jm2.19 - Mathbox for Stefan O'Rear

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Theorem jm2.19 41717

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 41677	. . . . . 6 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) → (𝑁 = 0 ↔ (𝐴 Y_rm 𝑁) = 0))
2	1	3adant2 1131	. . . . 5 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 = 0 ↔ (𝐴 Y_rm 𝑁) = 0))
3		0dvds 16216	. . . . . 6 ⊢ (𝑁 ∈ ℤ → (0 ∥ 𝑁 ↔ 𝑁 = 0))
4	3	3ad2ant3 1135	. . . . 5 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (0 ∥ 𝑁 ↔ 𝑁 = 0))
5		frmy 41638	. . . . . . . 8 ⊢ Y_rm :((ℤ_≥‘2) × ℤ)⟶ℤ
6	5	fovcl 7533	. . . . . . 7 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) → (𝐴 Y_rm 𝑁) ∈ ℤ)
7	6	3adant2 1131	. . . . . 6 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐴 Y_rm 𝑁) ∈ ℤ)
8		0dvds 16216	. . . . . 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 481	. . 3 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 = 0) → (0 ∥ 𝑁 ↔ 0 ∥ (𝐴 Y_rm 𝑁)))
12		simpr 485	. . . 4 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 = 0) → 𝑀 = 0)
13	12	breq1d 5157	. . 3 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 = 0) → (𝑀 ∥ 𝑁 ↔ 0 ∥ 𝑁))
14	12	oveq2d 7421	. . . . 5 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 = 0) → (𝐴 Y_rm 𝑀) = (𝐴 Y_rm 0))
15		simpl1 1191	. . . . . 6 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 = 0) → 𝐴 ∈ (ℤ_≥‘2))
16		rmy0 41653	. . . . . 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 2772	. . . 4 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 = 0) → (𝐴 Y_rm 𝑀) = 0)
19	18	breq1d 5157	. . 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 7533	. . . . . . . . 9 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ) → (𝐴 Y_rm 𝑀) ∈ ℤ)
22	21	3adant3 1132	. . . . . . . 8 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐴 Y_rm 𝑀) ∈ ℤ)
23		dvds0 16211	. . . . . . . 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 1133	. . . . . . 7 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐴 Y_rm 0) = 0)
26	24, 25	breqtrrd 5175	. . . . . 6 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐴 Y_rm 𝑀) ∥ (𝐴 Y_rm 0))
27		oveq2 7413	. . . . . . 7 ⊢ ((𝑁 mod (abs‘𝑀)) = 0 → (𝐴 Y_rm (𝑁 mod (abs‘𝑀))) = (𝐴 Y_rm 0))
28	27	breq2d 5159	. . . . . 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 481	. . . 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 12558	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ)
32	31	3ad2ant3 1135	. . . . . . . . . . . 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 12559	. . . . . . . . . . . . . 14 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℂ)
35	34	3ad2ant2 1134	. . . . . . . . . . . . 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 15391	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 ≠ 0) ∧ (𝑁 mod (abs‘𝑀)) ≠ 0) → (abs‘𝑀) ∈ ℝ⁺)
39		modlt 13841	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℝ ∧ (abs‘𝑀) ∈ ℝ⁺) → (𝑁 mod (abs‘𝑀)) < (abs‘𝑀))
40	33, 38, 39	syl2anc 584	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 ≠ 0) ∧ (𝑁 mod (abs‘𝑀)) ≠ 0) → (𝑁 mod (abs‘𝑀)) < (abs‘𝑀))
41		simpll1 1212	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 ≠ 0) ∧ (𝑁 mod (abs‘𝑀)) ≠ 0) → 𝐴 ∈ (ℤ_≥‘2))
42		simpll3 1214	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 ≠ 0) ∧ (𝑁 mod (abs‘𝑀)) ≠ 0) → 𝑁 ∈ ℤ)
43		simpll2 1213	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 ≠ 0) ∧ (𝑁 mod (abs‘𝑀)) ≠ 0) → 𝑀 ∈ ℤ)
44		nnabscl 15268	. . . . . . . . . . . . 13 ⊢ ((𝑀 ∈ ℤ ∧ 𝑀 ≠ 0) → (abs‘𝑀) ∈ ℕ)
45	43, 37, 44	syl2anc 584	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 ≠ 0) ∧ (𝑁 mod (abs‘𝑀)) ≠ 0) → (abs‘𝑀) ∈ ℕ)
46	42, 45	zmodcld 13853	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 ≠ 0) ∧ (𝑁 mod (abs‘𝑀)) ≠ 0) → (𝑁 mod (abs‘𝑀)) ∈ ℕ₀)
47		nn0abscl 15255	. . . . . . . . . . . . 13 ⊢ (𝑀 ∈ ℤ → (abs‘𝑀) ∈ ℕ₀)
48	47	3ad2ant2 1134	. . . . . . . . . . . 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 41672	. . . . . . . . . . 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 1371	. . . . . . . . . 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 12580	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 ≠ 0) ∧ (𝑁 mod (abs‘𝑀)) ≠ 0) → (𝑁 mod (abs‘𝑀)) ∈ ℤ)
54		rmyabs 41682	. . . . . . . . . . 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 584	. . . . . . . . . 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 13836	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 ≠ 0) ∧ (𝑁 mod (abs‘𝑀)) ≠ 0) → (𝑁 mod (abs‘𝑀)) ∈ ℝ)
57		modge0 13840	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℝ ∧ (abs‘𝑀) ∈ ℝ⁺) → 0 ≤ (𝑁 mod (abs‘𝑀)))
58	33, 38, 57	syl2anc 584	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 ≠ 0) ∧ (𝑁 mod (abs‘𝑀)) ≠ 0) → 0 ≤ (𝑁 mod (abs‘𝑀)))
59	56, 58	absidd 15365	. . . . . . . . . . 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 7421	. . . . . . . . . 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 2772	. . . . . . . . 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 41682	. . . . . . . . . 10 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ) → (abs‘(𝐴 Y_rm 𝑀)) = (𝐴 Y_rm (abs‘𝑀)))
63	41, 43, 62	syl2anc 584	. . . . . . . . 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 5179	. . . . . . . 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 7533	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ (𝑁 mod (abs‘𝑀)) ∈ ℤ) → (𝐴 Y_rm (𝑁 mod (abs‘𝑀))) ∈ ℤ)
66	41, 53, 65	syl2anc 584	. . . . . . . . . . 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 15255	. . . . . . . . . . 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 12529	. . . . . . . . 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 15255	. . . . . . . . . . 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 12529	. . . . . . . . 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 11357	. . . . . . . 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 485	. . . . . . . . 9 ⊢ ((((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 ≠ 0) ∧ (𝑁 mod (abs‘𝑀)) ≠ 0) → (𝑁 mod (abs‘𝑀)) ≠ 0)
77		rmyeq0 41677	. . . . . . . . . . 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 584	. . . . . . . . . 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 2985	. . . . . . . . 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 16251	. . . . . . . 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 1371	. . . . . . 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 413	. . . . 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 2959	. . . 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 1192	. . . 4 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 ≠ 0) → 𝑀 ∈ ℤ)
88		simpl3 1193	. . . 4 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 ≠ 0) → 𝑁 ∈ ℤ)
89		simpr 485	. . . 4 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 ≠ 0) → 𝑀 ≠ 0)
90		dvdsabsmod0 41711	. . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ≠ 0) → (𝑀 ∥ 𝑁 ↔ (𝑁 mod (abs‘𝑀)) = 0))
91	87, 88, 89, 90	syl3anc 1371	. . 3 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 ≠ 0) → (𝑀 ∥ 𝑁 ↔ (𝑁 mod (abs‘𝑀)) = 0))
92		simpl1 1191	. . . . 5 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 ≠ 0) → 𝐴 ∈ (ℤ_≥‘2))
93	32	adantr 481	. . . . . . 7 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 ≠ 0) → 𝑁 ∈ ℝ)
94		zre 12558	. . . . . . . . 9 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℝ)
95	94	3ad2ant2 1134	. . . . . . . 8 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 𝑀 ∈ ℝ)
96	95	adantr 481	. . . . . . 7 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 ≠ 0) → 𝑀 ∈ ℝ)
97		modabsdifz 41710	. . . . . . 7 ⊢ ((𝑁 ∈ ℝ ∧ 𝑀 ∈ ℝ ∧ 𝑀 ≠ 0) → ((𝑁 − (𝑁 mod (abs‘𝑀))) / 𝑀) ∈ ℤ)
98	93, 96, 89, 97	syl3anc 1371	. . . . . 6 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 ≠ 0) → ((𝑁 − (𝑁 mod (abs‘𝑀))) / 𝑀) ∈ ℤ)
99	98	znegcld 12664	. . . . 5 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 ≠ 0) → -((𝑁 − (𝑁 mod (abs‘𝑀))) / 𝑀) ∈ ℤ)
100		jm2.19lem4 41716	. . . . 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 1375	. . . 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 11238	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 𝑁 ∈ ℂ)
103	102	adantr 481	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 ≠ 0) → 𝑁 ∈ ℂ)
104	35	adantr 481	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 ≠ 0) → 𝑀 ∈ ℂ)
105	104, 89	absrpcld 15391	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 ≠ 0) → (abs‘𝑀) ∈ ℝ⁺)
106	93, 105	modcld 13836	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 ≠ 0) → (𝑁 mod (abs‘𝑀)) ∈ ℝ)
107	106	recnd 11238	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 ≠ 0) → (𝑁 mod (abs‘𝑀)) ∈ ℂ)
108	103, 107	subcld 11567	. . . . . . . . . 10 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 ≠ 0) → (𝑁 − (𝑁 mod (abs‘𝑀))) ∈ ℂ)
109	108, 104, 89	divcld 11986	. . . . . . . . 9 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 ≠ 0) → ((𝑁 − (𝑁 mod (abs‘𝑀))) / 𝑀) ∈ ℂ)
110	109, 104	mulneg1d 11663	. . . . . . . 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 7421	. . . . . . 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 11230	. . . . . . . 8 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 ≠ 0) → (((𝑁 − (𝑁 mod (abs‘𝑀))) / 𝑀) · 𝑀) ∈ ℂ)
113	103, 112	negsubd 11573	. . . . . . 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 11987	. . . . . . . . 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 7421	. . . . . . . 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 11572	. . . . . . . 8 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 ≠ 0) → (𝑁 − (𝑁 − (𝑁 mod (abs‘𝑀)))) = (𝑁 mod (abs‘𝑀)))
117	115, 116	eqtrd 2772	. . . . . . 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 2777	. . . . . 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 7421	. . . . 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 5159	. . . 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 3029	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 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ≠ wne 2940 class class class wbr 5147 ‘cfv 6540 (class class class)co 7405 ℂcc 11104 ℝcr 11105 0cc0 11106 + caddc 11109 · cmul 11111 < clt 11244 ≤ cle 11245 − cmin 11440 -cneg 11441 / cdiv 11867 ℕcn 12208 2c2 12263 ℕ₀cn0 12468 ℤcz 12554 ℤ_≥cuz 12818 ℝ⁺crp 12970 mod cmo 13830 abscabs 15177 ∥ cdvds 16193 Y_rm crmy 41624

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-inf2 9632 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 ax-addf 11185 ax-mulf 11186

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-iin 4999 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7666 df-om 7852 df-1st 7971 df-2nd 7972 df-supp 8143 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-2o 8463 df-oadd 8466 df-omul 8467 df-er 8699 df-map 8818 df-pm 8819 df-ixp 8888 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-fsupp 9358 df-fi 9402 df-sup 9433 df-inf 9434 df-oi 9501 df-card 9930 df-acn 9933 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-xnn0 12541 df-z 12555 df-dec 12674 df-uz 12819 df-q 12929 df-rp 12971 df-xneg 13088 df-xadd 13089 df-xmul 13090 df-ioo 13324 df-ioc 13325 df-ico 13326 df-icc 13327 df-fz 13481 df-fzo 13624 df-fl 13753 df-mod 13831 df-seq 13963 df-exp 14024 df-fac 14230 df-bc 14259 df-hash 14287 df-shft 15010 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-limsup 15411 df-clim 15428 df-rlim 15429 df-sum 15629 df-ef 16007 df-sin 16009 df-cos 16010 df-pi 16012 df-dvds 16194 df-gcd 16432 df-numer 16667 df-denom 16668 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-plusg 17206 df-mulr 17207 df-starv 17208 df-sca 17209 df-vsca 17210 df-ip 17211 df-tset 17212 df-ple 17213 df-ds 17215 df-unif 17216 df-hom 17217 df-cco 17218 df-rest 17364 df-topn 17365 df-0g 17383 df-gsum 17384 df-topgen 17385 df-pt 17386 df-prds 17389 df-xrs 17444 df-qtop 17449 df-imas 17450 df-xps 17452 df-mre 17526 df-mrc 17527 df-acs 17529 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-submnd 18668 df-mulg 18945 df-cntz 19175 df-cmn 19644 df-psmet 20928 df-xmet 20929 df-met 20930 df-bl 20931 df-mopn 20932 df-fbas 20933 df-fg 20934 df-cnfld 20937 df-top 22387 df-topon 22404 df-topsp 22426 df-bases 22440 df-cld 22514 df-ntr 22515 df-cls 22516 df-nei 22593 df-lp 22631 df-perf 22632 df-cn 22722 df-cnp 22723 df-haus 22810 df-tx 23057 df-hmeo 23250 df-fil 23341 df-fm 23433 df-flim 23434 df-flf 23435 df-xms 23817 df-ms 23818 df-tms 23819 df-cncf 24385 df-limc 25374 df-dv 25375 df-log 26056 df-squarenn 41564 df-pell1qr 41565 df-pell14qr 41566 df-pell1234qr 41567 df-pellfund 41568 df-rmx 41625 df-rmy 41626

This theorem is referenced by: jm2.20nn 41721

W3C validator