smflimsupmpt - Mathbox for Glauco Siliprandi

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Theorem smflimsupmpt 47257

Description: The superior limit of a sequence of sigma-measurable functions is sigma-measurable. Proposition 121F (d) 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 indices 𝐵(𝑚, 𝑥). (Contributed by Glauco Siliprandi, 23-Oct-2021.)

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

**Assertion**
Ref	Expression
smflimsupmpt	⊢ (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) 𝑊(𝑥,𝑚,𝑛)

**Proof of Theorem smflimsupmpt**
Step	Hyp	Ref	Expression
1		smflimsupmpt.g	. . . 4 ⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ (lim sup‘(𝑚 ∈ 𝑍 ↦ 𝐵)))
2	1	a1i 11	. . 3 ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐷 ↦ (lim sup‘(𝑚 ∈ 𝑍 ↦ 𝐵))))
3		smflimsupmpt.x	. . . 4 ⊢ Ⅎ𝑥𝜑
4		smflimsupmpt.d	. . . . . 6 ⊢ 𝐷 = {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴 ∣ (lim sup‘(𝑚 ∈ 𝑍 ↦ 𝐵)) ∈ ℝ}
5	4	a1i 11	. . . . 5 ⊢ (𝜑 → 𝐷 = {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴 ∣ (lim sup‘(𝑚 ∈ 𝑍 ↦ 𝐵)) ∈ ℝ})
6		simpr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴) → 𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴)
7		smflimsupmpt.n	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑛𝜑
8		smflimsupmpt.p	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑚𝜑
9		nfv 1916	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑚 𝑛 ∈ 𝑍
10	8, 9	nfan 1901	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑚(𝜑 ∧ 𝑛 ∈ 𝑍)
11		simpll 767	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → 𝜑)
12		smflimsupmpt.z	. . . . . . . . . . . . . . . . 17 ⊢ 𝑍 = (ℤ_≥‘𝑀)
13	12	uztrn2 12807	. . . . . . . . . . . . . . . 16 ⊢ ((𝑛 ∈ 𝑍 ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → 𝑚 ∈ 𝑍)
14	13	adantll 715	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → 𝑚 ∈ 𝑍)
15		simpr 484	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → 𝑚 ∈ 𝑍)
16		smflimsupmpt.f	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ (SMblFn‘𝑆))
17	16	elexd 3453	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V)
18		eqid 2736	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵)) = (𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))
19	18	fvmpt2 6959	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑚 ∈ 𝑍 ∧ (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V) → ((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚) = (𝑥 ∈ 𝐴 ↦ 𝐵))
20	15, 17, 19	syl2anc 585	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → ((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚) = (𝑥 ∈ 𝐴 ↦ 𝐵))
21	20	dmeqd 5860	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → dom ((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚) = dom (𝑥 ∈ 𝐴 ↦ 𝐵))
22		nfv 1916	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑥 𝑚 ∈ 𝑍
23	3, 22	nfan 1901	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑚 ∈ 𝑍)
24		eqid 2736	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
25		smflimsupmpt.b	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑊)
26	25	3expa 1119	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑚 ∈ 𝑍) ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑊)
27	23, 24, 26	dmmptdf 45653	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → dom (𝑥 ∈ 𝐴 ↦ 𝐵) = 𝐴)
28	21, 27	eqtr2d 2772	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → 𝐴 = dom ((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚))
29	11, 14, 28	syl2anc 585	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → 𝐴 = dom ((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚))
30	10, 29	iineq2d 4957	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴 = ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom ((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚))
31	7, 30	iuneq2df 45478	. . . . . . . . . . . 12 ⊢ (𝜑 → ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴 = ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom ((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚))
32	31	adantr 480	. . . . . . . . . . 11 ⊢ ((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 2838	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴) → 𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom ((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚))
34	33	adantrr 718	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴 ∧ (lim sup‘(𝑚 ∈ 𝑍 ↦ 𝐵)) ∈ ℝ)) → 𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom ((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚))
35		eliun 4937	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴 ↔ ∃𝑛 ∈ 𝑍 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴)
36	35	biimpi 216	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴 → ∃𝑛 ∈ 𝑍 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴)
37	36	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴) → ∃𝑛 ∈ 𝑍 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴)
38		nfv 1916	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑛(lim sup‘(𝑚 ∈ 𝑍 ↦ (((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚)‘𝑥))) = (lim sup‘(𝑚 ∈ 𝑍 ↦ 𝐵))
39		nfcv 2898	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑚𝑥
40		nfii1 4971	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑚∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴
41	39, 40	nfel 2913	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑚 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴
42	8, 9, 41	nf3an 1903	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑚(𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴)
43	20	fveq1d 6842	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚)‘𝑥) = ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥))
44	11, 14, 43	syl2anc 585	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → (((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚)‘𝑥) = ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥))
45	44	3adantl3 1170	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴) ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → (((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚)‘𝑥) = ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥))
46		eliinid 45541	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴 ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → 𝑥 ∈ 𝐴)
47	46	3ad2antl3 1189	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴) ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → 𝑥 ∈ 𝐴)
48		simpl1 1193	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴) ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → 𝜑)
49	14	3adantl3 1170	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴) ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → 𝑚 ∈ 𝑍)
50	48, 49, 47, 25	syl3anc 1374	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴) ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → 𝐵 ∈ 𝑊)
51	24	fvmpt2 6959	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝑊) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = 𝐵)
52	47, 50, 51	syl2anc 585	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴) ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = 𝐵)
53	45, 52	eqtrd 2771	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴) ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → (((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚)‘𝑥) = 𝐵)
54	42, 53	mpteq2da 5177	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴) → (𝑚 ∈ (ℤ_≥‘𝑛) ↦ (((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚)‘𝑥)) = (𝑚 ∈ (ℤ_≥‘𝑛) ↦ 𝐵))
55	54	fveq2d 6844	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴) → (lim sup‘(𝑚 ∈ (ℤ_≥‘𝑛) ↦ (((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚)‘𝑥))) = (lim sup‘(𝑚 ∈ (ℤ_≥‘𝑛) ↦ 𝐵)))
56		smflimsupmpt.m	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑀 ∈ ℤ)
57	56	3ad2ant1 1134	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴) → 𝑀 ∈ ℤ)
58	12	eluzelz2 45831	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ 𝑍 → 𝑛 ∈ ℤ)
59	58	3ad2ant2 1135	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴) → 𝑛 ∈ ℤ)
60		eqid 2736	. . . . . . . . . . . . . . . . 17 ⊢ (ℤ_≥‘𝑛) = (ℤ_≥‘𝑛)
61		fvexd 6855	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴) ∧ 𝑚 ∈ 𝑍) → (((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚)‘𝑥) ∈ V)
62	49, 61	syldan 592	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴) ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → (((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚)‘𝑥) ∈ V)
63	42, 57, 59, 12, 60, 61, 62	limsupequzmpt 46157	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴) → (lim sup‘(𝑚 ∈ 𝑍 ↦ (((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚)‘𝑥))) = (lim sup‘(𝑚 ∈ (ℤ_≥‘𝑛) ↦ (((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚)‘𝑥))))
64	9	nfci 2886	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑚𝑍
65		nfcv 2898	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑚(ℤ_≥‘𝑛)
66		simp2 1138	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴) → 𝑛 ∈ 𝑍)
67	59	uzidd 12804	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴) → 𝑛 ∈ (ℤ_≥‘𝑛))
68	42, 64, 65, 12, 60, 66, 67, 50	limsupequzmpt2 46146	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴) → (lim sup‘(𝑚 ∈ 𝑍 ↦ 𝐵)) = (lim sup‘(𝑚 ∈ (ℤ_≥‘𝑛) ↦ 𝐵)))
69	55, 63, 68	3eqtr4d 2781	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴) → (lim sup‘(𝑚 ∈ 𝑍 ↦ (((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚)‘𝑥))) = (lim sup‘(𝑚 ∈ 𝑍 ↦ 𝐵)))
70	69	3exp 1120	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑛 ∈ 𝑍 → (𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴 → (lim sup‘(𝑚 ∈ 𝑍 ↦ (((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚)‘𝑥))) = (lim sup‘(𝑚 ∈ 𝑍 ↦ 𝐵)))))
71	7, 38, 70	rexlimd 3244	. . . . . . . . . . . . 13 ⊢ (𝜑 → (∃𝑛 ∈ 𝑍 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴 → (lim sup‘(𝑚 ∈ 𝑍 ↦ (((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚)‘𝑥))) = (lim sup‘(𝑚 ∈ 𝑍 ↦ 𝐵))))
72	71	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴) → (∃𝑛 ∈ 𝑍 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴 → (lim sup‘(𝑚 ∈ 𝑍 ↦ (((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚)‘𝑥))) = (lim sup‘(𝑚 ∈ 𝑍 ↦ 𝐵))))
73	37, 72	mpd 15	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴) → (lim sup‘(𝑚 ∈ 𝑍 ↦ (((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚)‘𝑥))) = (lim sup‘(𝑚 ∈ 𝑍 ↦ 𝐵)))
74	73	adantrr 718	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴 ∧ (lim sup‘(𝑚 ∈ 𝑍 ↦ 𝐵)) ∈ ℝ)) → (lim sup‘(𝑚 ∈ 𝑍 ↦ (((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚)‘𝑥))) = (lim sup‘(𝑚 ∈ 𝑍 ↦ 𝐵)))
75		simprr 773	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴 ∧ (lim sup‘(𝑚 ∈ 𝑍 ↦ 𝐵)) ∈ ℝ)) → (lim sup‘(𝑚 ∈ 𝑍 ↦ 𝐵)) ∈ ℝ)
76	74, 75	eqeltrd 2836	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴 ∧ (lim sup‘(𝑚 ∈ 𝑍 ↦ 𝐵)) ∈ ℝ)) → (lim sup‘(𝑚 ∈ 𝑍 ↦ (((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚)‘𝑥))) ∈ ℝ)
77	34, 76	jca 511	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴 ∧ (lim sup‘(𝑚 ∈ 𝑍 ↦ 𝐵)) ∈ ℝ)) → (𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom ((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚) ∧ (lim sup‘(𝑚 ∈ 𝑍 ↦ (((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚)‘𝑥))) ∈ ℝ))
78	77	ex 412	. . . . . . 7 ⊢ (𝜑 → ((𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴 ∧ (lim sup‘(𝑚 ∈ 𝑍 ↦ 𝐵)) ∈ ℝ) → (𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom ((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚) ∧ (lim sup‘(𝑚 ∈ 𝑍 ↦ (((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚)‘𝑥))) ∈ ℝ)))
79		simpl 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom ((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚) ∧ (lim sup‘(𝑚 ∈ 𝑍 ↦ (((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚)‘𝑥))) ∈ ℝ)) → 𝜑)
80		simpr 484	. . . . . . . . . . 11 ⊢ ((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))
81	31	eqcomd 2742	. . . . . . . . . . . 12 ⊢ (𝜑 → ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom ((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚) = ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴)
82	81	adantr 480	. . . . . . . . . . 11 ⊢ ((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)
83	80, 82	eleqtrd 2838	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom ((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚)) → 𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴)
84	83	adantrr 718	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom ((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚) ∧ (lim sup‘(𝑚 ∈ 𝑍 ↦ (((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚)‘𝑥))) ∈ ℝ)) → 𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴)
85		simprr 773	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom ((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚) ∧ (lim sup‘(𝑚 ∈ 𝑍 ↦ (((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚)‘𝑥))) ∈ ℝ)) → (lim sup‘(𝑚 ∈ 𝑍 ↦ (((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚)‘𝑥))) ∈ ℝ)
86		simp2 1138	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴 ∧ (lim sup‘(𝑚 ∈ 𝑍 ↦ (((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚)‘𝑥))) ∈ ℝ) → 𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴)
87	73	eqcomd 2742	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴) → (lim sup‘(𝑚 ∈ 𝑍 ↦ 𝐵)) = (lim sup‘(𝑚 ∈ 𝑍 ↦ (((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚)‘𝑥))))
88	87	3adant3 1133	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴 ∧ (lim sup‘(𝑚 ∈ 𝑍 ↦ (((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚)‘𝑥))) ∈ ℝ) → (lim sup‘(𝑚 ∈ 𝑍 ↦ 𝐵)) = (lim sup‘(𝑚 ∈ 𝑍 ↦ (((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚)‘𝑥))))
89		simp3 1139	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴 ∧ (lim sup‘(𝑚 ∈ 𝑍 ↦ (((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚)‘𝑥))) ∈ ℝ) → (lim sup‘(𝑚 ∈ 𝑍 ↦ (((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚)‘𝑥))) ∈ ℝ)
90	88, 89	eqeltrd 2836	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴 ∧ (lim sup‘(𝑚 ∈ 𝑍 ↦ (((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚)‘𝑥))) ∈ ℝ) → (lim sup‘(𝑚 ∈ 𝑍 ↦ 𝐵)) ∈ ℝ)
91	86, 90	jca 511	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴 ∧ (lim sup‘(𝑚 ∈ 𝑍 ↦ (((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚)‘𝑥))) ∈ ℝ) → (𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴 ∧ (lim sup‘(𝑚 ∈ 𝑍 ↦ 𝐵)) ∈ ℝ))
92	79, 84, 85, 91	syl3anc 1374	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom ((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚) ∧ (lim sup‘(𝑚 ∈ 𝑍 ↦ (((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚)‘𝑥))) ∈ ℝ)) → (𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴 ∧ (lim sup‘(𝑚 ∈ 𝑍 ↦ 𝐵)) ∈ ℝ))
93	92	ex 412	. . . . . . 7 ⊢ (𝜑 → ((𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom ((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚) ∧ (lim sup‘(𝑚 ∈ 𝑍 ↦ (((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚)‘𝑥))) ∈ ℝ) → (𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴 ∧ (lim sup‘(𝑚 ∈ 𝑍 ↦ 𝐵)) ∈ ℝ)))
94	78, 93	impbid 212	. . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴 ∧ (lim sup‘(𝑚 ∈ 𝑍 ↦ 𝐵)) ∈ ℝ) ↔ (𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom ((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚) ∧ (lim sup‘(𝑚 ∈ 𝑍 ↦ (((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚)‘𝑥))) ∈ ℝ)))
95	3, 94	rabbida3 45565	. . . . 5 ⊢ (𝜑 → {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴 ∣ (lim sup‘(𝑚 ∈ 𝑍 ↦ 𝐵)) ∈ ℝ} = {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom ((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚) ∣ (lim sup‘(𝑚 ∈ 𝑍 ↦ (((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚)‘𝑥))) ∈ ℝ})
96	5, 95	eqtrd 2771	. . . 4 ⊢ (𝜑 → 𝐷 = {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom ((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚) ∣ (lim sup‘(𝑚 ∈ 𝑍 ↦ (((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚)‘𝑥))) ∈ ℝ})
97	4	eleq2i 2828	. . . . . . 7 ⊢ (𝑥 ∈ 𝐷 ↔ 𝑥 ∈ {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴 ∣ (lim sup‘(𝑚 ∈ 𝑍 ↦ 𝐵)) ∈ ℝ})
98	97	biimpi 216	. . . . . 6 ⊢ (𝑥 ∈ 𝐷 → 𝑥 ∈ {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴 ∣ (lim sup‘(𝑚 ∈ 𝑍 ↦ 𝐵)) ∈ ℝ})
99		rabidim1 3411	. . . . . 6 ⊢ (𝑥 ∈ {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴 ∣ (lim sup‘(𝑚 ∈ 𝑍 ↦ 𝐵)) ∈ ℝ} → 𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴)
100	98, 99	syl 17	. . . . 5 ⊢ (𝑥 ∈ 𝐷 → 𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)𝐴)
101	100, 87	sylan2 594	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (lim sup‘(𝑚 ∈ 𝑍 ↦ 𝐵)) = (lim sup‘(𝑚 ∈ 𝑍 ↦ (((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚)‘𝑥))))
102	3, 96, 101	mpteq12da 5168	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐷 ↦ (lim sup‘(𝑚 ∈ 𝑍 ↦ 𝐵))) = (𝑥 ∈ {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom ((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚) ∣ (lim sup‘(𝑚 ∈ 𝑍 ↦ (((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚)‘𝑥))) ∈ ℝ} ↦ (lim sup‘(𝑚 ∈ 𝑍 ↦ (((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚)‘𝑥)))))
103	2, 102	eqtrd 2771	. 2 ⊢ (𝜑 → 𝐺 = (𝑥 ∈ {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom ((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚) ∣ (lim sup‘(𝑚 ∈ 𝑍 ↦ (((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚)‘𝑥))) ∈ ℝ} ↦ (lim sup‘(𝑚 ∈ 𝑍 ↦ (((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚)‘𝑥)))))
104		nfmpt1 5184	. . 3 ⊢ Ⅎ𝑚(𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))
105		nfcv 2898	. . . 4 ⊢ Ⅎ𝑥𝑍
106		nfmpt1 5184	. . . 4 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐵)
107	105, 106	nfmpt 5183	. . 3 ⊢ Ⅎ𝑥(𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))
108		smflimsupmpt.s	. . 3 ⊢ (𝜑 → 𝑆 ∈ SAlg)
109	8, 16	fmptd2f 45664	. . 3 ⊢ (𝜑 → (𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵)):𝑍⟶(SMblFn‘𝑆))
110		eqid 2736	. . 3 ⊢ {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom ((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚) ∣ (lim sup‘(𝑚 ∈ 𝑍 ↦ (((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚)‘𝑥))) ∈ ℝ} = {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom ((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚) ∣ (lim sup‘(𝑚 ∈ 𝑍 ↦ (((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚)‘𝑥))) ∈ ℝ}
111		eqid 2736	. . 3 ⊢ (𝑥 ∈ {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom ((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚) ∣ (lim sup‘(𝑚 ∈ 𝑍 ↦ (((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚)‘𝑥))) ∈ ℝ} ↦ (lim sup‘(𝑚 ∈ 𝑍 ↦ (((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚)‘𝑥)))) = (𝑥 ∈ {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom ((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚) ∣ (lim sup‘(𝑚 ∈ 𝑍 ↦ (((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚)‘𝑥))) ∈ ℝ} ↦ (lim sup‘(𝑚 ∈ 𝑍 ↦ (((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚)‘𝑥))))
112	104, 107, 56, 12, 108, 109, 110, 111	smflimsup 47256	. 2 ⊢ (𝜑 → (𝑥 ∈ {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom ((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚) ∣ (lim sup‘(𝑚 ∈ 𝑍 ↦ (((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚)‘𝑥))) ∈ ℝ} ↦ (lim sup‘(𝑚 ∈ 𝑍 ↦ (((𝑚 ∈ 𝑍 ↦ (𝑥 ∈ 𝐴 ↦ 𝐵))‘𝑚)‘𝑥)))) ∈ (SMblFn‘𝑆))
113	103, 112	eqeltrd 2836	1 ⊢ (𝜑 → 𝐺 ∈ (SMblFn‘𝑆))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1542 Ⅎwnf 1785 ∈ wcel 2114 ∃wrex 3061 {crab 3389 Vcvv 3429 ∪ ciun 4933 ∩ ciin 4934 ↦ cmpt 5166 dom cdm 5631 ‘cfv 6498 ℝcr 11037 ℤcz 12524 ℤ_≥cuz 12788 lim supclsp 15432 SAlgcsalg 46736 SMblFncsmblfn 47123

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 2708 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 ax-inf2 9562 ax-cc 10357 ax-ac2 10385 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116

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 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3062 df-rmo 3342 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3909 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4851 df-int 4890 df-iun 4935 df-iin 4936 df-br 5086 df-opab 5148 df-mpt 5167 df-tr 5193 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-se 5585 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6265 df-ord 6326 df-on 6327 df-lim 6328 df-suc 6329 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-isom 6507 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-om 7818 df-1st 7942 df-2nd 7943 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-1o 8405 df-2o 8406 df-oadd 8409 df-omul 8410 df-er 8643 df-map 8775 df-pm 8776 df-en 8894 df-dom 8895 df-sdom 8896 df-fin 8897 df-sup 9355 df-inf 9356 df-oi 9425 df-card 9863 df-acn 9866 df-ac 10038 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-div 11808 df-nn 12175 df-2 12244 df-3 12245 df-n0 12438 df-z 12525 df-uz 12789 df-q 12899 df-rp 12943 df-ioo 13302 df-ioc 13303 df-ico 13304 df-fz 13462 df-fl 13751 df-ceil 13752 df-seq 13964 df-exp 14024 df-cj 15061 df-re 15062 df-im 15063 df-sqrt 15197 df-abs 15198 df-limsup 15433 df-clim 15450 df-rlim 15451 df-rest 17385 df-topgen 17406 df-top 22859 df-bases 22911 df-salg 46737 df-salgen 46741 df-smblfn 47124

This theorem is referenced by: smfliminflem 47258

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