smflimsuplem7 - Mathbox for Glauco Siliprandi

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Theorem smflimsuplem7 44753

Description: The superior limit of a sequence of sigma-measurable functions is sigma-measurable. Proposition 121F (d) of [Fremlin1] p. 39 . (Contributed by Glauco Siliprandi, 23-Oct-2021.)

**Hypotheses**
Ref	Expression
smflimsuplem7.m	⊢ (𝜑 → 𝑀 ∈ ℤ)
smflimsuplem7.z	⊢ 𝑍 = (ℤ_≥‘𝑀)
smflimsuplem7.s	⊢ (𝜑 → 𝑆 ∈ SAlg)
smflimsuplem7.f	⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆))
smflimsuplem7.d	⊢ 𝐷 = {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∣ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ}
smflimsuplem7.e	⊢ 𝐸 = (𝑘 ∈ 𝑍 ↦ {𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑘)dom (𝐹‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ_≥‘𝑘) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^*, < ) ∈ ℝ})
smflimsuplem7.h	⊢ 𝐻 = (𝑘 ∈ 𝑍 ↦ (𝑥 ∈ (𝐸‘𝑘) ↦ sup(ran (𝑚 ∈ (ℤ_≥‘𝑘) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^*, < )))

**Assertion**
Ref	Expression
smflimsuplem7	⊢ (𝜑 → 𝐷 = {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑘 ∈ (ℤ_≥‘𝑛)dom (𝐻‘𝑘) ∣ (𝑘 ∈ 𝑍 ↦ ((𝐻‘𝑘)‘𝑥)) ∈ dom ⇝ })

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

Proof of Theorem smflimsuplem7 Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		smflimsuplem7.d	. . 3 ⊢ 𝐷 = {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∣ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ}
2	1	a1i 11	. 2 ⊢ (𝜑 → 𝐷 = {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∣ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ})
3		simpl 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∣ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ}) → 𝜑)
4		rabidim2 43024	. . . . . . . . . 10 ⊢ (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))) ∈ ℝ)
5	4	adantl 483	. . . . . . . . 9 ⊢ ((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))) ∈ ℝ)
6		rabidim1 3425	. . . . . . . . . . 11 ⊢ (𝑥 ∈ {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))
7		eliun 4949	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ↔ ∃𝑛 ∈ 𝑍 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚))
8	6, 7	sylib 217	. . . . . . . . . 10 ⊢ (𝑥 ∈ {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∣ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ} → ∃𝑛 ∈ 𝑍 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚))
9	8	adantl 483	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∣ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ}) → ∃𝑛 ∈ 𝑍 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚))
10		nfv 1917	. . . . . . . . . . . 12 ⊢ Ⅎ𝑛(𝜑 ∧ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ)
11		nfv 1917	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑚𝜑
12		nfcv 2905	. . . . . . . . . . . . . . . . . . . . 21 ⊢ Ⅎ𝑚lim sup
13		nfmpt1 5204	. . . . . . . . . . . . . . . . . . . . 21 ⊢ Ⅎ𝑚(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))
14	12, 13	nffv 6839	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑚(lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)))
15		nfcv 2905	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑚ℝ
16	14, 15	nfel 2919	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑚(lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ
17	11, 16	nfan 1902	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑚(𝜑 ∧ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ)
18		nfv 1917	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑚 𝑛 ∈ 𝑍
19		nfcv 2905	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑚𝑥
20		nfii1 4980	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑚∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚)
21	19, 20	nfel 2919	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑚 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚)
22	17, 18, 21	nf3an 1904	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑚((𝜑 ∧ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ) ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚))
23		nfv 1917	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑚 𝑘 ∈ (ℤ_≥‘𝑛)
24	22, 23	nfan 1902	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑚(((𝜑 ∧ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ) ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚)) ∧ 𝑘 ∈ (ℤ_≥‘𝑛))
25		simpl1l 1224	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ) ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚)) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → 𝜑)
26		smflimsuplem7.m	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑀 ∈ ℤ)
27	25, 26	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ) ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚)) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → 𝑀 ∈ ℤ)
28		smflimsuplem7.z	. . . . . . . . . . . . . . . 16 ⊢ 𝑍 = (ℤ_≥‘𝑀)
29		smflimsuplem7.s	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑆 ∈ SAlg)
30	25, 29	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ) ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚)) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → 𝑆 ∈ SAlg)
31		smflimsuplem7.f	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆))
32	25, 31	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ) ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚)) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → 𝐹:𝑍⟶(SMblFn‘𝑆))
33		smflimsuplem7.e	. . . . . . . . . . . . . . . 16 ⊢ 𝐸 = (𝑘 ∈ 𝑍 ↦ {𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑘)dom (𝐹‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ_≥‘𝑘) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^*, < ) ∈ ℝ})
34		smflimsuplem7.h	. . . . . . . . . . . . . . . 16 ⊢ 𝐻 = (𝑘 ∈ 𝑍 ↦ (𝑥 ∈ (𝐸‘𝑘) ↦ sup(ran (𝑚 ∈ (ℤ_≥‘𝑘) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^*, < )))
35	28	uztrn2 12706	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑛 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → 𝑘 ∈ 𝑍)
36	35	3ad2antl2 1186	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ) ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚)) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → 𝑘 ∈ 𝑍)
37		simpl1r 1225	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ) ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚)) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ)
38		uzss 12710	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 ∈ (ℤ_≥‘𝑛) → (ℤ_≥‘𝑘) ⊆ (ℤ_≥‘𝑛))
39		iinss1 4960	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((ℤ_≥‘𝑘) ⊆ (ℤ_≥‘𝑛) → ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ⊆ ∩ 𝑚 ∈ (ℤ_≥‘𝑘)dom (𝐹‘𝑚))
40	38, 39	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 ∈ (ℤ_≥‘𝑛) → ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ⊆ ∩ 𝑚 ∈ (ℤ_≥‘𝑘)dom (𝐹‘𝑚))
41	40	adantl 483	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ⊆ ∩ 𝑚 ∈ (ℤ_≥‘𝑘)dom (𝐹‘𝑚))
42		simpl 484	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚))
43	41, 42	sseldd 3936	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑘)dom (𝐹‘𝑚))
44	43	3ad2antl3 1187	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ) ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚)) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑘)dom (𝐹‘𝑚))
45	24, 27, 28, 30, 32, 33, 34, 36, 37, 44	smflimsuplem2 44748	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ) ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚)) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → 𝑥 ∈ dom (𝐻‘𝑘))
46	45	ralrimiva 3140	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ) ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚)) → ∀𝑘 ∈ (ℤ_≥‘𝑛)𝑥 ∈ dom (𝐻‘𝑘))
47		vex 3446	. . . . . . . . . . . . . . 15 ⊢ 𝑥 ∈ V
48		eliin 4950	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ V → (𝑥 ∈ ∩ 𝑘 ∈ (ℤ_≥‘𝑛)dom (𝐻‘𝑘) ↔ ∀𝑘 ∈ (ℤ_≥‘𝑛)𝑥 ∈ dom (𝐻‘𝑘)))
49	47, 48	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ∩ 𝑘 ∈ (ℤ_≥‘𝑛)dom (𝐻‘𝑘) ↔ ∀𝑘 ∈ (ℤ_≥‘𝑛)𝑥 ∈ dom (𝐻‘𝑘))
50	46, 49	sylibr 233	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ) ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚)) → 𝑥 ∈ ∩ 𝑘 ∈ (ℤ_≥‘𝑛)dom (𝐻‘𝑘))
51	50	3exp 1119	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ) → (𝑛 ∈ 𝑍 → (𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) → 𝑥 ∈ ∩ 𝑘 ∈ (ℤ_≥‘𝑛)dom (𝐻‘𝑘))))
52	10, 51	reximdai 3241	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ) → (∃𝑛 ∈ 𝑍 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) → ∃𝑛 ∈ 𝑍 𝑥 ∈ ∩ 𝑘 ∈ (ℤ_≥‘𝑛)dom (𝐻‘𝑘)))
53	52	imp 408	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ) ∧ ∃𝑛 ∈ 𝑍 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚)) → ∃𝑛 ∈ 𝑍 𝑥 ∈ ∩ 𝑘 ∈ (ℤ_≥‘𝑛)dom (𝐻‘𝑘))
54		eliun 4949	. . . . . . . . . 10 ⊢ (𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑘 ∈ (ℤ_≥‘𝑛)dom (𝐻‘𝑘) ↔ ∃𝑛 ∈ 𝑍 𝑥 ∈ ∩ 𝑘 ∈ (ℤ_≥‘𝑛)dom (𝐻‘𝑘))
55	53, 54	sylibr 233	. . . . . . . . 9 ⊢ (((𝜑 ∧ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ) ∧ ∃𝑛 ∈ 𝑍 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚)) → 𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑘 ∈ (ℤ_≥‘𝑛)dom (𝐻‘𝑘))
56	3, 5, 9, 55	syl21anc 836	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∣ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ}) → 𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑘 ∈ (ℤ_≥‘𝑛)dom (𝐻‘𝑘))
57	7	biimpi 215	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) → ∃𝑛 ∈ 𝑍 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚))
58	6, 57	syl 17	. . . . . . . . . 10 ⊢ (𝑥 ∈ {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∣ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ} → ∃𝑛 ∈ 𝑍 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚))
59	58	adantl 483	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∣ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ}) → ∃𝑛 ∈ 𝑍 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚))
60		nfv 1917	. . . . . . . . . . 11 ⊢ Ⅎ𝑛𝜑
61		nfcv 2905	. . . . . . . . . . . 12 ⊢ Ⅎ𝑛𝑥
62		nfv 1917	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑛(lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ
63		nfiu1 4979	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑛∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚)
64	62, 63	nfrabw 3437	. . . . . . . . . . . 12 ⊢ Ⅎ𝑛{𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∣ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ}
65	61, 64	nfel 2919	. . . . . . . . . . 11 ⊢ Ⅎ𝑛 𝑥 ∈ {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∣ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ}
66	60, 65	nfan 1902	. . . . . . . . . 10 ⊢ Ⅎ𝑛(𝜑 ∧ 𝑥 ∈ {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∣ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ})
67		nfv 1917	. . . . . . . . . 10 ⊢ Ⅎ𝑛(𝑘 ∈ 𝑍 ↦ ((𝐻‘𝑘)‘𝑥)) ∈ dom ⇝
68		nfv 1917	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘((𝜑 ∧ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ) ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚))
69		simp1l 1197	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ) ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚)) → 𝜑)
70	69, 26	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ) ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚)) → 𝑀 ∈ ℤ)
71	69, 29	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ) ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚)) → 𝑆 ∈ SAlg)
72	69, 31	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ) ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚)) → 𝐹:𝑍⟶(SMblFn‘𝑆))
73		simp1r 1198	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ) ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚)) → (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ)
74		simp2 1137	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ) ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚)) → 𝑛 ∈ 𝑍)
75		simp3 1138	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ) ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚)) → 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚))
76	68, 22, 70, 28, 71, 72, 33, 34, 73, 74, 75	smflimsuplem6 44752	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ) ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚)) → (𝑘 ∈ 𝑍 ↦ ((𝐻‘𝑘)‘𝑥)) ∈ dom ⇝ )
77	76	3exp 1119	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ) → (𝑛 ∈ 𝑍 → (𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) → (𝑘 ∈ 𝑍 ↦ ((𝐻‘𝑘)‘𝑥)) ∈ dom ⇝ )))
78	5, 77	syldan 592	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∣ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ}) → (𝑛 ∈ 𝑍 → (𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) → (𝑘 ∈ 𝑍 ↦ ((𝐻‘𝑘)‘𝑥)) ∈ dom ⇝ )))
79	66, 67, 78	rexlimd 3246	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∣ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ}) → (∃𝑛 ∈ 𝑍 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) → (𝑘 ∈ 𝑍 ↦ ((𝐻‘𝑘)‘𝑥)) ∈ dom ⇝ ))
80	59, 79	mpd 15	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∣ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ}) → (𝑘 ∈ 𝑍 ↦ ((𝐻‘𝑘)‘𝑥)) ∈ dom ⇝ )
81	56, 80	jca 513	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∣ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ}) → (𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑘 ∈ (ℤ_≥‘𝑛)dom (𝐻‘𝑘) ∧ (𝑘 ∈ 𝑍 ↦ ((𝐻‘𝑘)‘𝑥)) ∈ dom ⇝ ))
82		rabid 3424	. . . . . . 7 ⊢ (𝑥 ∈ {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑘 ∈ (ℤ_≥‘𝑛)dom (𝐻‘𝑘) ∣ (𝑘 ∈ 𝑍 ↦ ((𝐻‘𝑘)‘𝑥)) ∈ dom ⇝ } ↔ (𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑘 ∈ (ℤ_≥‘𝑛)dom (𝐻‘𝑘) ∧ (𝑘 ∈ 𝑍 ↦ ((𝐻‘𝑘)‘𝑥)) ∈ dom ⇝ ))
83	81, 82	sylibr 233	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∣ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ}) → 𝑥 ∈ {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑘 ∈ (ℤ_≥‘𝑛)dom (𝐻‘𝑘) ∣ (𝑘 ∈ 𝑍 ↦ ((𝐻‘𝑘)‘𝑥)) ∈ dom ⇝ })
84	83	ex 414	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∣ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ} → 𝑥 ∈ {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑘 ∈ (ℤ_≥‘𝑛)dom (𝐻‘𝑘) ∣ (𝑘 ∈ 𝑍 ↦ ((𝐻‘𝑘)‘𝑥)) ∈ dom ⇝ }))
85		ssrab2 4028	. . . . . . . . . 10 ⊢ {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑘 ∈ (ℤ_≥‘𝑛)dom (𝐻‘𝑘) ∣ (𝑘 ∈ 𝑍 ↦ ((𝐻‘𝑘)‘𝑥)) ∈ dom ⇝ } ⊆ ∪ 𝑛 ∈ 𝑍 ∩ 𝑘 ∈ (ℤ_≥‘𝑛)dom (𝐻‘𝑘)
86	85	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑘 ∈ (ℤ_≥‘𝑛)dom (𝐻‘𝑘) ∣ (𝑘 ∈ 𝑍 ↦ ((𝐻‘𝑘)‘𝑥)) ∈ dom ⇝ } ⊆ ∪ 𝑛 ∈ 𝑍 ∩ 𝑘 ∈ (ℤ_≥‘𝑛)dom (𝐻‘𝑘))
87	28	eluzelz2 43330	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ 𝑍 → 𝑛 ∈ ℤ)
88	87	uzidd 12703	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ 𝑍 → 𝑛 ∈ (ℤ_≥‘𝑛))
89	88	adantl 483	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝑛 ∈ (ℤ_≥‘𝑛))
90		nfv 1917	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑛 ∈ 𝑍)
91		xrltso 12980	. . . . . . . . . . . . . . . . . . 19 ⊢ < Or ℝ^*
92	91	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ (𝐸‘𝑛)) → < Or ℝ^*)
93	92	supexd 9314	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ (𝐸‘𝑛)) → sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^*, < ) ∈ V)
94		eqid 2737	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ (𝐸‘𝑛) ↦ sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^, < )) = (𝑥 ∈ (𝐸‘𝑛) ↦ sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^, < ))
95	90, 93, 94	fnmptd 6629	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑥 ∈ (𝐸‘𝑛) ↦ sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^*, < )) Fn (𝐸‘𝑛))
96		fveq2 6829	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 = 𝑛 → (𝐸‘𝑘) = (𝐸‘𝑛))
97		fveq2 6829	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑘 = 𝑛 → (ℤ_≥‘𝑘) = (ℤ_≥‘𝑛))
98	97	mpteq1d 5191	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑘 = 𝑛 → (𝑚 ∈ (ℤ_≥‘𝑘) ↦ ((𝐹‘𝑚)‘𝑥)) = (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)))
99	98	rneqd 5883	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 = 𝑛 → ran (𝑚 ∈ (ℤ_≥‘𝑘) ↦ ((𝐹‘𝑚)‘𝑥)) = ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)))
100	99	supeq1d 9307	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 = 𝑛 → sup(ran (𝑚 ∈ (ℤ_≥‘𝑘) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^, < ) = sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^, < ))
101	96, 100	mpteq12dv 5187	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = 𝑛 → (𝑥 ∈ (𝐸‘𝑘) ↦ sup(ran (𝑚 ∈ (ℤ_≥‘𝑘) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^, < )) = (𝑥 ∈ (𝐸‘𝑛) ↦ sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^, < )))
102		fvex 6842	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐸‘𝑛) ∈ V
103	102	mptex 7159	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ (𝐸‘𝑛) ↦ sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^*, < )) ∈ V
104	101, 34, 103	fvmpt 6935	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ 𝑍 → (𝐻‘𝑛) = (𝑥 ∈ (𝐸‘𝑛) ↦ sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^*, < )))
105	104	adantl 483	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐻‘𝑛) = (𝑥 ∈ (𝐸‘𝑛) ↦ sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^*, < )))
106	105	fneq1d 6582	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ((𝐻‘𝑛) Fn (𝐸‘𝑛) ↔ (𝑥 ∈ (𝐸‘𝑛) ↦ sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^*, < )) Fn (𝐸‘𝑛)))
107	95, 106	mpbird 257	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐻‘𝑛) Fn (𝐸‘𝑛))
108	107	fndmd 6594	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → dom (𝐻‘𝑛) = (𝐸‘𝑛))
109	97	iineq1d 43012	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 = 𝑛 → ∩ 𝑚 ∈ (ℤ_≥‘𝑘)dom (𝐹‘𝑚) = ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚))
110	109	eleq2d 2823	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = 𝑛 → (𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑘)dom (𝐹‘𝑚) ↔ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚)))
111	100	eleq1d 2822	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = 𝑛 → (sup(ran (𝑚 ∈ (ℤ_≥‘𝑘) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^, < ) ∈ ℝ ↔ sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^, < ) ∈ ℝ))
112	110, 111	anbi12d 632	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑛 → ((𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑘)dom (𝐹‘𝑚) ∧ sup(ran (𝑚 ∈ (ℤ_≥‘𝑘) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^, < ) ∈ ℝ) ↔ (𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∧ sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^, < ) ∈ ℝ)))
113	112	rabbidva2 3406	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑛 → {𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑘)dom (𝐹‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ_≥‘𝑘) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^, < ) ∈ ℝ} = {𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^, < ) ∈ ℝ})
114		id 22	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ 𝑍 → 𝑛 ∈ 𝑍)
115		fveq2 6829	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 = 𝑦 → ((𝐹‘𝑚)‘𝑥) = ((𝐹‘𝑚)‘𝑦))
116	115	mpteq2dv 5198	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = 𝑦 → (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)) = (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑦)))
117	116	rneqd 5883	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 = 𝑦 → ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)) = ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑦)))
118	117	supeq1d 9307	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = 𝑦 → sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^, < ) = sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑦)), ℝ^, < ))
119	118	eleq1d 2822	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑦 → (sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^, < ) ∈ ℝ ↔ sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑦)), ℝ^, < ) ∈ ℝ))
120	119	cbvrabv 3414	. . . . . . . . . . . . . . . . . 18 ⊢ {𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^, < ) ∈ ℝ} = {𝑦 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑦)), ℝ^, < ) ∈ ℝ}
121	88	ne0d 4286	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ 𝑍 → (ℤ_≥‘𝑛) ≠ ∅)
122		fvex 6842	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐹‘𝑚) ∈ V
123	122	dmex 7830	. . . . . . . . . . . . . . . . . . . . 21 ⊢ dom (𝐹‘𝑚) ∈ V
124	123	rgenw 3066	. . . . . . . . . . . . . . . . . . . 20 ⊢ ∀𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∈ V
125	124	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ 𝑍 → ∀𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∈ V)
126	121, 125	iinexd 43055	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ 𝑍 → ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∈ V)
127	120, 126	rabexd 5281	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ 𝑍 → {𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^*, < ) ∈ ℝ} ∈ V)
128	33, 113, 114, 127	fvmptd3 6958	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ 𝑍 → (𝐸‘𝑛) = {𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^*, < ) ∈ ℝ})
129	128	adantl 483	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐸‘𝑛) = {𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^*, < ) ∈ ℝ})
130		ssrab2 4028	. . . . . . . . . . . . . . . 16 ⊢ {𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^*, < ) ∈ ℝ} ⊆ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚)
131	130	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → {𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^*, < ) ∈ ℝ} ⊆ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚))
132	129, 131	eqsstrd 3973	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐸‘𝑛) ⊆ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚))
133	108, 132	eqsstrd 3973	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → dom (𝐻‘𝑛) ⊆ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚))
134		fveq2 6829	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑛 → (𝐻‘𝑘) = (𝐻‘𝑛))
135	134	dmeqd 5851	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑛 → dom (𝐻‘𝑘) = dom (𝐻‘𝑛))
136	135	sseq1d 3966	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑛 → (dom (𝐻‘𝑘) ⊆ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ↔ dom (𝐻‘𝑛) ⊆ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚)))
137	136	rspcev 3573	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ (ℤ_≥‘𝑛) ∧ dom (𝐻‘𝑛) ⊆ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚)) → ∃𝑘 ∈ (ℤ_≥‘𝑛)dom (𝐻‘𝑘) ⊆ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚))
138	89, 133, 137	syl2anc 585	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ∃𝑘 ∈ (ℤ_≥‘𝑛)dom (𝐻‘𝑘) ⊆ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚))
139		iinss 5007	. . . . . . . . . . . 12 ⊢ (∃𝑘 ∈ (ℤ_≥‘𝑛)dom (𝐻‘𝑘) ⊆ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) → ∩ 𝑘 ∈ (ℤ_≥‘𝑛)dom (𝐻‘𝑘) ⊆ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚))
140	138, 139	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ∩ 𝑘 ∈ (ℤ_≥‘𝑛)dom (𝐻‘𝑘) ⊆ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚))
141	140	ralrimiva 3140	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑛 ∈ 𝑍 ∩ 𝑘 ∈ (ℤ_≥‘𝑛)dom (𝐻‘𝑘) ⊆ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚))
142		ss2iun 4963	. . . . . . . . . 10 ⊢ (∀𝑛 ∈ 𝑍 ∩ 𝑘 ∈ (ℤ_≥‘𝑛)dom (𝐻‘𝑘) ⊆ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) → ∪ 𝑛 ∈ 𝑍 ∩ 𝑘 ∈ (ℤ_≥‘𝑛)dom (𝐻‘𝑘) ⊆ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚))
143	141, 142	syl 17	. . . . . . . . 9 ⊢ (𝜑 → ∪ 𝑛 ∈ 𝑍 ∩ 𝑘 ∈ (ℤ_≥‘𝑛)dom (𝐻‘𝑘) ⊆ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚))
144	86, 143	sstrd 3945	. . . . . . . 8 ⊢ (𝜑 → {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑘 ∈ (ℤ_≥‘𝑛)dom (𝐻‘𝑘) ∣ (𝑘 ∈ 𝑍 ↦ ((𝐻‘𝑘)‘𝑥)) ∈ dom ⇝ } ⊆ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚))
145	82	simplbi 499	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑘 ∈ (ℤ_≥‘𝑛)dom (𝐻‘𝑘) ∣ (𝑘 ∈ 𝑍 ↦ ((𝐻‘𝑘)‘𝑥)) ∈ dom ⇝ } → 𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑘 ∈ (ℤ_≥‘𝑛)dom (𝐻‘𝑘))
146	54	biimpi 215	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑘 ∈ (ℤ_≥‘𝑛)dom (𝐻‘𝑘) → ∃𝑛 ∈ 𝑍 𝑥 ∈ ∩ 𝑘 ∈ (ℤ_≥‘𝑛)dom (𝐻‘𝑘))
147	145, 146	syl 17	. . . . . . . . . . 11 ⊢ (𝑥 ∈ {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑘 ∈ (ℤ_≥‘𝑛)dom (𝐻‘𝑘) ∣ (𝑘 ∈ 𝑍 ↦ ((𝐻‘𝑘)‘𝑥)) ∈ dom ⇝ } → ∃𝑛 ∈ 𝑍 𝑥 ∈ ∩ 𝑘 ∈ (ℤ_≥‘𝑛)dom (𝐻‘𝑘))
148	147	adantl 483	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑘 ∈ (ℤ_≥‘𝑛)dom (𝐻‘𝑘) ∣ (𝑘 ∈ 𝑍 ↦ ((𝐻‘𝑘)‘𝑥)) ∈ dom ⇝ }) → ∃𝑛 ∈ 𝑍 𝑥 ∈ ∩ 𝑘 ∈ (ℤ_≥‘𝑛)dom (𝐻‘𝑘))
149		nfiu1 4979	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑛∪ 𝑛 ∈ 𝑍 ∩ 𝑘 ∈ (ℤ_≥‘𝑛)dom (𝐻‘𝑘)
150	67, 149	nfrabw 3437	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑛{𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑘 ∈ (ℤ_≥‘𝑛)dom (𝐻‘𝑘) ∣ (𝑘 ∈ 𝑍 ↦ ((𝐻‘𝑘)‘𝑥)) ∈ dom ⇝ }
151	61, 150	nfel 2919	. . . . . . . . . . . 12 ⊢ Ⅎ𝑛 𝑥 ∈ {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑘 ∈ (ℤ_≥‘𝑛)dom (𝐻‘𝑘) ∣ (𝑘 ∈ 𝑍 ↦ ((𝐻‘𝑘)‘𝑥)) ∈ dom ⇝ }
152	60, 151	nfan 1902	. . . . . . . . . . 11 ⊢ Ⅎ𝑛(𝜑 ∧ 𝑥 ∈ {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑘 ∈ (ℤ_≥‘𝑛)dom (𝐻‘𝑘) ∣ (𝑘 ∈ 𝑍 ↦ ((𝐻‘𝑘)‘𝑥)) ∈ dom ⇝ })
153	82	simprbi 498	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑘 ∈ (ℤ_≥‘𝑛)dom (𝐻‘𝑘) ∣ (𝑘 ∈ 𝑍 ↦ ((𝐻‘𝑘)‘𝑥)) ∈ dom ⇝ } → (𝑘 ∈ 𝑍 ↦ ((𝐻‘𝑘)‘𝑥)) ∈ dom ⇝ )
154		nfv 1917	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑘𝜑
155		nfmpt1 5204	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑘(𝑘 ∈ 𝑍 ↦ ((𝐻‘𝑘)‘𝑥))
156		nfcv 2905	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑘dom ⇝
157	155, 156	nfel 2919	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑘(𝑘 ∈ 𝑍 ↦ ((𝐻‘𝑘)‘𝑥)) ∈ dom ⇝
158	154, 157	nfan 1902	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑘(𝜑 ∧ (𝑘 ∈ 𝑍 ↦ ((𝐻‘𝑘)‘𝑥)) ∈ dom ⇝ )
159		nfv 1917	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑘 𝑛 ∈ 𝑍
160		nfcv 2905	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑘𝑥
161		nfii1 4980	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑘∩ 𝑘 ∈ (ℤ_≥‘𝑛)dom (𝐻‘𝑘)
162	160, 161	nfel 2919	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑘 𝑥 ∈ ∩ 𝑘 ∈ (ℤ_≥‘𝑛)dom (𝐻‘𝑘)
163	158, 159, 162	nf3an 1904	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘((𝜑 ∧ (𝑘 ∈ 𝑍 ↦ ((𝐻‘𝑘)‘𝑥)) ∈ dom ⇝ ) ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ∩ 𝑘 ∈ (ℤ_≥‘𝑛)dom (𝐻‘𝑘))
164	26	adantr 482	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝑀 ∈ ℤ)
165	164	3adant3 1132	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ∩ 𝑘 ∈ (ℤ_≥‘𝑛)dom (𝐻‘𝑘)) → 𝑀 ∈ ℤ)
166	165	3adant1r 1177	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑘 ∈ 𝑍 ↦ ((𝐻‘𝑘)‘𝑥)) ∈ dom ⇝ ) ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ∩ 𝑘 ∈ (ℤ_≥‘𝑛)dom (𝐻‘𝑘)) → 𝑀 ∈ ℤ)
167	29	adantr 482	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝑆 ∈ SAlg)
168	167	3adant3 1132	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ∩ 𝑘 ∈ (ℤ_≥‘𝑛)dom (𝐻‘𝑘)) → 𝑆 ∈ SAlg)
169	168	3adant1r 1177	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑘 ∈ 𝑍 ↦ ((𝐻‘𝑘)‘𝑥)) ∈ dom ⇝ ) ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ∩ 𝑘 ∈ (ℤ_≥‘𝑛)dom (𝐻‘𝑘)) → 𝑆 ∈ SAlg)
170	31	adantr 482	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝐹:𝑍⟶(SMblFn‘𝑆))
171	170	3adant3 1132	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ∩ 𝑘 ∈ (ℤ_≥‘𝑛)dom (𝐻‘𝑘)) → 𝐹:𝑍⟶(SMblFn‘𝑆))
172	171	3adant1r 1177	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑘 ∈ 𝑍 ↦ ((𝐻‘𝑘)‘𝑥)) ∈ dom ⇝ ) ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ∩ 𝑘 ∈ (ℤ_≥‘𝑛)dom (𝐻‘𝑘)) → 𝐹:𝑍⟶(SMblFn‘𝑆))
173		simp2 1137	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑘 ∈ 𝑍 ↦ ((𝐻‘𝑘)‘𝑥)) ∈ dom ⇝ ) ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ∩ 𝑘 ∈ (ℤ_≥‘𝑛)dom (𝐻‘𝑘)) → 𝑛 ∈ 𝑍)
174		simp3 1138	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑘 ∈ 𝑍 ↦ ((𝐻‘𝑘)‘𝑥)) ∈ dom ⇝ ) ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ∩ 𝑘 ∈ (ℤ_≥‘𝑛)dom (𝐻‘𝑘)) → 𝑥 ∈ ∩ 𝑘 ∈ (ℤ_≥‘𝑛)dom (𝐻‘𝑘))
175		simp1r 1198	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑘 ∈ 𝑍 ↦ ((𝐻‘𝑘)‘𝑥)) ∈ dom ⇝ ) ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ∩ 𝑘 ∈ (ℤ_≥‘𝑛)dom (𝐻‘𝑘)) → (𝑘 ∈ 𝑍 ↦ ((𝐻‘𝑘)‘𝑥)) ∈ dom ⇝ )
176	163, 166, 28, 169, 172, 33, 34, 173, 174, 175	smflimsuplem4 44750	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑘 ∈ 𝑍 ↦ ((𝐻‘𝑘)‘𝑥)) ∈ dom ⇝ ) ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ ∩ 𝑘 ∈ (ℤ_≥‘𝑛)dom (𝐻‘𝑘)) → (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ)
177	176	3exp 1119	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑍 ↦ ((𝐻‘𝑘)‘𝑥)) ∈ dom ⇝ ) → (𝑛 ∈ 𝑍 → (𝑥 ∈ ∩ 𝑘 ∈ (ℤ_≥‘𝑛)dom (𝐻‘𝑘) → (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ)))
178	153, 177	sylan2 594	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑘 ∈ (ℤ_≥‘𝑛)dom (𝐻‘𝑘) ∣ (𝑘 ∈ 𝑍 ↦ ((𝐻‘𝑘)‘𝑥)) ∈ dom ⇝ }) → (𝑛 ∈ 𝑍 → (𝑥 ∈ ∩ 𝑘 ∈ (ℤ_≥‘𝑛)dom (𝐻‘𝑘) → (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ)))
179	152, 62, 178	rexlimd 3246	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑘 ∈ (ℤ_≥‘𝑛)dom (𝐻‘𝑘) ∣ (𝑘 ∈ 𝑍 ↦ ((𝐻‘𝑘)‘𝑥)) ∈ dom ⇝ }) → (∃𝑛 ∈ 𝑍 𝑥 ∈ ∩ 𝑘 ∈ (ℤ_≥‘𝑛)dom (𝐻‘𝑘) → (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ))
180	148, 179	mpd 15	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑘 ∈ (ℤ_≥‘𝑛)dom (𝐻‘𝑘) ∣ (𝑘 ∈ 𝑍 ↦ ((𝐻‘𝑘)‘𝑥)) ∈ dom ⇝ }) → (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ)
181	180	ralrimiva 3140	. . . . . . . 8 ⊢ (𝜑 → ∀𝑥 ∈ {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑘 ∈ (ℤ_≥‘𝑛)dom (𝐻‘𝑘) ∣ (𝑘 ∈ 𝑍 ↦ ((𝐻‘𝑘)‘𝑥)) ∈ dom ⇝ } (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ)
182	144, 181	jca 513	. . . . . . 7 ⊢ (𝜑 → ({𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑘 ∈ (ℤ_≥‘𝑛)dom (𝐻‘𝑘) ∣ (𝑘 ∈ 𝑍 ↦ ((𝐻‘𝑘)‘𝑥)) ∈ dom ⇝ } ⊆ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∧ ∀𝑥 ∈ {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑘 ∈ (ℤ_≥‘𝑛)dom (𝐻‘𝑘) ∣ (𝑘 ∈ 𝑍 ↦ ((𝐻‘𝑘)‘𝑥)) ∈ dom ⇝ } (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ))
183		nfrab1 3423	. . . . . . . 8 ⊢ Ⅎ𝑥{𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑘 ∈ (ℤ_≥‘𝑛)dom (𝐻‘𝑘) ∣ (𝑘 ∈ 𝑍 ↦ ((𝐻‘𝑘)‘𝑥)) ∈ dom ⇝ }
184		nfcv 2905	. . . . . . . 8 ⊢ Ⅎ𝑥∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚)
185	183, 184	ssrabf 43036	. . . . . . 7 ⊢ ({𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑘 ∈ (ℤ_≥‘𝑛)dom (𝐻‘𝑘) ∣ (𝑘 ∈ 𝑍 ↦ ((𝐻‘𝑘)‘𝑥)) ∈ dom ⇝ } ⊆ {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∣ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ} ↔ ({𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑘 ∈ (ℤ_≥‘𝑛)dom (𝐻‘𝑘) ∣ (𝑘 ∈ 𝑍 ↦ ((𝐻‘𝑘)‘𝑥)) ∈ dom ⇝ } ⊆ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∧ ∀𝑥 ∈ {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑘 ∈ (ℤ_≥‘𝑛)dom (𝐻‘𝑘) ∣ (𝑘 ∈ 𝑍 ↦ ((𝐻‘𝑘)‘𝑥)) ∈ dom ⇝ } (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ))
186	182, 185	sylibr 233	. . . . . 6 ⊢ (𝜑 → {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑘 ∈ (ℤ_≥‘𝑛)dom (𝐻‘𝑘) ∣ (𝑘 ∈ 𝑍 ↦ ((𝐻‘𝑘)‘𝑥)) ∈ dom ⇝ } ⊆ {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∣ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ})
187	186	sseld 3934	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑘 ∈ (ℤ_≥‘𝑛)dom (𝐻‘𝑘) ∣ (𝑘 ∈ 𝑍 ↦ ((𝐻‘𝑘)‘𝑥)) ∈ dom ⇝ } → 𝑥 ∈ {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∣ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ}))
188	84, 187	impbid 211	. . . 4 ⊢ (𝜑 → (𝑥 ∈ {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∣ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ} ↔ 𝑥 ∈ {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑘 ∈ (ℤ_≥‘𝑛)dom (𝐻‘𝑘) ∣ (𝑘 ∈ 𝑍 ↦ ((𝐻‘𝑘)‘𝑥)) ∈ dom ⇝ }))
189	188	alrimiv 1930	. . 3 ⊢ (𝜑 → ∀𝑥(𝑥 ∈ {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∣ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ} ↔ 𝑥 ∈ {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑘 ∈ (ℤ_≥‘𝑛)dom (𝐻‘𝑘) ∣ (𝑘 ∈ 𝑍 ↦ ((𝐻‘𝑘)‘𝑥)) ∈ dom ⇝ }))
190		nfrab1 3423	. . . 4 ⊢ Ⅎ𝑥{𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∣ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ}
191	190, 183	cleqf 2936	. . 3 ⊢ ({𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∣ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ} = {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑘 ∈ (ℤ_≥‘𝑛)dom (𝐻‘𝑘) ∣ (𝑘 ∈ 𝑍 ↦ ((𝐻‘𝑘)‘𝑥)) ∈ dom ⇝ } ↔ ∀𝑥(𝑥 ∈ {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∣ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ} ↔ 𝑥 ∈ {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑘 ∈ (ℤ_≥‘𝑛)dom (𝐻‘𝑘) ∣ (𝑘 ∈ 𝑍 ↦ ((𝐻‘𝑘)‘𝑥)) ∈ dom ⇝ }))
192	189, 191	sylibr 233	. 2 ⊢ (𝜑 → {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∣ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ} = {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑘 ∈ (ℤ_≥‘𝑛)dom (𝐻‘𝑘) ∣ (𝑘 ∈ 𝑍 ↦ ((𝐻‘𝑘)‘𝑥)) ∈ dom ⇝ })
193	2, 192	eqtrd 2777	1 ⊢ (𝜑 → 𝐷 = {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑘 ∈ (ℤ_≥‘𝑛)dom (𝐻‘𝑘) ∣ (𝑘 ∈ 𝑍 ↦ ((𝐻‘𝑘)‘𝑥)) ∈ dom ⇝ })

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∧ w3a 1087 ∀wal 1539 = wceq 1541 ∈ wcel 2106 ∀wral 3062 ∃wrex 3071 {crab 3404 Vcvv 3442 ⊆ wss 3901 ∪ ciun 4945 ∩ ciin 4946 ↦ cmpt 5179 Or wor 5535 dom cdm 5624 ran crn 5625 Fn wfn 6478 ⟶wf 6479 ‘cfv 6483 supcsup 9301 ℝcr 10975 ℝ^*cxr 11113 < clt 11114 ℤcz 12424 ℤ_≥cuz 12687 lim supclsp 15278 ⇝ cli 15292 SAlgcsalg 44237 SMblFncsmblfn 44622

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 2708 ax-rep 5233 ax-sep 5247 ax-nul 5254 ax-pow 5312 ax-pr 5376 ax-un 7654 ax-cnex 11032 ax-resscn 11033 ax-1cn 11034 ax-icn 11035 ax-addcl 11036 ax-addrcl 11037 ax-mulcl 11038 ax-mulrcl 11039 ax-mulcom 11040 ax-addass 11041 ax-mulass 11042 ax-distr 11043 ax-i2m1 11044 ax-1ne0 11045 ax-1rid 11046 ax-rnegex 11047 ax-rrecex 11048 ax-cnre 11049 ax-pre-lttri 11050 ax-pre-lttrn 11051 ax-pre-ltadd 11052 ax-pre-mulgt0 11053 ax-pre-sup 11054

This theorem depends on definitions: df-bi 206 df-an 398 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 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3350 df-reu 3351 df-rab 3405 df-v 3444 df-sbc 3731 df-csb 3847 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-pss 3920 df-nul 4274 df-if 4478 df-pw 4553 df-sn 4578 df-pr 4580 df-tp 4582 df-op 4584 df-uni 4857 df-int 4899 df-iun 4947 df-iin 4948 df-br 5097 df-opab 5159 df-mpt 5180 df-tr 5214 df-id 5522 df-eprel 5528 df-po 5536 df-so 5537 df-fr 5579 df-we 5581 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6242 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6435 df-fun 6485 df-fn 6486 df-f 6487 df-f1 6488 df-fo 6489 df-f1o 6490 df-fv 6491 df-riota 7297 df-ov 7344 df-oprab 7345 df-mpo 7346 df-om 7785 df-1st 7903 df-2nd 7904 df-frecs 8171 df-wrecs 8202 df-recs 8276 df-rdg 8315 df-1o 8371 df-er 8573 df-pm 8693 df-en 8809 df-dom 8810 df-sdom 8811 df-fin 8812 df-sup 9303 df-inf 9304 df-pnf 11116 df-mnf 11117 df-xr 11118 df-ltxr 11119 df-le 11120 df-sub 11312 df-neg 11313 df-div 11738 df-nn 12079 df-2 12141 df-3 12142 df-n0 12339 df-z 12425 df-uz 12688 df-q 12794 df-rp 12836 df-ioo 13188 df-ico 13190 df-fz 13345 df-fl 13617 df-ceil 13618 df-seq 13827 df-exp 13888 df-cj 14909 df-re 14910 df-im 14911 df-sqrt 15045 df-abs 15046 df-limsup 15279 df-clim 15296 df-rlim 15297 df-smblfn 44623

This theorem is referenced by: smflimsuplem8 44754

W3C validator