isomennd - Mathbox for Glauco Siliprandi

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Theorem isomennd 47225

Description: Sufficient condition to prove that 𝑂 is an outer measure. Definition 113A of [Fremlin1] p. 19 . (Contributed by Glauco Siliprandi, 11-Oct-2020.)

**Hypotheses**
Ref	Expression
isomennd.x	⊢ (𝜑 → 𝑋 ∈ 𝑉)
isomennd.o	⊢ (𝜑 → 𝑂:𝒫 𝑋⟶(0[,]+∞))
isomennd.o0	⊢ (𝜑 → (𝑂‘∅) = 0)
isomennd.le	⊢ ((𝜑 ∧ 𝑥 ⊆ 𝑋 ∧ 𝑦 ⊆ 𝑥) → (𝑂‘𝑦) ≤ (𝑂‘𝑥))
isomennd.sa	⊢ ((𝜑 ∧ 𝑎:ℕ⟶𝒫 𝑋) → (𝑂‘∪ 𝑛 ∈ ℕ (𝑎‘𝑛)) ≤ (Σ^{^}‘(𝑛 ∈ ℕ ↦ (𝑂‘(𝑎‘𝑛)))))

**Assertion**
Ref	Expression
isomennd	⊢ (𝜑 → 𝑂 ∈ OutMeas)

Distinct variable groups: 𝑂,𝑎,𝑛,𝑥 𝑦,𝑂,𝑥 𝑋,𝑎 𝜑,𝑎,𝑛,𝑥 𝜑,𝑦 Allowed substitution hints: 𝑉(𝑥,𝑦,𝑛,𝑎) 𝑋(𝑥,𝑦,𝑛)

Proof of Theorem isomennd Dummy variables 𝑓 𝑚 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		isomennd.o	. . . . 5 ⊢ (𝜑 → 𝑂:𝒫 𝑋⟶(0[,]+∞))
2		id 23	. . . . . 6 ⊢ (𝑂:𝒫 𝑋⟶(0[,]+∞) → 𝑂:𝒫 𝑋⟶(0[,]+∞))
3		fdm 6717	. . . . . . 7 ⊢ (𝑂:𝒫 𝑋⟶(0[,]+∞) → dom 𝑂 = 𝒫 𝑋)
4	3	feq2d 6691	. . . . . 6 ⊢ (𝑂:𝒫 𝑋⟶(0[,]+∞) → (𝑂:dom 𝑂⟶(0[,]+∞) ↔ 𝑂:𝒫 𝑋⟶(0[,]+∞)))
5	2, 4	mpbird 260	. . . . 5 ⊢ (𝑂:𝒫 𝑋⟶(0[,]+∞) → 𝑂:dom 𝑂⟶(0[,]+∞))
6	1, 5	syl 18	. . . 4 ⊢ (𝜑 → 𝑂:dom 𝑂⟶(0[,]+∞))
7		unipw 5433	. . . . . . 7 ⊢ ∪ 𝒫 𝑋 = 𝑋
8	7	pweqi 4579	. . . . . 6 ⊢ 𝒫 ∪ 𝒫 𝑋 = 𝒫 𝑋
9	8	a1i 11	. . . . 5 ⊢ (𝜑 → 𝒫 ∪ 𝒫 𝑋 = 𝒫 𝑋)
10	1, 3	syl 18	. . . . . . 7 ⊢ (𝜑 → dom 𝑂 = 𝒫 𝑋)
11	10	unieqd 4886	. . . . . 6 ⊢ (𝜑 → ∪ dom 𝑂 = ∪ 𝒫 𝑋)
12	11	pweqd 4580	. . . . 5 ⊢ (𝜑 → 𝒫 ∪ dom 𝑂 = 𝒫 ∪ 𝒫 𝑋)
13	9, 12, 10	3eqtr4rd 2809	. . . 4 ⊢ (𝜑 → dom 𝑂 = 𝒫 ∪ dom 𝑂)
14		isomennd.o0	. . . 4 ⊢ (𝜑 → (𝑂‘∅) = 0)
15	6, 13, 14	jca31 523	. . 3 ⊢ (𝜑 → ((𝑂:dom 𝑂⟶(0[,]+∞) ∧ dom 𝑂 = 𝒫 ∪ dom 𝑂) ∧ (𝑂‘∅) = 0))
16		simpl 487	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝒫 ∪ dom 𝑂 ∧ 𝑦 ∈ 𝒫 𝑥)) → 𝜑)
17		simpr 489	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝒫 ∪ dom 𝑂) → 𝑥 ∈ 𝒫 ∪ dom 𝑂)
18	12, 9	eqtrd 2798	. . . . . . . . 9 ⊢ (𝜑 → 𝒫 ∪ dom 𝑂 = 𝒫 𝑋)
19	18	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝒫 ∪ dom 𝑂) → 𝒫 ∪ dom 𝑂 = 𝒫 𝑋)
20	17, 19	eleqtrd 2865	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝒫 ∪ dom 𝑂) → 𝑥 ∈ 𝒫 𝑋)
21		elpwi 4570	. . . . . . 7 ⊢ (𝑥 ∈ 𝒫 𝑋 → 𝑥 ⊆ 𝑋)
22	20, 21	syl 18	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝒫 ∪ dom 𝑂) → 𝑥 ⊆ 𝑋)
23	22	adantrr 729	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝒫 ∪ dom 𝑂 ∧ 𝑦 ∈ 𝒫 𝑥)) → 𝑥 ⊆ 𝑋)
24		elpwi 4570	. . . . . . 7 ⊢ (𝑦 ∈ 𝒫 𝑥 → 𝑦 ⊆ 𝑥)
25	24	adantl 486	. . . . . 6 ⊢ ((𝑥 ∈ 𝒫 ∪ dom 𝑂 ∧ 𝑦 ∈ 𝒫 𝑥) → 𝑦 ⊆ 𝑥)
26	25	adantl 486	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝒫 ∪ dom 𝑂 ∧ 𝑦 ∈ 𝒫 𝑥)) → 𝑦 ⊆ 𝑥)
27		isomennd.le	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝑋 ∧ 𝑦 ⊆ 𝑥) → (𝑂‘𝑦) ≤ (𝑂‘𝑥))
28	16, 23, 26, 27	syl3anc 1398	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝒫 ∪ dom 𝑂 ∧ 𝑦 ∈ 𝒫 𝑥)) → (𝑂‘𝑦) ≤ (𝑂‘𝑥))
29	28	ralrimivva 3208	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝒫 ∪ dom 𝑂∀𝑦 ∈ 𝒫 𝑥(𝑂‘𝑦) ≤ (𝑂‘𝑥))
30		0le0 12343	. . . . . . . . 9 ⊢ 0 ≤ 0
31	30	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 = ∅) → 0 ≤ 0)
32		unieq 4884	. . . . . . . . . . . . 13 ⊢ (𝑥 = ∅ → ∪ 𝑥 = ∪ ∅)
33		uni0 4902	. . . . . . . . . . . . . 14 ⊢ ∪ ∅ = ∅
34	33	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝑥 = ∅ → ∪ ∅ = ∅)
35	32, 34	eqtrd 2798	. . . . . . . . . . . 12 ⊢ (𝑥 = ∅ → ∪ 𝑥 = ∅)
36	35	fveq2d 6887	. . . . . . . . . . 11 ⊢ (𝑥 = ∅ → (𝑂‘∪ 𝑥) = (𝑂‘∅))
37	36	adantl 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 = ∅) → (𝑂‘∪ 𝑥) = (𝑂‘∅))
38	14	adantr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 = ∅) → (𝑂‘∅) = 0)
39	37, 38	eqtrd 2798	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 = ∅) → (𝑂‘∪ 𝑥) = 0)
40		reseq2 5975	. . . . . . . . . . . . 13 ⊢ (𝑥 = ∅ → (𝑂 ↾ 𝑥) = (𝑂 ↾ ∅))
41		res0 5984	. . . . . . . . . . . . . 14 ⊢ (𝑂 ↾ ∅) = ∅
42	41	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝑥 = ∅ → (𝑂 ↾ ∅) = ∅)
43	40, 42	eqtrd 2798	. . . . . . . . . . . 12 ⊢ (𝑥 = ∅ → (𝑂 ↾ 𝑥) = ∅)
44	43	fveq2d 6887	. . . . . . . . . . 11 ⊢ (𝑥 = ∅ → (Σ^{^}‘(𝑂 ↾ 𝑥)) = (Σ^{^}‘∅))
45		sge00 47070	. . . . . . . . . . . 12 ⊢ (Σ^{^}‘∅) = 0
46	45	a1i 11	. . . . . . . . . . 11 ⊢ (𝑥 = ∅ → (Σ^{^}‘∅) = 0)
47	44, 46	eqtrd 2798	. . . . . . . . . 10 ⊢ (𝑥 = ∅ → (Σ^{^}‘(𝑂 ↾ 𝑥)) = 0)
48	47	adantl 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 = ∅) → (Σ^{^}‘(𝑂 ↾ 𝑥)) = 0)
49	39, 48	breq12d 5123	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 = ∅) → ((𝑂‘∪ 𝑥) ≤ (Σ^{^}‘(𝑂 ↾ 𝑥)) ↔ 0 ≤ 0))
50	31, 49	mpbird 260	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = ∅) → (𝑂‘∪ 𝑥) ≤ (Σ^{^}‘(𝑂 ↾ 𝑥)))
51	50	ad4ant14 764	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑂) ∧ 𝑥 ≼ ω) ∧ 𝑥 = ∅) → (𝑂‘∪ 𝑥) ≤ (Σ^{^}‘(𝑂 ↾ 𝑥)))
52		simpl 487	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑂) ∧ 𝑥 ≼ ω) ∧ ¬ 𝑥 = ∅) → ((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑂) ∧ 𝑥 ≼ ω))
53		neqne 2966	. . . . . . . 8 ⊢ (¬ 𝑥 = ∅ → 𝑥 ≠ ∅)
54	53	adantl 486	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑂) ∧ 𝑥 ≼ ω) ∧ ¬ 𝑥 = ∅) → 𝑥 ≠ ∅)
55		ssnnf1octb 45892	. . . . . . . . 9 ⊢ ((𝑥 ≼ ω ∧ 𝑥 ≠ ∅) → ∃𝑓(dom 𝑓 ⊆ ℕ ∧ 𝑓:dom 𝑓–1-1-onto→𝑥))
56	55	adantll 726	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑂) ∧ 𝑥 ≼ ω) ∧ 𝑥 ≠ ∅) → ∃𝑓(dom 𝑓 ⊆ ℕ ∧ 𝑓:dom 𝑓–1-1-onto→𝑥))
57	1	ad2antrr 738	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑂) ∧ (dom 𝑓 ⊆ ℕ ∧ 𝑓:dom 𝑓–1-1-onto→𝑥)) → 𝑂:𝒫 𝑋⟶(0[,]+∞))
58	14	ad2antrr 738	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑂) ∧ (dom 𝑓 ⊆ ℕ ∧ 𝑓:dom 𝑓–1-1-onto→𝑥)) → (𝑂‘∅) = 0)
59		simpr 489	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑂) → 𝑥 ∈ 𝒫 dom 𝑂)
60	10	pweqd 4580	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝒫 dom 𝑂 = 𝒫 𝒫 𝑋)
61	60	adantr 485	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑂) → 𝒫 dom 𝑂 = 𝒫 𝒫 𝑋)
62	59, 61	eleqtrd 2865	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑂) → 𝑥 ∈ 𝒫 𝒫 𝑋)
63		elpwi 4570	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ 𝒫 𝒫 𝑋 → 𝑥 ⊆ 𝒫 𝑋)
64	62, 63	syl 18	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑂) → 𝑥 ⊆ 𝒫 𝑋)
65	64	adantr 485	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑂) ∧ (dom 𝑓 ⊆ ℕ ∧ 𝑓:dom 𝑓–1-1-onto→𝑥)) → 𝑥 ⊆ 𝒫 𝑋)
66		simpl 487	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑂) → 𝜑)
67		isomennd.sa	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑎:ℕ⟶𝒫 𝑋) → (𝑂‘∪ 𝑛 ∈ ℕ (𝑎‘𝑛)) ≤ (Σ^{^}‘(𝑛 ∈ ℕ ↦ (𝑂‘(𝑎‘𝑛)))))
68	66, 67	sylan 591	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑂) ∧ 𝑎:ℕ⟶𝒫 𝑋) → (𝑂‘∪ 𝑛 ∈ ℕ (𝑎‘𝑛)) ≤ (Σ^{^}‘(𝑛 ∈ ℕ ↦ (𝑂‘(𝑎‘𝑛)))))
69	68	adantlr 727	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑂) ∧ (dom 𝑓 ⊆ ℕ ∧ 𝑓:dom 𝑓–1-1-onto→𝑥)) ∧ 𝑎:ℕ⟶𝒫 𝑋) → (𝑂‘∪ 𝑛 ∈ ℕ (𝑎‘𝑛)) ≤ (Σ^{^}‘(𝑛 ∈ ℕ ↦ (𝑂‘(𝑎‘𝑛)))))
70		simprl 782	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑂) ∧ (dom 𝑓 ⊆ ℕ ∧ 𝑓:dom 𝑓–1-1-onto→𝑥)) → dom 𝑓 ⊆ ℕ)
71		simprr 784	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑂) ∧ (dom 𝑓 ⊆ ℕ ∧ 𝑓:dom 𝑓–1-1-onto→𝑥)) → 𝑓:dom 𝑓–1-1-onto→𝑥)
72		eleq1w 2846	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 𝑛 → (𝑚 ∈ dom 𝑓 ↔ 𝑛 ∈ dom 𝑓))
73		fveq2 6883	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 𝑛 → (𝑓‘𝑚) = (𝑓‘𝑛))
74	72, 73	ifbieq1d 4513	. . . . . . . . . . . . 13 ⊢ (𝑚 = 𝑛 → if(𝑚 ∈ dom 𝑓, (𝑓‘𝑚), ∅) = if(𝑛 ∈ dom 𝑓, (𝑓‘𝑛), ∅))
75	74	cbvmptv 5216	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ ℕ ↦ if(𝑚 ∈ dom 𝑓, (𝑓‘𝑚), ∅)) = (𝑛 ∈ ℕ ↦ if(𝑛 ∈ dom 𝑓, (𝑓‘𝑛), ∅))
76	57, 58, 65, 69, 70, 71, 75	isomenndlem 47224	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑂) ∧ (dom 𝑓 ⊆ ℕ ∧ 𝑓:dom 𝑓–1-1-onto→𝑥)) → (𝑂‘∪ 𝑥) ≤ (Σ^{^}‘(𝑂 ↾ 𝑥)))
77	76	ex 417	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑂) → ((dom 𝑓 ⊆ ℕ ∧ 𝑓:dom 𝑓–1-1-onto→𝑥) → (𝑂‘∪ 𝑥) ≤ (Σ^{^}‘(𝑂 ↾ 𝑥))))
78	77	ad2antrr 738	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑂) ∧ 𝑥 ≼ ω) ∧ 𝑥 ≠ ∅) → ((dom 𝑓 ⊆ ℕ ∧ 𝑓:dom 𝑓–1-1-onto→𝑥) → (𝑂‘∪ 𝑥) ≤ (Σ^{^}‘(𝑂 ↾ 𝑥))))
79	78	exlimdv 1963	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑂) ∧ 𝑥 ≼ ω) ∧ 𝑥 ≠ ∅) → (∃𝑓(dom 𝑓 ⊆ ℕ ∧ 𝑓:dom 𝑓–1-1-onto→𝑥) → (𝑂‘∪ 𝑥) ≤ (Σ^{^}‘(𝑂 ↾ 𝑥))))
80	56, 79	mpd 16	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑂) ∧ 𝑥 ≼ ω) ∧ 𝑥 ≠ ∅) → (𝑂‘∪ 𝑥) ≤ (Σ^{^}‘(𝑂 ↾ 𝑥)))
81	52, 54, 80	syl2anc 595	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑂) ∧ 𝑥 ≼ ω) ∧ ¬ 𝑥 = ∅) → (𝑂‘∪ 𝑥) ≤ (Σ^{^}‘(𝑂 ↾ 𝑥)))
82	51, 81	pm2.61dan 824	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑂) ∧ 𝑥 ≼ ω) → (𝑂‘∪ 𝑥) ≤ (Σ^{^}‘(𝑂 ↾ 𝑥)))
83	82	ex 417	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑂) → (𝑥 ≼ ω → (𝑂‘∪ 𝑥) ≤ (Σ^{^}‘(𝑂 ↾ 𝑥))))
84	83	ralrimiva 3157	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝒫 dom 𝑂(𝑥 ≼ ω → (𝑂‘∪ 𝑥) ≤ (Σ^{^}‘(𝑂 ↾ 𝑥))))
85	15, 29, 84	jca31 523	. 2 ⊢ (𝜑 → ((((𝑂:dom 𝑂⟶(0[,]+∞) ∧ dom 𝑂 = 𝒫 ∪ dom 𝑂) ∧ (𝑂‘∅) = 0) ∧ ∀𝑥 ∈ 𝒫 ∪ dom 𝑂∀𝑦 ∈ 𝒫 𝑥(𝑂‘𝑦) ≤ (𝑂‘𝑥)) ∧ ∀𝑥 ∈ 𝒫 dom 𝑂(𝑥 ≼ ω → (𝑂‘∪ 𝑥) ≤ (Σ^{^}‘(𝑂 ↾ 𝑥)))))
86		isomennd.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
87	86	pwexd 5352	. . . 4 ⊢ (𝜑 → 𝒫 𝑋 ∈ V)
88	1, 87	fexd 7227	. . 3 ⊢ (𝜑 → 𝑂 ∈ V)
89		isome 47188	. . 3 ⊢ (𝑂 ∈ V → (𝑂 ∈ OutMeas ↔ ((((𝑂:dom 𝑂⟶(0[,]+∞) ∧ dom 𝑂 = 𝒫 ∪ dom 𝑂) ∧ (𝑂‘∅) = 0) ∧ ∀𝑥 ∈ 𝒫 ∪ dom 𝑂∀𝑦 ∈ 𝒫 𝑥(𝑂‘𝑦) ≤ (𝑂‘𝑥)) ∧ ∀𝑥 ∈ 𝒫 dom 𝑂(𝑥 ≼ ω → (𝑂‘∪ 𝑥) ≤ (Σ^{^}‘(𝑂 ↾ 𝑥))))))
90	88, 89	syl 18	. 2 ⊢ (𝜑 → (𝑂 ∈ OutMeas ↔ ((((𝑂:dom 𝑂⟶(0[,]+∞) ∧ dom 𝑂 = 𝒫 ∪ dom 𝑂) ∧ (𝑂‘∅) = 0) ∧ ∀𝑥 ∈ 𝒫 ∪ dom 𝑂∀𝑦 ∈ 𝒫 𝑥(𝑂‘𝑦) ≤ (𝑂‘𝑥)) ∧ ∀𝑥 ∈ 𝒫 dom 𝑂(𝑥 ≼ ω → (𝑂‘∪ 𝑥) ≤ (Σ^{^}‘(𝑂 ↾ 𝑥))))))
91	85, 90	mpbird 260	1 ⊢ (𝜑 → 𝑂 ∈ OutMeas)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 400 ∧ w3a 1103 = wceq 1570 ∃wex 1809 ∈ wcel 2143 ≠ wne 2958 ∀wral 3079 Vcvv 3455 ⊆ wss 3906 ∅c0 4287 ifcif 4488 𝒫 cpw 4563 ∪ cuni 4873 ∪ ciun 4957 class class class wbr 5110 ↦ cmpt 5193 dom cdm 5663 ↾ cres 5665 ⟶wf 6534 –1-1-onto→wf1o 6537 ‘cfv 6538 (class class class)co 7412 ωcom 7863 ≼ cdom 8942 0cc0 11101 +∞cpnf 11241 ≤ cle 11245 ℕcn 12234 [,]cicc 13376 Σ^{^}csumge0 47056 OutMeascome 47183

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5239 ax-sep 5258 ax-nul 5270 ax-pow 5338 ax-pr 5406 ax-un 7734 ax-inf2 9611 ax-cnex 11157 ax-resscn 11158 ax-1cn 11159 ax-icn 11160 ax-addcl 11161 ax-addrcl 11162 ax-mulcl 11163 ax-mulrcl 11164 ax-mulcom 11165 ax-addass 11166 ax-mulass 11167 ax-distr 11168 ax-i2m1 11169 ax-1ne0 11170 ax-1rid 11171 ax-rnegex 11172 ax-rrecex 11173 ax-cnre 11174 ax-pre-lttri 11175 ax-pre-lttrn 11176 ax-pre-ltadd 11177 ax-pre-mulgt0 11178 ax-pre-sup 11179

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3746 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4288 df-if 4489 df-pw 4565 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4874 df-int 4914 df-iun 4959 df-br 5111 df-opab 5175 df-mpt 5194 df-tr 5220 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-se 5617 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-isom 6547 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7864 df-1st 7987 df-2nd 7988 df-frecs 8279 df-wrecs 8310 df-recs 8359 df-rdg 8398 df-1o 8454 df-er 8695 df-en 8945 df-dom 8946 df-sdom 8947 df-fin 8948 df-sup 9403 df-oi 9473 df-card 9926 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11444 df-neg 11445 df-div 11873 df-nn 12235 df-2 12304 df-3 12305 df-n0 12506 df-z 12593 df-uz 12864 df-rp 13018 df-xadd 13139 df-ico 13379 df-icc 13380 df-fz 13537 df-fzo 13685 df-seq 14040 df-exp 14100 df-hash 14369 df-cj 15152 df-re 15153 df-im 15154 df-sqrt 15288 df-abs 15289 df-clim 15541 df-sum 15740 df-sumge0 47057 df-ome 47184

This theorem is referenced by: ovnome 47267

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