isomennd - Mathbox for Glauco Siliprandi

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Theorem isomennd 46513

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 22	. . . . . 6 ⊢ (𝑂:𝒫 𝑋⟶(0[,]+∞) → 𝑂:𝒫 𝑋⟶(0[,]+∞))
3		fdm 6665	. . . . . . 7 ⊢ (𝑂:𝒫 𝑋⟶(0[,]+∞) → dom 𝑂 = 𝒫 𝑋)
4	3	feq2d 6640	. . . . . 6 ⊢ (𝑂:𝒫 𝑋⟶(0[,]+∞) → (𝑂:dom 𝑂⟶(0[,]+∞) ↔ 𝑂:𝒫 𝑋⟶(0[,]+∞)))
5	2, 4	mpbird 257	. . . . 5 ⊢ (𝑂:𝒫 𝑋⟶(0[,]+∞) → 𝑂:dom 𝑂⟶(0[,]+∞))
6	1, 5	syl 17	. . . 4 ⊢ (𝜑 → 𝑂:dom 𝑂⟶(0[,]+∞))
7		unipw 5397	. . . . . . 7 ⊢ ∪ 𝒫 𝑋 = 𝑋
8	7	pweqi 4569	. . . . . 6 ⊢ 𝒫 ∪ 𝒫 𝑋 = 𝒫 𝑋
9	8	a1i 11	. . . . 5 ⊢ (𝜑 → 𝒫 ∪ 𝒫 𝑋 = 𝒫 𝑋)
10	1, 3	syl 17	. . . . . . 7 ⊢ (𝜑 → dom 𝑂 = 𝒫 𝑋)
11	10	unieqd 4874	. . . . . 6 ⊢ (𝜑 → ∪ dom 𝑂 = ∪ 𝒫 𝑋)
12	11	pweqd 4570	. . . . 5 ⊢ (𝜑 → 𝒫 ∪ dom 𝑂 = 𝒫 ∪ 𝒫 𝑋)
13	9, 12, 10	3eqtr4rd 2775	. . . 4 ⊢ (𝜑 → dom 𝑂 = 𝒫 ∪ dom 𝑂)
14		isomennd.o0	. . . 4 ⊢ (𝜑 → (𝑂‘∅) = 0)
15	6, 13, 14	jca31 514	. . 3 ⊢ (𝜑 → ((𝑂:dom 𝑂⟶(0[,]+∞) ∧ dom 𝑂 = 𝒫 ∪ dom 𝑂) ∧ (𝑂‘∅) = 0))
16		simpl 482	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝒫 ∪ dom 𝑂 ∧ 𝑦 ∈ 𝒫 𝑥)) → 𝜑)
17		simpr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝒫 ∪ dom 𝑂) → 𝑥 ∈ 𝒫 ∪ dom 𝑂)
18	12, 9	eqtrd 2764	. . . . . . . . 9 ⊢ (𝜑 → 𝒫 ∪ dom 𝑂 = 𝒫 𝑋)
19	18	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝒫 ∪ dom 𝑂) → 𝒫 ∪ dom 𝑂 = 𝒫 𝑋)
20	17, 19	eleqtrd 2830	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝒫 ∪ dom 𝑂) → 𝑥 ∈ 𝒫 𝑋)
21		elpwi 4560	. . . . . . 7 ⊢ (𝑥 ∈ 𝒫 𝑋 → 𝑥 ⊆ 𝑋)
22	20, 21	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝒫 ∪ dom 𝑂) → 𝑥 ⊆ 𝑋)
23	22	adantrr 717	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝒫 ∪ dom 𝑂 ∧ 𝑦 ∈ 𝒫 𝑥)) → 𝑥 ⊆ 𝑋)
24		elpwi 4560	. . . . . . 7 ⊢ (𝑦 ∈ 𝒫 𝑥 → 𝑦 ⊆ 𝑥)
25	24	adantl 481	. . . . . 6 ⊢ ((𝑥 ∈ 𝒫 ∪ dom 𝑂 ∧ 𝑦 ∈ 𝒫 𝑥) → 𝑦 ⊆ 𝑥)
26	25	adantl 481	. . . . 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 1373	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝒫 ∪ dom 𝑂 ∧ 𝑦 ∈ 𝒫 𝑥)) → (𝑂‘𝑦) ≤ (𝑂‘𝑥))
29	28	ralrimivva 3172	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝒫 ∪ dom 𝑂∀𝑦 ∈ 𝒫 𝑥(𝑂‘𝑦) ≤ (𝑂‘𝑥))
30		0le0 12247	. . . . . . . . 9 ⊢ 0 ≤ 0
31	30	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 = ∅) → 0 ≤ 0)
32		unieq 4872	. . . . . . . . . . . . 13 ⊢ (𝑥 = ∅ → ∪ 𝑥 = ∪ ∅)
33		uni0 4889	. . . . . . . . . . . . . 14 ⊢ ∪ ∅ = ∅
34	33	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝑥 = ∅ → ∪ ∅ = ∅)
35	32, 34	eqtrd 2764	. . . . . . . . . . . 12 ⊢ (𝑥 = ∅ → ∪ 𝑥 = ∅)
36	35	fveq2d 6830	. . . . . . . . . . 11 ⊢ (𝑥 = ∅ → (𝑂‘∪ 𝑥) = (𝑂‘∅))
37	36	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 = ∅) → (𝑂‘∪ 𝑥) = (𝑂‘∅))
38	14	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 = ∅) → (𝑂‘∅) = 0)
39	37, 38	eqtrd 2764	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 = ∅) → (𝑂‘∪ 𝑥) = 0)
40		reseq2 5929	. . . . . . . . . . . . 13 ⊢ (𝑥 = ∅ → (𝑂 ↾ 𝑥) = (𝑂 ↾ ∅))
41		res0 5938	. . . . . . . . . . . . . 14 ⊢ (𝑂 ↾ ∅) = ∅
42	41	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝑥 = ∅ → (𝑂 ↾ ∅) = ∅)
43	40, 42	eqtrd 2764	. . . . . . . . . . . 12 ⊢ (𝑥 = ∅ → (𝑂 ↾ 𝑥) = ∅)
44	43	fveq2d 6830	. . . . . . . . . . 11 ⊢ (𝑥 = ∅ → (Σ^{^}‘(𝑂 ↾ 𝑥)) = (Σ^{^}‘∅))
45		sge00 46358	. . . . . . . . . . . 12 ⊢ (Σ^{^}‘∅) = 0
46	45	a1i 11	. . . . . . . . . . 11 ⊢ (𝑥 = ∅ → (Σ^{^}‘∅) = 0)
47	44, 46	eqtrd 2764	. . . . . . . . . 10 ⊢ (𝑥 = ∅ → (Σ^{^}‘(𝑂 ↾ 𝑥)) = 0)
48	47	adantl 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 = ∅) → (Σ^{^}‘(𝑂 ↾ 𝑥)) = 0)
49	39, 48	breq12d 5108	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 = ∅) → ((𝑂‘∪ 𝑥) ≤ (Σ^{^}‘(𝑂 ↾ 𝑥)) ↔ 0 ≤ 0))
50	31, 49	mpbird 257	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = ∅) → (𝑂‘∪ 𝑥) ≤ (Σ^{^}‘(𝑂 ↾ 𝑥)))
51	50	ad4ant14 752	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑂) ∧ 𝑥 ≼ ω) ∧ 𝑥 = ∅) → (𝑂‘∪ 𝑥) ≤ (Σ^{^}‘(𝑂 ↾ 𝑥)))
52		simpl 482	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑂) ∧ 𝑥 ≼ ω) ∧ ¬ 𝑥 = ∅) → ((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑂) ∧ 𝑥 ≼ ω))
53		neqne 2933	. . . . . . . 8 ⊢ (¬ 𝑥 = ∅ → 𝑥 ≠ ∅)
54	53	adantl 481	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑂) ∧ 𝑥 ≼ ω) ∧ ¬ 𝑥 = ∅) → 𝑥 ≠ ∅)
55		ssnnf1octb 45172	. . . . . . . . 9 ⊢ ((𝑥 ≼ ω ∧ 𝑥 ≠ ∅) → ∃𝑓(dom 𝑓 ⊆ ℕ ∧ 𝑓:dom 𝑓–1-1-onto→𝑥))
56	55	adantll 714	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑂) ∧ 𝑥 ≼ ω) ∧ 𝑥 ≠ ∅) → ∃𝑓(dom 𝑓 ⊆ ℕ ∧ 𝑓:dom 𝑓–1-1-onto→𝑥))
57	1	ad2antrr 726	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑂) ∧ (dom 𝑓 ⊆ ℕ ∧ 𝑓:dom 𝑓–1-1-onto→𝑥)) → 𝑂:𝒫 𝑋⟶(0[,]+∞))
58	14	ad2antrr 726	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑂) ∧ (dom 𝑓 ⊆ ℕ ∧ 𝑓:dom 𝑓–1-1-onto→𝑥)) → (𝑂‘∅) = 0)
59		simpr 484	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑂) → 𝑥 ∈ 𝒫 dom 𝑂)
60	10	pweqd 4570	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝒫 dom 𝑂 = 𝒫 𝒫 𝑋)
61	60	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑂) → 𝒫 dom 𝑂 = 𝒫 𝒫 𝑋)
62	59, 61	eleqtrd 2830	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑂) → 𝑥 ∈ 𝒫 𝒫 𝑋)
63		elpwi 4560	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ 𝒫 𝒫 𝑋 → 𝑥 ⊆ 𝒫 𝑋)
64	62, 63	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑂) → 𝑥 ⊆ 𝒫 𝑋)
65	64	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑂) ∧ (dom 𝑓 ⊆ ℕ ∧ 𝑓:dom 𝑓–1-1-onto→𝑥)) → 𝑥 ⊆ 𝒫 𝑋)
66		simpl 482	. . . . . . . . . . . . . 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 580	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑂) ∧ 𝑎:ℕ⟶𝒫 𝑋) → (𝑂‘∪ 𝑛 ∈ ℕ (𝑎‘𝑛)) ≤ (Σ^{^}‘(𝑛 ∈ ℕ ↦ (𝑂‘(𝑎‘𝑛)))))
69	68	adantlr 715	. . . . . . . . . . . 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 770	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑂) ∧ (dom 𝑓 ⊆ ℕ ∧ 𝑓:dom 𝑓–1-1-onto→𝑥)) → dom 𝑓 ⊆ ℕ)
71		simprr 772	. . . . . . . . . . . 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 2811	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 𝑛 → (𝑚 ∈ dom 𝑓 ↔ 𝑛 ∈ dom 𝑓))
73		fveq2 6826	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 𝑛 → (𝑓‘𝑚) = (𝑓‘𝑛))
74	72, 73	ifbieq1d 4503	. . . . . . . . . . . . 13 ⊢ (𝑚 = 𝑛 → if(𝑚 ∈ dom 𝑓, (𝑓‘𝑚), ∅) = if(𝑛 ∈ dom 𝑓, (𝑓‘𝑛), ∅))
75	74	cbvmptv 5199	. . . . . . . . . . . 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 46512	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑂) ∧ (dom 𝑓 ⊆ ℕ ∧ 𝑓:dom 𝑓–1-1-onto→𝑥)) → (𝑂‘∪ 𝑥) ≤ (Σ^{^}‘(𝑂 ↾ 𝑥)))
77	76	ex 412	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑂) → ((dom 𝑓 ⊆ ℕ ∧ 𝑓:dom 𝑓–1-1-onto→𝑥) → (𝑂‘∪ 𝑥) ≤ (Σ^{^}‘(𝑂 ↾ 𝑥))))
78	77	ad2antrr 726	. . . . . . . . 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 1933	. . . . . . . 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 15	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑂) ∧ 𝑥 ≼ ω) ∧ 𝑥 ≠ ∅) → (𝑂‘∪ 𝑥) ≤ (Σ^{^}‘(𝑂 ↾ 𝑥)))
81	52, 54, 80	syl2anc 584	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑂) ∧ 𝑥 ≼ ω) ∧ ¬ 𝑥 = ∅) → (𝑂‘∪ 𝑥) ≤ (Σ^{^}‘(𝑂 ↾ 𝑥)))
82	51, 81	pm2.61dan 812	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑂) ∧ 𝑥 ≼ ω) → (𝑂‘∪ 𝑥) ≤ (Σ^{^}‘(𝑂 ↾ 𝑥)))
83	82	ex 412	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑂) → (𝑥 ≼ ω → (𝑂‘∪ 𝑥) ≤ (Σ^{^}‘(𝑂 ↾ 𝑥))))
84	83	ralrimiva 3121	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝒫 dom 𝑂(𝑥 ≼ ω → (𝑂‘∪ 𝑥) ≤ (Σ^{^}‘(𝑂 ↾ 𝑥))))
85	15, 29, 84	jca31 514	. 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 5321	. . . 4 ⊢ (𝜑 → 𝒫 𝑋 ∈ V)
88	1, 87	fexd 7167	. . 3 ⊢ (𝜑 → 𝑂 ∈ V)
89		isome 46476	. . 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 17	. 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 257	1 ⊢ (𝜑 → 𝑂 ∈ OutMeas)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∃wex 1779 ∈ wcel 2109 ≠ wne 2925 ∀wral 3044 Vcvv 3438 ⊆ wss 3905 ∅c0 4286 ifcif 4478 𝒫 cpw 4553 ∪ cuni 4861 ∪ ciun 4944 class class class wbr 5095 ↦ cmpt 5176 dom cdm 5623 ↾ cres 5625 ⟶wf 6482 –1-1-onto→wf1o 6485 ‘cfv 6486 (class class class)co 7353 ωcom 7806 ≼ cdom 8877 0cc0 11028 +∞cpnf 11165 ≤ cle 11169 ℕcn 12146 [,]cicc 13269 Σ^{^}csumge0 46344 OutMeascome 46471

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5221 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7675 ax-inf2 9556 ax-cnex 11084 ax-resscn 11085 ax-1cn 11086 ax-icn 11087 ax-addcl 11088 ax-addrcl 11089 ax-mulcl 11090 ax-mulrcl 11091 ax-mulcom 11092 ax-addass 11093 ax-mulass 11094 ax-distr 11095 ax-i2m1 11096 ax-1ne0 11097 ax-1rid 11098 ax-rnegex 11099 ax-rrecex 11100 ax-cnre 11101 ax-pre-lttri 11102 ax-pre-lttrn 11103 ax-pre-ltadd 11104 ax-pre-mulgt0 11105 ax-pre-sup 11106

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3345 df-reu 3346 df-rab 3397 df-v 3440 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4479 df-pw 4555 df-sn 4580 df-pr 4582 df-op 4586 df-uni 4862 df-int 4900 df-iun 4946 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 df-id 5518 df-eprel 5523 df-po 5531 df-so 5532 df-fr 5576 df-se 5577 df-we 5578 df-xp 5629 df-rel 5630 df-cnv 5631 df-co 5632 df-dm 5633 df-rn 5634 df-res 5635 df-ima 5636 df-pred 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-isom 6495 df-riota 7310 df-ov 7356 df-oprab 7357 df-mpo 7358 df-om 7807 df-1st 7931 df-2nd 7932 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-er 8632 df-en 8880 df-dom 8881 df-sdom 8882 df-fin 8883 df-sup 9351 df-oi 9421 df-card 9854 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11367 df-neg 11368 df-div 11796 df-nn 12147 df-2 12209 df-3 12210 df-n0 12403 df-z 12490 df-uz 12754 df-rp 12912 df-xadd 13033 df-ico 13272 df-icc 13273 df-fz 13429 df-fzo 13576 df-seq 13927 df-exp 13987 df-hash 14256 df-cj 15024 df-re 15025 df-im 15026 df-sqrt 15160 df-abs 15161 df-clim 15413 df-sum 15612 df-sumge0 46345 df-ome 46472

This theorem is referenced by: ovnome 46555

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