Theorem meaiunlelem

Description: The measure of the union of countable sets is less than or equal to the sum of the measures, Property 112C (d) of Fremlin1 p. 16. (Contributed by Glauco Siliprandi, 17-Aug-2020)

	Ref	Expression
Hypotheses	meaiunlelem.1	⊢ Ⅎ 𝑛 𝜑
	meaiunlelem.m	⊢ ( 𝜑 → 𝑀 ∈ Meas )
	meaiunlelem.s	⊢ 𝑆 = dom 𝑀
	meaiunlelem.z	⊢ 𝑍 = ( ℤ≥ ‘ 𝑁 )
	meaiunlelem.e	⊢ ( 𝜑 → 𝐸 : 𝑍 ⟶ 𝑆 )
	meaiunlelem.f	⊢ 𝐹 = ( 𝑛 ∈ 𝑍 ↦ ( ( 𝐸 ‘ 𝑛 ) ∖ ∪ 𝑖 ∈ ( 𝑁 ..^ 𝑛 ) ( 𝐸 ‘ 𝑖 ) ) )
Assertion	meaiunlelem	⊢ ( 𝜑 → ( 𝑀 ‘ ∪ 𝑛 ∈ 𝑍 ( 𝐸 ‘ 𝑛 ) ) ≤ ( Σ^ ‘ ( 𝑛 ∈ 𝑍 ↦ ( 𝑀 ‘ ( 𝐸 ‘ 𝑛 ) ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		meaiunlelem.1	⊢ Ⅎ 𝑛 𝜑
2		meaiunlelem.m	⊢ ( 𝜑 → 𝑀 ∈ Meas )
3		meaiunlelem.s	⊢ 𝑆 = dom 𝑀
4		meaiunlelem.z	⊢ 𝑍 = ( ℤ≥ ‘ 𝑁 )
5		meaiunlelem.e	⊢ ( 𝜑 → 𝐸 : 𝑍 ⟶ 𝑆 )
6		meaiunlelem.f	⊢ 𝐹 = ( 𝑛 ∈ 𝑍 ↦ ( ( 𝐸 ‘ 𝑛 ) ∖ ∪ 𝑖 ∈ ( 𝑁 ..^ 𝑛 ) ( 𝐸 ‘ 𝑖 ) ) )
7	1 4 5 6	iundjiun	⊢ ( 𝜑 → ( ( ∀ 𝑥 ∈ 𝑍 ∪ 𝑛 ∈ ( 𝑁 ... 𝑥 ) ( 𝐹 ‘ 𝑛 ) = ∪ 𝑛 ∈ ( 𝑁 ... 𝑥 ) ( 𝐸 ‘ 𝑛 ) ∧ ∪ 𝑛 ∈ 𝑍 ( 𝐹 ‘ 𝑛 ) = ∪ 𝑛 ∈ 𝑍 ( 𝐸 ‘ 𝑛 ) ) ∧ Disj 𝑛 ∈ 𝑍 ( 𝐹 ‘ 𝑛 ) ) )
8	7	simplrd	⊢ ( 𝜑 → ∪ 𝑛 ∈ 𝑍 ( 𝐹 ‘ 𝑛 ) = ∪ 𝑛 ∈ 𝑍 ( 𝐸 ‘ 𝑛 ) )
9	8	eqcomd	⊢ ( 𝜑 → ∪ 𝑛 ∈ 𝑍 ( 𝐸 ‘ 𝑛 ) = ∪ 𝑛 ∈ 𝑍 ( 𝐹 ‘ 𝑛 ) )
10	9	fveq2d	⊢ ( 𝜑 → ( 𝑀 ‘ ∪ 𝑛 ∈ 𝑍 ( 𝐸 ‘ 𝑛 ) ) = ( 𝑀 ‘ ∪ 𝑛 ∈ 𝑍 ( 𝐹 ‘ 𝑛 ) ) )
11	2 3	dmmeasal	⊢ ( 𝜑 → 𝑆 ∈ SAlg )
12	11	adantr	⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → 𝑆 ∈ SAlg )
13	5	ffvelcdmda	⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ( 𝐸 ‘ 𝑛 ) ∈ 𝑆 )
14		fzofi	⊢ ( 𝑁 ..^ 𝑛 ) ∈ Fin
15		isfinite	⊢ ( ( 𝑁 ..^ 𝑛 ) ∈ Fin ↔ ( 𝑁 ..^ 𝑛 ) ≺ ω )
16	15	biimpi	⊢ ( ( 𝑁 ..^ 𝑛 ) ∈ Fin → ( 𝑁 ..^ 𝑛 ) ≺ ω )
17		sdomdom	⊢ ( ( 𝑁 ..^ 𝑛 ) ≺ ω → ( 𝑁 ..^ 𝑛 ) ≼ ω )
18	16 17	syl	⊢ ( ( 𝑁 ..^ 𝑛 ) ∈ Fin → ( 𝑁 ..^ 𝑛 ) ≼ ω )
19	14 18	ax-mp	⊢ ( 𝑁 ..^ 𝑛 ) ≼ ω
20	19	a1i	⊢ ( 𝜑 → ( 𝑁 ..^ 𝑛 ) ≼ ω )
21	5	adantr	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 𝑁 ..^ 𝑛 ) ) → 𝐸 : 𝑍 ⟶ 𝑆 )
22		elfzouz	⊢ ( 𝑖 ∈ ( 𝑁 ..^ 𝑛 ) → 𝑖 ∈ ( ℤ≥ ‘ 𝑁 ) )
23	4	eqcomi	⊢ ( ℤ≥ ‘ 𝑁 ) = 𝑍
24	22 23	eleqtrdi	⊢ ( 𝑖 ∈ ( 𝑁 ..^ 𝑛 ) → 𝑖 ∈ 𝑍 )
25	24	adantl	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 𝑁 ..^ 𝑛 ) ) → 𝑖 ∈ 𝑍 )
26	21 25	ffvelcdmd	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 𝑁 ..^ 𝑛 ) ) → ( 𝐸 ‘ 𝑖 ) ∈ 𝑆 )
27	11 20 26	saliuncl	⊢ ( 𝜑 → ∪ 𝑖 ∈ ( 𝑁 ..^ 𝑛 ) ( 𝐸 ‘ 𝑖 ) ∈ 𝑆 )
28	27	adantr	⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ∪ 𝑖 ∈ ( 𝑁 ..^ 𝑛 ) ( 𝐸 ‘ 𝑖 ) ∈ 𝑆 )
29		saldifcl2	⊢ ( ( 𝑆 ∈ SAlg ∧ ( 𝐸 ‘ 𝑛 ) ∈ 𝑆 ∧ ∪ 𝑖 ∈ ( 𝑁 ..^ 𝑛 ) ( 𝐸 ‘ 𝑖 ) ∈ 𝑆 ) → ( ( 𝐸 ‘ 𝑛 ) ∖ ∪ 𝑖 ∈ ( 𝑁 ..^ 𝑛 ) ( 𝐸 ‘ 𝑖 ) ) ∈ 𝑆 )
30	12 13 28 29	syl3anc	⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ( ( 𝐸 ‘ 𝑛 ) ∖ ∪ 𝑖 ∈ ( 𝑁 ..^ 𝑛 ) ( 𝐸 ‘ 𝑖 ) ) ∈ 𝑆 )
31	1 30 6	fmptdf	⊢ ( 𝜑 → 𝐹 : 𝑍 ⟶ 𝑆 )
32	31	ffvelcdmda	⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ( 𝐹 ‘ 𝑛 ) ∈ 𝑆 )
33		eqid	⊢ ( ℤ≥ ‘ 𝑁 ) = ( ℤ≥ ‘ 𝑁 )
34	33	uzct	⊢ ( ℤ≥ ‘ 𝑁 ) ≼ ω
35	4 34	eqbrtri	⊢ 𝑍 ≼ ω
36	35	a1i	⊢ ( 𝜑 → 𝑍 ≼ ω )
37	7	simprd	⊢ ( 𝜑 → Disj 𝑛 ∈ 𝑍 ( 𝐹 ‘ 𝑛 ) )
38	1 2 3 32 36 37	meadjiun	⊢ ( 𝜑 → ( 𝑀 ‘ ∪ 𝑛 ∈ 𝑍 ( 𝐹 ‘ 𝑛 ) ) = ( Σ^ ‘ ( 𝑛 ∈ 𝑍 ↦ ( 𝑀 ‘ ( 𝐹 ‘ 𝑛 ) ) ) ) )
39		eqidd	⊢ ( 𝜑 → ( Σ^ ‘ ( 𝑛 ∈ 𝑍 ↦ ( 𝑀 ‘ ( 𝐹 ‘ 𝑛 ) ) ) ) = ( Σ^ ‘ ( 𝑛 ∈ 𝑍 ↦ ( 𝑀 ‘ ( 𝐹 ‘ 𝑛 ) ) ) ) )
40	10 38 39	3eqtrd	⊢ ( 𝜑 → ( 𝑀 ‘ ∪ 𝑛 ∈ 𝑍 ( 𝐸 ‘ 𝑛 ) ) = ( Σ^ ‘ ( 𝑛 ∈ 𝑍 ↦ ( 𝑀 ‘ ( 𝐹 ‘ 𝑛 ) ) ) ) )
41	4	fvexi	⊢ 𝑍 ∈ V
42	41	a1i	⊢ ( 𝜑 → 𝑍 ∈ V )
43	2	adantr	⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → 𝑀 ∈ Meas )
44	43 3 32	meacl	⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ( 𝑀 ‘ ( 𝐹 ‘ 𝑛 ) ) ∈ ( 0 [,] +∞ ) )
45	43 3 13	meacl	⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ( 𝑀 ‘ ( 𝐸 ‘ 𝑛 ) ) ∈ ( 0 [,] +∞ ) )
46		simpr	⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → 𝑛 ∈ 𝑍 )
47	13	difexd	⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ( ( 𝐸 ‘ 𝑛 ) ∖ ∪ 𝑖 ∈ ( 𝑁 ..^ 𝑛 ) ( 𝐸 ‘ 𝑖 ) ) ∈ V )
48	6	fvmpt2	⊢ ( ( 𝑛 ∈ 𝑍 ∧ ( ( 𝐸 ‘ 𝑛 ) ∖ ∪ 𝑖 ∈ ( 𝑁 ..^ 𝑛 ) ( 𝐸 ‘ 𝑖 ) ) ∈ V ) → ( 𝐹 ‘ 𝑛 ) = ( ( 𝐸 ‘ 𝑛 ) ∖ ∪ 𝑖 ∈ ( 𝑁 ..^ 𝑛 ) ( 𝐸 ‘ 𝑖 ) ) )
49	46 47 48	syl2anc	⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ( 𝐹 ‘ 𝑛 ) = ( ( 𝐸 ‘ 𝑛 ) ∖ ∪ 𝑖 ∈ ( 𝑁 ..^ 𝑛 ) ( 𝐸 ‘ 𝑖 ) ) )
50		difssd	⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ( ( 𝐸 ‘ 𝑛 ) ∖ ∪ 𝑖 ∈ ( 𝑁 ..^ 𝑛 ) ( 𝐸 ‘ 𝑖 ) ) ⊆ ( 𝐸 ‘ 𝑛 ) )
51	49 50	eqsstrd	⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ( 𝐹 ‘ 𝑛 ) ⊆ ( 𝐸 ‘ 𝑛 ) )
52	43 3 32 13 51	meassle	⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ( 𝑀 ‘ ( 𝐹 ‘ 𝑛 ) ) ≤ ( 𝑀 ‘ ( 𝐸 ‘ 𝑛 ) ) )
53	1 42 44 45 52	sge0lempt	⊢ ( 𝜑 → ( Σ^ ‘ ( 𝑛 ∈ 𝑍 ↦ ( 𝑀 ‘ ( 𝐹 ‘ 𝑛 ) ) ) ) ≤ ( Σ^ ‘ ( 𝑛 ∈ 𝑍 ↦ ( 𝑀 ‘ ( 𝐸 ‘ 𝑛 ) ) ) ) )
54	40 53	eqbrtrd	⊢ ( 𝜑 → ( 𝑀 ‘ ∪ 𝑛 ∈ 𝑍 ( 𝐸 ‘ 𝑛 ) ) ≤ ( Σ^ ‘ ( 𝑛 ∈ 𝑍 ↦ ( 𝑀 ‘ ( 𝐸 ‘ 𝑛 ) ) ) ) )