meaiininclem - Mathbox for Glauco Siliprandi

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Theorem meaiininclem 47301

Description: Measures are continuous from above: if 𝐸 is a nonincreasing sequence of measurable sets, and any of the sets has finite measure, then the measure of the intersection is the limit of the measures. This is Proposition 112C (f) of [Fremlin1] p. 16. (Contributed by Glauco Siliprandi, 8-Apr-2021.)

**Hypotheses**
Ref	Expression
meaiininclem.m	⊢ (𝜑 → 𝑀 ∈ Meas)
meaiininclem.n	⊢ (𝜑 → 𝑁 ∈ ℤ)
meaiininclem.z	⊢ 𝑍 = (ℤ_≥‘𝑁)
meaiininclem.e	⊢ (𝜑 → 𝐸:𝑍⟶dom 𝑀)
meaiininclem.i	⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐸‘(𝑛 + 1)) ⊆ (𝐸‘𝑛))
meaiininclem.k	⊢ (𝜑 → 𝐾 ∈ (ℤ_≥‘𝑁))
meaiininclem.r	⊢ (𝜑 → (𝑀‘(𝐸‘𝐾)) ∈ ℝ)
meaiininclem.s	⊢ 𝑆 = (𝑛 ∈ 𝑍 ↦ (𝑀‘(𝐸‘𝑛)))
meaiininclem.g	⊢ 𝐺 = (𝑛 ∈ 𝑍 ↦ ((𝐸‘𝐾) ∖ (𝐸‘𝑛)))
meaiininclem.f	⊢ 𝐹 = ∪ 𝑛 ∈ 𝑍 (𝐺‘𝑛)

**Assertion**
Ref	Expression
meaiininclem	⊢ (𝜑 → 𝑆 ⇝ (𝑀‘∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛)))

Distinct variable groups: 𝑛,𝐸 𝑛,𝐹 𝑛,𝐺 𝑛,𝐾 𝑛,𝑀 𝑛,𝑁 𝑛,𝑍 𝜑,𝑛 Allowed substitution hint: 𝑆(𝑛)

Proof of Theorem meaiininclem Dummy variables 𝑚 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		meaiininclem.k	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐾 ∈ (ℤ_≥‘𝑁))
2		uzss 12913	. . . . . . . . . . . . . 14 ⊢ (𝐾 ∈ (ℤ_≥‘𝑁) → (ℤ_≥‘𝐾) ⊆ (ℤ_≥‘𝑁))
3	1, 2	syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → (ℤ_≥‘𝐾) ⊆ (ℤ_≥‘𝑁))
4		meaiininclem.z	. . . . . . . . . . . . 13 ⊢ 𝑍 = (ℤ_≥‘𝑁)
5	3, 4	sseqtrrdi 3975	. . . . . . . . . . . 12 ⊢ (𝜑 → (ℤ_≥‘𝐾) ⊆ 𝑍)
6	5	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝐾)) → (ℤ_≥‘𝐾) ⊆ 𝑍)
7		simpr 490	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝐾)) → 𝑛 ∈ (ℤ_≥‘𝐾))
8	6, 7	sseldd 3935	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝐾)) → 𝑛 ∈ 𝑍)
9		meaiininclem.g	. . . . . . . . . . . 12 ⊢ 𝐺 = (𝑛 ∈ 𝑍 ↦ ((𝐸‘𝐾) ∖ (𝐸‘𝑛)))
10	9	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐺 = (𝑛 ∈ 𝑍 ↦ ((𝐸‘𝐾) ∖ (𝐸‘𝑛))))
11		meaiininclem.m	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑀 ∈ Meas)
12		eqid 2762	. . . . . . . . . . . . . . 15 ⊢ dom 𝑀 = dom 𝑀
13	11, 12	dmmeasal 47267	. . . . . . . . . . . . . 14 ⊢ (𝜑 → dom 𝑀 ∈ SAlg)
14	13	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → dom 𝑀 ∈ SAlg)
15	1, 4	eleqtrrdi 2873	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐾 ∈ 𝑍)
16		meaiininclem.e	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐸:𝑍⟶dom 𝑀)
17	16	ffvelcdmda 7080	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝐾 ∈ 𝑍) → (𝐸‘𝐾) ∈ dom 𝑀)
18	15, 17	mpdan 700	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐸‘𝐾) ∈ dom 𝑀)
19	18	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐸‘𝐾) ∈ dom 𝑀)
20	16	ffvelcdmda 7080	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐸‘𝑛) ∈ dom 𝑀)
21		saldifcl2 47143	. . . . . . . . . . . . 13 ⊢ ((dom 𝑀 ∈ SAlg ∧ (𝐸‘𝐾) ∈ dom 𝑀 ∧ (𝐸‘𝑛) ∈ dom 𝑀) → ((𝐸‘𝐾) ∖ (𝐸‘𝑛)) ∈ dom 𝑀)
22	14, 19, 20, 21	syl3anc 1398	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ((𝐸‘𝐾) ∖ (𝐸‘𝑛)) ∈ dom 𝑀)
23	22	elexd 3476	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ((𝐸‘𝐾) ∖ (𝐸‘𝑛)) ∈ V)
24	10, 23	fvmpt2d 7004	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐺‘𝑛) = ((𝐸‘𝐾) ∖ (𝐸‘𝑛)))
25	8, 24	syldan 603	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝐾)) → (𝐺‘𝑛) = ((𝐸‘𝐾) ∖ (𝐸‘𝑛)))
26	25	fveq2d 6886	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝐾)) → (𝑀‘(𝐺‘𝑛)) = (𝑀‘((𝐸‘𝐾) ∖ (𝐸‘𝑛))))
27	11	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝐾)) → 𝑀 ∈ Meas)
28	18	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝐾)) → (𝐸‘𝐾) ∈ dom 𝑀)
29		meaiininclem.r	. . . . . . . . . 10 ⊢ (𝜑 → (𝑀‘(𝐸‘𝐾)) ∈ ℝ)
30	29	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝐾)) → (𝑀‘(𝐸‘𝐾)) ∈ ℝ)
31	8, 20	syldan 603	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝐾)) → (𝐸‘𝑛) ∈ dom 𝑀)
32		simpl 488	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ (𝐾..^𝑛)) → 𝜑)
33	32, 5	syl 18	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ (𝐾..^𝑛)) → (ℤ_≥‘𝐾) ⊆ 𝑍)
34		elfzouz 13721	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ (𝐾..^𝑛) → 𝑚 ∈ (ℤ_≥‘𝐾))
35	34	adantl 487	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ (𝐾..^𝑛)) → 𝑚 ∈ (ℤ_≥‘𝐾))
36	33, 35	sseldd 3935	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ (𝐾..^𝑛)) → 𝑚 ∈ 𝑍)
37		eleq1w 2845	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑚 → (𝑛 ∈ 𝑍 ↔ 𝑚 ∈ 𝑍))
38	37	anbi2d 642	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑚 → ((𝜑 ∧ 𝑛 ∈ 𝑍) ↔ (𝜑 ∧ 𝑚 ∈ 𝑍)))
39		fvoveq1 7439	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑚 → (𝐸‘(𝑛 + 1)) = (𝐸‘(𝑚 + 1)))
40		fveq2 6882	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑚 → (𝐸‘𝑛) = (𝐸‘𝑚))
41	39, 40	sseq12d 3967	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑚 → ((𝐸‘(𝑛 + 1)) ⊆ (𝐸‘𝑛) ↔ (𝐸‘(𝑚 + 1)) ⊆ (𝐸‘𝑚)))
42	38, 41	imbi12d 347	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑚 → (((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐸‘(𝑛 + 1)) ⊆ (𝐸‘𝑛)) ↔ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (𝐸‘(𝑚 + 1)) ⊆ (𝐸‘𝑚))))
43		meaiininclem.i	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐸‘(𝑛 + 1)) ⊆ (𝐸‘𝑛))
44	42, 43	chvarvv 2022	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (𝐸‘(𝑚 + 1)) ⊆ (𝐸‘𝑚))
45	32, 36, 44	syl2anc 596	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ (𝐾..^𝑛)) → (𝐸‘(𝑚 + 1)) ⊆ (𝐸‘𝑚))
46	45	adantlr 728	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝐾)) ∧ 𝑚 ∈ (𝐾..^𝑛)) → (𝐸‘(𝑚 + 1)) ⊆ (𝐸‘𝑚))
47	7, 46	ssdec 45907	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝐾)) → (𝐸‘𝑛) ⊆ (𝐸‘𝐾))
48	27, 28, 30, 31, 47	meadif 47294	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝐾)) → (𝑀‘((𝐸‘𝐾) ∖ (𝐸‘𝑛))) = ((𝑀‘(𝐸‘𝐾)) − (𝑀‘(𝐸‘𝑛))))
49	26, 48	eqtrd 2797	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝐾)) → (𝑀‘(𝐺‘𝑛)) = ((𝑀‘(𝐸‘𝐾)) − (𝑀‘(𝐸‘𝑛))))
50	49	oveq2d 7432	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝐾)) → ((𝑀‘(𝐸‘𝐾)) − (𝑀‘(𝐺‘𝑛))) = ((𝑀‘(𝐸‘𝐾)) − ((𝑀‘(𝐸‘𝐾)) − (𝑀‘(𝐸‘𝑛)))))
51	29	recnd 11264	. . . . . . . 8 ⊢ (𝜑 → (𝑀‘(𝐸‘𝐾)) ∈ ℂ)
52	51	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝐾)) → (𝑀‘(𝐸‘𝐾)) ∈ ℂ)
53	27, 28, 30, 47, 31	meassre 47292	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝐾)) → (𝑀‘(𝐸‘𝑛)) ∈ ℝ)
54	53	recnd 11264	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝐾)) → (𝑀‘(𝐸‘𝑛)) ∈ ℂ)
55	52, 54	nncand 11601	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝐾)) → ((𝑀‘(𝐸‘𝐾)) − ((𝑀‘(𝐸‘𝐾)) − (𝑀‘(𝐸‘𝑛)))) = (𝑀‘(𝐸‘𝑛)))
56	50, 55	eqtr2d 2798	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝐾)) → (𝑀‘(𝐸‘𝑛)) = ((𝑀‘(𝐸‘𝐾)) − (𝑀‘(𝐺‘𝑛))))
57	56	mpteq2dva 5202	. . . 4 ⊢ (𝜑 → (𝑛 ∈ (ℤ_≥‘𝐾) ↦ (𝑀‘(𝐸‘𝑛))) = (𝑛 ∈ (ℤ_≥‘𝐾) ↦ ((𝑀‘(𝐸‘𝐾)) − (𝑀‘(𝐺‘𝑛)))))
58		nfv 1947	. . . . 5 ⊢ Ⅎ𝑛𝜑
59		eqid 2762	. . . . 5 ⊢ (ℤ_≥‘𝐾) = (ℤ_≥‘𝐾)
60	1	eluzelzd 46191	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ ℤ)
61		difssd 4087	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ((𝐸‘𝐾) ∖ (𝐸‘𝑛)) ⊆ (𝐸‘𝐾))
62	24, 61	eqsstrd 3968	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐺‘𝑛) ⊆ (𝐸‘𝐾))
63	8, 62	syldan 603	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝐾)) → (𝐺‘𝑛) ⊆ (𝐸‘𝐾))
64	22, 9	fmptd 7110	. . . . . . . . 9 ⊢ (𝜑 → 𝐺:𝑍⟶dom 𝑀)
65	64	ffvelcdmda 7080	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐺‘𝑛) ∈ dom 𝑀)
66	8, 65	syldan 603	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝐾)) → (𝐺‘𝑛) ∈ dom 𝑀)
67	27, 28, 30, 63, 66	meassre 47292	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝐾)) → (𝑀‘(𝐺‘𝑛)) ∈ ℝ)
68	67	recnd 11264	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝐾)) → (𝑀‘(𝐺‘𝑛)) ∈ ℂ)
69		meaiininclem.n	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℤ)
70	43	sscond 4096	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ((𝐸‘𝐾) ∖ (𝐸‘𝑛)) ⊆ ((𝐸‘𝐾) ∖ (𝐸‘(𝑛 + 1))))
71	40	difeq2d 4077	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑚 → ((𝐸‘𝐾) ∖ (𝐸‘𝑛)) = ((𝐸‘𝐾) ∖ (𝐸‘𝑚)))
72	71	cbvmptv 5213	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ 𝑍 ↦ ((𝐸‘𝐾) ∖ (𝐸‘𝑛))) = (𝑚 ∈ 𝑍 ↦ ((𝐸‘𝐾) ∖ (𝐸‘𝑚)))
73	9, 72	eqtri 2785	. . . . . . . . . . 11 ⊢ 𝐺 = (𝑚 ∈ 𝑍 ↦ ((𝐸‘𝐾) ∖ (𝐸‘𝑚)))
74		fveq2 6882	. . . . . . . . . . . 12 ⊢ (𝑚 = (𝑛 + 1) → (𝐸‘𝑚) = (𝐸‘(𝑛 + 1)))
75	74	difeq2d 4077	. . . . . . . . . . 11 ⊢ (𝑚 = (𝑛 + 1) → ((𝐸‘𝐾) ∖ (𝐸‘𝑚)) = ((𝐸‘𝐾) ∖ (𝐸‘(𝑛 + 1))))
76	4	peano2uzs 12954	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ 𝑍 → (𝑛 + 1) ∈ 𝑍)
77	76	adantl 487	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑛 + 1) ∈ 𝑍)
78		fvex 6895	. . . . . . . . . . . . 13 ⊢ (𝐸‘𝐾) ∈ V
79	78	difexi 5299	. . . . . . . . . . . 12 ⊢ ((𝐸‘𝐾) ∖ (𝐸‘(𝑛 + 1))) ∈ V
80	79	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ((𝐸‘𝐾) ∖ (𝐸‘(𝑛 + 1))) ∈ V)
81	73, 75, 77, 80	fvmptd3 7014	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐺‘(𝑛 + 1)) = ((𝐸‘𝐾) ∖ (𝐸‘(𝑛 + 1))))
82	24, 81	sseq12d 3967	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ((𝐺‘𝑛) ⊆ (𝐺‘(𝑛 + 1)) ↔ ((𝐸‘𝐾) ∖ (𝐸‘𝑛)) ⊆ ((𝐸‘𝐾) ∖ (𝐸‘(𝑛 + 1)))))
83	70, 82	mpbird 260	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐺‘𝑛) ⊆ (𝐺‘(𝑛 + 1)))
84	11	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝑀 ∈ Meas)
85	84, 12, 65, 19, 62	meassle 47278	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑀‘(𝐺‘𝑛)) ≤ (𝑀‘(𝐸‘𝐾)))
86		eqid 2762	. . . . . . . 8 ⊢ (𝑛 ∈ 𝑍 ↦ (𝑀‘(𝐺‘𝑛))) = (𝑛 ∈ 𝑍 ↦ (𝑀‘(𝐺‘𝑛)))
87	11, 69, 4, 64, 83, 29, 85, 86	meaiuninc2 47297	. . . . . . 7 ⊢ (𝜑 → (𝑛 ∈ 𝑍 ↦ (𝑀‘(𝐺‘𝑛))) ⇝ (𝑀‘∪ 𝑛 ∈ 𝑍 (𝐺‘𝑛)))
88		eqid 2762	. . . . . . . 8 ⊢ (𝑛 ∈ (ℤ_≥‘𝐾) ↦ (𝑀‘(𝐺‘𝑛))) = (𝑛 ∈ (ℤ_≥‘𝐾) ↦ (𝑀‘(𝐺‘𝑛)))
89	4, 86, 15, 88	climresmpt 46474	. . . . . . 7 ⊢ (𝜑 → ((𝑛 ∈ (ℤ_≥‘𝐾) ↦ (𝑀‘(𝐺‘𝑛))) ⇝ (𝑀‘∪ 𝑛 ∈ 𝑍 (𝐺‘𝑛)) ↔ (𝑛 ∈ 𝑍 ↦ (𝑀‘(𝐺‘𝑛))) ⇝ (𝑀‘∪ 𝑛 ∈ 𝑍 (𝐺‘𝑛))))
90	87, 89	mpbird 260	. . . . . 6 ⊢ (𝜑 → (𝑛 ∈ (ℤ_≥‘𝐾) ↦ (𝑀‘(𝐺‘𝑛))) ⇝ (𝑀‘∪ 𝑛 ∈ 𝑍 (𝐺‘𝑛)))
91		meaiininclem.f	. . . . . . . . 9 ⊢ 𝐹 = ∪ 𝑛 ∈ 𝑍 (𝐺‘𝑛)
92	91	eqcomi 2771	. . . . . . . 8 ⊢ ∪ 𝑛 ∈ 𝑍 (𝐺‘𝑛) = 𝐹
93	92	fveq2i 6885	. . . . . . 7 ⊢ (𝑀‘∪ 𝑛 ∈ 𝑍 (𝐺‘𝑛)) = (𝑀‘𝐹)
94	93	a1i 11	. . . . . 6 ⊢ (𝜑 → (𝑀‘∪ 𝑛 ∈ 𝑍 (𝐺‘𝑛)) = (𝑀‘𝐹))
95	90, 94	breqtrd 5135	. . . . 5 ⊢ (𝜑 → (𝑛 ∈ (ℤ_≥‘𝐾) ↦ (𝑀‘(𝐺‘𝑛))) ⇝ (𝑀‘𝐹))
96	58, 59, 60, 51, 68, 95	climsubc1mpt 46477	. . . 4 ⊢ (𝜑 → (𝑛 ∈ (ℤ_≥‘𝐾) ↦ ((𝑀‘(𝐸‘𝐾)) − (𝑀‘(𝐺‘𝑛)))) ⇝ ((𝑀‘(𝐸‘𝐾)) − (𝑀‘𝐹)))
97	57, 96	eqbrtrd 5131	. . 3 ⊢ (𝜑 → (𝑛 ∈ (ℤ_≥‘𝐾) ↦ (𝑀‘(𝐸‘𝑛))) ⇝ ((𝑀‘(𝐸‘𝐾)) − (𝑀‘𝐹)))
98		eqid 2762	. . . 4 ⊢ (𝑛 ∈ 𝑍 ↦ (𝑀‘(𝐸‘𝑛))) = (𝑛 ∈ 𝑍 ↦ (𝑀‘(𝐸‘𝑛)))
99		eqid 2762	. . . 4 ⊢ (𝑛 ∈ (ℤ_≥‘𝐾) ↦ (𝑀‘(𝐸‘𝑛))) = (𝑛 ∈ (ℤ_≥‘𝐾) ↦ (𝑀‘(𝐸‘𝑛)))
100	4, 98, 15, 99	climresmpt 46474	. . 3 ⊢ (𝜑 → ((𝑛 ∈ (ℤ_≥‘𝐾) ↦ (𝑀‘(𝐸‘𝑛))) ⇝ ((𝑀‘(𝐸‘𝐾)) − (𝑀‘𝐹)) ↔ (𝑛 ∈ 𝑍 ↦ (𝑀‘(𝐸‘𝑛))) ⇝ ((𝑀‘(𝐸‘𝐾)) − (𝑀‘𝐹))))
101	97, 100	mpbid 235	. 2 ⊢ (𝜑 → (𝑛 ∈ 𝑍 ↦ (𝑀‘(𝐸‘𝑛))) ⇝ ((𝑀‘(𝐸‘𝐾)) − (𝑀‘𝐹)))
102		meaiininclem.s	. . . 4 ⊢ 𝑆 = (𝑛 ∈ 𝑍 ↦ (𝑀‘(𝐸‘𝑛)))
103	102	a1i 11	. . 3 ⊢ (𝜑 → 𝑆 = (𝑛 ∈ 𝑍 ↦ (𝑀‘(𝐸‘𝑛))))
104		eqidd 2763	. . . . . 6 ⊢ (𝜑 → (𝑀‘(𝐹 ∪ ((𝐸‘𝐾) ∖ 𝐹))) = (𝑀‘(𝐹 ∪ ((𝐸‘𝐾) ∖ 𝐹))))
105	4	uzct 45884	. . . . . . . . . 10 ⊢ 𝑍 ≼ ω
106	105	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 𝑍 ≼ ω)
107	13, 106, 65	saliuncl 47138	. . . . . . . 8 ⊢ (𝜑 → ∪ 𝑛 ∈ 𝑍 (𝐺‘𝑛) ∈ dom 𝑀)
108	91, 107	eqeltrid 2866	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ dom 𝑀)
109		saldifcl2 47143	. . . . . . . 8 ⊢ ((dom 𝑀 ∈ SAlg ∧ (𝐸‘𝐾) ∈ dom 𝑀 ∧ 𝐹 ∈ dom 𝑀) → ((𝐸‘𝐾) ∖ 𝐹) ∈ dom 𝑀)
110	13, 18, 108, 109	syl3anc 1398	. . . . . . 7 ⊢ (𝜑 → ((𝐸‘𝐾) ∖ 𝐹) ∈ dom 𝑀)
111		disjdif 4429	. . . . . . . 8 ⊢ (𝐹 ∩ ((𝐸‘𝐾) ∖ 𝐹)) = ∅
112	111	a1i 11	. . . . . . 7 ⊢ (𝜑 → (𝐹 ∩ ((𝐸‘𝐾) ∖ 𝐹)) = ∅)
113	62	iunssd 5013	. . . . . . . . 9 ⊢ (𝜑 → ∪ 𝑛 ∈ 𝑍 (𝐺‘𝑛) ⊆ (𝐸‘𝐾))
114	91, 113	eqsstrid 3972	. . . . . . . 8 ⊢ (𝜑 → 𝐹 ⊆ (𝐸‘𝐾))
115	11, 18, 29, 114, 108	meassre 47292	. . . . . . 7 ⊢ (𝜑 → (𝑀‘𝐹) ∈ ℝ)
116		difssd 4087	. . . . . . . 8 ⊢ (𝜑 → ((𝐸‘𝐾) ∖ 𝐹) ⊆ (𝐸‘𝐾))
117	11, 18, 29, 116, 110	meassre 47292	. . . . . . 7 ⊢ (𝜑 → (𝑀‘((𝐸‘𝐾) ∖ 𝐹)) ∈ ℝ)
118	11, 12, 108, 110, 112, 115, 117	meadjunre 47291	. . . . . 6 ⊢ (𝜑 → (𝑀‘(𝐹 ∪ ((𝐸‘𝐾) ∖ 𝐹))) = ((𝑀‘𝐹) + (𝑀‘((𝐸‘𝐾) ∖ 𝐹))))
119		undif 4441	. . . . . . . 8 ⊢ (𝐹 ⊆ (𝐸‘𝐾) ↔ (𝐹 ∪ ((𝐸‘𝐾) ∖ 𝐹)) = (𝐸‘𝐾))
120	114, 119	sylib 221	. . . . . . 7 ⊢ (𝜑 → (𝐹 ∪ ((𝐸‘𝐾) ∖ 𝐹)) = (𝐸‘𝐾))
121	120	fveq2d 6886	. . . . . 6 ⊢ (𝜑 → (𝑀‘(𝐹 ∪ ((𝐸‘𝐾) ∖ 𝐹))) = (𝑀‘(𝐸‘𝐾)))
122	104, 118, 121	3eqtr3d 2805	. . . . 5 ⊢ (𝜑 → ((𝑀‘𝐹) + (𝑀‘((𝐸‘𝐾) ∖ 𝐹))) = (𝑀‘(𝐸‘𝐾)))
123	115	recnd 11264	. . . . . 6 ⊢ (𝜑 → (𝑀‘𝐹) ∈ ℂ)
124	117	recnd 11264	. . . . . 6 ⊢ (𝜑 → (𝑀‘((𝐸‘𝐾) ∖ 𝐹)) ∈ ℂ)
125	51, 123, 124	subaddd 11614	. . . . 5 ⊢ (𝜑 → (((𝑀‘(𝐸‘𝐾)) − (𝑀‘𝐹)) = (𝑀‘((𝐸‘𝐾) ∖ 𝐹)) ↔ ((𝑀‘𝐹) + (𝑀‘((𝐸‘𝐾) ∖ 𝐹))) = (𝑀‘(𝐸‘𝐾))))
126	122, 125	mpbird 260	. . . 4 ⊢ (𝜑 → ((𝑀‘(𝐸‘𝐾)) − (𝑀‘𝐹)) = (𝑀‘((𝐸‘𝐾) ∖ 𝐹)))
127		simpllr 788	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ ((𝐸‘𝐾) ∖ 𝐹)) ∧ 𝑛 ∈ 𝑍) ∧ ¬ 𝑥 ∈ (𝐸‘𝑛)) → 𝑥 ∈ ((𝐸‘𝐾) ∖ 𝐹))
128		simplr 781	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ((𝐸‘𝐾) ∖ 𝐹) ∧ 𝑛 ∈ 𝑍) ∧ ¬ 𝑥 ∈ (𝐸‘𝑛)) → 𝑛 ∈ 𝑍)
129		eldifi 4081	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ ((𝐸‘𝐾) ∖ 𝐹) → 𝑥 ∈ (𝐸‘𝐾))
130	129	ad2antrr 739	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ ((𝐸‘𝐾) ∖ 𝐹) ∧ 𝑛 ∈ 𝑍) ∧ ¬ 𝑥 ∈ (𝐸‘𝑛)) → 𝑥 ∈ (𝐸‘𝐾))
131		simpr 490	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ ((𝐸‘𝐾) ∖ 𝐹) ∧ 𝑛 ∈ 𝑍) ∧ ¬ 𝑥 ∈ (𝐸‘𝑛)) → ¬ 𝑥 ∈ (𝐸‘𝑛))
132	130, 131	eldifd 3913	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ((𝐸‘𝐾) ∖ 𝐹) ∧ 𝑛 ∈ 𝑍) ∧ ¬ 𝑥 ∈ (𝐸‘𝑛)) → 𝑥 ∈ ((𝐸‘𝐾) ∖ (𝐸‘𝑛)))
133		rspe 3254	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ((𝐸‘𝐾) ∖ (𝐸‘𝑛))) → ∃𝑛 ∈ 𝑍 𝑥 ∈ ((𝐸‘𝐾) ∖ (𝐸‘𝑛)))
134	128, 132, 133	syl2anc 596	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ ((𝐸‘𝐾) ∖ 𝐹) ∧ 𝑛 ∈ 𝑍) ∧ ¬ 𝑥 ∈ (𝐸‘𝑛)) → ∃𝑛 ∈ 𝑍 𝑥 ∈ ((𝐸‘𝐾) ∖ (𝐸‘𝑛)))
135		eliun 4958	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ((𝐸‘𝐾) ∖ (𝐸‘𝑛)) ↔ ∃𝑛 ∈ 𝑍 𝑥 ∈ ((𝐸‘𝐾) ∖ (𝐸‘𝑛)))
136	134, 135	sylibr 237	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ((𝐸‘𝐾) ∖ 𝐹) ∧ 𝑛 ∈ 𝑍) ∧ ¬ 𝑥 ∈ (𝐸‘𝑛)) → 𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ((𝐸‘𝐾) ∖ (𝐸‘𝑛)))
137	136	adantlll 731	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ ((𝐸‘𝐾) ∖ 𝐹)) ∧ 𝑛 ∈ 𝑍) ∧ ¬ 𝑥 ∈ (𝐸‘𝑛)) → 𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ((𝐸‘𝐾) ∖ (𝐸‘𝑛)))
138	91	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐹 = ∪ 𝑛 ∈ 𝑍 (𝐺‘𝑛))
139	24	iuneq2dv 4979	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ∪ 𝑛 ∈ 𝑍 (𝐺‘𝑛) = ∪ 𝑛 ∈ 𝑍 ((𝐸‘𝐾) ∖ (𝐸‘𝑛)))
140	138, 139	eqtrd 2797	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐹 = ∪ 𝑛 ∈ 𝑍 ((𝐸‘𝐾) ∖ (𝐸‘𝑛)))
141	140	eqcomd 2768	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∪ 𝑛 ∈ 𝑍 ((𝐸‘𝐾) ∖ (𝐸‘𝑛)) = 𝐹)
142	141	ad3antrrr 743	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ ((𝐸‘𝐾) ∖ 𝐹)) ∧ 𝑛 ∈ 𝑍) ∧ ¬ 𝑥 ∈ (𝐸‘𝑛)) → ∪ 𝑛 ∈ 𝑍 ((𝐸‘𝐾) ∖ (𝐸‘𝑛)) = 𝐹)
143	137, 142	eleqtrd 2864	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ ((𝐸‘𝐾) ∖ 𝐹)) ∧ 𝑛 ∈ 𝑍) ∧ ¬ 𝑥 ∈ (𝐸‘𝑛)) → 𝑥 ∈ 𝐹)
144		elndif 4083	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐹 → ¬ 𝑥 ∈ ((𝐸‘𝐾) ∖ 𝐹))
145	143, 144	syl 18	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ ((𝐸‘𝐾) ∖ 𝐹)) ∧ 𝑛 ∈ 𝑍) ∧ ¬ 𝑥 ∈ (𝐸‘𝑛)) → ¬ 𝑥 ∈ ((𝐸‘𝐾) ∖ 𝐹))
146	127, 145	condan 830	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝐸‘𝐾) ∖ 𝐹)) ∧ 𝑛 ∈ 𝑍) → 𝑥 ∈ (𝐸‘𝑛))
147	146	ralrimiva 3156	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐸‘𝐾) ∖ 𝐹)) → ∀𝑛 ∈ 𝑍 𝑥 ∈ (𝐸‘𝑛))
148		vex 3457	. . . . . . . . 9 ⊢ 𝑥 ∈ V
149		eliin 4959	. . . . . . . . 9 ⊢ (𝑥 ∈ V → (𝑥 ∈ ∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛) ↔ ∀𝑛 ∈ 𝑍 𝑥 ∈ (𝐸‘𝑛)))
150	148, 149	ax-mp 5	. . . . . . . 8 ⊢ (𝑥 ∈ ∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛) ↔ ∀𝑛 ∈ 𝑍 𝑥 ∈ (𝐸‘𝑛))
151	147, 150	sylibr 237	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐸‘𝐾) ∖ 𝐹)) → 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛))
152	151	ssd 45901	. . . . . 6 ⊢ (𝜑 → ((𝐸‘𝐾) ∖ 𝐹) ⊆ ∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛))
153		ssid 3956	. . . . . . . . . . . 12 ⊢ (𝐸‘𝐾) ⊆ (𝐸‘𝐾)
154	153	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐸‘𝐾) ⊆ (𝐸‘𝐾))
155		fveq2 6882	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝐾 → (𝐸‘𝑛) = (𝐸‘𝐾))
156	155	sseq1d 3965	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝐾 → ((𝐸‘𝑛) ⊆ (𝐸‘𝐾) ↔ (𝐸‘𝐾) ⊆ (𝐸‘𝐾)))
157	156	rspcev 3579	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ 𝑍 ∧ (𝐸‘𝐾) ⊆ (𝐸‘𝐾)) → ∃𝑛 ∈ 𝑍 (𝐸‘𝑛) ⊆ (𝐸‘𝐾))
158	15, 154, 157	syl2anc 596	. . . . . . . . . 10 ⊢ (𝜑 → ∃𝑛 ∈ 𝑍 (𝐸‘𝑛) ⊆ (𝐸‘𝐾))
159		iinss 5019	. . . . . . . . . 10 ⊢ (∃𝑛 ∈ 𝑍 (𝐸‘𝑛) ⊆ (𝐸‘𝐾) → ∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛) ⊆ (𝐸‘𝐾))
160	158, 159	syl 18	. . . . . . . . 9 ⊢ (𝜑 → ∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛) ⊆ (𝐸‘𝐾))
161	160	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) → ∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛) ⊆ (𝐸‘𝐾))
162		simpr 490	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) → 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛))
163	161, 162	sseldd 3935	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) → 𝑥 ∈ (𝐸‘𝐾))
164		nfcv 2924	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑛𝑥
165		nfii1 4991	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑛∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛)
166	164, 165	nfel 2938	. . . . . . . . . . . 12 ⊢ Ⅎ𝑛 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛)
167		iinss2 5020	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ 𝑍 → ∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛) ⊆ (𝐸‘𝑛))
168	167	adantl 487	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛) ∧ 𝑛 ∈ 𝑍) → ∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛) ⊆ (𝐸‘𝑛))
169		simpl 488	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛) ∧ 𝑛 ∈ 𝑍) → 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛))
170	168, 169	sseldd 3935	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛) ∧ 𝑛 ∈ 𝑍) → 𝑥 ∈ (𝐸‘𝑛))
171		elndif 4083	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (𝐸‘𝑛) → ¬ 𝑥 ∈ ((𝐸‘𝐾) ∖ (𝐸‘𝑛)))
172	170, 171	syl 18	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛) ∧ 𝑛 ∈ 𝑍) → ¬ 𝑥 ∈ ((𝐸‘𝐾) ∖ (𝐸‘𝑛)))
173	172	ex 418	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛) → (𝑛 ∈ 𝑍 → ¬ 𝑥 ∈ ((𝐸‘𝐾) ∖ (𝐸‘𝑛))))
174	166, 173	ralrimi 3262	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛) → ∀𝑛 ∈ 𝑍 ¬ 𝑥 ∈ ((𝐸‘𝐾) ∖ (𝐸‘𝑛)))
175		ralnex 3090	. . . . . . . . . . 11 ⊢ (∀𝑛 ∈ 𝑍 ¬ 𝑥 ∈ ((𝐸‘𝐾) ∖ (𝐸‘𝑛)) ↔ ¬ ∃𝑛 ∈ 𝑍 𝑥 ∈ ((𝐸‘𝐾) ∖ (𝐸‘𝑛)))
176	174, 175	sylib 221	. . . . . . . . . 10 ⊢ (𝑥 ∈ ∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛) → ¬ ∃𝑛 ∈ 𝑍 𝑥 ∈ ((𝐸‘𝐾) ∖ (𝐸‘𝑛)))
177	176, 135	sylnibr 332	. . . . . . . . 9 ⊢ (𝑥 ∈ ∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛) → ¬ 𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ((𝐸‘𝐾) ∖ (𝐸‘𝑛)))
178	177	adantl 487	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) → ¬ 𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ((𝐸‘𝐾) ∖ (𝐸‘𝑛)))
179	140	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) → 𝐹 = ∪ 𝑛 ∈ 𝑍 ((𝐸‘𝐾) ∖ (𝐸‘𝑛)))
180	178, 179	neleqtrrd 2885	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) → ¬ 𝑥 ∈ 𝐹)
181	163, 180	eldifd 3913	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) → 𝑥 ∈ ((𝐸‘𝐾) ∖ 𝐹))
182	152, 181	eqelssd 3955	. . . . 5 ⊢ (𝜑 → ((𝐸‘𝐾) ∖ 𝐹) = ∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛))
183	182	fveq2d 6886	. . . 4 ⊢ (𝜑 → (𝑀‘((𝐸‘𝐾) ∖ 𝐹)) = (𝑀‘∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛)))
184	126, 183	eqtr2d 2798	. . 3 ⊢ (𝜑 → (𝑀‘∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) = ((𝑀‘(𝐸‘𝐾)) − (𝑀‘𝐹)))
185	103, 184	breq12d 5120	. 2 ⊢ (𝜑 → (𝑆 ⇝ (𝑀‘∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) ↔ (𝑛 ∈ 𝑍 ↦ (𝑀‘(𝐸‘𝑛))) ⇝ ((𝑀‘(𝐸‘𝐾)) − (𝑀‘𝐹))))
186	101, 185	mpbird 260	1 ⊢ (𝜑 → 𝑆 ⇝ (𝑀‘∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1570 ∈ wcel 2145 ∀wral 3078 ∃wrex 3088 Vcvv 3453 ∖ cdif 3899 ∪ cun 3900 ∩ cin 3901 ⊆ wss 3902 ∅c0 4282 ∪ ciun 4954 ∩ ciin 4955 class class class wbr 5107 ↦ cmpt 5190 dom cdm 5659 ⟶wf 6533 ‘cfv 6537 (class class class)co 7416 ωcom 7865 ≼ cdom 8953 ℂcc 11125 ℝcr 11126 1c1 11128 + caddc 11130 − cmin 11468 ℤcz 12618 ℤ_≥cuz 12890 ..^cfzo 13711 ⇝ cli 15573 SAlgcsalg 47123 Meascmea 47264

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2734 ax-rep 5236 ax-sep 5255 ax-nul 5267 ax-pow 5334 ax-pr 5402 ax-un 7739 ax-inf2 9623 ax-cnex 11183 ax-resscn 11184 ax-1cn 11185 ax-icn 11186 ax-addcl 11187 ax-addrcl 11188 ax-mulcl 11189 ax-mulrcl 11190 ax-mulcom 11191 ax-addass 11192 ax-mulass 11193 ax-distr 11194 ax-i2m1 11195 ax-1ne0 11196 ax-1rid 11197 ax-rnegex 11198 ax-rrecex 11199 ax-cnre 11200 ax-pre-lttri 11201 ax-pre-lttrn 11202 ax-pre-ltadd 11203 ax-pre-mulgt0 11204 ax-pre-sup 11205

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4283 df-if 4486 df-pw 4562 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4871 df-int 4911 df-iun 4956 df-iin 4957 df-disj 5075 df-br 5108 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5554 df-eprel 5559 df-po 5567 df-so 5568 df-fr 5612 df-se 5613 df-we 5614 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7373 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7866 df-1st 7989 df-2nd 7990 df-frecs 8283 df-wrecs 8314 df-recs 8363 df-rdg 8402 df-1o 8458 df-2o 8459 df-oadd 8462 df-omul 8463 df-er 8699 df-map 8831 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-sup 9415 df-oi 9485 df-card 9947 df-acn 9950 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11470 df-neg 11471 df-div 11899 df-nn 12261 df-2 12330 df-3 12331 df-n0 12532 df-z 12619 df-uz 12891 df-rp 13045 df-xadd 13166 df-ico 13406 df-icc 13407 df-fz 13564 df-fzo 13712 df-seq 14068 df-exp 14128 df-hash 14397 df-cj 15188 df-re 15189 df-im 15190 df-sqrt 15324 df-abs 15325 df-clim 15577 df-sum 15776 df-salg 47124 df-sumge0 47178 df-mea 47265

This theorem is used by: meaiininc 47302

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