meaiininclem - Mathbox for Glauco Siliprandi

	Mathbox for Glauco Siliprandi		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > meaiininclem		Structured version Visualization version GIF version

Theorem meaiininclem 46797

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 12778	. . . . . . . . . . . . . 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 3976	. . . . . . . . . . . 12 ⊢ (𝜑 → (ℤ_≥‘𝐾) ⊆ 𝑍)
6	5	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝐾)) → (ℤ_≥‘𝐾) ⊆ 𝑍)
7		simpr 484	. . . . . . . . . . 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 2737	. . . . . . . . . . . . . . 15 ⊢ dom 𝑀 = dom 𝑀
13	11, 12	dmmeasal 46763	. . . . . . . . . . . . . 14 ⊢ (𝜑 → dom 𝑀 ∈ SAlg)
14	13	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → dom 𝑀 ∈ SAlg)
15	1, 4	eleqtrrdi 2848	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐾 ∈ 𝑍)
16		meaiininclem.e	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐸:𝑍⟶dom 𝑀)
17	16	ffvelcdmda 7031	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝐾 ∈ 𝑍) → (𝐸‘𝐾) ∈ dom 𝑀)
18	15, 17	mpdan 688	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐸‘𝐾) ∈ dom 𝑀)
19	18	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐸‘𝐾) ∈ dom 𝑀)
20	16	ffvelcdmda 7031	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐸‘𝑛) ∈ dom 𝑀)
21		saldifcl2 46639	. . . . . . . . . . . . 13 ⊢ ((dom 𝑀 ∈ SAlg ∧ (𝐸‘𝐾) ∈ dom 𝑀 ∧ (𝐸‘𝑛) ∈ dom 𝑀) → ((𝐸‘𝐾) ∖ (𝐸‘𝑛)) ∈ dom 𝑀)
22	14, 19, 20, 21	syl3anc 1374	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ((𝐸‘𝐾) ∖ (𝐸‘𝑛)) ∈ dom 𝑀)
23	22	elexd 3465	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ((𝐸‘𝐾) ∖ (𝐸‘𝑛)) ∈ V)
24	10, 23	fvmpt2d 6956	. . . . . . . . . 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 6839	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝐾)) → (𝑀‘(𝐺‘𝑛)) = (𝑀‘((𝐸‘𝐾) ∖ (𝐸‘𝑛))))
27	11	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝐾)) → 𝑀 ∈ Meas)
28	18	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝐾)) → (𝐸‘𝐾) ∈ dom 𝑀)
29		meaiininclem.r	. . . . . . . . . 10 ⊢ (𝜑 → (𝑀‘(𝐸‘𝐾)) ∈ ℝ)
30	29	adantr 480	. . . . . . . . 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 482	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ (𝐾..^𝑛)) → 𝜑)
33	32, 5	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ (𝐾..^𝑛)) → (ℤ_≥‘𝐾) ⊆ 𝑍)
34		elfzouz 13583	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ (𝐾..^𝑛) → 𝑚 ∈ (ℤ_≥‘𝐾))
35	34	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ (𝐾..^𝑛)) → 𝑚 ∈ (ℤ_≥‘𝐾))
36	33, 35	sseldd 3935	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ (𝐾..^𝑛)) → 𝑚 ∈ 𝑍)
37		eleq1w 2820	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑚 → (𝑛 ∈ 𝑍 ↔ 𝑚 ∈ 𝑍))
38	37	anbi2d 631	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑚 → ((𝜑 ∧ 𝑛 ∈ 𝑍) ↔ (𝜑 ∧ 𝑚 ∈ 𝑍)))
39		fvoveq1 7383	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑚 → (𝐸‘(𝑛 + 1)) = (𝐸‘(𝑚 + 1)))
40		fveq2 6835	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑚 → (𝐸‘𝑛) = (𝐸‘𝑚))
41	39, 40	sseq12d 3968	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑚 → ((𝐸‘(𝑛 + 1)) ⊆ (𝐸‘𝑛) ↔ (𝐸‘(𝑚 + 1)) ⊆ (𝐸‘𝑚)))
42	38, 41	imbi12d 344	. . . . . . . . . . . . 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 1991	. . . . . . . . . . . 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 716	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝐾)) ∧ 𝑚 ∈ (𝐾..^𝑛)) → (𝐸‘(𝑚 + 1)) ⊆ (𝐸‘𝑚))
47	7, 46	ssdec 45399	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝐾)) → (𝐸‘𝑛) ⊆ (𝐸‘𝐾))
48	27, 28, 30, 31, 47	meadif 46790	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝐾)) → (𝑀‘((𝐸‘𝐾) ∖ (𝐸‘𝑛))) = ((𝑀‘(𝐸‘𝐾)) − (𝑀‘(𝐸‘𝑛))))
49	26, 48	eqtrd 2772	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝐾)) → (𝑀‘(𝐺‘𝑛)) = ((𝑀‘(𝐸‘𝐾)) − (𝑀‘(𝐸‘𝑛))))
50	49	oveq2d 7376	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝐾)) → ((𝑀‘(𝐸‘𝐾)) − (𝑀‘(𝐺‘𝑛))) = ((𝑀‘(𝐸‘𝐾)) − ((𝑀‘(𝐸‘𝐾)) − (𝑀‘(𝐸‘𝑛)))))
51	29	recnd 11164	. . . . . . . 8 ⊢ (𝜑 → (𝑀‘(𝐸‘𝐾)) ∈ ℂ)
52	51	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝐾)) → (𝑀‘(𝐸‘𝐾)) ∈ ℂ)
53	27, 28, 30, 47, 31	meassre 46788	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝐾)) → (𝑀‘(𝐸‘𝑛)) ∈ ℝ)
54	53	recnd 11164	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝐾)) → (𝑀‘(𝐸‘𝑛)) ∈ ℂ)
55	52, 54	nncand 11501	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝐾)) → ((𝑀‘(𝐸‘𝐾)) − ((𝑀‘(𝐸‘𝐾)) − (𝑀‘(𝐸‘𝑛)))) = (𝑀‘(𝐸‘𝑛)))
56	50, 55	eqtr2d 2773	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝐾)) → (𝑀‘(𝐸‘𝑛)) = ((𝑀‘(𝐸‘𝐾)) − (𝑀‘(𝐺‘𝑛))))
57	56	mpteq2dva 5192	. . . 4 ⊢ (𝜑 → (𝑛 ∈ (ℤ_≥‘𝐾) ↦ (𝑀‘(𝐸‘𝑛))) = (𝑛 ∈ (ℤ_≥‘𝐾) ↦ ((𝑀‘(𝐸‘𝐾)) − (𝑀‘(𝐺‘𝑛)))))
58		nfv 1916	. . . . 5 ⊢ Ⅎ𝑛𝜑
59		eqid 2737	. . . . 5 ⊢ (ℤ_≥‘𝐾) = (ℤ_≥‘𝐾)
60	1	eluzelzd 45686	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ ℤ)
61		difssd 4090	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ((𝐸‘𝐾) ∖ (𝐸‘𝑛)) ⊆ (𝐸‘𝐾))
62	24, 61	eqsstrd 3969	. . . . . . . 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 7061	. . . . . . . . 9 ⊢ (𝜑 → 𝐺:𝑍⟶dom 𝑀)
65	64	ffvelcdmda 7031	. . . . . . . 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 46788	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝐾)) → (𝑀‘(𝐺‘𝑛)) ∈ ℝ)
68	67	recnd 11164	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝐾)) → (𝑀‘(𝐺‘𝑛)) ∈ ℂ)
69		meaiininclem.n	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℤ)
70	43	sscond 4099	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ((𝐸‘𝐾) ∖ (𝐸‘𝑛)) ⊆ ((𝐸‘𝐾) ∖ (𝐸‘(𝑛 + 1))))
71	40	difeq2d 4079	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑚 → ((𝐸‘𝐾) ∖ (𝐸‘𝑛)) = ((𝐸‘𝐾) ∖ (𝐸‘𝑚)))
72	71	cbvmptv 5203	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ 𝑍 ↦ ((𝐸‘𝐾) ∖ (𝐸‘𝑛))) = (𝑚 ∈ 𝑍 ↦ ((𝐸‘𝐾) ∖ (𝐸‘𝑚)))
73	9, 72	eqtri 2760	. . . . . . . . . . 11 ⊢ 𝐺 = (𝑚 ∈ 𝑍 ↦ ((𝐸‘𝐾) ∖ (𝐸‘𝑚)))
74		fveq2 6835	. . . . . . . . . . . 12 ⊢ (𝑚 = (𝑛 + 1) → (𝐸‘𝑚) = (𝐸‘(𝑛 + 1)))
75	74	difeq2d 4079	. . . . . . . . . . 11 ⊢ (𝑚 = (𝑛 + 1) → ((𝐸‘𝐾) ∖ (𝐸‘𝑚)) = ((𝐸‘𝐾) ∖ (𝐸‘(𝑛 + 1))))
76	4	peano2uzs 12819	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ 𝑍 → (𝑛 + 1) ∈ 𝑍)
77	76	adantl 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑛 + 1) ∈ 𝑍)
78		fvex 6848	. . . . . . . . . . . . 13 ⊢ (𝐸‘𝐾) ∈ V
79	78	difexi 5276	. . . . . . . . . . . 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 6966	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐺‘(𝑛 + 1)) = ((𝐸‘𝐾) ∖ (𝐸‘(𝑛 + 1))))
82	24, 81	sseq12d 3968	. . . . . . . . 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 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝑀 ∈ Meas)
85	84, 12, 65, 19, 62	meassle 46774	. . . . . . . 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 46793	. . . . . . 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 45970	. . . . . . 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 2746	. . . . . . . 8 ⊢ ∪ 𝑛 ∈ 𝑍 (𝐺‘𝑛) = 𝐹
93	92	fveq2i 6838	. . . . . . 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 5125	. . . . 5 ⊢ (𝜑 → (𝑛 ∈ (ℤ_≥‘𝐾) ↦ (𝑀‘(𝐺‘𝑛))) ⇝ (𝑀‘𝐹))
96	58, 59, 60, 51, 68, 95	climsubc1mpt 45973	. . . 4 ⊢ (𝜑 → (𝑛 ∈ (ℤ_≥‘𝐾) ↦ ((𝑀‘(𝐸‘𝐾)) − (𝑀‘(𝐺‘𝑛)))) ⇝ ((𝑀‘(𝐸‘𝐾)) − (𝑀‘𝐹)))
97	57, 96	eqbrtrd 5121	. . 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 45970	. . 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 232	. 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 45375	. . . . . . . . . 10 ⊢ 𝑍 ≼ ω
106	105	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 𝑍 ≼ ω)
107	13, 106, 65	saliuncl 46634	. . . . . . . 8 ⊢ (𝜑 → ∪ 𝑛 ∈ 𝑍 (𝐺‘𝑛) ∈ dom 𝑀)
108	91, 107	eqeltrid 2841	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ dom 𝑀)
109		saldifcl2 46639	. . . . . . . 8 ⊢ ((dom 𝑀 ∈ SAlg ∧ (𝐸‘𝐾) ∈ dom 𝑀 ∧ 𝐹 ∈ dom 𝑀) → ((𝐸‘𝐾) ∖ 𝐹) ∈ dom 𝑀)
110	13, 18, 108, 109	syl3anc 1374	. . . . . . 7 ⊢ (𝜑 → ((𝐸‘𝐾) ∖ 𝐹) ∈ dom 𝑀)
111		disjdif 4425	. . . . . . . 8 ⊢ (𝐹 ∩ ((𝐸‘𝐾) ∖ 𝐹)) = ∅
112	111	a1i 11	. . . . . . 7 ⊢ (𝜑 → (𝐹 ∩ ((𝐸‘𝐾) ∖ 𝐹)) = ∅)
113	62	iunssd 5007	. . . . . . . . 9 ⊢ (𝜑 → ∪ 𝑛 ∈ 𝑍 (𝐺‘𝑛) ⊆ (𝐸‘𝐾))
114	91, 113	eqsstrid 3973	. . . . . . . 8 ⊢ (𝜑 → 𝐹 ⊆ (𝐸‘𝐾))
115	11, 18, 29, 114, 108	meassre 46788	. . . . . . 7 ⊢ (𝜑 → (𝑀‘𝐹) ∈ ℝ)
116		difssd 4090	. . . . . . . 8 ⊢ (𝜑 → ((𝐸‘𝐾) ∖ 𝐹) ⊆ (𝐸‘𝐾))
117	11, 18, 29, 116, 110	meassre 46788	. . . . . . 7 ⊢ (𝜑 → (𝑀‘((𝐸‘𝐾) ∖ 𝐹)) ∈ ℝ)
118	11, 12, 108, 110, 112, 115, 117	meadjunre 46787	. . . . . 6 ⊢ (𝜑 → (𝑀‘(𝐹 ∪ ((𝐸‘𝐾) ∖ 𝐹))) = ((𝑀‘𝐹) + (𝑀‘((𝐸‘𝐾) ∖ 𝐹))))
119		undif 4435	. . . . . . . 8 ⊢ (𝐹 ⊆ (𝐸‘𝐾) ↔ (𝐹 ∪ ((𝐸‘𝐾) ∖ 𝐹)) = (𝐸‘𝐾))
120	114, 119	sylib 218	. . . . . . 7 ⊢ (𝜑 → (𝐹 ∪ ((𝐸‘𝐾) ∖ 𝐹)) = (𝐸‘𝐾))
121	120	fveq2d 6839	. . . . . 6 ⊢ (𝜑 → (𝑀‘(𝐹 ∪ ((𝐸‘𝐾) ∖ 𝐹))) = (𝑀‘(𝐸‘𝐾)))
122	104, 118, 121	3eqtr3d 2780	. . . . 5 ⊢ (𝜑 → ((𝑀‘𝐹) + (𝑀‘((𝐸‘𝐾) ∖ 𝐹))) = (𝑀‘(𝐸‘𝐾)))
123	115	recnd 11164	. . . . . 6 ⊢ (𝜑 → (𝑀‘𝐹) ∈ ℂ)
124	117	recnd 11164	. . . . . 6 ⊢ (𝜑 → (𝑀‘((𝐸‘𝐾) ∖ 𝐹)) ∈ ℂ)
125	51, 123, 124	subaddd 11514	. . . . 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 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 4084	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ ((𝐸‘𝐾) ∖ 𝐹) → 𝑥 ∈ (𝐸‘𝐾))
130	129	ad2antrr 727	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ ((𝐸‘𝐾) ∖ 𝐹) ∧ 𝑛 ∈ 𝑍) ∧ ¬ 𝑥 ∈ (𝐸‘𝑛)) → 𝑥 ∈ (𝐸‘𝐾))
131		simpr 484	. . . . . . . . . . . . . . . 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 3227	. . . . . . . . . . . . . . 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 4951	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ((𝐸‘𝐾) ∖ (𝐸‘𝑛)) ↔ ∃𝑛 ∈ 𝑍 𝑥 ∈ ((𝐸‘𝐾) ∖ (𝐸‘𝑛)))
136	134, 135	sylibr 234	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ((𝐸‘𝐾) ∖ 𝐹) ∧ 𝑛 ∈ 𝑍) ∧ ¬ 𝑥 ∈ (𝐸‘𝑛)) → 𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ((𝐸‘𝐾) ∖ (𝐸‘𝑛)))
137	136	adantlll 719	. . . . . . . . . . . 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 4972	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ∪ 𝑛 ∈ 𝑍 (𝐺‘𝑛) = ∪ 𝑛 ∈ 𝑍 ((𝐸‘𝐾) ∖ (𝐸‘𝑛)))
140	138, 139	eqtrd 2772	. . . . . . . . . . . . . 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 731	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ ((𝐸‘𝐾) ∖ 𝐹)) ∧ 𝑛 ∈ 𝑍) ∧ ¬ 𝑥 ∈ (𝐸‘𝑛)) → ∪ 𝑛 ∈ 𝑍 ((𝐸‘𝐾) ∖ (𝐸‘𝑛)) = 𝐹)
143	137, 142	eleqtrd 2839	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ ((𝐸‘𝐾) ∖ 𝐹)) ∧ 𝑛 ∈ 𝑍) ∧ ¬ 𝑥 ∈ (𝐸‘𝑛)) → 𝑥 ∈ 𝐹)
144		elndif 4086	. . . . . . . . . . 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 3129	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐸‘𝐾) ∖ 𝐹)) → ∀𝑛 ∈ 𝑍 𝑥 ∈ (𝐸‘𝑛))
148		vex 3445	. . . . . . . . 9 ⊢ 𝑥 ∈ V
149		eliin 4952	. . . . . . . . 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 234	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐸‘𝐾) ∖ 𝐹)) → 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛))
152	151	ssd 45392	. . . . . 6 ⊢ (𝜑 → ((𝐸‘𝐾) ∖ 𝐹) ⊆ ∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛))
153		ssid 3957	. . . . . . . . . . . 12 ⊢ (𝐸‘𝐾) ⊆ (𝐸‘𝐾)
154	153	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐸‘𝐾) ⊆ (𝐸‘𝐾))
155		fveq2 6835	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝐾 → (𝐸‘𝑛) = (𝐸‘𝐾))
156	155	sseq1d 3966	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝐾 → ((𝐸‘𝑛) ⊆ (𝐸‘𝐾) ↔ (𝐸‘𝐾) ⊆ (𝐸‘𝐾)))
157	156	rspcev 3577	. . . . . . . . . . 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 5013	. . . . . . . . . 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 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) → ∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛) ⊆ (𝐸‘𝐾))
162		simpr 484	. . . . . . . 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 2899	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑛𝑥
165		nfii1 4985	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑛∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛)
166	164, 165	nfel 2914	. . . . . . . . . . . 12 ⊢ Ⅎ𝑛 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛)
167		iinss2 5014	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ 𝑍 → ∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛) ⊆ (𝐸‘𝑛))
168	167	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛) ∧ 𝑛 ∈ 𝑍) → ∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛) ⊆ (𝐸‘𝑛))
169		simpl 482	. . . . . . . . . . . . . . 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 4086	. . . . . . . . . . . . . 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 412	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛) → (𝑛 ∈ 𝑍 → ¬ 𝑥 ∈ ((𝐸‘𝐾) ∖ (𝐸‘𝑛))))
174	166, 173	ralrimi 3235	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛) → ∀𝑛 ∈ 𝑍 ¬ 𝑥 ∈ ((𝐸‘𝐾) ∖ (𝐸‘𝑛)))
175		ralnex 3063	. . . . . . . . . . 11 ⊢ (∀𝑛 ∈ 𝑍 ¬ 𝑥 ∈ ((𝐸‘𝐾) ∖ (𝐸‘𝑛)) ↔ ¬ ∃𝑛 ∈ 𝑍 𝑥 ∈ ((𝐸‘𝐾) ∖ (𝐸‘𝑛)))
176	174, 175	sylib 218	. . . . . . . . . 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 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) → ¬ 𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ((𝐸‘𝐾) ∖ (𝐸‘𝑛)))
179	140	adantr 480	. . . . . . . 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 3913	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) → 𝑥 ∈ ((𝐸‘𝐾) ∖ 𝐹))
182	152, 181	eqelssd 3956	. . . . 5 ⊢ (𝜑 → ((𝐸‘𝐾) ∖ 𝐹) = ∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛))
183	182	fveq2d 6839	. . . 4 ⊢ (𝜑 → (𝑀‘((𝐸‘𝐾) ∖ 𝐹)) = (𝑀‘∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛)))
184	126, 183	eqtr2d 2773	. . 3 ⊢ (𝜑 → (𝑀‘∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) = ((𝑀‘(𝐸‘𝐾)) − (𝑀‘𝐹)))
185	103, 184	breq12d 5112	. 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 206 ∧ wa 395 = wceq 1542 ∈ wcel 2114 ∀wral 3052 ∃wrex 3061 Vcvv 3441 ∖ cdif 3899 ∪ cun 3900 ∩ cin 3901 ⊆ wss 3902 ∅c0 4286 ∪ ciun 4947 ∩ ciin 4948 class class class wbr 5099 ↦ cmpt 5180 dom cdm 5625 ⟶wf 6489 ‘cfv 6493 (class class class)co 7360 ωcom 7810 ≼ cdom 8885 ℂcc 11028 ℝcr 11029 1c1 11031 + caddc 11033 − cmin 11368 ℤcz 12492 ℤ_≥cuz 12755 ..^cfzo 13574 ⇝ cli 15411 SAlgcsalg 46619 Meascmea 46760

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5225 ax-sep 5242 ax-nul 5252 ax-pow 5311 ax-pr 5378 ax-un 7682 ax-inf2 9554 ax-cnex 11086 ax-resscn 11087 ax-1cn 11088 ax-icn 11089 ax-addcl 11090 ax-addrcl 11091 ax-mulcl 11092 ax-mulrcl 11093 ax-mulcom 11094 ax-addass 11095 ax-mulass 11096 ax-distr 11097 ax-i2m1 11098 ax-1ne0 11099 ax-1rid 11100 ax-rnegex 11101 ax-rrecex 11102 ax-cnre 11103 ax-pre-lttri 11104 ax-pre-lttrn 11105 ax-pre-ltadd 11106 ax-pre-mulgt0 11107 ax-pre-sup 11108

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3062 df-rmo 3351 df-reu 3352 df-rab 3401 df-v 3443 df-sbc 3742 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4287 df-if 4481 df-pw 4557 df-sn 4582 df-pr 4584 df-op 4588 df-uni 4865 df-int 4904 df-iun 4949 df-iin 4950 df-disj 5067 df-br 5100 df-opab 5162 df-mpt 5181 df-tr 5207 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-se 5579 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6260 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-isom 6502 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-om 7811 df-1st 7935 df-2nd 7936 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-1o 8399 df-2o 8400 df-oadd 8403 df-omul 8404 df-er 8637 df-map 8769 df-en 8888 df-dom 8889 df-sdom 8890 df-fin 8891 df-sup 9349 df-oi 9419 df-card 9855 df-acn 9858 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-div 11799 df-nn 12150 df-2 12212 df-3 12213 df-n0 12406 df-z 12493 df-uz 12756 df-rp 12910 df-xadd 13031 df-ico 13271 df-icc 13272 df-fz 13428 df-fzo 13575 df-seq 13929 df-exp 13989 df-hash 14258 df-cj 15026 df-re 15027 df-im 15028 df-sqrt 15162 df-abs 15163 df-clim 15415 df-sum 15614 df-salg 46620 df-sumge0 46674 df-mea 46761

This theorem is referenced by: meaiininc 46798

W3C validator