Theorem 4sqlem11 12964

Description: Lemma for 4sq 12973. Use the pigeonhole principle to show that the sets {𝑚↑2 ∣ 𝑚 ∈ (0...𝑁)} and {-1 − 𝑛↑2 ∣ 𝑛 ∈ (0...𝑁)} have a common element, mod 𝑃. Note that although the conclusion is stated in terms of 𝐴 ∩ ran 𝐹 being nonempty, it is also inhabited by 4sqleminfi 12960 and fin0 7067. (Contributed by Mario Carneiro, 15-Jul-2014.)

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

Assertion

Ref	Expression
4sqlem11	⊢ (𝜑 → (𝐴 ∩ ran 𝐹) ≠ ∅)

Distinct variable groups:   𝑣,𝐴   𝑚,𝑁,𝑢   𝑣,𝑃   𝑃,𝑚,𝑢   𝜑,𝑣   𝜑,𝑚,𝑢

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

Proof of Theorem 4sqlem11

Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		4sq.2	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℕ)
2		4sq.4	. . . . . . . 8 ⊢ (𝜑 → 𝑃 ∈ ℙ)
3		prmnn 12672	. . . . . . . 8 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
4	2, 3	syl 14	. . . . . . 7 ⊢ (𝜑 → 𝑃 ∈ ℕ)
5		4sqlem11.5	. . . . . . 7 ⊢ 𝐴 = {𝑢 ∣ ∃𝑚 ∈ (0...𝑁)𝑢 = ((𝑚↑2) mod 𝑃)}
6	1, 4, 5	4sqlemafi 12958	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ Fin)
7		4sqlem11.6	. . . . . . 7 ⊢ 𝐹 = (𝑣 ∈ 𝐴 ↦ ((𝑃 − 1) − 𝑣))
8	1, 4, 5, 7	4sqlemffi 12959	. . . . . 6 ⊢ (𝜑 → ran 𝐹 ∈ Fin)
9	1, 4, 5, 7	4sqleminfi 12960	. . . . . 6 ⊢ (𝜑 → (𝐴 ∩ ran 𝐹) ∈ Fin)
10		unfiin 7111	. . . . . 6 ⊢ ((𝐴 ∈ Fin ∧ ran 𝐹 ∈ Fin ∧ (𝐴 ∩ ran 𝐹) ∈ Fin) → (𝐴 ∪ ran 𝐹) ∈ Fin)
11	6, 8, 9, 10	syl3anc 1271	. . . . 5 ⊢ (𝜑 → (𝐴 ∪ ran 𝐹) ∈ Fin)
12		hashcl 11033	. . . . 5 ⊢ ((𝐴 ∪ ran 𝐹) ∈ Fin → (♯‘(𝐴 ∪ ran 𝐹)) ∈ ℕ0)
13	11, 12	syl 14	. . . 4 ⊢ (𝜑 → (♯‘(𝐴 ∪ ran 𝐹)) ∈ ℕ0)
14	13	nn0red 9446	. . 3 ⊢ (𝜑 → (♯‘(𝐴 ∪ ran 𝐹)) ∈ ℝ)
15		prmz 12673	. . . . 5 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℤ)
16	2, 15	syl 14	. . . 4 ⊢ (𝜑 → 𝑃 ∈ ℤ)
17	16	zred 9592	. . 3 ⊢ (𝜑 → 𝑃 ∈ ℝ)
18		0zd 9481	. . . . . . 7 ⊢ (𝜑 → 0 ∈ ℤ)
19		peano2zm 9507	. . . . . . . 8 ⊢ (𝑃 ∈ ℤ → (𝑃 − 1) ∈ ℤ)
20	16, 19	syl 14	. . . . . . 7 ⊢ (𝜑 → (𝑃 − 1) ∈ ℤ)
21	18, 20	fzfigd 10683	. . . . . 6 ⊢ (𝜑 → (0...(𝑃 − 1)) ∈ Fin)
22		elfzelz 10250	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ (0...𝑁) → 𝑚 ∈ ℤ)
23		zsqcl 10862	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ ℤ → (𝑚↑2) ∈ ℤ)
24	22, 23	syl 14	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ (0...𝑁) → (𝑚↑2) ∈ ℤ)
25		zmodfz 10598	. . . . . . . . . . . 12 ⊢ (((𝑚↑2) ∈ ℤ ∧ 𝑃 ∈ ℕ) → ((𝑚↑2) mod 𝑃) ∈ (0...(𝑃 − 1)))
26	24, 4, 25	syl2anr 290	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...𝑁)) → ((𝑚↑2) mod 𝑃) ∈ (0...(𝑃 − 1)))
27		eleq1a 2301	. . . . . . . . . . 11 ⊢ (((𝑚↑2) mod 𝑃) ∈ (0...(𝑃 − 1)) → (𝑢 = ((𝑚↑2) mod 𝑃) → 𝑢 ∈ (0...(𝑃 − 1))))
28	26, 27	syl 14	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...𝑁)) → (𝑢 = ((𝑚↑2) mod 𝑃) → 𝑢 ∈ (0...(𝑃 − 1))))
29	28	rexlimdva 2648	. . . . . . . . 9 ⊢ (𝜑 → (∃𝑚 ∈ (0...𝑁)𝑢 = ((𝑚↑2) mod 𝑃) → 𝑢 ∈ (0...(𝑃 − 1))))
30	29	abssdv 3299	. . . . . . . 8 ⊢ (𝜑 → {𝑢 ∣ ∃𝑚 ∈ (0...𝑁)𝑢 = ((𝑚↑2) mod 𝑃)} ⊆ (0...(𝑃 − 1)))
31	5, 30	eqsstrid 3271	. . . . . . 7 ⊢ (𝜑 → 𝐴 ⊆ (0...(𝑃 − 1)))
32	20	zcnd 9593	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑃 − 1) ∈ ℂ)
33	32	addlidd 8319	. . . . . . . . . . . 12 ⊢ (𝜑 → (0 + (𝑃 − 1)) = (𝑃 − 1))
34	33	oveq1d 6028	. . . . . . . . . . 11 ⊢ (𝜑 → ((0 + (𝑃 − 1)) − 𝑣) = ((𝑃 − 1) − 𝑣))
35	34	adantr 276	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → ((0 + (𝑃 − 1)) − 𝑣) = ((𝑃 − 1) − 𝑣))
36	31	sselda 3225	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → 𝑣 ∈ (0...(𝑃 − 1)))
37		fzrev3i 10313	. . . . . . . . . . 11 ⊢ (𝑣 ∈ (0...(𝑃 − 1)) → ((0 + (𝑃 − 1)) − 𝑣) ∈ (0...(𝑃 − 1)))
38	36, 37	syl 14	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → ((0 + (𝑃 − 1)) − 𝑣) ∈ (0...(𝑃 − 1)))
39	35, 38	eqeltrrd 2307	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → ((𝑃 − 1) − 𝑣) ∈ (0...(𝑃 − 1)))
40	39, 7	fmptd 5797	. . . . . . . 8 ⊢ (𝜑 → 𝐹:𝐴⟶(0...(𝑃 − 1)))
41	40	frnd 5489	. . . . . . 7 ⊢ (𝜑 → ran 𝐹 ⊆ (0...(𝑃 − 1)))
42	31, 41	unssd 3381	. . . . . 6 ⊢ (𝜑 → (𝐴 ∪ ran 𝐹) ⊆ (0...(𝑃 − 1)))
43		ssdomg 6947	. . . . . 6 ⊢ ((0...(𝑃 − 1)) ∈ Fin → ((𝐴 ∪ ran 𝐹) ⊆ (0...(𝑃 − 1)) → (𝐴 ∪ ran 𝐹) ≼ (0...(𝑃 − 1))))
44	21, 42, 43	sylc 62	. . . . 5 ⊢ (𝜑 → (𝐴 ∪ ran 𝐹) ≼ (0...(𝑃 − 1)))
45		fihashdom 11056	. . . . . 6 ⊢ (((𝐴 ∪ ran 𝐹) ∈ Fin ∧ (0...(𝑃 − 1)) ∈ Fin) → ((♯‘(𝐴 ∪ ran 𝐹)) ≤ (♯‘(0...(𝑃 − 1))) ↔ (𝐴 ∪ ran 𝐹) ≼ (0...(𝑃 − 1))))
46	11, 21, 45	syl2anc 411	. . . . 5 ⊢ (𝜑 → ((♯‘(𝐴 ∪ ran 𝐹)) ≤ (♯‘(0...(𝑃 − 1))) ↔ (𝐴 ∪ ran 𝐹) ≼ (0...(𝑃 − 1))))
47	44, 46	mpbird 167	. . . 4 ⊢ (𝜑 → (♯‘(𝐴 ∪ ran 𝐹)) ≤ (♯‘(0...(𝑃 − 1))))
48		fz01en 10278	. . . . . . 7 ⊢ (𝑃 ∈ ℤ → (0...(𝑃 − 1)) ≈ (1...𝑃))
49	16, 48	syl 14	. . . . . 6 ⊢ (𝜑 → (0...(𝑃 − 1)) ≈ (1...𝑃))
50		1zzd 9496	. . . . . . . 8 ⊢ (𝜑 → 1 ∈ ℤ)
51	50, 16	fzfigd 10683	. . . . . . 7 ⊢ (𝜑 → (1...𝑃) ∈ Fin)
52		hashen 11036	. . . . . . 7 ⊢ (((0...(𝑃 − 1)) ∈ Fin ∧ (1...𝑃) ∈ Fin) → ((♯‘(0...(𝑃 − 1))) = (♯‘(1...𝑃)) ↔ (0...(𝑃 − 1)) ≈ (1...𝑃)))
53	21, 51, 52	syl2anc 411	. . . . . 6 ⊢ (𝜑 → ((♯‘(0...(𝑃 − 1))) = (♯‘(1...𝑃)) ↔ (0...(𝑃 − 1)) ≈ (1...𝑃)))
54	49, 53	mpbird 167	. . . . 5 ⊢ (𝜑 → (♯‘(0...(𝑃 − 1))) = (♯‘(1...𝑃)))
55	4	nnnn0d 9445	. . . . . 6 ⊢ (𝜑 → 𝑃 ∈ ℕ0)
56		hashfz1 11035	. . . . . 6 ⊢ (𝑃 ∈ ℕ0 → (♯‘(1...𝑃)) = 𝑃)
57	55, 56	syl 14	. . . . 5 ⊢ (𝜑 → (♯‘(1...𝑃)) = 𝑃)
58	54, 57	eqtrd 2262	. . . 4 ⊢ (𝜑 → (♯‘(0...(𝑃 − 1))) = 𝑃)
59	47, 58	breqtrd 4112	. . 3 ⊢ (𝜑 → (♯‘(𝐴 ∪ ran 𝐹)) ≤ 𝑃)
60	14, 17, 59	lensymd 8291	. 2 ⊢ (𝜑 → ¬ 𝑃 < (♯‘(𝐴 ∪ ran 𝐹)))
61	17	adantr 276	. . . . . 6 ⊢ ((𝜑 ∧ (𝐴 ∩ ran 𝐹) = ∅) → 𝑃 ∈ ℝ)
62	61	ltp1d 9100	. . . . 5 ⊢ ((𝜑 ∧ (𝐴 ∩ ran 𝐹) = ∅) → 𝑃 < (𝑃 + 1))
63	1	nncnd 9147	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℂ)
64		1cnd 8185	. . . . . . . . 9 ⊢ (𝜑 → 1 ∈ ℂ)
65	63, 63, 64, 64	add4d 8338	. . . . . . . 8 ⊢ (𝜑 → ((𝑁 + 𝑁) + (1 + 1)) = ((𝑁 + 1) + (𝑁 + 1)))
66		4sq.3	. . . . . . . . . 10 ⊢ (𝜑 → 𝑃 = ((2 · 𝑁) + 1))
67	66	oveq1d 6028	. . . . . . . . 9 ⊢ (𝜑 → (𝑃 + 1) = (((2 · 𝑁) + 1) + 1))
68		2cn 9204	. . . . . . . . . . 11 ⊢ 2 ∈ ℂ
69		mulcl 8149	. . . . . . . . . . 11 ⊢ ((2 ∈ ℂ ∧ 𝑁 ∈ ℂ) → (2 · 𝑁) ∈ ℂ)
70	68, 63, 69	sylancr 414	. . . . . . . . . 10 ⊢ (𝜑 → (2 · 𝑁) ∈ ℂ)
71	70, 64, 64	addassd 8192	. . . . . . . . 9 ⊢ (𝜑 → (((2 · 𝑁) + 1) + 1) = ((2 · 𝑁) + (1 + 1)))
72	63	2timesd 9377	. . . . . . . . . 10 ⊢ (𝜑 → (2 · 𝑁) = (𝑁 + 𝑁))
73	72	oveq1d 6028	. . . . . . . . 9 ⊢ (𝜑 → ((2 · 𝑁) + (1 + 1)) = ((𝑁 + 𝑁) + (1 + 1)))
74	67, 71, 73	3eqtrd 2266	. . . . . . . 8 ⊢ (𝜑 → (𝑃 + 1) = ((𝑁 + 𝑁) + (1 + 1)))
75	1	nnzd 9591	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑁 ∈ ℤ)
76	18, 75	fzfigd 10683	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (0...𝑁) ∈ Fin)
77	26	ex 115	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑚 ∈ (0...𝑁) → ((𝑚↑2) mod 𝑃) ∈ (0...(𝑃 − 1))))
78	4	adantr 276	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → 𝑃 ∈ ℕ)
79	22	ad2antrl 490	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → 𝑚 ∈ ℤ)
80	79, 23	syl 14	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (𝑚↑2) ∈ ℤ)
81		elfzelz 10250	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑢 ∈ (0...𝑁) → 𝑢 ∈ ℤ)
82	81	ad2antll 491	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → 𝑢 ∈ ℤ)
83		zsqcl 10862	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑢 ∈ ℤ → (𝑢↑2) ∈ ℤ)
84	82, 83	syl 14	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (𝑢↑2) ∈ ℤ)
85		moddvds 12350	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑃 ∈ ℕ ∧ (𝑚↑2) ∈ ℤ ∧ (𝑢↑2) ∈ ℤ) → (((𝑚↑2) mod 𝑃) = ((𝑢↑2) mod 𝑃) ↔ 𝑃 ∥ ((𝑚↑2) − (𝑢↑2))))
86	78, 80, 84, 85	syl3anc 1271	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (((𝑚↑2) mod 𝑃) = ((𝑢↑2) mod 𝑃) ↔ 𝑃 ∥ ((𝑚↑2) − (𝑢↑2))))
87	79	zcnd 9593	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → 𝑚 ∈ ℂ)
88	82	zcnd 9593	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → 𝑢 ∈ ℂ)
89		subsq 10898	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑚 ∈ ℂ ∧ 𝑢 ∈ ℂ) → ((𝑚↑2) − (𝑢↑2)) = ((𝑚 + 𝑢) · (𝑚 − 𝑢)))
90	87, 88, 89	syl2anc 411	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → ((𝑚↑2) − (𝑢↑2)) = ((𝑚 + 𝑢) · (𝑚 − 𝑢)))
91	90	breq2d 4098	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (𝑃 ∥ ((𝑚↑2) − (𝑢↑2)) ↔ 𝑃 ∥ ((𝑚 + 𝑢) · (𝑚 − 𝑢))))
92	2	adantr 276	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → 𝑃 ∈ ℙ)
93	79, 82	zaddcld 9596	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (𝑚 + 𝑢) ∈ ℤ)
94	79, 82	zsubcld 9597	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (𝑚 − 𝑢) ∈ ℤ)
95		euclemma 12708	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑃 ∈ ℙ ∧ (𝑚 + 𝑢) ∈ ℤ ∧ (𝑚 − 𝑢) ∈ ℤ) → (𝑃 ∥ ((𝑚 + 𝑢) · (𝑚 − 𝑢)) ↔ (𝑃 ∥ (𝑚 + 𝑢) ∨ 𝑃 ∥ (𝑚 − 𝑢))))
96	92, 93, 94, 95	syl3anc 1271	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (𝑃 ∥ ((𝑚 + 𝑢) · (𝑚 − 𝑢)) ↔ (𝑃 ∥ (𝑚 + 𝑢) ∨ 𝑃 ∥ (𝑚 − 𝑢))))
97	86, 91, 96	3bitrd 214	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (((𝑚↑2) mod 𝑃) = ((𝑢↑2) mod 𝑃) ↔ (𝑃 ∥ (𝑚 + 𝑢) ∨ 𝑃 ∥ (𝑚 − 𝑢))))
98		zdceq 9545	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑚 ∈ ℤ ∧ 𝑢 ∈ ℤ) → DECID 𝑚 = 𝑢)
99	79, 82, 98	syl2anc 411	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → DECID 𝑚 = 𝑢)
100	93	zred 9592	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (𝑚 + 𝑢) ∈ ℝ)
101		2re 9203	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ 2 ∈ ℝ
102	1	nnred 9146	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝜑 → 𝑁 ∈ ℝ)
103		remulcl 8150	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((2 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (2 · 𝑁) ∈ ℝ)
104	101, 102, 103	sylancr 414	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝜑 → (2 · 𝑁) ∈ ℝ)
105	104	adantr 276	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (2 · 𝑁) ∈ ℝ)
106	92, 15	syl 14	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → 𝑃 ∈ ℤ)
107	106	zred 9592	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → 𝑃 ∈ ℝ)
108	79	zred 9592	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → 𝑚 ∈ ℝ)
109	82	zred 9592	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → 𝑢 ∈ ℝ)
110	102	adantr 276	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → 𝑁 ∈ ℝ)
111		elfzle2 10253	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑚 ∈ (0...𝑁) → 𝑚 ≤ 𝑁)
112	111	ad2antrl 490	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → 𝑚 ≤ 𝑁)
113		elfzle2 10253	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑢 ∈ (0...𝑁) → 𝑢 ≤ 𝑁)
114	113	ad2antll 491	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → 𝑢 ≤ 𝑁)
115	108, 109, 110, 110, 112, 114	le2addd 8733	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (𝑚 + 𝑢) ≤ (𝑁 + 𝑁))
116	63	adantr 276	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → 𝑁 ∈ ℂ)
117	116	2timesd 9377	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (2 · 𝑁) = (𝑁 + 𝑁))
118	115, 117	breqtrrd 4114	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (𝑚 + 𝑢) ≤ (2 · 𝑁))
119	104	ltp1d 9100	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝜑 → (2 · 𝑁) < ((2 · 𝑁) + 1))
120	119, 66	breqtrrd 4114	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝜑 → (2 · 𝑁) < 𝑃)
121	120	adantr 276	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (2 · 𝑁) < 𝑃)
122	100, 105, 107, 118, 121	lelttrd 8294	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (𝑚 + 𝑢) < 𝑃)
123		zltnle 9515	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝑚 + 𝑢) ∈ ℤ ∧ 𝑃 ∈ ℤ) → ((𝑚 + 𝑢) < 𝑃 ↔ ¬ 𝑃 ≤ (𝑚 + 𝑢)))
124	93, 106, 123	syl2anc 411	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → ((𝑚 + 𝑢) < 𝑃 ↔ ¬ 𝑃 ≤ (𝑚 + 𝑢)))
125	122, 124	mpbid 147	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → ¬ 𝑃 ≤ (𝑚 + 𝑢))
126	125	adantr 276	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) ∧ 𝑚 ≠ 𝑢) → ¬ 𝑃 ≤ (𝑚 + 𝑢))
127	16	ad2antrr 488	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) ∧ 𝑚 ≠ 𝑢) → 𝑃 ∈ ℤ)
128	93	adantr 276	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) ∧ 𝑚 ≠ 𝑢) → (𝑚 + 𝑢) ∈ ℤ)
129		1red 8184	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) ∧ 𝑚 ≠ 𝑢) → 1 ∈ ℝ)
130		nn0abscl 11636	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑚 − 𝑢) ∈ ℤ → (abs‘(𝑚 − 𝑢)) ∈ ℕ0)
131	94, 130	syl 14	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (abs‘(𝑚 − 𝑢)) ∈ ℕ0)
132	131	nn0red 9446	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (abs‘(𝑚 − 𝑢)) ∈ ℝ)
133	132	adantr 276	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) ∧ 𝑚 ≠ 𝑢) → (abs‘(𝑚 − 𝑢)) ∈ ℝ)
134	128	zred 9592	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) ∧ 𝑚 ≠ 𝑢) → (𝑚 + 𝑢) ∈ ℝ)
135	131	adantr 276	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) ∧ 𝑚 ≠ 𝑢) → (abs‘(𝑚 − 𝑢)) ∈ ℕ0)
136	135	nn0zd 9590	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) ∧ 𝑚 ≠ 𝑢) → (abs‘(𝑚 − 𝑢)) ∈ ℤ)
137	94	zcnd 9593	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (𝑚 − 𝑢) ∈ ℂ)
138	137	adantr 276	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) ∧ 𝑚 ≠ 𝑢) → (𝑚 − 𝑢) ∈ ℂ)
139	87, 88	subeq0ad 8490	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → ((𝑚 − 𝑢) = 0 ↔ 𝑚 = 𝑢))
140	139	necon3bid 2441	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → ((𝑚 − 𝑢) ≠ 0 ↔ 𝑚 ≠ 𝑢))
141	140	biimpar 297	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) ∧ 𝑚 ≠ 𝑢) → (𝑚 − 𝑢) ≠ 0)
142		0zd 9481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) ∧ 𝑚 ≠ 𝑢) → 0 ∈ ℤ)
143		zapne 9544	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((𝑚 − 𝑢) ∈ ℤ ∧ 0 ∈ ℤ) → ((𝑚 − 𝑢) # 0 ↔ (𝑚 − 𝑢) ≠ 0))
144	94, 142, 143	syl2an2r 597	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) ∧ 𝑚 ≠ 𝑢) → ((𝑚 − 𝑢) # 0 ↔ (𝑚 − 𝑢) ≠ 0))
145	141, 144	mpbird 167	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) ∧ 𝑚 ≠ 𝑢) → (𝑚 − 𝑢) # 0)
146	138, 145	absrpclapd 11739	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) ∧ 𝑚 ≠ 𝑢) → (abs‘(𝑚 − 𝑢)) ∈ ℝ+)
147	146	rpgt0d 9924	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) ∧ 𝑚 ≠ 𝑢) → 0 < (abs‘(𝑚 − 𝑢)))
148		elnnz 9479	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((abs‘(𝑚 − 𝑢)) ∈ ℕ ↔ ((abs‘(𝑚 − 𝑢)) ∈ ℤ ∧ 0 < (abs‘(𝑚 − 𝑢))))
149	136, 147, 148	sylanbrc 417	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) ∧ 𝑚 ≠ 𝑢) → (abs‘(𝑚 − 𝑢)) ∈ ℕ)
150	149	nnge1d 9176	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) ∧ 𝑚 ≠ 𝑢) → 1 ≤ (abs‘(𝑚 − 𝑢)))
151		0cnd 8162	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → 0 ∈ ℂ)
152	87, 88, 151	abs3difd 11751	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (abs‘(𝑚 − 𝑢)) ≤ ((abs‘(𝑚 − 0)) + (abs‘(0 − 𝑢))))
153	87	subid1d 8469	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (𝑚 − 0) = 𝑚)
154	153	fveq2d 5639	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (abs‘(𝑚 − 0)) = (abs‘𝑚))
155		elfzle1 10252	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑚 ∈ (0...𝑁) → 0 ≤ 𝑚)
156	155	ad2antrl 490	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → 0 ≤ 𝑚)
157	108, 156	absidd 11718	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (abs‘𝑚) = 𝑚)
158	154, 157	eqtrd 2262	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (abs‘(𝑚 − 0)) = 𝑚)
159		0cn 8161	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ 0 ∈ ℂ
160		abssub 11652	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((0 ∈ ℂ ∧ 𝑢 ∈ ℂ) → (abs‘(0 − 𝑢)) = (abs‘(𝑢 − 0)))
161	159, 88, 160	sylancr 414	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (abs‘(0 − 𝑢)) = (abs‘(𝑢 − 0)))
162	88	subid1d 8469	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (𝑢 − 0) = 𝑢)
163	162	fveq2d 5639	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (abs‘(𝑢 − 0)) = (abs‘𝑢))
164		elfzle1 10252	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑢 ∈ (0...𝑁) → 0 ≤ 𝑢)
165	164	ad2antll 491	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → 0 ≤ 𝑢)
166	109, 165	absidd 11718	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (abs‘𝑢) = 𝑢)
167	161, 163, 166	3eqtrd 2266	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (abs‘(0 − 𝑢)) = 𝑢)
168	158, 167	oveq12d 6031	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → ((abs‘(𝑚 − 0)) + (abs‘(0 − 𝑢))) = (𝑚 + 𝑢))
169	152, 168	breqtrd 4112	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (abs‘(𝑚 − 𝑢)) ≤ (𝑚 + 𝑢))
170	169	adantr 276	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) ∧ 𝑚 ≠ 𝑢) → (abs‘(𝑚 − 𝑢)) ≤ (𝑚 + 𝑢))
171	129, 133, 134, 150, 170	letrd 8293	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) ∧ 𝑚 ≠ 𝑢) → 1 ≤ (𝑚 + 𝑢))
172		elnnz1 9492	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑚 + 𝑢) ∈ ℕ ↔ ((𝑚 + 𝑢) ∈ ℤ ∧ 1 ≤ (𝑚 + 𝑢)))
173	128, 171, 172	sylanbrc 417	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) ∧ 𝑚 ≠ 𝑢) → (𝑚 + 𝑢) ∈ ℕ)
174		dvdsle 12395	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑃 ∈ ℤ ∧ (𝑚 + 𝑢) ∈ ℕ) → (𝑃 ∥ (𝑚 + 𝑢) → 𝑃 ≤ (𝑚 + 𝑢)))
175	127, 173, 174	syl2anc 411	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) ∧ 𝑚 ≠ 𝑢) → (𝑃 ∥ (𝑚 + 𝑢) → 𝑃 ≤ (𝑚 + 𝑢)))
176	126, 175	mtod 667	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) ∧ 𝑚 ≠ 𝑢) → ¬ 𝑃 ∥ (𝑚 + 𝑢))
177	176	ex 115	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (𝑚 ≠ 𝑢 → ¬ 𝑃 ∥ (𝑚 + 𝑢)))
178	177	a1d 22	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (DECID 𝑚 = 𝑢 → (𝑚 ≠ 𝑢 → ¬ 𝑃 ∥ (𝑚 + 𝑢))))
179	178	necon4addc 2470	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (DECID 𝑚 = 𝑢 → (𝑃 ∥ (𝑚 + 𝑢) → 𝑚 = 𝑢)))
180	99, 179	mpd 13	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (𝑃 ∥ (𝑚 + 𝑢) → 𝑚 = 𝑢))
181		dvdsabsb 12361	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑃 ∈ ℤ ∧ (𝑚 − 𝑢) ∈ ℤ) → (𝑃 ∥ (𝑚 − 𝑢) ↔ 𝑃 ∥ (abs‘(𝑚 − 𝑢))))
182	106, 94, 181	syl2anc 411	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (𝑃 ∥ (𝑚 − 𝑢) ↔ 𝑃 ∥ (abs‘(𝑚 − 𝑢))))
183		letr 8252	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑃 ∈ ℝ ∧ (abs‘(𝑚 − 𝑢)) ∈ ℝ ∧ (𝑚 + 𝑢) ∈ ℝ) → ((𝑃 ≤ (abs‘(𝑚 − 𝑢)) ∧ (abs‘(𝑚 − 𝑢)) ≤ (𝑚 + 𝑢)) → 𝑃 ≤ (𝑚 + 𝑢)))
184	107, 132, 100, 183	syl3anc 1271	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → ((𝑃 ≤ (abs‘(𝑚 − 𝑢)) ∧ (abs‘(𝑚 − 𝑢)) ≤ (𝑚 + 𝑢)) → 𝑃 ≤ (𝑚 + 𝑢)))
185	169, 184	mpan2d 428	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (𝑃 ≤ (abs‘(𝑚 − 𝑢)) → 𝑃 ≤ (𝑚 + 𝑢)))
186	125, 185	mtod 667	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → ¬ 𝑃 ≤ (abs‘(𝑚 − 𝑢)))
187	186	adantr 276	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) ∧ 𝑚 ≠ 𝑢) → ¬ 𝑃 ≤ (abs‘(𝑚 − 𝑢)))
188		dvdsle 12395	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑃 ∈ ℤ ∧ (abs‘(𝑚 − 𝑢)) ∈ ℕ) → (𝑃 ∥ (abs‘(𝑚 − 𝑢)) → 𝑃 ≤ (abs‘(𝑚 − 𝑢))))
189	106, 149, 188	syl2an2r 597	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) ∧ 𝑚 ≠ 𝑢) → (𝑃 ∥ (abs‘(𝑚 − 𝑢)) → 𝑃 ≤ (abs‘(𝑚 − 𝑢))))
190	187, 189	mtod 667	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) ∧ 𝑚 ≠ 𝑢) → ¬ 𝑃 ∥ (abs‘(𝑚 − 𝑢)))
191	190	ex 115	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (𝑚 ≠ 𝑢 → ¬ 𝑃 ∥ (abs‘(𝑚 − 𝑢))))
192	191	a1d 22	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (DECID 𝑚 = 𝑢 → (𝑚 ≠ 𝑢 → ¬ 𝑃 ∥ (abs‘(𝑚 − 𝑢)))))
193	192	necon4addc 2470	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (DECID 𝑚 = 𝑢 → (𝑃 ∥ (abs‘(𝑚 − 𝑢)) → 𝑚 = 𝑢)))
194	99, 193	mpd 13	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (𝑃 ∥ (abs‘(𝑚 − 𝑢)) → 𝑚 = 𝑢))
195	182, 194	sylbid 150	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (𝑃 ∥ (𝑚 − 𝑢) → 𝑚 = 𝑢))
196	180, 195	jaod 722	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → ((𝑃 ∥ (𝑚 + 𝑢) ∨ 𝑃 ∥ (𝑚 − 𝑢)) → 𝑚 = 𝑢))
197	97, 196	sylbid 150	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (((𝑚↑2) mod 𝑃) = ((𝑢↑2) mod 𝑃) → 𝑚 = 𝑢))
198		oveq1 6020	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑚 = 𝑢 → (𝑚↑2) = (𝑢↑2))
199	198	oveq1d 6028	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑚 = 𝑢 → ((𝑚↑2) mod 𝑃) = ((𝑢↑2) mod 𝑃))
200	197, 199	impbid1 142	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (((𝑚↑2) mod 𝑃) = ((𝑢↑2) mod 𝑃) ↔ 𝑚 = 𝑢))
201	200	ex 115	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁)) → (((𝑚↑2) mod 𝑃) = ((𝑢↑2) mod 𝑃) ↔ 𝑚 = 𝑢)))
202	77, 201	dom2lem 6940	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑚 ∈ (0...𝑁) ↦ ((𝑚↑2) mod 𝑃)):(0...𝑁)–1-1→(0...(𝑃 − 1)))
203		f1f1orn 5591	. . . . . . . . . . . . . . . 16 ⊢ ((𝑚 ∈ (0...𝑁) ↦ ((𝑚↑2) mod 𝑃)):(0...𝑁)–1-1→(0...(𝑃 − 1)) → (𝑚 ∈ (0...𝑁) ↦ ((𝑚↑2) mod 𝑃)):(0...𝑁)–1-1-onto→ran (𝑚 ∈ (0...𝑁) ↦ ((𝑚↑2) mod 𝑃)))
204	202, 203	syl 14	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑚 ∈ (0...𝑁) ↦ ((𝑚↑2) mod 𝑃)):(0...𝑁)–1-1-onto→ran (𝑚 ∈ (0...𝑁) ↦ ((𝑚↑2) mod 𝑃)))
205		eqid 2229	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑚 ∈ (0...𝑁) ↦ ((𝑚↑2) mod 𝑃)) = (𝑚 ∈ (0...𝑁) ↦ ((𝑚↑2) mod 𝑃))
206	205	rnmpt 4978	. . . . . . . . . . . . . . . . 17 ⊢ ran (𝑚 ∈ (0...𝑁) ↦ ((𝑚↑2) mod 𝑃)) = {𝑢 ∣ ∃𝑚 ∈ (0...𝑁)𝑢 = ((𝑚↑2) mod 𝑃)}
207	5, 206	eqtr4i 2253	. . . . . . . . . . . . . . . 16 ⊢ 𝐴 = ran (𝑚 ∈ (0...𝑁) ↦ ((𝑚↑2) mod 𝑃))
208		f1oeq3 5570	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 = ran (𝑚 ∈ (0...𝑁) ↦ ((𝑚↑2) mod 𝑃)) → ((𝑚 ∈ (0...𝑁) ↦ ((𝑚↑2) mod 𝑃)):(0...𝑁)–1-1-onto→𝐴 ↔ (𝑚 ∈ (0...𝑁) ↦ ((𝑚↑2) mod 𝑃)):(0...𝑁)–1-1-onto→ran (𝑚 ∈ (0...𝑁) ↦ ((𝑚↑2) mod 𝑃))))
209	207, 208	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ ((𝑚 ∈ (0...𝑁) ↦ ((𝑚↑2) mod 𝑃)):(0...𝑁)–1-1-onto→𝐴 ↔ (𝑚 ∈ (0...𝑁) ↦ ((𝑚↑2) mod 𝑃)):(0...𝑁)–1-1-onto→ran (𝑚 ∈ (0...𝑁) ↦ ((𝑚↑2) mod 𝑃)))
210	204, 209	sylibr 134	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑚 ∈ (0...𝑁) ↦ ((𝑚↑2) mod 𝑃)):(0...𝑁)–1-1-onto→𝐴)
211		f1oeng 6925	. . . . . . . . . . . . . 14 ⊢ (((0...𝑁) ∈ Fin ∧ (𝑚 ∈ (0...𝑁) ↦ ((𝑚↑2) mod 𝑃)):(0...𝑁)–1-1-onto→𝐴) → (0...𝑁) ≈ 𝐴)
212	76, 210, 211	syl2anc 411	. . . . . . . . . . . . 13 ⊢ (𝜑 → (0...𝑁) ≈ 𝐴)
213	212	ensymd 6952	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ≈ (0...𝑁))
214		ax-1cn 8115	. . . . . . . . . . . . . . 15 ⊢ 1 ∈ ℂ
215		pncan 8375	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 + 1) − 1) = 𝑁)
216	63, 214, 215	sylancl 413	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝑁 + 1) − 1) = 𝑁)
217	216	oveq2d 6029	. . . . . . . . . . . . 13 ⊢ (𝜑 → (0...((𝑁 + 1) − 1)) = (0...𝑁))
218	1	nnnn0d 9445	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑁 ∈ ℕ0)
219		peano2nn0 9432	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ0)
220	218, 219	syl 14	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑁 + 1) ∈ ℕ0)
221	220	nn0zd 9590	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑁 + 1) ∈ ℤ)
222		fz01en 10278	. . . . . . . . . . . . . 14 ⊢ ((𝑁 + 1) ∈ ℤ → (0...((𝑁 + 1) − 1)) ≈ (1...(𝑁 + 1)))
223	221, 222	syl 14	. . . . . . . . . . . . 13 ⊢ (𝜑 → (0...((𝑁 + 1) − 1)) ≈ (1...(𝑁 + 1)))
224	217, 223	eqbrtrrd 4110	. . . . . . . . . . . 12 ⊢ (𝜑 → (0...𝑁) ≈ (1...(𝑁 + 1)))
225		entr 6953	. . . . . . . . . . . 12 ⊢ ((𝐴 ≈ (0...𝑁) ∧ (0...𝑁) ≈ (1...(𝑁 + 1))) → 𝐴 ≈ (1...(𝑁 + 1)))
226	213, 224, 225	syl2anc 411	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ≈ (1...(𝑁 + 1)))
227	50, 221	fzfigd 10683	. . . . . . . . . . . 12 ⊢ (𝜑 → (1...(𝑁 + 1)) ∈ Fin)
228		hashen 11036	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ Fin ∧ (1...(𝑁 + 1)) ∈ Fin) → ((♯‘𝐴) = (♯‘(1...(𝑁 + 1))) ↔ 𝐴 ≈ (1...(𝑁 + 1))))
229	6, 227, 228	syl2anc 411	. . . . . . . . . . 11 ⊢ (𝜑 → ((♯‘𝐴) = (♯‘(1...(𝑁 + 1))) ↔ 𝐴 ≈ (1...(𝑁 + 1))))
230	226, 229	mpbird 167	. . . . . . . . . 10 ⊢ (𝜑 → (♯‘𝐴) = (♯‘(1...(𝑁 + 1))))
231		hashfz1 11035	. . . . . . . . . . 11 ⊢ ((𝑁 + 1) ∈ ℕ0 → (♯‘(1...(𝑁 + 1))) = (𝑁 + 1))
232	220, 231	syl 14	. . . . . . . . . 10 ⊢ (𝜑 → (♯‘(1...(𝑁 + 1))) = (𝑁 + 1))
233	230, 232	eqtrd 2262	. . . . . . . . 9 ⊢ (𝜑 → (♯‘𝐴) = (𝑁 + 1))
234	39	ex 115	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑣 ∈ 𝐴 → ((𝑃 − 1) − 𝑣) ∈ (0...(𝑃 − 1))))
235	32	adantr 276	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝐴 ∧ 𝑘 ∈ 𝐴)) → (𝑃 − 1) ∈ ℂ)
236		fzssuz 10290	. . . . . . . . . . . . . . . . . . . 20 ⊢ (0...(𝑃 − 1)) ⊆ (ℤ≥‘0)
237		uzssz 9766	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (ℤ≥‘0) ⊆ ℤ
238		zsscn 9477	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ℤ ⊆ ℂ
239	237, 238	sstri 3234	. . . . . . . . . . . . . . . . . . . 20 ⊢ (ℤ≥‘0) ⊆ ℂ
240	236, 239	sstri 3234	. . . . . . . . . . . . . . . . . . 19 ⊢ (0...(𝑃 − 1)) ⊆ ℂ
241	31, 240	sstrdi 3237	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐴 ⊆ ℂ)
242	241	sselda 3225	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → 𝑣 ∈ ℂ)
243	242	adantrr 479	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝐴 ∧ 𝑘 ∈ 𝐴)) → 𝑣 ∈ ℂ)
244	241	sselda 3225	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑘 ∈ ℂ)
245	244	adantrl 478	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝐴 ∧ 𝑘 ∈ 𝐴)) → 𝑘 ∈ ℂ)
246	235, 243, 245	subcanad 8523	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝐴 ∧ 𝑘 ∈ 𝐴)) → (((𝑃 − 1) − 𝑣) = ((𝑃 − 1) − 𝑘) ↔ 𝑣 = 𝑘))
247	246	ex 115	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝑣 ∈ 𝐴 ∧ 𝑘 ∈ 𝐴) → (((𝑃 − 1) − 𝑣) = ((𝑃 − 1) − 𝑘) ↔ 𝑣 = 𝑘)))
248	234, 247	dom2lem 6940	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑣 ∈ 𝐴 ↦ ((𝑃 − 1) − 𝑣)):𝐴–1-1→(0...(𝑃 − 1)))
249		f1eq1 5534	. . . . . . . . . . . . . 14 ⊢ (𝐹 = (𝑣 ∈ 𝐴 ↦ ((𝑃 − 1) − 𝑣)) → (𝐹:𝐴–1-1→(0...(𝑃 − 1)) ↔ (𝑣 ∈ 𝐴 ↦ ((𝑃 − 1) − 𝑣)):𝐴–1-1→(0...(𝑃 − 1))))
250	7, 249	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (𝐹:𝐴–1-1→(0...(𝑃 − 1)) ↔ (𝑣 ∈ 𝐴 ↦ ((𝑃 − 1) − 𝑣)):𝐴–1-1→(0...(𝑃 − 1)))
251	248, 250	sylibr 134	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹:𝐴–1-1→(0...(𝑃 − 1)))
252		f1f1orn 5591	. . . . . . . . . . . 12 ⊢ (𝐹:𝐴–1-1→(0...(𝑃 − 1)) → 𝐹:𝐴–1-1-onto→ran 𝐹)
253	251, 252	syl 14	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹:𝐴–1-1-onto→ran 𝐹)
254	6, 253	fihasheqf1od 11041	. . . . . . . . . 10 ⊢ (𝜑 → (♯‘𝐴) = (♯‘ran 𝐹))
255	254, 233	eqtr3d 2264	. . . . . . . . 9 ⊢ (𝜑 → (♯‘ran 𝐹) = (𝑁 + 1))
256	233, 255	oveq12d 6031	. . . . . . . 8 ⊢ (𝜑 → ((♯‘𝐴) + (♯‘ran 𝐹)) = ((𝑁 + 1) + (𝑁 + 1)))
257	65, 74, 256	3eqtr4d 2272	. . . . . . 7 ⊢ (𝜑 → (𝑃 + 1) = ((♯‘𝐴) + (♯‘ran 𝐹)))
258	257	adantr 276	. . . . . 6 ⊢ ((𝜑 ∧ (𝐴 ∩ ran 𝐹) = ∅) → (𝑃 + 1) = ((♯‘𝐴) + (♯‘ran 𝐹)))
259	6	adantr 276	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐴 ∩ ran 𝐹) = ∅) → 𝐴 ∈ Fin)
260	8	adantr 276	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐴 ∩ ran 𝐹) = ∅) → ran 𝐹 ∈ Fin)
261		simpr 110	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐴 ∩ ran 𝐹) = ∅) → (𝐴 ∩ ran 𝐹) = ∅)
262		hashun 11058	. . . . . . 7 ⊢ ((𝐴 ∈ Fin ∧ ran 𝐹 ∈ Fin ∧ (𝐴 ∩ ran 𝐹) = ∅) → (♯‘(𝐴 ∪ ran 𝐹)) = ((♯‘𝐴) + (♯‘ran 𝐹)))
263	259, 260, 261, 262	syl3anc 1271	. . . . . 6 ⊢ ((𝜑 ∧ (𝐴 ∩ ran 𝐹) = ∅) → (♯‘(𝐴 ∪ ran 𝐹)) = ((♯‘𝐴) + (♯‘ran 𝐹)))
264	258, 263	eqtr4d 2265	. . . . 5 ⊢ ((𝜑 ∧ (𝐴 ∩ ran 𝐹) = ∅) → (𝑃 + 1) = (♯‘(𝐴 ∪ ran 𝐹)))
265	62, 264	breqtrd 4112	. . . 4 ⊢ ((𝜑 ∧ (𝐴 ∩ ran 𝐹) = ∅) → 𝑃 < (♯‘(𝐴 ∪ ran 𝐹)))
266	265	ex 115	. . 3 ⊢ (𝜑 → ((𝐴 ∩ ran 𝐹) = ∅ → 𝑃 < (♯‘(𝐴 ∪ ran 𝐹))))
267	266	necon3bd 2443	. 2 ⊢ (𝜑 → (¬ 𝑃 < (♯‘(𝐴 ∪ ran 𝐹)) → (𝐴 ∩ ran 𝐹) ≠ ∅))
268	60, 267	mpd 13	1 ⊢ (𝜑 → (𝐴 ∩ ran 𝐹) ≠ ∅)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 713  DECID wdc 839   = wceq 1395   ∈ wcel 2200  {cab 2215   ≠ wne 2400  ∃wrex 2509   ∪ cun 3196   ∩ cin 3197   ⊆ wss 3198  ∅c0 3492   class class class wbr 4086   ↦ cmpt 4148  ran crn 4724  –1-1→wf1 5321  –1-1-onto→wf1o 5323  ‘cfv 5324  (class class class)co 6013   ≈ cen 6902   ≼ cdom 6903  Fincfn 6904  ℂcc 8020  ℝcr 8021  0cc0 8022  1c1 8023   + caddc 8025   · cmul 8027   < clt 8204   ≤ cle 8205   − cmin 8340   # cap 8751  ℕcn 9133  2c2 9184  ℕ₀cn0 9392  ℤcz 9469  ℤ_≥cuz 9745  ...cfz 10233   mod cmo 10574  ↑cexp 10790  ♯chash 11027  abscabs 11548   ∥ cdvds 12338  ℙcprime 12669

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-iinf 4684  ax-cnex 8113  ax-resscn 8114  ax-1cn 8115  ax-1re 8116  ax-icn 8117  ax-addcl 8118  ax-addrcl 8119  ax-mulcl 8120  ax-mulrcl 8121  ax-addcom 8122  ax-mulcom 8123  ax-addass 8124  ax-mulass 8125  ax-distr 8126  ax-i2m1 8127  ax-0lt1 8128  ax-1rid 8129  ax-0id 8130  ax-rnegex 8131  ax-precex 8132  ax-cnre 8133  ax-pre-ltirr 8134  ax-pre-ltwlin 8135  ax-pre-lttrn 8136  ax-pre-apti 8137  ax-pre-ltadd 8138  ax-pre-mulgt0 8139  ax-pre-mulext 8140  ax-arch 8141  ax-caucvg 8142

This theorem depends on definitions:  df-bi 117  df-stab 836  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-if 3604  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-tr 4186  df-id 4388  df-po 4391  df-iso 4392  df-iord 4461  df-on 4463  df-ilim 4464  df-suc 4466  df-iom 4687  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-recs 6466  df-irdg 6531  df-frec 6552  df-1o 6577  df-2o 6578  df-oadd 6581  df-er 6697  df-en 6905  df-dom 6906  df-fin 6907  df-sup 7174  df-pnf 8206  df-mnf 8207  df-xr 8208  df-ltxr 8209  df-le 8210  df-sub 8342  df-neg 8343  df-reap 8745  df-ap 8752  df-div 8843  df-inn 9134  df-2 9192  df-3 9193  df-4 9194  df-n0 9393  df-z 9470  df-uz 9746  df-q 9844  df-rp 9879  df-fz 10234  df-fzo 10368  df-fl 10520  df-mod 10575  df-seqfrec 10700  df-exp 10791  df-ihash 11028  df-cj 11393  df-re 11394  df-im 11395  df-rsqrt 11549  df-abs 11550  df-dvds 12339  df-gcd 12515  df-prm 12670

This theorem is referenced by:  4sqlem12  12965