jm2.24 - Mathbox for Stefan O'Rear

	Mathbox for Stefan O'Rear		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > jm2.24		Structured version Visualization version GIF version

Theorem jm2.24 42261

Description: Lemma 2.24 of [JonesMatijasevic] p. 697 extended to ℤ. Could be eliminated with a more careful proof of jm2.26lem3 42299. (Contributed by Stefan O'Rear, 3-Oct-2014.)

**Assertion**
Ref	Expression
jm2.24	⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) → ((𝐴 Y_rm (𝑁 − 1)) + (𝐴 Y_rm 𝑁)) < (𝐴 X_rm 𝑁))

**Proof of Theorem jm2.24**
Step	Hyp	Ref	Expression
1		simpll 764	. . . . . 6 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ 𝑁 ≤ 0) → 𝐴 ∈ (ℤ_≥‘2))
2		peano2zm 12606	. . . . . . 7 ⊢ (𝑁 ∈ ℤ → (𝑁 − 1) ∈ ℤ)
3	2	ad2antlr 724	. . . . . 6 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ 𝑁 ≤ 0) → (𝑁 − 1) ∈ ℤ)
4		frmy 42212	. . . . . . 7 ⊢ Y_rm :((ℤ_≥‘2) × ℤ)⟶ℤ
5	4	fovcl 7532	. . . . . 6 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ (𝑁 − 1) ∈ ℤ) → (𝐴 Y_rm (𝑁 − 1)) ∈ ℤ)
6	1, 3, 5	syl2anc 583	. . . . 5 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ 𝑁 ≤ 0) → (𝐴 Y_rm (𝑁 − 1)) ∈ ℤ)
7	6	zred 12667	. . . 4 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ 𝑁 ≤ 0) → (𝐴 Y_rm (𝑁 − 1)) ∈ ℝ)
8	4	fovcl 7532	. . . . . 6 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) → (𝐴 Y_rm 𝑁) ∈ ℤ)
9	8	zred 12667	. . . . 5 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) → (𝐴 Y_rm 𝑁) ∈ ℝ)
10	9	adantr 480	. . . 4 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ 𝑁 ≤ 0) → (𝐴 Y_rm 𝑁) ∈ ℝ)
11	7, 10	readdcld 11244	. . 3 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ 𝑁 ≤ 0) → ((𝐴 Y_rm (𝑁 − 1)) + (𝐴 Y_rm 𝑁)) ∈ ℝ)
12		0red 11218	. . 3 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ 𝑁 ≤ 0) → 0 ∈ ℝ)
13		frmx 42211	. . . . . 6 ⊢ X_rm :((ℤ_≥‘2) × ℤ)⟶ℕ₀
14	13	fovcl 7532	. . . . 5 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) → (𝐴 X_rm 𝑁) ∈ ℕ₀)
15	14	adantr 480	. . . 4 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ 𝑁 ≤ 0) → (𝐴 X_rm 𝑁) ∈ ℕ₀)
16	15	nn0red 12534	. . 3 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ 𝑁 ≤ 0) → (𝐴 X_rm 𝑁) ∈ ℝ)
17		znegcl 12598	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℤ → -𝑁 ∈ ℤ)
18	17	ad2antlr 724	. . . . . . . . 9 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ 𝑁 ≤ 0) → -𝑁 ∈ ℤ)
19	18	peano2zd 12670	. . . . . . . 8 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ 𝑁 ≤ 0) → (-𝑁 + 1) ∈ ℤ)
20	4	fovcl 7532	. . . . . . . 8 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ (-𝑁 + 1) ∈ ℤ) → (𝐴 Y_rm (-𝑁 + 1)) ∈ ℤ)
21	1, 19, 20	syl2anc 583	. . . . . . 7 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ 𝑁 ≤ 0) → (𝐴 Y_rm (-𝑁 + 1)) ∈ ℤ)
22	21	zred 12667	. . . . . 6 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ 𝑁 ≤ 0) → (𝐴 Y_rm (-𝑁 + 1)) ∈ ℝ)
23	4	fovcl 7532	. . . . . . . 8 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ -𝑁 ∈ ℤ) → (𝐴 Y_rm -𝑁) ∈ ℤ)
24	1, 18, 23	syl2anc 583	. . . . . . 7 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ 𝑁 ≤ 0) → (𝐴 Y_rm -𝑁) ∈ ℤ)
25	24	zred 12667	. . . . . 6 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ 𝑁 ≤ 0) → (𝐴 Y_rm -𝑁) ∈ ℝ)
26		rmy0 42227	. . . . . . . 8 ⊢ (𝐴 ∈ (ℤ_≥‘2) → (𝐴 Y_rm 0) = 0)
27	26	ad2antrr 723	. . . . . . 7 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ 𝑁 ≤ 0) → (𝐴 Y_rm 0) = 0)
28		simpr 484	. . . . . . . . . 10 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ 𝑁 ≤ 0) → 𝑁 ≤ 0)
29		zre 12563	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ)
30	29	ad2antlr 724	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ 𝑁 ≤ 0) → 𝑁 ∈ ℝ)
31	30	le0neg1d 11786	. . . . . . . . . 10 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ 𝑁 ≤ 0) → (𝑁 ≤ 0 ↔ 0 ≤ -𝑁))
32	28, 31	mpbid 231	. . . . . . . . 9 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ 𝑁 ≤ 0) → 0 ≤ -𝑁)
33		0zd 12571	. . . . . . . . . 10 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ 𝑁 ≤ 0) → 0 ∈ ℤ)
34		zleltp1 12614	. . . . . . . . . 10 ⊢ ((0 ∈ ℤ ∧ -𝑁 ∈ ℤ) → (0 ≤ -𝑁 ↔ 0 < (-𝑁 + 1)))
35	33, 18, 34	syl2anc 583	. . . . . . . . 9 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ 𝑁 ≤ 0) → (0 ≤ -𝑁 ↔ 0 < (-𝑁 + 1)))
36	32, 35	mpbid 231	. . . . . . . 8 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ 𝑁 ≤ 0) → 0 < (-𝑁 + 1))
37		ltrmy 42250	. . . . . . . . 9 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 0 ∈ ℤ ∧ (-𝑁 + 1) ∈ ℤ) → (0 < (-𝑁 + 1) ↔ (𝐴 Y_rm 0) < (𝐴 Y_rm (-𝑁 + 1))))
38	1, 33, 19, 37	syl3anc 1368	. . . . . . . 8 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ 𝑁 ≤ 0) → (0 < (-𝑁 + 1) ↔ (𝐴 Y_rm 0) < (𝐴 Y_rm (-𝑁 + 1))))
39	36, 38	mpbid 231	. . . . . . 7 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ 𝑁 ≤ 0) → (𝐴 Y_rm 0) < (𝐴 Y_rm (-𝑁 + 1)))
40	27, 39	eqbrtrrd 5165	. . . . . 6 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ 𝑁 ≤ 0) → 0 < (𝐴 Y_rm (-𝑁 + 1)))
41		lermy 42253	. . . . . . . . 9 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 0 ∈ ℤ ∧ -𝑁 ∈ ℤ) → (0 ≤ -𝑁 ↔ (𝐴 Y_rm 0) ≤ (𝐴 Y_rm -𝑁)))
42	1, 33, 18, 41	syl3anc 1368	. . . . . . . 8 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ 𝑁 ≤ 0) → (0 ≤ -𝑁 ↔ (𝐴 Y_rm 0) ≤ (𝐴 Y_rm -𝑁)))
43	32, 42	mpbid 231	. . . . . . 7 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ 𝑁 ≤ 0) → (𝐴 Y_rm 0) ≤ (𝐴 Y_rm -𝑁))
44	27, 43	eqbrtrrd 5165	. . . . . 6 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ 𝑁 ≤ 0) → 0 ≤ (𝐴 Y_rm -𝑁))
45	22, 25, 40, 44	addgtge0d 11789	. . . . 5 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ 𝑁 ≤ 0) → 0 < ((𝐴 Y_rm (-𝑁 + 1)) + (𝐴 Y_rm -𝑁)))
46	7	recnd 11243	. . . . . . 7 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ 𝑁 ≤ 0) → (𝐴 Y_rm (𝑁 − 1)) ∈ ℂ)
47	10	recnd 11243	. . . . . . 7 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ 𝑁 ≤ 0) → (𝐴 Y_rm 𝑁) ∈ ℂ)
48	46, 47	negdid 11585	. . . . . 6 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ 𝑁 ≤ 0) → -((𝐴 Y_rm (𝑁 − 1)) + (𝐴 Y_rm 𝑁)) = (-(𝐴 Y_rm (𝑁 − 1)) + -(𝐴 Y_rm 𝑁)))
49		rmyneg 42226	. . . . . . . 8 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ (𝑁 − 1) ∈ ℤ) → (𝐴 Y_rm -(𝑁 − 1)) = -(𝐴 Y_rm (𝑁 − 1)))
50	1, 3, 49	syl2anc 583	. . . . . . 7 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ 𝑁 ≤ 0) → (𝐴 Y_rm -(𝑁 − 1)) = -(𝐴 Y_rm (𝑁 − 1)))
51		rmyneg 42226	. . . . . . . 8 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) → (𝐴 Y_rm -𝑁) = -(𝐴 Y_rm 𝑁))
52	51	adantr 480	. . . . . . 7 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ 𝑁 ≤ 0) → (𝐴 Y_rm -𝑁) = -(𝐴 Y_rm 𝑁))
53	50, 52	oveq12d 7422	. . . . . 6 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ 𝑁 ≤ 0) → ((𝐴 Y_rm -(𝑁 − 1)) + (𝐴 Y_rm -𝑁)) = (-(𝐴 Y_rm (𝑁 − 1)) + -(𝐴 Y_rm 𝑁)))
54		zcn 12564	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ)
55	54	ad2antlr 724	. . . . . . . . 9 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ 𝑁 ≤ 0) → 𝑁 ∈ ℂ)
56		ax-1cn 11167	. . . . . . . . 9 ⊢ 1 ∈ ℂ
57		negsubdi 11517	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → -(𝑁 − 1) = (-𝑁 + 1))
58	55, 56, 57	sylancl 585	. . . . . . . 8 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ 𝑁 ≤ 0) → -(𝑁 − 1) = (-𝑁 + 1))
59	58	oveq2d 7420	. . . . . . 7 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ 𝑁 ≤ 0) → (𝐴 Y_rm -(𝑁 − 1)) = (𝐴 Y_rm (-𝑁 + 1)))
60	59	oveq1d 7419	. . . . . 6 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ 𝑁 ≤ 0) → ((𝐴 Y_rm -(𝑁 − 1)) + (𝐴 Y_rm -𝑁)) = ((𝐴 Y_rm (-𝑁 + 1)) + (𝐴 Y_rm -𝑁)))
61	48, 53, 60	3eqtr2d 2772	. . . . 5 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ 𝑁 ≤ 0) → -((𝐴 Y_rm (𝑁 − 1)) + (𝐴 Y_rm 𝑁)) = ((𝐴 Y_rm (-𝑁 + 1)) + (𝐴 Y_rm -𝑁)))
62	45, 61	breqtrrd 5169	. . . 4 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ 𝑁 ≤ 0) → 0 < -((𝐴 Y_rm (𝑁 − 1)) + (𝐴 Y_rm 𝑁)))
63	11	lt0neg1d 11784	. . . 4 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ 𝑁 ≤ 0) → (((𝐴 Y_rm (𝑁 − 1)) + (𝐴 Y_rm 𝑁)) < 0 ↔ 0 < -((𝐴 Y_rm (𝑁 − 1)) + (𝐴 Y_rm 𝑁))))
64	62, 63	mpbird 257	. . 3 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ 𝑁 ≤ 0) → ((𝐴 Y_rm (𝑁 − 1)) + (𝐴 Y_rm 𝑁)) < 0)
65	15	nn0ge0d 12536	. . 3 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ 𝑁 ≤ 0) → 0 ≤ (𝐴 X_rm 𝑁))
66	11, 12, 16, 64, 65	ltletrd 11375	. 2 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ 𝑁 ≤ 0) → ((𝐴 Y_rm (𝑁 − 1)) + (𝐴 Y_rm 𝑁)) < (𝐴 X_rm 𝑁))
67		simpll 764	. . 3 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ 0 < 𝑁) → 𝐴 ∈ (ℤ_≥‘2))
68		elnnz 12569	. . . . 5 ⊢ (𝑁 ∈ ℕ ↔ (𝑁 ∈ ℤ ∧ 0 < 𝑁))
69	68	biimpri 227	. . . 4 ⊢ ((𝑁 ∈ ℤ ∧ 0 < 𝑁) → 𝑁 ∈ ℕ)
70	69	adantll 711	. . 3 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ 0 < 𝑁) → 𝑁 ∈ ℕ)
71		jm2.24nn 42257	. . 3 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℕ) → ((𝐴 Y_rm (𝑁 − 1)) + (𝐴 Y_rm 𝑁)) < (𝐴 X_rm 𝑁))
72	67, 70, 71	syl2anc 583	. 2 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ 0 < 𝑁) → ((𝐴 Y_rm (𝑁 − 1)) + (𝐴 Y_rm 𝑁)) < (𝐴 X_rm 𝑁))
73	29	adantl 481	. . 3 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) → 𝑁 ∈ ℝ)
74		0re 11217	. . 3 ⊢ 0 ∈ ℝ
75		lelttric 11322	. . 3 ⊢ ((𝑁 ∈ ℝ ∧ 0 ∈ ℝ) → (𝑁 ≤ 0 ∨ 0 < 𝑁))
76	73, 74, 75	sylancl 585	. 2 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) → (𝑁 ≤ 0 ∨ 0 < 𝑁))
77	66, 72, 76	mpjaodan 955	1 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) → ((𝐴 Y_rm (𝑁 − 1)) + (𝐴 Y_rm 𝑁)) < (𝐴 X_rm 𝑁))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∨ wo 844 = wceq 1533 ∈ wcel 2098 class class class wbr 5141 ‘cfv 6536 (class class class)co 7404 ℂcc 11107 ℝcr 11108 0cc0 11109 1c1 11110 + caddc 11112 < clt 11249 ≤ cle 11250 − cmin 11445 -cneg 11446 ℕcn 12213 2c2 12268 ℕ₀cn0 12473 ℤcz 12559 ℤ_≥cuz 12823 X_rm crmx 42197 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: jm2.26lem3 42299

W3C validator