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Theorem measvuni 33890

Description: The measure of a countable disjoint union is the sum of the measures. This theorem uses a collection rather than a set of subsets of 𝑆. (Contributed by Thierry Arnoux, 7-Mar-2017.)

**Assertion**
Ref	Expression
measvuni	⊢ ((𝑀 ∈ (measures‘𝑆) ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑆 ∧ (𝐴 ≼ ω ∧ Disj 𝑥 ∈ 𝐴 𝐵)) → (𝑀‘∪ 𝑥 ∈ 𝐴 𝐵) = Σ^*𝑥 ∈ 𝐴(𝑀‘𝐵))

Distinct variable groups: 𝑥,𝐴 𝑥,𝑀 𝑥,𝑆 Allowed substitution hint: 𝐵(𝑥)

**Proof of Theorem measvuni**
Step	Hyp	Ref	Expression
1		simp1 1133	. . . 4 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑆 ∧ (𝐴 ≼ ω ∧ Disj 𝑥 ∈ 𝐴 𝐵)) → 𝑀 ∈ (measures‘𝑆))
2		rabid 3440	. . . . . . . 8 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ {∅}} ↔ (𝑥 ∈ 𝐴 ∧ 𝐵 ∈ {∅}))
3	2	simprbi 495	. . . . . . 7 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ {∅}} → 𝐵 ∈ {∅})
4	3	adantl 480	. . . . . 6 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ {∅}}) → 𝐵 ∈ {∅})
5	4	ralrimiva 3136	. . . . 5 ⊢ (𝑀 ∈ (measures‘𝑆) → ∀𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ {∅}}𝐵 ∈ {∅})
6	5	3ad2ant1 1130	. . . 4 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑆 ∧ (𝐴 ≼ ω ∧ Disj 𝑥 ∈ 𝐴 𝐵)) → ∀𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ {∅}}𝐵 ∈ {∅})
7		ssrab2 4069	. . . . . . 7 ⊢ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ {∅}} ⊆ 𝐴
8		ssct 9074	. . . . . . 7 ⊢ (({𝑥 ∈ 𝐴 ∣ 𝐵 ∈ {∅}} ⊆ 𝐴 ∧ 𝐴 ≼ ω) → {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ {∅}} ≼ ω)
9	7, 8	mpan 688	. . . . . 6 ⊢ (𝐴 ≼ ω → {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ {∅}} ≼ ω)
10	9	adantr 479	. . . . 5 ⊢ ((𝐴 ≼ ω ∧ Disj 𝑥 ∈ 𝐴 𝐵) → {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ {∅}} ≼ ω)
11	10	3ad2ant3 1132	. . . 4 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑆 ∧ (𝐴 ≼ ω ∧ Disj 𝑥 ∈ 𝐴 𝐵)) → {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ {∅}} ≼ ω)
12		simp3r 1199	. . . . 5 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑆 ∧ (𝐴 ≼ ω ∧ Disj 𝑥 ∈ 𝐴 𝐵)) → Disj 𝑥 ∈ 𝐴 𝐵)
13		nfrab1 3439	. . . . . 6 ⊢ Ⅎ𝑥{𝑥 ∈ 𝐴 ∣ 𝐵 ∈ {∅}}
14		nfcv 2892	. . . . . 6 ⊢ Ⅎ𝑥𝐴
15	13, 14	disjss1f 32407	. . . . 5 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝐵 ∈ {∅}} ⊆ 𝐴 → (Disj 𝑥 ∈ 𝐴 𝐵 → Disj 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ {∅}}𝐵))
16	7, 12, 15	mpsyl 68	. . . 4 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑆 ∧ (𝐴 ≼ ω ∧ Disj 𝑥 ∈ 𝐴 𝐵)) → Disj 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ {∅}}𝐵)
17	13	measvunilem0 33889	. . . 4 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ ∀𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ {∅}}𝐵 ∈ {∅} ∧ ({𝑥 ∈ 𝐴 ∣ 𝐵 ∈ {∅}} ≼ ω ∧ Disj 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ {∅}}𝐵)) → (𝑀‘∪ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ {∅}}𝐵) = Σ^*𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ {∅}} (𝑀‘𝐵))
18	1, 6, 11, 16, 17	syl112anc 1371	. . 3 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑆 ∧ (𝐴 ≼ ω ∧ Disj 𝑥 ∈ 𝐴 𝐵)) → (𝑀‘∪ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ {∅}}𝐵) = Σ^*𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ {∅}} (𝑀‘𝐵))
19		rabid 3440	. . . . . . . 8 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ (𝑆 ∖ {∅})} ↔ (𝑥 ∈ 𝐴 ∧ 𝐵 ∈ (𝑆 ∖ {∅})))
20	19	simprbi 495	. . . . . . 7 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ (𝑆 ∖ {∅})} → 𝐵 ∈ (𝑆 ∖ {∅}))
21	20	adantl 480	. . . . . 6 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ (𝑆 ∖ {∅})}) → 𝐵 ∈ (𝑆 ∖ {∅}))
22	21	ralrimiva 3136	. . . . 5 ⊢ (𝑀 ∈ (measures‘𝑆) → ∀𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ (𝑆 ∖ {∅})}𝐵 ∈ (𝑆 ∖ {∅}))
23	22	3ad2ant1 1130	. . . 4 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑆 ∧ (𝐴 ≼ ω ∧ Disj 𝑥 ∈ 𝐴 𝐵)) → ∀𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ (𝑆 ∖ {∅})}𝐵 ∈ (𝑆 ∖ {∅}))
24		ssrab2 4069	. . . . . . 7 ⊢ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ (𝑆 ∖ {∅})} ⊆ 𝐴
25		ssct 9074	. . . . . . 7 ⊢ (({𝑥 ∈ 𝐴 ∣ 𝐵 ∈ (𝑆 ∖ {∅})} ⊆ 𝐴 ∧ 𝐴 ≼ ω) → {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ (𝑆 ∖ {∅})} ≼ ω)
26	24, 25	mpan 688	. . . . . 6 ⊢ (𝐴 ≼ ω → {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ (𝑆 ∖ {∅})} ≼ ω)
27	26	adantr 479	. . . . 5 ⊢ ((𝐴 ≼ ω ∧ Disj 𝑥 ∈ 𝐴 𝐵) → {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ (𝑆 ∖ {∅})} ≼ ω)
28	27	3ad2ant3 1132	. . . 4 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑆 ∧ (𝐴 ≼ ω ∧ Disj 𝑥 ∈ 𝐴 𝐵)) → {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ (𝑆 ∖ {∅})} ≼ ω)
29		nfrab1 3439	. . . . . 6 ⊢ Ⅎ𝑥{𝑥 ∈ 𝐴 ∣ 𝐵 ∈ (𝑆 ∖ {∅})}
30	29, 14	disjss1f 32407	. . . . 5 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝐵 ∈ (𝑆 ∖ {∅})} ⊆ 𝐴 → (Disj 𝑥 ∈ 𝐴 𝐵 → Disj 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ (𝑆 ∖ {∅})}𝐵))
31	24, 12, 30	mpsyl 68	. . . 4 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑆 ∧ (𝐴 ≼ ω ∧ Disj 𝑥 ∈ 𝐴 𝐵)) → Disj 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ (𝑆 ∖ {∅})}𝐵)
32	29	measvunilem 33888	. . . 4 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ ∀𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ (𝑆 ∖ {∅})}𝐵 ∈ (𝑆 ∖ {∅}) ∧ ({𝑥 ∈ 𝐴 ∣ 𝐵 ∈ (𝑆 ∖ {∅})} ≼ ω ∧ Disj 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ (𝑆 ∖ {∅})}𝐵)) → (𝑀‘∪ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ (𝑆 ∖ {∅})}𝐵) = Σ^*𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ (𝑆 ∖ {∅})} (𝑀‘𝐵))
33	1, 23, 28, 31, 32	syl112anc 1371	. . 3 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑆 ∧ (𝐴 ≼ ω ∧ Disj 𝑥 ∈ 𝐴 𝐵)) → (𝑀‘∪ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ (𝑆 ∖ {∅})}𝐵) = Σ^*𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ (𝑆 ∖ {∅})} (𝑀‘𝐵))
34	18, 33	oveq12d 7434	. 2 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑆 ∧ (𝐴 ≼ ω ∧ Disj 𝑥 ∈ 𝐴 𝐵)) → ((𝑀‘∪ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ {∅}}𝐵) +_𝑒 (𝑀‘∪ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ (𝑆 ∖ {∅})}𝐵)) = (Σ^𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ {∅}} (𝑀‘𝐵) +_𝑒 Σ^𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ (𝑆 ∖ {∅})} (𝑀‘𝐵)))
35		nfv 1909	. . . . . . 7 ⊢ Ⅎ𝑥 𝑀 ∈ (measures‘𝑆)
36		nfra1 3272	. . . . . . 7 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑆
37		nfv 1909	. . . . . . . 8 ⊢ Ⅎ𝑥 𝐴 ≼ ω
38		nfdisj1 5122	. . . . . . . 8 ⊢ Ⅎ𝑥Disj 𝑥 ∈ 𝐴 𝐵
39	37, 38	nfan 1894	. . . . . . 7 ⊢ Ⅎ𝑥(𝐴 ≼ ω ∧ Disj 𝑥 ∈ 𝐴 𝐵)
40	35, 36, 39	nf3an 1896	. . . . . 6 ⊢ Ⅎ𝑥(𝑀 ∈ (measures‘𝑆) ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑆 ∧ (𝐴 ≼ ω ∧ Disj 𝑥 ∈ 𝐴 𝐵))
41	13, 29	nfun 4158	. . . . . 6 ⊢ Ⅎ𝑥({𝑥 ∈ 𝐴 ∣ 𝐵 ∈ {∅}} ∪ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ (𝑆 ∖ {∅})})
42		simp2 1134	. . . . . . . . 9 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑆 ∧ (𝐴 ≼ ω ∧ Disj 𝑥 ∈ 𝐴 𝐵)) → ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑆)
43		rabid2 3453	. . . . . . . . 9 ⊢ (𝐴 = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ 𝑆} ↔ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑆)
44	42, 43	sylibr 233	. . . . . . . 8 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑆 ∧ (𝐴 ≼ ω ∧ Disj 𝑥 ∈ 𝐴 𝐵)) → 𝐴 = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ 𝑆})
45		elun 4141	. . . . . . . . . . 11 ⊢ (𝐵 ∈ ({∅} ∪ (𝑆 ∖ {∅})) ↔ (𝐵 ∈ {∅} ∨ 𝐵 ∈ (𝑆 ∖ {∅})))
46		measbase 33873	. . . . . . . . . . . . . 14 ⊢ (𝑀 ∈ (measures‘𝑆) → 𝑆 ∈ ∪ ran sigAlgebra)
47		0elsiga 33790	. . . . . . . . . . . . . 14 ⊢ (𝑆 ∈ ∪ ran sigAlgebra → ∅ ∈ 𝑆)
48		snssi 4807	. . . . . . . . . . . . . 14 ⊢ (∅ ∈ 𝑆 → {∅} ⊆ 𝑆)
49	46, 47, 48	3syl 18	. . . . . . . . . . . . 13 ⊢ (𝑀 ∈ (measures‘𝑆) → {∅} ⊆ 𝑆)
50		undif 4477	. . . . . . . . . . . . 13 ⊢ ({∅} ⊆ 𝑆 ↔ ({∅} ∪ (𝑆 ∖ {∅})) = 𝑆)
51	49, 50	sylib 217	. . . . . . . . . . . 12 ⊢ (𝑀 ∈ (measures‘𝑆) → ({∅} ∪ (𝑆 ∖ {∅})) = 𝑆)
52	51	eleq2d 2811	. . . . . . . . . . 11 ⊢ (𝑀 ∈ (measures‘𝑆) → (𝐵 ∈ ({∅} ∪ (𝑆 ∖ {∅})) ↔ 𝐵 ∈ 𝑆))
53	45, 52	bitr3id 284	. . . . . . . . . 10 ⊢ (𝑀 ∈ (measures‘𝑆) → ((𝐵 ∈ {∅} ∨ 𝐵 ∈ (𝑆 ∖ {∅})) ↔ 𝐵 ∈ 𝑆))
54	53	rabbidv 3427	. . . . . . . . 9 ⊢ (𝑀 ∈ (measures‘𝑆) → {𝑥 ∈ 𝐴 ∣ (𝐵 ∈ {∅} ∨ 𝐵 ∈ (𝑆 ∖ {∅}))} = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ 𝑆})
55	54	3ad2ant1 1130	. . . . . . . 8 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑆 ∧ (𝐴 ≼ ω ∧ Disj 𝑥 ∈ 𝐴 𝐵)) → {𝑥 ∈ 𝐴 ∣ (𝐵 ∈ {∅} ∨ 𝐵 ∈ (𝑆 ∖ {∅}))} = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ 𝑆})
56	44, 55	eqtr4d 2768	. . . . . . 7 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑆 ∧ (𝐴 ≼ ω ∧ Disj 𝑥 ∈ 𝐴 𝐵)) → 𝐴 = {𝑥 ∈ 𝐴 ∣ (𝐵 ∈ {∅} ∨ 𝐵 ∈ (𝑆 ∖ {∅}))})
57		unrab 4300	. . . . . . 7 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝐵 ∈ {∅}} ∪ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ (𝑆 ∖ {∅})}) = {𝑥 ∈ 𝐴 ∣ (𝐵 ∈ {∅} ∨ 𝐵 ∈ (𝑆 ∖ {∅}))}
58	56, 57	eqtr4di 2783	. . . . . 6 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑆 ∧ (𝐴 ≼ ω ∧ Disj 𝑥 ∈ 𝐴 𝐵)) → 𝐴 = ({𝑥 ∈ 𝐴 ∣ 𝐵 ∈ {∅}} ∪ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ (𝑆 ∖ {∅})}))
59		eqidd 2726	. . . . . 6 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑆 ∧ (𝐴 ≼ ω ∧ Disj 𝑥 ∈ 𝐴 𝐵)) → 𝐵 = 𝐵)
60	40, 14, 41, 58, 59	iuneq12df 5017	. . . . 5 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑆 ∧ (𝐴 ≼ ω ∧ Disj 𝑥 ∈ 𝐴 𝐵)) → ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑥 ∈ ({𝑥 ∈ 𝐴 ∣ 𝐵 ∈ {∅}} ∪ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ (𝑆 ∖ {∅})})𝐵)
61	60	fveq2d 6896	. . . 4 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑆 ∧ (𝐴 ≼ ω ∧ Disj 𝑥 ∈ 𝐴 𝐵)) → (𝑀‘∪ 𝑥 ∈ 𝐴 𝐵) = (𝑀‘∪ 𝑥 ∈ ({𝑥 ∈ 𝐴 ∣ 𝐵 ∈ {∅}} ∪ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ (𝑆 ∖ {∅})})𝐵))
62		iunxun 5092	. . . . 5 ⊢ ∪ 𝑥 ∈ ({𝑥 ∈ 𝐴 ∣ 𝐵 ∈ {∅}} ∪ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ (𝑆 ∖ {∅})})𝐵 = (∪ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ {∅}}𝐵 ∪ ∪ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ (𝑆 ∖ {∅})}𝐵)
63	62	fveq2i 6895	. . . 4 ⊢ (𝑀‘∪ 𝑥 ∈ ({𝑥 ∈ 𝐴 ∣ 𝐵 ∈ {∅}} ∪ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ (𝑆 ∖ {∅})})𝐵) = (𝑀‘(∪ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ {∅}}𝐵 ∪ ∪ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ (𝑆 ∖ {∅})}𝐵))
64	61, 63	eqtrdi 2781	. . 3 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑆 ∧ (𝐴 ≼ ω ∧ Disj 𝑥 ∈ 𝐴 𝐵)) → (𝑀‘∪ 𝑥 ∈ 𝐴 𝐵) = (𝑀‘(∪ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ {∅}}𝐵 ∪ ∪ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ (𝑆 ∖ {∅})}𝐵)))
65	46	3ad2ant1 1130	. . . . 5 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑆 ∧ (𝐴 ≼ ω ∧ Disj 𝑥 ∈ 𝐴 𝐵)) → 𝑆 ∈ ∪ ran sigAlgebra)
66	47	adantr 479	. . . . . . . . 9 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐵 ∈ {∅}) → ∅ ∈ 𝑆)
67		elsni 4641	. . . . . . . . . . 11 ⊢ (𝐵 ∈ {∅} → 𝐵 = ∅)
68	67	eleq1d 2810	. . . . . . . . . 10 ⊢ (𝐵 ∈ {∅} → (𝐵 ∈ 𝑆 ↔ ∅ ∈ 𝑆))
69	68	adantl 480	. . . . . . . . 9 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐵 ∈ {∅}) → (𝐵 ∈ 𝑆 ↔ ∅ ∈ 𝑆))
70	66, 69	mpbird 256	. . . . . . . 8 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐵 ∈ {∅}) → 𝐵 ∈ 𝑆)
71	46, 3, 70	syl2an 594	. . . . . . 7 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ {∅}}) → 𝐵 ∈ 𝑆)
72	71	ralrimiva 3136	. . . . . 6 ⊢ (𝑀 ∈ (measures‘𝑆) → ∀𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ {∅}}𝐵 ∈ 𝑆)
73	72	3ad2ant1 1130	. . . . 5 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑆 ∧ (𝐴 ≼ ω ∧ Disj 𝑥 ∈ 𝐴 𝐵)) → ∀𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ {∅}}𝐵 ∈ 𝑆)
74	13	sigaclcuni 33794	. . . . 5 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ ∀𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ {∅}}𝐵 ∈ 𝑆 ∧ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ {∅}} ≼ ω) → ∪ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ {∅}}𝐵 ∈ 𝑆)
75	65, 73, 11, 74	syl3anc 1368	. . . 4 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑆 ∧ (𝐴 ≼ ω ∧ Disj 𝑥 ∈ 𝐴 𝐵)) → ∪ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ {∅}}𝐵 ∈ 𝑆)
76	21	eldifad 3951	. . . . . . 7 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ (𝑆 ∖ {∅})}) → 𝐵 ∈ 𝑆)
77	76	ralrimiva 3136	. . . . . 6 ⊢ (𝑀 ∈ (measures‘𝑆) → ∀𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ (𝑆 ∖ {∅})}𝐵 ∈ 𝑆)
78	77	3ad2ant1 1130	. . . . 5 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑆 ∧ (𝐴 ≼ ω ∧ Disj 𝑥 ∈ 𝐴 𝐵)) → ∀𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ (𝑆 ∖ {∅})}𝐵 ∈ 𝑆)
79	29	sigaclcuni 33794	. . . . 5 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ ∀𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ (𝑆 ∖ {∅})}𝐵 ∈ 𝑆 ∧ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ (𝑆 ∖ {∅})} ≼ ω) → ∪ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ (𝑆 ∖ {∅})}𝐵 ∈ 𝑆)
80	65, 78, 28, 79	syl3anc 1368	. . . 4 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑆 ∧ (𝐴 ≼ ω ∧ Disj 𝑥 ∈ 𝐴 𝐵)) → ∪ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ (𝑆 ∖ {∅})}𝐵 ∈ 𝑆)
81	3, 67	syl 17	. . . . . . 7 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ {∅}} → 𝐵 = ∅)
82	81	iuneq2i 5012	. . . . . 6 ⊢ ∪ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ {∅}}𝐵 = ∪ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ {∅}}∅
83		iun0 5060	. . . . . 6 ⊢ ∪ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ {∅}}∅ = ∅
84	82, 83	eqtri 2753	. . . . 5 ⊢ ∪ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ {∅}}𝐵 = ∅
85		ineq1 4199	. . . . . 6 ⊢ (∪ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ {∅}}𝐵 = ∅ → (∪ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ {∅}}𝐵 ∩ ∪ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ (𝑆 ∖ {∅})}𝐵) = (∅ ∩ ∪ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ (𝑆 ∖ {∅})}𝐵))
86		0in 4389	. . . . . 6 ⊢ (∅ ∩ ∪ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ (𝑆 ∖ {∅})}𝐵) = ∅
87	85, 86	eqtrdi 2781	. . . . 5 ⊢ (∪ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ {∅}}𝐵 = ∅ → (∪ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ {∅}}𝐵 ∩ ∪ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ (𝑆 ∖ {∅})}𝐵) = ∅)
88	84, 87	mp1i 13	. . . 4 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑆 ∧ (𝐴 ≼ ω ∧ Disj 𝑥 ∈ 𝐴 𝐵)) → (∪ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ {∅}}𝐵 ∩ ∪ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ (𝑆 ∖ {∅})}𝐵) = ∅)
89		measun 33887	. . . 4 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ (∪ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ {∅}}𝐵 ∈ 𝑆 ∧ ∪ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ (𝑆 ∖ {∅})}𝐵 ∈ 𝑆) ∧ (∪ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ {∅}}𝐵 ∩ ∪ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ (𝑆 ∖ {∅})}𝐵) = ∅) → (𝑀‘(∪ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ {∅}}𝐵 ∪ ∪ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ (𝑆 ∖ {∅})}𝐵)) = ((𝑀‘∪ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ {∅}}𝐵) +_𝑒 (𝑀‘∪ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ (𝑆 ∖ {∅})}𝐵)))
90	1, 75, 80, 88, 89	syl121anc 1372	. . 3 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑆 ∧ (𝐴 ≼ ω ∧ Disj 𝑥 ∈ 𝐴 𝐵)) → (𝑀‘(∪ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ {∅}}𝐵 ∪ ∪ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ (𝑆 ∖ {∅})}𝐵)) = ((𝑀‘∪ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ {∅}}𝐵) +_𝑒 (𝑀‘∪ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ (𝑆 ∖ {∅})}𝐵)))
91	64, 90	eqtrd 2765	. 2 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑆 ∧ (𝐴 ≼ ω ∧ Disj 𝑥 ∈ 𝐴 𝐵)) → (𝑀‘∪ 𝑥 ∈ 𝐴 𝐵) = ((𝑀‘∪ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ {∅}}𝐵) +_𝑒 (𝑀‘∪ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ (𝑆 ∖ {∅})}𝐵)))
92	40, 58	esumeq1d 33711	. . 3 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑆 ∧ (𝐴 ≼ ω ∧ Disj 𝑥 ∈ 𝐴 𝐵)) → Σ^𝑥 ∈ 𝐴(𝑀‘𝐵) = Σ^𝑥 ∈ ({𝑥 ∈ 𝐴 ∣ 𝐵 ∈ {∅}} ∪ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ (𝑆 ∖ {∅})})(𝑀‘𝐵))
93		ctex 8982	. . . . 5 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝐵 ∈ {∅}} ≼ ω → {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ {∅}} ∈ V)
94	11, 93	syl 17	. . . 4 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑆 ∧ (𝐴 ≼ ω ∧ Disj 𝑥 ∈ 𝐴 𝐵)) → {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ {∅}} ∈ V)
95		ctex 8982	. . . . 5 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝐵 ∈ (𝑆 ∖ {∅})} ≼ ω → {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ (𝑆 ∖ {∅})} ∈ V)
96	28, 95	syl 17	. . . 4 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑆 ∧ (𝐴 ≼ ω ∧ Disj 𝑥 ∈ 𝐴 𝐵)) → {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ (𝑆 ∖ {∅})} ∈ V)
97		inrab 4301	. . . . . 6 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝐵 ∈ {∅}} ∩ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ (𝑆 ∖ {∅})}) = {𝑥 ∈ 𝐴 ∣ (𝐵 ∈ {∅} ∧ 𝐵 ∈ (𝑆 ∖ {∅}))}
98		noel 4326	. . . . . . . . . 10 ⊢ ¬ 𝐵 ∈ ∅
99		disjdif 4467	. . . . . . . . . . 11 ⊢ ({∅} ∩ (𝑆 ∖ {∅})) = ∅
100	99	eleq2i 2817	. . . . . . . . . 10 ⊢ (𝐵 ∈ ({∅} ∩ (𝑆 ∖ {∅})) ↔ 𝐵 ∈ ∅)
101	98, 100	mtbir 322	. . . . . . . . 9 ⊢ ¬ 𝐵 ∈ ({∅} ∩ (𝑆 ∖ {∅}))
102		elin 3955	. . . . . . . . 9 ⊢ (𝐵 ∈ ({∅} ∩ (𝑆 ∖ {∅})) ↔ (𝐵 ∈ {∅} ∧ 𝐵 ∈ (𝑆 ∖ {∅})))
103	101, 102	mtbi 321	. . . . . . . 8 ⊢ ¬ (𝐵 ∈ {∅} ∧ 𝐵 ∈ (𝑆 ∖ {∅}))
104	103	rgenw 3055	. . . . . . 7 ⊢ ∀𝑥 ∈ 𝐴 ¬ (𝐵 ∈ {∅} ∧ 𝐵 ∈ (𝑆 ∖ {∅}))
105		rabeq0 4380	. . . . . . 7 ⊢ ({𝑥 ∈ 𝐴 ∣ (𝐵 ∈ {∅} ∧ 𝐵 ∈ (𝑆 ∖ {∅}))} = ∅ ↔ ∀𝑥 ∈ 𝐴 ¬ (𝐵 ∈ {∅} ∧ 𝐵 ∈ (𝑆 ∖ {∅})))
106	104, 105	mpbir 230	. . . . . 6 ⊢ {𝑥 ∈ 𝐴 ∣ (𝐵 ∈ {∅} ∧ 𝐵 ∈ (𝑆 ∖ {∅}))} = ∅
107	97, 106	eqtri 2753	. . . . 5 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝐵 ∈ {∅}} ∩ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ (𝑆 ∖ {∅})}) = ∅
108	107	a1i 11	. . . 4 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑆 ∧ (𝐴 ≼ ω ∧ Disj 𝑥 ∈ 𝐴 𝐵)) → ({𝑥 ∈ 𝐴 ∣ 𝐵 ∈ {∅}} ∩ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ (𝑆 ∖ {∅})}) = ∅)
109	1	adantr 479	. . . . 5 ⊢ (((𝑀 ∈ (measures‘𝑆) ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑆 ∧ (𝐴 ≼ ω ∧ Disj 𝑥 ∈ 𝐴 𝐵)) ∧ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ {∅}}) → 𝑀 ∈ (measures‘𝑆))
110	1, 71	sylan 578	. . . . 5 ⊢ (((𝑀 ∈ (measures‘𝑆) ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑆 ∧ (𝐴 ≼ ω ∧ Disj 𝑥 ∈ 𝐴 𝐵)) ∧ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ {∅}}) → 𝐵 ∈ 𝑆)
111		measvxrge0 33881	. . . . 5 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝐵 ∈ 𝑆) → (𝑀‘𝐵) ∈ (0[,]+∞))
112	109, 110, 111	syl2anc 582	. . . 4 ⊢ (((𝑀 ∈ (measures‘𝑆) ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑆 ∧ (𝐴 ≼ ω ∧ Disj 𝑥 ∈ 𝐴 𝐵)) ∧ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ {∅}}) → (𝑀‘𝐵) ∈ (0[,]+∞))
113	1	adantr 479	. . . . 5 ⊢ (((𝑀 ∈ (measures‘𝑆) ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑆 ∧ (𝐴 ≼ ω ∧ Disj 𝑥 ∈ 𝐴 𝐵)) ∧ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ (𝑆 ∖ {∅})}) → 𝑀 ∈ (measures‘𝑆))
114	20	adantl 480	. . . . . 6 ⊢ (((𝑀 ∈ (measures‘𝑆) ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑆 ∧ (𝐴 ≼ ω ∧ Disj 𝑥 ∈ 𝐴 𝐵)) ∧ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ (𝑆 ∖ {∅})}) → 𝐵 ∈ (𝑆 ∖ {∅}))
115	114	eldifad 3951	. . . . 5 ⊢ (((𝑀 ∈ (measures‘𝑆) ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑆 ∧ (𝐴 ≼ ω ∧ Disj 𝑥 ∈ 𝐴 𝐵)) ∧ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ (𝑆 ∖ {∅})}) → 𝐵 ∈ 𝑆)
116	113, 115, 111	syl2anc 582	. . . 4 ⊢ (((𝑀 ∈ (measures‘𝑆) ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑆 ∧ (𝐴 ≼ ω ∧ Disj 𝑥 ∈ 𝐴 𝐵)) ∧ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ (𝑆 ∖ {∅})}) → (𝑀‘𝐵) ∈ (0[,]+∞))
117	40, 13, 29, 94, 96, 108, 112, 116	esumsplit 33729	. . 3 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑆 ∧ (𝐴 ≼ ω ∧ Disj 𝑥 ∈ 𝐴 𝐵)) → Σ^𝑥 ∈ ({𝑥 ∈ 𝐴 ∣ 𝐵 ∈ {∅}} ∪ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ (𝑆 ∖ {∅})})(𝑀‘𝐵) = (Σ^𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ {∅}} (𝑀‘𝐵) +_𝑒 Σ^*𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ (𝑆 ∖ {∅})} (𝑀‘𝐵)))
118	92, 117	eqtrd 2765	. 2 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑆 ∧ (𝐴 ≼ ω ∧ Disj 𝑥 ∈ 𝐴 𝐵)) → Σ^𝑥 ∈ 𝐴(𝑀‘𝐵) = (Σ^𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ {∅}} (𝑀‘𝐵) +_𝑒 Σ^*𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ (𝑆 ∖ {∅})} (𝑀‘𝐵)))
119	34, 91, 118	3eqtr4d 2775	1 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑆 ∧ (𝐴 ≼ ω ∧ Disj 𝑥 ∈ 𝐴 𝐵)) → (𝑀‘∪ 𝑥 ∈ 𝐴 𝐵) = Σ^*𝑥 ∈ 𝐴(𝑀‘𝐵))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 394 ∨ wo 845 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ∀wral 3051 {crab 3419 Vcvv 3463 ∖ cdif 3936 ∪ cun 3937 ∩ cin 3938 ⊆ wss 3939 ∅c0 4318 {csn 4624 ∪ cuni 4903 ∪ ciun 4991 Disj wdisj 5108 class class class wbr 5143 ran crn 5673 ‘cfv 6543 (class class class)co 7416 ωcom 7868 ≼ cdom 8960 0cc0 11138 +∞cpnf 11275 +_𝑒 cxad 13122 [,]cicc 13359 Σ^*cesum 33703 sigAlgebracsiga 33784 measurescmeas 33871

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-inf2 9664 ax-ac2 10486 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 ax-pre-sup 11216 ax-addf 11217 ax-mulf 11218

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-iin 4994 df-disj 5109 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 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 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-of 7682 df-om 7869 df-1st 7991 df-2nd 7992 df-supp 8164 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-2o 8486 df-er 8723 df-map 8845 df-pm 8846 df-ixp 8915 df-en 8963 df-dom 8964 df-sdom 8965 df-fin 8966 df-fsupp 9386 df-fi 9434 df-sup 9465 df-inf 9466 df-oi 9533 df-dju 9924 df-card 9962 df-acn 9965 df-ac 10139 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-div 11902 df-nn 12243 df-2 12305 df-3 12306 df-4 12307 df-5 12308 df-6 12309 df-7 12310 df-8 12311 df-9 12312 df-n0 12503 df-z 12589 df-dec 12708 df-uz 12853 df-q 12963 df-rp 13007 df-xneg 13124 df-xadd 13125 df-xmul 13126 df-ioo 13360 df-ioc 13361 df-ico 13362 df-icc 13363 df-fz 13517 df-fzo 13660 df-fl 13789 df-mod 13867 df-seq 13999 df-exp 14059 df-fac 14265 df-bc 14294 df-hash 14322 df-shft 15046 df-cj 15078 df-re 15079 df-im 15080 df-sqrt 15214 df-abs 15215 df-limsup 15447 df-clim 15464 df-rlim 15465 df-sum 15665 df-ef 16043 df-sin 16045 df-cos 16046 df-pi 16048 df-struct 17115 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17180 df-ress 17209 df-plusg 17245 df-mulr 17246 df-starv 17247 df-sca 17248 df-vsca 17249 df-ip 17250 df-tset 17251 df-ple 17252 df-ds 17254 df-unif 17255 df-hom 17256 df-cco 17257 df-rest 17403 df-topn 17404 df-0g 17422 df-gsum 17423 df-topgen 17424 df-pt 17425 df-prds 17428 df-ordt 17482 df-xrs 17483 df-qtop 17488 df-imas 17489 df-xps 17491 df-mre 17565 df-mrc 17566 df-acs 17568 df-ps 18557 df-tsr 18558 df-plusf 18598 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-mhm 18739 df-submnd 18740 df-grp 18897 df-minusg 18898 df-sbg 18899 df-mulg 19028 df-subg 19082 df-cntz 19272 df-cmn 19741 df-abl 19742 df-mgp 20079 df-rng 20097 df-ur 20126 df-ring 20179 df-cring 20180 df-subrng 20487 df-subrg 20512 df-abv 20701 df-lmod 20749 df-scaf 20750 df-sra 21062 df-rgmod 21063 df-psmet 21275 df-xmet 21276 df-met 21277 df-bl 21278 df-mopn 21279 df-fbas 21280 df-fg 21281 df-cnfld 21284 df-top 22814 df-topon 22831 df-topsp 22853 df-bases 22867 df-cld 22941 df-ntr 22942 df-cls 22943 df-nei 23020 df-lp 23058 df-perf 23059 df-cn 23149 df-cnp 23150 df-haus 23237 df-tx 23484 df-hmeo 23677 df-fil 23768 df-fm 23860 df-flim 23861 df-flf 23862 df-tmd 23994 df-tgp 23995 df-tsms 24049 df-trg 24082 df-xms 24244 df-ms 24245 df-tms 24246 df-nm 24509 df-ngp 24510 df-nrg 24512 df-nlm 24513 df-ii 24815 df-cncf 24816 df-limc 25813 df-dv 25814 df-log 26508 df-esum 33704 df-siga 33785 df-meas 33872

This theorem is referenced by: measiuns 33893 measinblem 33896 sibfof 34017 dstrvprob 34148

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