ismeannd - Mathbox for Glauco Siliprandi

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Theorem ismeannd 46825

Description: Sufficient condition to prove that 𝑀 is a measure. (Contributed by Glauco Siliprandi, 17-Aug-2020.)

**Hypotheses**
Ref	Expression
ismeannd.sal	⊢ (𝜑 → 𝑆 ∈ SAlg)
ismeannd.mf	⊢ (𝜑 → 𝑀:𝑆⟶(0[,]+∞))
ismeannd.m0	⊢ (𝜑 → (𝑀‘∅) = 0)
ismeannd.iun	⊢ ((𝜑 ∧ 𝑒:ℕ⟶𝑆 ∧ Disj 𝑛 ∈ ℕ (𝑒‘𝑛)) → (𝑀‘∪ 𝑛 ∈ ℕ (𝑒‘𝑛)) = (Σ^{^}‘(𝑛 ∈ ℕ ↦ (𝑀‘(𝑒‘𝑛)))))

**Assertion**
Ref	Expression
ismeannd	⊢ (𝜑 → 𝑀 ∈ Meas)

Distinct variable groups: 𝑒,𝑀,𝑛 𝜑,𝑒,𝑛 Allowed substitution hints: 𝑆(𝑒,𝑛)

Proof of Theorem ismeannd Dummy variables 𝑥 𝑦 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ismeannd.mf	. . . . 5 ⊢ (𝜑 → 𝑀:𝑆⟶(0[,]+∞))
2	1	fdmd 6680	. . . . . 6 ⊢ (𝜑 → dom 𝑀 = 𝑆)
3	2	feq2d 6654	. . . . 5 ⊢ (𝜑 → (𝑀:dom 𝑀⟶(0[,]+∞) ↔ 𝑀:𝑆⟶(0[,]+∞)))
4	1, 3	mpbird 257	. . . 4 ⊢ (𝜑 → 𝑀:dom 𝑀⟶(0[,]+∞))
5		ismeannd.sal	. . . . 5 ⊢ (𝜑 → 𝑆 ∈ SAlg)
6	2, 5	eqeltrd 2837	. . . 4 ⊢ (𝜑 → dom 𝑀 ∈ SAlg)
7	4, 6	jca 511	. . 3 ⊢ (𝜑 → (𝑀:dom 𝑀⟶(0[,]+∞) ∧ dom 𝑀 ∈ SAlg))
8		ismeannd.m0	. . 3 ⊢ (𝜑 → (𝑀‘∅) = 0)
9		unieq 4876	. . . . . . . . . . . 12 ⊢ (𝑥 = ∅ → ∪ 𝑥 = ∪ ∅)
10		uni0 4893	. . . . . . . . . . . . 13 ⊢ ∪ ∅ = ∅
11	10	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑥 = ∅ → ∪ ∅ = ∅)
12	9, 11	eqtrd 2772	. . . . . . . . . . 11 ⊢ (𝑥 = ∅ → ∪ 𝑥 = ∅)
13	12	fveq2d 6846	. . . . . . . . . 10 ⊢ (𝑥 = ∅ → (𝑀‘∪ 𝑥) = (𝑀‘∅))
14	13, 8	sylan9eqr 2794	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 = ∅) → (𝑀‘∪ 𝑥) = 0)
15		reseq2 5941	. . . . . . . . . . . . 13 ⊢ (𝑥 = ∅ → (𝑀 ↾ 𝑥) = (𝑀 ↾ ∅))
16		res0 5950	. . . . . . . . . . . . . 14 ⊢ (𝑀 ↾ ∅) = ∅
17	16	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝑥 = ∅ → (𝑀 ↾ ∅) = ∅)
18	15, 17	eqtrd 2772	. . . . . . . . . . . 12 ⊢ (𝑥 = ∅ → (𝑀 ↾ 𝑥) = ∅)
19	18	fveq2d 6846	. . . . . . . . . . 11 ⊢ (𝑥 = ∅ → (Σ^{^}‘(𝑀 ↾ 𝑥)) = (Σ^{^}‘∅))
20	19	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 = ∅) → (Σ^{^}‘(𝑀 ↾ 𝑥)) = (Σ^{^}‘∅))
21		sge00 46734	. . . . . . . . . . 11 ⊢ (Σ^{^}‘∅) = 0
22	21	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 = ∅) → (Σ^{^}‘∅) = 0)
23	20, 22	eqtrd 2772	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 = ∅) → (Σ^{^}‘(𝑀 ↾ 𝑥)) = 0)
24	14, 23	eqtr4d 2775	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 = ∅) → (𝑀‘∪ 𝑥) = (Σ^{^}‘(𝑀 ↾ 𝑥)))
25	24	adantlr 716	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ 𝑥 = ∅) → (𝑀‘∪ 𝑥) = (Σ^{^}‘(𝑀 ↾ 𝑥)))
26	25	adantlr 716	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ (𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦)) ∧ 𝑥 = ∅) → (𝑀‘∪ 𝑥) = (Σ^{^}‘(𝑀 ↾ 𝑥)))
27		simpll 767	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ (𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦)) ∧ ¬ 𝑥 = ∅) → (𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀))
28		simplrr 778	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ (𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦)) ∧ ¬ 𝑥 = ∅) → Disj 𝑦 ∈ 𝑥 𝑦)
29	27, 28	jca 511	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ (𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦)) ∧ ¬ 𝑥 = ∅) → ((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ Disj 𝑦 ∈ 𝑥 𝑦))
30		simplrl 777	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ (𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦)) ∧ ¬ 𝑥 = ∅) → 𝑥 ≼ ω)
31		neqne 2941	. . . . . . . . 9 ⊢ (¬ 𝑥 = ∅ → 𝑥 ≠ ∅)
32	31	adantl 481	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ (𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦)) ∧ ¬ 𝑥 = ∅) → 𝑥 ≠ ∅)
33		id 22	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑤 → 𝑦 = 𝑤)
34	33	cbvdisjv 5078	. . . . . . . . . . 11 ⊢ (Disj 𝑦 ∈ 𝑥 𝑦 ↔ Disj 𝑤 ∈ 𝑥 𝑤)
35	34	biimpi 216	. . . . . . . . . 10 ⊢ (Disj 𝑦 ∈ 𝑥 𝑦 → Disj 𝑤 ∈ 𝑥 𝑤)
36	35	adantl 481	. . . . . . . . 9 ⊢ ((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → Disj 𝑤 ∈ 𝑥 𝑤)
37	36	ad2antlr 728	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ (𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦)) ∧ ¬ 𝑥 = ∅) → Disj 𝑤 ∈ 𝑥 𝑤)
38	30, 32, 37	nnfoctbdj 46814	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ (𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦)) ∧ ¬ 𝑥 = ∅) → ∃𝑒(𝑒:ℕ–onto→(𝑥 ∪ {∅}) ∧ Disj 𝑛 ∈ ℕ (𝑒‘𝑛)))
39		simpl 482	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ Disj 𝑦 ∈ 𝑥 𝑦) ∧ (𝑒:ℕ–onto→(𝑥 ∪ {∅}) ∧ Disj 𝑛 ∈ ℕ (𝑒‘𝑛))) → ((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ Disj 𝑦 ∈ 𝑥 𝑦))
40		simprl 771	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ Disj 𝑦 ∈ 𝑥 𝑦) ∧ (𝑒:ℕ–onto→(𝑥 ∪ {∅}) ∧ Disj 𝑛 ∈ ℕ (𝑒‘𝑛))) → 𝑒:ℕ–onto→(𝑥 ∪ {∅}))
41		simprr 773	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ Disj 𝑦 ∈ 𝑥 𝑦) ∧ (𝑒:ℕ–onto→(𝑥 ∪ {∅}) ∧ Disj 𝑛 ∈ ℕ (𝑒‘𝑛))) → Disj 𝑛 ∈ ℕ (𝑒‘𝑛))
42		founiiun0 45549	. . . . . . . . . . . . 13 ⊢ (𝑒:ℕ–onto→(𝑥 ∪ {∅}) → ∪ 𝑥 = ∪ 𝑛 ∈ ℕ (𝑒‘𝑛))
43	42	fveq2d 6846	. . . . . . . . . . . 12 ⊢ (𝑒:ℕ–onto→(𝑥 ∪ {∅}) → (𝑀‘∪ 𝑥) = (𝑀‘∪ 𝑛 ∈ ℕ (𝑒‘𝑛)))
44	43	ad2antlr 728	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ Disj 𝑦 ∈ 𝑥 𝑦) ∧ 𝑒:ℕ–onto→(𝑥 ∪ {∅})) ∧ Disj 𝑛 ∈ ℕ (𝑒‘𝑛)) → (𝑀‘∪ 𝑥) = (𝑀‘∪ 𝑛 ∈ ℕ (𝑒‘𝑛)))
45		simplll 775	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ 𝑒:ℕ–onto→(𝑥 ∪ {∅})) ∧ Disj 𝑛 ∈ ℕ (𝑒‘𝑛)) → 𝜑)
46		fof 6754	. . . . . . . . . . . . . . . 16 ⊢ (𝑒:ℕ–onto→(𝑥 ∪ {∅}) → 𝑒:ℕ⟶(𝑥 ∪ {∅}))
47	46	adantl 481	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ 𝑒:ℕ–onto→(𝑥 ∪ {∅})) → 𝑒:ℕ⟶(𝑥 ∪ {∅}))
48		elpwi 4563	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ 𝒫 dom 𝑀 → 𝑥 ⊆ dom 𝑀)
49	48	adantl 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) → 𝑥 ⊆ dom 𝑀)
50	2	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) → dom 𝑀 = 𝑆)
51	49, 50	sseqtrd 3972	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) → 𝑥 ⊆ 𝑆)
52		0sal 46678	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑆 ∈ SAlg → ∅ ∈ 𝑆)
53	5, 52	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ∅ ∈ 𝑆)
54		snssi 4766	. . . . . . . . . . . . . . . . . . 19 ⊢ (∅ ∈ 𝑆 → {∅} ⊆ 𝑆)
55	53, 54	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → {∅} ⊆ 𝑆)
56	55	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) → {∅} ⊆ 𝑆)
57	51, 56	unssd 4146	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) → (𝑥 ∪ {∅}) ⊆ 𝑆)
58	57	adantr 480	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ 𝑒:ℕ–onto→(𝑥 ∪ {∅})) → (𝑥 ∪ {∅}) ⊆ 𝑆)
59	47, 58	fssd 6687	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ 𝑒:ℕ–onto→(𝑥 ∪ {∅})) → 𝑒:ℕ⟶𝑆)
60	59	adantr 480	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ 𝑒:ℕ–onto→(𝑥 ∪ {∅})) ∧ Disj 𝑛 ∈ ℕ (𝑒‘𝑛)) → 𝑒:ℕ⟶𝑆)
61		simpr 484	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ 𝑒:ℕ–onto→(𝑥 ∪ {∅})) ∧ Disj 𝑛 ∈ ℕ (𝑒‘𝑛)) → Disj 𝑛 ∈ ℕ (𝑒‘𝑛))
62		ismeannd.iun	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑒:ℕ⟶𝑆 ∧ Disj 𝑛 ∈ ℕ (𝑒‘𝑛)) → (𝑀‘∪ 𝑛 ∈ ℕ (𝑒‘𝑛)) = (Σ^{^}‘(𝑛 ∈ ℕ ↦ (𝑀‘(𝑒‘𝑛)))))
63	45, 60, 61, 62	syl3anc 1374	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ 𝑒:ℕ–onto→(𝑥 ∪ {∅})) ∧ Disj 𝑛 ∈ ℕ (𝑒‘𝑛)) → (𝑀‘∪ 𝑛 ∈ ℕ (𝑒‘𝑛)) = (Σ^{^}‘(𝑛 ∈ ℕ ↦ (𝑀‘(𝑒‘𝑛)))))
64	63	adantllr 720	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ Disj 𝑦 ∈ 𝑥 𝑦) ∧ 𝑒:ℕ–onto→(𝑥 ∪ {∅})) ∧ Disj 𝑛 ∈ ℕ (𝑒‘𝑛)) → (𝑀‘∪ 𝑛 ∈ ℕ (𝑒‘𝑛)) = (Σ^{^}‘(𝑛 ∈ ℕ ↦ (𝑀‘(𝑒‘𝑛)))))
65	1	feqmptd 6910	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝑀 = (𝑦 ∈ 𝑆 ↦ (𝑀‘𝑦)))
66	65	reseq1d 5945	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝑀 ↾ 𝑥) = ((𝑦 ∈ 𝑆 ↦ (𝑀‘𝑦)) ↾ 𝑥))
67	66	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) → (𝑀 ↾ 𝑥) = ((𝑦 ∈ 𝑆 ↦ (𝑀‘𝑦)) ↾ 𝑥))
68	67	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ ∅ ∈ 𝑥) → (𝑀 ↾ 𝑥) = ((𝑦 ∈ 𝑆 ↦ (𝑀‘𝑦)) ↾ 𝑥))
69	51	resmptd 6007	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) → ((𝑦 ∈ 𝑆 ↦ (𝑀‘𝑦)) ↾ 𝑥) = (𝑦 ∈ 𝑥 ↦ (𝑀‘𝑦)))
70	69	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ ∅ ∈ 𝑥) → ((𝑦 ∈ 𝑆 ↦ (𝑀‘𝑦)) ↾ 𝑥) = (𝑦 ∈ 𝑥 ↦ (𝑀‘𝑦)))
71		snssi 4766	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (∅ ∈ 𝑥 → {∅} ⊆ 𝑥)
72		ssequn2 4143	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ({∅} ⊆ 𝑥 ↔ (𝑥 ∪ {∅}) = 𝑥)
73	71, 72	sylib 218	. . . . . . . . . . . . . . . . . . . 20 ⊢ (∅ ∈ 𝑥 → (𝑥 ∪ {∅}) = 𝑥)
74	73	eqcomd 2743	. . . . . . . . . . . . . . . . . . 19 ⊢ (∅ ∈ 𝑥 → 𝑥 = (𝑥 ∪ {∅}))
75	74	mpteq1d 5190	. . . . . . . . . . . . . . . . . 18 ⊢ (∅ ∈ 𝑥 → (𝑦 ∈ 𝑥 ↦ (𝑀‘𝑦)) = (𝑦 ∈ (𝑥 ∪ {∅}) ↦ (𝑀‘𝑦)))
76	75	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ ∅ ∈ 𝑥) → (𝑦 ∈ 𝑥 ↦ (𝑀‘𝑦)) = (𝑦 ∈ (𝑥 ∪ {∅}) ↦ (𝑀‘𝑦)))
77	68, 70, 76	3eqtrd 2776	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ ∅ ∈ 𝑥) → (𝑀 ↾ 𝑥) = (𝑦 ∈ (𝑥 ∪ {∅}) ↦ (𝑀‘𝑦)))
78	77	fveq2d 6846	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ ∅ ∈ 𝑥) → (Σ^{^}‘(𝑀 ↾ 𝑥)) = (Σ^{^}‘(𝑦 ∈ (𝑥 ∪ {∅}) ↦ (𝑀‘𝑦))))
79		nfv 1916	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑦((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ ¬ ∅ ∈ 𝑥)
80		simplr 769	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ ¬ ∅ ∈ 𝑥) → 𝑥 ∈ 𝒫 dom 𝑀)
81		p0ex 5331	. . . . . . . . . . . . . . . . . 18 ⊢ {∅} ∈ V
82	81	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ ¬ ∅ ∈ 𝑥) → {∅} ∈ V)
83		disjsn 4670	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∩ {∅}) = ∅ ↔ ¬ ∅ ∈ 𝑥)
84	83	biimpri 228	. . . . . . . . . . . . . . . . . 18 ⊢ (¬ ∅ ∈ 𝑥 → (𝑥 ∩ {∅}) = ∅)
85	84	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ ¬ ∅ ∈ 𝑥) → (𝑥 ∩ {∅}) = ∅)
86	1	ad2antrr 727	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ 𝑦 ∈ 𝑥) → 𝑀:𝑆⟶(0[,]+∞))
87	51	sselda 3935	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ 𝑆)
88	86, 87	ffvelcdmd 7039	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ 𝑦 ∈ 𝑥) → (𝑀‘𝑦) ∈ (0[,]+∞))
89	88	adantlr 716	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ ¬ ∅ ∈ 𝑥) ∧ 𝑦 ∈ 𝑥) → (𝑀‘𝑦) ∈ (0[,]+∞))
90		elsni 4599	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 ∈ {∅} → 𝑦 = ∅)
91	90	fveq2d 6846	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 ∈ {∅} → (𝑀‘𝑦) = (𝑀‘∅))
92	91	adantl 481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑦 ∈ {∅}) → (𝑀‘𝑦) = (𝑀‘∅))
93	1, 53	ffvelcdmd 7039	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝑀‘∅) ∈ (0[,]+∞))
94	93	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑦 ∈ {∅}) → (𝑀‘∅) ∈ (0[,]+∞))
95	92, 94	eqeltrd 2837	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑦 ∈ {∅}) → (𝑀‘𝑦) ∈ (0[,]+∞))
96	95	ad4ant14 753	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ ¬ ∅ ∈ 𝑥) ∧ 𝑦 ∈ {∅}) → (𝑀‘𝑦) ∈ (0[,]+∞))
97	79, 80, 82, 85, 89, 96	sge0splitmpt 46769	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ ¬ ∅ ∈ 𝑥) → (Σ^{^}‘(𝑦 ∈ (𝑥 ∪ {∅}) ↦ (𝑀‘𝑦))) = ((Σ^{^}‘(𝑦 ∈ 𝑥 ↦ (𝑀‘𝑦))) +_𝑒 (Σ^{^}‘(𝑦 ∈ {∅} ↦ (𝑀‘𝑦)))))
98		fveq2 6842	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 = ∅ → (𝑀‘𝑦) = (𝑀‘∅))
99	98	adantl 481	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑦 = ∅) → (𝑀‘𝑦) = (𝑀‘∅))
100	8	adantr 480	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑦 = ∅) → (𝑀‘∅) = 0)
101	99, 100	eqtrd 2772	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑦 = ∅) → (𝑀‘𝑦) = 0)
102	90, 101	sylan2 594	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑦 ∈ {∅}) → (𝑀‘𝑦) = 0)
103	102	mpteq2dva 5193	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝑦 ∈ {∅} ↦ (𝑀‘𝑦)) = (𝑦 ∈ {∅} ↦ 0))
104	103	fveq2d 6846	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (Σ^{^}‘(𝑦 ∈ {∅} ↦ (𝑀‘𝑦))) = (Σ^{^}‘(𝑦 ∈ {∅} ↦ 0)))
105		nfv 1916	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑦𝜑
106	81	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → {∅} ∈ V)
107	105, 106	sge0z 46733	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (Σ^{^}‘(𝑦 ∈ {∅} ↦ 0)) = 0)
108	104, 107	eqtrd 2772	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (Σ^{^}‘(𝑦 ∈ {∅} ↦ (𝑀‘𝑦))) = 0)
109	108	oveq2d 7384	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((Σ^{^}‘(𝑦 ∈ 𝑥 ↦ (𝑀‘𝑦))) +_𝑒 (Σ^{^}‘(𝑦 ∈ {∅} ↦ (𝑀‘𝑦)))) = ((Σ^{^}‘(𝑦 ∈ 𝑥 ↦ (𝑀‘𝑦))) +_𝑒 0))
110	109	ad2antrr 727	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ ¬ ∅ ∈ 𝑥) → ((Σ^{^}‘(𝑦 ∈ 𝑥 ↦ (𝑀‘𝑦))) +_𝑒 (Σ^{^}‘(𝑦 ∈ {∅} ↦ (𝑀‘𝑦)))) = ((Σ^{^}‘(𝑦 ∈ 𝑥 ↦ (𝑀‘𝑦))) +_𝑒 0))
111		simpr 484	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) → 𝑥 ∈ 𝒫 dom 𝑀)
112	67, 69	eqtrd 2772	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) → (𝑀 ↾ 𝑥) = (𝑦 ∈ 𝑥 ↦ (𝑀‘𝑦)))
113	1	adantr 480	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) → 𝑀:𝑆⟶(0[,]+∞))
114	113, 51	fssresd 6709	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) → (𝑀 ↾ 𝑥):𝑥⟶(0[,]+∞))
115	112, 114	feq1dd 6653	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) → (𝑦 ∈ 𝑥 ↦ (𝑀‘𝑦)):𝑥⟶(0[,]+∞))
116	111, 115	sge0xrcl 46743	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) → (Σ^{^}‘(𝑦 ∈ 𝑥 ↦ (𝑀‘𝑦))) ∈ ℝ^*)
117	116	xaddridd 13170	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) → ((Σ^{^}‘(𝑦 ∈ 𝑥 ↦ (𝑀‘𝑦))) +_𝑒 0) = (Σ^{^}‘(𝑦 ∈ 𝑥 ↦ (𝑀‘𝑦))))
118	112	fveq2d 6846	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) → (Σ^{^}‘(𝑀 ↾ 𝑥)) = (Σ^{^}‘(𝑦 ∈ 𝑥 ↦ (𝑀‘𝑦))))
119	118	eqcomd 2743	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) → (Σ^{^}‘(𝑦 ∈ 𝑥 ↦ (𝑀‘𝑦))) = (Σ^{^}‘(𝑀 ↾ 𝑥)))
120	117, 119	eqtrd 2772	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) → ((Σ^{^}‘(𝑦 ∈ 𝑥 ↦ (𝑀‘𝑦))) +_𝑒 0) = (Σ^{^}‘(𝑀 ↾ 𝑥)))
121	120	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ ¬ ∅ ∈ 𝑥) → ((Σ^{^}‘(𝑦 ∈ 𝑥 ↦ (𝑀‘𝑦))) +_𝑒 0) = (Σ^{^}‘(𝑀 ↾ 𝑥)))
122	97, 110, 121	3eqtrrd 2777	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ ¬ ∅ ∈ 𝑥) → (Σ^{^}‘(𝑀 ↾ 𝑥)) = (Σ^{^}‘(𝑦 ∈ (𝑥 ∪ {∅}) ↦ (𝑀‘𝑦))))
123	78, 122	pm2.61dan 813	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) → (Σ^{^}‘(𝑀 ↾ 𝑥)) = (Σ^{^}‘(𝑦 ∈ (𝑥 ∪ {∅}) ↦ (𝑀‘𝑦))))
124	123	ad2antrr 727	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ 𝑒:ℕ–onto→(𝑥 ∪ {∅})) ∧ Disj 𝑛 ∈ ℕ (𝑒‘𝑛)) → (Σ^{^}‘(𝑀 ↾ 𝑥)) = (Σ^{^}‘(𝑦 ∈ (𝑥 ∪ {∅}) ↦ (𝑀‘𝑦))))
125		nfv 1916	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑦(((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ 𝑒:ℕ–onto→(𝑥 ∪ {∅})) ∧ Disj 𝑛 ∈ ℕ (𝑒‘𝑛))
126		nfv 1916	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑛((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ 𝑒:ℕ–onto→(𝑥 ∪ {∅}))
127		nfdisj1 5081	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑛Disj 𝑛 ∈ ℕ (𝑒‘𝑛)
128	126, 127	nfan 1901	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑛(((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ 𝑒:ℕ–onto→(𝑥 ∪ {∅})) ∧ Disj 𝑛 ∈ ℕ (𝑒‘𝑛))
129		fveq2 6842	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝑒‘𝑛) → (𝑀‘𝑦) = (𝑀‘(𝑒‘𝑛)))
130		nnex 12163	. . . . . . . . . . . . . . 15 ⊢ ℕ ∈ V
131	130	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ 𝑒:ℕ–onto→(𝑥 ∪ {∅})) ∧ Disj 𝑛 ∈ ℕ (𝑒‘𝑛)) → ℕ ∈ V)
132		simplr 769	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ 𝑒:ℕ–onto→(𝑥 ∪ {∅})) ∧ Disj 𝑛 ∈ ℕ (𝑒‘𝑛)) → 𝑒:ℕ–onto→(𝑥 ∪ {∅}))
133		eqidd 2738	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ 𝑒:ℕ–onto→(𝑥 ∪ {∅})) ∧ Disj 𝑛 ∈ ℕ (𝑒‘𝑛)) ∧ 𝑛 ∈ ℕ) → (𝑒‘𝑛) = (𝑒‘𝑛))
134	1	ad2antrr 727	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ 𝑦 ∈ (𝑥 ∪ {∅})) → 𝑀:𝑆⟶(0[,]+∞))
135	57	sselda 3935	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ 𝑦 ∈ (𝑥 ∪ {∅})) → 𝑦 ∈ 𝑆)
136	134, 135	ffvelcdmd 7039	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ 𝑦 ∈ (𝑥 ∪ {∅})) → (𝑀‘𝑦) ∈ (0[,]+∞))
137	136	ad4ant14 753	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ 𝑒:ℕ–onto→(𝑥 ∪ {∅})) ∧ Disj 𝑛 ∈ ℕ (𝑒‘𝑛)) ∧ 𝑦 ∈ (𝑥 ∪ {∅})) → (𝑀‘𝑦) ∈ (0[,]+∞))
138	45, 101	sylan 581	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ 𝑒:ℕ–onto→(𝑥 ∪ {∅})) ∧ Disj 𝑛 ∈ ℕ (𝑒‘𝑛)) ∧ 𝑦 = ∅) → (𝑀‘𝑦) = 0)
139	125, 128, 129, 131, 132, 61, 133, 137, 138	sge0fodjrn 46775	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ 𝑒:ℕ–onto→(𝑥 ∪ {∅})) ∧ Disj 𝑛 ∈ ℕ (𝑒‘𝑛)) → (Σ^{^}‘(𝑦 ∈ (𝑥 ∪ {∅}) ↦ (𝑀‘𝑦))) = (Σ^{^}‘(𝑛 ∈ ℕ ↦ (𝑀‘(𝑒‘𝑛)))))
140	124, 139	eqtr2d 2773	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ 𝑒:ℕ–onto→(𝑥 ∪ {∅})) ∧ Disj 𝑛 ∈ ℕ (𝑒‘𝑛)) → (Σ^{^}‘(𝑛 ∈ ℕ ↦ (𝑀‘(𝑒‘𝑛)))) = (Σ^{^}‘(𝑀 ↾ 𝑥)))
141	140	adantllr 720	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ Disj 𝑦 ∈ 𝑥 𝑦) ∧ 𝑒:ℕ–onto→(𝑥 ∪ {∅})) ∧ Disj 𝑛 ∈ ℕ (𝑒‘𝑛)) → (Σ^{^}‘(𝑛 ∈ ℕ ↦ (𝑀‘(𝑒‘𝑛)))) = (Σ^{^}‘(𝑀 ↾ 𝑥)))
142	44, 64, 141	3eqtrd 2776	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ Disj 𝑦 ∈ 𝑥 𝑦) ∧ 𝑒:ℕ–onto→(𝑥 ∪ {∅})) ∧ Disj 𝑛 ∈ ℕ (𝑒‘𝑛)) → (𝑀‘∪ 𝑥) = (Σ^{^}‘(𝑀 ↾ 𝑥)))
143	39, 40, 41, 142	syl21anc 838	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ Disj 𝑦 ∈ 𝑥 𝑦) ∧ (𝑒:ℕ–onto→(𝑥 ∪ {∅}) ∧ Disj 𝑛 ∈ ℕ (𝑒‘𝑛))) → (𝑀‘∪ 𝑥) = (Σ^{^}‘(𝑀 ↾ 𝑥)))
144	143	ex 412	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ Disj 𝑦 ∈ 𝑥 𝑦) → ((𝑒:ℕ–onto→(𝑥 ∪ {∅}) ∧ Disj 𝑛 ∈ ℕ (𝑒‘𝑛)) → (𝑀‘∪ 𝑥) = (Σ^{^}‘(𝑀 ↾ 𝑥))))
145	144	exlimdv 1935	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ Disj 𝑦 ∈ 𝑥 𝑦) → (∃𝑒(𝑒:ℕ–onto→(𝑥 ∪ {∅}) ∧ Disj 𝑛 ∈ ℕ (𝑒‘𝑛)) → (𝑀‘∪ 𝑥) = (Σ^{^}‘(𝑀 ↾ 𝑥))))
146	29, 38, 145	sylc 65	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ (𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦)) ∧ ¬ 𝑥 = ∅) → (𝑀‘∪ 𝑥) = (Σ^{^}‘(𝑀 ↾ 𝑥)))
147	26, 146	pm2.61dan 813	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ (𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦)) → (𝑀‘∪ 𝑥) = (Σ^{^}‘(𝑀 ↾ 𝑥)))
148	147	ex 412	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) → ((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → (𝑀‘∪ 𝑥) = (Σ^{^}‘(𝑀 ↾ 𝑥))))
149	148	ralrimiva 3130	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝒫 dom 𝑀((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → (𝑀‘∪ 𝑥) = (Σ^{^}‘(𝑀 ↾ 𝑥))))
150	7, 8, 149	jca31 514	. 2 ⊢ (𝜑 → (((𝑀:dom 𝑀⟶(0[,]+∞) ∧ dom 𝑀 ∈ SAlg) ∧ (𝑀‘∅) = 0) ∧ ∀𝑥 ∈ 𝒫 dom 𝑀((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → (𝑀‘∪ 𝑥) = (Σ^{^}‘(𝑀 ↾ 𝑥)))))
151		ismea 46809	. 2 ⊢ (𝑀 ∈ Meas ↔ (((𝑀:dom 𝑀⟶(0[,]+∞) ∧ dom 𝑀 ∈ SAlg) ∧ (𝑀‘∅) = 0) ∧ ∀𝑥 ∈ 𝒫 dom 𝑀((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → (𝑀‘∪ 𝑥) = (Σ^{^}‘(𝑀 ↾ 𝑥)))))
152	150, 151	sylibr 234	1 ⊢ (𝜑 → 𝑀 ∈ Meas)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1542 ∃wex 1781 ∈ wcel 2114 ≠ wne 2933 ∀wral 3052 Vcvv 3442 ∪ cun 3901 ∩ cin 3902 ⊆ wss 3903 ∅c0 4287 𝒫 cpw 4556 {csn 4582 ∪ cuni 4865 ∪ ciun 4948 Disj wdisj 5067 class class class wbr 5100 ↦ cmpt 5181 dom cdm 5632 ↾ cres 5634 ⟶wf 6496 –onto→wfo 6498 ‘cfv 6500 (class class class)co 7368 ωcom 7818 ≼ cdom 8893 0cc0 11038 +∞cpnf 11175 ℕcn 12157 +_𝑒 cxad 13036 [,]cicc 13276 SAlgcsalg 46666 Σ^{^}csumge0 46720 Meascmea 46807

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5226 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 ax-inf2 9562 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-int 4905 df-iun 4950 df-disj 5068 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-se 5586 df-we 5587 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6267 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-isom 6509 df-riota 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-om 7819 df-1st 7943 df-2nd 7944 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-1o 8407 df-er 8645 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-sup 9357 df-oi 9427 df-card 9863 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-le 11184 df-sub 11378 df-neg 11379 df-div 11807 df-nn 12158 df-2 12220 df-3 12221 df-n0 12414 df-z 12501 df-uz 12764 df-rp 12918 df-xadd 13039 df-ico 13279 df-icc 13280 df-fz 13436 df-fzo 13583 df-seq 13937 df-exp 13997 df-hash 14266 df-cj 15034 df-re 15035 df-im 15036 df-sqrt 15170 df-abs 15171 df-clim 15423 df-sum 15622 df-salg 46667 df-sumge0 46721 df-mea 46808

This theorem is referenced by: volmea 46832 caratheodory 46886

W3C validator