Theorem 0ome 40743

Description: The map that assigns 0 to every subset, is an outer measure. (Contributed by Glauco Siliprandi, 11-Oct-2020.)

Hypotheses

Ref	Expression
0ome.1	⊢ (𝜑 → 𝑋 ∈ 𝑉)
0ome.2	⊢ 𝑂 = (𝑥 ∈ 𝒫 𝑋 ↦ 0)

Assertion

Ref	Expression
0ome	⊢ (𝜑 → 𝑂 ∈ OutMeas)

Distinct variable group:   𝑥,𝑋

Allowed substitution hints:   𝜑(𝑥)   𝑂(𝑥)   𝑉(𝑥)

Proof of Theorem 0ome

Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2622	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝒫 𝑋 ↦ 0) = (𝑦 ∈ 𝒫 𝑋 ↦ 0)
2		0e0iccpnf 12283	. . . . . . . . . 10 ⊢ 0 ∈ (0[,]+∞)
3	2	a1i 11	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝒫 𝑋 → 0 ∈ (0[,]+∞))
4	1, 3	fmpti 6383	. . . . . . . 8 ⊢ (𝑦 ∈ 𝒫 𝑋 ↦ 0):𝒫 𝑋⟶(0[,]+∞)
5		0ome.2	. . . . . . . . . . 11 ⊢ 𝑂 = (𝑥 ∈ 𝒫 𝑋 ↦ 0)
6		eqidd 2623	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → 0 = 0)
7	6	cbvmptv 4750	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝒫 𝑋 ↦ 0) = (𝑦 ∈ 𝒫 𝑋 ↦ 0)
8	5, 7	eqtri 2644	. . . . . . . . . 10 ⊢ 𝑂 = (𝑦 ∈ 𝒫 𝑋 ↦ 0)
9	8	feq1i 6036	. . . . . . . . 9 ⊢ (𝑂:dom 𝑂⟶(0[,]+∞) ↔ (𝑦 ∈ 𝒫 𝑋 ↦ 0):dom 𝑂⟶(0[,]+∞))
10	8	dmeqi 5325	. . . . . . . . . . 11 ⊢ dom 𝑂 = dom (𝑦 ∈ 𝒫 𝑋 ↦ 0)
11		0re 10040	. . . . . . . . . . . . 13 ⊢ 0 ∈ ℝ
12	11	rgenw 2924	. . . . . . . . . . . 12 ⊢ ∀𝑦 ∈ 𝒫 𝑋0 ∈ ℝ
13		dmmptg 5632	. . . . . . . . . . . 12 ⊢ (∀𝑦 ∈ 𝒫 𝑋0 ∈ ℝ → dom (𝑦 ∈ 𝒫 𝑋 ↦ 0) = 𝒫 𝑋)
14	12, 13	ax-mp 5	. . . . . . . . . . 11 ⊢ dom (𝑦 ∈ 𝒫 𝑋 ↦ 0) = 𝒫 𝑋
15	10, 14	eqtri 2644	. . . . . . . . . 10 ⊢ dom 𝑂 = 𝒫 𝑋
16	15	feq2i 6037	. . . . . . . . 9 ⊢ ((𝑦 ∈ 𝒫 𝑋 ↦ 0):dom 𝑂⟶(0[,]+∞) ↔ (𝑦 ∈ 𝒫 𝑋 ↦ 0):𝒫 𝑋⟶(0[,]+∞))
17	9, 16	bitri 264	. . . . . . . 8 ⊢ (𝑂:dom 𝑂⟶(0[,]+∞) ↔ (𝑦 ∈ 𝒫 𝑋 ↦ 0):𝒫 𝑋⟶(0[,]+∞))
18	4, 17	mpbir 221	. . . . . . 7 ⊢ 𝑂:dom 𝑂⟶(0[,]+∞)
19		unipw 4918	. . . . . . . . . 10 ⊢ ∪ 𝒫 𝑋 = 𝑋
20	19	pweqi 4162	. . . . . . . . 9 ⊢ 𝒫 ∪ 𝒫 𝑋 = 𝒫 𝑋
21	20	eqcomi 2631	. . . . . . . 8 ⊢ 𝒫 𝑋 = 𝒫 ∪ 𝒫 𝑋
22	15	eqcomi 2631	. . . . . . . . . 10 ⊢ 𝒫 𝑋 = dom 𝑂
23	22	unieqi 4445	. . . . . . . . 9 ⊢ ∪ 𝒫 𝑋 = ∪ dom 𝑂
24	23	pweqi 4162	. . . . . . . 8 ⊢ 𝒫 ∪ 𝒫 𝑋 = 𝒫 ∪ dom 𝑂
25	15, 21, 24	3eqtri 2648	. . . . . . 7 ⊢ dom 𝑂 = 𝒫 ∪ dom 𝑂
26	18, 25	pm3.2i 471	. . . . . 6 ⊢ (𝑂:dom 𝑂⟶(0[,]+∞) ∧ dom 𝑂 = 𝒫 ∪ dom 𝑂)
27		0elpw 4834	. . . . . . 7 ⊢ ∅ ∈ 𝒫 𝑋
28		eqidd 2623	. . . . . . . 8 ⊢ (𝑦 = ∅ → 0 = 0)
29	11	elexi 3213	. . . . . . . 8 ⊢ 0 ∈ V
30	28, 8, 29	fvmpt 6282	. . . . . . 7 ⊢ (∅ ∈ 𝒫 𝑋 → (𝑂‘∅) = 0)
31	27, 30	ax-mp 5	. . . . . 6 ⊢ (𝑂‘∅) = 0
32	26, 31	pm3.2i 471	. . . . 5 ⊢ ((𝑂:dom 𝑂⟶(0[,]+∞) ∧ dom 𝑂 = 𝒫 ∪ dom 𝑂) ∧ (𝑂‘∅) = 0)
33	11	leidi 10562	. . . . . . . . 9 ⊢ 0 ≤ 0
34	33	a1i 11	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝒫 ∪ dom 𝑂 ∧ 𝑧 ∈ 𝒫 𝑦) → 0 ≤ 0)
35		elpwi 4168	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ 𝒫 𝑦 → 𝑧 ⊆ 𝑦)
36	35	adantl 482	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ 𝒫 ∪ dom 𝑂 ∧ 𝑧 ∈ 𝒫 𝑦) → 𝑧 ⊆ 𝑦)
37		id 22	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ 𝒫 ∪ dom 𝑂 → 𝑦 ∈ 𝒫 ∪ dom 𝑂)
38	21, 24	eqtr2i 2645	. . . . . . . . . . . . . . . 16 ⊢ 𝒫 ∪ dom 𝑂 = 𝒫 𝑋
39	38	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ 𝒫 ∪ dom 𝑂 → 𝒫 ∪ dom 𝑂 = 𝒫 𝑋)
40	37, 39	eleqtrd 2703	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ 𝒫 ∪ dom 𝑂 → 𝑦 ∈ 𝒫 𝑋)
41		elpwi 4168	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ 𝒫 𝑋 → 𝑦 ⊆ 𝑋)
42	40, 41	syl 17	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ 𝒫 ∪ dom 𝑂 → 𝑦 ⊆ 𝑋)
43	42	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ 𝒫 ∪ dom 𝑂 ∧ 𝑧 ∈ 𝒫 𝑦) → 𝑦 ⊆ 𝑋)
44	36, 43	sstrd 3613	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ 𝒫 ∪ dom 𝑂 ∧ 𝑧 ∈ 𝒫 𝑦) → 𝑧 ⊆ 𝑋)
45		simpr 477	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ 𝒫 ∪ dom 𝑂 ∧ 𝑧 ∈ 𝒫 𝑦) → 𝑧 ∈ 𝒫 𝑦)
46		elpwg 4166	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ 𝒫 𝑦 → (𝑧 ∈ 𝒫 𝑋 ↔ 𝑧 ⊆ 𝑋))
47	45, 46	syl 17	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ 𝒫 ∪ dom 𝑂 ∧ 𝑧 ∈ 𝒫 𝑦) → (𝑧 ∈ 𝒫 𝑋 ↔ 𝑧 ⊆ 𝑋))
48	44, 47	mpbird 247	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝒫 ∪ dom 𝑂 ∧ 𝑧 ∈ 𝒫 𝑦) → 𝑧 ∈ 𝒫 𝑋)
49	11	a1i 11	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝒫 ∪ dom 𝑂 ∧ 𝑧 ∈ 𝒫 𝑦) → 0 ∈ ℝ)
50		eqidd 2623	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑧 → 0 = 0)
51	50	cbvmptv 4750	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ 𝒫 𝑋 ↦ 0) = (𝑧 ∈ 𝒫 𝑋 ↦ 0)
52	8, 51	eqtri 2644	. . . . . . . . . . 11 ⊢ 𝑂 = (𝑧 ∈ 𝒫 𝑋 ↦ 0)
53	52	fvmpt2 6291	. . . . . . . . . 10 ⊢ ((𝑧 ∈ 𝒫 𝑋 ∧ 0 ∈ ℝ) → (𝑂‘𝑧) = 0)
54	48, 49, 53	syl2anc 693	. . . . . . . . 9 ⊢ ((𝑦 ∈ 𝒫 ∪ dom 𝑂 ∧ 𝑧 ∈ 𝒫 𝑦) → (𝑂‘𝑧) = 0)
55	8	fvmpt2 6291	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ 𝒫 𝑋 ∧ 0 ∈ ℝ) → (𝑂‘𝑦) = 0)
56	40, 11, 55	sylancl 694	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝒫 ∪ dom 𝑂 → (𝑂‘𝑦) = 0)
57	56	adantr 481	. . . . . . . . 9 ⊢ ((𝑦 ∈ 𝒫 ∪ dom 𝑂 ∧ 𝑧 ∈ 𝒫 𝑦) → (𝑂‘𝑦) = 0)
58	54, 57	breq12d 4666	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝒫 ∪ dom 𝑂 ∧ 𝑧 ∈ 𝒫 𝑦) → ((𝑂‘𝑧) ≤ (𝑂‘𝑦) ↔ 0 ≤ 0))
59	34, 58	mpbird 247	. . . . . . 7 ⊢ ((𝑦 ∈ 𝒫 ∪ dom 𝑂 ∧ 𝑧 ∈ 𝒫 𝑦) → (𝑂‘𝑧) ≤ (𝑂‘𝑦))
60	59	ralrimiva 2966	. . . . . 6 ⊢ (𝑦 ∈ 𝒫 ∪ dom 𝑂 → ∀𝑧 ∈ 𝒫 𝑦(𝑂‘𝑧) ≤ (𝑂‘𝑦))
61	60	rgen 2922	. . . . 5 ⊢ ∀𝑦 ∈ 𝒫 ∪ dom 𝑂∀𝑧 ∈ 𝒫 𝑦(𝑂‘𝑧) ≤ (𝑂‘𝑦)
62	32, 61	pm3.2i 471	. . . 4 ⊢ (((𝑂:dom 𝑂⟶(0[,]+∞) ∧ dom 𝑂 = 𝒫 ∪ dom 𝑂) ∧ (𝑂‘∅) = 0) ∧ ∀𝑦 ∈ 𝒫 ∪ dom 𝑂∀𝑧 ∈ 𝒫 𝑦(𝑂‘𝑧) ≤ (𝑂‘𝑦))
63	33	a1i 11	. . . . . . 7 ⊢ (𝑦 ∈ 𝒫 dom 𝑂 → 0 ≤ 0)
64	52	a1i 11	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝒫 dom 𝑂 → 𝑂 = (𝑧 ∈ 𝒫 𝑋 ↦ 0))
65		eqidd 2623	. . . . . . . . 9 ⊢ ((𝑦 ∈ 𝒫 dom 𝑂 ∧ 𝑧 = ∪ 𝑦) → 0 = 0)
66		id 22	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ 𝒫 dom 𝑂 → 𝑦 ∈ 𝒫 dom 𝑂)
67	15	pweqi 4162	. . . . . . . . . . . . . 14 ⊢ 𝒫 dom 𝑂 = 𝒫 𝒫 𝑋
68	67	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ 𝒫 dom 𝑂 → 𝒫 dom 𝑂 = 𝒫 𝒫 𝑋)
69	66, 68	eleqtrd 2703	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ 𝒫 dom 𝑂 → 𝑦 ∈ 𝒫 𝒫 𝑋)
70		elpwi 4168	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ 𝒫 𝒫 𝑋 → 𝑦 ⊆ 𝒫 𝑋)
71	69, 70	syl 17	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝒫 dom 𝑂 → 𝑦 ⊆ 𝒫 𝑋)
72		sspwuni 4611	. . . . . . . . . . 11 ⊢ (𝑦 ⊆ 𝒫 𝑋 ↔ ∪ 𝑦 ⊆ 𝑋)
73	71, 72	sylib 208	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝒫 dom 𝑂 → ∪ 𝑦 ⊆ 𝑋)
74		vuniex 6954	. . . . . . . . . . . 12 ⊢ ∪ 𝑦 ∈ V
75	74	a1i 11	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝒫 dom 𝑂 → ∪ 𝑦 ∈ V)
76		elpwg 4166	. . . . . . . . . . 11 ⊢ (∪ 𝑦 ∈ V → (∪ 𝑦 ∈ 𝒫 𝑋 ↔ ∪ 𝑦 ⊆ 𝑋))
77	75, 76	syl 17	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝒫 dom 𝑂 → (∪ 𝑦 ∈ 𝒫 𝑋 ↔ ∪ 𝑦 ⊆ 𝑋))
78	73, 77	mpbird 247	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝒫 dom 𝑂 → ∪ 𝑦 ∈ 𝒫 𝑋)
79	11	a1i 11	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝒫 dom 𝑂 → 0 ∈ ℝ)
80	64, 65, 78, 79	fvmptd 6288	. . . . . . . 8 ⊢ (𝑦 ∈ 𝒫 dom 𝑂 → (𝑂‘∪ 𝑦) = 0)
81	64	reseq1d 5395	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝒫 dom 𝑂 → (𝑂 ↾ 𝑦) = ((𝑧 ∈ 𝒫 𝑋 ↦ 0) ↾ 𝑦))
82		resmpt 5449	. . . . . . . . . . . 12 ⊢ (𝑦 ⊆ 𝒫 𝑋 → ((𝑧 ∈ 𝒫 𝑋 ↦ 0) ↾ 𝑦) = (𝑧 ∈ 𝑦 ↦ 0))
83	71, 82	syl 17	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝒫 dom 𝑂 → ((𝑧 ∈ 𝒫 𝑋 ↦ 0) ↾ 𝑦) = (𝑧 ∈ 𝑦 ↦ 0))
84	81, 83	eqtrd 2656	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝒫 dom 𝑂 → (𝑂 ↾ 𝑦) = (𝑧 ∈ 𝑦 ↦ 0))
85	84	fveq2d 6195	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝒫 dom 𝑂 → (Σ^‘(𝑂 ↾ 𝑦)) = (Σ^‘(𝑧 ∈ 𝑦 ↦ 0)))
86		nfv 1843	. . . . . . . . . 10 ⊢ Ⅎ𝑧 𝑦 ∈ 𝒫 dom 𝑂
87	86, 66	sge0z 40592	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝒫 dom 𝑂 → (Σ^‘(𝑧 ∈ 𝑦 ↦ 0)) = 0)
88	85, 87	eqtrd 2656	. . . . . . . 8 ⊢ (𝑦 ∈ 𝒫 dom 𝑂 → (Σ^‘(𝑂 ↾ 𝑦)) = 0)
89	80, 88	breq12d 4666	. . . . . . 7 ⊢ (𝑦 ∈ 𝒫 dom 𝑂 → ((𝑂‘∪ 𝑦) ≤ (Σ^‘(𝑂 ↾ 𝑦)) ↔ 0 ≤ 0))
90	63, 89	mpbird 247	. . . . . 6 ⊢ (𝑦 ∈ 𝒫 dom 𝑂 → (𝑂‘∪ 𝑦) ≤ (Σ^‘(𝑂 ↾ 𝑦)))
91	90	a1d 25	. . . . 5 ⊢ (𝑦 ∈ 𝒫 dom 𝑂 → (𝑦 ≼ ω → (𝑂‘∪ 𝑦) ≤ (Σ^‘(𝑂 ↾ 𝑦))))
92	91	rgen 2922	. . . 4 ⊢ ∀𝑦 ∈ 𝒫 dom 𝑂(𝑦 ≼ ω → (𝑂‘∪ 𝑦) ≤ (Σ^‘(𝑂 ↾ 𝑦)))
93	62, 92	pm3.2i 471	. . 3 ⊢ ((((𝑂:dom 𝑂⟶(0[,]+∞) ∧ dom 𝑂 = 𝒫 ∪ dom 𝑂) ∧ (𝑂‘∅) = 0) ∧ ∀𝑦 ∈ 𝒫 ∪ dom 𝑂∀𝑧 ∈ 𝒫 𝑦(𝑂‘𝑧) ≤ (𝑂‘𝑦)) ∧ ∀𝑦 ∈ 𝒫 dom 𝑂(𝑦 ≼ ω → (𝑂‘∪ 𝑦) ≤ (Σ^‘(𝑂 ↾ 𝑦))))
94	93	a1i 11	. 2 ⊢ (𝜑 → ((((𝑂:dom 𝑂⟶(0[,]+∞) ∧ dom 𝑂 = 𝒫 ∪ dom 𝑂) ∧ (𝑂‘∅) = 0) ∧ ∀𝑦 ∈ 𝒫 ∪ dom 𝑂∀𝑧 ∈ 𝒫 𝑦(𝑂‘𝑧) ≤ (𝑂‘𝑦)) ∧ ∀𝑦 ∈ 𝒫 dom 𝑂(𝑦 ≼ ω → (𝑂‘∪ 𝑦) ≤ (Σ^‘(𝑂 ↾ 𝑦)))))
95	8	a1i 11	. . . 4 ⊢ (𝜑 → 𝑂 = (𝑦 ∈ 𝒫 𝑋 ↦ 0))
96		0ome.1	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
97		pwexg 4850	. . . . . 6 ⊢ (𝑋 ∈ 𝑉 → 𝒫 𝑋 ∈ V)
98	96, 97	syl 17	. . . . 5 ⊢ (𝜑 → 𝒫 𝑋 ∈ V)
99		mptexg 6484	. . . . 5 ⊢ (𝒫 𝑋 ∈ V → (𝑦 ∈ 𝒫 𝑋 ↦ 0) ∈ V)
100	98, 99	syl 17	. . . 4 ⊢ (𝜑 → (𝑦 ∈ 𝒫 𝑋 ↦ 0) ∈ V)
101	95, 100	eqeltrd 2701	. . 3 ⊢ (𝜑 → 𝑂 ∈ V)
102		isome 40708	. . 3 ⊢ (𝑂 ∈ V → (𝑂 ∈ OutMeas ↔ ((((𝑂:dom 𝑂⟶(0[,]+∞) ∧ dom 𝑂 = 𝒫 ∪ dom 𝑂) ∧ (𝑂‘∅) = 0) ∧ ∀𝑦 ∈ 𝒫 ∪ dom 𝑂∀𝑧 ∈ 𝒫 𝑦(𝑂‘𝑧) ≤ (𝑂‘𝑦)) ∧ ∀𝑦 ∈ 𝒫 dom 𝑂(𝑦 ≼ ω → (𝑂‘∪ 𝑦) ≤ (Σ^‘(𝑂 ↾ 𝑦))))))
103	101, 102	syl 17	. 2 ⊢ (𝜑 → (𝑂 ∈ OutMeas ↔ ((((𝑂:dom 𝑂⟶(0[,]+∞) ∧ dom 𝑂 = 𝒫 ∪ dom 𝑂) ∧ (𝑂‘∅) = 0) ∧ ∀𝑦 ∈ 𝒫 ∪ dom 𝑂∀𝑧 ∈ 𝒫 𝑦(𝑂‘𝑧) ≤ (𝑂‘𝑦)) ∧ ∀𝑦 ∈ 𝒫 dom 𝑂(𝑦 ≼ ω → (𝑂‘∪ 𝑦) ≤ (Σ^‘(𝑂 ↾ 𝑦))))))
104	94, 103	mpbird 247	1 ⊢ (𝜑 → 𝑂 ∈ OutMeas)

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∀wral 2912  Vcvv 3200   ⊆ wss 3574  ∅c0 3915  𝒫 cpw 4158  ∪ cuni 4436   class class class wbr 4653   ↦ cmpt 4729  dom cdm 5114   ↾ cres 5116  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650  ωcom 7065   ≼ cdom 7953  ℝcr 9935  0cc0 9936  +∞cpnf 10071   ≤ cle 10075  [,]cicc 12178  Σ^{^}csumge0 40579  OutMeascome 40703

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-oi 8415  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-sum 14417  df-sumge0 40580  df-ome 40704

This theorem is referenced by: (None)