smfliminfmpt - Mathbox for Glauco Siliprandi

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Theorem smfliminfmpt 41038

Description: The inferior limit of a countable set of sigma-measurable functions is sigma-measurable. Proposition 121F (e) of [Fremlin1] p. 39 . 𝐴 can contain 𝑚 as a free variable, in other words it can be thought of as an indexed collection 𝐴(𝑚). 𝐵 can be thought of as a collection with two indexes 𝐵(𝑚, 𝑥). (Contributed by Glauco Siliprandi, 2-Jan-2022.)

**Hypotheses**
Ref	Expression
smfliminfmpt.p	⊢ Ⅎ𝑚𝜑
smfliminfmpt.x	⊢ Ⅎ𝑥𝜑
smfliminfmpt.n	⊢ Ⅎ𝑛𝜑
smfliminfmpt.m	⊢ (𝜑 → 𝑀 ∈ ℤ)
smfliminfmpt.z	⊢ 𝑍 = (ℤ_≥‘𝑀)
smfliminfmpt.s	⊢ (𝜑 → 𝑆 ∈ SAlg)
smfliminfmpt.b	⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)
smfliminfmpt.f	⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ (SMblFn‘𝑆))
smfliminfmpt.d	⊢ 𝐷 = {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴 ∣ (lim inf‘(𝑚 ∈ 𝑍 ↦ 𝐵)) ∈ ℝ}
smfliminfmpt.g	⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ (lim inf‘(𝑚 ∈ 𝑍 ↦ 𝐵)))

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

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

**Proof of Theorem smfliminfmpt**
Step	Hyp	Ref	Expression
1		smfliminfmpt.g	. . . 4 ⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ (lim inf‘(𝑚 ∈ 𝑍 ↦ 𝐵)))
2	1	a1i 11	. . 3 ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐷 ↦ (lim inf‘(𝑚 ∈ 𝑍 ↦ 𝐵))))
3		smfliminfmpt.x	. . . 4 ⊢ Ⅎ𝑥𝜑
4		smfliminfmpt.d	. . . . . 6 ⊢ 𝐷 = {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴 ∣ (lim inf‘(𝑚 ∈ 𝑍 ↦ 𝐵)) ∈ ℝ}
5	4	a1i 11	. . . . 5 ⊢ (𝜑 → 𝐷 = {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴 ∣ (lim inf‘(𝑚 ∈ 𝑍 ↦ 𝐵)) ∈ ℝ})
6		simpr 477	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴) → 𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴)
7		smfliminfmpt.n	. . . . . . . . . . . 12 ⊢ Ⅎ𝑛𝜑
8		smfliminfmpt.p	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑚𝜑
9		nfv 1843	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑚 𝑛 ∈ 𝑍
10	8, 9	nfan 1828	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑚(𝜑 ∧ 𝑛 ∈ 𝑍)
11		simpll 790	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → 𝜑)
12		smfliminfmpt.z	. . . . . . . . . . . . . . . 16 ⊢ 𝑍 = (ℤ_≥‘𝑀)
13	12	uztrn2 11705	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 ∈ 𝑍 ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → 𝑚 ∈ 𝑍)
14	13	adantll 750	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → 𝑚 ∈ 𝑍)
15		simpr 477	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → 𝑚 ∈ 𝑍)
16		smfliminfmpt.f	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ (SMblFn‘𝑆))
17	16	elexd 3214	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V)
18		eqid 2622	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵)) = (𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))
19	18	fvmpt2 6291	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑚 ∈ 𝑍 ∧ (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V) → ((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚) = (𝑥 ∈ 𝐴 ↦ 𝐵))
20	15, 17, 19	syl2anc 693	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → ((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚) = (𝑥 ∈ 𝐴 ↦ 𝐵))
21	20	dmeqd 5326	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → dom ((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚) = dom (𝑥 ∈ 𝐴 ↦ 𝐵))
22		nfv 1843	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑥 𝑚 ∈ 𝑍
23	3, 22	nfan 1828	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑚 ∈ 𝑍)
24		eqid 2622	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
25		smfliminfmpt.b	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)
26	25	3expa 1265	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑚 ∈ 𝑍) ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)
27	23, 24, 26	dmmptdf 39417	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → dom (𝑥 ∈ 𝐴 ↦ 𝐵) = 𝐴)
28	21, 27	eqtr2d 2657	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → 𝐴 = dom ((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚))
29	11, 14, 28	syl2anc 693	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → 𝐴 = dom ((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚))
30	10, 29	iineq2d 4541	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴 = ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom ((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚))
31	7, 30	iuneq2df 39212	. . . . . . . . . . 11 ⊢ (𝜑 → ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴 = ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom ((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚))
32	31	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴) → ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴 = ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom ((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚))
33	6, 32	eleqtrd 2703	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴) → 𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom ((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚))
34	33	adantrr 753	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴 ∧ (lim inf‘(𝑚 ∈ 𝑍 ↦ 𝐵)) ∈ ℝ)) → 𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom ((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚))
35		eliun 4524	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴 ↔ ∃𝑛 ∈ 𝑍 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴)
36	35	biimpi 206	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴 → ∃𝑛 ∈ 𝑍 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴)
37	36	adantl 482	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴) → ∃𝑛 ∈ 𝑍 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴)
38		nfv 1843	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑛(lim inf‘(𝑚 ∈ 𝑍 ↦ (((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚)‘𝑥))) = (lim inf‘(𝑚 ∈ 𝑍 ↦ 𝐵))
39		nfcv 2764	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑚𝑥
40		nfii1 4551	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑚∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴
41	39, 40	nfel 2777	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑚 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴
42	8, 9, 41	nf3an 1831	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑚(𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴)
43	20	fveq1d 6193	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚)‘𝑥) = ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥))
44	11, 14, 43	syl2anc 693	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → (((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚)‘𝑥) = ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥))
45	44	3adantl3 1219	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴) ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → (((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚)‘𝑥) = ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥))
46		eliinid 39294	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴 ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → 𝑥 ∈ 𝐴)
47	46	3ad2antl3 1225	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴) ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → 𝑥 ∈ 𝐴)
48		simpl1 1064	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴) ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → 𝜑)
49		simp2 1062	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴) → 𝑛 ∈ 𝑍)
50	49, 13	sylan 488	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴) ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → 𝑚 ∈ 𝑍)
51	48, 50, 47, 25	syl3anc 1326	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴) ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → 𝐵 ∈ 𝑉)
52	24	fvmpt2 6291	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝑉) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = 𝐵)
53	47, 51, 52	syl2anc 693	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴) ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = 𝐵)
54	45, 53	eqtrd 2656	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴) ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → (((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚)‘𝑥) = 𝐵)
55	42, 54	mpteq2da 4743	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴) → (𝑚 ∈ (ℤ_≥‘𝑛) ↦ (((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚)‘𝑥)) = (𝑚 ∈ (ℤ_≥‘𝑛) ↦ 𝐵))
56	55	fveq2d 6195	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴) → (lim inf‘(𝑚 ∈ (ℤ_≥‘𝑛) ↦ (((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚)‘𝑥))) = (lim inf‘(𝑚 ∈ (ℤ_≥‘𝑛) ↦ 𝐵)))
57		nfcv 2764	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑚𝑍
58		nfcv 2764	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑚(ℤ_≥‘𝑛)
59		eqid 2622	. . . . . . . . . . . . . . . 16 ⊢ (ℤ_≥‘𝑛) = (ℤ_≥‘𝑛)
60	12	eluzelz2 39627	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ 𝑍 → 𝑛 ∈ ℤ)
61	60	uzidd 39631	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ 𝑍 → 𝑛 ∈ (ℤ_≥‘𝑛))
62	61	3ad2ant2 1083	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴) → 𝑛 ∈ (ℤ_≥‘𝑛))
63		fvexd 6203	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴) ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → (((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚)‘𝑥) ∈ V)
64	42, 57, 58, 12, 59, 49, 62, 63	liminfequzmpt2 40023	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴) → (lim inf‘(𝑚 ∈ 𝑍 ↦ (((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚)‘𝑥))) = (lim inf‘(𝑚 ∈ (ℤ_≥‘𝑛) ↦ (((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚)‘𝑥))))
65	42, 57, 58, 12, 59, 49, 62, 51	liminfequzmpt2 40023	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴) → (lim inf‘(𝑚 ∈ 𝑍 ↦ 𝐵)) = (lim inf‘(𝑚 ∈ (ℤ_≥‘𝑛) ↦ 𝐵)))
66	56, 64, 65	3eqtr4d 2666	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴) → (lim inf‘(𝑚 ∈ 𝑍 ↦ (((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚)‘𝑥))) = (lim inf‘(𝑚 ∈ 𝑍 ↦ 𝐵)))
67	66	3exp 1264	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑛 ∈ 𝑍 → (𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴 → (lim inf‘(𝑚 ∈ 𝑍 ↦ (((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚)‘𝑥))) = (lim inf‘(𝑚 ∈ 𝑍 ↦ 𝐵)))))
68	7, 38, 67	rexlimd 3026	. . . . . . . . . . . 12 ⊢ (𝜑 → (∃𝑛 ∈ 𝑍 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴 → (lim inf‘(𝑚 ∈ 𝑍 ↦ (((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚)‘𝑥))) = (lim inf‘(𝑚 ∈ 𝑍 ↦ 𝐵))))
69	68	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴) → (∃𝑛 ∈ 𝑍 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴 → (lim inf‘(𝑚 ∈ 𝑍 ↦ (((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚)‘𝑥))) = (lim inf‘(𝑚 ∈ 𝑍 ↦ 𝐵))))
70	37, 69	mpd 15	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴) → (lim inf‘(𝑚 ∈ 𝑍 ↦ (((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚)‘𝑥))) = (lim inf‘(𝑚 ∈ 𝑍 ↦ 𝐵)))
71	70	adantrr 753	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴 ∧ (lim inf‘(𝑚 ∈ 𝑍 ↦ 𝐵)) ∈ ℝ)) → (lim inf‘(𝑚 ∈ 𝑍 ↦ (((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚)‘𝑥))) = (lim inf‘(𝑚 ∈ 𝑍 ↦ 𝐵)))
72		simprr 796	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴 ∧ (lim inf‘(𝑚 ∈ 𝑍 ↦ 𝐵)) ∈ ℝ)) → (lim inf‘(𝑚 ∈ 𝑍 ↦ 𝐵)) ∈ ℝ)
73	71, 72	eqeltrd 2701	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴 ∧ (lim inf‘(𝑚 ∈ 𝑍 ↦ 𝐵)) ∈ ℝ)) → (lim inf‘(𝑚 ∈ 𝑍 ↦ (((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚)‘𝑥))) ∈ ℝ)
74	34, 73	jca 554	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴 ∧ (lim inf‘(𝑚 ∈ 𝑍 ↦ 𝐵)) ∈ ℝ)) → (𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom ((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚) ∧ (lim inf‘(𝑚 ∈ 𝑍 ↦ (((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚)‘𝑥))) ∈ ℝ))
75		simpl 473	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom ((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚) ∧ (lim inf‘(𝑚 ∈ 𝑍 ↦ (((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚)‘𝑥))) ∈ ℝ)) → 𝜑)
76		simpr 477	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom ((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚)) → 𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom ((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚))
77	31	eqcomd 2628	. . . . . . . . . . 11 ⊢ (𝜑 → ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom ((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚) = ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴)
78	77	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom ((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚)) → ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom ((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚) = ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴)
79	76, 78	eleqtrd 2703	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom ((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚)) → 𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴)
80	79	adantrr 753	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom ((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚) ∧ (lim inf‘(𝑚 ∈ 𝑍 ↦ (((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚)‘𝑥))) ∈ ℝ)) → 𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴)
81		simprr 796	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom ((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚) ∧ (lim inf‘(𝑚 ∈ 𝑍 ↦ (((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚)‘𝑥))) ∈ ℝ)) → (lim inf‘(𝑚 ∈ 𝑍 ↦ (((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚)‘𝑥))) ∈ ℝ)
82		simp2 1062	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴 ∧ (lim inf‘(𝑚 ∈ 𝑍 ↦ (((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚)‘𝑥))) ∈ ℝ) → 𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴)
83	70	eqcomd 2628	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴) → (lim inf‘(𝑚 ∈ 𝑍 ↦ 𝐵)) = (lim inf‘(𝑚 ∈ 𝑍 ↦ (((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚)‘𝑥))))
84	83	3adant3 1081	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴 ∧ (lim inf‘(𝑚 ∈ 𝑍 ↦ (((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚)‘𝑥))) ∈ ℝ) → (lim inf‘(𝑚 ∈ 𝑍 ↦ 𝐵)) = (lim inf‘(𝑚 ∈ 𝑍 ↦ (((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚)‘𝑥))))
85		simp3 1063	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴 ∧ (lim inf‘(𝑚 ∈ 𝑍 ↦ (((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚)‘𝑥))) ∈ ℝ) → (lim inf‘(𝑚 ∈ 𝑍 ↦ (((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚)‘𝑥))) ∈ ℝ)
86	84, 85	eqeltrd 2701	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴 ∧ (lim inf‘(𝑚 ∈ 𝑍 ↦ (((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚)‘𝑥))) ∈ ℝ) → (lim inf‘(𝑚 ∈ 𝑍 ↦ 𝐵)) ∈ ℝ)
87	82, 86	jca 554	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴 ∧ (lim inf‘(𝑚 ∈ 𝑍 ↦ (((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚)‘𝑥))) ∈ ℝ) → (𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴 ∧ (lim inf‘(𝑚 ∈ 𝑍 ↦ 𝐵)) ∈ ℝ))
88	75, 80, 81, 87	syl3anc 1326	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom ((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚) ∧ (lim inf‘(𝑚 ∈ 𝑍 ↦ (((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚)‘𝑥))) ∈ ℝ)) → (𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴 ∧ (lim inf‘(𝑚 ∈ 𝑍 ↦ 𝐵)) ∈ ℝ))
89	74, 88	impbida 877	. . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴 ∧ (lim inf‘(𝑚 ∈ 𝑍 ↦ 𝐵)) ∈ ℝ) ↔ (𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom ((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚) ∧ (lim inf‘(𝑚 ∈ 𝑍 ↦ (((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚)‘𝑥))) ∈ ℝ)))
90	3, 89	rabbida3 39320	. . . . 5 ⊢ (𝜑 → {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴 ∣ (lim inf‘(𝑚 ∈ 𝑍 ↦ 𝐵)) ∈ ℝ} = {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom ((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚) ∣ (lim inf‘(𝑚 ∈ 𝑍 ↦ (((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚)‘𝑥))) ∈ ℝ})
91	5, 90	eqtrd 2656	. . . 4 ⊢ (𝜑 → 𝐷 = {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom ((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚) ∣ (lim inf‘(𝑚 ∈ 𝑍 ↦ (((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚)‘𝑥))) ∈ ℝ})
92	4	eleq2i 2693	. . . . . . 7 ⊢ (𝑥 ∈ 𝐷 ↔ 𝑥 ∈ {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴 ∣ (lim inf‘(𝑚 ∈ 𝑍 ↦ 𝐵)) ∈ ℝ})
93	92	biimpi 206	. . . . . 6 ⊢ (𝑥 ∈ 𝐷 → 𝑥 ∈ {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴 ∣ (lim inf‘(𝑚 ∈ 𝑍 ↦ 𝐵)) ∈ ℝ})
94		rabidim1 3117	. . . . . 6 ⊢ (𝑥 ∈ {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴 ∣ (lim inf‘(𝑚 ∈ 𝑍 ↦ 𝐵)) ∈ ℝ} → 𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴)
95	93, 94	syl 17	. . . . 5 ⊢ (𝑥 ∈ 𝐷 → 𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴)
96	95, 83	sylan2 491	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (lim inf‘(𝑚 ∈ 𝑍 ↦ 𝐵)) = (lim inf‘(𝑚 ∈ 𝑍 ↦ (((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚)‘𝑥))))
97	3, 91, 96	mpteq12da 39452	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐷 ↦ (lim inf‘(𝑚 ∈ 𝑍 ↦ 𝐵))) = (𝑥 ∈ {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom ((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚) ∣ (lim inf‘(𝑚 ∈ 𝑍 ↦ (((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚)‘𝑥))) ∈ ℝ} ↦ (lim inf‘(𝑚 ∈ 𝑍 ↦ (((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚)‘𝑥)))))
98	2, 97	eqtrd 2656	. 2 ⊢ (𝜑 → 𝐺 = (𝑥 ∈ {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom ((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚) ∣ (lim inf‘(𝑚 ∈ 𝑍 ↦ (((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚)‘𝑥))) ∈ ℝ} ↦ (lim inf‘(𝑚 ∈ 𝑍 ↦ (((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚)‘𝑥)))))
99		nfmpt1 4747	. . 3 ⊢ Ⅎ𝑚(𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))
100		nfcv 2764	. . . 4 ⊢ Ⅎ𝑥𝑍
101		nfmpt1 4747	. . . 4 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐵)
102	100, 101	nfmpt 4746	. . 3 ⊢ Ⅎ𝑥(𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))
103		smfliminfmpt.m	. . 3 ⊢ (𝜑 → 𝑀 ∈ ℤ)
104		smfliminfmpt.s	. . 3 ⊢ (𝜑 → 𝑆 ∈ SAlg)
105	8, 16	fmptd2f 39442	. . 3 ⊢ (𝜑 → (𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵)):𝑍⟶(SMblFn‘𝑆))
106		eqid 2622	. . 3 ⊢ {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom ((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚) ∣ (lim inf‘(𝑚 ∈ 𝑍 ↦ (((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚)‘𝑥))) ∈ ℝ} = {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom ((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚) ∣ (lim inf‘(𝑚 ∈ 𝑍 ↦ (((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚)‘𝑥))) ∈ ℝ}
107		eqid 2622	. . 3 ⊢ (𝑥 ∈ {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom ((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚) ∣ (lim inf‘(𝑚 ∈ 𝑍 ↦ (((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚)‘𝑥))) ∈ ℝ} ↦ (lim inf‘(𝑚 ∈ 𝑍 ↦ (((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚)‘𝑥)))) = (𝑥 ∈ {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom ((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚) ∣ (lim inf‘(𝑚 ∈ 𝑍 ↦ (((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚)‘𝑥))) ∈ ℝ} ↦ (lim inf‘(𝑚 ∈ 𝑍 ↦ (((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚)‘𝑥))))
108	99, 102, 103, 12, 104, 105, 106, 107	smfliminf 41037	. 2 ⊢ (𝜑 → (𝑥 ∈ {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom ((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚) ∣ (lim inf‘(𝑚 ∈ 𝑍 ↦ (((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚)‘𝑥))) ∈ ℝ} ↦ (lim inf‘(𝑚 ∈ 𝑍 ↦ (((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚)‘𝑥)))) ∈ (SMblFn‘𝑆))
109	98, 108	eqeltrd 2701	1 ⊢ (𝜑 → 𝐺 ∈ (SMblFn‘𝑆))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 384 ∧ w3a 1037 = wceq 1483 Ⅎwnf 1708 ∈ wcel 1990 ∃wrex 2913 {crab 2916 Vcvv 3200 ∪ ciun 4520 ∩ ciin 4521 ↦ cmpt 4729 dom cdm 5114 ‘cfv 5888 ℝcr 9935 ℤcz 11377 ℤ_≥cuz 11687 lim infclsi 39983 SAlgcsalg 40528 SMblFncsmblfn 40909

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cc 9257 ax-ac2 9285 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-omul 7565 df-er 7742 df-map 7859 df-pm 7860 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-acn 8768 df-ac 8939 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-n0 11293 df-z 11378 df-uz 11688 df-q 11789 df-rp 11833 df-xneg 11946 df-ioo 12179 df-ioc 12180 df-ico 12181 df-icc 12182 df-fz 12327 df-fzo 12466 df-fl 12593 df-ceil 12594 df-seq 12802 df-exp 12861 df-hash 13118 df-word 13299 df-concat 13301 df-s1 13302 df-s2 13593 df-s3 13594 df-s4 13595 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-limsup 14202 df-clim 14219 df-rlim 14220 df-rest 16083 df-topgen 16104 df-top 20699 df-bases 20750 df-liminf 39984 df-salg 40529 df-salgen 40533 df-smblfn 40910

This theorem is referenced by: (None)

W3C validator