smfliminflem - Mathbox for Glauco Siliprandi

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Theorem smfliminflem 45536

Description: The inferior limit of a countable set of sigma-measurable functions is sigma-measurable. Proposition 121F (e) of [Fremlin1] p. 39 . (Contributed by Glauco Siliprandi, 2-Jan-2022.)

**Hypotheses**
Ref	Expression
smfliminflem.m	⊢ (𝜑 → 𝑀 ∈ ℤ)
smfliminflem.z	⊢ 𝑍 = (ℤ_≥‘𝑀)
smfliminflem.s	⊢ (𝜑 → 𝑆 ∈ SAlg)
smfliminflem.f	⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆))
smfliminflem.d	⊢ 𝐷 = {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∣ (lim inf‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ}
smfliminflem.g	⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ (lim inf‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))))

**Assertion**
Ref	Expression
smfliminflem	⊢ (𝜑 → 𝐺 ∈ (SMblFn‘𝑆))

Distinct variable groups: 𝑥,𝐷 𝑛,𝐹,𝑥 𝑚,𝑀 𝑆,𝑚 𝑚,𝑍,𝑛,𝑥 𝜑,𝑚,𝑛,𝑥 Allowed substitution hints: 𝐷(𝑚,𝑛) 𝑆(𝑥,𝑛) 𝐹(𝑚) 𝐺(𝑥,𝑚,𝑛) 𝑀(𝑥,𝑛)

**Proof of Theorem smfliminflem**
Step	Hyp	Ref	Expression
1		smfliminflem.g	. . . 4 ⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ (lim inf‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))))
2	1	a1i 11	. . 3 ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐷 ↦ (lim inf‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)))))
3		smfliminflem.d	. . . . . . . . . 10 ⊢ 𝐷 = {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∣ (lim inf‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ}
4		ssrab2 4077	. . . . . . . . . 10 ⊢ {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∣ (lim inf‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ} ⊆ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚)
5	3, 4	eqsstri 4016	. . . . . . . . 9 ⊢ 𝐷 ⊆ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚)
6		id 22	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐷 → 𝑥 ∈ 𝐷)
7	5, 6	sselid 3980	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐷 → 𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚))
8		smfliminflem.z	. . . . . . . . 9 ⊢ 𝑍 = (ℤ_≥‘𝑀)
9		eqid 2732	. . . . . . . . 9 ⊢ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) = ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚)
10	8, 9	allbutfi 44093	. . . . . . . 8 ⊢ (𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ↔ ∃𝑛 ∈ 𝑍 ∀𝑚 ∈ (ℤ_≥‘𝑛)𝑥 ∈ dom (𝐹‘𝑚))
11	7, 10	sylib 217	. . . . . . 7 ⊢ (𝑥 ∈ 𝐷 → ∃𝑛 ∈ 𝑍 ∀𝑚 ∈ (ℤ_≥‘𝑛)𝑥 ∈ dom (𝐹‘𝑚))
12	11	adantl 482	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ∃𝑛 ∈ 𝑍 ∀𝑚 ∈ (ℤ_≥‘𝑛)𝑥 ∈ dom (𝐹‘𝑚))
13		nfv 1917	. . . . . . . . . 10 ⊢ Ⅎ𝑚(𝜑 ∧ 𝑛 ∈ 𝑍)
14		nfra1 3281	. . . . . . . . . 10 ⊢ Ⅎ𝑚∀𝑚 ∈ (ℤ_≥‘𝑛)𝑥 ∈ dom (𝐹‘𝑚)
15	13, 14	nfan 1902	. . . . . . . . 9 ⊢ Ⅎ𝑚((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ ∀𝑚 ∈ (ℤ_≥‘𝑛)𝑥 ∈ dom (𝐹‘𝑚))
16	8	fvexi 6905	. . . . . . . . . 10 ⊢ 𝑍 ∈ V
17	16	a1i 11	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ ∀𝑚 ∈ (ℤ_≥‘𝑛)𝑥 ∈ dom (𝐹‘𝑚)) → 𝑍 ∈ V)
18	8	eluzelz2 44103	. . . . . . . . . . 11 ⊢ (𝑛 ∈ 𝑍 → 𝑛 ∈ ℤ)
19	18	zred 12665	. . . . . . . . . 10 ⊢ (𝑛 ∈ 𝑍 → 𝑛 ∈ ℝ)
20	19	ad2antlr 725	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ ∀𝑚 ∈ (ℤ_≥‘𝑛)𝑥 ∈ dom (𝐹‘𝑚)) → 𝑛 ∈ ℝ)
21		simpll 765	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ ∀𝑚 ∈ (ℤ_≥‘𝑛)𝑥 ∈ dom (𝐹‘𝑚)) → 𝜑)
22		elinel1 4195	. . . . . . . . . . 11 ⊢ (𝑚 ∈ (𝑍 ∩ (𝑛[,)+∞)) → 𝑚 ∈ 𝑍)
23		smfliminflem.s	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑆 ∈ SAlg)
24	23	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → 𝑆 ∈ SAlg)
25		smfliminflem.f	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆))
26	25	ffvelcdmda 7086	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (𝐹‘𝑚) ∈ (SMblFn‘𝑆))
27		eqid 2732	. . . . . . . . . . . 12 ⊢ dom (𝐹‘𝑚) = dom (𝐹‘𝑚)
28	24, 26, 27	smff 45438	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (𝐹‘𝑚):dom (𝐹‘𝑚)⟶ℝ)
29	21, 22, 28	syl2an 596	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ ∀𝑚 ∈ (ℤ_≥‘𝑛)𝑥 ∈ dom (𝐹‘𝑚)) ∧ 𝑚 ∈ (𝑍 ∩ (𝑛[,)+∞))) → (𝐹‘𝑚):dom (𝐹‘𝑚)⟶ℝ)
30		simplr 767	. . . . . . . . . . . 12 ⊢ (((𝑛 ∈ 𝑍 ∧ ∀𝑚 ∈ (ℤ_≥‘𝑛)𝑥 ∈ dom (𝐹‘𝑚)) ∧ 𝑚 ∈ (𝑍 ∩ (𝑛[,)+∞))) → ∀𝑚 ∈ (ℤ_≥‘𝑛)𝑥 ∈ dom (𝐹‘𝑚))
31		eqid 2732	. . . . . . . . . . . . . 14 ⊢ (ℤ_≥‘𝑛) = (ℤ_≥‘𝑛)
32	18	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ 𝑍 ∧ 𝑚 ∈ (𝑍 ∩ (𝑛[,)+∞))) → 𝑛 ∈ ℤ)
33	8, 22	eluzelz2d 44113	. . . . . . . . . . . . . . 15 ⊢ (𝑚 ∈ (𝑍 ∩ (𝑛[,)+∞)) → 𝑚 ∈ ℤ)
34	33	adantl 482	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ 𝑍 ∧ 𝑚 ∈ (𝑍 ∩ (𝑛[,)+∞))) → 𝑚 ∈ ℤ)
35	19	rexrd 11263	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ 𝑍 → 𝑛 ∈ ℝ^*)
36	35	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 ∈ 𝑍 ∧ 𝑚 ∈ (𝑍 ∩ (𝑛[,)+∞))) → 𝑛 ∈ ℝ^*)
37		pnfxr 11267	. . . . . . . . . . . . . . . 16 ⊢ +∞ ∈ ℝ^*
38	37	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 ∈ 𝑍 ∧ 𝑚 ∈ (𝑍 ∩ (𝑛[,)+∞))) → +∞ ∈ ℝ^*)
39		elinel2 4196	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 ∈ (𝑍 ∩ (𝑛[,)+∞)) → 𝑚 ∈ (𝑛[,)+∞))
40	39	adantl 482	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 ∈ 𝑍 ∧ 𝑚 ∈ (𝑍 ∩ (𝑛[,)+∞))) → 𝑚 ∈ (𝑛[,)+∞))
41	36, 38, 40	icogelbd 44261	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ 𝑍 ∧ 𝑚 ∈ (𝑍 ∩ (𝑛[,)+∞))) → 𝑛 ≤ 𝑚)
42	31, 32, 34, 41	eluzd 44109	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ 𝑍 ∧ 𝑚 ∈ (𝑍 ∩ (𝑛[,)+∞))) → 𝑚 ∈ (ℤ_≥‘𝑛))
43	42	adantlr 713	. . . . . . . . . . . 12 ⊢ (((𝑛 ∈ 𝑍 ∧ ∀𝑚 ∈ (ℤ_≥‘𝑛)𝑥 ∈ dom (𝐹‘𝑚)) ∧ 𝑚 ∈ (𝑍 ∩ (𝑛[,)+∞))) → 𝑚 ∈ (ℤ_≥‘𝑛))
44		rspa 3245	. . . . . . . . . . . 12 ⊢ ((∀𝑚 ∈ (ℤ_≥‘𝑛)𝑥 ∈ dom (𝐹‘𝑚) ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → 𝑥 ∈ dom (𝐹‘𝑚))
45	30, 43, 44	syl2anc 584	. . . . . . . . . . 11 ⊢ (((𝑛 ∈ 𝑍 ∧ ∀𝑚 ∈ (ℤ_≥‘𝑛)𝑥 ∈ dom (𝐹‘𝑚)) ∧ 𝑚 ∈ (𝑍 ∩ (𝑛[,)+∞))) → 𝑥 ∈ dom (𝐹‘𝑚))
46	45	adantlll 716	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ ∀𝑚 ∈ (ℤ_≥‘𝑛)𝑥 ∈ dom (𝐹‘𝑚)) ∧ 𝑚 ∈ (𝑍 ∩ (𝑛[,)+∞))) → 𝑥 ∈ dom (𝐹‘𝑚))
47	29, 46	ffvelcdmd 7087	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ ∀𝑚 ∈ (ℤ_≥‘𝑛)𝑥 ∈ dom (𝐹‘𝑚)) ∧ 𝑚 ∈ (𝑍 ∩ (𝑛[,)+∞))) → ((𝐹‘𝑚)‘𝑥) ∈ ℝ)
48	15, 17, 20, 47	liminfval4 44495	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ ∀𝑚 ∈ (ℤ_≥‘𝑛)𝑥 ∈ dom (𝐹‘𝑚)) → (lim inf‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) = -_𝑒(lim sup‘(𝑚 ∈ 𝑍 ↦ -((𝐹‘𝑚)‘𝑥))))
49	48	rexlimdva2 3157	. . . . . . 7 ⊢ (𝜑 → (∃𝑛 ∈ 𝑍 ∀𝑚 ∈ (ℤ_≥‘𝑛)𝑥 ∈ dom (𝐹‘𝑚) → (lim inf‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) = -_𝑒(lim sup‘(𝑚 ∈ 𝑍 ↦ -((𝐹‘𝑚)‘𝑥)))))
50	49	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (∃𝑛 ∈ 𝑍 ∀𝑚 ∈ (ℤ_≥‘𝑛)𝑥 ∈ dom (𝐹‘𝑚) → (lim inf‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) = -_𝑒(lim sup‘(𝑚 ∈ 𝑍 ↦ -((𝐹‘𝑚)‘𝑥)))))
51	12, 50	mpd 15	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (lim inf‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) = -_𝑒(lim sup‘(𝑚 ∈ 𝑍 ↦ -((𝐹‘𝑚)‘𝑥))))
52	51	xnegeqd 44137	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → -_𝑒(lim inf‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) = -_𝑒-_𝑒(lim sup‘(𝑚 ∈ 𝑍 ↦ -((𝐹‘𝑚)‘𝑥))))
53	16	mptex 7224	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ 𝑍 ↦ -((𝐹‘𝑚)‘𝑥)) ∈ V
54	53	limsupcli 44463	. . . . . . . . . . 11 ⊢ (lim sup‘(𝑚 ∈ 𝑍 ↦ -((𝐹‘𝑚)‘𝑥))) ∈ ℝ^*
55	54	xnegnegi 44139	. . . . . . . . . 10 ⊢ -_𝑒-_𝑒(lim sup‘(𝑚 ∈ 𝑍 ↦ -((𝐹‘𝑚)‘𝑥))) = (lim sup‘(𝑚 ∈ 𝑍 ↦ -((𝐹‘𝑚)‘𝑥)))
56	55	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → -_𝑒-_𝑒(lim sup‘(𝑚 ∈ 𝑍 ↦ -((𝐹‘𝑚)‘𝑥))) = (lim sup‘(𝑚 ∈ 𝑍 ↦ -((𝐹‘𝑚)‘𝑥))))
57	52, 56	eqtr2d 2773	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (lim sup‘(𝑚 ∈ 𝑍 ↦ -((𝐹‘𝑚)‘𝑥))) = -_𝑒(lim inf‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))))
58	3	reqabi 3454	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐷 ↔ (𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∧ (lim inf‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ))
59	58	simprbi 497	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐷 → (lim inf‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ)
60	59	adantl 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (lim inf‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ)
61	60	rexnegd 43822	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → -_𝑒(lim inf‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) = -(lim inf‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))))
62	57, 61	eqtr2d 2773	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → -(lim inf‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) = (lim sup‘(𝑚 ∈ 𝑍 ↦ -((𝐹‘𝑚)‘𝑥))))
63	60	renegcld 11640	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → -(lim inf‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ)
64	62, 63	eqeltrrd 2834	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (lim sup‘(𝑚 ∈ 𝑍 ↦ -((𝐹‘𝑚)‘𝑥))) ∈ ℝ)
65	64	rexnegd 43822	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → -_𝑒(lim sup‘(𝑚 ∈ 𝑍 ↦ -((𝐹‘𝑚)‘𝑥))) = -(lim sup‘(𝑚 ∈ 𝑍 ↦ -((𝐹‘𝑚)‘𝑥))))
66	51, 65	eqtrd 2772	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (lim inf‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) = -(lim sup‘(𝑚 ∈ 𝑍 ↦ -((𝐹‘𝑚)‘𝑥))))
67	66	mpteq2dva 5248	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐷 ↦ (lim inf‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)))) = (𝑥 ∈ 𝐷 ↦ -(lim sup‘(𝑚 ∈ 𝑍 ↦ -((𝐹‘𝑚)‘𝑥)))))
68	2, 67	eqtrd 2772	. 2 ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐷 ↦ -(lim sup‘(𝑚 ∈ 𝑍 ↦ -((𝐹‘𝑚)‘𝑥)))))
69		nfv 1917	. . 3 ⊢ Ⅎ𝑥𝜑
70	18, 31	uzn0d 44125	. . . . . . . 8 ⊢ (𝑛 ∈ 𝑍 → (ℤ_≥‘𝑛) ≠ ∅)
71		fvex 6904	. . . . . . . . . . 11 ⊢ (𝐹‘𝑚) ∈ V
72	71	dmex 7901	. . . . . . . . . 10 ⊢ dom (𝐹‘𝑚) ∈ V
73	72	rgenw 3065	. . . . . . . . 9 ⊢ ∀𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∈ V
74	73	a1i 11	. . . . . . . 8 ⊢ (𝑛 ∈ 𝑍 → ∀𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∈ V)
75		iinexg 5341	. . . . . . . 8 ⊢ (((ℤ_≥‘𝑛) ≠ ∅ ∧ ∀𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∈ V) → ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∈ V)
76	70, 74, 75	syl2anc 584	. . . . . . 7 ⊢ (𝑛 ∈ 𝑍 → ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∈ V)
77	76	rgen 3063	. . . . . 6 ⊢ ∀𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∈ V
78		iunexg 7949	. . . . . 6 ⊢ ((𝑍 ∈ V ∧ ∀𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∈ V) → ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∈ V)
79	16, 77, 78	mp2an 690	. . . . 5 ⊢ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∈ V
80	79, 5	ssexi 5322	. . . 4 ⊢ 𝐷 ∈ V
81	80	a1i 11	. . 3 ⊢ (𝜑 → 𝐷 ∈ V)
82	3	a1i 11	. . . . . 6 ⊢ (𝜑 → 𝐷 = {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∣ (lim inf‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ})
83	10	biimpi 215	. . . . . . . . 9 ⊢ (𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) → ∃𝑛 ∈ 𝑍 ∀𝑚 ∈ (ℤ_≥‘𝑛)𝑥 ∈ dom (𝐹‘𝑚))
84	49	imp 407	. . . . . . . . 9 ⊢ ((𝜑 ∧ ∃𝑛 ∈ 𝑍 ∀𝑚 ∈ (ℤ_≥‘𝑛)𝑥 ∈ dom (𝐹‘𝑚)) → (lim inf‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) = -_𝑒(lim sup‘(𝑚 ∈ 𝑍 ↦ -((𝐹‘𝑚)‘𝑥))))
85	83, 84	sylan2 593	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚)) → (lim inf‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) = -_𝑒(lim sup‘(𝑚 ∈ 𝑍 ↦ -((𝐹‘𝑚)‘𝑥))))
86	54	a1i 11	. . . . . . . . . 10 ⊢ (((lim inf‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) = -_𝑒(lim sup‘(𝑚 ∈ 𝑍 ↦ -((𝐹‘𝑚)‘𝑥))) ∧ (lim inf‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ) → (lim sup‘(𝑚 ∈ 𝑍 ↦ -((𝐹‘𝑚)‘𝑥))) ∈ ℝ^*)
87		simpl 483	. . . . . . . . . . 11 ⊢ (((lim inf‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) = -_𝑒(lim sup‘(𝑚 ∈ 𝑍 ↦ -((𝐹‘𝑚)‘𝑥))) ∧ (lim inf‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ) → (lim inf‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) = -_𝑒(lim sup‘(𝑚 ∈ 𝑍 ↦ -((𝐹‘𝑚)‘𝑥))))
88		simpr 485	. . . . . . . . . . 11 ⊢ (((lim inf‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) = -_𝑒(lim sup‘(𝑚 ∈ 𝑍 ↦ -((𝐹‘𝑚)‘𝑥))) ∧ (lim inf‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ) → (lim inf‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ)
89	87, 88	eqeltrrd 2834	. . . . . . . . . 10 ⊢ (((lim inf‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) = -_𝑒(lim sup‘(𝑚 ∈ 𝑍 ↦ -((𝐹‘𝑚)‘𝑥))) ∧ (lim inf‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ) → -_𝑒(lim sup‘(𝑚 ∈ 𝑍 ↦ -((𝐹‘𝑚)‘𝑥))) ∈ ℝ)
90		xnegrecl2 44160	. . . . . . . . . 10 ⊢ (((lim sup‘(𝑚 ∈ 𝑍 ↦ -((𝐹‘𝑚)‘𝑥))) ∈ ℝ^* ∧ -_𝑒(lim sup‘(𝑚 ∈ 𝑍 ↦ -((𝐹‘𝑚)‘𝑥))) ∈ ℝ) → (lim sup‘(𝑚 ∈ 𝑍 ↦ -((𝐹‘𝑚)‘𝑥))) ∈ ℝ)
91	86, 89, 90	syl2anc 584	. . . . . . . . 9 ⊢ (((lim inf‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) = -_𝑒(lim sup‘(𝑚 ∈ 𝑍 ↦ -((𝐹‘𝑚)‘𝑥))) ∧ (lim inf‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ) → (lim sup‘(𝑚 ∈ 𝑍 ↦ -((𝐹‘𝑚)‘𝑥))) ∈ ℝ)
92		simpl 483	. . . . . . . . . 10 ⊢ (((lim inf‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) = -_𝑒(lim sup‘(𝑚 ∈ 𝑍 ↦ -((𝐹‘𝑚)‘𝑥))) ∧ (lim sup‘(𝑚 ∈ 𝑍 ↦ -((𝐹‘𝑚)‘𝑥))) ∈ ℝ) → (lim inf‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) = -_𝑒(lim sup‘(𝑚 ∈ 𝑍 ↦ -((𝐹‘𝑚)‘𝑥))))
93		xnegrecl 44138	. . . . . . . . . . 11 ⊢ ((lim sup‘(𝑚 ∈ 𝑍 ↦ -((𝐹‘𝑚)‘𝑥))) ∈ ℝ → -_𝑒(lim sup‘(𝑚 ∈ 𝑍 ↦ -((𝐹‘𝑚)‘𝑥))) ∈ ℝ)
94	93	adantl 482	. . . . . . . . . 10 ⊢ (((lim inf‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) = -_𝑒(lim sup‘(𝑚 ∈ 𝑍 ↦ -((𝐹‘𝑚)‘𝑥))) ∧ (lim sup‘(𝑚 ∈ 𝑍 ↦ -((𝐹‘𝑚)‘𝑥))) ∈ ℝ) → -_𝑒(lim sup‘(𝑚 ∈ 𝑍 ↦ -((𝐹‘𝑚)‘𝑥))) ∈ ℝ)
95	92, 94	eqeltrd 2833	. . . . . . . . 9 ⊢ (((lim inf‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) = -_𝑒(lim sup‘(𝑚 ∈ 𝑍 ↦ -((𝐹‘𝑚)‘𝑥))) ∧ (lim sup‘(𝑚 ∈ 𝑍 ↦ -((𝐹‘𝑚)‘𝑥))) ∈ ℝ) → (lim inf‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ)
96	91, 95	impbida 799	. . . . . . . 8 ⊢ ((lim inf‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) = -_𝑒(lim sup‘(𝑚 ∈ 𝑍 ↦ -((𝐹‘𝑚)‘𝑥))) → ((lim inf‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ ↔ (lim sup‘(𝑚 ∈ 𝑍 ↦ -((𝐹‘𝑚)‘𝑥))) ∈ ℝ))
97	85, 96	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚)) → ((lim inf‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ ↔ (lim sup‘(𝑚 ∈ 𝑍 ↦ -((𝐹‘𝑚)‘𝑥))) ∈ ℝ))
98	97	rabbidva 3439	. . . . . 6 ⊢ (𝜑 → {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∣ (lim inf‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ} = {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∣ (lim sup‘(𝑚 ∈ 𝑍 ↦ -((𝐹‘𝑚)‘𝑥))) ∈ ℝ})
99	82, 98	eqtrd 2772	. . . . 5 ⊢ (𝜑 → 𝐷 = {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∣ (lim sup‘(𝑚 ∈ 𝑍 ↦ -((𝐹‘𝑚)‘𝑥))) ∈ ℝ})
100	69, 99	mpteq1df 43928	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐷 ↦ (lim sup‘(𝑚 ∈ 𝑍 ↦ -((𝐹‘𝑚)‘𝑥)))) = (𝑥 ∈ {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∣ (lim sup‘(𝑚 ∈ 𝑍 ↦ -((𝐹‘𝑚)‘𝑥))) ∈ ℝ} ↦ (lim sup‘(𝑚 ∈ 𝑍 ↦ -((𝐹‘𝑚)‘𝑥)))))
101		nfv 1917	. . . . 5 ⊢ Ⅎ𝑚𝜑
102		nfv 1917	. . . . 5 ⊢ Ⅎ𝑛𝜑
103		smfliminflem.m	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℤ)
104		negex 11457	. . . . . 6 ⊢ -((𝐹‘𝑚)‘𝑥) ∈ V
105	104	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍 ∧ 𝑥 ∈ dom (𝐹‘𝑚)) → -((𝐹‘𝑚)‘𝑥) ∈ V)
106		nfv 1917	. . . . . 6 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑚 ∈ 𝑍)
107	72	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → dom (𝐹‘𝑚) ∈ V)
108	28	ffvelcdmda 7086	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ 𝑍) ∧ 𝑥 ∈ dom (𝐹‘𝑚)) → ((𝐹‘𝑚)‘𝑥) ∈ ℝ)
109	28	feqmptd 6960	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (𝐹‘𝑚) = (𝑥 ∈ dom (𝐹‘𝑚) ↦ ((𝐹‘𝑚)‘𝑥)))
110	109, 26	eqeltrrd 2834	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (𝑥 ∈ dom (𝐹‘𝑚) ↦ ((𝐹‘𝑚)‘𝑥)) ∈ (SMblFn‘𝑆))
111	106, 24, 107, 108, 110	smfneg 45509	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (𝑥 ∈ dom (𝐹‘𝑚) ↦ -((𝐹‘𝑚)‘𝑥)) ∈ (SMblFn‘𝑆))
112		eqid 2732	. . . . 5 ⊢ {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∣ (lim sup‘(𝑚 ∈ 𝑍 ↦ -((𝐹‘𝑚)‘𝑥))) ∈ ℝ} = {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∣ (lim sup‘(𝑚 ∈ 𝑍 ↦ -((𝐹‘𝑚)‘𝑥))) ∈ ℝ}
113		eqid 2732	. . . . 5 ⊢ (𝑥 ∈ {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∣ (lim sup‘(𝑚 ∈ 𝑍 ↦ -((𝐹‘𝑚)‘𝑥))) ∈ ℝ} ↦ (lim sup‘(𝑚 ∈ 𝑍 ↦ -((𝐹‘𝑚)‘𝑥)))) = (𝑥 ∈ {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∣ (lim sup‘(𝑚 ∈ 𝑍 ↦ -((𝐹‘𝑚)‘𝑥))) ∈ ℝ} ↦ (lim sup‘(𝑚 ∈ 𝑍 ↦ -((𝐹‘𝑚)‘𝑥))))
114	101, 69, 102, 103, 8, 23, 105, 111, 112, 113	smflimsupmpt 45535	. . . 4 ⊢ (𝜑 → (𝑥 ∈ {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∣ (lim sup‘(𝑚 ∈ 𝑍 ↦ -((𝐹‘𝑚)‘𝑥))) ∈ ℝ} ↦ (lim sup‘(𝑚 ∈ 𝑍 ↦ -((𝐹‘𝑚)‘𝑥)))) ∈ (SMblFn‘𝑆))
115	100, 114	eqeltrd 2833	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐷 ↦ (lim sup‘(𝑚 ∈ 𝑍 ↦ -((𝐹‘𝑚)‘𝑥)))) ∈ (SMblFn‘𝑆))
116	69, 23, 81, 64, 115	smfneg 45509	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐷 ↦ -(lim sup‘(𝑚 ∈ 𝑍 ↦ -((𝐹‘𝑚)‘𝑥)))) ∈ (SMblFn‘𝑆))
117	68, 116	eqeltrd 2833	1 ⊢ (𝜑 → 𝐺 ∈ (SMblFn‘𝑆))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ≠ wne 2940 ∀wral 3061 ∃wrex 3070 {crab 3432 Vcvv 3474 ∩ cin 3947 ∅c0 4322 ∪ ciun 4997 ∩ ciin 4998 ↦ cmpt 5231 dom cdm 5676 ⟶wf 6539 ‘cfv 6543 (class class class)co 7408 ℝcr 11108 +∞cpnf 11244 ℝ^*cxr 11246 -cneg 11444 ℤcz 12557 ℤ_≥cuz 12821 -_𝑒cxne 13088 [,)cico 13325 lim supclsp 15413 lim infclsi 44457 SAlgcsalg 45014 SMblFncsmblfn 45401

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 ax-inf2 9635 ax-cc 10429 ax-ac2 10457 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 ax-pre-sup 11187

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7364 df-ov 7411 df-oprab 7412 df-mpo 7413 df-om 7855 df-1st 7974 df-2nd 7975 df-frecs 8265 df-wrecs 8296 df-recs 8370 df-rdg 8409 df-1o 8465 df-oadd 8469 df-omul 8470 df-er 8702 df-map 8821 df-pm 8822 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-sup 9436 df-inf 9437 df-oi 9504 df-card 9933 df-acn 9936 df-ac 10110 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-div 11871 df-nn 12212 df-2 12274 df-3 12275 df-4 12276 df-n0 12472 df-z 12558 df-uz 12822 df-q 12932 df-rp 12974 df-xneg 13091 df-ioo 13327 df-ioc 13328 df-ico 13329 df-icc 13330 df-fz 13484 df-fzo 13627 df-fl 13756 df-ceil 13757 df-seq 13966 df-exp 14027 df-hash 14290 df-word 14464 df-concat 14520 df-s1 14545 df-s2 14798 df-s3 14799 df-s4 14800 df-cj 15045 df-re 15046 df-im 15047 df-sqrt 15181 df-abs 15182 df-limsup 15414 df-clim 15431 df-rlim 15432 df-rest 17367 df-topgen 17388 df-top 22395 df-bases 22448 df-liminf 44458 df-salg 45015 df-salgen 45019 df-smblfn 45402

This theorem is referenced by: smfliminf 45537

W3C validator