ismeannd - Mathbox for Glauco Siliprandi

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Theorem ismeannd 46426

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 6712	. . . . . 6 ⊢ (𝜑 → dom 𝑀 = 𝑆)
3	2	feq2d 6688	. . . . 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 2833	. . . 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 4891	. . . . . . . . . . . 12 ⊢ (𝑥 = ∅ → ∪ 𝑥 = ∪ ∅)
10		uni0 4908	. . . . . . . . . . . . 13 ⊢ ∪ ∅ = ∅
11	10	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑥 = ∅ → ∪ ∅ = ∅)
12	9, 11	eqtrd 2769	. . . . . . . . . . 11 ⊢ (𝑥 = ∅ → ∪ 𝑥 = ∅)
13	12	fveq2d 6876	. . . . . . . . . 10 ⊢ (𝑥 = ∅ → (𝑀‘∪ 𝑥) = (𝑀‘∅))
14	13, 8	sylan9eqr 2791	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 = ∅) → (𝑀‘∪ 𝑥) = 0)
15		reseq2 5958	. . . . . . . . . . . . 13 ⊢ (𝑥 = ∅ → (𝑀 ↾ 𝑥) = (𝑀 ↾ ∅))
16		res0 5967	. . . . . . . . . . . . . 14 ⊢ (𝑀 ↾ ∅) = ∅
17	16	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝑥 = ∅ → (𝑀 ↾ ∅) = ∅)
18	15, 17	eqtrd 2769	. . . . . . . . . . . 12 ⊢ (𝑥 = ∅ → (𝑀 ↾ 𝑥) = ∅)
19	18	fveq2d 6876	. . . . . . . . . . 11 ⊢ (𝑥 = ∅ → (Σ^{^}‘(𝑀 ↾ 𝑥)) = (Σ^{^}‘∅))
20	19	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 = ∅) → (Σ^{^}‘(𝑀 ↾ 𝑥)) = (Σ^{^}‘∅))
21		sge00 46335	. . . . . . . . . . 11 ⊢ (Σ^{^}‘∅) = 0
22	21	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 = ∅) → (Σ^{^}‘∅) = 0)
23	20, 22	eqtrd 2769	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 = ∅) → (Σ^{^}‘(𝑀 ↾ 𝑥)) = 0)
24	14, 23	eqtr4d 2772	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 = ∅) → (𝑀‘∪ 𝑥) = (Σ^{^}‘(𝑀 ↾ 𝑥)))
25	24	adantlr 715	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ 𝑥 = ∅) → (𝑀‘∪ 𝑥) = (Σ^{^}‘(𝑀 ↾ 𝑥)))
26	25	adantlr 715	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ (𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦)) ∧ 𝑥 = ∅) → (𝑀‘∪ 𝑥) = (Σ^{^}‘(𝑀 ↾ 𝑥)))
27		simpll 766	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ (𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦)) ∧ ¬ 𝑥 = ∅) → (𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀))
28		simplrr 777	. . . . . . . 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 776	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ (𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦)) ∧ ¬ 𝑥 = ∅) → 𝑥 ≼ ω)
31		neqne 2939	. . . . . . . . 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 5094	. . . . . . . . . . 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 727	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ (𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦)) ∧ ¬ 𝑥 = ∅) → Disj 𝑤 ∈ 𝑥 𝑤)
38	30, 32, 37	nnfoctbdj 46415	. . . . . . 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 770	. . . . . . . . . 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 772	. . . . . . . . . 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 45141	. . . . . . . . . . . . 13 ⊢ (𝑒:ℕ–onto→(𝑥 ∪ {∅}) → ∪ 𝑥 = ∪ 𝑛 ∈ ℕ (𝑒‘𝑛))
43	42	fveq2d 6876	. . . . . . . . . . . 12 ⊢ (𝑒:ℕ–onto→(𝑥 ∪ {∅}) → (𝑀‘∪ 𝑥) = (𝑀‘∪ 𝑛 ∈ ℕ (𝑒‘𝑛)))
44	43	ad2antlr 727	. . . . . . . . . . 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 774	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ 𝑒:ℕ–onto→(𝑥 ∪ {∅})) ∧ Disj 𝑛 ∈ ℕ (𝑒‘𝑛)) → 𝜑)
46		fof 6786	. . . . . . . . . . . . . . . 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 4580	. . . . . . . . . . . . . . . . . . 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 3993	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) → 𝑥 ⊆ 𝑆)
52		0sal 46279	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑆 ∈ SAlg → ∅ ∈ 𝑆)
53	5, 52	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ∅ ∈ 𝑆)
54		snssi 4781	. . . . . . . . . . . . . . . . . . 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 4165	. . . . . . . . . . . . . . . 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 6719	. . . . . . . . . . . . . 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 1372	. . . . . . . . . . . 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 719	. . . . . . . . . . 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 6943	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝑀 = (𝑦 ∈ 𝑆 ↦ (𝑀‘𝑦)))
66	65	reseq1d 5962	. . . . . . . . . . . . . . . . . . 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 6024	. . . . . . . . . . . . . . . . . 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 4781	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (∅ ∈ 𝑥 → {∅} ⊆ 𝑥)
72		ssequn2 4162	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ({∅} ⊆ 𝑥 ↔ (𝑥 ∪ {∅}) = 𝑥)
73	71, 72	sylib 218	. . . . . . . . . . . . . . . . . . . 20 ⊢ (∅ ∈ 𝑥 → (𝑥 ∪ {∅}) = 𝑥)
74	73	eqcomd 2740	. . . . . . . . . . . . . . . . . . 19 ⊢ (∅ ∈ 𝑥 → 𝑥 = (𝑥 ∪ {∅}))
75	74	mpteq1d 5207	. . . . . . . . . . . . . . . . . 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 2773	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ ∅ ∈ 𝑥) → (𝑀 ↾ 𝑥) = (𝑦 ∈ (𝑥 ∪ {∅}) ↦ (𝑀‘𝑦)))
78	77	fveq2d 6876	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ ∅ ∈ 𝑥) → (Σ^{^}‘(𝑀 ↾ 𝑥)) = (Σ^{^}‘(𝑦 ∈ (𝑥 ∪ {∅}) ↦ (𝑀‘𝑦))))
79		nfv 1913	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑦((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ ¬ ∅ ∈ 𝑥)
80		simplr 768	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ ¬ ∅ ∈ 𝑥) → 𝑥 ∈ 𝒫 dom 𝑀)
81		p0ex 5351	. . . . . . . . . . . . . . . . . 18 ⊢ {∅} ∈ V
82	81	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ ¬ ∅ ∈ 𝑥) → {∅} ∈ V)
83		disjsn 4684	. . . . . . . . . . . . . . . . . . 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 726	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ 𝑦 ∈ 𝑥) → 𝑀:𝑆⟶(0[,]+∞))
87	51	sselda 3956	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ 𝑆)
88	86, 87	ffvelcdmd 7071	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ 𝑦 ∈ 𝑥) → (𝑀‘𝑦) ∈ (0[,]+∞))
89	88	adantlr 715	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ ¬ ∅ ∈ 𝑥) ∧ 𝑦 ∈ 𝑥) → (𝑀‘𝑦) ∈ (0[,]+∞))
90		elsni 4616	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 ∈ {∅} → 𝑦 = ∅)
91	90	fveq2d 6876	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 ∈ {∅} → (𝑀‘𝑦) = (𝑀‘∅))
92	91	adantl 481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑦 ∈ {∅}) → (𝑀‘𝑦) = (𝑀‘∅))
93	1, 53	ffvelcdmd 7071	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝑀‘∅) ∈ (0[,]+∞))
94	93	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑦 ∈ {∅}) → (𝑀‘∅) ∈ (0[,]+∞))
95	92, 94	eqeltrd 2833	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑦 ∈ {∅}) → (𝑀‘𝑦) ∈ (0[,]+∞))
96	95	ad4ant14 752	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ ¬ ∅ ∈ 𝑥) ∧ 𝑦 ∈ {∅}) → (𝑀‘𝑦) ∈ (0[,]+∞))
97	79, 80, 82, 85, 89, 96	sge0splitmpt 46370	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ ¬ ∅ ∈ 𝑥) → (Σ^{^}‘(𝑦 ∈ (𝑥 ∪ {∅}) ↦ (𝑀‘𝑦))) = ((Σ^{^}‘(𝑦 ∈ 𝑥 ↦ (𝑀‘𝑦))) +_𝑒 (Σ^{^}‘(𝑦 ∈ {∅} ↦ (𝑀‘𝑦)))))
98		fveq2 6872	. . . . . . . . . . . . . . . . . . . . . . . 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 2769	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑦 = ∅) → (𝑀‘𝑦) = 0)
102	90, 101	sylan2 593	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑦 ∈ {∅}) → (𝑀‘𝑦) = 0)
103	102	mpteq2dva 5211	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝑦 ∈ {∅} ↦ (𝑀‘𝑦)) = (𝑦 ∈ {∅} ↦ 0))
104	103	fveq2d 6876	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (Σ^{^}‘(𝑦 ∈ {∅} ↦ (𝑀‘𝑦))) = (Σ^{^}‘(𝑦 ∈ {∅} ↦ 0)))
105		nfv 1913	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑦𝜑
106	81	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → {∅} ∈ V)
107	105, 106	sge0z 46334	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (Σ^{^}‘(𝑦 ∈ {∅} ↦ 0)) = 0)
108	104, 107	eqtrd 2769	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (Σ^{^}‘(𝑦 ∈ {∅} ↦ (𝑀‘𝑦))) = 0)
109	108	oveq2d 7415	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((Σ^{^}‘(𝑦 ∈ 𝑥 ↦ (𝑀‘𝑦))) +_𝑒 (Σ^{^}‘(𝑦 ∈ {∅} ↦ (𝑀‘𝑦)))) = ((Σ^{^}‘(𝑦 ∈ 𝑥 ↦ (𝑀‘𝑦))) +_𝑒 0))
110	109	ad2antrr 726	. . . . . . . . . . . . . . . 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 2769	. . . . . . . . . . . . . . . . . . . . 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 6741	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) → (𝑀 ↾ 𝑥):𝑥⟶(0[,]+∞))
115	112, 114	feq1dd 6687	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) → (𝑦 ∈ 𝑥 ↦ (𝑀‘𝑦)):𝑥⟶(0[,]+∞))
116	111, 115	sge0xrcl 46344	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) → (Σ^{^}‘(𝑦 ∈ 𝑥 ↦ (𝑀‘𝑦))) ∈ ℝ^*)
117	116	xaddridd 13251	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) → ((Σ^{^}‘(𝑦 ∈ 𝑥 ↦ (𝑀‘𝑦))) +_𝑒 0) = (Σ^{^}‘(𝑦 ∈ 𝑥 ↦ (𝑀‘𝑦))))
118	112	fveq2d 6876	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) → (Σ^{^}‘(𝑀 ↾ 𝑥)) = (Σ^{^}‘(𝑦 ∈ 𝑥 ↦ (𝑀‘𝑦))))
119	118	eqcomd 2740	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) → (Σ^{^}‘(𝑦 ∈ 𝑥 ↦ (𝑀‘𝑦))) = (Σ^{^}‘(𝑀 ↾ 𝑥)))
120	117, 119	eqtrd 2769	. . . . . . . . . . . . . . . . 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 2774	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ ¬ ∅ ∈ 𝑥) → (Σ^{^}‘(𝑀 ↾ 𝑥)) = (Σ^{^}‘(𝑦 ∈ (𝑥 ∪ {∅}) ↦ (𝑀‘𝑦))))
123	78, 122	pm2.61dan 812	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) → (Σ^{^}‘(𝑀 ↾ 𝑥)) = (Σ^{^}‘(𝑦 ∈ (𝑥 ∪ {∅}) ↦ (𝑀‘𝑦))))
124	123	ad2antrr 726	. . . . . . . . . . . . 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 1913	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑦(((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ 𝑒:ℕ–onto→(𝑥 ∪ {∅})) ∧ Disj 𝑛 ∈ ℕ (𝑒‘𝑛))
126		nfv 1913	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑛((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ 𝑒:ℕ–onto→(𝑥 ∪ {∅}))
127		nfdisj1 5097	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑛Disj 𝑛 ∈ ℕ (𝑒‘𝑛)
128	126, 127	nfan 1898	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑛(((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ 𝑒:ℕ–onto→(𝑥 ∪ {∅})) ∧ Disj 𝑛 ∈ ℕ (𝑒‘𝑛))
129		fveq2 6872	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝑒‘𝑛) → (𝑀‘𝑦) = (𝑀‘(𝑒‘𝑛)))
130		nnex 12238	. . . . . . . . . . . . . . 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 768	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ 𝑒:ℕ–onto→(𝑥 ∪ {∅})) ∧ Disj 𝑛 ∈ ℕ (𝑒‘𝑛)) → 𝑒:ℕ–onto→(𝑥 ∪ {∅}))
133		eqidd 2735	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ 𝑒:ℕ–onto→(𝑥 ∪ {∅})) ∧ Disj 𝑛 ∈ ℕ (𝑒‘𝑛)) ∧ 𝑛 ∈ ℕ) → (𝑒‘𝑛) = (𝑒‘𝑛))
134	1	ad2antrr 726	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ 𝑦 ∈ (𝑥 ∪ {∅})) → 𝑀:𝑆⟶(0[,]+∞))
135	57	sselda 3956	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ 𝑦 ∈ (𝑥 ∪ {∅})) → 𝑦 ∈ 𝑆)
136	134, 135	ffvelcdmd 7071	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ 𝑦 ∈ (𝑥 ∪ {∅})) → (𝑀‘𝑦) ∈ (0[,]+∞))
137	136	ad4ant14 752	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ 𝑒:ℕ–onto→(𝑥 ∪ {∅})) ∧ Disj 𝑛 ∈ ℕ (𝑒‘𝑛)) ∧ 𝑦 ∈ (𝑥 ∪ {∅})) → (𝑀‘𝑦) ∈ (0[,]+∞))
138	45, 101	sylan 580	. . . . . . . . . . . . . 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 46376	. . . . . . . . . . . . 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 2770	. . . . . . . . . . . 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 719	. . . . . . . . . . 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 2773	. . . . . . . . . 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 837	. . . . . . . . 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 1932	. . . . . . 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 812	. . . . 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 46410	. 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 1086 = wceq 1539 ∃wex 1778 ∈ wcel 2107 ≠ wne 2931 ∀wral 3050 Vcvv 3457 ∪ cun 3922 ∩ cin 3923 ⊆ wss 3924 ∅c0 4306 𝒫 cpw 4573 {csn 4599 ∪ cuni 4880 ∪ ciun 4964 Disj wdisj 5083 class class class wbr 5116 ↦ cmpt 5198 dom cdm 5651 ↾ cres 5653 ⟶wf 6523 –onto→wfo 6525 ‘cfv 6527 (class class class)co 7399 ωcom 7855 ≼ cdom 8951 0cc0 11121 +∞cpnf 11258 ℕcn 12232 +_𝑒 cxad 13118 [,]cicc 13356 SAlgcsalg 46267 Σ^{^}csumge0 46321 Meascmea 46408

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2706 ax-rep 5246 ax-sep 5263 ax-nul 5273 ax-pow 5332 ax-pr 5399 ax-un 7723 ax-inf2 9647 ax-cnex 11177 ax-resscn 11178 ax-1cn 11179 ax-icn 11180 ax-addcl 11181 ax-addrcl 11182 ax-mulcl 11183 ax-mulrcl 11184 ax-mulcom 11185 ax-addass 11186 ax-mulass 11187 ax-distr 11188 ax-i2m1 11189 ax-1ne0 11190 ax-1rid 11191 ax-rnegex 11192 ax-rrecex 11193 ax-cnre 11194 ax-pre-lttri 11195 ax-pre-lttrn 11196 ax-pre-ltadd 11197 ax-pre-mulgt0 11198 ax-pre-sup 11199

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2808 df-nfc 2884 df-ne 2932 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3357 df-reu 3358 df-rab 3414 df-v 3459 df-sbc 3764 df-csb 3873 df-dif 3927 df-un 3929 df-in 3931 df-ss 3941 df-pss 3944 df-nul 4307 df-if 4499 df-pw 4575 df-sn 4600 df-pr 4602 df-op 4606 df-uni 4881 df-int 4920 df-iun 4966 df-disj 5084 df-br 5117 df-opab 5179 df-mpt 5199 df-tr 5227 df-id 5545 df-eprel 5550 df-po 5558 df-so 5559 df-fr 5603 df-se 5604 df-we 5605 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6287 df-ord 6352 df-on 6353 df-lim 6354 df-suc 6355 df-iota 6480 df-fun 6529 df-fn 6530 df-f 6531 df-f1 6532 df-fo 6533 df-f1o 6534 df-fv 6535 df-isom 6536 df-riota 7356 df-ov 7402 df-oprab 7403 df-mpo 7404 df-om 7856 df-1st 7982 df-2nd 7983 df-frecs 8274 df-wrecs 8305 df-recs 8379 df-rdg 8418 df-1o 8474 df-er 8713 df-en 8954 df-dom 8955 df-sdom 8956 df-fin 8957 df-sup 9448 df-oi 9516 df-card 9945 df-pnf 11263 df-mnf 11264 df-xr 11265 df-ltxr 11266 df-le 11267 df-sub 11460 df-neg 11461 df-div 11887 df-nn 12233 df-2 12295 df-3 12296 df-n0 12494 df-z 12581 df-uz 12845 df-rp 13001 df-xadd 13121 df-ico 13359 df-icc 13360 df-fz 13514 df-fzo 13661 df-seq 14009 df-exp 14069 df-hash 14337 df-cj 15105 df-re 15106 df-im 15107 df-sqrt 15241 df-abs 15242 df-clim 15491 df-sum 15690 df-salg 46268 df-sumge0 46322 df-mea 46409

This theorem is referenced by: volmea 46433 caratheodory 46487

W3C validator