ismrer1 - Mathbox for Jeff Madsen

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Theorem ismrer1 37010

Description: An isometry between ℝ and ℝ↑1. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 22-Sep-2015.)

**Hypotheses**
Ref	Expression
ismrer1.1	⊢ 𝑅 = ((abs ∘ − ) ↾ (ℝ × ℝ))
ismrer1.2	⊢ 𝐹 = (𝑥 ∈ ℝ ↦ ({𝐴} × {𝑥}))

**Assertion**
Ref	Expression
ismrer1	⊢ (𝐴 ∈ 𝑉 → 𝐹 ∈ (𝑅 Ismty (ℝⁿ‘{𝐴})))

Distinct variable group: 𝑥,𝐴 Allowed substitution hints: 𝑅(𝑥) 𝐹(𝑥) 𝑉(𝑥)

Proof of Theorem ismrer1 Dummy variables 𝑘 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		sneq 4638	. . . . . . . 8 ⊢ (𝑦 = 𝐴 → {𝑦} = {𝐴})
2	1	xpeq1d 5705	. . . . . . 7 ⊢ (𝑦 = 𝐴 → ({𝑦} × {𝑥}) = ({𝐴} × {𝑥}))
3	2	mpteq2dv 5250	. . . . . 6 ⊢ (𝑦 = 𝐴 → (𝑥 ∈ ℝ ↦ ({𝑦} × {𝑥})) = (𝑥 ∈ ℝ ↦ ({𝐴} × {𝑥})))
4		ismrer1.2	. . . . . 6 ⊢ 𝐹 = (𝑥 ∈ ℝ ↦ ({𝐴} × {𝑥}))
5	3, 4	eqtr4di 2789	. . . . 5 ⊢ (𝑦 = 𝐴 → (𝑥 ∈ ℝ ↦ ({𝑦} × {𝑥})) = 𝐹)
6	5	f1oeq1d 6828	. . . 4 ⊢ (𝑦 = 𝐴 → ((𝑥 ∈ ℝ ↦ ({𝑦} × {𝑥})):ℝ–1-1-onto→(ℝ ↑_m {𝑦}) ↔ 𝐹:ℝ–1-1-onto→(ℝ ↑_m {𝑦})))
7	1	oveq2d 7428	. . . . 5 ⊢ (𝑦 = 𝐴 → (ℝ ↑_m {𝑦}) = (ℝ ↑_m {𝐴}))
8		f1oeq3 6823	. . . . 5 ⊢ ((ℝ ↑_m {𝑦}) = (ℝ ↑_m {𝐴}) → (𝐹:ℝ–1-1-onto→(ℝ ↑_m {𝑦}) ↔ 𝐹:ℝ–1-1-onto→(ℝ ↑_m {𝐴})))
9	7, 8	syl 17	. . . 4 ⊢ (𝑦 = 𝐴 → (𝐹:ℝ–1-1-onto→(ℝ ↑_m {𝑦}) ↔ 𝐹:ℝ–1-1-onto→(ℝ ↑_m {𝐴})))
10	6, 9	bitrd 279	. . 3 ⊢ (𝑦 = 𝐴 → ((𝑥 ∈ ℝ ↦ ({𝑦} × {𝑥})):ℝ–1-1-onto→(ℝ ↑_m {𝑦}) ↔ 𝐹:ℝ–1-1-onto→(ℝ ↑_m {𝐴})))
11		eqid 2731	. . . 4 ⊢ {𝑦} = {𝑦}
12		reex 11204	. . . 4 ⊢ ℝ ∈ V
13		vex 3477	. . . 4 ⊢ 𝑦 ∈ V
14		eqid 2731	. . . 4 ⊢ (𝑥 ∈ ℝ ↦ ({𝑦} × {𝑥})) = (𝑥 ∈ ℝ ↦ ({𝑦} × {𝑥}))
15	11, 12, 13, 14	mapsnf1o3 8892	. . 3 ⊢ (𝑥 ∈ ℝ ↦ ({𝑦} × {𝑥})):ℝ–1-1-onto→(ℝ ↑_m {𝑦})
16	10, 15	vtoclg 3542	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐹:ℝ–1-1-onto→(ℝ ↑_m {𝐴}))
17		sneq 4638	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑦 → {𝑥} = {𝑦})
18	17	xpeq2d 5706	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑦 → ({𝐴} × {𝑥}) = ({𝐴} × {𝑦}))
19		snex 5431	. . . . . . . . . . . . . . . . 17 ⊢ {𝐴} ∈ V
20		snex 5431	. . . . . . . . . . . . . . . . 17 ⊢ {𝑥} ∈ V
21	19, 20	xpex 7743	. . . . . . . . . . . . . . . 16 ⊢ ({𝐴} × {𝑥}) ∈ V
22	18, 4, 21	fvmpt3i 7003	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ ℝ → (𝐹‘𝑦) = ({𝐴} × {𝑦}))
23	22	fveq1d 6893	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ℝ → ((𝐹‘𝑦)‘𝐴) = (({𝐴} × {𝑦})‘𝐴))
24	23	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ) → ((𝐹‘𝑦)‘𝐴) = (({𝐴} × {𝑦})‘𝐴))
25		snidg 4662	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴})
26		fvconst2g 7205	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ V ∧ 𝐴 ∈ {𝐴}) → (({𝐴} × {𝑦})‘𝐴) = 𝑦)
27	13, 25, 26	sylancr 586	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ 𝑉 → (({𝐴} × {𝑦})‘𝐴) = 𝑦)
28	24, 27	sylan9eqr 2793	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ)) → ((𝐹‘𝑦)‘𝐴) = 𝑦)
29		sneq 4638	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑧 → {𝑥} = {𝑧})
30	29	xpeq2d 5706	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑧 → ({𝐴} × {𝑥}) = ({𝐴} × {𝑧}))
31	30, 4, 21	fvmpt3i 7003	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ ℝ → (𝐹‘𝑧) = ({𝐴} × {𝑧}))
32	31	fveq1d 6893	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ ℝ → ((𝐹‘𝑧)‘𝐴) = (({𝐴} × {𝑧})‘𝐴))
33	32	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ) → ((𝐹‘𝑧)‘𝐴) = (({𝐴} × {𝑧})‘𝐴))
34		vex 3477	. . . . . . . . . . . . . 14 ⊢ 𝑧 ∈ V
35		fvconst2g 7205	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ V ∧ 𝐴 ∈ {𝐴}) → (({𝐴} × {𝑧})‘𝐴) = 𝑧)
36	34, 25, 35	sylancr 586	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ 𝑉 → (({𝐴} × {𝑧})‘𝐴) = 𝑧)
37	33, 36	sylan9eqr 2793	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ)) → ((𝐹‘𝑧)‘𝐴) = 𝑧)
38	28, 37	oveq12d 7430	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ)) → (((𝐹‘𝑦)‘𝐴) − ((𝐹‘𝑧)‘𝐴)) = (𝑦 − 𝑧))
39	38	oveq1d 7427	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ)) → ((((𝐹‘𝑦)‘𝐴) − ((𝐹‘𝑧)‘𝐴))↑2) = ((𝑦 − 𝑧)↑2))
40		resubcl 11529	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ) → (𝑦 − 𝑧) ∈ ℝ)
41	40	adantl 481	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ)) → (𝑦 − 𝑧) ∈ ℝ)
42		absresq 15254	. . . . . . . . . . 11 ⊢ ((𝑦 − 𝑧) ∈ ℝ → ((abs‘(𝑦 − 𝑧))↑2) = ((𝑦 − 𝑧)↑2))
43	41, 42	syl 17	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ)) → ((abs‘(𝑦 − 𝑧))↑2) = ((𝑦 − 𝑧)↑2))
44	39, 43	eqtr4d 2774	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ)) → ((((𝐹‘𝑦)‘𝐴) − ((𝐹‘𝑧)‘𝐴))↑2) = ((abs‘(𝑦 − 𝑧))↑2))
45	41	recnd 11247	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ)) → (𝑦 − 𝑧) ∈ ℂ)
46	45	abscld 15388	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ)) → (abs‘(𝑦 − 𝑧)) ∈ ℝ)
47	46	recnd 11247	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ)) → (abs‘(𝑦 − 𝑧)) ∈ ℂ)
48	47	sqcld 14114	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ)) → ((abs‘(𝑦 − 𝑧))↑2) ∈ ℂ)
49	44, 48	eqeltrd 2832	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ)) → ((((𝐹‘𝑦)‘𝐴) − ((𝐹‘𝑧)‘𝐴))↑2) ∈ ℂ)
50		fveq2 6891	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐴 → ((𝐹‘𝑦)‘𝑘) = ((𝐹‘𝑦)‘𝐴))
51		fveq2 6891	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐴 → ((𝐹‘𝑧)‘𝑘) = ((𝐹‘𝑧)‘𝐴))
52	50, 51	oveq12d 7430	. . . . . . . . . 10 ⊢ (𝑘 = 𝐴 → (((𝐹‘𝑦)‘𝑘) − ((𝐹‘𝑧)‘𝑘)) = (((𝐹‘𝑦)‘𝐴) − ((𝐹‘𝑧)‘𝐴)))
53	52	oveq1d 7427	. . . . . . . . 9 ⊢ (𝑘 = 𝐴 → ((((𝐹‘𝑦)‘𝑘) − ((𝐹‘𝑧)‘𝑘))↑2) = ((((𝐹‘𝑦)‘𝐴) − ((𝐹‘𝑧)‘𝐴))↑2))
54	53	sumsn 15697	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ ((((𝐹‘𝑦)‘𝐴) − ((𝐹‘𝑧)‘𝐴))↑2) ∈ ℂ) → Σ𝑘 ∈ {𝐴} ((((𝐹‘𝑦)‘𝑘) − ((𝐹‘𝑧)‘𝑘))↑2) = ((((𝐹‘𝑦)‘𝐴) − ((𝐹‘𝑧)‘𝐴))↑2))
55	49, 54	syldan 590	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ)) → Σ𝑘 ∈ {𝐴} ((((𝐹‘𝑦)‘𝑘) − ((𝐹‘𝑧)‘𝑘))↑2) = ((((𝐹‘𝑦)‘𝐴) − ((𝐹‘𝑧)‘𝐴))↑2))
56	55, 44	eqtrd 2771	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ)) → Σ𝑘 ∈ {𝐴} ((((𝐹‘𝑦)‘𝑘) − ((𝐹‘𝑧)‘𝑘))↑2) = ((abs‘(𝑦 − 𝑧))↑2))
57	56	fveq2d 6895	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ)) → (√‘Σ𝑘 ∈ {𝐴} ((((𝐹‘𝑦)‘𝑘) − ((𝐹‘𝑧)‘𝑘))↑2)) = (√‘((abs‘(𝑦 − 𝑧))↑2)))
58	45	absge0d 15396	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ)) → 0 ≤ (abs‘(𝑦 − 𝑧)))
59	46, 58	sqrtsqd 15371	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ)) → (√‘((abs‘(𝑦 − 𝑧))↑2)) = (abs‘(𝑦 − 𝑧)))
60	57, 59	eqtrd 2771	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ)) → (√‘Σ𝑘 ∈ {𝐴} ((((𝐹‘𝑦)‘𝑘) − ((𝐹‘𝑧)‘𝑘))↑2)) = (abs‘(𝑦 − 𝑧)))
61		f1of 6833	. . . . . . . 8 ⊢ (𝐹:ℝ–1-1-onto→(ℝ ↑_m {𝐴}) → 𝐹:ℝ⟶(ℝ ↑_m {𝐴}))
62	16, 61	syl 17	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → 𝐹:ℝ⟶(ℝ ↑_m {𝐴}))
63	62	ffvelcdmda 7086	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ ℝ) → (𝐹‘𝑦) ∈ (ℝ ↑_m {𝐴}))
64	62	ffvelcdmda 7086	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑧 ∈ ℝ) → (𝐹‘𝑧) ∈ (ℝ ↑_m {𝐴}))
65	63, 64	anim12dan 618	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ)) → ((𝐹‘𝑦) ∈ (ℝ ↑_m {𝐴}) ∧ (𝐹‘𝑧) ∈ (ℝ ↑_m {𝐴})))
66		snfi 9047	. . . . . 6 ⊢ {𝐴} ∈ Fin
67		eqid 2731	. . . . . . 7 ⊢ (ℝ ↑_m {𝐴}) = (ℝ ↑_m {𝐴})
68	67	rrnmval 37000	. . . . . 6 ⊢ (({𝐴} ∈ Fin ∧ (𝐹‘𝑦) ∈ (ℝ ↑_m {𝐴}) ∧ (𝐹‘𝑧) ∈ (ℝ ↑_m {𝐴})) → ((𝐹‘𝑦)(ℝⁿ‘{𝐴})(𝐹‘𝑧)) = (√‘Σ𝑘 ∈ {𝐴} ((((𝐹‘𝑦)‘𝑘) − ((𝐹‘𝑧)‘𝑘))↑2)))
69	66, 68	mp3an1 1447	. . . . 5 ⊢ (((𝐹‘𝑦) ∈ (ℝ ↑_m {𝐴}) ∧ (𝐹‘𝑧) ∈ (ℝ ↑_m {𝐴})) → ((𝐹‘𝑦)(ℝⁿ‘{𝐴})(𝐹‘𝑧)) = (√‘Σ𝑘 ∈ {𝐴} ((((𝐹‘𝑦)‘𝑘) − ((𝐹‘𝑧)‘𝑘))↑2)))
70	65, 69	syl 17	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ)) → ((𝐹‘𝑦)(ℝⁿ‘{𝐴})(𝐹‘𝑧)) = (√‘Σ𝑘 ∈ {𝐴} ((((𝐹‘𝑦)‘𝑘) − ((𝐹‘𝑧)‘𝑘))↑2)))
71		ismrer1.1	. . . . . 6 ⊢ 𝑅 = ((abs ∘ − ) ↾ (ℝ × ℝ))
72	71	remetdval 24526	. . . . 5 ⊢ ((𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ) → (𝑦𝑅𝑧) = (abs‘(𝑦 − 𝑧)))
73	72	adantl 481	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ)) → (𝑦𝑅𝑧) = (abs‘(𝑦 − 𝑧)))
74	60, 70, 73	3eqtr4rd 2782	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ)) → (𝑦𝑅𝑧) = ((𝐹‘𝑦)(ℝⁿ‘{𝐴})(𝐹‘𝑧)))
75	74	ralrimivva 3199	. 2 ⊢ (𝐴 ∈ 𝑉 → ∀𝑦 ∈ ℝ ∀𝑧 ∈ ℝ (𝑦𝑅𝑧) = ((𝐹‘𝑦)(ℝⁿ‘{𝐴})(𝐹‘𝑧)))
76	71	rexmet 24528	. . 3 ⊢ 𝑅 ∈ (∞Met‘ℝ)
77	67	rrnmet 37001	. . . 4 ⊢ ({𝐴} ∈ Fin → (ℝⁿ‘{𝐴}) ∈ (Met‘(ℝ ↑_m {𝐴})))
78		metxmet 24061	. . . 4 ⊢ ((ℝⁿ‘{𝐴}) ∈ (Met‘(ℝ ↑_m {𝐴})) → (ℝⁿ‘{𝐴}) ∈ (∞Met‘(ℝ ↑_m {𝐴})))
79	66, 77, 78	mp2b 10	. . 3 ⊢ (ℝⁿ‘{𝐴}) ∈ (∞Met‘(ℝ ↑_m {𝐴}))
80		isismty 36973	. . 3 ⊢ ((𝑅 ∈ (∞Met‘ℝ) ∧ (ℝⁿ‘{𝐴}) ∈ (∞Met‘(ℝ ↑_m {𝐴}))) → (𝐹 ∈ (𝑅 Ismty (ℝⁿ‘{𝐴})) ↔ (𝐹:ℝ–1-1-onto→(ℝ ↑_m {𝐴}) ∧ ∀𝑦 ∈ ℝ ∀𝑧 ∈ ℝ (𝑦𝑅𝑧) = ((𝐹‘𝑦)(ℝⁿ‘{𝐴})(𝐹‘𝑧)))))
81	76, 79, 80	mp2an 689	. 2 ⊢ (𝐹 ∈ (𝑅 Ismty (ℝⁿ‘{𝐴})) ↔ (𝐹:ℝ–1-1-onto→(ℝ ↑_m {𝐴}) ∧ ∀𝑦 ∈ ℝ ∀𝑧 ∈ ℝ (𝑦𝑅𝑧) = ((𝐹‘𝑦)(ℝⁿ‘{𝐴})(𝐹‘𝑧))))
82	16, 75, 81	sylanbrc 582	1 ⊢ (𝐴 ∈ 𝑉 → 𝐹 ∈ (𝑅 Ismty (ℝⁿ‘{𝐴})))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1540 ∈ wcel 2105 ∀wral 3060 Vcvv 3473 {csn 4628 ↦ cmpt 5231 × cxp 5674 ↾ cres 5678 ∘ ccom 5680 ⟶wf 6539 –1-1-onto→wf1o 6542 ‘cfv 6543 (class class class)co 7412 ↑_m cmap 8823 Fincfn 8942 ℂcc 11111 ℝcr 11112 − cmin 11449 2c2 12272 ↑cexp 14032 √csqrt 15185 abscabs 15186 Σcsu 15637 ∞Metcxmet 21130 Metcmet 21131 Ismty cismty 36970 ℝⁿcrrn 36997

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7728 ax-inf2 9639 ax-cnex 11169 ax-resscn 11170 ax-1cn 11171 ax-icn 11172 ax-addcl 11173 ax-addrcl 11174 ax-mulcl 11175 ax-mulrcl 11176 ax-mulcom 11177 ax-addass 11178 ax-mulass 11179 ax-distr 11180 ax-i2m1 11181 ax-1ne0 11182 ax-1rid 11183 ax-rnegex 11184 ax-rrecex 11185 ax-cnre 11186 ax-pre-lttri 11187 ax-pre-lttrn 11188 ax-pre-ltadd 11189 ax-pre-mulgt0 11190 ax-pre-sup 11191

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7859 df-1st 7978 df-2nd 7979 df-frecs 8269 df-wrecs 8300 df-recs 8374 df-rdg 8413 df-1o 8469 df-er 8706 df-map 8825 df-en 8943 df-dom 8944 df-sdom 8945 df-fin 8946 df-sup 9440 df-oi 9508 df-card 9937 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 df-le 11259 df-sub 11451 df-neg 11452 df-div 11877 df-nn 12218 df-2 12280 df-3 12281 df-4 12282 df-n0 12478 df-z 12564 df-uz 12828 df-rp 12980 df-xadd 13098 df-ico 13335 df-fz 13490 df-fzo 13633 df-seq 13972 df-exp 14033 df-hash 14296 df-cj 15051 df-re 15052 df-im 15053 df-sqrt 15187 df-abs 15188 df-clim 15437 df-sum 15638 df-xmet 21138 df-met 21139 df-ismty 36971 df-rrn 36998

This theorem is referenced by: reheibor 37011

W3C validator