jm2.19 - Mathbox for Stefan O'Rear

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Theorem jm2.19 42291

Description: Lemma 2.19 of [JonesMatijasevic] p. 696. Transfer divisibility constraints between Y-values and their indices. (Contributed by Stefan O'Rear, 24-Sep-2014.)

**Assertion**
Ref	Expression
jm2.19	⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ∥ 𝑁 ↔ (𝐴 Y_rm 𝑀) ∥ (𝐴 Y_rm 𝑁)))

**Proof of Theorem jm2.19**
Step	Hyp	Ref	Expression
1		rmyeq0 42251	. . . . . 6 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) → (𝑁 = 0 ↔ (𝐴 Y_rm 𝑁) = 0))
2	1	3adant2 1128	. . . . 5 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 = 0 ↔ (𝐴 Y_rm 𝑁) = 0))
3		0dvds 16225	. . . . . 6 ⊢ (𝑁 ∈ ℤ → (0 ∥ 𝑁 ↔ 𝑁 = 0))
4	3	3ad2ant3 1132	. . . . 5 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (0 ∥ 𝑁 ↔ 𝑁 = 0))
5		frmy 42212	. . . . . . . 8 ⊢ Y_rm :((ℤ_≥‘2) × ℤ)⟶ℤ
6	5	fovcl 7532	. . . . . . 7 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) → (𝐴 Y_rm 𝑁) ∈ ℤ)
7	6	3adant2 1128	. . . . . 6 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐴 Y_rm 𝑁) ∈ ℤ)
8		0dvds 16225	. . . . . 6 ⊢ ((𝐴 Y_rm 𝑁) ∈ ℤ → (0 ∥ (𝐴 Y_rm 𝑁) ↔ (𝐴 Y_rm 𝑁) = 0))
9	7, 8	syl 17	. . . . 5 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (0 ∥ (𝐴 Y_rm 𝑁) ↔ (𝐴 Y_rm 𝑁) = 0))
10	2, 4, 9	3bitr4d 311	. . . 4 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (0 ∥ 𝑁 ↔ 0 ∥ (𝐴 Y_rm 𝑁)))
11	10	adantr 480	. . 3 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 = 0) → (0 ∥ 𝑁 ↔ 0 ∥ (𝐴 Y_rm 𝑁)))
12		simpr 484	. . . 4 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 = 0) → 𝑀 = 0)
13	12	breq1d 5151	. . 3 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 = 0) → (𝑀 ∥ 𝑁 ↔ 0 ∥ 𝑁))
14	12	oveq2d 7420	. . . . 5 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 = 0) → (𝐴 Y_rm 𝑀) = (𝐴 Y_rm 0))
15		simpl1 1188	. . . . . 6 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 = 0) → 𝐴 ∈ (ℤ_≥‘2))
16		rmy0 42227	. . . . . 6 ⊢ (𝐴 ∈ (ℤ_≥‘2) → (𝐴 Y_rm 0) = 0)
17	15, 16	syl 17	. . . . 5 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 = 0) → (𝐴 Y_rm 0) = 0)
18	14, 17	eqtrd 2766	. . . 4 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 = 0) → (𝐴 Y_rm 𝑀) = 0)
19	18	breq1d 5151	. . 3 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 = 0) → ((𝐴 Y_rm 𝑀) ∥ (𝐴 Y_rm 𝑁) ↔ 0 ∥ (𝐴 Y_rm 𝑁)))
20	11, 13, 19	3bitr4d 311	. 2 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 = 0) → (𝑀 ∥ 𝑁 ↔ (𝐴 Y_rm 𝑀) ∥ (𝐴 Y_rm 𝑁)))
21	5	fovcl 7532	. . . . . . . . 9 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ) → (𝐴 Y_rm 𝑀) ∈ ℤ)
22	21	3adant3 1129	. . . . . . . 8 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐴 Y_rm 𝑀) ∈ ℤ)
23		dvds0 16220	. . . . . . . 8 ⊢ ((𝐴 Y_rm 𝑀) ∈ ℤ → (𝐴 Y_rm 𝑀) ∥ 0)
24	22, 23	syl 17	. . . . . . 7 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐴 Y_rm 𝑀) ∥ 0)
25	16	3ad2ant1 1130	. . . . . . 7 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐴 Y_rm 0) = 0)
26	24, 25	breqtrrd 5169	. . . . . 6 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐴 Y_rm 𝑀) ∥ (𝐴 Y_rm 0))
27		oveq2 7412	. . . . . . 7 ⊢ ((𝑁 mod (abs‘𝑀)) = 0 → (𝐴 Y_rm (𝑁 mod (abs‘𝑀))) = (𝐴 Y_rm 0))
28	27	breq2d 5153	. . . . . 6 ⊢ ((𝑁 mod (abs‘𝑀)) = 0 → ((𝐴 Y_rm 𝑀) ∥ (𝐴 Y_rm (𝑁 mod (abs‘𝑀))) ↔ (𝐴 Y_rm 𝑀) ∥ (𝐴 Y_rm 0)))
29	26, 28	syl5ibrcom 246	. . . . 5 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑁 mod (abs‘𝑀)) = 0 → (𝐴 Y_rm 𝑀) ∥ (𝐴 Y_rm (𝑁 mod (abs‘𝑀)))))
30	29	adantr 480	. . . 4 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 ≠ 0) → ((𝑁 mod (abs‘𝑀)) = 0 → (𝐴 Y_rm 𝑀) ∥ (𝐴 Y_rm (𝑁 mod (abs‘𝑀)))))
31		zre 12563	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ)
32	31	3ad2ant3 1132	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 𝑁 ∈ ℝ)
33	32	ad2antrr 723	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 ≠ 0) ∧ (𝑁 mod (abs‘𝑀)) ≠ 0) → 𝑁 ∈ ℝ)
34		zcn 12564	. . . . . . . . . . . . . 14 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℂ)
35	34	3ad2ant2 1131	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 𝑀 ∈ ℂ)
36	35	ad2antrr 723	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 ≠ 0) ∧ (𝑁 mod (abs‘𝑀)) ≠ 0) → 𝑀 ∈ ℂ)
37		simplr 766	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 ≠ 0) ∧ (𝑁 mod (abs‘𝑀)) ≠ 0) → 𝑀 ≠ 0)
38	36, 37	absrpcld 15399	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 ≠ 0) ∧ (𝑁 mod (abs‘𝑀)) ≠ 0) → (abs‘𝑀) ∈ ℝ⁺)
39		modlt 13848	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℝ ∧ (abs‘𝑀) ∈ ℝ⁺) → (𝑁 mod (abs‘𝑀)) < (abs‘𝑀))
40	33, 38, 39	syl2anc 583	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 ≠ 0) ∧ (𝑁 mod (abs‘𝑀)) ≠ 0) → (𝑁 mod (abs‘𝑀)) < (abs‘𝑀))
41		simpll1 1209	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 ≠ 0) ∧ (𝑁 mod (abs‘𝑀)) ≠ 0) → 𝐴 ∈ (ℤ_≥‘2))
42		simpll3 1211	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 ≠ 0) ∧ (𝑁 mod (abs‘𝑀)) ≠ 0) → 𝑁 ∈ ℤ)
43		simpll2 1210	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 ≠ 0) ∧ (𝑁 mod (abs‘𝑀)) ≠ 0) → 𝑀 ∈ ℤ)
44		nnabscl 15276	. . . . . . . . . . . . 13 ⊢ ((𝑀 ∈ ℤ ∧ 𝑀 ≠ 0) → (abs‘𝑀) ∈ ℕ)
45	43, 37, 44	syl2anc 583	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 ≠ 0) ∧ (𝑁 mod (abs‘𝑀)) ≠ 0) → (abs‘𝑀) ∈ ℕ)
46	42, 45	zmodcld 13860	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 ≠ 0) ∧ (𝑁 mod (abs‘𝑀)) ≠ 0) → (𝑁 mod (abs‘𝑀)) ∈ ℕ₀)
47		nn0abscl 15263	. . . . . . . . . . . . 13 ⊢ (𝑀 ∈ ℤ → (abs‘𝑀) ∈ ℕ₀)
48	47	3ad2ant2 1131	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (abs‘𝑀) ∈ ℕ₀)
49	48	ad2antrr 723	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 ≠ 0) ∧ (𝑁 mod (abs‘𝑀)) ≠ 0) → (abs‘𝑀) ∈ ℕ₀)
50		ltrmynn0 42246	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ (𝑁 mod (abs‘𝑀)) ∈ ℕ₀ ∧ (abs‘𝑀) ∈ ℕ₀) → ((𝑁 mod (abs‘𝑀)) < (abs‘𝑀) ↔ (𝐴 Y_rm (𝑁 mod (abs‘𝑀))) < (𝐴 Y_rm (abs‘𝑀))))
51	41, 46, 49, 50	syl3anc 1368	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 ≠ 0) ∧ (𝑁 mod (abs‘𝑀)) ≠ 0) → ((𝑁 mod (abs‘𝑀)) < (abs‘𝑀) ↔ (𝐴 Y_rm (𝑁 mod (abs‘𝑀))) < (𝐴 Y_rm (abs‘𝑀))))
52	40, 51	mpbid 231	. . . . . . . . 9 ⊢ ((((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 ≠ 0) ∧ (𝑁 mod (abs‘𝑀)) ≠ 0) → (𝐴 Y_rm (𝑁 mod (abs‘𝑀))) < (𝐴 Y_rm (abs‘𝑀)))
53	46	nn0zd 12585	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 ≠ 0) ∧ (𝑁 mod (abs‘𝑀)) ≠ 0) → (𝑁 mod (abs‘𝑀)) ∈ ℤ)
54		rmyabs 42256	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ (𝑁 mod (abs‘𝑀)) ∈ ℤ) → (abs‘(𝐴 Y_rm (𝑁 mod (abs‘𝑀)))) = (𝐴 Y_rm (abs‘(𝑁 mod (abs‘𝑀)))))
55	41, 53, 54	syl2anc 583	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 ≠ 0) ∧ (𝑁 mod (abs‘𝑀)) ≠ 0) → (abs‘(𝐴 Y_rm (𝑁 mod (abs‘𝑀)))) = (𝐴 Y_rm (abs‘(𝑁 mod (abs‘𝑀)))))
56	33, 38	modcld 13843	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 ≠ 0) ∧ (𝑁 mod (abs‘𝑀)) ≠ 0) → (𝑁 mod (abs‘𝑀)) ∈ ℝ)
57		modge0 13847	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℝ ∧ (abs‘𝑀) ∈ ℝ⁺) → 0 ≤ (𝑁 mod (abs‘𝑀)))
58	33, 38, 57	syl2anc 583	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 ≠ 0) ∧ (𝑁 mod (abs‘𝑀)) ≠ 0) → 0 ≤ (𝑁 mod (abs‘𝑀)))
59	56, 58	absidd 15373	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 ≠ 0) ∧ (𝑁 mod (abs‘𝑀)) ≠ 0) → (abs‘(𝑁 mod (abs‘𝑀))) = (𝑁 mod (abs‘𝑀)))
60	59	oveq2d 7420	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 ≠ 0) ∧ (𝑁 mod (abs‘𝑀)) ≠ 0) → (𝐴 Y_rm (abs‘(𝑁 mod (abs‘𝑀)))) = (𝐴 Y_rm (𝑁 mod (abs‘𝑀))))
61	55, 60	eqtrd 2766	. . . . . . . . 9 ⊢ ((((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 ≠ 0) ∧ (𝑁 mod (abs‘𝑀)) ≠ 0) → (abs‘(𝐴 Y_rm (𝑁 mod (abs‘𝑀)))) = (𝐴 Y_rm (𝑁 mod (abs‘𝑀))))
62		rmyabs 42256	. . . . . . . . . 10 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ) → (abs‘(𝐴 Y_rm 𝑀)) = (𝐴 Y_rm (abs‘𝑀)))
63	41, 43, 62	syl2anc 583	. . . . . . . . 9 ⊢ ((((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 ≠ 0) ∧ (𝑁 mod (abs‘𝑀)) ≠ 0) → (abs‘(𝐴 Y_rm 𝑀)) = (𝐴 Y_rm (abs‘𝑀)))
64	52, 61, 63	3brtr4d 5173	. . . . . . . 8 ⊢ ((((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 ≠ 0) ∧ (𝑁 mod (abs‘𝑀)) ≠ 0) → (abs‘(𝐴 Y_rm (𝑁 mod (abs‘𝑀)))) < (abs‘(𝐴 Y_rm 𝑀)))
65	5	fovcl 7532	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ (𝑁 mod (abs‘𝑀)) ∈ ℤ) → (𝐴 Y_rm (𝑁 mod (abs‘𝑀))) ∈ ℤ)
66	41, 53, 65	syl2anc 583	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 ≠ 0) ∧ (𝑁 mod (abs‘𝑀)) ≠ 0) → (𝐴 Y_rm (𝑁 mod (abs‘𝑀))) ∈ ℤ)
67		nn0abscl 15263	. . . . . . . . . . 11 ⊢ ((𝐴 Y_rm (𝑁 mod (abs‘𝑀))) ∈ ℤ → (abs‘(𝐴 Y_rm (𝑁 mod (abs‘𝑀)))) ∈ ℕ₀)
68	66, 67	syl 17	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 ≠ 0) ∧ (𝑁 mod (abs‘𝑀)) ≠ 0) → (abs‘(𝐴 Y_rm (𝑁 mod (abs‘𝑀)))) ∈ ℕ₀)
69	68	nn0red 12534	. . . . . . . . 9 ⊢ ((((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 ≠ 0) ∧ (𝑁 mod (abs‘𝑀)) ≠ 0) → (abs‘(𝐴 Y_rm (𝑁 mod (abs‘𝑀)))) ∈ ℝ)
70	22	ad2antrr 723	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 ≠ 0) ∧ (𝑁 mod (abs‘𝑀)) ≠ 0) → (𝐴 Y_rm 𝑀) ∈ ℤ)
71		nn0abscl 15263	. . . . . . . . . . 11 ⊢ ((𝐴 Y_rm 𝑀) ∈ ℤ → (abs‘(𝐴 Y_rm 𝑀)) ∈ ℕ₀)
72	70, 71	syl 17	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 ≠ 0) ∧ (𝑁 mod (abs‘𝑀)) ≠ 0) → (abs‘(𝐴 Y_rm 𝑀)) ∈ ℕ₀)
73	72	nn0red 12534	. . . . . . . . 9 ⊢ ((((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 ≠ 0) ∧ (𝑁 mod (abs‘𝑀)) ≠ 0) → (abs‘(𝐴 Y_rm 𝑀)) ∈ ℝ)
74	69, 73	ltnled 11362	. . . . . . . 8 ⊢ ((((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 ≠ 0) ∧ (𝑁 mod (abs‘𝑀)) ≠ 0) → ((abs‘(𝐴 Y_rm (𝑁 mod (abs‘𝑀)))) < (abs‘(𝐴 Y_rm 𝑀)) ↔ ¬ (abs‘(𝐴 Y_rm 𝑀)) ≤ (abs‘(𝐴 Y_rm (𝑁 mod (abs‘𝑀))))))
75	64, 74	mpbid 231	. . . . . . 7 ⊢ ((((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 ≠ 0) ∧ (𝑁 mod (abs‘𝑀)) ≠ 0) → ¬ (abs‘(𝐴 Y_rm 𝑀)) ≤ (abs‘(𝐴 Y_rm (𝑁 mod (abs‘𝑀)))))
76		simpr 484	. . . . . . . . 9 ⊢ ((((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 ≠ 0) ∧ (𝑁 mod (abs‘𝑀)) ≠ 0) → (𝑁 mod (abs‘𝑀)) ≠ 0)
77		rmyeq0 42251	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ (𝑁 mod (abs‘𝑀)) ∈ ℤ) → ((𝑁 mod (abs‘𝑀)) = 0 ↔ (𝐴 Y_rm (𝑁 mod (abs‘𝑀))) = 0))
78	41, 53, 77	syl2anc 583	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 ≠ 0) ∧ (𝑁 mod (abs‘𝑀)) ≠ 0) → ((𝑁 mod (abs‘𝑀)) = 0 ↔ (𝐴 Y_rm (𝑁 mod (abs‘𝑀))) = 0))
79	78	necon3bid 2979	. . . . . . . . 9 ⊢ ((((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 ≠ 0) ∧ (𝑁 mod (abs‘𝑀)) ≠ 0) → ((𝑁 mod (abs‘𝑀)) ≠ 0 ↔ (𝐴 Y_rm (𝑁 mod (abs‘𝑀))) ≠ 0))
80	76, 79	mpbid 231	. . . . . . . 8 ⊢ ((((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 ≠ 0) ∧ (𝑁 mod (abs‘𝑀)) ≠ 0) → (𝐴 Y_rm (𝑁 mod (abs‘𝑀))) ≠ 0)
81		dvdsleabs2 16260	. . . . . . . 8 ⊢ (((𝐴 Y_rm 𝑀) ∈ ℤ ∧ (𝐴 Y_rm (𝑁 mod (abs‘𝑀))) ∈ ℤ ∧ (𝐴 Y_rm (𝑁 mod (abs‘𝑀))) ≠ 0) → ((𝐴 Y_rm 𝑀) ∥ (𝐴 Y_rm (𝑁 mod (abs‘𝑀))) → (abs‘(𝐴 Y_rm 𝑀)) ≤ (abs‘(𝐴 Y_rm (𝑁 mod (abs‘𝑀))))))
82	70, 66, 80, 81	syl3anc 1368	. . . . . . 7 ⊢ ((((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 ≠ 0) ∧ (𝑁 mod (abs‘𝑀)) ≠ 0) → ((𝐴 Y_rm 𝑀) ∥ (𝐴 Y_rm (𝑁 mod (abs‘𝑀))) → (abs‘(𝐴 Y_rm 𝑀)) ≤ (abs‘(𝐴 Y_rm (𝑁 mod (abs‘𝑀))))))
83	75, 82	mtod 197	. . . . . 6 ⊢ ((((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 ≠ 0) ∧ (𝑁 mod (abs‘𝑀)) ≠ 0) → ¬ (𝐴 Y_rm 𝑀) ∥ (𝐴 Y_rm (𝑁 mod (abs‘𝑀))))
84	83	ex 412	. . . . 5 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 ≠ 0) → ((𝑁 mod (abs‘𝑀)) ≠ 0 → ¬ (𝐴 Y_rm 𝑀) ∥ (𝐴 Y_rm (𝑁 mod (abs‘𝑀)))))
85	84	necon4ad 2953	. . . 4 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 ≠ 0) → ((𝐴 Y_rm 𝑀) ∥ (𝐴 Y_rm (𝑁 mod (abs‘𝑀))) → (𝑁 mod (abs‘𝑀)) = 0))
86	30, 85	impbid 211	. . 3 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 ≠ 0) → ((𝑁 mod (abs‘𝑀)) = 0 ↔ (𝐴 Y_rm 𝑀) ∥ (𝐴 Y_rm (𝑁 mod (abs‘𝑀)))))
87		simpl2 1189	. . . 4 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 ≠ 0) → 𝑀 ∈ ℤ)
88		simpl3 1190	. . . 4 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 ≠ 0) → 𝑁 ∈ ℤ)
89		simpr 484	. . . 4 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 ≠ 0) → 𝑀 ≠ 0)
90		dvdsabsmod0 42285	. . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ≠ 0) → (𝑀 ∥ 𝑁 ↔ (𝑁 mod (abs‘𝑀)) = 0))
91	87, 88, 89, 90	syl3anc 1368	. . 3 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 ≠ 0) → (𝑀 ∥ 𝑁 ↔ (𝑁 mod (abs‘𝑀)) = 0))
92		simpl1 1188	. . . . 5 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 ≠ 0) → 𝐴 ∈ (ℤ_≥‘2))
93	32	adantr 480	. . . . . . 7 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 ≠ 0) → 𝑁 ∈ ℝ)
94		zre 12563	. . . . . . . . 9 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℝ)
95	94	3ad2ant2 1131	. . . . . . . 8 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 𝑀 ∈ ℝ)
96	95	adantr 480	. . . . . . 7 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 ≠ 0) → 𝑀 ∈ ℝ)
97		modabsdifz 42284	. . . . . . 7 ⊢ ((𝑁 ∈ ℝ ∧ 𝑀 ∈ ℝ ∧ 𝑀 ≠ 0) → ((𝑁 − (𝑁 mod (abs‘𝑀))) / 𝑀) ∈ ℤ)
98	93, 96, 89, 97	syl3anc 1368	. . . . . 6 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 ≠ 0) → ((𝑁 − (𝑁 mod (abs‘𝑀))) / 𝑀) ∈ ℤ)
99	98	znegcld 12669	. . . . 5 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 ≠ 0) → -((𝑁 − (𝑁 mod (abs‘𝑀))) / 𝑀) ∈ ℤ)
100		jm2.19lem4 42290	. . . . 5 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ -((𝑁 − (𝑁 mod (abs‘𝑀))) / 𝑀) ∈ ℤ) → ((𝐴 Y_rm 𝑀) ∥ (𝐴 Y_rm 𝑁) ↔ (𝐴 Y_rm 𝑀) ∥ (𝐴 Y_rm (𝑁 + (-((𝑁 − (𝑁 mod (abs‘𝑀))) / 𝑀) · 𝑀)))))
101	92, 87, 88, 99, 100	syl121anc 1372	. . . 4 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 ≠ 0) → ((𝐴 Y_rm 𝑀) ∥ (𝐴 Y_rm 𝑁) ↔ (𝐴 Y_rm 𝑀) ∥ (𝐴 Y_rm (𝑁 + (-((𝑁 − (𝑁 mod (abs‘𝑀))) / 𝑀) · 𝑀)))))
102	32	recnd 11243	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 𝑁 ∈ ℂ)
103	102	adantr 480	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 ≠ 0) → 𝑁 ∈ ℂ)
104	35	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 ≠ 0) → 𝑀 ∈ ℂ)
105	104, 89	absrpcld 15399	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 ≠ 0) → (abs‘𝑀) ∈ ℝ⁺)
106	93, 105	modcld 13843	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 ≠ 0) → (𝑁 mod (abs‘𝑀)) ∈ ℝ)
107	106	recnd 11243	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 ≠ 0) → (𝑁 mod (abs‘𝑀)) ∈ ℂ)
108	103, 107	subcld 11572	. . . . . . . . . 10 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 ≠ 0) → (𝑁 − (𝑁 mod (abs‘𝑀))) ∈ ℂ)
109	108, 104, 89	divcld 11991	. . . . . . . . 9 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 ≠ 0) → ((𝑁 − (𝑁 mod (abs‘𝑀))) / 𝑀) ∈ ℂ)
110	109, 104	mulneg1d 11668	. . . . . . . 8 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 ≠ 0) → (-((𝑁 − (𝑁 mod (abs‘𝑀))) / 𝑀) · 𝑀) = -(((𝑁 − (𝑁 mod (abs‘𝑀))) / 𝑀) · 𝑀))
111	110	oveq2d 7420	. . . . . . 7 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 ≠ 0) → (𝑁 + (-((𝑁 − (𝑁 mod (abs‘𝑀))) / 𝑀) · 𝑀)) = (𝑁 + -(((𝑁 − (𝑁 mod (abs‘𝑀))) / 𝑀) · 𝑀)))
112	109, 104	mulcld 11235	. . . . . . . 8 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 ≠ 0) → (((𝑁 − (𝑁 mod (abs‘𝑀))) / 𝑀) · 𝑀) ∈ ℂ)
113	103, 112	negsubd 11578	. . . . . . 7 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 ≠ 0) → (𝑁 + -(((𝑁 − (𝑁 mod (abs‘𝑀))) / 𝑀) · 𝑀)) = (𝑁 − (((𝑁 − (𝑁 mod (abs‘𝑀))) / 𝑀) · 𝑀)))
114	108, 104, 89	divcan1d 11992	. . . . . . . . 9 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 ≠ 0) → (((𝑁 − (𝑁 mod (abs‘𝑀))) / 𝑀) · 𝑀) = (𝑁 − (𝑁 mod (abs‘𝑀))))
115	114	oveq2d 7420	. . . . . . . 8 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 ≠ 0) → (𝑁 − (((𝑁 − (𝑁 mod (abs‘𝑀))) / 𝑀) · 𝑀)) = (𝑁 − (𝑁 − (𝑁 mod (abs‘𝑀)))))
116	103, 107	nncand 11577	. . . . . . . 8 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 ≠ 0) → (𝑁 − (𝑁 − (𝑁 mod (abs‘𝑀)))) = (𝑁 mod (abs‘𝑀)))
117	115, 116	eqtrd 2766	. . . . . . 7 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 ≠ 0) → (𝑁 − (((𝑁 − (𝑁 mod (abs‘𝑀))) / 𝑀) · 𝑀)) = (𝑁 mod (abs‘𝑀)))
118	111, 113, 117	3eqtrrd 2771	. . . . . 6 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 ≠ 0) → (𝑁 mod (abs‘𝑀)) = (𝑁 + (-((𝑁 − (𝑁 mod (abs‘𝑀))) / 𝑀) · 𝑀)))
119	118	oveq2d 7420	. . . . 5 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 ≠ 0) → (𝐴 Y_rm (𝑁 mod (abs‘𝑀))) = (𝐴 Y_rm (𝑁 + (-((𝑁 − (𝑁 mod (abs‘𝑀))) / 𝑀) · 𝑀))))
120	119	breq2d 5153	. . . 4 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 ≠ 0) → ((𝐴 Y_rm 𝑀) ∥ (𝐴 Y_rm (𝑁 mod (abs‘𝑀))) ↔ (𝐴 Y_rm 𝑀) ∥ (𝐴 Y_rm (𝑁 + (-((𝑁 − (𝑁 mod (abs‘𝑀))) / 𝑀) · 𝑀)))))
121	101, 120	bitr4d 282	. . 3 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 ≠ 0) → ((𝐴 Y_rm 𝑀) ∥ (𝐴 Y_rm 𝑁) ↔ (𝐴 Y_rm 𝑀) ∥ (𝐴 Y_rm (𝑁 mod (abs‘𝑀)))))
122	86, 91, 121	3bitr4d 311	. 2 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 ≠ 0) → (𝑀 ∥ 𝑁 ↔ (𝐴 Y_rm 𝑀) ∥ (𝐴 Y_rm 𝑁)))
123	20, 122	pm2.61dane 3023	1 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ∥ 𝑁 ↔ (𝐴 Y_rm 𝑀) ∥ (𝐴 Y_rm 𝑁)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ≠ wne 2934 class class class wbr 5141 ‘cfv 6536 (class class class)co 7404 ℂcc 11107 ℝcr 11108 0cc0 11109 + caddc 11112 · cmul 11114 < clt 11249 ≤ cle 11250 − cmin 11445 -cneg 11446 / cdiv 11872 ℕcn 12213 2c2 12268 ℕ₀cn0 12473 ℤcz 12559 ℤ_≥cuz 12823 ℝ⁺crp 12977 mod cmo 13837 abscabs 15185 ∥ cdvds 16202 Y_rm crmy 42198

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-inf2 9635 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 ax-pre-sup 11187 ax-addf 11188

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-iin 4993 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-of 7666 df-om 7852 df-1st 7971 df-2nd 7972 df-supp 8144 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-1o 8464 df-2o 8465 df-oadd 8468 df-omul 8469 df-er 8702 df-map 8821 df-pm 8822 df-ixp 8891 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-fsupp 9361 df-fi 9405 df-sup 9436 df-inf 9437 df-oi 9504 df-card 9933 df-acn 9936 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-div 11873 df-nn 12214 df-2 12276 df-3 12277 df-4 12278 df-5 12279 df-6 12280 df-7 12281 df-8 12282 df-9 12283 df-n0 12474 df-xnn0 12546 df-z 12560 df-dec 12679 df-uz 12824 df-q 12934 df-rp 12978 df-xneg 13095 df-xadd 13096 df-xmul 13097 df-ioo 13331 df-ioc 13332 df-ico 13333 df-icc 13334 df-fz 13488 df-fzo 13631 df-fl 13760 df-mod 13838 df-seq 13970 df-exp 14031 df-fac 14237 df-bc 14266 df-hash 14294 df-shft 15018 df-cj 15050 df-re 15051 df-im 15052 df-sqrt 15186 df-abs 15187 df-limsup 15419 df-clim 15436 df-rlim 15437 df-sum 15637 df-ef 16015 df-sin 16017 df-cos 16018 df-pi 16020 df-dvds 16203 df-gcd 16441 df-numer 16678 df-denom 16679 df-struct 17087 df-sets 17104 df-slot 17122 df-ndx 17134 df-base 17152 df-ress 17181 df-plusg 17217 df-mulr 17218 df-starv 17219 df-sca 17220 df-vsca 17221 df-ip 17222 df-tset 17223 df-ple 17224 df-ds 17226 df-unif 17227 df-hom 17228 df-cco 17229 df-rest 17375 df-topn 17376 df-0g 17394 df-gsum 17395 df-topgen 17396 df-pt 17397 df-prds 17400 df-xrs 17455 df-qtop 17460 df-imas 17461 df-xps 17463 df-mre 17537 df-mrc 17538 df-acs 17540 df-mgm 18571 df-sgrp 18650 df-mnd 18666 df-submnd 18712 df-mulg 18994 df-cntz 19231 df-cmn 19700 df-psmet 21228 df-xmet 21229 df-met 21230 df-bl 21231 df-mopn 21232 df-fbas 21233 df-fg 21234 df-cnfld 21237 df-top 22747 df-topon 22764 df-topsp 22786 df-bases 22800 df-cld 22874 df-ntr 22875 df-cls 22876 df-nei 22953 df-lp 22991 df-perf 22992 df-cn 23082 df-cnp 23083 df-haus 23170 df-tx 23417 df-hmeo 23610 df-fil 23701 df-fm 23793 df-flim 23794 df-flf 23795 df-xms 24177 df-ms 24178 df-tms 24179 df-cncf 24749 df-limc 25746 df-dv 25747 df-log 26441 df-squarenn 42138 df-pell1qr 42139 df-pell14qr 42140 df-pell1234qr 42141 df-pellfund 42142 df-rmx 42199 df-rmy 42200

This theorem is referenced by: jm2.20nn 42295

W3C validator