jm2.24 - Mathbox for Stefan O'Rear

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Theorem jm2.24 42384

Description: Lemma 2.24 of [JonesMatijasevic] p. 697 extended to ℤ. Could be eliminated with a more careful proof of jm2.26lem3 42422. (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 766	. . . . . 6 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ 𝑁 ≤ 0) → 𝐴 ∈ (ℤ_≥‘2))
2		peano2zm 12636	. . . . . . 7 ⊢ (𝑁 ∈ ℤ → (𝑁 − 1) ∈ ℤ)
3	2	ad2antlr 726	. . . . . 6 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ 𝑁 ≤ 0) → (𝑁 − 1) ∈ ℤ)
4		frmy 42335	. . . . . . 7 ⊢ Y_rm :((ℤ_≥‘2) × ℤ)⟶ℤ
5	4	fovcl 7549	. . . . . 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 12697	. . . 4 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ 𝑁 ≤ 0) → (𝐴 Y_rm (𝑁 − 1)) ∈ ℝ)
8	4	fovcl 7549	. . . . . 6 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) → (𝐴 Y_rm 𝑁) ∈ ℤ)
9	8	zred 12697	. . . . 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 11274	. . 3 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ 𝑁 ≤ 0) → ((𝐴 Y_rm (𝑁 − 1)) + (𝐴 Y_rm 𝑁)) ∈ ℝ)
12		0red 11248	. . 3 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ 𝑁 ≤ 0) → 0 ∈ ℝ)
13		frmx 42334	. . . . . 6 ⊢ X_rm :((ℤ_≥‘2) × ℤ)⟶ℕ₀
14	13	fovcl 7549	. . . . 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 12564	. . 3 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ 𝑁 ≤ 0) → (𝐴 X_rm 𝑁) ∈ ℝ)
17		znegcl 12628	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℤ → -𝑁 ∈ ℤ)
18	17	ad2antlr 726	. . . . . . . . 9 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ 𝑁 ≤ 0) → -𝑁 ∈ ℤ)
19	18	peano2zd 12700	. . . . . . . 8 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ 𝑁 ≤ 0) → (-𝑁 + 1) ∈ ℤ)
20	4	fovcl 7549	. . . . . . . 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 12697	. . . . . 6 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ 𝑁 ≤ 0) → (𝐴 Y_rm (-𝑁 + 1)) ∈ ℝ)
23	4	fovcl 7549	. . . . . . . 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 12697	. . . . . 6 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ 𝑁 ≤ 0) → (𝐴 Y_rm -𝑁) ∈ ℝ)
26		rmy0 42350	. . . . . . . 8 ⊢ (𝐴 ∈ (ℤ_≥‘2) → (𝐴 Y_rm 0) = 0)
27	26	ad2antrr 725	. . . . . . 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 12593	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ)
30	29	ad2antlr 726	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ 𝑁 ≤ 0) → 𝑁 ∈ ℝ)
31	30	le0neg1d 11816	. . . . . . . . . 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 12601	. . . . . . . . . 10 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ 𝑁 ≤ 0) → 0 ∈ ℤ)
34		zleltp1 12644	. . . . . . . . . 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 42373	. . . . . . . . 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 1369	. . . . . . . 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 5172	. . . . . 6 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ 𝑁 ≤ 0) → 0 < (𝐴 Y_rm (-𝑁 + 1)))
41		lermy 42376	. . . . . . . . 9 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 0 ∈ ℤ ∧ -𝑁 ∈ ℤ) → (0 ≤ -𝑁 ↔ (𝐴 Y_rm 0) ≤ (𝐴 Y_rm -𝑁)))
42	1, 33, 18, 41	syl3anc 1369	. . . . . . . 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 5172	. . . . . 6 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ 𝑁 ≤ 0) → 0 ≤ (𝐴 Y_rm -𝑁))
45	22, 25, 40, 44	addgtge0d 11819	. . . . 5 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ 𝑁 ≤ 0) → 0 < ((𝐴 Y_rm (-𝑁 + 1)) + (𝐴 Y_rm -𝑁)))
46	7	recnd 11273	. . . . . . 7 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ 𝑁 ≤ 0) → (𝐴 Y_rm (𝑁 − 1)) ∈ ℂ)
47	10	recnd 11273	. . . . . . 7 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ 𝑁 ≤ 0) → (𝐴 Y_rm 𝑁) ∈ ℂ)
48	46, 47	negdid 11615	. . . . . 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 42349	. . . . . . . 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 42349	. . . . . . . 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 7438	. . . . . 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 12594	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ)
55	54	ad2antlr 726	. . . . . . . . 9 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ 𝑁 ≤ 0) → 𝑁 ∈ ℂ)
56		ax-1cn 11197	. . . . . . . . 9 ⊢ 1 ∈ ℂ
57		negsubdi 11547	. . . . . . . . 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 7436	. . . . . . 7 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ 𝑁 ≤ 0) → (𝐴 Y_rm -(𝑁 − 1)) = (𝐴 Y_rm (-𝑁 + 1)))
60	59	oveq1d 7435	. . . . . 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 2774	. . . . 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 5176	. . . 4 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ 𝑁 ≤ 0) → 0 < -((𝐴 Y_rm (𝑁 − 1)) + (𝐴 Y_rm 𝑁)))
63	11	lt0neg1d 11814	. . . 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 12566	. . 3 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ 𝑁 ≤ 0) → 0 ≤ (𝐴 X_rm 𝑁))
66	11, 12, 16, 64, 65	ltletrd 11405	. 2 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ 𝑁 ≤ 0) → ((𝐴 Y_rm (𝑁 − 1)) + (𝐴 Y_rm 𝑁)) < (𝐴 X_rm 𝑁))
67		simpll 766	. . 3 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ 0 < 𝑁) → 𝐴 ∈ (ℤ_≥‘2))
68		elnnz 12599	. . . . 5 ⊢ (𝑁 ∈ ℕ ↔ (𝑁 ∈ ℤ ∧ 0 < 𝑁))
69	68	biimpri 227	. . . 4 ⊢ ((𝑁 ∈ ℤ ∧ 0 < 𝑁) → 𝑁 ∈ ℕ)
70	69	adantll 713	. . 3 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ 0 < 𝑁) → 𝑁 ∈ ℕ)
71		jm2.24nn 42380	. . 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 11247	. . 3 ⊢ 0 ∈ ℝ
75		lelttric 11352	. . 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 957	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 846 = wceq 1534 ∈ wcel 2099 class class class wbr 5148 ‘cfv 6548 (class class class)co 7420 ℂcc 11137 ℝcr 11138 0cc0 11139 1c1 11140 + caddc 11142 < clt 11279 ≤ cle 11280 − cmin 11475 -cneg 11476 ℕcn 12243 2c2 12298 ℕ₀cn0 12503 ℤcz 12589 ℤ_≥cuz 12853 X_rm crmx 42320 Y_rm crmy 42321

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-inf2 9665 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 ax-pre-sup 11217 ax-addf 11218

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 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 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-isom 6557 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-of 7685 df-om 7871 df-1st 7993 df-2nd 7994 df-supp 8166 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-2o 8488 df-oadd 8491 df-omul 8492 df-er 8725 df-map 8847 df-pm 8848 df-ixp 8917 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fsupp 9387 df-fi 9435 df-sup 9466 df-inf 9467 df-oi 9534 df-card 9963 df-acn 9966 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-div 11903 df-nn 12244 df-2 12306 df-3 12307 df-4 12308 df-5 12309 df-6 12310 df-7 12311 df-8 12312 df-9 12313 df-n0 12504 df-xnn0 12576 df-z 12590 df-dec 12709 df-uz 12854 df-q 12964 df-rp 13008 df-xneg 13125 df-xadd 13126 df-xmul 13127 df-ioo 13361 df-ioc 13362 df-ico 13363 df-icc 13364 df-fz 13518 df-fzo 13661 df-fl 13790 df-mod 13868 df-seq 14000 df-exp 14060 df-fac 14266 df-bc 14295 df-hash 14323 df-shft 15047 df-cj 15079 df-re 15080 df-im 15081 df-sqrt 15215 df-abs 15216 df-limsup 15448 df-clim 15465 df-rlim 15466 df-sum 15666 df-ef 16044 df-sin 16046 df-cos 16047 df-pi 16049 df-dvds 16232 df-gcd 16470 df-numer 16707 df-denom 16708 df-struct 17116 df-sets 17133 df-slot 17151 df-ndx 17163 df-base 17181 df-ress 17210 df-plusg 17246 df-mulr 17247 df-starv 17248 df-sca 17249 df-vsca 17250 df-ip 17251 df-tset 17252 df-ple 17253 df-ds 17255 df-unif 17256 df-hom 17257 df-cco 17258 df-rest 17404 df-topn 17405 df-0g 17423 df-gsum 17424 df-topgen 17425 df-pt 17426 df-prds 17429 df-xrs 17484 df-qtop 17489 df-imas 17490 df-xps 17492 df-mre 17566 df-mrc 17567 df-acs 17569 df-mgm 18600 df-sgrp 18679 df-mnd 18695 df-submnd 18741 df-mulg 19024 df-cntz 19268 df-cmn 19737 df-psmet 21271 df-xmet 21272 df-met 21273 df-bl 21274 df-mopn 21275 df-fbas 21276 df-fg 21277 df-cnfld 21280 df-top 22809 df-topon 22826 df-topsp 22848 df-bases 22862 df-cld 22936 df-ntr 22937 df-cls 22938 df-nei 23015 df-lp 23053 df-perf 23054 df-cn 23144 df-cnp 23145 df-haus 23232 df-tx 23479 df-hmeo 23672 df-fil 23763 df-fm 23855 df-flim 23856 df-flf 23857 df-xms 24239 df-ms 24240 df-tms 24241 df-cncf 24811 df-limc 25808 df-dv 25809 df-log 26503 df-squarenn 42261 df-pell1qr 42262 df-pell14qr 42263 df-pell1234qr 42264 df-pellfund 42265 df-rmx 42322 df-rmy 42323

This theorem is referenced by: jm2.26lem3 42422

W3C validator