smfliminflem - Mathbox for Glauco Siliprandi

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

Theorem smfliminflem 43989

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 3983	. . . . . . . . . 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 3925	. . . . . . . . 9 ⊢ 𝐷 ⊆ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚)
6		id 22	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐷 → 𝑥 ∈ 𝐷)
7	5, 6	sseldi 3889	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐷 → 𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚))
8		smfliminflem.z	. . . . . . . . 9 ⊢ 𝑍 = (ℤ_≥‘𝑀)
9		eqid 2734	. . . . . . . . 9 ⊢ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) = ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚)
10	8, 9	allbutfi 42558	. . . . . . . 8 ⊢ (𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ↔ ∃𝑛 ∈ 𝑍 ∀𝑚 ∈ (ℤ_≥‘𝑛)𝑥 ∈ dom (𝐹‘𝑚))
11	7, 10	sylib 221	. . . . . . 7 ⊢ (𝑥 ∈ 𝐷 → ∃𝑛 ∈ 𝑍 ∀𝑚 ∈ (ℤ_≥‘𝑛)𝑥 ∈ dom (𝐹‘𝑚))
12	11	adantl 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ∃𝑛 ∈ 𝑍 ∀𝑚 ∈ (ℤ_≥‘𝑛)𝑥 ∈ dom (𝐹‘𝑚))
13		nfv 1922	. . . . . . . . . 10 ⊢ Ⅎ𝑚(𝜑 ∧ 𝑛 ∈ 𝑍)
14		nfra1 3133	. . . . . . . . . 10 ⊢ Ⅎ𝑚∀𝑚 ∈ (ℤ_≥‘𝑛)𝑥 ∈ dom (𝐹‘𝑚)
15	13, 14	nfan 1907	. . . . . . . . 9 ⊢ Ⅎ𝑚((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ ∀𝑚 ∈ (ℤ_≥‘𝑛)𝑥 ∈ dom (𝐹‘𝑚))
16	8	fvexi 6720	. . . . . . . . . 10 ⊢ 𝑍 ∈ V
17	16	a1i 11	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ ∀𝑚 ∈ (ℤ_≥‘𝑛)𝑥 ∈ dom (𝐹‘𝑚)) → 𝑍 ∈ V)
18	8	eluzelz2 42568	. . . . . . . . . . 11 ⊢ (𝑛 ∈ 𝑍 → 𝑛 ∈ ℤ)
19	18	zred 12265	. . . . . . . . . 10 ⊢ (𝑛 ∈ 𝑍 → 𝑛 ∈ ℝ)
20	19	ad2antlr 727	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ ∀𝑚 ∈ (ℤ_≥‘𝑛)𝑥 ∈ dom (𝐹‘𝑚)) → 𝑛 ∈ ℝ)
21		simpll 767	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ ∀𝑚 ∈ (ℤ_≥‘𝑛)𝑥 ∈ dom (𝐹‘𝑚)) → 𝜑)
22		elinel1 4099	. . . . . . . . . . 11 ⊢ (𝑚 ∈ (𝑍 ∩ (𝑛[,)+∞)) → 𝑚 ∈ 𝑍)
23		smfliminflem.s	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑆 ∈ SAlg)
24	23	adantr 484	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → 𝑆 ∈ SAlg)
25		smfliminflem.f	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆))
26	25	ffvelrnda 6893	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (𝐹‘𝑚) ∈ (SMblFn‘𝑆))
27		eqid 2734	. . . . . . . . . . . 12 ⊢ dom (𝐹‘𝑚) = dom (𝐹‘𝑚)
28	24, 26, 27	smff 43894	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (𝐹‘𝑚):dom (𝐹‘𝑚)⟶ℝ)
29	21, 22, 28	syl2an 599	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ ∀𝑚 ∈ (ℤ_≥‘𝑛)𝑥 ∈ dom (𝐹‘𝑚)) ∧ 𝑚 ∈ (𝑍 ∩ (𝑛[,)+∞))) → (𝐹‘𝑚):dom (𝐹‘𝑚)⟶ℝ)
30		simplr 769	. . . . . . . . . . . 12 ⊢ (((𝑛 ∈ 𝑍 ∧ ∀𝑚 ∈ (ℤ_≥‘𝑛)𝑥 ∈ dom (𝐹‘𝑚)) ∧ 𝑚 ∈ (𝑍 ∩ (𝑛[,)+∞))) → ∀𝑚 ∈ (ℤ_≥‘𝑛)𝑥 ∈ dom (𝐹‘𝑚))
31		eqid 2734	. . . . . . . . . . . . . 14 ⊢ (ℤ_≥‘𝑛) = (ℤ_≥‘𝑛)
32	18	adantr 484	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ 𝑍 ∧ 𝑚 ∈ (𝑍 ∩ (𝑛[,)+∞))) → 𝑛 ∈ ℤ)
33	8, 22	eluzelz2d 42578	. . . . . . . . . . . . . . 15 ⊢ (𝑚 ∈ (𝑍 ∩ (𝑛[,)+∞)) → 𝑚 ∈ ℤ)
34	33	adantl 485	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ 𝑍 ∧ 𝑚 ∈ (𝑍 ∩ (𝑛[,)+∞))) → 𝑚 ∈ ℤ)
35	19	rexrd 10866	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ 𝑍 → 𝑛 ∈ ℝ^*)
36	35	adantr 484	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 ∈ 𝑍 ∧ 𝑚 ∈ (𝑍 ∩ (𝑛[,)+∞))) → 𝑛 ∈ ℝ^*)
37		pnfxr 10870	. . . . . . . . . . . . . . . 16 ⊢ +∞ ∈ ℝ^*
38	37	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 ∈ 𝑍 ∧ 𝑚 ∈ (𝑍 ∩ (𝑛[,)+∞))) → +∞ ∈ ℝ^*)
39		elinel2 4100	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 ∈ (𝑍 ∩ (𝑛[,)+∞)) → 𝑚 ∈ (𝑛[,)+∞))
40	39	adantl 485	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 ∈ 𝑍 ∧ 𝑚 ∈ (𝑍 ∩ (𝑛[,)+∞))) → 𝑚 ∈ (𝑛[,)+∞))
41	36, 38, 40	icogelbd 42723	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ 𝑍 ∧ 𝑚 ∈ (𝑍 ∩ (𝑛[,)+∞))) → 𝑛 ≤ 𝑚)
42	31, 32, 34, 41	eluzd 42574	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ 𝑍 ∧ 𝑚 ∈ (𝑍 ∩ (𝑛[,)+∞))) → 𝑚 ∈ (ℤ_≥‘𝑛))
43	42	adantlr 715	. . . . . . . . . . . 12 ⊢ (((𝑛 ∈ 𝑍 ∧ ∀𝑚 ∈ (ℤ_≥‘𝑛)𝑥 ∈ dom (𝐹‘𝑚)) ∧ 𝑚 ∈ (𝑍 ∩ (𝑛[,)+∞))) → 𝑚 ∈ (ℤ_≥‘𝑛))
44		rspa 3121	. . . . . . . . . . . 12 ⊢ ((∀𝑚 ∈ (ℤ_≥‘𝑛)𝑥 ∈ dom (𝐹‘𝑚) ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → 𝑥 ∈ dom (𝐹‘𝑚))
45	30, 43, 44	syl2anc 587	. . . . . . . . . . 11 ⊢ (((𝑛 ∈ 𝑍 ∧ ∀𝑚 ∈ (ℤ_≥‘𝑛)𝑥 ∈ dom (𝐹‘𝑚)) ∧ 𝑚 ∈ (𝑍 ∩ (𝑛[,)+∞))) → 𝑥 ∈ dom (𝐹‘𝑚))
46	45	adantlll 718	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ ∀𝑚 ∈ (ℤ_≥‘𝑛)𝑥 ∈ dom (𝐹‘𝑚)) ∧ 𝑚 ∈ (𝑍 ∩ (𝑛[,)+∞))) → 𝑥 ∈ dom (𝐹‘𝑚))
47	29, 46	ffvelrnd 6894	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ ∀𝑚 ∈ (ℤ_≥‘𝑛)𝑥 ∈ dom (𝐹‘𝑚)) ∧ 𝑚 ∈ (𝑍 ∩ (𝑛[,)+∞))) → ((𝐹‘𝑚)‘𝑥) ∈ ℝ)
48	15, 17, 20, 47	liminfval4 42959	. . . . . . . 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 3199	. . . . . . 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 484	. . . . . 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 42602	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → -_𝑒(lim inf‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) = -_𝑒-_𝑒(lim sup‘(𝑚 ∈ 𝑍 ↦ -((𝐹‘𝑚)‘𝑥))))
53	16	mptex 7028	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ 𝑍 ↦ -((𝐹‘𝑚)‘𝑥)) ∈ V
54	53	limsupcli 42927	. . . . . . . . . . 11 ⊢ (lim sup‘(𝑚 ∈ 𝑍 ↦ -((𝐹‘𝑚)‘𝑥))) ∈ ℝ^*
55	54	xnegnegi 42604	. . . . . . . . . 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 2775	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (lim sup‘(𝑚 ∈ 𝑍 ↦ -((𝐹‘𝑚)‘𝑥))) = -_𝑒(lim inf‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))))
58	3	rabeq2i 3391	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐷 ↔ (𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∧ (lim inf‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ))
59	58	simprbi 500	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐷 → (lim inf‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ)
60	59	adantl 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (lim inf‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ)
61	60	rexnegd 42317	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → -_𝑒(lim inf‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) = -(lim inf‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))))
62	57, 61	eqtr2d 2775	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → -(lim inf‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) = (lim sup‘(𝑚 ∈ 𝑍 ↦ -((𝐹‘𝑚)‘𝑥))))
63	60	renegcld 11242	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → -(lim inf‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ)
64	62, 63	eqeltrrd 2835	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (lim sup‘(𝑚 ∈ 𝑍 ↦ -((𝐹‘𝑚)‘𝑥))) ∈ ℝ)
65	64	rexnegd 42317	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → -_𝑒(lim sup‘(𝑚 ∈ 𝑍 ↦ -((𝐹‘𝑚)‘𝑥))) = -(lim sup‘(𝑚 ∈ 𝑍 ↦ -((𝐹‘𝑚)‘𝑥))))
66	51, 65	eqtrd 2774	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (lim inf‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) = -(lim sup‘(𝑚 ∈ 𝑍 ↦ -((𝐹‘𝑚)‘𝑥))))
67	66	mpteq2dva 5139	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐷 ↦ (lim inf‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)))) = (𝑥 ∈ 𝐷 ↦ -(lim sup‘(𝑚 ∈ 𝑍 ↦ -((𝐹‘𝑚)‘𝑥)))))
68	2, 67	eqtrd 2774	. 2 ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐷 ↦ -(lim sup‘(𝑚 ∈ 𝑍 ↦ -((𝐹‘𝑚)‘𝑥)))))
69		nfv 1922	. . 3 ⊢ Ⅎ𝑥𝜑
70	18, 31	uzn0d 42590	. . . . . . . 8 ⊢ (𝑛 ∈ 𝑍 → (ℤ_≥‘𝑛) ≠ ∅)
71		fvex 6719	. . . . . . . . . . 11 ⊢ (𝐹‘𝑚) ∈ V
72	71	dmex 7678	. . . . . . . . . 10 ⊢ dom (𝐹‘𝑚) ∈ V
73	72	rgenw 3066	. . . . . . . . 9 ⊢ ∀𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∈ V
74	73	a1i 11	. . . . . . . 8 ⊢ (𝑛 ∈ 𝑍 → ∀𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∈ V)
75		iinexg 5223	. . . . . . . 8 ⊢ (((ℤ_≥‘𝑛) ≠ ∅ ∧ ∀𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∈ V) → ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∈ V)
76	70, 74, 75	syl2anc 587	. . . . . . 7 ⊢ (𝑛 ∈ 𝑍 → ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∈ V)
77	76	rgen 3064	. . . . . 6 ⊢ ∀𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∈ V
78		iunexg 7725	. . . . . 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 692	. . . . 5 ⊢ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∈ V
80	79, 5	ssexi 5204	. . . 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 219	. . . . . . . . 9 ⊢ (𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) → ∃𝑛 ∈ 𝑍 ∀𝑚 ∈ (ℤ_≥‘𝑛)𝑥 ∈ dom (𝐹‘𝑚))
84	49	imp 410	. . . . . . . . 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 596	. . . . . . . 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 486	. . . . . . . . . . 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 488	. . . . . . . . . . 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 2835	. . . . . . . . . 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 42627	. . . . . . . . . 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 587	. . . . . . . . 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 486	. . . . . . . . . 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 42603	. . . . . . . . . . 11 ⊢ ((lim sup‘(𝑚 ∈ 𝑍 ↦ -((𝐹‘𝑚)‘𝑥))) ∈ ℝ → -_𝑒(lim sup‘(𝑚 ∈ 𝑍 ↦ -((𝐹‘𝑚)‘𝑥))) ∈ ℝ)
94	93	adantl 485	. . . . . . . . . 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 2834	. . . . . . . . 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 801	. . . . . . . 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 3381	. . . . . 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 2774	. . . . 5 ⊢ (𝜑 → 𝐷 = {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∣ (lim sup‘(𝑚 ∈ 𝑍 ↦ -((𝐹‘𝑚)‘𝑥))) ∈ ℝ})
100	69, 99	mpteq1df 42404	. . . 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 1922	. . . . 5 ⊢ Ⅎ𝑚𝜑
102		nfv 1922	. . . . 5 ⊢ Ⅎ𝑛𝜑
103		smfliminflem.m	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℤ)
104		negex 11059	. . . . . 6 ⊢ -((𝐹‘𝑚)‘𝑥) ∈ V
105	104	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍 ∧ 𝑥 ∈ dom (𝐹‘𝑚)) → -((𝐹‘𝑚)‘𝑥) ∈ V)
106		nfv 1922	. . . . . 6 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑚 ∈ 𝑍)
107	72	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → dom (𝐹‘𝑚) ∈ V)
108	28	ffvelrnda 6893	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ 𝑍) ∧ 𝑥 ∈ dom (𝐹‘𝑚)) → ((𝐹‘𝑚)‘𝑥) ∈ ℝ)
109	28	feqmptd 6769	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (𝐹‘𝑚) = (𝑥 ∈ dom (𝐹‘𝑚) ↦ ((𝐹‘𝑚)‘𝑥)))
110	109, 26	eqeltrrd 2835	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (𝑥 ∈ dom (𝐹‘𝑚) ↦ ((𝐹‘𝑚)‘𝑥)) ∈ (SMblFn‘𝑆))
111	106, 24, 107, 108, 110	smfneg 43963	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (𝑥 ∈ dom (𝐹‘𝑚) ↦ -((𝐹‘𝑚)‘𝑥)) ∈ (SMblFn‘𝑆))
112		eqid 2734	. . . . 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 2734	. . . . 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 43988	. . . 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 2834	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐷 ↦ (lim sup‘(𝑚 ∈ 𝑍 ↦ -((𝐹‘𝑚)‘𝑥)))) ∈ (SMblFn‘𝑆))
116	69, 23, 81, 64, 115	smfneg 43963	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐷 ↦ -(lim sup‘(𝑚 ∈ 𝑍 ↦ -((𝐹‘𝑚)‘𝑥)))) ∈ (SMblFn‘𝑆))
117	68, 116	eqeltrd 2834	1 ⊢ (𝜑 → 𝐺 ∈ (SMblFn‘𝑆))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1089 = wceq 1543 ∈ wcel 2110 ≠ wne 2935 ∀wral 3054 ∃wrex 3055 {crab 3058 Vcvv 3401 ∩ cin 3856 ∅c0 4227 ∪ ciun 4894 ∩ ciin 4895 ↦ cmpt 5124 dom cdm 5540 ⟶wf 6365 ‘cfv 6369 (class class class)co 7202 ℝcr 10711 +∞cpnf 10847 ℝ^*cxr 10849 -cneg 11046 ℤcz 12159 ℤ_≥cuz 12421 -_𝑒cxne 12684 [,)cico 12920 lim supclsp 15014 lim infclsi 42921 SAlgcsalg 43478 SMblFncsmblfn 43862

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 2706 ax-rep 5168 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 ax-inf2 9245 ax-cc 10032 ax-ac2 10060 ax-cnex 10768 ax-resscn 10769 ax-1cn 10770 ax-icn 10771 ax-addcl 10772 ax-addrcl 10773 ax-mulcl 10774 ax-mulrcl 10775 ax-mulcom 10776 ax-addass 10777 ax-mulass 10778 ax-distr 10779 ax-i2m1 10780 ax-1ne0 10781 ax-1rid 10782 ax-rnegex 10783 ax-rrecex 10784 ax-cnre 10785 ax-pre-lttri 10786 ax-pre-lttrn 10787 ax-pre-ltadd 10788 ax-pre-mulgt0 10789 ax-pre-sup 10790

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 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-pss 3876 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-tp 4536 df-op 4538 df-uni 4810 df-int 4850 df-iun 4896 df-iin 4897 df-br 5044 df-opab 5106 df-mpt 5125 df-tr 5151 df-id 5444 df-eprel 5449 df-po 5457 df-so 5458 df-fr 5498 df-se 5499 df-we 5500 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-pred 6149 df-ord 6205 df-on 6206 df-lim 6207 df-suc 6208 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-isom 6378 df-riota 7159 df-ov 7205 df-oprab 7206 df-mpo 7207 df-om 7634 df-1st 7750 df-2nd 7751 df-wrecs 8036 df-recs 8097 df-rdg 8135 df-1o 8191 df-oadd 8195 df-omul 8196 df-er 8380 df-map 8499 df-pm 8500 df-en 8616 df-dom 8617 df-sdom 8618 df-fin 8619 df-sup 9047 df-inf 9048 df-oi 9115 df-card 9538 df-acn 9541 df-ac 9713 df-pnf 10852 df-mnf 10853 df-xr 10854 df-ltxr 10855 df-le 10856 df-sub 11047 df-neg 11048 df-div 11473 df-nn 11814 df-2 11876 df-3 11877 df-4 11878 df-n0 12074 df-z 12160 df-uz 12422 df-q 12528 df-rp 12570 df-xneg 12687 df-ioo 12922 df-ioc 12923 df-ico 12924 df-icc 12925 df-fz 13079 df-fzo 13222 df-fl 13350 df-ceil 13351 df-seq 13558 df-exp 13619 df-hash 13880 df-word 14053 df-concat 14109 df-s1 14136 df-s2 14396 df-s3 14397 df-s4 14398 df-cj 14645 df-re 14646 df-im 14647 df-sqrt 14781 df-abs 14782 df-limsup 15015 df-clim 15032 df-rlim 15033 df-rest 16899 df-topgen 16920 df-top 21763 df-bases 21815 df-liminf 42922 df-salg 43479 df-salgen 43483 df-smblfn 43863

This theorem is referenced by: smfliminf 43990

W3C validator