 𝑁))) + 1))
185	177, 184	breqtrrd 5135	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑛 ∈ (0...𝑁)) ∧ ((𝑚↑2) mod 𝑃) = ((𝑃 − 1) − ((𝑛↑2) mod 𝑃))) → (((𝑚↑2) + (𝑛↑2)) + 1) < (𝑃 · 𝑃))
186	101	nnred 12360	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑛 ∈ (0...𝑁)) ∧ ((𝑚↑2) mod 𝑃) = ((𝑃 − 1) − ((𝑛↑2) mod 𝑃))) → (((𝑚↑2) + (𝑛↑2)) + 1) ∈ ℝ)
187	45	nngt0d 12397	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑛 ∈ (0...𝑁)) ∧ ((𝑚↑2) mod 𝑃) = ((𝑃 − 1) − ((𝑛↑2) mod 𝑃))) → 0 < 𝑃)
188		ltdivmul 12202	. . . . . . . . . . 11 ⊢ (((((𝑚↑2) + (𝑛↑2)) + 1) ∈ ℝ ∧ 𝑃 ∈ ℝ ∧ (𝑃 ∈ ℝ ∧ 0 < 𝑃)) → (((((𝑚↑2) + (𝑛↑2)) + 1) / 𝑃) < 𝑃 ↔ (((𝑚↑2) + (𝑛↑2)) + 1) < (𝑃 · 𝑃)))
189	186, 51, 51, 187, 188	syl112anc 1401	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑛 ∈ (0...𝑁)) ∧ ((𝑚↑2) mod 𝑃) = ((𝑃 − 1) − ((𝑛↑2) mod 𝑃))) → (((((𝑚↑2) + (𝑛↑2)) + 1) / 𝑃) < 𝑃 ↔ (((𝑚↑2) + (𝑛↑2)) + 1) < (𝑃 · 𝑃)))
190	185, 189	mpbird 260	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑛 ∈ (0...𝑁)) ∧ ((𝑚↑2) mod 𝑃) = ((𝑃 − 1) − ((𝑛↑2) mod 𝑃))) → ((((𝑚↑2) + (𝑛↑2)) + 1) / 𝑃) < 𝑃)
191		1z 12736	. . . . . . . . . 10 ⊢ 1 ∈ ℤ
192		elfzm11 13739	. . . . . . . . . 10 ⊢ ((1 ∈ ℤ ∧ 𝑃 ∈ ℤ) → (((((𝑚↑2) + (𝑛↑2)) + 1) / 𝑃) ∈ (1...(𝑃 − 1)) ↔ (((((𝑚↑2) + (𝑛↑2)) + 1) / 𝑃) ∈ ℤ ∧ 1 ≤ ((((𝑚↑2) + (𝑛↑2)) + 1) / 𝑃) ∧ ((((𝑚↑2) + (𝑛↑2)) + 1) / 𝑃) < 𝑃)))
193	191, 68, 192	sylancr 599	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑛 ∈ (0...𝑁)) ∧ ((𝑚↑2) mod 𝑃) = ((𝑃 − 1) − ((𝑛↑2) mod 𝑃))) → (((((𝑚↑2) + (𝑛↑2)) + 1) / 𝑃) ∈ (1...(𝑃 − 1)) ↔ (((((𝑚↑2) + (𝑛↑2)) + 1) / 𝑃) ∈ ℤ ∧ 1 ≤ ((((𝑚↑2) + (𝑛↑2)) + 1) / 𝑃) ∧ ((((𝑚↑2) + (𝑛↑2)) + 1) / 𝑃) < 𝑃)))
194	109, 118, 190, 193	mpbir3and 1361	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑛 ∈ (0...𝑁)) ∧ ((𝑚↑2) mod 𝑃) = ((𝑃 − 1) − ((𝑛↑2) mod 𝑃))) → ((((𝑚↑2) + (𝑛↑2)) + 1) / 𝑃) ∈ (1...(𝑃 − 1)))
195		gzreim 17129	. . . . . . . . 9 ⊢ ((𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (𝑚 + (i · 𝑛)) ∈ ℤ[i])
196	79, 57, 195	syl2anc 596	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑛 ∈ (0...𝑁)) ∧ ((𝑚↑2) mod 𝑃) = ((𝑃 − 1) − ((𝑛↑2) mod 𝑃))) → (𝑚 + (i · 𝑛)) ∈ ℤ[i])
197		gzcn 17122	. . . . . . . . . . . . 13 ⊢ ((𝑚 + (i · 𝑛)) ∈ ℤ[i] → (𝑚 + (i · 𝑛)) ∈ ℂ)
198	196, 197	syl 18	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑛 ∈ (0...𝑁)) ∧ ((𝑚↑2) mod 𝑃) = ((𝑃 − 1) − ((𝑛↑2) mod 𝑃))) → (𝑚 + (i · 𝑛)) ∈ ℂ)
199	198	absvalsq2d 15625	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑛 ∈ (0...𝑁)) ∧ ((𝑚↑2) mod 𝑃) = ((𝑃 − 1) − ((𝑛↑2) mod 𝑃))) → ((abs‘(𝑚 + (i · 𝑛)))↑2) = (((ℜ‘(𝑚 + (i · 𝑛)))↑2) + ((ℑ‘(𝑚 + (i · 𝑛)))↑2)))
200	137, 145	crred 15410	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑛 ∈ (0...𝑁)) ∧ ((𝑚↑2) mod 𝑃) = ((𝑃 − 1) − ((𝑛↑2) mod 𝑃))) → (ℜ‘(𝑚 + (i · 𝑛))) = 𝑚)
201	200	oveq1d 7442	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑛 ∈ (0...𝑁)) ∧ ((𝑚↑2) mod 𝑃) = ((𝑃 − 1) − ((𝑛↑2) mod 𝑃))) → ((ℜ‘(𝑚 + (i · 𝑛)))↑2) = (𝑚↑2))
202	137, 145	crimd 15411	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑛 ∈ (0...𝑁)) ∧ ((𝑚↑2) mod 𝑃) = ((𝑃 − 1) − ((𝑛↑2) mod 𝑃))) → (ℑ‘(𝑚 + (i · 𝑛))) = 𝑛)
203	202	oveq1d 7442	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑛 ∈ (0...𝑁)) ∧ ((𝑚↑2) mod 𝑃) = ((𝑃 − 1) − ((𝑛↑2) mod 𝑃))) → ((ℑ‘(𝑚 + (i · 𝑛)))↑2) = (𝑛↑2))
204	201, 203	oveq12d 7445	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑛 ∈ (0...𝑁)) ∧ ((𝑚↑2) mod 𝑃) = ((𝑃 − 1) − ((𝑛↑2) mod 𝑃))) → (((ℜ‘(𝑚 + (i · 𝑛)))↑2) + ((ℑ‘(𝑚 + (i · 𝑛)))↑2)) = ((𝑚↑2) + (𝑛↑2)))
205	199, 204	eqtrd 2798	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑛 ∈ (0...𝑁)) ∧ ((𝑚↑2) mod 𝑃) = ((𝑃 − 1) − ((𝑛↑2) mod 𝑃))) → ((abs‘(𝑚 + (i · 𝑛)))↑2) = ((𝑚↑2) + (𝑛↑2)))
206	205	oveq1d 7442	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑛 ∈ (0...𝑁)) ∧ ((𝑚↑2) mod 𝑃) = ((𝑃 − 1) − ((𝑛↑2) mod 𝑃))) → (((abs‘(𝑚 + (i · 𝑛)))↑2) + 1) = (((𝑚↑2) + (𝑛↑2)) + 1))
207	101	nncnd 12361	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑛 ∈ (0...𝑁)) ∧ ((𝑚↑2) mod 𝑃) = ((𝑃 − 1) − ((𝑛↑2) mod 𝑃))) → (((𝑚↑2) + (𝑛↑2)) + 1) ∈ ℂ)
208	207, 95, 106	divcan1d 12104	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑛 ∈ (0...𝑁)) ∧ ((𝑚↑2) mod 𝑃) = ((𝑃 − 1) − ((𝑛↑2) mod 𝑃))) → (((((𝑚↑2) + (𝑛↑2)) + 1) / 𝑃) · 𝑃) = (((𝑚↑2) + (𝑛↑2)) + 1))
209	206, 208	eqtr4d 2801	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑛 ∈ (0...𝑁)) ∧ ((𝑚↑2) mod 𝑃) = ((𝑃 − 1) − ((𝑛↑2) mod 𝑃))) → (((abs‘(𝑚 + (i · 𝑛)))↑2) + 1) = (((((𝑚↑2) + (𝑛↑2)) + 1) / 𝑃) · 𝑃))
210		oveq1 7434	. . . . . . . . . 10 ⊢ (𝑘 = ((((𝑚↑2) + (𝑛↑2)) + 1) / 𝑃) → (𝑘 · 𝑃) = (((((𝑚↑2) + (𝑛↑2)) + 1) / 𝑃) · 𝑃))
211	210	eqeq2d 2774	. . . . . . . . 9 ⊢ (𝑘 = ((((𝑚↑2) + (𝑛↑2)) + 1) / 𝑃) → ((((abs‘𝑢)↑2) + 1) = (𝑘 · 𝑃) ↔ (((abs‘𝑢)↑2) + 1) = (((((𝑚↑2) + (𝑛↑2)) + 1) / 𝑃) · 𝑃)))
212		fveq2 6893	. . . . . . . . . . . 12 ⊢ (𝑢 = (𝑚 + (i · 𝑛)) → (abs‘𝑢) = (abs‘(𝑚 + (i · 𝑛))))
213	212	oveq1d 7442	. . . . . . . . . . 11 ⊢ (𝑢 = (𝑚 + (i · 𝑛)) → ((abs‘𝑢)↑2) = ((abs‘(𝑚 + (i · 𝑛)))↑2))
214	213	oveq1d 7442	. . . . . . . . . 10 ⊢ (𝑢 = (𝑚 + (i · 𝑛)) → (((abs‘𝑢)↑2) + 1) = (((abs‘(𝑚 + (i · 𝑛)))↑2) + 1))
215	214	eqeq1d 2765	. . . . . . . . 9 ⊢ (𝑢 = (𝑚 + (i · 𝑛)) → ((((abs‘𝑢)↑2) + 1) = (((((𝑚↑2) + (𝑛↑2)) + 1) / 𝑃) · 𝑃) ↔ (((abs‘(𝑚 + (i · 𝑛)))↑2) + 1) = (((((𝑚↑2) + (𝑛↑2)) + 1) / 𝑃) · 𝑃)))
216	211, 215	rspc2ev 3591	. . . . . . . 8 ⊢ ((((((𝑚↑2) + (𝑛↑2)) + 1) / 𝑃) ∈ (1...(𝑃 − 1)) ∧ (𝑚 + (i · 𝑛)) ∈ ℤ[i] ∧ (((abs‘(𝑚 + (i · 𝑛)))↑2) + 1) = (((((𝑚↑2) + (𝑛↑2)) + 1) / 𝑃) · 𝑃)) → ∃𝑘 ∈ (1...(𝑃 − 1))∃𝑢 ∈ ℤ[i] (((abs‘𝑢)↑2) + 1) = (𝑘 · 𝑃))
217	194, 196, 209, 216	syl3anc 1398	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑛 ∈ (0...𝑁)) ∧ ((𝑚↑2) mod 𝑃) = ((𝑃 − 1) − ((𝑛↑2) mod 𝑃))) → ∃𝑘 ∈ (1...(𝑃 − 1))∃𝑢 ∈ ℤ[i] (((abs‘𝑢)↑2) + 1) = (𝑘 · 𝑃))
218	217	3expia 1139	. . . . . 6 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑛 ∈ (0...𝑁))) → (((𝑚↑2) mod 𝑃) = ((𝑃 − 1) − ((𝑛↑2) mod 𝑃)) → ∃𝑘 ∈ (1...(𝑃 − 1))∃𝑢 ∈ ℤ[i] (((abs‘𝑢)↑2) + 1) = (𝑘 · 𝑃)))
219	42, 218	syl5 35	. . . . 5 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑛 ∈ (0...𝑁))) → ((𝑗 = ((𝑚↑2) mod 𝑃) ∧ 𝑗 = ((𝑃 − 1) − ((𝑛↑2) mod 𝑃))) → ∃𝑘 ∈ (1...(𝑃 − 1))∃𝑢 ∈ ℤ[i] (((abs‘𝑢)↑2) + 1) = (𝑘 · 𝑃)))
220	219	rexlimdvva 3222	. . . 4 ⊢ (𝜑 → (∃𝑚 ∈ (0...𝑁)∃𝑛 ∈ (0...𝑁)(𝑗 = ((𝑚↑2) mod 𝑃) ∧ 𝑗 = ((𝑃 − 1) − ((𝑛↑2) mod 𝑃))) → ∃𝑘 ∈ (1...(𝑃 − 1))∃𝑢 ∈ ℤ[i] (((abs‘𝑢)↑2) + 1) = (𝑘 · 𝑃)))
221	41, 220	biimtrid 245	. . 3 ⊢ (𝜑 → (𝑗 ∈ (𝐴 ∩ ran 𝐹) → ∃𝑘 ∈ (1...(𝑃 − 1))∃𝑢 ∈ ℤ[i] (((abs‘𝑢)↑2) + 1) = (𝑘 · 𝑃)))
222	221	exlimdv 1967	. 2 ⊢ (𝜑 → (∃𝑗 𝑗 ∈ (𝐴 ∩ ran 𝐹) → ∃𝑘 ∈ (1...(𝑃 − 1))∃𝑢 ∈ ℤ[i] (((abs‘𝑢)↑2) + 1) = (𝑘 · 𝑃)))
223	9, 222	mpd 16	1 ⊢ (𝜑 → ∃𝑘 ∈ (1...(𝑃 − 1))∃𝑢 ∈ ℤ[i] (((abs‘𝑢)↑2) + 1) = (𝑘 · 𝑃))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1569 ∃wex 1811 ∈ wcel 2146 {cab 2741 ≠ wne 2958 ∃wrex 3089 ∩ cin 4091 ∅c0 4280 class class class wbr 5105 ↦ cmpt 5188 ran crn 5655 ‘cfv 6546 (class class class)co 7427 ℂcc 11208 ℝcr 11209 0cc0 11210 1c1 11211 ici 11212 + caddc 11213 · cmul 11215 < clt 11353 ≤ cle 11354 − cmin 11551 / cdiv 11983 ℕcn 12345 2c2 12407 4c4 12409 ℕ₀cn0 12616 ℤcz 12703 ℝ⁺crp 13130 ...cfz 13649 mod cmo 14019 ↑cexp 14214 ℜcre 15276 ℑcim 15277 abscabs 15413 ∥ cdvds 16434 ℙcprime 16858 ℤ[i]cgz 17119

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1944 ax-6 2001 ax-7 2042 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2196 ax-12 2215 ax-ext 2735 ax-rep 5234 ax-sep 5251 ax-nul 5262 ax-pow 5330 ax-pr 5394 ax-un 7758 ax-cnex 11266 ax-resscn 11267 ax-1cn 11268 ax-icn 11269 ax-addcl 11270 ax-addrcl 11271 ax-mulcl 11272 ax-mulrcl 11273 ax-mulcom 11274 ax-addass 11275 ax-mulass 11276 ax-distr 11277 ax-i2m1 11278 ax-1ne0 11279 ax-1rid 11280 ax-rnegex 11281 ax-rrecex 11282 ax-cnre 11283 ax-pre-lttri 11284 ax-pre-lttrn 11285 ax-pre-ltadd 11286 ax-pre-mulgt0 11287 ax-pre-sup 11288

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2101 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3368 df-reu 3369 df-rab 3416 df-v 3455 df-sbc 3742 df-csb 3851 df-ss 3900 df-pss 4016 df-dif 4049 df-in 4093 df-un 4153 df-nul 4281 df-if 4485 df-pw 4561 df-sn 4588 df-pr 4590 df-op 4838 df-uni 4870 df-int 4910 df-iun 4955 df-br 5106 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5549 df-eprel 5554 df-po 5562 df-so 5563 df-fr 5607 df-we 5609 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-pred 6310 df-ord 6371 df-on 6372 df-lim 6373 df-suc 6374 df-iota 6500 df-fun 6548 df-fn 6549 df-f 6550 df-f1 6551 df-fo 6552 df-f1o 6553 df-fv 6554 df-riota 7384 df-ov 7430 df-oprab 7431 df-mpo 7432 df-om 7885 df-1st 8008 df-2nd 8009 df-frecs 8310 df-wrecs 8341 df-recs 8390 df-rdg 8429 df-1o 8487 df-2o 8488 df-oadd 8491 df-er 8728 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-sup 9444 df-inf 9445 df-dju 9992 df-card 10030 df-pnf 11355 df-mnf 11356 df-xr 11357 df-ltxr 11358 df-le 11359 df-sub 11553 df-neg 11554 df-div 11984 df-nn 12346 df-2 12415 df-3 12416 df-4 12417 df-n0 12617 df-xnn0 12690 df-z 12704 df-uz 12976 df-rp 13131 df-fz 13650 df-fl 13942 df-mod 14020 df-seq 14155 df-exp 14215 df-hash 14485 df-cj 15278 df-re 15279 df-im 15280 df-sqrt 15414 df-abs 15415 df-dvds 16435 df-gcd 16677 df-prm 16859 df-gz 17120

This theorem is used by: 4sqlem13 17147

W3C validator