jm2.24 - Mathbox for Stefan O'Rear

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Theorem jm2.24 41702

Description: Lemma 2.24 of [JonesMatijasevic] p. 697 extended to ℤ. Could be eliminated with a more careful proof of jm2.26lem3 41740. (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 12605	. . . . . . 7 ⊢ (𝑁 ∈ ℤ → (𝑁 − 1) ∈ ℤ)
3	2	ad2antlr 726	. . . . . 6 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ 𝑁 ≤ 0) → (𝑁 − 1) ∈ ℤ)
4		frmy 41653	. . . . . . 7 ⊢ Y_rm :((ℤ_≥‘2) × ℤ)⟶ℤ
5	4	fovcl 7537	. . . . . 6 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ (𝑁 − 1) ∈ ℤ) → (𝐴 Y_rm (𝑁 − 1)) ∈ ℤ)
6	1, 3, 5	syl2anc 585	. . . . 5 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ 𝑁 ≤ 0) → (𝐴 Y_rm (𝑁 − 1)) ∈ ℤ)
7	6	zred 12666	. . . 4 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ 𝑁 ≤ 0) → (𝐴 Y_rm (𝑁 − 1)) ∈ ℝ)
8	4	fovcl 7537	. . . . . 6 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) → (𝐴 Y_rm 𝑁) ∈ ℤ)
9	8	zred 12666	. . . . 5 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) → (𝐴 Y_rm 𝑁) ∈ ℝ)
10	9	adantr 482	. . . 4 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ 𝑁 ≤ 0) → (𝐴 Y_rm 𝑁) ∈ ℝ)
11	7, 10	readdcld 11243	. . 3 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ 𝑁 ≤ 0) → ((𝐴 Y_rm (𝑁 − 1)) + (𝐴 Y_rm 𝑁)) ∈ ℝ)
12		0red 11217	. . 3 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ 𝑁 ≤ 0) → 0 ∈ ℝ)
13		frmx 41652	. . . . . 6 ⊢ X_rm :((ℤ_≥‘2) × ℤ)⟶ℕ₀
14	13	fovcl 7537	. . . . 5 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) → (𝐴 X_rm 𝑁) ∈ ℕ₀)
15	14	adantr 482	. . . 4 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ 𝑁 ≤ 0) → (𝐴 X_rm 𝑁) ∈ ℕ₀)
16	15	nn0red 12533	. . 3 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ 𝑁 ≤ 0) → (𝐴 X_rm 𝑁) ∈ ℝ)
17		znegcl 12597	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℤ → -𝑁 ∈ ℤ)
18	17	ad2antlr 726	. . . . . . . . 9 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ 𝑁 ≤ 0) → -𝑁 ∈ ℤ)
19	18	peano2zd 12669	. . . . . . . 8 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ 𝑁 ≤ 0) → (-𝑁 + 1) ∈ ℤ)
20	4	fovcl 7537	. . . . . . . 8 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ (-𝑁 + 1) ∈ ℤ) → (𝐴 Y_rm (-𝑁 + 1)) ∈ ℤ)
21	1, 19, 20	syl2anc 585	. . . . . . 7 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ 𝑁 ≤ 0) → (𝐴 Y_rm (-𝑁 + 1)) ∈ ℤ)
22	21	zred 12666	. . . . . 6 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ 𝑁 ≤ 0) → (𝐴 Y_rm (-𝑁 + 1)) ∈ ℝ)
23	4	fovcl 7537	. . . . . . . 8 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ -𝑁 ∈ ℤ) → (𝐴 Y_rm -𝑁) ∈ ℤ)
24	1, 18, 23	syl2anc 585	. . . . . . 7 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ 𝑁 ≤ 0) → (𝐴 Y_rm -𝑁) ∈ ℤ)
25	24	zred 12666	. . . . . 6 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ 𝑁 ≤ 0) → (𝐴 Y_rm -𝑁) ∈ ℝ)
26		rmy0 41668	. . . . . . . 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 486	. . . . . . . . . 10 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ 𝑁 ≤ 0) → 𝑁 ≤ 0)
29		zre 12562	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ)
30	29	ad2antlr 726	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ 𝑁 ≤ 0) → 𝑁 ∈ ℝ)
31	30	le0neg1d 11785	. . . . . . . . . 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 12570	. . . . . . . . . 10 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ 𝑁 ≤ 0) → 0 ∈ ℤ)
34		zleltp1 12613	. . . . . . . . . 10 ⊢ ((0 ∈ ℤ ∧ -𝑁 ∈ ℤ) → (0 ≤ -𝑁 ↔ 0 < (-𝑁 + 1)))
35	33, 18, 34	syl2anc 585	. . . . . . . . 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 41691	. . . . . . . . 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 1372	. . . . . . . 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 5173	. . . . . 6 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ 𝑁 ≤ 0) → 0 < (𝐴 Y_rm (-𝑁 + 1)))
41		lermy 41694	. . . . . . . . 9 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 0 ∈ ℤ ∧ -𝑁 ∈ ℤ) → (0 ≤ -𝑁 ↔ (𝐴 Y_rm 0) ≤ (𝐴 Y_rm -𝑁)))
42	1, 33, 18, 41	syl3anc 1372	. . . . . . . 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 5173	. . . . . 6 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ 𝑁 ≤ 0) → 0 ≤ (𝐴 Y_rm -𝑁))
45	22, 25, 40, 44	addgtge0d 11788	. . . . 5 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ 𝑁 ≤ 0) → 0 < ((𝐴 Y_rm (-𝑁 + 1)) + (𝐴 Y_rm -𝑁)))
46	7	recnd 11242	. . . . . . 7 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ 𝑁 ≤ 0) → (𝐴 Y_rm (𝑁 − 1)) ∈ ℂ)
47	10	recnd 11242	. . . . . . 7 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ 𝑁 ≤ 0) → (𝐴 Y_rm 𝑁) ∈ ℂ)
48	46, 47	negdid 11584	. . . . . 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 41667	. . . . . . . 8 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ (𝑁 − 1) ∈ ℤ) → (𝐴 Y_rm -(𝑁 − 1)) = -(𝐴 Y_rm (𝑁 − 1)))
50	1, 3, 49	syl2anc 585	. . . . . . 7 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ 𝑁 ≤ 0) → (𝐴 Y_rm -(𝑁 − 1)) = -(𝐴 Y_rm (𝑁 − 1)))
51		rmyneg 41667	. . . . . . . 8 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) → (𝐴 Y_rm -𝑁) = -(𝐴 Y_rm 𝑁))
52	51	adantr 482	. . . . . . 7 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ 𝑁 ≤ 0) → (𝐴 Y_rm -𝑁) = -(𝐴 Y_rm 𝑁))
53	50, 52	oveq12d 7427	. . . . . 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 12563	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ)
55	54	ad2antlr 726	. . . . . . . . 9 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ 𝑁 ≤ 0) → 𝑁 ∈ ℂ)
56		ax-1cn 11168	. . . . . . . . 9 ⊢ 1 ∈ ℂ
57		negsubdi 11516	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → -(𝑁 − 1) = (-𝑁 + 1))
58	55, 56, 57	sylancl 587	. . . . . . . 8 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ 𝑁 ≤ 0) → -(𝑁 − 1) = (-𝑁 + 1))
59	58	oveq2d 7425	. . . . . . 7 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ 𝑁 ≤ 0) → (𝐴 Y_rm -(𝑁 − 1)) = (𝐴 Y_rm (-𝑁 + 1)))
60	59	oveq1d 7424	. . . . . 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 2779	. . . . 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 5177	. . . 4 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ 𝑁 ≤ 0) → 0 < -((𝐴 Y_rm (𝑁 − 1)) + (𝐴 Y_rm 𝑁)))
63	11	lt0neg1d 11783	. . . 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 12535	. . 3 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ 𝑁 ≤ 0) → 0 ≤ (𝐴 X_rm 𝑁))
66	11, 12, 16, 64, 65	ltletrd 11374	. 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 12568	. . . . 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 41698	. . 3 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℕ) → ((𝐴 Y_rm (𝑁 − 1)) + (𝐴 Y_rm 𝑁)) < (𝐴 X_rm 𝑁))
72	67, 70, 71	syl2anc 585	. 2 ⊢ (((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) ∧ 0 < 𝑁) → ((𝐴 Y_rm (𝑁 − 1)) + (𝐴 Y_rm 𝑁)) < (𝐴 X_rm 𝑁))
73	29	adantl 483	. . 3 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) → 𝑁 ∈ ℝ)
74		0re 11216	. . 3 ⊢ 0 ∈ ℝ
75		lelttric 11321	. . 3 ⊢ ((𝑁 ∈ ℝ ∧ 0 ∈ ℝ) → (𝑁 ≤ 0 ∨ 0 < 𝑁))
76	73, 74, 75	sylancl 587	. 2 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) → (𝑁 ≤ 0 ∨ 0 < 𝑁))
77	66, 72, 76	mpjaodan 958	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 397 ∨ wo 846 = wceq 1542 ∈ wcel 2107 class class class wbr 5149 ‘cfv 6544 (class class class)co 7409 ℂcc 11108 ℝcr 11109 0cc0 11110 1c1 11111 + caddc 11113 < clt 11248 ≤ cle 11249 − cmin 11444 -cneg 11445 ℕcn 12212 2c2 12267 ℕ₀cn0 12472 ℤcz 12558 ℤ_≥cuz 12822 X_rm crmx 41638 Y_rm crmy 41639

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-inf2 9636 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 ax-pre-sup 11188 ax-addf 11189 ax-mulf 11190

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-iin 5001 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-of 7670 df-om 7856 df-1st 7975 df-2nd 7976 df-supp 8147 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-2o 8467 df-oadd 8470 df-omul 8471 df-er 8703 df-map 8822 df-pm 8823 df-ixp 8892 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-fsupp 9362 df-fi 9406 df-sup 9437 df-inf 9438 df-oi 9505 df-card 9934 df-acn 9937 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-div 11872 df-nn 12213 df-2 12275 df-3 12276 df-4 12277 df-5 12278 df-6 12279 df-7 12280 df-8 12281 df-9 12282 df-n0 12473 df-xnn0 12545 df-z 12559 df-dec 12678 df-uz 12823 df-q 12933 df-rp 12975 df-xneg 13092 df-xadd 13093 df-xmul 13094 df-ioo 13328 df-ioc 13329 df-ico 13330 df-icc 13331 df-fz 13485 df-fzo 13628 df-fl 13757 df-mod 13835 df-seq 13967 df-exp 14028 df-fac 14234 df-bc 14263 df-hash 14291 df-shft 15014 df-cj 15046 df-re 15047 df-im 15048 df-sqrt 15182 df-abs 15183 df-limsup 15415 df-clim 15432 df-rlim 15433 df-sum 15633 df-ef 16011 df-sin 16013 df-cos 16014 df-pi 16016 df-dvds 16198 df-gcd 16436 df-numer 16671 df-denom 16672 df-struct 17080 df-sets 17097 df-slot 17115 df-ndx 17127 df-base 17145 df-ress 17174 df-plusg 17210 df-mulr 17211 df-starv 17212 df-sca 17213 df-vsca 17214 df-ip 17215 df-tset 17216 df-ple 17217 df-ds 17219 df-unif 17220 df-hom 17221 df-cco 17222 df-rest 17368 df-topn 17369 df-0g 17387 df-gsum 17388 df-topgen 17389 df-pt 17390 df-prds 17393 df-xrs 17448 df-qtop 17453 df-imas 17454 df-xps 17456 df-mre 17530 df-mrc 17531 df-acs 17533 df-mgm 18561 df-sgrp 18610 df-mnd 18626 df-submnd 18672 df-mulg 18951 df-cntz 19181 df-cmn 19650 df-psmet 20936 df-xmet 20937 df-met 20938 df-bl 20939 df-mopn 20940 df-fbas 20941 df-fg 20942 df-cnfld 20945 df-top 22396 df-topon 22413 df-topsp 22435 df-bases 22449 df-cld 22523 df-ntr 22524 df-cls 22525 df-nei 22602 df-lp 22640 df-perf 22641 df-cn 22731 df-cnp 22732 df-haus 22819 df-tx 23066 df-hmeo 23259 df-fil 23350 df-fm 23442 df-flim 23443 df-flf 23444 df-xms 23826 df-ms 23827 df-tms 23828 df-cncf 24394 df-limc 25383 df-dv 25384 df-log 26065 df-squarenn 41579 df-pell1qr 41580 df-pell14qr 41581 df-pell1234qr 41582 df-pellfund 41583 df-rmx 41640 df-rmy 41641

This theorem is referenced by: jm2.26lem3 41740

W3C validator