ismeannd - Mathbox for Glauco Siliprandi

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Theorem ismeannd 43976

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 6609	. . . . . 6 ⊢ (𝜑 → dom 𝑀 = 𝑆)
3	2	feq2d 6584	. . . . 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 2841	. . . 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 4856	. . . . . . . . . . . 12 ⊢ (𝑥 = ∅ → ∪ 𝑥 = ∪ ∅)
10		uni0 4875	. . . . . . . . . . . . 13 ⊢ ∪ ∅ = ∅
11	10	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑥 = ∅ → ∪ ∅ = ∅)
12	9, 11	eqtrd 2780	. . . . . . . . . . 11 ⊢ (𝑥 = ∅ → ∪ 𝑥 = ∅)
13	12	fveq2d 6775	. . . . . . . . . 10 ⊢ (𝑥 = ∅ → (𝑀‘∪ 𝑥) = (𝑀‘∅))
14	13, 8	sylan9eqr 2802	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 = ∅) → (𝑀‘∪ 𝑥) = 0)
15		reseq2 5885	. . . . . . . . . . . . 13 ⊢ (𝑥 = ∅ → (𝑀 ↾ 𝑥) = (𝑀 ↾ ∅))
16		res0 5894	. . . . . . . . . . . . . 14 ⊢ (𝑀 ↾ ∅) = ∅
17	16	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝑥 = ∅ → (𝑀 ↾ ∅) = ∅)
18	15, 17	eqtrd 2780	. . . . . . . . . . . 12 ⊢ (𝑥 = ∅ → (𝑀 ↾ 𝑥) = ∅)
19	18	fveq2d 6775	. . . . . . . . . . 11 ⊢ (𝑥 = ∅ → (Σ^{^}‘(𝑀 ↾ 𝑥)) = (Σ^{^}‘∅))
20	19	adantl 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 = ∅) → (Σ^{^}‘(𝑀 ↾ 𝑥)) = (Σ^{^}‘∅))
21		sge00 43885	. . . . . . . . . . 11 ⊢ (Σ^{^}‘∅) = 0
22	21	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 = ∅) → (Σ^{^}‘∅) = 0)
23	20, 22	eqtrd 2780	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 = ∅) → (Σ^{^}‘(𝑀 ↾ 𝑥)) = 0)
24	14, 23	eqtr4d 2783	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 = ∅) → (𝑀‘∪ 𝑥) = (Σ^{^}‘(𝑀 ↾ 𝑥)))
25	24	adantlr 712	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ 𝑥 = ∅) → (𝑀‘∪ 𝑥) = (Σ^{^}‘(𝑀 ↾ 𝑥)))
26	25	adantlr 712	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ (𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦)) ∧ 𝑥 = ∅) → (𝑀‘∪ 𝑥) = (Σ^{^}‘(𝑀 ↾ 𝑥)))
27		simpll 764	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ (𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦)) ∧ ¬ 𝑥 = ∅) → (𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀))
28		simplrr 775	. . . . . . . 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 774	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ (𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦)) ∧ ¬ 𝑥 = ∅) → 𝑥 ≼ ω)
31		neqne 2953	. . . . . . . . 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 5055	. . . . . . . . . . 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 724	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ (𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦)) ∧ ¬ 𝑥 = ∅) → Disj 𝑤 ∈ 𝑥 𝑤)
38	30, 32, 37	nnfoctbdj 43965	. . . . . . 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 768	. . . . . . . . . 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 770	. . . . . . . . . 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 42698	. . . . . . . . . . . . 13 ⊢ (𝑒:ℕ–onto→(𝑥 ∪ {∅}) → ∪ 𝑥 = ∪ 𝑛 ∈ ℕ (𝑒‘𝑛))
43	42	fveq2d 6775	. . . . . . . . . . . 12 ⊢ (𝑒:ℕ–onto→(𝑥 ∪ {∅}) → (𝑀‘∪ 𝑥) = (𝑀‘∪ 𝑛 ∈ ℕ (𝑒‘𝑛)))
44	43	ad2antlr 724	. . . . . . . . . . 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 772	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ 𝑒:ℕ–onto→(𝑥 ∪ {∅})) ∧ Disj 𝑛 ∈ ℕ (𝑒‘𝑛)) → 𝜑)
46		fof 6686	. . . . . . . . . . . . . . . 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 4548	. . . . . . . . . . . . . . . . . . 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 3966	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) → 𝑥 ⊆ 𝑆)
52		0sal 43832	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑆 ∈ SAlg → ∅ ∈ 𝑆)
53	5, 52	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ∅ ∈ 𝑆)
54		snssi 4747	. . . . . . . . . . . . . . . . . . 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 4125	. . . . . . . . . . . . . . . 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 6616	. . . . . . . . . . . . . 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 1370	. . . . . . . . . . . 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 716	. . . . . . . . . . 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 6834	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝑀 = (𝑦 ∈ 𝑆 ↦ (𝑀‘𝑦)))
66	65	reseq1d 5889	. . . . . . . . . . . . . . . . . . 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 5947	. . . . . . . . . . . . . . . . . 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 4747	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (∅ ∈ 𝑥 → {∅} ⊆ 𝑥)
72		ssequn2 4122	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ({∅} ⊆ 𝑥 ↔ (𝑥 ∪ {∅}) = 𝑥)
73	71, 72	sylib 217	. . . . . . . . . . . . . . . . . . . 20 ⊢ (∅ ∈ 𝑥 → (𝑥 ∪ {∅}) = 𝑥)
74	73	eqcomd 2746	. . . . . . . . . . . . . . . . . . 19 ⊢ (∅ ∈ 𝑥 → 𝑥 = (𝑥 ∪ {∅}))
75	74	mpteq1d 5174	. . . . . . . . . . . . . . . . . 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 2784	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ ∅ ∈ 𝑥) → (𝑀 ↾ 𝑥) = (𝑦 ∈ (𝑥 ∪ {∅}) ↦ (𝑀‘𝑦)))
78	77	fveq2d 6775	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ ∅ ∈ 𝑥) → (Σ^{^}‘(𝑀 ↾ 𝑥)) = (Σ^{^}‘(𝑦 ∈ (𝑥 ∪ {∅}) ↦ (𝑀‘𝑦))))
79		nfv 1921	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑦((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ ¬ ∅ ∈ 𝑥)
80		simplr 766	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ ¬ ∅ ∈ 𝑥) → 𝑥 ∈ 𝒫 dom 𝑀)
81		p0ex 5311	. . . . . . . . . . . . . . . . . 18 ⊢ {∅} ∈ V
82	81	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ ¬ ∅ ∈ 𝑥) → {∅} ∈ V)
83		disjsn 4653	. . . . . . . . . . . . . . . . . . 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 723	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ 𝑦 ∈ 𝑥) → 𝑀:𝑆⟶(0[,]+∞))
87	51	sselda 3926	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ 𝑆)
88	86, 87	ffvelrnd 6959	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ 𝑦 ∈ 𝑥) → (𝑀‘𝑦) ∈ (0[,]+∞))
89	88	adantlr 712	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ ¬ ∅ ∈ 𝑥) ∧ 𝑦 ∈ 𝑥) → (𝑀‘𝑦) ∈ (0[,]+∞))
90		elsni 4584	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 ∈ {∅} → 𝑦 = ∅)
91	90	fveq2d 6775	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 ∈ {∅} → (𝑀‘𝑦) = (𝑀‘∅))
92	91	adantl 482	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑦 ∈ {∅}) → (𝑀‘𝑦) = (𝑀‘∅))
93	1, 53	ffvelrnd 6959	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝑀‘∅) ∈ (0[,]+∞))
94	93	adantr 481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑦 ∈ {∅}) → (𝑀‘∅) ∈ (0[,]+∞))
95	92, 94	eqeltrd 2841	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑦 ∈ {∅}) → (𝑀‘𝑦) ∈ (0[,]+∞))
96	95	ad4ant14 749	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ ¬ ∅ ∈ 𝑥) ∧ 𝑦 ∈ {∅}) → (𝑀‘𝑦) ∈ (0[,]+∞))
97	79, 80, 82, 85, 89, 96	sge0splitmpt 43920	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ ¬ ∅ ∈ 𝑥) → (Σ^{^}‘(𝑦 ∈ (𝑥 ∪ {∅}) ↦ (𝑀‘𝑦))) = ((Σ^{^}‘(𝑦 ∈ 𝑥 ↦ (𝑀‘𝑦))) +_𝑒 (Σ^{^}‘(𝑦 ∈ {∅} ↦ (𝑀‘𝑦)))))
98		fveq2 6771	. . . . . . . . . . . . . . . . . . . . . . . 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 2780	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑦 = ∅) → (𝑀‘𝑦) = 0)
102	90, 101	sylan2 593	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑦 ∈ {∅}) → (𝑀‘𝑦) = 0)
103	102	mpteq2dva 5179	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝑦 ∈ {∅} ↦ (𝑀‘𝑦)) = (𝑦 ∈ {∅} ↦ 0))
104	103	fveq2d 6775	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (Σ^{^}‘(𝑦 ∈ {∅} ↦ (𝑀‘𝑦))) = (Σ^{^}‘(𝑦 ∈ {∅} ↦ 0)))
105		nfv 1921	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑦𝜑
106	81	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → {∅} ∈ V)
107	105, 106	sge0z 43884	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (Σ^{^}‘(𝑦 ∈ {∅} ↦ 0)) = 0)
108	104, 107	eqtrd 2780	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (Σ^{^}‘(𝑦 ∈ {∅} ↦ (𝑀‘𝑦))) = 0)
109	108	oveq2d 7287	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((Σ^{^}‘(𝑦 ∈ 𝑥 ↦ (𝑀‘𝑦))) +_𝑒 (Σ^{^}‘(𝑦 ∈ {∅} ↦ (𝑀‘𝑦)))) = ((Σ^{^}‘(𝑦 ∈ 𝑥 ↦ (𝑀‘𝑦))) +_𝑒 0))
110	109	ad2antrr 723	. . . . . . . . . . . . . . . 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 2780	. . . . . . . . . . . . . . . . . . . . 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 6639	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) → (𝑀 ↾ 𝑥):𝑥⟶(0[,]+∞))
115	112, 114	feq1dd 42673	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) → (𝑦 ∈ 𝑥 ↦ (𝑀‘𝑦)):𝑥⟶(0[,]+∞))
116	111, 115	sge0xrcl 43894	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) → (Σ^{^}‘(𝑦 ∈ 𝑥 ↦ (𝑀‘𝑦))) ∈ ℝ^*)
117	116	xaddid1d 12976	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) → ((Σ^{^}‘(𝑦 ∈ 𝑥 ↦ (𝑀‘𝑦))) +_𝑒 0) = (Σ^{^}‘(𝑦 ∈ 𝑥 ↦ (𝑀‘𝑦))))
118	112	fveq2d 6775	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) → (Σ^{^}‘(𝑀 ↾ 𝑥)) = (Σ^{^}‘(𝑦 ∈ 𝑥 ↦ (𝑀‘𝑦))))
119	118	eqcomd 2746	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) → (Σ^{^}‘(𝑦 ∈ 𝑥 ↦ (𝑀‘𝑦))) = (Σ^{^}‘(𝑀 ↾ 𝑥)))
120	117, 119	eqtrd 2780	. . . . . . . . . . . . . . . . 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 2785	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ ¬ ∅ ∈ 𝑥) → (Σ^{^}‘(𝑀 ↾ 𝑥)) = (Σ^{^}‘(𝑦 ∈ (𝑥 ∪ {∅}) ↦ (𝑀‘𝑦))))
123	78, 122	pm2.61dan 810	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) → (Σ^{^}‘(𝑀 ↾ 𝑥)) = (Σ^{^}‘(𝑦 ∈ (𝑥 ∪ {∅}) ↦ (𝑀‘𝑦))))
124	123	ad2antrr 723	. . . . . . . . . . . . 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 1921	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑦(((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ 𝑒:ℕ–onto→(𝑥 ∪ {∅})) ∧ Disj 𝑛 ∈ ℕ (𝑒‘𝑛))
126		nfv 1921	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑛((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ 𝑒:ℕ–onto→(𝑥 ∪ {∅}))
127		nfdisj1 5058	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑛Disj 𝑛 ∈ ℕ (𝑒‘𝑛)
128	126, 127	nfan 1906	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑛(((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ 𝑒:ℕ–onto→(𝑥 ∪ {∅})) ∧ Disj 𝑛 ∈ ℕ (𝑒‘𝑛))
129		fveq2 6771	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝑒‘𝑛) → (𝑀‘𝑦) = (𝑀‘(𝑒‘𝑛)))
130		nnex 11979	. . . . . . . . . . . . . . 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 766	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ 𝑒:ℕ–onto→(𝑥 ∪ {∅})) ∧ Disj 𝑛 ∈ ℕ (𝑒‘𝑛)) → 𝑒:ℕ–onto→(𝑥 ∪ {∅}))
133		eqidd 2741	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ 𝑒:ℕ–onto→(𝑥 ∪ {∅})) ∧ Disj 𝑛 ∈ ℕ (𝑒‘𝑛)) ∧ 𝑛 ∈ ℕ) → (𝑒‘𝑛) = (𝑒‘𝑛))
134	1	ad2antrr 723	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ 𝑦 ∈ (𝑥 ∪ {∅})) → 𝑀:𝑆⟶(0[,]+∞))
135	57	sselda 3926	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ 𝑦 ∈ (𝑥 ∪ {∅})) → 𝑦 ∈ 𝑆)
136	134, 135	ffvelrnd 6959	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝒫 dom 𝑀) ∧ 𝑦 ∈ (𝑥 ∪ {∅})) → (𝑀‘𝑦) ∈ (0[,]+∞))
137	136	ad4ant14 749	. . . . . . . . . . . . . 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 43926	. . . . . . . . . . . . 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 2781	. . . . . . . . . . . 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 716	. . . . . . . . . . 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 2784	. . . . . . . . . 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 835	. . . . . . . . 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 1940	. . . . . . 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 810	. . . . 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 3110	. . 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 43960	. 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 1086 = wceq 1542 ∃wex 1786 ∈ wcel 2110 ≠ wne 2945 ∀wral 3066 Vcvv 3431 ∪ cun 3890 ∩ cin 3891 ⊆ wss 3892 ∅c0 4262 𝒫 cpw 4539 {csn 4567 ∪ cuni 4845 ∪ ciun 4930 Disj wdisj 5044 class class class wbr 5079 ↦ cmpt 5162 dom cdm 5590 ↾ cres 5592 ⟶wf 6428 –onto→wfo 6430 ‘cfv 6432 (class class class)co 7271 ωcom 7706 ≼ cdom 8714 0cc0 10872 +∞cpnf 11007 ℕcn 11973 +_𝑒 cxad 12845 [,]cicc 13081 SAlgcsalg 43820 Σ^{^}csumge0 43871 Meascmea 43958

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7582 ax-inf2 9377 ax-cnex 10928 ax-resscn 10929 ax-1cn 10930 ax-icn 10931 ax-addcl 10932 ax-addrcl 10933 ax-mulcl 10934 ax-mulrcl 10935 ax-mulcom 10936 ax-addass 10937 ax-mulass 10938 ax-distr 10939 ax-i2m1 10940 ax-1ne0 10941 ax-1rid 10942 ax-rnegex 10943 ax-rrecex 10944 ax-cnre 10945 ax-pre-lttri 10946 ax-pre-lttrn 10947 ax-pre-ltadd 10948 ax-pre-mulgt0 10949 ax-pre-sup 10950

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rmo 3074 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-int 4886 df-iun 4932 df-disj 5045 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5490 df-eprel 5496 df-po 5504 df-so 5505 df-fr 5545 df-se 5546 df-we 5547 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-pred 6201 df-ord 6268 df-on 6269 df-lim 6270 df-suc 6271 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-isom 6441 df-riota 7228 df-ov 7274 df-oprab 7275 df-mpo 7276 df-om 7707 df-1st 7824 df-2nd 7825 df-frecs 8088 df-wrecs 8119 df-recs 8193 df-rdg 8232 df-1o 8288 df-er 8481 df-en 8717 df-dom 8718 df-sdom 8719 df-fin 8720 df-sup 9179 df-oi 9247 df-card 9698 df-pnf 11012 df-mnf 11013 df-xr 11014 df-ltxr 11015 df-le 11016 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-n0 12234 df-z 12320 df-uz 12582 df-rp 12730 df-xadd 12848 df-ico 13084 df-icc 13085 df-fz 13239 df-fzo 13382 df-seq 13720 df-exp 13781 df-hash 14043 df-cj 14808 df-re 14809 df-im 14810 df-sqrt 14944 df-abs 14945 df-clim 15195 df-sum 15396 df-salg 43821 df-sumge0 43872 df-mea 43959

This theorem is referenced by: volmea 43983 caratheodory 44037

W3C validator