meaiininclem - Mathbox for Glauco Siliprandi

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Theorem meaiininclem 43699

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 12461	. . . . . . . . . . . . . 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 3952	. . . . . . . . . . . 12 ⊢ (𝜑 → (ℤ_≥‘𝐾) ⊆ 𝑍)
6	5	adantr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝐾)) → (ℤ_≥‘𝐾) ⊆ 𝑍)
7		simpr 488	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝐾)) → 𝑛 ∈ (ℤ_≥‘𝐾))
8	6, 7	sseldd 3902	. . . . . . . . . 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 2737	. . . . . . . . . . . . . . 15 ⊢ dom 𝑀 = dom 𝑀
13	11, 12	dmmeasal 43665	. . . . . . . . . . . . . 14 ⊢ (𝜑 → dom 𝑀 ∈ SAlg)
14	13	adantr 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → dom 𝑀 ∈ SAlg)
15	1, 4	eleqtrrdi 2849	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐾 ∈ 𝑍)
16		meaiininclem.e	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐸:𝑍⟶dom 𝑀)
17	16	ffvelrnda 6904	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝐾 ∈ 𝑍) → (𝐸‘𝐾) ∈ dom 𝑀)
18	15, 17	mpdan 687	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐸‘𝐾) ∈ dom 𝑀)
19	18	adantr 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐸‘𝐾) ∈ dom 𝑀)
20	16	ffvelrnda 6904	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐸‘𝑛) ∈ dom 𝑀)
21		saldifcl2 43542	. . . . . . . . . . . . 13 ⊢ ((dom 𝑀 ∈ SAlg ∧ (𝐸‘𝐾) ∈ dom 𝑀 ∧ (𝐸‘𝑛) ∈ dom 𝑀) → ((𝐸‘𝐾) ∖ (𝐸‘𝑛)) ∈ dom 𝑀)
22	14, 19, 20, 21	syl3anc 1373	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ((𝐸‘𝐾) ∖ (𝐸‘𝑛)) ∈ dom 𝑀)
23	22	elexd 3428	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ((𝐸‘𝐾) ∖ (𝐸‘𝑛)) ∈ V)
24	10, 23	fvmpt2d 6831	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐺‘𝑛) = ((𝐸‘𝐾) ∖ (𝐸‘𝑛)))
25	8, 24	syldan 594	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝐾)) → (𝐺‘𝑛) = ((𝐸‘𝐾) ∖ (𝐸‘𝑛)))
26	25	fveq2d 6721	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝐾)) → (𝑀‘(𝐺‘𝑛)) = (𝑀‘((𝐸‘𝐾) ∖ (𝐸‘𝑛))))
27	11	adantr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝐾)) → 𝑀 ∈ Meas)
28	18	adantr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝐾)) → (𝐸‘𝐾) ∈ dom 𝑀)
29		meaiininclem.r	. . . . . . . . . 10 ⊢ (𝜑 → (𝑀‘(𝐸‘𝐾)) ∈ ℝ)
30	29	adantr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝐾)) → (𝑀‘(𝐸‘𝐾)) ∈ ℝ)
31	8, 20	syldan 594	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝐾)) → (𝐸‘𝑛) ∈ dom 𝑀)
32		simpl 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ (𝐾..^𝑛)) → 𝜑)
33	32, 5	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ (𝐾..^𝑛)) → (ℤ_≥‘𝐾) ⊆ 𝑍)
34		elfzouz 13247	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ (𝐾..^𝑛) → 𝑚 ∈ (ℤ_≥‘𝐾))
35	34	adantl 485	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ (𝐾..^𝑛)) → 𝑚 ∈ (ℤ_≥‘𝐾))
36	33, 35	sseldd 3902	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ (𝐾..^𝑛)) → 𝑚 ∈ 𝑍)
37		eleq1w 2820	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑚 → (𝑛 ∈ 𝑍 ↔ 𝑚 ∈ 𝑍))
38	37	anbi2d 632	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑚 → ((𝜑 ∧ 𝑛 ∈ 𝑍) ↔ (𝜑 ∧ 𝑚 ∈ 𝑍)))
39		fvoveq1 7236	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑚 → (𝐸‘(𝑛 + 1)) = (𝐸‘(𝑚 + 1)))
40		fveq2 6717	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑚 → (𝐸‘𝑛) = (𝐸‘𝑚))
41	39, 40	sseq12d 3934	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑚 → ((𝐸‘(𝑛 + 1)) ⊆ (𝐸‘𝑛) ↔ (𝐸‘(𝑚 + 1)) ⊆ (𝐸‘𝑚)))
42	38, 41	imbi12d 348	. . . . . . . . . . . . 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 2007	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (𝐸‘(𝑚 + 1)) ⊆ (𝐸‘𝑚))
45	32, 36, 44	syl2anc 587	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ (𝐾..^𝑛)) → (𝐸‘(𝑚 + 1)) ⊆ (𝐸‘𝑚))
46	45	adantlr 715	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝐾)) ∧ 𝑚 ∈ (𝐾..^𝑛)) → (𝐸‘(𝑚 + 1)) ⊆ (𝐸‘𝑚))
47	7, 46	ssdec 42311	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝐾)) → (𝐸‘𝑛) ⊆ (𝐸‘𝐾))
48	27, 28, 30, 31, 47	meadif 43692	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝐾)) → (𝑀‘((𝐸‘𝐾) ∖ (𝐸‘𝑛))) = ((𝑀‘(𝐸‘𝐾)) − (𝑀‘(𝐸‘𝑛))))
49	26, 48	eqtrd 2777	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝐾)) → (𝑀‘(𝐺‘𝑛)) = ((𝑀‘(𝐸‘𝐾)) − (𝑀‘(𝐸‘𝑛))))
50	49	oveq2d 7229	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝐾)) → ((𝑀‘(𝐸‘𝐾)) − (𝑀‘(𝐺‘𝑛))) = ((𝑀‘(𝐸‘𝐾)) − ((𝑀‘(𝐸‘𝐾)) − (𝑀‘(𝐸‘𝑛)))))
51	29	recnd 10861	. . . . . . . 8 ⊢ (𝜑 → (𝑀‘(𝐸‘𝐾)) ∈ ℂ)
52	51	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝐾)) → (𝑀‘(𝐸‘𝐾)) ∈ ℂ)
53	27, 28, 30, 47, 31	meassre 43690	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝐾)) → (𝑀‘(𝐸‘𝑛)) ∈ ℝ)
54	53	recnd 10861	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝐾)) → (𝑀‘(𝐸‘𝑛)) ∈ ℂ)
55	52, 54	nncand 11194	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝐾)) → ((𝑀‘(𝐸‘𝐾)) − ((𝑀‘(𝐸‘𝐾)) − (𝑀‘(𝐸‘𝑛)))) = (𝑀‘(𝐸‘𝑛)))
56	50, 55	eqtr2d 2778	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝐾)) → (𝑀‘(𝐸‘𝑛)) = ((𝑀‘(𝐸‘𝐾)) − (𝑀‘(𝐺‘𝑛))))
57	56	mpteq2dva 5150	. . . 4 ⊢ (𝜑 → (𝑛 ∈ (ℤ_≥‘𝐾) ↦ (𝑀‘(𝐸‘𝑛))) = (𝑛 ∈ (ℤ_≥‘𝐾) ↦ ((𝑀‘(𝐸‘𝐾)) − (𝑀‘(𝐺‘𝑛)))))
58		nfv 1922	. . . . 5 ⊢ Ⅎ𝑛𝜑
59		eqid 2737	. . . . 5 ⊢ (ℤ_≥‘𝐾) = (ℤ_≥‘𝐾)
60	1	eluzelzd 42587	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ ℤ)
61		difssd 4047	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ((𝐸‘𝐾) ∖ (𝐸‘𝑛)) ⊆ (𝐸‘𝐾))
62	24, 61	eqsstrd 3939	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐺‘𝑛) ⊆ (𝐸‘𝐾))
63	8, 62	syldan 594	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝐾)) → (𝐺‘𝑛) ⊆ (𝐸‘𝐾))
64	22, 9	fmptd 6931	. . . . . . . . 9 ⊢ (𝜑 → 𝐺:𝑍⟶dom 𝑀)
65	64	ffvelrnda 6904	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐺‘𝑛) ∈ dom 𝑀)
66	8, 65	syldan 594	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝐾)) → (𝐺‘𝑛) ∈ dom 𝑀)
67	27, 28, 30, 63, 66	meassre 43690	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝐾)) → (𝑀‘(𝐺‘𝑛)) ∈ ℝ)
68	67	recnd 10861	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝐾)) → (𝑀‘(𝐺‘𝑛)) ∈ ℂ)
69		meaiininclem.n	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℤ)
70	43	sscond 4056	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ((𝐸‘𝐾) ∖ (𝐸‘𝑛)) ⊆ ((𝐸‘𝐾) ∖ (𝐸‘(𝑛 + 1))))
71	40	difeq2d 4037	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑚 → ((𝐸‘𝐾) ∖ (𝐸‘𝑛)) = ((𝐸‘𝐾) ∖ (𝐸‘𝑚)))
72	71	cbvmptv 5158	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ 𝑍 ↦ ((𝐸‘𝐾) ∖ (𝐸‘𝑛))) = (𝑚 ∈ 𝑍 ↦ ((𝐸‘𝐾) ∖ (𝐸‘𝑚)))
73	9, 72	eqtri 2765	. . . . . . . . . . 11 ⊢ 𝐺 = (𝑚 ∈ 𝑍 ↦ ((𝐸‘𝐾) ∖ (𝐸‘𝑚)))
74		fveq2 6717	. . . . . . . . . . . 12 ⊢ (𝑚 = (𝑛 + 1) → (𝐸‘𝑚) = (𝐸‘(𝑛 + 1)))
75	74	difeq2d 4037	. . . . . . . . . . 11 ⊢ (𝑚 = (𝑛 + 1) → ((𝐸‘𝐾) ∖ (𝐸‘𝑚)) = ((𝐸‘𝐾) ∖ (𝐸‘(𝑛 + 1))))
76	4	peano2uzs 12498	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ 𝑍 → (𝑛 + 1) ∈ 𝑍)
77	76	adantl 485	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑛 + 1) ∈ 𝑍)
78		fvex 6730	. . . . . . . . . . . . 13 ⊢ (𝐸‘𝐾) ∈ V
79	78	difexi 5221	. . . . . . . . . . . 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 6841	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐺‘(𝑛 + 1)) = ((𝐸‘𝐾) ∖ (𝐸‘(𝑛 + 1))))
82	24, 81	sseq12d 3934	. . . . . . . . 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 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝑀 ∈ Meas)
85	84, 12, 65, 19, 62	meassle 43676	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑀‘(𝐺‘𝑛)) ≤ (𝑀‘(𝐸‘𝐾)))
86		eqid 2737	. . . . . . . 8 ⊢ (𝑛 ∈ 𝑍 ↦ (𝑀‘(𝐺‘𝑛))) = (𝑛 ∈ 𝑍 ↦ (𝑀‘(𝐺‘𝑛)))
87	11, 69, 4, 64, 83, 29, 85, 86	meaiuninc2 43695	. . . . . . 7 ⊢ (𝜑 → (𝑛 ∈ 𝑍 ↦ (𝑀‘(𝐺‘𝑛))) ⇝ (𝑀‘∪ 𝑛 ∈ 𝑍 (𝐺‘𝑛)))
88		eqid 2737	. . . . . . . 8 ⊢ (𝑛 ∈ (ℤ_≥‘𝐾) ↦ (𝑀‘(𝐺‘𝑛))) = (𝑛 ∈ (ℤ_≥‘𝐾) ↦ (𝑀‘(𝐺‘𝑛)))
89	4, 86, 15, 88	climresmpt 42875	. . . . . . 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 2746	. . . . . . . 8 ⊢ ∪ 𝑛 ∈ 𝑍 (𝐺‘𝑛) = 𝐹
93	92	fveq2i 6720	. . . . . . 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 5079	. . . . 5 ⊢ (𝜑 → (𝑛 ∈ (ℤ_≥‘𝐾) ↦ (𝑀‘(𝐺‘𝑛))) ⇝ (𝑀‘𝐹))
96	58, 59, 60, 51, 68, 95	climsubc1mpt 42878	. . . 4 ⊢ (𝜑 → (𝑛 ∈ (ℤ_≥‘𝐾) ↦ ((𝑀‘(𝐸‘𝐾)) − (𝑀‘(𝐺‘𝑛)))) ⇝ ((𝑀‘(𝐸‘𝐾)) − (𝑀‘𝐹)))
97	57, 96	eqbrtrd 5075	. . 3 ⊢ (𝜑 → (𝑛 ∈ (ℤ_≥‘𝐾) ↦ (𝑀‘(𝐸‘𝑛))) ⇝ ((𝑀‘(𝐸‘𝐾)) − (𝑀‘𝐹)))
98		eqid 2737	. . . 4 ⊢ (𝑛 ∈ 𝑍 ↦ (𝑀‘(𝐸‘𝑛))) = (𝑛 ∈ 𝑍 ↦ (𝑀‘(𝐸‘𝑛)))
99		eqid 2737	. . . 4 ⊢ (𝑛 ∈ (ℤ_≥‘𝐾) ↦ (𝑀‘(𝐸‘𝑛))) = (𝑛 ∈ (ℤ_≥‘𝐾) ↦ (𝑀‘(𝐸‘𝑛)))
100	4, 98, 15, 99	climresmpt 42875	. . 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 2738	. . . . . 6 ⊢ (𝜑 → (𝑀‘(𝐹 ∪ ((𝐸‘𝐾) ∖ 𝐹))) = (𝑀‘(𝐹 ∪ ((𝐸‘𝐾) ∖ 𝐹))))
105	4	uzct 42284	. . . . . . . . . 10 ⊢ 𝑍 ≼ ω
106	105	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 𝑍 ≼ ω)
107	13, 106, 65	saliuncl 43538	. . . . . . . 8 ⊢ (𝜑 → ∪ 𝑛 ∈ 𝑍 (𝐺‘𝑛) ∈ dom 𝑀)
108	91, 107	eqeltrid 2842	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ dom 𝑀)
109		saldifcl2 43542	. . . . . . . 8 ⊢ ((dom 𝑀 ∈ SAlg ∧ (𝐸‘𝐾) ∈ dom 𝑀 ∧ 𝐹 ∈ dom 𝑀) → ((𝐸‘𝐾) ∖ 𝐹) ∈ dom 𝑀)
110	13, 18, 108, 109	syl3anc 1373	. . . . . . 7 ⊢ (𝜑 → ((𝐸‘𝐾) ∖ 𝐹) ∈ dom 𝑀)
111		disjdif 4386	. . . . . . . 8 ⊢ (𝐹 ∩ ((𝐸‘𝐾) ∖ 𝐹)) = ∅
112	111	a1i 11	. . . . . . 7 ⊢ (𝜑 → (𝐹 ∩ ((𝐸‘𝐾) ∖ 𝐹)) = ∅)
113	62	iunssd 4959	. . . . . . . . 9 ⊢ (𝜑 → ∪ 𝑛 ∈ 𝑍 (𝐺‘𝑛) ⊆ (𝐸‘𝐾))
114	91, 113	eqsstrid 3949	. . . . . . . 8 ⊢ (𝜑 → 𝐹 ⊆ (𝐸‘𝐾))
115	11, 18, 29, 114, 108	meassre 43690	. . . . . . 7 ⊢ (𝜑 → (𝑀‘𝐹) ∈ ℝ)
116		difssd 4047	. . . . . . . 8 ⊢ (𝜑 → ((𝐸‘𝐾) ∖ 𝐹) ⊆ (𝐸‘𝐾))
117	11, 18, 29, 116, 110	meassre 43690	. . . . . . 7 ⊢ (𝜑 → (𝑀‘((𝐸‘𝐾) ∖ 𝐹)) ∈ ℝ)
118	11, 12, 108, 110, 112, 115, 117	meadjunre 43689	. . . . . 6 ⊢ (𝜑 → (𝑀‘(𝐹 ∪ ((𝐸‘𝐾) ∖ 𝐹))) = ((𝑀‘𝐹) + (𝑀‘((𝐸‘𝐾) ∖ 𝐹))))
119		undif 4396	. . . . . . . 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 6721	. . . . . 6 ⊢ (𝜑 → (𝑀‘(𝐹 ∪ ((𝐸‘𝐾) ∖ 𝐹))) = (𝑀‘(𝐸‘𝐾)))
122	104, 118, 121	3eqtr3d 2785	. . . . 5 ⊢ (𝜑 → ((𝑀‘𝐹) + (𝑀‘((𝐸‘𝐾) ∖ 𝐹))) = (𝑀‘(𝐸‘𝐾)))
123	115	recnd 10861	. . . . . 6 ⊢ (𝜑 → (𝑀‘𝐹) ∈ ℂ)
124	117	recnd 10861	. . . . . 6 ⊢ (𝜑 → (𝑀‘((𝐸‘𝐾) ∖ 𝐹)) ∈ ℂ)
125	51, 123, 124	subaddd 11207	. . . . 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 776	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ ((𝐸‘𝐾) ∖ 𝐹)) ∧ 𝑛 ∈ 𝑍) ∧ ¬ 𝑥 ∈ (𝐸‘𝑛)) → 𝑥 ∈ ((𝐸‘𝐾) ∖ 𝐹))
128		simplr 769	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ((𝐸‘𝐾) ∖ 𝐹) ∧ 𝑛 ∈ 𝑍) ∧ ¬ 𝑥 ∈ (𝐸‘𝑛)) → 𝑛 ∈ 𝑍)
129		eldifi 4041	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ ((𝐸‘𝐾) ∖ 𝐹) → 𝑥 ∈ (𝐸‘𝐾))
130	129	ad2antrr 726	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ ((𝐸‘𝐾) ∖ 𝐹) ∧ 𝑛 ∈ 𝑍) ∧ ¬ 𝑥 ∈ (𝐸‘𝑛)) → 𝑥 ∈ (𝐸‘𝐾))
131		simpr 488	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ ((𝐸‘𝐾) ∖ 𝐹) ∧ 𝑛 ∈ 𝑍) ∧ ¬ 𝑥 ∈ (𝐸‘𝑛)) → ¬ 𝑥 ∈ (𝐸‘𝑛))
132	130, 131	eldifd 3877	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ((𝐸‘𝐾) ∖ 𝐹) ∧ 𝑛 ∈ 𝑍) ∧ ¬ 𝑥 ∈ (𝐸‘𝑛)) → 𝑥 ∈ ((𝐸‘𝐾) ∖ (𝐸‘𝑛)))
133		rspe 3223	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ((𝐸‘𝐾) ∖ (𝐸‘𝑛))) → ∃𝑛 ∈ 𝑍 𝑥 ∈ ((𝐸‘𝐾) ∖ (𝐸‘𝑛)))
134	128, 132, 133	syl2anc 587	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ ((𝐸‘𝐾) ∖ 𝐹) ∧ 𝑛 ∈ 𝑍) ∧ ¬ 𝑥 ∈ (𝐸‘𝑛)) → ∃𝑛 ∈ 𝑍 𝑥 ∈ ((𝐸‘𝐾) ∖ (𝐸‘𝑛)))
135		eliun 4908	. . . . . . . . . . . . . 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 718	. . . . . . . . . . . 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 4928	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ∪ 𝑛 ∈ 𝑍 (𝐺‘𝑛) = ∪ 𝑛 ∈ 𝑍 ((𝐸‘𝐾) ∖ (𝐸‘𝑛)))
140	138, 139	eqtrd 2777	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐹 = ∪ 𝑛 ∈ 𝑍 ((𝐸‘𝐾) ∖ (𝐸‘𝑛)))
141	140	eqcomd 2743	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∪ 𝑛 ∈ 𝑍 ((𝐸‘𝐾) ∖ (𝐸‘𝑛)) = 𝐹)
142	141	ad3antrrr 730	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ ((𝐸‘𝐾) ∖ 𝐹)) ∧ 𝑛 ∈ 𝑍) ∧ ¬ 𝑥 ∈ (𝐸‘𝑛)) → ∪ 𝑛 ∈ 𝑍 ((𝐸‘𝐾) ∖ (𝐸‘𝑛)) = 𝐹)
143	137, 142	eleqtrd 2840	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ ((𝐸‘𝐾) ∖ 𝐹)) ∧ 𝑛 ∈ 𝑍) ∧ ¬ 𝑥 ∈ (𝐸‘𝑛)) → 𝑥 ∈ 𝐹)
144		elndif 4043	. . . . . . . . . . 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 818	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝐸‘𝐾) ∖ 𝐹)) ∧ 𝑛 ∈ 𝑍) → 𝑥 ∈ (𝐸‘𝑛))
147	146	ralrimiva 3105	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐸‘𝐾) ∖ 𝐹)) → ∀𝑛 ∈ 𝑍 𝑥 ∈ (𝐸‘𝑛))
148		vex 3412	. . . . . . . . 9 ⊢ 𝑥 ∈ V
149		eliin 4909	. . . . . . . . 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 42303	. . . . . 6 ⊢ (𝜑 → ((𝐸‘𝐾) ∖ 𝐹) ⊆ ∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛))
153		ssid 3923	. . . . . . . . . . . 12 ⊢ (𝐸‘𝐾) ⊆ (𝐸‘𝐾)
154	153	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐸‘𝐾) ⊆ (𝐸‘𝐾))
155		fveq2 6717	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝐾 → (𝐸‘𝑛) = (𝐸‘𝐾))
156	155	sseq1d 3932	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝐾 → ((𝐸‘𝑛) ⊆ (𝐸‘𝐾) ↔ (𝐸‘𝐾) ⊆ (𝐸‘𝐾)))
157	156	rspcev 3537	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ 𝑍 ∧ (𝐸‘𝐾) ⊆ (𝐸‘𝐾)) → ∃𝑛 ∈ 𝑍 (𝐸‘𝑛) ⊆ (𝐸‘𝐾))
158	15, 154, 157	syl2anc 587	. . . . . . . . . 10 ⊢ (𝜑 → ∃𝑛 ∈ 𝑍 (𝐸‘𝑛) ⊆ (𝐸‘𝐾))
159		iinss 4965	. . . . . . . . . 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 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) → ∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛) ⊆ (𝐸‘𝐾))
162		simpr 488	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) → 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛))
163	161, 162	sseldd 3902	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) → 𝑥 ∈ (𝐸‘𝐾))
164		nfcv 2904	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑛𝑥
165		nfii1 4939	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑛∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛)
166	164, 165	nfel 2918	. . . . . . . . . . . 12 ⊢ Ⅎ𝑛 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛)
167		iinss2 4966	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ 𝑍 → ∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛) ⊆ (𝐸‘𝑛))
168	167	adantl 485	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛) ∧ 𝑛 ∈ 𝑍) → ∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛) ⊆ (𝐸‘𝑛))
169		simpl 486	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛) ∧ 𝑛 ∈ 𝑍) → 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛))
170	168, 169	sseldd 3902	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛) ∧ 𝑛 ∈ 𝑍) → 𝑥 ∈ (𝐸‘𝑛))
171		elndif 4043	. . . . . . . . . . . . . 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 416	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛) → (𝑛 ∈ 𝑍 → ¬ 𝑥 ∈ ((𝐸‘𝐾) ∖ (𝐸‘𝑛))))
174	166, 173	ralrimi 3137	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛) → ∀𝑛 ∈ 𝑍 ¬ 𝑥 ∈ ((𝐸‘𝐾) ∖ (𝐸‘𝑛)))
175		ralnex 3158	. . . . . . . . . . 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 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) → ¬ 𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ((𝐸‘𝐾) ∖ (𝐸‘𝑛)))
179	140	adantr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) → 𝐹 = ∪ 𝑛 ∈ 𝑍 ((𝐸‘𝐾) ∖ (𝐸‘𝑛)))
180	178, 179	neleqtrrd 2860	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) → ¬ 𝑥 ∈ 𝐹)
181	163, 180	eldifd 3877	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) → 𝑥 ∈ ((𝐸‘𝐾) ∖ 𝐹))
182	152, 181	eqelssd 3922	. . . . 5 ⊢ (𝜑 → ((𝐸‘𝐾) ∖ 𝐹) = ∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛))
183	182	fveq2d 6721	. . . 4 ⊢ (𝜑 → (𝑀‘((𝐸‘𝐾) ∖ 𝐹)) = (𝑀‘∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛)))
184	126, 183	eqtr2d 2778	. . 3 ⊢ (𝜑 → (𝑀‘∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) = ((𝑀‘(𝐸‘𝐾)) − (𝑀‘𝐹)))
185	103, 184	breq12d 5066	. 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

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1543 ∈ wcel 2110 ∀wral 3061 ∃wrex 3062 Vcvv 3408 ∖ cdif 3863 ∪ cun 3864 ∩ cin 3865 ⊆ wss 3866 ∅c0 4237 ∪ ciun 4904 ∩ ciin 4905 class class class wbr 5053 ↦ cmpt 5135 dom cdm 5551 ⟶wf 6376 ‘cfv 6380 (class class class)co 7213 ωcom 7644 ≼ cdom 8624 ℂcc 10727 ℝcr 10728 1c1 10730 + caddc 10732 − cmin 11062 ℤcz 12176 ℤ_≥cuz 12438 ..^cfzo 13238 ⇝ cli 15045 SAlgcsalg 43524 Meascmea 43662

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-inf2 9256 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 ax-pre-sup 10807

This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-int 4860 df-iun 4906 df-iin 4907 df-disj 5019 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-se 5510 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-isom 6389 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-om 7645 df-1st 7761 df-2nd 7762 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-1o 8202 df-oadd 8206 df-omul 8207 df-er 8391 df-map 8510 df-en 8627 df-dom 8628 df-sdom 8629 df-fin 8630 df-sup 9058 df-oi 9126 df-card 9555 df-acn 9558 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-div 11490 df-nn 11831 df-2 11893 df-3 11894 df-n0 12091 df-z 12177 df-uz 12439 df-rp 12587 df-xadd 12705 df-ico 12941 df-icc 12942 df-fz 13096 df-fzo 13239 df-seq 13575 df-exp 13636 df-hash 13897 df-cj 14662 df-re 14663 df-im 14664 df-sqrt 14798 df-abs 14799 df-clim 15049 df-sum 15250 df-salg 43525 df-sumge0 43576 df-mea 43663

This theorem is referenced by: meaiininc 43700

W3C validator