ismeannd - Mathbox for Glauco Siliprandi

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Theorem ismeannd 45262

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 6728	. . . . . 6 ⊢ (𝜑 → dom 𝑀 = 𝑆)
3	2	feq2d 6703	. . . . 5 ⊢ (𝜑 → (𝑀:dom 𝑀⟶(0[,]+∞) ↔ 𝑀:𝑆⟶(0[,]+∞)))
4	1, 3	mpbird 256	. . . 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 512	. . 3 ⊢ (𝜑 → (𝑀:dom 𝑀⟶(0[,]+∞) ∧ dom 𝑀 ∈ SAlg))
8		ismeannd.m0	. . 3 ⊢ (𝜑 → (𝑀‘∅) = 0)
9		unieq 4919	. . . . . . . . . . . 12 ⊢ (𝑥 = ∅ → ∪ 𝑥 = ∪ ∅)
10		uni0 4939	. . . . . . . . . . . . 13 ⊢ ∪ ∅ = ∅
11	10	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑥 = ∅ → ∪ ∅ = ∅)
12	9, 11	eqtrd 2772	. . . . . . . . . . 11 ⊢ (𝑥 = ∅ → ∪ 𝑥 = ∅)
13	12	fveq2d 6895	. . . . . . . . . 10 ⊢ (𝑥 = ∅ → (𝑀‘∪ 𝑥) = (𝑀‘∅))
14	13, 8	sylan9eqr 2794	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 = ∅) → (𝑀‘∪ 𝑥) = 0)
15		reseq2 5976	. . . . . . . . . . . . 13 ⊢ (𝑥 = ∅ → (𝑀 ↾ 𝑥) = (𝑀 ↾ ∅))
16		res0 5985	. . . . . . . . . . . . . 14 ⊢ (𝑀 ↾ ∅) = ∅
17	16	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝑥 = ∅ → (𝑀 ↾ ∅) = ∅)
18	15, 17	eqtrd 2772	. . . . . . . . . . . 12 ⊢ (𝑥 = ∅ → (𝑀 ↾ 𝑥) = ∅)
19	18	fveq2d 6895	. . . . . . . . . . 11 ⊢ (𝑥 = ∅ → (Σ^{^}‘(𝑀 ↾ 𝑥)) = (Σ^{^}‘∅))
20	19	adantl 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 = ∅) → (Σ^{^}‘(𝑀 ↾ 𝑥)) = (Σ^{^}‘∅))
21		sge00 45171	. . . . . . . . . . 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 713	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ 𝑥 = ∅) → (𝑀‘∪ 𝑥) = (Σ^{^}‘(𝑀 ↾ 𝑥)))
26	25	adantlr 713	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ (𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦)) ∧ 𝑥 = ∅) → (𝑀‘∪ 𝑥) = (Σ^{^}‘(𝑀 ↾ 𝑥)))
27		simpll 765	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ (𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦)) ∧ ¬ 𝑥 = ∅) → (𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀))
28		simplrr 776	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ (𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦)) ∧ ¬ 𝑥 = ∅) → Disj 𝑦 ∈ 𝑥 𝑦)
29	27, 28	jca 512	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ (𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦)) ∧ ¬ 𝑥 = ∅) → ((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ Disj 𝑦 ∈ 𝑥 𝑦))
30		simplrl 775	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ (𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦)) ∧ ¬ 𝑥 = ∅) → 𝑥 ≼ ω)
31		neqne 2948	. . . . . . . . 9 ⊢ (¬ 𝑥 = ∅ → 𝑥 ≠ ∅)
32	31	adantl 482	. . . . . . . 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 5124	. . . . . . . . . . 11 ⊢ (Disj 𝑦 ∈ 𝑥 𝑦 ↔ Disj 𝑤 ∈ 𝑥 𝑤)
35	34	biimpi 215	. . . . . . . . . 10 ⊢ (Disj 𝑦 ∈ 𝑥 𝑦 → Disj 𝑤 ∈ 𝑥 𝑤)
36	35	adantl 482	. . . . . . . . 9 ⊢ ((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → Disj 𝑤 ∈ 𝑥 𝑤)
37	36	ad2antlr 725	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ (𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦)) ∧ ¬ 𝑥 = ∅) → Disj 𝑤 ∈ 𝑥 𝑤)
38	30, 32, 37	nnfoctbdj 45251	. . . . . . 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 483	. . . . . . . . . 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 769	. . . . . . . . . 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 771	. . . . . . . . . 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 43968	. . . . . . . . . . . . 13 ⊢ (𝑒:ℕ–onto→(𝑥 ∪ {∅}) → ∪ 𝑥 = ∪ 𝑛 ∈ ℕ (𝑒‘𝑛))
43	42	fveq2d 6895	. . . . . . . . . . . 12 ⊢ (𝑒:ℕ–onto→(𝑥 ∪ {∅}) → (𝑀‘∪ 𝑥) = (𝑀‘∪ 𝑛 ∈ ℕ (𝑒‘𝑛)))
44	43	ad2antlr 725	. . . . . . . . . . 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 773	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ 𝑒:ℕ–onto→(𝑥 ∪ {∅})) ∧ Disj 𝑛 ∈ ℕ (𝑒‘𝑛)) → 𝜑)
46		fof 6805	. . . . . . . . . . . . . . . 16 ⊢ (𝑒:ℕ–onto→(𝑥 ∪ {∅}) → 𝑒:ℕ⟶(𝑥 ∪ {∅}))
47	46	adantl 482	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ 𝑒:ℕ–onto→(𝑥 ∪ {∅})) → 𝑒:ℕ⟶(𝑥 ∪ {∅}))
48		elpwi 4609	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ 𝒫 dom 𝑀 → 𝑥 ⊆ dom 𝑀)
49	48	adantl 482	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) → 𝑥 ⊆ dom 𝑀)
50	2	adantr 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) → dom 𝑀 = 𝑆)
51	49, 50	sseqtrd 4022	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) → 𝑥 ⊆ 𝑆)
52		0sal 45115	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑆 ∈ SAlg → ∅ ∈ 𝑆)
53	5, 52	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ∅ ∈ 𝑆)
54		snssi 4811	. . . . . . . . . . . . . . . . . . 19 ⊢ (∅ ∈ 𝑆 → {∅} ⊆ 𝑆)
55	53, 54	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → {∅} ⊆ 𝑆)
56	55	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) → {∅} ⊆ 𝑆)
57	51, 56	unssd 4186	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) → (𝑥 ∪ {∅}) ⊆ 𝑆)
58	57	adantr 481	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ 𝑒:ℕ–onto→(𝑥 ∪ {∅})) → (𝑥 ∪ {∅}) ⊆ 𝑆)
59	47, 58	fssd 6735	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ 𝑒:ℕ–onto→(𝑥 ∪ {∅})) → 𝑒:ℕ⟶𝑆)
60	59	adantr 481	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ 𝑒:ℕ–onto→(𝑥 ∪ {∅})) ∧ Disj 𝑛 ∈ ℕ (𝑒‘𝑛)) → 𝑒:ℕ⟶𝑆)
61		simpr 485	. . . . . . . . . . . . 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 1371	. . . . . . . . . . . 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 717	. . . . . . . . . . 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 6960	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝑀 = (𝑦 ∈ 𝑆 ↦ (𝑀‘𝑦)))
66	65	reseq1d 5980	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝑀 ↾ 𝑥) = ((𝑦 ∈ 𝑆 ↦ (𝑀‘𝑦)) ↾ 𝑥))
67	66	adantr 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) → (𝑀 ↾ 𝑥) = ((𝑦 ∈ 𝑆 ↦ (𝑀‘𝑦)) ↾ 𝑥))
68	67	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ ∅ ∈ 𝑥) → (𝑀 ↾ 𝑥) = ((𝑦 ∈ 𝑆 ↦ (𝑀‘𝑦)) ↾ 𝑥))
69	51	resmptd 6040	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) → ((𝑦 ∈ 𝑆 ↦ (𝑀‘𝑦)) ↾ 𝑥) = (𝑦 ∈ 𝑥 ↦ (𝑀‘𝑦)))
70	69	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ ∅ ∈ 𝑥) → ((𝑦 ∈ 𝑆 ↦ (𝑀‘𝑦)) ↾ 𝑥) = (𝑦 ∈ 𝑥 ↦ (𝑀‘𝑦)))
71		snssi 4811	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (∅ ∈ 𝑥 → {∅} ⊆ 𝑥)
72		ssequn2 4183	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ({∅} ⊆ 𝑥 ↔ (𝑥 ∪ {∅}) = 𝑥)
73	71, 72	sylib 217	. . . . . . . . . . . . . . . . . . . 20 ⊢ (∅ ∈ 𝑥 → (𝑥 ∪ {∅}) = 𝑥)
74	73	eqcomd 2738	. . . . . . . . . . . . . . . . . . 19 ⊢ (∅ ∈ 𝑥 → 𝑥 = (𝑥 ∪ {∅}))
75	74	mpteq1d 5243	. . . . . . . . . . . . . . . . . 18 ⊢ (∅ ∈ 𝑥 → (𝑦 ∈ 𝑥 ↦ (𝑀‘𝑦)) = (𝑦 ∈ (𝑥 ∪ {∅}) ↦ (𝑀‘𝑦)))
76	75	adantl 482	. . . . . . . . . . . . . . . . 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 6895	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ ∅ ∈ 𝑥) → (Σ^{^}‘(𝑀 ↾ 𝑥)) = (Σ^{^}‘(𝑦 ∈ (𝑥 ∪ {∅}) ↦ (𝑀‘𝑦))))
79		nfv 1917	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑦((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ ¬ ∅ ∈ 𝑥)
80		simplr 767	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ ¬ ∅ ∈ 𝑥) → 𝑥 ∈ 𝒫 dom 𝑀)
81		p0ex 5382	. . . . . . . . . . . . . . . . . 18 ⊢ {∅} ∈ V
82	81	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ ¬ ∅ ∈ 𝑥) → {∅} ∈ V)
83		disjsn 4715	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∩ {∅}) = ∅ ↔ ¬ ∅ ∈ 𝑥)
84	83	biimpri 227	. . . . . . . . . . . . . . . . . 18 ⊢ (¬ ∅ ∈ 𝑥 → (𝑥 ∩ {∅}) = ∅)
85	84	adantl 482	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ ¬ ∅ ∈ 𝑥) → (𝑥 ∩ {∅}) = ∅)
86	1	ad2antrr 724	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ 𝑦 ∈ 𝑥) → 𝑀:𝑆⟶(0[,]+∞))
87	51	sselda 3982	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ 𝑆)
88	86, 87	ffvelcdmd 7087	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ 𝑦 ∈ 𝑥) → (𝑀‘𝑦) ∈ (0[,]+∞))
89	88	adantlr 713	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ ¬ ∅ ∈ 𝑥) ∧ 𝑦 ∈ 𝑥) → (𝑀‘𝑦) ∈ (0[,]+∞))
90		elsni 4645	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 ∈ {∅} → 𝑦 = ∅)
91	90	fveq2d 6895	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 ∈ {∅} → (𝑀‘𝑦) = (𝑀‘∅))
92	91	adantl 482	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑦 ∈ {∅}) → (𝑀‘𝑦) = (𝑀‘∅))
93	1, 53	ffvelcdmd 7087	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝑀‘∅) ∈ (0[,]+∞))
94	93	adantr 481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑦 ∈ {∅}) → (𝑀‘∅) ∈ (0[,]+∞))
95	92, 94	eqeltrd 2833	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑦 ∈ {∅}) → (𝑀‘𝑦) ∈ (0[,]+∞))
96	95	ad4ant14 750	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ ¬ ∅ ∈ 𝑥) ∧ 𝑦 ∈ {∅}) → (𝑀‘𝑦) ∈ (0[,]+∞))
97	79, 80, 82, 85, 89, 96	sge0splitmpt 45206	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ ¬ ∅ ∈ 𝑥) → (Σ^{^}‘(𝑦 ∈ (𝑥 ∪ {∅}) ↦ (𝑀‘𝑦))) = ((Σ^{^}‘(𝑦 ∈ 𝑥 ↦ (𝑀‘𝑦))) +_𝑒 (Σ^{^}‘(𝑦 ∈ {∅} ↦ (𝑀‘𝑦)))))
98		fveq2 6891	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 = ∅ → (𝑀‘𝑦) = (𝑀‘∅))
99	98	adantl 482	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑦 = ∅) → (𝑀‘𝑦) = (𝑀‘∅))
100	8	adantr 481	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑦 = ∅) → (𝑀‘∅) = 0)
101	99, 100	eqtrd 2772	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑦 = ∅) → (𝑀‘𝑦) = 0)
102	90, 101	sylan2 593	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑦 ∈ {∅}) → (𝑀‘𝑦) = 0)
103	102	mpteq2dva 5248	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝑦 ∈ {∅} ↦ (𝑀‘𝑦)) = (𝑦 ∈ {∅} ↦ 0))
104	103	fveq2d 6895	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (Σ^{^}‘(𝑦 ∈ {∅} ↦ (𝑀‘𝑦))) = (Σ^{^}‘(𝑦 ∈ {∅} ↦ 0)))
105		nfv 1917	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑦𝜑
106	81	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → {∅} ∈ V)
107	105, 106	sge0z 45170	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (Σ^{^}‘(𝑦 ∈ {∅} ↦ 0)) = 0)
108	104, 107	eqtrd 2772	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (Σ^{^}‘(𝑦 ∈ {∅} ↦ (𝑀‘𝑦))) = 0)
109	108	oveq2d 7427	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((Σ^{^}‘(𝑦 ∈ 𝑥 ↦ (𝑀‘𝑦))) +_𝑒 (Σ^{^}‘(𝑦 ∈ {∅} ↦ (𝑀‘𝑦)))) = ((Σ^{^}‘(𝑦 ∈ 𝑥 ↦ (𝑀‘𝑦))) +_𝑒 0))
110	109	ad2antrr 724	. . . . . . . . . . . . . . . 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 485	. . . . . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) → 𝑀:𝑆⟶(0[,]+∞))
114	113, 51	fssresd 6758	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) → (𝑀 ↾ 𝑥):𝑥⟶(0[,]+∞))
115	112, 114	feq1dd 43945	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) → (𝑦 ∈ 𝑥 ↦ (𝑀‘𝑦)):𝑥⟶(0[,]+∞))
116	111, 115	sge0xrcl 45180	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) → (Σ^{^}‘(𝑦 ∈ 𝑥 ↦ (𝑀‘𝑦))) ∈ ℝ^*)
117	116	xaddridd 13224	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) → ((Σ^{^}‘(𝑦 ∈ 𝑥 ↦ (𝑀‘𝑦))) +_𝑒 0) = (Σ^{^}‘(𝑦 ∈ 𝑥 ↦ (𝑀‘𝑦))))
118	112	fveq2d 6895	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) → (Σ^{^}‘(𝑀 ↾ 𝑥)) = (Σ^{^}‘(𝑦 ∈ 𝑥 ↦ (𝑀‘𝑦))))
119	118	eqcomd 2738	. . . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . 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 811	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) → (Σ^{^}‘(𝑀 ↾ 𝑥)) = (Σ^{^}‘(𝑦 ∈ (𝑥 ∪ {∅}) ↦ (𝑀‘𝑦))))
124	123	ad2antrr 724	. . . . . . . . . . . . 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 1917	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑦(((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ 𝑒:ℕ–onto→(𝑥 ∪ {∅})) ∧ Disj 𝑛 ∈ ℕ (𝑒‘𝑛))
126		nfv 1917	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑛((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ 𝑒:ℕ–onto→(𝑥 ∪ {∅}))
127		nfdisj1 5127	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑛Disj 𝑛 ∈ ℕ (𝑒‘𝑛)
128	126, 127	nfan 1902	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑛(((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ 𝑒:ℕ–onto→(𝑥 ∪ {∅})) ∧ Disj 𝑛 ∈ ℕ (𝑒‘𝑛))
129		fveq2 6891	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝑒‘𝑛) → (𝑀‘𝑦) = (𝑀‘(𝑒‘𝑛)))
130		nnex 12220	. . . . . . . . . . . . . . 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 767	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ 𝑒:ℕ–onto→(𝑥 ∪ {∅})) ∧ Disj 𝑛 ∈ ℕ (𝑒‘𝑛)) → 𝑒:ℕ–onto→(𝑥 ∪ {∅}))
133		eqidd 2733	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ 𝑒:ℕ–onto→(𝑥 ∪ {∅})) ∧ Disj 𝑛 ∈ ℕ (𝑒‘𝑛)) ∧ 𝑛 ∈ ℕ) → (𝑒‘𝑛) = (𝑒‘𝑛))
134	1	ad2antrr 724	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ 𝑦 ∈ (𝑥 ∪ {∅})) → 𝑀:𝑆⟶(0[,]+∞))
135	57	sselda 3982	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ 𝑦 ∈ (𝑥 ∪ {∅})) → 𝑦 ∈ 𝑆)
136	134, 135	ffvelcdmd 7087	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ 𝑦 ∈ (𝑥 ∪ {∅})) → (𝑀‘𝑦) ∈ (0[,]+∞))
137	136	ad4ant14 750	. . . . . . . . . . . . . 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 45212	. . . . . . . . . . . . 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 717	. . . . . . . . . . 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 836	. . . . . . . . 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 413	. . . . . . . 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 1936	. . . . . . 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 811	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ (𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦)) → (𝑀‘∪ 𝑥) = (Σ^{^}‘(𝑀 ↾ 𝑥)))
148	147	ex 413	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) → ((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → (𝑀‘∪ 𝑥) = (Σ^{^}‘(𝑀 ↾ 𝑥))))
149	148	ralrimiva 3146	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝒫 dom 𝑀((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → (𝑀‘∪ 𝑥) = (Σ^{^}‘(𝑀 ↾ 𝑥))))
150	7, 8, 149	jca31 515	. 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 45246	. 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 233	1 ⊢ (𝜑 → 𝑀 ∈ Meas)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∃wex 1781 ∈ wcel 2106 ≠ wne 2940 ∀wral 3061 Vcvv 3474 ∪ cun 3946 ∩ cin 3947 ⊆ wss 3948 ∅c0 4322 𝒫 cpw 4602 {csn 4628 ∪ cuni 4908 ∪ ciun 4997 Disj wdisj 5113 class class class wbr 5148 ↦ cmpt 5231 dom cdm 5676 ↾ cres 5678 ⟶wf 6539 –onto→wfo 6541 ‘cfv 6543 (class class class)co 7411 ωcom 7857 ≼ cdom 8939 0cc0 11112 +∞cpnf 11247 ℕcn 12214 +_𝑒 cxad 13092 [,]cicc 13329 SAlgcsalg 45103 Σ^{^}csumge0 45157 Meascmea 45244

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 ax-inf2 9638 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-disj 5114 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-1st 7977 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-sup 9439 df-oi 9507 df-card 9936 df-pnf 11252 df-mnf 11253 df-xr 11254 df-ltxr 11255 df-le 11256 df-sub 11448 df-neg 11449 df-div 11874 df-nn 12215 df-2 12277 df-3 12278 df-n0 12475 df-z 12561 df-uz 12825 df-rp 12977 df-xadd 13095 df-ico 13332 df-icc 13333 df-fz 13487 df-fzo 13630 df-seq 13969 df-exp 14030 df-hash 14293 df-cj 15048 df-re 15049 df-im 15050 df-sqrt 15184 df-abs 15185 df-clim 15434 df-sum 15635 df-salg 45104 df-sumge0 45158 df-mea 45245

This theorem is referenced by: volmea 45269 caratheodory 45323

W3C validator