jm2.19 - Mathbox for Stefan O'Rear

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Theorem jm2.19 42445

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 42405	. . . . . 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 16261	. . . . . 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 42366	. . . . . . . 8 ⊢ Y_rm :((ℤ_≥‘2) × ℤ)⟶ℤ
6	5	fovcl 7555	. . . . . . 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 16261	. . . . . 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 310	. . . 4 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (0 ∥ 𝑁 ↔ 0 ∥ (𝐴 Y_rm 𝑁)))
11	10	adantr 479	. . 3 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 = 0) → (0 ∥ 𝑁 ↔ 0 ∥ (𝐴 Y_rm 𝑁)))
12		simpr 483	. . . 4 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 = 0) → 𝑀 = 0)
13	12	breq1d 5162	. . 3 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 = 0) → (𝑀 ∥ 𝑁 ↔ 0 ∥ 𝑁))
14	12	oveq2d 7442	. . . . 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 42381	. . . . . 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 2768	. . . 4 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 = 0) → (𝐴 Y_rm 𝑀) = 0)
19	18	breq1d 5162	. . 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 310	. 2 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 = 0) → (𝑀 ∥ 𝑁 ↔ (𝐴 Y_rm 𝑀) ∥ (𝐴 Y_rm 𝑁)))
21	5	fovcl 7555	. . . . . . . . 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 16256	. . . . . . . 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 5180	. . . . . 6 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐴 Y_rm 𝑀) ∥ (𝐴 Y_rm 0))
27		oveq2 7434	. . . . . . 7 ⊢ ((𝑁 mod (abs‘𝑀)) = 0 → (𝐴 Y_rm (𝑁 mod (abs‘𝑀))) = (𝐴 Y_rm 0))
28	27	breq2d 5164	. . . . . 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 479	. . . 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 12600	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ)
32	31	3ad2ant3 1132	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 𝑁 ∈ ℝ)
33	32	ad2antrr 724	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 ≠ 0) ∧ (𝑁 mod (abs‘𝑀)) ≠ 0) → 𝑁 ∈ ℝ)
34		zcn 12601	. . . . . . . . . . . . . 14 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℂ)
35	34	3ad2ant2 1131	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 𝑀 ∈ ℂ)
36	35	ad2antrr 724	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 ≠ 0) ∧ (𝑁 mod (abs‘𝑀)) ≠ 0) → 𝑀 ∈ ℂ)
37		simplr 767	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 ≠ 0) ∧ (𝑁 mod (abs‘𝑀)) ≠ 0) → 𝑀 ≠ 0)
38	36, 37	absrpcld 15435	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 ≠ 0) ∧ (𝑁 mod (abs‘𝑀)) ≠ 0) → (abs‘𝑀) ∈ ℝ⁺)
39		modlt 13885	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℝ ∧ (abs‘𝑀) ∈ ℝ⁺) → (𝑁 mod (abs‘𝑀)) < (abs‘𝑀))
40	33, 38, 39	syl2anc 582	. . . . . . . . . 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 15312	. . . . . . . . . . . . 13 ⊢ ((𝑀 ∈ ℤ ∧ 𝑀 ≠ 0) → (abs‘𝑀) ∈ ℕ)
45	43, 37, 44	syl2anc 582	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 ≠ 0) ∧ (𝑁 mod (abs‘𝑀)) ≠ 0) → (abs‘𝑀) ∈ ℕ)
46	42, 45	zmodcld 13897	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 ≠ 0) ∧ (𝑁 mod (abs‘𝑀)) ≠ 0) → (𝑁 mod (abs‘𝑀)) ∈ ℕ₀)
47		nn0abscl 15299	. . . . . . . . . . . . 13 ⊢ (𝑀 ∈ ℤ → (abs‘𝑀) ∈ ℕ₀)
48	47	3ad2ant2 1131	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (abs‘𝑀) ∈ ℕ₀)
49	48	ad2antrr 724	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 ≠ 0) ∧ (𝑁 mod (abs‘𝑀)) ≠ 0) → (abs‘𝑀) ∈ ℕ₀)
50		ltrmynn0 42400	. . . . . . . . . . 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 12622	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 ≠ 0) ∧ (𝑁 mod (abs‘𝑀)) ≠ 0) → (𝑁 mod (abs‘𝑀)) ∈ ℤ)
54		rmyabs 42410	. . . . . . . . . . 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 582	. . . . . . . . . 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 13880	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 ≠ 0) ∧ (𝑁 mod (abs‘𝑀)) ≠ 0) → (𝑁 mod (abs‘𝑀)) ∈ ℝ)
57		modge0 13884	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℝ ∧ (abs‘𝑀) ∈ ℝ⁺) → 0 ≤ (𝑁 mod (abs‘𝑀)))
58	33, 38, 57	syl2anc 582	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 ≠ 0) ∧ (𝑁 mod (abs‘𝑀)) ≠ 0) → 0 ≤ (𝑁 mod (abs‘𝑀)))
59	56, 58	absidd 15409	. . . . . . . . . . 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 7442	. . . . . . . . . 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 2768	. . . . . . . . 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 42410	. . . . . . . . . 10 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ) → (abs‘(𝐴 Y_rm 𝑀)) = (𝐴 Y_rm (abs‘𝑀)))
63	41, 43, 62	syl2anc 582	. . . . . . . . 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 5184	. . . . . . . 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 7555	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ (𝑁 mod (abs‘𝑀)) ∈ ℤ) → (𝐴 Y_rm (𝑁 mod (abs‘𝑀))) ∈ ℤ)
66	41, 53, 65	syl2anc 582	. . . . . . . . . . 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 15299	. . . . . . . . . . 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 12571	. . . . . . . . 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 724	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 ≠ 0) ∧ (𝑁 mod (abs‘𝑀)) ≠ 0) → (𝐴 Y_rm 𝑀) ∈ ℤ)
71		nn0abscl 15299	. . . . . . . . . . 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 12571	. . . . . . . . 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 11399	. . . . . . . 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 483	. . . . . . . . 9 ⊢ ((((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 ≠ 0) ∧ (𝑁 mod (abs‘𝑀)) ≠ 0) → (𝑁 mod (abs‘𝑀)) ≠ 0)
77		rmyeq0 42405	. . . . . . . . . . 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 582	. . . . . . . . . 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 2982	. . . . . . . . 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 16296	. . . . . . . 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 411	. . . . 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 2956	. . . 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 483	. . . 4 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 ≠ 0) → 𝑀 ≠ 0)
90		dvdsabsmod0 42439	. . . 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 479	. . . . . . 7 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 ≠ 0) → 𝑁 ∈ ℝ)
94		zre 12600	. . . . . . . . 9 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℝ)
95	94	3ad2ant2 1131	. . . . . . . 8 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 𝑀 ∈ ℝ)
96	95	adantr 479	. . . . . . 7 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 ≠ 0) → 𝑀 ∈ ℝ)
97		modabsdifz 42438	. . . . . . 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 12706	. . . . 5 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 ≠ 0) → -((𝑁 − (𝑁 mod (abs‘𝑀))) / 𝑀) ∈ ℤ)
100		jm2.19lem4 42444	. . . . 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 11280	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 𝑁 ∈ ℂ)
103	102	adantr 479	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 ≠ 0) → 𝑁 ∈ ℂ)
104	35	adantr 479	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 ≠ 0) → 𝑀 ∈ ℂ)
105	104, 89	absrpcld 15435	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 ≠ 0) → (abs‘𝑀) ∈ ℝ⁺)
106	93, 105	modcld 13880	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 ≠ 0) → (𝑁 mod (abs‘𝑀)) ∈ ℝ)
107	106	recnd 11280	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 ≠ 0) → (𝑁 mod (abs‘𝑀)) ∈ ℂ)
108	103, 107	subcld 11609	. . . . . . . . . 10 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 ≠ 0) → (𝑁 − (𝑁 mod (abs‘𝑀))) ∈ ℂ)
109	108, 104, 89	divcld 12028	. . . . . . . . 9 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 ≠ 0) → ((𝑁 − (𝑁 mod (abs‘𝑀))) / 𝑀) ∈ ℂ)
110	109, 104	mulneg1d 11705	. . . . . . . 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 7442	. . . . . . 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 11272	. . . . . . . 8 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 ≠ 0) → (((𝑁 − (𝑁 mod (abs‘𝑀))) / 𝑀) · 𝑀) ∈ ℂ)
113	103, 112	negsubd 11615	. . . . . . 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 12029	. . . . . . . . 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 7442	. . . . . . . 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 11614	. . . . . . . 8 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 ≠ 0) → (𝑁 − (𝑁 − (𝑁 mod (abs‘𝑀)))) = (𝑁 mod (abs‘𝑀)))
117	115, 116	eqtrd 2768	. . . . . . 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 2773	. . . . . 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 7442	. . . . 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 5164	. . . 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 281	. . 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 310	. 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 3026	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 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ≠ wne 2937 class class class wbr 5152 ‘cfv 6553 (class class class)co 7426 ℂcc 11144 ℝcr 11145 0cc0 11146 + caddc 11149 · cmul 11151 < clt 11286 ≤ cle 11287 − cmin 11482 -cneg 11483 / cdiv 11909 ℕcn 12250 2c2 12305 ℕ₀cn0 12510 ℤcz 12596 ℤ_≥cuz 12860 ℝ⁺crp 13014 mod cmo 13874 abscabs 15221 ∥ cdvds 16238 Y_rm crmy 42352

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 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-inf2 9672 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 ax-pre-sup 11224 ax-addf 11225

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-tp 4637 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-iin 5003 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-isom 6562 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-of 7691 df-om 7877 df-1st 7999 df-2nd 8000 df-supp 8172 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-2o 8494 df-oadd 8497 df-omul 8498 df-er 8731 df-map 8853 df-pm 8854 df-ixp 8923 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-fsupp 9394 df-fi 9442 df-sup 9473 df-inf 9474 df-oi 9541 df-card 9970 df-acn 9973 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-div 11910 df-nn 12251 df-2 12313 df-3 12314 df-4 12315 df-5 12316 df-6 12317 df-7 12318 df-8 12319 df-9 12320 df-n0 12511 df-xnn0 12583 df-z 12597 df-dec 12716 df-uz 12861 df-q 12971 df-rp 13015 df-xneg 13132 df-xadd 13133 df-xmul 13134 df-ioo 13368 df-ioc 13369 df-ico 13370 df-icc 13371 df-fz 13525 df-fzo 13668 df-fl 13797 df-mod 13875 df-seq 14007 df-exp 14067 df-fac 14273 df-bc 14302 df-hash 14330 df-shft 15054 df-cj 15086 df-re 15087 df-im 15088 df-sqrt 15222 df-abs 15223 df-limsup 15455 df-clim 15472 df-rlim 15473 df-sum 15673 df-ef 16051 df-sin 16053 df-cos 16054 df-pi 16056 df-dvds 16239 df-gcd 16477 df-numer 16714 df-denom 16715 df-struct 17123 df-sets 17140 df-slot 17158 df-ndx 17170 df-base 17188 df-ress 17217 df-plusg 17253 df-mulr 17254 df-starv 17255 df-sca 17256 df-vsca 17257 df-ip 17258 df-tset 17259 df-ple 17260 df-ds 17262 df-unif 17263 df-hom 17264 df-cco 17265 df-rest 17411 df-topn 17412 df-0g 17430 df-gsum 17431 df-topgen 17432 df-pt 17433 df-prds 17436 df-xrs 17491 df-qtop 17496 df-imas 17497 df-xps 17499 df-mre 17573 df-mrc 17574 df-acs 17576 df-mgm 18607 df-sgrp 18686 df-mnd 18702 df-submnd 18748 df-mulg 19031 df-cntz 19275 df-cmn 19744 df-psmet 21278 df-xmet 21279 df-met 21280 df-bl 21281 df-mopn 21282 df-fbas 21283 df-fg 21284 df-cnfld 21287 df-top 22816 df-topon 22833 df-topsp 22855 df-bases 22869 df-cld 22943 df-ntr 22944 df-cls 22945 df-nei 23022 df-lp 23060 df-perf 23061 df-cn 23151 df-cnp 23152 df-haus 23239 df-tx 23486 df-hmeo 23679 df-fil 23770 df-fm 23862 df-flim 23863 df-flf 23864 df-xms 24246 df-ms 24247 df-tms 24248 df-cncf 24818 df-limc 25815 df-dv 25816 df-log 26510 df-squarenn 42292 df-pell1qr 42293 df-pell14qr 42294 df-pell1234qr 42295 df-pellfund 42296 df-rmx 42353 df-rmy 42354

This theorem is referenced by: jm2.20nn 42449

W3C validator