4sqlem11 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > 4sqlem11		Structured version Visualization version GIF version

Theorem 4sqlem11 15497

Description: Lemma for 4sq 15506. 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. (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
4sqlem11	⊢ (𝜑 → (𝐴 ∩ ran 𝐹) ≠ ∅)

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

Proof of Theorem 4sqlem11 Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fzfid 12634	. . . . . 6 ⊢ (𝜑 → (0...(𝑃 − 1)) ∈ Fin)
2		4sqlem11.5	. . . . . . . 8 ⊢ 𝐴 = {𝑢 ∣ ∃𝑚 ∈ (0...𝑁)𝑢 = ((𝑚↑2) mod 𝑃)}
3		elfzelz 12213	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ (0...𝑁) → 𝑚 ∈ ℤ)
4		zsqcl 12796	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ ℤ → (𝑚↑2) ∈ ℤ)
5	3, 4	syl 17	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ (0...𝑁) → (𝑚↑2) ∈ ℤ)
6		4sq.4	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑃 ∈ ℙ)
7		prmnn 15226	. . . . . . . . . . . . 13 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
8	6, 7	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑃 ∈ ℕ)
9		zmodfz 12554	. . . . . . . . . . . 12 ⊢ (((𝑚↑2) ∈ ℤ ∧ 𝑃 ∈ ℕ) → ((𝑚↑2) mod 𝑃) ∈ (0...(𝑃 − 1)))
10	5, 8, 9	syl2anr 494	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...𝑁)) → ((𝑚↑2) mod 𝑃) ∈ (0...(𝑃 − 1)))
11		eleq1a 2683	. . . . . . . . . . 11 ⊢ (((𝑚↑2) mod 𝑃) ∈ (0...(𝑃 − 1)) → (𝑢 = ((𝑚↑2) mod 𝑃) → 𝑢 ∈ (0...(𝑃 − 1))))
12	10, 11	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...𝑁)) → (𝑢 = ((𝑚↑2) mod 𝑃) → 𝑢 ∈ (0...(𝑃 − 1))))
13	12	rexlimdva 3013	. . . . . . . . 9 ⊢ (𝜑 → (∃𝑚 ∈ (0...𝑁)𝑢 = ((𝑚↑2) mod 𝑃) → 𝑢 ∈ (0...(𝑃 − 1))))
14	13	abssdv 3639	. . . . . . . 8 ⊢ (𝜑 → {𝑢 ∣ ∃𝑚 ∈ (0...𝑁)𝑢 = ((𝑚↑2) mod 𝑃)} ⊆ (0...(𝑃 − 1)))
15	2, 14	syl5eqss 3612	. . . . . . 7 ⊢ (𝜑 → 𝐴 ⊆ (0...(𝑃 − 1)))
16		prmz 15227	. . . . . . . . . . . . . . . 16 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℤ)
17	6, 16	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑃 ∈ ℤ)
18		peano2zm 11297	. . . . . . . . . . . . . . 15 ⊢ (𝑃 ∈ ℤ → (𝑃 − 1) ∈ ℤ)
19	17, 18	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑃 − 1) ∈ ℤ)
20	19	zcnd 11359	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑃 − 1) ∈ ℂ)
21	20	addid2d 10116	. . . . . . . . . . . 12 ⊢ (𝜑 → (0 + (𝑃 − 1)) = (𝑃 − 1))
22	21	oveq1d 6564	. . . . . . . . . . 11 ⊢ (𝜑 → ((0 + (𝑃 − 1)) − 𝑣) = ((𝑃 − 1) − 𝑣))
23	22	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → ((0 + (𝑃 − 1)) − 𝑣) = ((𝑃 − 1) − 𝑣))
24	15	sselda 3568	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → 𝑣 ∈ (0...(𝑃 − 1)))
25		fzrev3i 12277	. . . . . . . . . . 11 ⊢ (𝑣 ∈ (0...(𝑃 − 1)) → ((0 + (𝑃 − 1)) − 𝑣) ∈ (0...(𝑃 − 1)))
26	24, 25	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → ((0 + (𝑃 − 1)) − 𝑣) ∈ (0...(𝑃 − 1)))
27	23, 26	eqeltrrd 2689	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → ((𝑃 − 1) − 𝑣) ∈ (0...(𝑃 − 1)))
28		4sqlem11.6	. . . . . . . . 9 ⊢ 𝐹 = (𝑣 ∈ 𝐴 ↦ ((𝑃 − 1) − 𝑣))
29	27, 28	fmptd 6292	. . . . . . . 8 ⊢ (𝜑 → 𝐹:𝐴⟶(0...(𝑃 − 1)))
30		frn 5966	. . . . . . . 8 ⊢ (𝐹:𝐴⟶(0...(𝑃 − 1)) → ran 𝐹 ⊆ (0...(𝑃 − 1)))
31	29, 30	syl 17	. . . . . . 7 ⊢ (𝜑 → ran 𝐹 ⊆ (0...(𝑃 − 1)))
32	15, 31	unssd 3751	. . . . . 6 ⊢ (𝜑 → (𝐴 ∪ ran 𝐹) ⊆ (0...(𝑃 − 1)))
33		ssdomg 7887	. . . . . 6 ⊢ ((0...(𝑃 − 1)) ∈ Fin → ((𝐴 ∪ ran 𝐹) ⊆ (0...(𝑃 − 1)) → (𝐴 ∪ ran 𝐹) ≼ (0...(𝑃 − 1))))
34	1, 32, 33	sylc 63	. . . . 5 ⊢ (𝜑 → (𝐴 ∪ ran 𝐹) ≼ (0...(𝑃 − 1)))
35		ssfi 8065	. . . . . . 7 ⊢ (((0...(𝑃 − 1)) ∈ Fin ∧ (𝐴 ∪ ran 𝐹) ⊆ (0...(𝑃 − 1))) → (𝐴 ∪ ran 𝐹) ∈ Fin)
36	1, 32, 35	syl2anc 691	. . . . . 6 ⊢ (𝜑 → (𝐴 ∪ ran 𝐹) ∈ Fin)
37		hashdom 13029	. . . . . 6 ⊢ (((𝐴 ∪ ran 𝐹) ∈ Fin ∧ (0...(𝑃 − 1)) ∈ Fin) → ((#‘(𝐴 ∪ ran 𝐹)) ≤ (#‘(0...(𝑃 − 1))) ↔ (𝐴 ∪ ran 𝐹) ≼ (0...(𝑃 − 1))))
38	36, 1, 37	syl2anc 691	. . . . 5 ⊢ (𝜑 → ((#‘(𝐴 ∪ ran 𝐹)) ≤ (#‘(0...(𝑃 − 1))) ↔ (𝐴 ∪ ran 𝐹) ≼ (0...(𝑃 − 1))))
39	34, 38	mpbird 246	. . . 4 ⊢ (𝜑 → (#‘(𝐴 ∪ ran 𝐹)) ≤ (#‘(0...(𝑃 − 1))))
40		fz01en 12240	. . . . . . 7 ⊢ (𝑃 ∈ ℤ → (0...(𝑃 − 1)) ≈ (1...𝑃))
41	17, 40	syl 17	. . . . . 6 ⊢ (𝜑 → (0...(𝑃 − 1)) ≈ (1...𝑃))
42		fzfid 12634	. . . . . . 7 ⊢ (𝜑 → (1...𝑃) ∈ Fin)
43		hashen 12997	. . . . . . 7 ⊢ (((0...(𝑃 − 1)) ∈ Fin ∧ (1...𝑃) ∈ Fin) → ((#‘(0...(𝑃 − 1))) = (#‘(1...𝑃)) ↔ (0...(𝑃 − 1)) ≈ (1...𝑃)))
44	1, 42, 43	syl2anc 691	. . . . . 6 ⊢ (𝜑 → ((#‘(0...(𝑃 − 1))) = (#‘(1...𝑃)) ↔ (0...(𝑃 − 1)) ≈ (1...𝑃)))
45	41, 44	mpbird 246	. . . . 5 ⊢ (𝜑 → (#‘(0...(𝑃 − 1))) = (#‘(1...𝑃)))
46	8	nnnn0d 11228	. . . . . 6 ⊢ (𝜑 → 𝑃 ∈ ℕ₀)
47		hashfz1 12996	. . . . . 6 ⊢ (𝑃 ∈ ℕ₀ → (#‘(1...𝑃)) = 𝑃)
48	46, 47	syl 17	. . . . 5 ⊢ (𝜑 → (#‘(1...𝑃)) = 𝑃)
49	45, 48	eqtrd 2644	. . . 4 ⊢ (𝜑 → (#‘(0...(𝑃 − 1))) = 𝑃)
50	39, 49	breqtrd 4609	. . 3 ⊢ (𝜑 → (#‘(𝐴 ∪ ran 𝐹)) ≤ 𝑃)
51		hashcl 13009	. . . . . 6 ⊢ ((𝐴 ∪ ran 𝐹) ∈ Fin → (#‘(𝐴 ∪ ran 𝐹)) ∈ ℕ₀)
52	36, 51	syl 17	. . . . 5 ⊢ (𝜑 → (#‘(𝐴 ∪ ran 𝐹)) ∈ ℕ₀)
53	52	nn0red 11229	. . . 4 ⊢ (𝜑 → (#‘(𝐴 ∪ ran 𝐹)) ∈ ℝ)
54	17	zred 11358	. . . 4 ⊢ (𝜑 → 𝑃 ∈ ℝ)
55	53, 54	lenltd 10062	. . 3 ⊢ (𝜑 → ((#‘(𝐴 ∪ ran 𝐹)) ≤ 𝑃 ↔ ¬ 𝑃 < (#‘(𝐴 ∪ ran 𝐹))))
56	50, 55	mpbid 221	. 2 ⊢ (𝜑 → ¬ 𝑃 < (#‘(𝐴 ∪ ran 𝐹)))
57	54	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝐴 ∩ ran 𝐹) = ∅) → 𝑃 ∈ ℝ)
58	57	ltp1d 10833	. . . . 5 ⊢ ((𝜑 ∧ (𝐴 ∩ ran 𝐹) = ∅) → 𝑃 < (𝑃 + 1))
59		4sq.2	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ ℕ)
60	59	nncnd 10913	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℂ)
61		1cnd 9935	. . . . . . . . 9 ⊢ (𝜑 → 1 ∈ ℂ)
62	60, 60, 61, 61	add4d 10143	. . . . . . . 8 ⊢ (𝜑 → ((𝑁 + 𝑁) + (1 + 1)) = ((𝑁 + 1) + (𝑁 + 1)))
63		4sq.3	. . . . . . . . . 10 ⊢ (𝜑 → 𝑃 = ((2 · 𝑁) + 1))
64	63	oveq1d 6564	. . . . . . . . 9 ⊢ (𝜑 → (𝑃 + 1) = (((2 · 𝑁) + 1) + 1))
65		2cn 10968	. . . . . . . . . . 11 ⊢ 2 ∈ ℂ
66		mulcl 9899	. . . . . . . . . . 11 ⊢ ((2 ∈ ℂ ∧ 𝑁 ∈ ℂ) → (2 · 𝑁) ∈ ℂ)
67	65, 60, 66	sylancr 694	. . . . . . . . . 10 ⊢ (𝜑 → (2 · 𝑁) ∈ ℂ)
68	67, 61, 61	addassd 9941	. . . . . . . . 9 ⊢ (𝜑 → (((2 · 𝑁) + 1) + 1) = ((2 · 𝑁) + (1 + 1)))
69	60	2timesd 11152	. . . . . . . . . 10 ⊢ (𝜑 → (2 · 𝑁) = (𝑁 + 𝑁))
70	69	oveq1d 6564	. . . . . . . . 9 ⊢ (𝜑 → ((2 · 𝑁) + (1 + 1)) = ((𝑁 + 𝑁) + (1 + 1)))
71	64, 68, 70	3eqtrd 2648	. . . . . . . 8 ⊢ (𝜑 → (𝑃 + 1) = ((𝑁 + 𝑁) + (1 + 1)))
72	10	ex 449	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑚 ∈ (0...𝑁) → ((𝑚↑2) mod 𝑃) ∈ (0...(𝑃 − 1))))
73	8	adantr 480	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → 𝑃 ∈ ℕ)
74	3	ad2antrl 760	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → 𝑚 ∈ ℤ)
75	74, 4	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (𝑚↑2) ∈ ℤ)
76		elfzelz 12213	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑢 ∈ (0...𝑁) → 𝑢 ∈ ℤ)
77	76	ad2antll 761	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → 𝑢 ∈ ℤ)
78		zsqcl 12796	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑢 ∈ ℤ → (𝑢↑2) ∈ ℤ)
79	77, 78	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (𝑢↑2) ∈ ℤ)
80		moddvds 14829	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑃 ∈ ℕ ∧ (𝑚↑2) ∈ ℤ ∧ (𝑢↑2) ∈ ℤ) → (((𝑚↑2) mod 𝑃) = ((𝑢↑2) mod 𝑃) ↔ 𝑃 ∥ ((𝑚↑2) − (𝑢↑2))))
81	73, 75, 79, 80	syl3anc 1318	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (((𝑚↑2) mod 𝑃) = ((𝑢↑2) mod 𝑃) ↔ 𝑃 ∥ ((𝑚↑2) − (𝑢↑2))))
82	74	zcnd 11359	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → 𝑚 ∈ ℂ)
83	77	zcnd 11359	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → 𝑢 ∈ ℂ)
84		subsq 12834	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑚 ∈ ℂ ∧ 𝑢 ∈ ℂ) → ((𝑚↑2) − (𝑢↑2)) = ((𝑚 + 𝑢) · (𝑚 − 𝑢)))
85	82, 83, 84	syl2anc 691	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → ((𝑚↑2) − (𝑢↑2)) = ((𝑚 + 𝑢) · (𝑚 − 𝑢)))
86	85	breq2d 4595	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (𝑃 ∥ ((𝑚↑2) − (𝑢↑2)) ↔ 𝑃 ∥ ((𝑚 + 𝑢) · (𝑚 − 𝑢))))
87	6	adantr 480	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → 𝑃 ∈ ℙ)
88	74, 77	zaddcld 11362	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (𝑚 + 𝑢) ∈ ℤ)
89	74, 77	zsubcld 11363	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (𝑚 − 𝑢) ∈ ℤ)
90		euclemma 15263	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑃 ∈ ℙ ∧ (𝑚 + 𝑢) ∈ ℤ ∧ (𝑚 − 𝑢) ∈ ℤ) → (𝑃 ∥ ((𝑚 + 𝑢) · (𝑚 − 𝑢)) ↔ (𝑃 ∥ (𝑚 + 𝑢) ∨ 𝑃 ∥ (𝑚 − 𝑢))))
91	87, 88, 89, 90	syl3anc 1318	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (𝑃 ∥ ((𝑚 + 𝑢) · (𝑚 − 𝑢)) ↔ (𝑃 ∥ (𝑚 + 𝑢) ∨ 𝑃 ∥ (𝑚 − 𝑢))))
92	81, 86, 91	3bitrd 293	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (((𝑚↑2) mod 𝑃) = ((𝑢↑2) mod 𝑃) ↔ (𝑃 ∥ (𝑚 + 𝑢) ∨ 𝑃 ∥ (𝑚 − 𝑢))))
93	88	zred 11358	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (𝑚 + 𝑢) ∈ ℝ)
94		2re 10967	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ 2 ∈ ℝ
95	59	nnred 10912	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝜑 → 𝑁 ∈ ℝ)
96		remulcl 9900	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((2 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (2 · 𝑁) ∈ ℝ)
97	94, 95, 96	sylancr 694	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝜑 → (2 · 𝑁) ∈ ℝ)
98	97	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (2 · 𝑁) ∈ ℝ)
99	87, 16	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → 𝑃 ∈ ℤ)
100	99	zred 11358	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → 𝑃 ∈ ℝ)
101	74	zred 11358	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → 𝑚 ∈ ℝ)
102	77	zred 11358	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → 𝑢 ∈ ℝ)
103	95	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → 𝑁 ∈ ℝ)
104		elfzle2 12216	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑚 ∈ (0...𝑁) → 𝑚 ≤ 𝑁)
105	104	ad2antrl 760	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → 𝑚 ≤ 𝑁)
106		elfzle2 12216	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑢 ∈ (0...𝑁) → 𝑢 ≤ 𝑁)
107	106	ad2antll 761	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → 𝑢 ≤ 𝑁)
108	101, 102, 103, 103, 105, 107	le2addd 10525	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (𝑚 + 𝑢) ≤ (𝑁 + 𝑁))
109	60	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → 𝑁 ∈ ℂ)
110	109	2timesd 11152	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (2 · 𝑁) = (𝑁 + 𝑁))
111	108, 110	breqtrrd 4611	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (𝑚 + 𝑢) ≤ (2 · 𝑁))
112	97	ltp1d 10833	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝜑 → (2 · 𝑁) < ((2 · 𝑁) + 1))
113	112, 63	breqtrrd 4611	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝜑 → (2 · 𝑁) < 𝑃)
114	113	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (2 · 𝑁) < 𝑃)
115	93, 98, 100, 111, 114	lelttrd 10074	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (𝑚 + 𝑢) < 𝑃)
116	93, 100	ltnled 10063	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → ((𝑚 + 𝑢) < 𝑃 ↔ ¬ 𝑃 ≤ (𝑚 + 𝑢)))
117	115, 116	mpbid 221	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → ¬ 𝑃 ≤ (𝑚 + 𝑢))
118	117	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) ∧ 𝑚 ≠ 𝑢) → ¬ 𝑃 ≤ (𝑚 + 𝑢))
119	17	ad2antrr 758	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) ∧ 𝑚 ≠ 𝑢) → 𝑃 ∈ ℤ)
120	88	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) ∧ 𝑚 ≠ 𝑢) → (𝑚 + 𝑢) ∈ ℤ)
121		1red 9934	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) ∧ 𝑚 ≠ 𝑢) → 1 ∈ ℝ)
122		nn0abscl 13900	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑚 − 𝑢) ∈ ℤ → (abs‘(𝑚 − 𝑢)) ∈ ℕ₀)
123	89, 122	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (abs‘(𝑚 − 𝑢)) ∈ ℕ₀)
124	123	nn0red 11229	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (abs‘(𝑚 − 𝑢)) ∈ ℝ)
125	124	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) ∧ 𝑚 ≠ 𝑢) → (abs‘(𝑚 − 𝑢)) ∈ ℝ)
126	120	zred 11358	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) ∧ 𝑚 ≠ 𝑢) → (𝑚 + 𝑢) ∈ ℝ)
127	123	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) ∧ 𝑚 ≠ 𝑢) → (abs‘(𝑚 − 𝑢)) ∈ ℕ₀)
128	127	nn0zd 11356	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) ∧ 𝑚 ≠ 𝑢) → (abs‘(𝑚 − 𝑢)) ∈ ℤ)
129	89	zcnd 11359	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (𝑚 − 𝑢) ∈ ℂ)
130	129	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) ∧ 𝑚 ≠ 𝑢) → (𝑚 − 𝑢) ∈ ℂ)
131	82, 83	subeq0ad 10281	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → ((𝑚 − 𝑢) = 0 ↔ 𝑚 = 𝑢))
132	131	necon3bid 2826	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → ((𝑚 − 𝑢) ≠ 0 ↔ 𝑚 ≠ 𝑢))
133	132	biimpar 501	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) ∧ 𝑚 ≠ 𝑢) → (𝑚 − 𝑢) ≠ 0)
134	130, 133	absrpcld 14035	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) ∧ 𝑚 ≠ 𝑢) → (abs‘(𝑚 − 𝑢)) ∈ ℝ⁺)
135	134	rpgt0d 11751	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) ∧ 𝑚 ≠ 𝑢) → 0 < (abs‘(𝑚 − 𝑢)))
136		elnnz 11264	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((abs‘(𝑚 − 𝑢)) ∈ ℕ ↔ ((abs‘(𝑚 − 𝑢)) ∈ ℤ ∧ 0 < (abs‘(𝑚 − 𝑢))))
137	128, 135, 136	sylanbrc 695	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) ∧ 𝑚 ≠ 𝑢) → (abs‘(𝑚 − 𝑢)) ∈ ℕ)
138	137	nnge1d 10940	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) ∧ 𝑚 ≠ 𝑢) → 1 ≤ (abs‘(𝑚 − 𝑢)))
139		0cnd 9912	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → 0 ∈ ℂ)
140	82, 83, 139	abs3difd 14047	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (abs‘(𝑚 − 𝑢)) ≤ ((abs‘(𝑚 − 0)) + (abs‘(0 − 𝑢))))
141	82	subid1d 10260	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (𝑚 − 0) = 𝑚)
142	141	fveq2d 6107	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (abs‘(𝑚 − 0)) = (abs‘𝑚))
143		elfzle1 12215	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑚 ∈ (0...𝑁) → 0 ≤ 𝑚)
144	143	ad2antrl 760	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → 0 ≤ 𝑚)
145	101, 144	absidd 14009	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (abs‘𝑚) = 𝑚)
146	142, 145	eqtrd 2644	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (abs‘(𝑚 − 0)) = 𝑚)
147		0cn 9911	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ 0 ∈ ℂ
148		abssub 13914	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((0 ∈ ℂ ∧ 𝑢 ∈ ℂ) → (abs‘(0 − 𝑢)) = (abs‘(𝑢 − 0)))
149	147, 83, 148	sylancr 694	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (abs‘(0 − 𝑢)) = (abs‘(𝑢 − 0)))
150	83	subid1d 10260	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (𝑢 − 0) = 𝑢)
151	150	fveq2d 6107	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (abs‘(𝑢 − 0)) = (abs‘𝑢))
152		elfzle1 12215	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑢 ∈ (0...𝑁) → 0 ≤ 𝑢)
153	152	ad2antll 761	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → 0 ≤ 𝑢)
154	102, 153	absidd 14009	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (abs‘𝑢) = 𝑢)
155	149, 151, 154	3eqtrd 2648	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (abs‘(0 − 𝑢)) = 𝑢)
156	146, 155	oveq12d 6567	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → ((abs‘(𝑚 − 0)) + (abs‘(0 − 𝑢))) = (𝑚 + 𝑢))
157	140, 156	breqtrd 4609	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (abs‘(𝑚 − 𝑢)) ≤ (𝑚 + 𝑢))
158	157	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) ∧ 𝑚 ≠ 𝑢) → (abs‘(𝑚 − 𝑢)) ≤ (𝑚 + 𝑢))
159	121, 125, 126, 138, 158	letrd 10073	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) ∧ 𝑚 ≠ 𝑢) → 1 ≤ (𝑚 + 𝑢))
160		elnnz1 11280	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑚 + 𝑢) ∈ ℕ ↔ ((𝑚 + 𝑢) ∈ ℤ ∧ 1 ≤ (𝑚 + 𝑢)))
161	120, 159, 160	sylanbrc 695	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) ∧ 𝑚 ≠ 𝑢) → (𝑚 + 𝑢) ∈ ℕ)
162		dvdsle 14870	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑃 ∈ ℤ ∧ (𝑚 + 𝑢) ∈ ℕ) → (𝑃 ∥ (𝑚 + 𝑢) → 𝑃 ≤ (𝑚 + 𝑢)))
163	119, 161, 162	syl2anc 691	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) ∧ 𝑚 ≠ 𝑢) → (𝑃 ∥ (𝑚 + 𝑢) → 𝑃 ≤ (𝑚 + 𝑢)))
164	118, 163	mtod 188	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) ∧ 𝑚 ≠ 𝑢) → ¬ 𝑃 ∥ (𝑚 + 𝑢))
165	164	ex 449	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (𝑚 ≠ 𝑢 → ¬ 𝑃 ∥ (𝑚 + 𝑢)))
166	165	necon4ad 2801	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (𝑃 ∥ (𝑚 + 𝑢) → 𝑚 = 𝑢))
167		dvdsabsb 14839	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑃 ∈ ℤ ∧ (𝑚 − 𝑢) ∈ ℤ) → (𝑃 ∥ (𝑚 − 𝑢) ↔ 𝑃 ∥ (abs‘(𝑚 − 𝑢))))
168	99, 89, 167	syl2anc 691	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (𝑃 ∥ (𝑚 − 𝑢) ↔ 𝑃 ∥ (abs‘(𝑚 − 𝑢))))
169		letr 10010	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑃 ∈ ℝ ∧ (abs‘(𝑚 − 𝑢)) ∈ ℝ ∧ (𝑚 + 𝑢) ∈ ℝ) → ((𝑃 ≤ (abs‘(𝑚 − 𝑢)) ∧ (abs‘(𝑚 − 𝑢)) ≤ (𝑚 + 𝑢)) → 𝑃 ≤ (𝑚 + 𝑢)))
170	100, 124, 93, 169	syl3anc 1318	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → ((𝑃 ≤ (abs‘(𝑚 − 𝑢)) ∧ (abs‘(𝑚 − 𝑢)) ≤ (𝑚 + 𝑢)) → 𝑃 ≤ (𝑚 + 𝑢)))
171	157, 170	mpan2d 706	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (𝑃 ≤ (abs‘(𝑚 − 𝑢)) → 𝑃 ≤ (𝑚 + 𝑢)))
172	117, 171	mtod 188	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → ¬ 𝑃 ≤ (abs‘(𝑚 − 𝑢)))
173	172	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) ∧ 𝑚 ≠ 𝑢) → ¬ 𝑃 ≤ (abs‘(𝑚 − 𝑢)))
174	99	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) ∧ 𝑚 ≠ 𝑢) → 𝑃 ∈ ℤ)
175		dvdsle 14870	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑃 ∈ ℤ ∧ (abs‘(𝑚 − 𝑢)) ∈ ℕ) → (𝑃 ∥ (abs‘(𝑚 − 𝑢)) → 𝑃 ≤ (abs‘(𝑚 − 𝑢))))
176	174, 137, 175	syl2anc 691	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) ∧ 𝑚 ≠ 𝑢) → (𝑃 ∥ (abs‘(𝑚 − 𝑢)) → 𝑃 ≤ (abs‘(𝑚 − 𝑢))))
177	173, 176	mtod 188	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) ∧ 𝑚 ≠ 𝑢) → ¬ 𝑃 ∥ (abs‘(𝑚 − 𝑢)))
178	177	ex 449	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (𝑚 ≠ 𝑢 → ¬ 𝑃 ∥ (abs‘(𝑚 − 𝑢))))
179	178	necon4ad 2801	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (𝑃 ∥ (abs‘(𝑚 − 𝑢)) → 𝑚 = 𝑢))
180	168, 179	sylbid 229	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (𝑃 ∥ (𝑚 − 𝑢) → 𝑚 = 𝑢))
181	166, 180	jaod 394	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → ((𝑃 ∥ (𝑚 + 𝑢) ∨ 𝑃 ∥ (𝑚 − 𝑢)) → 𝑚 = 𝑢))
182	92, 181	sylbid 229	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (((𝑚↑2) mod 𝑃) = ((𝑢↑2) mod 𝑃) → 𝑚 = 𝑢))
183		oveq1 6556	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑚 = 𝑢 → (𝑚↑2) = (𝑢↑2))
184	183	oveq1d 6564	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑚 = 𝑢 → ((𝑚↑2) mod 𝑃) = ((𝑢↑2) mod 𝑃))
185	182, 184	impbid1 214	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (((𝑚↑2) mod 𝑃) = ((𝑢↑2) mod 𝑃) ↔ 𝑚 = 𝑢))
186	185	ex 449	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁)) → (((𝑚↑2) mod 𝑃) = ((𝑢↑2) mod 𝑃) ↔ 𝑚 = 𝑢)))
187	72, 186	dom2lem 7881	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑚 ∈ (0...𝑁) ↦ ((𝑚↑2) mod 𝑃)):(0...𝑁)–1-1→(0...(𝑃 − 1)))
188		f1f1orn 6061	. . . . . . . . . . . . . . . 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)))
189	187, 188	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑚 ∈ (0...𝑁) ↦ ((𝑚↑2) mod 𝑃)):(0...𝑁)–1-1-onto→ran (𝑚 ∈ (0...𝑁) ↦ ((𝑚↑2) mod 𝑃)))
190		eqid 2610	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑚 ∈ (0...𝑁) ↦ ((𝑚↑2) mod 𝑃)) = (𝑚 ∈ (0...𝑁) ↦ ((𝑚↑2) mod 𝑃))
191	190	rnmpt 5292	. . . . . . . . . . . . . . . . 17 ⊢ ran (𝑚 ∈ (0...𝑁) ↦ ((𝑚↑2) mod 𝑃)) = {𝑢 ∣ ∃𝑚 ∈ (0...𝑁)𝑢 = ((𝑚↑2) mod 𝑃)}
192	2, 191	eqtr4i 2635	. . . . . . . . . . . . . . . 16 ⊢ 𝐴 = ran (𝑚 ∈ (0...𝑁) ↦ ((𝑚↑2) mod 𝑃))
193		f1oeq3 6042	. . . . . . . . . . . . . . . 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))))
194	192, 193	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)))
195	189, 194	sylibr 223	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑚 ∈ (0...𝑁) ↦ ((𝑚↑2) mod 𝑃)):(0...𝑁)–1-1-onto→𝐴)
196		ovex 6577	. . . . . . . . . . . . . . 15 ⊢ (0...𝑁) ∈ V
197	196	f1oen 7862	. . . . . . . . . . . . . 14 ⊢ ((𝑚 ∈ (0...𝑁) ↦ ((𝑚↑2) mod 𝑃)):(0...𝑁)–1-1-onto→𝐴 → (0...𝑁) ≈ 𝐴)
198	195, 197	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (0...𝑁) ≈ 𝐴)
199	198	ensymd 7893	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ≈ (0...𝑁))
200		ax-1cn 9873	. . . . . . . . . . . . . . 15 ⊢ 1 ∈ ℂ
201		pncan 10166	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 + 1) − 1) = 𝑁)
202	60, 200, 201	sylancl 693	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝑁 + 1) − 1) = 𝑁)
203	202	oveq2d 6565	. . . . . . . . . . . . 13 ⊢ (𝜑 → (0...((𝑁 + 1) − 1)) = (0...𝑁))
204	59	nnnn0d 11228	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
205		peano2nn0 11210	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ ℕ₀ → (𝑁 + 1) ∈ ℕ₀)
206	204, 205	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑁 + 1) ∈ ℕ₀)
207	206	nn0zd 11356	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑁 + 1) ∈ ℤ)
208		fz01en 12240	. . . . . . . . . . . . . 14 ⊢ ((𝑁 + 1) ∈ ℤ → (0...((𝑁 + 1) − 1)) ≈ (1...(𝑁 + 1)))
209	207, 208	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (0...((𝑁 + 1) − 1)) ≈ (1...(𝑁 + 1)))
210	203, 209	eqbrtrrd 4607	. . . . . . . . . . . 12 ⊢ (𝜑 → (0...𝑁) ≈ (1...(𝑁 + 1)))
211		entr 7894	. . . . . . . . . . . 12 ⊢ ((𝐴 ≈ (0...𝑁) ∧ (0...𝑁) ≈ (1...(𝑁 + 1))) → 𝐴 ≈ (1...(𝑁 + 1)))
212	199, 210, 211	syl2anc 691	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ≈ (1...(𝑁 + 1)))
213		ssfi 8065	. . . . . . . . . . . . 13 ⊢ (((0...(𝑃 − 1)) ∈ Fin ∧ 𝐴 ⊆ (0...(𝑃 − 1))) → 𝐴 ∈ Fin)
214	1, 15, 213	syl2anc 691	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ∈ Fin)
215		fzfid 12634	. . . . . . . . . . . 12 ⊢ (𝜑 → (1...(𝑁 + 1)) ∈ Fin)
216		hashen 12997	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ Fin ∧ (1...(𝑁 + 1)) ∈ Fin) → ((#‘𝐴) = (#‘(1...(𝑁 + 1))) ↔ 𝐴 ≈ (1...(𝑁 + 1))))
217	214, 215, 216	syl2anc 691	. . . . . . . . . . 11 ⊢ (𝜑 → ((#‘𝐴) = (#‘(1...(𝑁 + 1))) ↔ 𝐴 ≈ (1...(𝑁 + 1))))
218	212, 217	mpbird 246	. . . . . . . . . 10 ⊢ (𝜑 → (#‘𝐴) = (#‘(1...(𝑁 + 1))))
219		hashfz1 12996	. . . . . . . . . . 11 ⊢ ((𝑁 + 1) ∈ ℕ₀ → (#‘(1...(𝑁 + 1))) = (𝑁 + 1))
220	206, 219	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (#‘(1...(𝑁 + 1))) = (𝑁 + 1))
221	218, 220	eqtrd 2644	. . . . . . . . 9 ⊢ (𝜑 → (#‘𝐴) = (𝑁 + 1))
222	27	ex 449	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑣 ∈ 𝐴 → ((𝑃 − 1) − 𝑣) ∈ (0...(𝑃 − 1))))
223	20	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝐴 ∧ 𝑘 ∈ 𝐴)) → (𝑃 − 1) ∈ ℂ)
224		fzssuz 12253	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (0...(𝑃 − 1)) ⊆ (ℤ_≥‘0)
225		uzssz 11583	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (ℤ_≥‘0) ⊆ ℤ
226		zsscn 11262	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ℤ ⊆ ℂ
227	225, 226	sstri 3577	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (ℤ_≥‘0) ⊆ ℂ
228	224, 227	sstri 3577	. . . . . . . . . . . . . . . . . . . 20 ⊢ (0...(𝑃 − 1)) ⊆ ℂ
229	15, 228	syl6ss 3580	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐴 ⊆ ℂ)
230	229	sselda 3568	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → 𝑣 ∈ ℂ)
231	230	adantrr 749	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝐴 ∧ 𝑘 ∈ 𝐴)) → 𝑣 ∈ ℂ)
232	229	sselda 3568	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑘 ∈ ℂ)
233	232	adantrl 748	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝐴 ∧ 𝑘 ∈ 𝐴)) → 𝑘 ∈ ℂ)
234	223, 231, 233	subcanad 10314	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝐴 ∧ 𝑘 ∈ 𝐴)) → (((𝑃 − 1) − 𝑣) = ((𝑃 − 1) − 𝑘) ↔ 𝑣 = 𝑘))
235	234	ex 449	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((𝑣 ∈ 𝐴 ∧ 𝑘 ∈ 𝐴) → (((𝑃 − 1) − 𝑣) = ((𝑃 − 1) − 𝑘) ↔ 𝑣 = 𝑘)))
236	222, 235	dom2lem 7881	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑣 ∈ 𝐴 ↦ ((𝑃 − 1) − 𝑣)):𝐴–1-1→(0...(𝑃 − 1)))
237		f1eq1 6009	. . . . . . . . . . . . . . 15 ⊢ (𝐹 = (𝑣 ∈ 𝐴 ↦ ((𝑃 − 1) − 𝑣)) → (𝐹:𝐴–1-1→(0...(𝑃 − 1)) ↔ (𝑣 ∈ 𝐴 ↦ ((𝑃 − 1) − 𝑣)):𝐴–1-1→(0...(𝑃 − 1))))
238	28, 237	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ (𝐹:𝐴–1-1→(0...(𝑃 − 1)) ↔ (𝑣 ∈ 𝐴 ↦ ((𝑃 − 1) − 𝑣)):𝐴–1-1→(0...(𝑃 − 1)))
239	236, 238	sylibr 223	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹:𝐴–1-1→(0...(𝑃 − 1)))
240		f1f1orn 6061	. . . . . . . . . . . . 13 ⊢ (𝐹:𝐴–1-1→(0...(𝑃 − 1)) → 𝐹:𝐴–1-1-onto→ran 𝐹)
241	239, 240	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹:𝐴–1-1-onto→ran 𝐹)
242		f1oeng 7860	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ Fin ∧ 𝐹:𝐴–1-1-onto→ran 𝐹) → 𝐴 ≈ ran 𝐹)
243	214, 241, 242	syl2anc 691	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ≈ ran 𝐹)
244		ssfi 8065	. . . . . . . . . . . . 13 ⊢ (((0...(𝑃 − 1)) ∈ Fin ∧ ran 𝐹 ⊆ (0...(𝑃 − 1))) → ran 𝐹 ∈ Fin)
245	1, 31, 244	syl2anc 691	. . . . . . . . . . . 12 ⊢ (𝜑 → ran 𝐹 ∈ Fin)
246		hashen 12997	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ Fin ∧ ran 𝐹 ∈ Fin) → ((#‘𝐴) = (#‘ran 𝐹) ↔ 𝐴 ≈ ran 𝐹))
247	214, 245, 246	syl2anc 691	. . . . . . . . . . 11 ⊢ (𝜑 → ((#‘𝐴) = (#‘ran 𝐹) ↔ 𝐴 ≈ ran 𝐹))
248	243, 247	mpbird 246	. . . . . . . . . 10 ⊢ (𝜑 → (#‘𝐴) = (#‘ran 𝐹))
249	248, 221	eqtr3d 2646	. . . . . . . . 9 ⊢ (𝜑 → (#‘ran 𝐹) = (𝑁 + 1))
250	221, 249	oveq12d 6567	. . . . . . . 8 ⊢ (𝜑 → ((#‘𝐴) + (#‘ran 𝐹)) = ((𝑁 + 1) + (𝑁 + 1)))
251	62, 71, 250	3eqtr4d 2654	. . . . . . 7 ⊢ (𝜑 → (𝑃 + 1) = ((#‘𝐴) + (#‘ran 𝐹)))
252	251	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝐴 ∩ ran 𝐹) = ∅) → (𝑃 + 1) = ((#‘𝐴) + (#‘ran 𝐹)))
253	214	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐴 ∩ ran 𝐹) = ∅) → 𝐴 ∈ Fin)
254	245	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐴 ∩ ran 𝐹) = ∅) → ran 𝐹 ∈ Fin)
255		simpr 476	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐴 ∩ ran 𝐹) = ∅) → (𝐴 ∩ ran 𝐹) = ∅)
256		hashun 13032	. . . . . . 7 ⊢ ((𝐴 ∈ Fin ∧ ran 𝐹 ∈ Fin ∧ (𝐴 ∩ ran 𝐹) = ∅) → (#‘(𝐴 ∪ ran 𝐹)) = ((#‘𝐴) + (#‘ran 𝐹)))
257	253, 254, 255, 256	syl3anc 1318	. . . . . 6 ⊢ ((𝜑 ∧ (𝐴 ∩ ran 𝐹) = ∅) → (#‘(𝐴 ∪ ran 𝐹)) = ((#‘𝐴) + (#‘ran 𝐹)))
258	252, 257	eqtr4d 2647	. . . . 5 ⊢ ((𝜑 ∧ (𝐴 ∩ ran 𝐹) = ∅) → (𝑃 + 1) = (#‘(𝐴 ∪ ran 𝐹)))
259	58, 258	breqtrd 4609	. . . 4 ⊢ ((𝜑 ∧ (𝐴 ∩ ran 𝐹) = ∅) → 𝑃 < (#‘(𝐴 ∪ ran 𝐹)))
260	259	ex 449	. . 3 ⊢ (𝜑 → ((𝐴 ∩ ran 𝐹) = ∅ → 𝑃 < (#‘(𝐴 ∪ ran 𝐹))))
261	260	necon3bd 2796	. 2 ⊢ (𝜑 → (¬ 𝑃 < (#‘(𝐴 ∪ ran 𝐹)) → (𝐴 ∩ ran 𝐹) ≠ ∅))
262	56, 261	mpd 15	1 ⊢ (𝜑 → (𝐴 ∩ ran 𝐹) ≠ ∅)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∨ wo 382 ∧ wa 383 = wceq 1475 ∈ wcel 1977 {cab 2596 ≠ wne 2780 ∃wrex 2897 ∪ cun 3538 ∩ cin 3539 ⊆ wss 3540 ∅c0 3874 class class class wbr 4583 ↦ cmpt 4643 ran crn 5039 ⟶wf 5800 –1-1→wf1 5801 –1-1-onto→wf1o 5803 ‘cfv 5804 (class class class)co 6549 ≈ cen 7838 ≼ cdom 7839 Fincfn 7841 ℂcc 9813 ℝcr 9814 0cc0 9815 1c1 9816 + caddc 9818 · cmul 9820 < clt 9953 ≤ cle 9954 − cmin 10145 ℕcn 10897 2c2 10947 ℕ₀cn0 11169 ℤcz 11254 ℤ_≥cuz 11563 ...cfz 12197 mod cmo 12530 ↑cexp 12722 #chash 12979 abscabs 13822 ∥ cdvds 14821 ℙcprime 15223

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-sup 8231 df-inf 8232 df-card 8648 df-cda 8873 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-n0 11170 df-xnn0 11241 df-z 11255 df-uz 11564 df-rp 11709 df-fz 12198 df-fl 12455 df-mod 12531 df-seq 12664 df-exp 12723 df-hash 12980 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-dvds 14822 df-gcd 15055 df-prm 15224

This theorem is referenced by: 4sqlem12 15498

W3C validator