meaiininclem - Mathbox for Glauco Siliprandi

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Theorem meaiininclem 45202

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 12845	. . . . . . . . . . . . . 14 ⊢ (𝐾 ∈ (ℤ_≥‘𝑁) → (ℤ_≥‘𝐾) ⊆ (ℤ_≥‘𝑁))
3	1, 2	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (ℤ_≥‘𝐾) ⊆ (ℤ_≥‘𝑁))
4		meaiininclem.z	. . . . . . . . . . . . 13 ⊢ 𝑍 = (ℤ_≥‘𝑁)
5	3, 4	sseqtrrdi 4034	. . . . . . . . . . . 12 ⊢ (𝜑 → (ℤ_≥‘𝐾) ⊆ 𝑍)
6	5	adantr 482	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝐾)) → (ℤ_≥‘𝐾) ⊆ 𝑍)
7		simpr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝐾)) → 𝑛 ∈ (ℤ_≥‘𝐾))
8	6, 7	sseldd 3984	. . . . . . . . . 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 2733	. . . . . . . . . . . . . . 15 ⊢ dom 𝑀 = dom 𝑀
13	11, 12	dmmeasal 45168	. . . . . . . . . . . . . 14 ⊢ (𝜑 → dom 𝑀 ∈ SAlg)
14	13	adantr 482	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → dom 𝑀 ∈ SAlg)
15	1, 4	eleqtrrdi 2845	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐾 ∈ 𝑍)
16		meaiininclem.e	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐸:𝑍⟶dom 𝑀)
17	16	ffvelcdmda 7087	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝐾 ∈ 𝑍) → (𝐸‘𝐾) ∈ dom 𝑀)
18	15, 17	mpdan 686	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐸‘𝐾) ∈ dom 𝑀)
19	18	adantr 482	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐸‘𝐾) ∈ dom 𝑀)
20	16	ffvelcdmda 7087	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐸‘𝑛) ∈ dom 𝑀)
21		saldifcl2 45044	. . . . . . . . . . . . 13 ⊢ ((dom 𝑀 ∈ SAlg ∧ (𝐸‘𝐾) ∈ dom 𝑀 ∧ (𝐸‘𝑛) ∈ dom 𝑀) → ((𝐸‘𝐾) ∖ (𝐸‘𝑛)) ∈ dom 𝑀)
22	14, 19, 20, 21	syl3anc 1372	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ((𝐸‘𝐾) ∖ (𝐸‘𝑛)) ∈ dom 𝑀)
23	22	elexd 3495	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ((𝐸‘𝐾) ∖ (𝐸‘𝑛)) ∈ V)
24	10, 23	fvmpt2d 7012	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐺‘𝑛) = ((𝐸‘𝐾) ∖ (𝐸‘𝑛)))
25	8, 24	syldan 592	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝐾)) → (𝐺‘𝑛) = ((𝐸‘𝐾) ∖ (𝐸‘𝑛)))
26	25	fveq2d 6896	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝐾)) → (𝑀‘(𝐺‘𝑛)) = (𝑀‘((𝐸‘𝐾) ∖ (𝐸‘𝑛))))
27	11	adantr 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝐾)) → 𝑀 ∈ Meas)
28	18	adantr 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝐾)) → (𝐸‘𝐾) ∈ dom 𝑀)
29		meaiininclem.r	. . . . . . . . . 10 ⊢ (𝜑 → (𝑀‘(𝐸‘𝐾)) ∈ ℝ)
30	29	adantr 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝐾)) → (𝑀‘(𝐸‘𝐾)) ∈ ℝ)
31	8, 20	syldan 592	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝐾)) → (𝐸‘𝑛) ∈ dom 𝑀)
32		simpl 484	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ (𝐾..^𝑛)) → 𝜑)
33	32, 5	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ (𝐾..^𝑛)) → (ℤ_≥‘𝐾) ⊆ 𝑍)
34		elfzouz 13636	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ (𝐾..^𝑛) → 𝑚 ∈ (ℤ_≥‘𝐾))
35	34	adantl 483	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ (𝐾..^𝑛)) → 𝑚 ∈ (ℤ_≥‘𝐾))
36	33, 35	sseldd 3984	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ (𝐾..^𝑛)) → 𝑚 ∈ 𝑍)
37		eleq1w 2817	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑚 → (𝑛 ∈ 𝑍 ↔ 𝑚 ∈ 𝑍))
38	37	anbi2d 630	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑚 → ((𝜑 ∧ 𝑛 ∈ 𝑍) ↔ (𝜑 ∧ 𝑚 ∈ 𝑍)))
39		fvoveq1 7432	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑚 → (𝐸‘(𝑛 + 1)) = (𝐸‘(𝑚 + 1)))
40		fveq2 6892	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑚 → (𝐸‘𝑛) = (𝐸‘𝑚))
41	39, 40	sseq12d 4016	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑚 → ((𝐸‘(𝑛 + 1)) ⊆ (𝐸‘𝑛) ↔ (𝐸‘(𝑚 + 1)) ⊆ (𝐸‘𝑚)))
42	38, 41	imbi12d 345	. . . . . . . . . . . . 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 2003	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (𝐸‘(𝑚 + 1)) ⊆ (𝐸‘𝑚))
45	32, 36, 44	syl2anc 585	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ (𝐾..^𝑛)) → (𝐸‘(𝑚 + 1)) ⊆ (𝐸‘𝑚))
46	45	adantlr 714	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝐾)) ∧ 𝑚 ∈ (𝐾..^𝑛)) → (𝐸‘(𝑚 + 1)) ⊆ (𝐸‘𝑚))
47	7, 46	ssdec 43777	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝐾)) → (𝐸‘𝑛) ⊆ (𝐸‘𝐾))
48	27, 28, 30, 31, 47	meadif 45195	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝐾)) → (𝑀‘((𝐸‘𝐾) ∖ (𝐸‘𝑛))) = ((𝑀‘(𝐸‘𝐾)) − (𝑀‘(𝐸‘𝑛))))
49	26, 48	eqtrd 2773	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝐾)) → (𝑀‘(𝐺‘𝑛)) = ((𝑀‘(𝐸‘𝐾)) − (𝑀‘(𝐸‘𝑛))))
50	49	oveq2d 7425	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝐾)) → ((𝑀‘(𝐸‘𝐾)) − (𝑀‘(𝐺‘𝑛))) = ((𝑀‘(𝐸‘𝐾)) − ((𝑀‘(𝐸‘𝐾)) − (𝑀‘(𝐸‘𝑛)))))
51	29	recnd 11242	. . . . . . . 8 ⊢ (𝜑 → (𝑀‘(𝐸‘𝐾)) ∈ ℂ)
52	51	adantr 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝐾)) → (𝑀‘(𝐸‘𝐾)) ∈ ℂ)
53	27, 28, 30, 47, 31	meassre 45193	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝐾)) → (𝑀‘(𝐸‘𝑛)) ∈ ℝ)
54	53	recnd 11242	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝐾)) → (𝑀‘(𝐸‘𝑛)) ∈ ℂ)
55	52, 54	nncand 11576	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝐾)) → ((𝑀‘(𝐸‘𝐾)) − ((𝑀‘(𝐸‘𝐾)) − (𝑀‘(𝐸‘𝑛)))) = (𝑀‘(𝐸‘𝑛)))
56	50, 55	eqtr2d 2774	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝐾)) → (𝑀‘(𝐸‘𝑛)) = ((𝑀‘(𝐸‘𝐾)) − (𝑀‘(𝐺‘𝑛))))
57	56	mpteq2dva 5249	. . . 4 ⊢ (𝜑 → (𝑛 ∈ (ℤ_≥‘𝐾) ↦ (𝑀‘(𝐸‘𝑛))) = (𝑛 ∈ (ℤ_≥‘𝐾) ↦ ((𝑀‘(𝐸‘𝐾)) − (𝑀‘(𝐺‘𝑛)))))
58		nfv 1918	. . . . 5 ⊢ Ⅎ𝑛𝜑
59		eqid 2733	. . . . 5 ⊢ (ℤ_≥‘𝐾) = (ℤ_≥‘𝐾)
60	1	eluzelzd 44085	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ ℤ)
61		difssd 4133	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ((𝐸‘𝐾) ∖ (𝐸‘𝑛)) ⊆ (𝐸‘𝐾))
62	24, 61	eqsstrd 4021	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐺‘𝑛) ⊆ (𝐸‘𝐾))
63	8, 62	syldan 592	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝐾)) → (𝐺‘𝑛) ⊆ (𝐸‘𝐾))
64	22, 9	fmptd 7114	. . . . . . . . 9 ⊢ (𝜑 → 𝐺:𝑍⟶dom 𝑀)
65	64	ffvelcdmda 7087	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐺‘𝑛) ∈ dom 𝑀)
66	8, 65	syldan 592	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝐾)) → (𝐺‘𝑛) ∈ dom 𝑀)
67	27, 28, 30, 63, 66	meassre 45193	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝐾)) → (𝑀‘(𝐺‘𝑛)) ∈ ℝ)
68	67	recnd 11242	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝐾)) → (𝑀‘(𝐺‘𝑛)) ∈ ℂ)
69		meaiininclem.n	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℤ)
70	43	sscond 4142	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ((𝐸‘𝐾) ∖ (𝐸‘𝑛)) ⊆ ((𝐸‘𝐾) ∖ (𝐸‘(𝑛 + 1))))
71	40	difeq2d 4123	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑚 → ((𝐸‘𝐾) ∖ (𝐸‘𝑛)) = ((𝐸‘𝐾) ∖ (𝐸‘𝑚)))
72	71	cbvmptv 5262	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ 𝑍 ↦ ((𝐸‘𝐾) ∖ (𝐸‘𝑛))) = (𝑚 ∈ 𝑍 ↦ ((𝐸‘𝐾) ∖ (𝐸‘𝑚)))
73	9, 72	eqtri 2761	. . . . . . . . . . 11 ⊢ 𝐺 = (𝑚 ∈ 𝑍 ↦ ((𝐸‘𝐾) ∖ (𝐸‘𝑚)))
74		fveq2 6892	. . . . . . . . . . . 12 ⊢ (𝑚 = (𝑛 + 1) → (𝐸‘𝑚) = (𝐸‘(𝑛 + 1)))
75	74	difeq2d 4123	. . . . . . . . . . 11 ⊢ (𝑚 = (𝑛 + 1) → ((𝐸‘𝐾) ∖ (𝐸‘𝑚)) = ((𝐸‘𝐾) ∖ (𝐸‘(𝑛 + 1))))
76	4	peano2uzs 12886	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ 𝑍 → (𝑛 + 1) ∈ 𝑍)
77	76	adantl 483	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑛 + 1) ∈ 𝑍)
78		fvex 6905	. . . . . . . . . . . . 13 ⊢ (𝐸‘𝐾) ∈ V
79	78	difexi 5329	. . . . . . . . . . . 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 7022	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐺‘(𝑛 + 1)) = ((𝐸‘𝐾) ∖ (𝐸‘(𝑛 + 1))))
82	24, 81	sseq12d 4016	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ((𝐺‘𝑛) ⊆ (𝐺‘(𝑛 + 1)) ↔ ((𝐸‘𝐾) ∖ (𝐸‘𝑛)) ⊆ ((𝐸‘𝐾) ∖ (𝐸‘(𝑛 + 1)))))
83	70, 82	mpbird 257	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐺‘𝑛) ⊆ (𝐺‘(𝑛 + 1)))
84	11	adantr 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝑀 ∈ Meas)
85	84, 12, 65, 19, 62	meassle 45179	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑀‘(𝐺‘𝑛)) ≤ (𝑀‘(𝐸‘𝐾)))
86		eqid 2733	. . . . . . . 8 ⊢ (𝑛 ∈ 𝑍 ↦ (𝑀‘(𝐺‘𝑛))) = (𝑛 ∈ 𝑍 ↦ (𝑀‘(𝐺‘𝑛)))
87	11, 69, 4, 64, 83, 29, 85, 86	meaiuninc2 45198	. . . . . . 7 ⊢ (𝜑 → (𝑛 ∈ 𝑍 ↦ (𝑀‘(𝐺‘𝑛))) ⇝ (𝑀‘∪ 𝑛 ∈ 𝑍 (𝐺‘𝑛)))
88		eqid 2733	. . . . . . . 8 ⊢ (𝑛 ∈ (ℤ_≥‘𝐾) ↦ (𝑀‘(𝐺‘𝑛))) = (𝑛 ∈ (ℤ_≥‘𝐾) ↦ (𝑀‘(𝐺‘𝑛)))
89	4, 86, 15, 88	climresmpt 44375	. . . . . . 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 257	. . . . . 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 2742	. . . . . . . 8 ⊢ ∪ 𝑛 ∈ 𝑍 (𝐺‘𝑛) = 𝐹
93	92	fveq2i 6895	. . . . . . 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 5175	. . . . 5 ⊢ (𝜑 → (𝑛 ∈ (ℤ_≥‘𝐾) ↦ (𝑀‘(𝐺‘𝑛))) ⇝ (𝑀‘𝐹))
96	58, 59, 60, 51, 68, 95	climsubc1mpt 44378	. . . 4 ⊢ (𝜑 → (𝑛 ∈ (ℤ_≥‘𝐾) ↦ ((𝑀‘(𝐸‘𝐾)) − (𝑀‘(𝐺‘𝑛)))) ⇝ ((𝑀‘(𝐸‘𝐾)) − (𝑀‘𝐹)))
97	57, 96	eqbrtrd 5171	. . 3 ⊢ (𝜑 → (𝑛 ∈ (ℤ_≥‘𝐾) ↦ (𝑀‘(𝐸‘𝑛))) ⇝ ((𝑀‘(𝐸‘𝐾)) − (𝑀‘𝐹)))
98		eqid 2733	. . . 4 ⊢ (𝑛 ∈ 𝑍 ↦ (𝑀‘(𝐸‘𝑛))) = (𝑛 ∈ 𝑍 ↦ (𝑀‘(𝐸‘𝑛)))
99		eqid 2733	. . . 4 ⊢ (𝑛 ∈ (ℤ_≥‘𝐾) ↦ (𝑀‘(𝐸‘𝑛))) = (𝑛 ∈ (ℤ_≥‘𝐾) ↦ (𝑀‘(𝐸‘𝑛)))
100	4, 98, 15, 99	climresmpt 44375	. . 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 231	. 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 2734	. . . . . 6 ⊢ (𝜑 → (𝑀‘(𝐹 ∪ ((𝐸‘𝐾) ∖ 𝐹))) = (𝑀‘(𝐹 ∪ ((𝐸‘𝐾) ∖ 𝐹))))
105	4	uzct 43750	. . . . . . . . . 10 ⊢ 𝑍 ≼ ω
106	105	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 𝑍 ≼ ω)
107	13, 106, 65	saliuncl 45039	. . . . . . . 8 ⊢ (𝜑 → ∪ 𝑛 ∈ 𝑍 (𝐺‘𝑛) ∈ dom 𝑀)
108	91, 107	eqeltrid 2838	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ dom 𝑀)
109		saldifcl2 45044	. . . . . . . 8 ⊢ ((dom 𝑀 ∈ SAlg ∧ (𝐸‘𝐾) ∈ dom 𝑀 ∧ 𝐹 ∈ dom 𝑀) → ((𝐸‘𝐾) ∖ 𝐹) ∈ dom 𝑀)
110	13, 18, 108, 109	syl3anc 1372	. . . . . . 7 ⊢ (𝜑 → ((𝐸‘𝐾) ∖ 𝐹) ∈ dom 𝑀)
111		disjdif 4472	. . . . . . . 8 ⊢ (𝐹 ∩ ((𝐸‘𝐾) ∖ 𝐹)) = ∅
112	111	a1i 11	. . . . . . 7 ⊢ (𝜑 → (𝐹 ∩ ((𝐸‘𝐾) ∖ 𝐹)) = ∅)
113	62	iunssd 5054	. . . . . . . . 9 ⊢ (𝜑 → ∪ 𝑛 ∈ 𝑍 (𝐺‘𝑛) ⊆ (𝐸‘𝐾))
114	91, 113	eqsstrid 4031	. . . . . . . 8 ⊢ (𝜑 → 𝐹 ⊆ (𝐸‘𝐾))
115	11, 18, 29, 114, 108	meassre 45193	. . . . . . 7 ⊢ (𝜑 → (𝑀‘𝐹) ∈ ℝ)
116		difssd 4133	. . . . . . . 8 ⊢ (𝜑 → ((𝐸‘𝐾) ∖ 𝐹) ⊆ (𝐸‘𝐾))
117	11, 18, 29, 116, 110	meassre 45193	. . . . . . 7 ⊢ (𝜑 → (𝑀‘((𝐸‘𝐾) ∖ 𝐹)) ∈ ℝ)
118	11, 12, 108, 110, 112, 115, 117	meadjunre 45192	. . . . . 6 ⊢ (𝜑 → (𝑀‘(𝐹 ∪ ((𝐸‘𝐾) ∖ 𝐹))) = ((𝑀‘𝐹) + (𝑀‘((𝐸‘𝐾) ∖ 𝐹))))
119		undif 4482	. . . . . . . 8 ⊢ (𝐹 ⊆ (𝐸‘𝐾) ↔ (𝐹 ∪ ((𝐸‘𝐾) ∖ 𝐹)) = (𝐸‘𝐾))
120	114, 119	sylib 217	. . . . . . 7 ⊢ (𝜑 → (𝐹 ∪ ((𝐸‘𝐾) ∖ 𝐹)) = (𝐸‘𝐾))
121	120	fveq2d 6896	. . . . . 6 ⊢ (𝜑 → (𝑀‘(𝐹 ∪ ((𝐸‘𝐾) ∖ 𝐹))) = (𝑀‘(𝐸‘𝐾)))
122	104, 118, 121	3eqtr3d 2781	. . . . 5 ⊢ (𝜑 → ((𝑀‘𝐹) + (𝑀‘((𝐸‘𝐾) ∖ 𝐹))) = (𝑀‘(𝐸‘𝐾)))
123	115	recnd 11242	. . . . . 6 ⊢ (𝜑 → (𝑀‘𝐹) ∈ ℂ)
124	117	recnd 11242	. . . . . 6 ⊢ (𝜑 → (𝑀‘((𝐸‘𝐾) ∖ 𝐹)) ∈ ℂ)
125	51, 123, 124	subaddd 11589	. . . . 5 ⊢ (𝜑 → (((𝑀‘(𝐸‘𝐾)) − (𝑀‘𝐹)) = (𝑀‘((𝐸‘𝐾) ∖ 𝐹)) ↔ ((𝑀‘𝐹) + (𝑀‘((𝐸‘𝐾) ∖ 𝐹))) = (𝑀‘(𝐸‘𝐾))))
126	122, 125	mpbird 257	. . . 4 ⊢ (𝜑 → ((𝑀‘(𝐸‘𝐾)) − (𝑀‘𝐹)) = (𝑀‘((𝐸‘𝐾) ∖ 𝐹)))
127		simpllr 775	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ ((𝐸‘𝐾) ∖ 𝐹)) ∧ 𝑛 ∈ 𝑍) ∧ ¬ 𝑥 ∈ (𝐸‘𝑛)) → 𝑥 ∈ ((𝐸‘𝐾) ∖ 𝐹))
128		simplr 768	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ((𝐸‘𝐾) ∖ 𝐹) ∧ 𝑛 ∈ 𝑍) ∧ ¬ 𝑥 ∈ (𝐸‘𝑛)) → 𝑛 ∈ 𝑍)
129		eldifi 4127	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ ((𝐸‘𝐾) ∖ 𝐹) → 𝑥 ∈ (𝐸‘𝐾))
130	129	ad2antrr 725	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ ((𝐸‘𝐾) ∖ 𝐹) ∧ 𝑛 ∈ 𝑍) ∧ ¬ 𝑥 ∈ (𝐸‘𝑛)) → 𝑥 ∈ (𝐸‘𝐾))
131		simpr 486	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ ((𝐸‘𝐾) ∖ 𝐹) ∧ 𝑛 ∈ 𝑍) ∧ ¬ 𝑥 ∈ (𝐸‘𝑛)) → ¬ 𝑥 ∈ (𝐸‘𝑛))
132	130, 131	eldifd 3960	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ((𝐸‘𝐾) ∖ 𝐹) ∧ 𝑛 ∈ 𝑍) ∧ ¬ 𝑥 ∈ (𝐸‘𝑛)) → 𝑥 ∈ ((𝐸‘𝐾) ∖ (𝐸‘𝑛)))
133		rspe 3247	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ((𝐸‘𝐾) ∖ (𝐸‘𝑛))) → ∃𝑛 ∈ 𝑍 𝑥 ∈ ((𝐸‘𝐾) ∖ (𝐸‘𝑛)))
134	128, 132, 133	syl2anc 585	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ ((𝐸‘𝐾) ∖ 𝐹) ∧ 𝑛 ∈ 𝑍) ∧ ¬ 𝑥 ∈ (𝐸‘𝑛)) → ∃𝑛 ∈ 𝑍 𝑥 ∈ ((𝐸‘𝐾) ∖ (𝐸‘𝑛)))
135		eliun 5002	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ((𝐸‘𝐾) ∖ (𝐸‘𝑛)) ↔ ∃𝑛 ∈ 𝑍 𝑥 ∈ ((𝐸‘𝐾) ∖ (𝐸‘𝑛)))
136	134, 135	sylibr 233	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ((𝐸‘𝐾) ∖ 𝐹) ∧ 𝑛 ∈ 𝑍) ∧ ¬ 𝑥 ∈ (𝐸‘𝑛)) → 𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ((𝐸‘𝐾) ∖ (𝐸‘𝑛)))
137	136	adantlll 717	. . . . . . . . . . . 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 5022	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ∪ 𝑛 ∈ 𝑍 (𝐺‘𝑛) = ∪ 𝑛 ∈ 𝑍 ((𝐸‘𝐾) ∖ (𝐸‘𝑛)))
140	138, 139	eqtrd 2773	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐹 = ∪ 𝑛 ∈ 𝑍 ((𝐸‘𝐾) ∖ (𝐸‘𝑛)))
141	140	eqcomd 2739	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∪ 𝑛 ∈ 𝑍 ((𝐸‘𝐾) ∖ (𝐸‘𝑛)) = 𝐹)
142	141	ad3antrrr 729	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ ((𝐸‘𝐾) ∖ 𝐹)) ∧ 𝑛 ∈ 𝑍) ∧ ¬ 𝑥 ∈ (𝐸‘𝑛)) → ∪ 𝑛 ∈ 𝑍 ((𝐸‘𝐾) ∖ (𝐸‘𝑛)) = 𝐹)
143	137, 142	eleqtrd 2836	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ ((𝐸‘𝐾) ∖ 𝐹)) ∧ 𝑛 ∈ 𝑍) ∧ ¬ 𝑥 ∈ (𝐸‘𝑛)) → 𝑥 ∈ 𝐹)
144		elndif 4129	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐹 → ¬ 𝑥 ∈ ((𝐸‘𝐾) ∖ 𝐹))
145	143, 144	syl 17	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ ((𝐸‘𝐾) ∖ 𝐹)) ∧ 𝑛 ∈ 𝑍) ∧ ¬ 𝑥 ∈ (𝐸‘𝑛)) → ¬ 𝑥 ∈ ((𝐸‘𝐾) ∖ 𝐹))
146	127, 145	condan 817	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝐸‘𝐾) ∖ 𝐹)) ∧ 𝑛 ∈ 𝑍) → 𝑥 ∈ (𝐸‘𝑛))
147	146	ralrimiva 3147	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐸‘𝐾) ∖ 𝐹)) → ∀𝑛 ∈ 𝑍 𝑥 ∈ (𝐸‘𝑛))
148		vex 3479	. . . . . . . . 9 ⊢ 𝑥 ∈ V
149		eliin 5003	. . . . . . . . 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 233	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐸‘𝐾) ∖ 𝐹)) → 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛))
152	151	ssd 43769	. . . . . 6 ⊢ (𝜑 → ((𝐸‘𝐾) ∖ 𝐹) ⊆ ∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛))
153		ssid 4005	. . . . . . . . . . . 12 ⊢ (𝐸‘𝐾) ⊆ (𝐸‘𝐾)
154	153	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐸‘𝐾) ⊆ (𝐸‘𝐾))
155		fveq2 6892	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝐾 → (𝐸‘𝑛) = (𝐸‘𝐾))
156	155	sseq1d 4014	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝐾 → ((𝐸‘𝑛) ⊆ (𝐸‘𝐾) ↔ (𝐸‘𝐾) ⊆ (𝐸‘𝐾)))
157	156	rspcev 3613	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ 𝑍 ∧ (𝐸‘𝐾) ⊆ (𝐸‘𝐾)) → ∃𝑛 ∈ 𝑍 (𝐸‘𝑛) ⊆ (𝐸‘𝐾))
158	15, 154, 157	syl2anc 585	. . . . . . . . . 10 ⊢ (𝜑 → ∃𝑛 ∈ 𝑍 (𝐸‘𝑛) ⊆ (𝐸‘𝐾))
159		iinss 5060	. . . . . . . . . 10 ⊢ (∃𝑛 ∈ 𝑍 (𝐸‘𝑛) ⊆ (𝐸‘𝐾) → ∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛) ⊆ (𝐸‘𝐾))
160	158, 159	syl 17	. . . . . . . . 9 ⊢ (𝜑 → ∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛) ⊆ (𝐸‘𝐾))
161	160	adantr 482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) → ∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛) ⊆ (𝐸‘𝐾))
162		simpr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) → 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛))
163	161, 162	sseldd 3984	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) → 𝑥 ∈ (𝐸‘𝐾))
164		nfcv 2904	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑛𝑥
165		nfii1 5033	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑛∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛)
166	164, 165	nfel 2918	. . . . . . . . . . . 12 ⊢ Ⅎ𝑛 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛)
167		iinss2 5061	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ 𝑍 → ∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛) ⊆ (𝐸‘𝑛))
168	167	adantl 483	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛) ∧ 𝑛 ∈ 𝑍) → ∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛) ⊆ (𝐸‘𝑛))
169		simpl 484	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛) ∧ 𝑛 ∈ 𝑍) → 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛))
170	168, 169	sseldd 3984	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛) ∧ 𝑛 ∈ 𝑍) → 𝑥 ∈ (𝐸‘𝑛))
171		elndif 4129	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (𝐸‘𝑛) → ¬ 𝑥 ∈ ((𝐸‘𝐾) ∖ (𝐸‘𝑛)))
172	170, 171	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛) ∧ 𝑛 ∈ 𝑍) → ¬ 𝑥 ∈ ((𝐸‘𝐾) ∖ (𝐸‘𝑛)))
173	172	ex 414	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛) → (𝑛 ∈ 𝑍 → ¬ 𝑥 ∈ ((𝐸‘𝐾) ∖ (𝐸‘𝑛))))
174	166, 173	ralrimi 3255	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛) → ∀𝑛 ∈ 𝑍 ¬ 𝑥 ∈ ((𝐸‘𝐾) ∖ (𝐸‘𝑛)))
175		ralnex 3073	. . . . . . . . . . 11 ⊢ (∀𝑛 ∈ 𝑍 ¬ 𝑥 ∈ ((𝐸‘𝐾) ∖ (𝐸‘𝑛)) ↔ ¬ ∃𝑛 ∈ 𝑍 𝑥 ∈ ((𝐸‘𝐾) ∖ (𝐸‘𝑛)))
176	174, 175	sylib 217	. . . . . . . . . 10 ⊢ (𝑥 ∈ ∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛) → ¬ ∃𝑛 ∈ 𝑍 𝑥 ∈ ((𝐸‘𝐾) ∖ (𝐸‘𝑛)))
177	176, 135	sylnibr 329	. . . . . . . . 9 ⊢ (𝑥 ∈ ∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛) → ¬ 𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ((𝐸‘𝐾) ∖ (𝐸‘𝑛)))
178	177	adantl 483	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) → ¬ 𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ((𝐸‘𝐾) ∖ (𝐸‘𝑛)))
179	140	adantr 482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) → 𝐹 = ∪ 𝑛 ∈ 𝑍 ((𝐸‘𝐾) ∖ (𝐸‘𝑛)))
180	178, 179	neleqtrrd 2857	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) → ¬ 𝑥 ∈ 𝐹)
181	163, 180	eldifd 3960	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) → 𝑥 ∈ ((𝐸‘𝐾) ∖ 𝐹))
182	152, 181	eqelssd 4004	. . . . 5 ⊢ (𝜑 → ((𝐸‘𝐾) ∖ 𝐹) = ∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛))
183	182	fveq2d 6896	. . . 4 ⊢ (𝜑 → (𝑀‘((𝐸‘𝐾) ∖ 𝐹)) = (𝑀‘∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛)))
184	126, 183	eqtr2d 2774	. . 3 ⊢ (𝜑 → (𝑀‘∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) = ((𝑀‘(𝐸‘𝐾)) − (𝑀‘𝐹)))
185	103, 184	breq12d 5162	. 2 ⊢ (𝜑 → (𝑆 ⇝ (𝑀‘∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) ↔ (𝑛 ∈ 𝑍 ↦ (𝑀‘(𝐸‘𝑛))) ⇝ ((𝑀‘(𝐸‘𝐾)) − (𝑀‘𝐹))))
186	101, 185	mpbird 257	1 ⊢ (𝜑 → 𝑆 ⇝ (𝑀‘∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∀wral 3062 ∃wrex 3071 Vcvv 3475 ∖ cdif 3946 ∪ cun 3947 ∩ cin 3948 ⊆ wss 3949 ∅c0 4323 ∪ ciun 4998 ∩ ciin 4999 class class class wbr 5149 ↦ cmpt 5232 dom cdm 5677 ⟶wf 6540 ‘cfv 6544 (class class class)co 7409 ωcom 7855 ≼ cdom 8937 ℂcc 11108 ℝcr 11109 1c1 11111 + caddc 11113 − cmin 11444 ℤcz 12558 ℤ_≥cuz 12822 ..^cfzo 13627 ⇝ cli 15428 SAlgcsalg 45024 Meascmea 45165

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-inf2 9636 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 ax-pre-sup 11188

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-iin 5001 df-disj 5115 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-1st 7975 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-oadd 8470 df-omul 8471 df-er 8703 df-map 8822 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-sup 9437 df-oi 9505 df-card 9934 df-acn 9937 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-div 11872 df-nn 12213 df-2 12275 df-3 12276 df-n0 12473 df-z 12559 df-uz 12823 df-rp 12975 df-xadd 13093 df-ico 13330 df-icc 13331 df-fz 13485 df-fzo 13628 df-seq 13967 df-exp 14028 df-hash 14291 df-cj 15046 df-re 15047 df-im 15048 df-sqrt 15182 df-abs 15183 df-clim 15432 df-sum 15633 df-salg 45025 df-sumge0 45079 df-mea 45166

This theorem is referenced by: meaiininc 45203

W3C validator