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Theorem mbflimsup 24261

Description: The limit supremum of a sequence of measurable real-valued functions is measurable. (Contributed by Mario Carneiro, 7-Sep-2014.) (Revised by AV, 12-Sep-2020.)

**Hypotheses**
Ref	Expression
mbflimsup.1	⊢ 𝑍 = (ℤ_≥‘𝑀)
mbflimsup.2	⊢ 𝐺 = (𝑥 ∈ 𝐴 ↦ (lim sup‘(𝑛 ∈ 𝑍 ↦ 𝐵)))
mbflimsup.h	⊢ 𝐻 = (𝑚 ∈ ℝ ↦ sup((((𝑛 ∈ 𝑍 ↦ 𝐵) “ (𝑚[,)+∞)) ∩ ℝ^), ℝ^, < ))
mbflimsup.3	⊢ (𝜑 → 𝑀 ∈ ℤ)
mbflimsup.4	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (lim sup‘(𝑛 ∈ 𝑍 ↦ 𝐵)) ∈ ℝ)
mbflimsup.5	⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn)
mbflimsup.6	⊢ ((𝜑 ∧ (𝑛 ∈ 𝑍 ∧ 𝑥 ∈ 𝐴)) → 𝐵 ∈ ℝ)

**Assertion**
Ref	Expression
mbflimsup	⊢ (𝜑 → 𝐺 ∈ MblFn)

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

Proof of Theorem mbflimsup Dummy variables 𝑖 𝑘 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mbflimsup.2	. . 3 ⊢ 𝐺 = (𝑥 ∈ 𝐴 ↦ (lim sup‘(𝑛 ∈ 𝑍 ↦ 𝐵)))
2		mbflimsup.h	. . . . . 6 ⊢ 𝐻 = (𝑚 ∈ ℝ ↦ sup((((𝑛 ∈ 𝑍 ↦ 𝐵) “ (𝑚[,)+∞)) ∩ ℝ^), ℝ^, < ))
3		mbflimsup.1	. . . . . . . . 9 ⊢ 𝑍 = (ℤ_≥‘𝑀)
4	3	fvexi 6678	. . . . . . . 8 ⊢ 𝑍 ∈ V
5	4	mptex 6980	. . . . . . 7 ⊢ (𝑛 ∈ 𝑍 ↦ 𝐵) ∈ V
6	5	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑛 ∈ 𝑍 ↦ 𝐵) ∈ V)
7		uzssz 12258	. . . . . . . . 9 ⊢ (ℤ_≥‘𝑀) ⊆ ℤ
8	3, 7	eqsstri 4000	. . . . . . . 8 ⊢ 𝑍 ⊆ ℤ
9		zssre 11982	. . . . . . . 8 ⊢ ℤ ⊆ ℝ
10	8, 9	sstri 3975	. . . . . . 7 ⊢ 𝑍 ⊆ ℝ
11	10	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑍 ⊆ ℝ)
12		mbflimsup.3	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ℤ)
13	3	uzsup 13225	. . . . . . . 8 ⊢ (𝑀 ∈ ℤ → sup(𝑍, ℝ^*, < ) = +∞)
14	12, 13	syl 17	. . . . . . 7 ⊢ (𝜑 → sup(𝑍, ℝ^*, < ) = +∞)
15	14	adantr 483	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → sup(𝑍, ℝ^*, < ) = +∞)
16	2, 6, 11, 15	limsupval2 14831	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (lim sup‘(𝑛 ∈ 𝑍 ↦ 𝐵)) = inf((𝐻 “ 𝑍), ℝ^*, < ))
17		imassrn 5934	. . . . . . 7 ⊢ (𝐻 “ 𝑍) ⊆ ran 𝐻
18	12	adantr 483	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑀 ∈ ℤ)
19		mbflimsup.6	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑛 ∈ 𝑍 ∧ 𝑥 ∈ 𝐴)) → 𝐵 ∈ ℝ)
20	19	anass1rs 653	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑛 ∈ 𝑍) → 𝐵 ∈ ℝ)
21	20	fmpttd 6873	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑛 ∈ 𝑍 ↦ 𝐵):𝑍⟶ℝ)
22		mbflimsup.4	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (lim sup‘(𝑛 ∈ 𝑍 ↦ 𝐵)) ∈ ℝ)
23	22	ltpnfd 12510	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (lim sup‘(𝑛 ∈ 𝑍 ↦ 𝐵)) < +∞)
24	2, 3	limsupgre 14832	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℤ ∧ (𝑛 ∈ 𝑍 ↦ 𝐵):𝑍⟶ℝ ∧ (lim sup‘(𝑛 ∈ 𝑍 ↦ 𝐵)) < +∞) → 𝐻:ℝ⟶ℝ)
25	18, 21, 23, 24	syl3anc 1367	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐻:ℝ⟶ℝ)
26	25	frnd 6515	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ran 𝐻 ⊆ ℝ)
27	17, 26	sstrid 3977	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐻 “ 𝑍) ⊆ ℝ)
28	25	fdmd 6517	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → dom 𝐻 = ℝ)
29	28	ineq1d 4187	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (dom 𝐻 ∩ 𝑍) = (ℝ ∩ 𝑍))
30		sseqin2 4191	. . . . . . . . . 10 ⊢ (𝑍 ⊆ ℝ ↔ (ℝ ∩ 𝑍) = 𝑍)
31	10, 30	mpbi 232	. . . . . . . . 9 ⊢ (ℝ ∩ 𝑍) = 𝑍
32	29, 31	syl6eq 2872	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (dom 𝐻 ∩ 𝑍) = 𝑍)
33		uzid 12252	. . . . . . . . . . . 12 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ_≥‘𝑀))
34	12, 33	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ∈ (ℤ_≥‘𝑀))
35	34, 3	eleqtrrdi 2924	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ 𝑍)
36	35	adantr 483	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑀 ∈ 𝑍)
37	36	ne0d 4300	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑍 ≠ ∅)
38	32, 37	eqnetrd 3083	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (dom 𝐻 ∩ 𝑍) ≠ ∅)
39		imadisj 5942	. . . . . . . 8 ⊢ ((𝐻 “ 𝑍) = ∅ ↔ (dom 𝐻 ∩ 𝑍) = ∅)
40	39	necon3bii 3068	. . . . . . 7 ⊢ ((𝐻 “ 𝑍) ≠ ∅ ↔ (dom 𝐻 ∩ 𝑍) ≠ ∅)
41	38, 40	sylibr 236	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐻 “ 𝑍) ≠ ∅)
42	22	leidd 11200	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (lim sup‘(𝑛 ∈ 𝑍 ↦ 𝐵)) ≤ (lim sup‘(𝑛 ∈ 𝑍 ↦ 𝐵)))
43	20	rexrd 10685	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑛 ∈ 𝑍) → 𝐵 ∈ ℝ^*)
44	43	fmpttd 6873	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑛 ∈ 𝑍 ↦ 𝐵):𝑍⟶ℝ^*)
45	22	rexrd 10685	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (lim sup‘(𝑛 ∈ 𝑍 ↦ 𝐵)) ∈ ℝ^*)
46	2	limsuple 14829	. . . . . . . . . . 11 ⊢ ((𝑍 ⊆ ℝ ∧ (𝑛 ∈ 𝑍 ↦ 𝐵):𝑍⟶ℝ^* ∧ (lim sup‘(𝑛 ∈ 𝑍 ↦ 𝐵)) ∈ ℝ^*) → ((lim sup‘(𝑛 ∈ 𝑍 ↦ 𝐵)) ≤ (lim sup‘(𝑛 ∈ 𝑍 ↦ 𝐵)) ↔ ∀𝑦 ∈ ℝ (lim sup‘(𝑛 ∈ 𝑍 ↦ 𝐵)) ≤ (𝐻‘𝑦)))
47	11, 44, 45, 46	syl3anc 1367	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((lim sup‘(𝑛 ∈ 𝑍 ↦ 𝐵)) ≤ (lim sup‘(𝑛 ∈ 𝑍 ↦ 𝐵)) ↔ ∀𝑦 ∈ ℝ (lim sup‘(𝑛 ∈ 𝑍 ↦ 𝐵)) ≤ (𝐻‘𝑦)))
48	42, 47	mpbid 234	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∀𝑦 ∈ ℝ (lim sup‘(𝑛 ∈ 𝑍 ↦ 𝐵)) ≤ (𝐻‘𝑦))
49		ssralv 4032	. . . . . . . . 9 ⊢ (𝑍 ⊆ ℝ → (∀𝑦 ∈ ℝ (lim sup‘(𝑛 ∈ 𝑍 ↦ 𝐵)) ≤ (𝐻‘𝑦) → ∀𝑦 ∈ 𝑍 (lim sup‘(𝑛 ∈ 𝑍 ↦ 𝐵)) ≤ (𝐻‘𝑦)))
50	10, 48, 49	mpsyl 68	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∀𝑦 ∈ 𝑍 (lim sup‘(𝑛 ∈ 𝑍 ↦ 𝐵)) ≤ (𝐻‘𝑦))
51	2	limsupgf 14826	. . . . . . . . . 10 ⊢ 𝐻:ℝ⟶ℝ^*
52		ffn 6508	. . . . . . . . . 10 ⊢ (𝐻:ℝ⟶ℝ^* → 𝐻 Fn ℝ)
53	51, 52	ax-mp 5	. . . . . . . . 9 ⊢ 𝐻 Fn ℝ
54		breq2 5062	. . . . . . . . . 10 ⊢ (𝑧 = (𝐻‘𝑦) → ((lim sup‘(𝑛 ∈ 𝑍 ↦ 𝐵)) ≤ 𝑧 ↔ (lim sup‘(𝑛 ∈ 𝑍 ↦ 𝐵)) ≤ (𝐻‘𝑦)))
55	54	ralima 6994	. . . . . . . . 9 ⊢ ((𝐻 Fn ℝ ∧ 𝑍 ⊆ ℝ) → (∀𝑧 ∈ (𝐻 “ 𝑍)(lim sup‘(𝑛 ∈ 𝑍 ↦ 𝐵)) ≤ 𝑧 ↔ ∀𝑦 ∈ 𝑍 (lim sup‘(𝑛 ∈ 𝑍 ↦ 𝐵)) ≤ (𝐻‘𝑦)))
56	53, 11, 55	sylancr 589	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (∀𝑧 ∈ (𝐻 “ 𝑍)(lim sup‘(𝑛 ∈ 𝑍 ↦ 𝐵)) ≤ 𝑧 ↔ ∀𝑦 ∈ 𝑍 (lim sup‘(𝑛 ∈ 𝑍 ↦ 𝐵)) ≤ (𝐻‘𝑦)))
57	50, 56	mpbird 259	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∀𝑧 ∈ (𝐻 “ 𝑍)(lim sup‘(𝑛 ∈ 𝑍 ↦ 𝐵)) ≤ 𝑧)
58		breq1 5061	. . . . . . . . 9 ⊢ (𝑦 = (lim sup‘(𝑛 ∈ 𝑍 ↦ 𝐵)) → (𝑦 ≤ 𝑧 ↔ (lim sup‘(𝑛 ∈ 𝑍 ↦ 𝐵)) ≤ 𝑧))
59	58	ralbidv 3197	. . . . . . . 8 ⊢ (𝑦 = (lim sup‘(𝑛 ∈ 𝑍 ↦ 𝐵)) → (∀𝑧 ∈ (𝐻 “ 𝑍)𝑦 ≤ 𝑧 ↔ ∀𝑧 ∈ (𝐻 “ 𝑍)(lim sup‘(𝑛 ∈ 𝑍 ↦ 𝐵)) ≤ 𝑧))
60	59	rspcev 3622	. . . . . . 7 ⊢ (((lim sup‘(𝑛 ∈ 𝑍 ↦ 𝐵)) ∈ ℝ ∧ ∀𝑧 ∈ (𝐻 “ 𝑍)(lim sup‘(𝑛 ∈ 𝑍 ↦ 𝐵)) ≤ 𝑧) → ∃𝑦 ∈ ℝ ∀𝑧 ∈ (𝐻 “ 𝑍)𝑦 ≤ 𝑧)
61	22, 57, 60	syl2anc 586	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∃𝑦 ∈ ℝ ∀𝑧 ∈ (𝐻 “ 𝑍)𝑦 ≤ 𝑧)
62		infxrre 12723	. . . . . 6 ⊢ (((𝐻 “ 𝑍) ⊆ ℝ ∧ (𝐻 “ 𝑍) ≠ ∅ ∧ ∃𝑦 ∈ ℝ ∀𝑧 ∈ (𝐻 “ 𝑍)𝑦 ≤ 𝑧) → inf((𝐻 “ 𝑍), ℝ^*, < ) = inf((𝐻 “ 𝑍), ℝ, < ))
63	27, 41, 61, 62	syl3anc 1367	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → inf((𝐻 “ 𝑍), ℝ^*, < ) = inf((𝐻 “ 𝑍), ℝ, < ))
64		df-ima 5562	. . . . . . 7 ⊢ (𝐻 “ 𝑍) = ran (𝐻 ↾ 𝑍)
65	25	feqmptd 6727	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐻 = (𝑖 ∈ ℝ ↦ (𝐻‘𝑖)))
66	65	reseq1d 5846	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐻 ↾ 𝑍) = ((𝑖 ∈ ℝ ↦ (𝐻‘𝑖)) ↾ 𝑍))
67		resmpt 5899	. . . . . . . . . . 11 ⊢ (𝑍 ⊆ ℝ → ((𝑖 ∈ ℝ ↦ (𝐻‘𝑖)) ↾ 𝑍) = (𝑖 ∈ 𝑍 ↦ (𝐻‘𝑖)))
68	10, 67	ax-mp 5	. . . . . . . . . 10 ⊢ ((𝑖 ∈ ℝ ↦ (𝐻‘𝑖)) ↾ 𝑍) = (𝑖 ∈ 𝑍 ↦ (𝐻‘𝑖))
69	66, 68	syl6eq 2872	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐻 ↾ 𝑍) = (𝑖 ∈ 𝑍 ↦ (𝐻‘𝑖)))
70	10	sseli 3962	. . . . . . . . . . . . 13 ⊢ (𝑖 ∈ 𝑍 → 𝑖 ∈ ℝ)
71		ffvelrn 6843	. . . . . . . . . . . . 13 ⊢ ((𝐻:ℝ⟶ℝ ∧ 𝑖 ∈ ℝ) → (𝐻‘𝑖) ∈ ℝ)
72	25, 70, 71	syl2an 597	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑖 ∈ 𝑍) → (𝐻‘𝑖) ∈ ℝ)
73	72	rexrd 10685	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑖 ∈ 𝑍) → (𝐻‘𝑖) ∈ ℝ^*)
74		simplll 773	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑖 ∈ 𝑍) ∧ 𝑛 ∈ (ℤ_≥‘𝑖)) → 𝜑)
75	3	uztrn2 12256	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑖 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑖)) → 𝑛 ∈ 𝑍)
76	75	adantll 712	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑖 ∈ 𝑍) ∧ 𝑛 ∈ (ℤ_≥‘𝑖)) → 𝑛 ∈ 𝑍)
77		simpllr 774	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑖 ∈ 𝑍) ∧ 𝑛 ∈ (ℤ_≥‘𝑖)) → 𝑥 ∈ 𝐴)
78	74, 76, 77, 19	syl12anc 834	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑖 ∈ 𝑍) ∧ 𝑛 ∈ (ℤ_≥‘𝑖)) → 𝐵 ∈ ℝ)
79	78	fmpttd 6873	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑖 ∈ 𝑍) → (𝑛 ∈ (ℤ_≥‘𝑖) ↦ 𝐵):(ℤ_≥‘𝑖)⟶ℝ)
80	79	frnd 6515	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑖 ∈ 𝑍) → ran (𝑛 ∈ (ℤ_≥‘𝑖) ↦ 𝐵) ⊆ ℝ)
81		eqid 2821	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ (ℤ_≥‘𝑖) ↦ 𝐵) = (𝑛 ∈ (ℤ_≥‘𝑖) ↦ 𝐵)
82	81, 78	dmmptd 6487	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑖 ∈ 𝑍) → dom (𝑛 ∈ (ℤ_≥‘𝑖) ↦ 𝐵) = (ℤ_≥‘𝑖))
83		simpr 487	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → 𝑖 ∈ 𝑍)
84	83, 3	eleqtrdi 2923	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → 𝑖 ∈ (ℤ_≥‘𝑀))
85		eluzelz 12247	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 ∈ (ℤ_≥‘𝑀) → 𝑖 ∈ ℤ)
86	84, 85	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → 𝑖 ∈ ℤ)
87	86	adantlr 713	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑖 ∈ 𝑍) → 𝑖 ∈ ℤ)
88		uzid 12252	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ ℤ → 𝑖 ∈ (ℤ_≥‘𝑖))
89		ne0i 4299	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ (ℤ_≥‘𝑖) → (ℤ_≥‘𝑖) ≠ ∅)
90	87, 88, 89	3syl 18	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑖 ∈ 𝑍) → (ℤ_≥‘𝑖) ≠ ∅)
91	82, 90	eqnetrd 3083	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑖 ∈ 𝑍) → dom (𝑛 ∈ (ℤ_≥‘𝑖) ↦ 𝐵) ≠ ∅)
92		dm0rn0 5789	. . . . . . . . . . . . . . 15 ⊢ (dom (𝑛 ∈ (ℤ_≥‘𝑖) ↦ 𝐵) = ∅ ↔ ran (𝑛 ∈ (ℤ_≥‘𝑖) ↦ 𝐵) = ∅)
93	92	necon3bii 3068	. . . . . . . . . . . . . 14 ⊢ (dom (𝑛 ∈ (ℤ_≥‘𝑖) ↦ 𝐵) ≠ ∅ ↔ ran (𝑛 ∈ (ℤ_≥‘𝑖) ↦ 𝐵) ≠ ∅)
94	91, 93	sylib 220	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑖 ∈ 𝑍) → ran (𝑛 ∈ (ℤ_≥‘𝑖) ↦ 𝐵) ≠ ∅)
95	84	adantlr 713	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑖 ∈ 𝑍) → 𝑖 ∈ (ℤ_≥‘𝑀))
96		uzss 12259	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 ∈ (ℤ_≥‘𝑀) → (ℤ_≥‘𝑖) ⊆ (ℤ_≥‘𝑀))
97	95, 96	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑖 ∈ 𝑍) → (ℤ_≥‘𝑖) ⊆ (ℤ_≥‘𝑀))
98	97, 3	sseqtrrdi 4017	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑖 ∈ 𝑍) → (ℤ_≥‘𝑖) ⊆ 𝑍)
99	72	leidd 11200	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑖 ∈ 𝑍) → (𝐻‘𝑖) ≤ (𝐻‘𝑖))
100	10	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑖 ∈ 𝑍) → 𝑍 ⊆ ℝ)
101	44	adantr 483	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑖 ∈ 𝑍) → (𝑛 ∈ 𝑍 ↦ 𝐵):𝑍⟶ℝ^*)
102		simpr 487	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑖 ∈ 𝑍) → 𝑖 ∈ 𝑍)
103	10, 102	sseldi 3964	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑖 ∈ 𝑍) → 𝑖 ∈ ℝ)
104	2	limsupgle 14828	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑍 ⊆ ℝ ∧ (𝑛 ∈ 𝑍 ↦ 𝐵):𝑍⟶ℝ^) ∧ 𝑖 ∈ ℝ ∧ (𝐻‘𝑖) ∈ ℝ^) → ((𝐻‘𝑖) ≤ (𝐻‘𝑖) ↔ ∀𝑘 ∈ 𝑍 (𝑖 ≤ 𝑘 → ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑘) ≤ (𝐻‘𝑖))))
105	100, 101, 103, 73, 104	syl211anc 1372	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑖 ∈ 𝑍) → ((𝐻‘𝑖) ≤ (𝐻‘𝑖) ↔ ∀𝑘 ∈ 𝑍 (𝑖 ≤ 𝑘 → ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑘) ≤ (𝐻‘𝑖))))
106	99, 105	mpbid 234	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑖 ∈ 𝑍) → ∀𝑘 ∈ 𝑍 (𝑖 ≤ 𝑘 → ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑘) ≤ (𝐻‘𝑖)))
107		ssralv 4032	. . . . . . . . . . . . . . . . 17 ⊢ ((ℤ_≥‘𝑖) ⊆ 𝑍 → (∀𝑘 ∈ 𝑍 (𝑖 ≤ 𝑘 → ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑘) ≤ (𝐻‘𝑖)) → ∀𝑘 ∈ (ℤ_≥‘𝑖)(𝑖 ≤ 𝑘 → ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑘) ≤ (𝐻‘𝑖))))
108	98, 106, 107	sylc 65	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑖 ∈ 𝑍) → ∀𝑘 ∈ (ℤ_≥‘𝑖)(𝑖 ≤ 𝑘 → ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑘) ≤ (𝐻‘𝑖)))
109	98	adantr 483	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑖 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑖)) → (ℤ_≥‘𝑖) ⊆ 𝑍)
110	109	resmptd 5902	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑖 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑖)) → ((𝑛 ∈ 𝑍 ↦ 𝐵) ↾ (ℤ_≥‘𝑖)) = (𝑛 ∈ (ℤ_≥‘𝑖) ↦ 𝐵))
111	110	fveq1d 6666	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑖 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑖)) → (((𝑛 ∈ 𝑍 ↦ 𝐵) ↾ (ℤ_≥‘𝑖))‘𝑘) = ((𝑛 ∈ (ℤ_≥‘𝑖) ↦ 𝐵)‘𝑘))
112		fvres 6683	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 ∈ (ℤ_≥‘𝑖) → (((𝑛 ∈ 𝑍 ↦ 𝐵) ↾ (ℤ_≥‘𝑖))‘𝑘) = ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑘))
113	112	adantl 484	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑖 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑖)) → (((𝑛 ∈ 𝑍 ↦ 𝐵) ↾ (ℤ_≥‘𝑖))‘𝑘) = ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑘))
114	111, 113	eqtr3d 2858	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑖 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑖)) → ((𝑛 ∈ (ℤ_≥‘𝑖) ↦ 𝐵)‘𝑘) = ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑘))
115	114	breq1d 5068	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑖 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑖)) → (((𝑛 ∈ (ℤ_≥‘𝑖) ↦ 𝐵)‘𝑘) ≤ (𝐻‘𝑖) ↔ ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑘) ≤ (𝐻‘𝑖)))
116		eluzle 12250	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 ∈ (ℤ_≥‘𝑖) → 𝑖 ≤ 𝑘)
117	116	adantl 484	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑖 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑖)) → 𝑖 ≤ 𝑘)
118		biimt 363	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 ≤ 𝑘 → (((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑘) ≤ (𝐻‘𝑖) ↔ (𝑖 ≤ 𝑘 → ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑘) ≤ (𝐻‘𝑖))))
119	117, 118	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑖 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑖)) → (((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑘) ≤ (𝐻‘𝑖) ↔ (𝑖 ≤ 𝑘 → ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑘) ≤ (𝐻‘𝑖))))
120	115, 119	bitrd 281	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑖 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑖)) → (((𝑛 ∈ (ℤ_≥‘𝑖) ↦ 𝐵)‘𝑘) ≤ (𝐻‘𝑖) ↔ (𝑖 ≤ 𝑘 → ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑘) ≤ (𝐻‘𝑖))))
121	120	ralbidva 3196	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑖 ∈ 𝑍) → (∀𝑘 ∈ (ℤ_≥‘𝑖)((𝑛 ∈ (ℤ_≥‘𝑖) ↦ 𝐵)‘𝑘) ≤ (𝐻‘𝑖) ↔ ∀𝑘 ∈ (ℤ_≥‘𝑖)(𝑖 ≤ 𝑘 → ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑘) ≤ (𝐻‘𝑖))))
122	108, 121	mpbird 259	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑖 ∈ 𝑍) → ∀𝑘 ∈ (ℤ_≥‘𝑖)((𝑛 ∈ (ℤ_≥‘𝑖) ↦ 𝐵)‘𝑘) ≤ (𝐻‘𝑖))
123		ffn 6508	. . . . . . . . . . . . . . . 16 ⊢ ((𝑛 ∈ (ℤ_≥‘𝑖) ↦ 𝐵):(ℤ_≥‘𝑖)⟶ℝ → (𝑛 ∈ (ℤ_≥‘𝑖) ↦ 𝐵) Fn (ℤ_≥‘𝑖))
124		breq1 5061	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = ((𝑛 ∈ (ℤ_≥‘𝑖) ↦ 𝐵)‘𝑘) → (𝑧 ≤ (𝐻‘𝑖) ↔ ((𝑛 ∈ (ℤ_≥‘𝑖) ↦ 𝐵)‘𝑘) ≤ (𝐻‘𝑖)))
125	124	ralrn 6848	. . . . . . . . . . . . . . . 16 ⊢ ((𝑛 ∈ (ℤ_≥‘𝑖) ↦ 𝐵) Fn (ℤ_≥‘𝑖) → (∀𝑧 ∈ ran (𝑛 ∈ (ℤ_≥‘𝑖) ↦ 𝐵)𝑧 ≤ (𝐻‘𝑖) ↔ ∀𝑘 ∈ (ℤ_≥‘𝑖)((𝑛 ∈ (ℤ_≥‘𝑖) ↦ 𝐵)‘𝑘) ≤ (𝐻‘𝑖)))
126	79, 123, 125	3syl 18	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑖 ∈ 𝑍) → (∀𝑧 ∈ ran (𝑛 ∈ (ℤ_≥‘𝑖) ↦ 𝐵)𝑧 ≤ (𝐻‘𝑖) ↔ ∀𝑘 ∈ (ℤ_≥‘𝑖)((𝑛 ∈ (ℤ_≥‘𝑖) ↦ 𝐵)‘𝑘) ≤ (𝐻‘𝑖)))
127	122, 126	mpbird 259	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑖 ∈ 𝑍) → ∀𝑧 ∈ ran (𝑛 ∈ (ℤ_≥‘𝑖) ↦ 𝐵)𝑧 ≤ (𝐻‘𝑖))
128		brralrspcev 5118	. . . . . . . . . . . . . 14 ⊢ (((𝐻‘𝑖) ∈ ℝ ∧ ∀𝑧 ∈ ran (𝑛 ∈ (ℤ_≥‘𝑖) ↦ 𝐵)𝑧 ≤ (𝐻‘𝑖)) → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑛 ∈ (ℤ_≥‘𝑖) ↦ 𝐵)𝑧 ≤ 𝑦)
129	72, 127, 128	syl2anc 586	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑖 ∈ 𝑍) → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑛 ∈ (ℤ_≥‘𝑖) ↦ 𝐵)𝑧 ≤ 𝑦)
130	80, 94, 129	suprcld 11598	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑖 ∈ 𝑍) → sup(ran (𝑛 ∈ (ℤ_≥‘𝑖) ↦ 𝐵), ℝ, < ) ∈ ℝ)
131	130	rexrd 10685	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑖 ∈ 𝑍) → sup(ran (𝑛 ∈ (ℤ_≥‘𝑖) ↦ 𝐵), ℝ, < ) ∈ ℝ^*)
132	80	adantr 483	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑖 ∈ 𝑍) ∧ (𝑘 ∈ 𝑍 ∧ 𝑖 ≤ 𝑘)) → ran (𝑛 ∈ (ℤ_≥‘𝑖) ↦ 𝐵) ⊆ ℝ)
133	94	adantr 483	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑖 ∈ 𝑍) ∧ (𝑘 ∈ 𝑍 ∧ 𝑖 ≤ 𝑘)) → ran (𝑛 ∈ (ℤ_≥‘𝑖) ↦ 𝐵) ≠ ∅)
134	129	adantr 483	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑖 ∈ 𝑍) ∧ (𝑘 ∈ 𝑍 ∧ 𝑖 ≤ 𝑘)) → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑛 ∈ (ℤ_≥‘𝑖) ↦ 𝐵)𝑧 ≤ 𝑦)
135	8	sseli 3962	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 ∈ 𝑍 → 𝑘 ∈ ℤ)
136		eluz 12251	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑖 ∈ ℤ ∧ 𝑘 ∈ ℤ) → (𝑘 ∈ (ℤ_≥‘𝑖) ↔ 𝑖 ≤ 𝑘))
137	87, 135, 136	syl2an 597	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑖 ∈ 𝑍) ∧ 𝑘 ∈ 𝑍) → (𝑘 ∈ (ℤ_≥‘𝑖) ↔ 𝑖 ≤ 𝑘))
138	137	biimprd 250	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑖 ∈ 𝑍) ∧ 𝑘 ∈ 𝑍) → (𝑖 ≤ 𝑘 → 𝑘 ∈ (ℤ_≥‘𝑖)))
139	138	impr 457	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑖 ∈ 𝑍) ∧ (𝑘 ∈ 𝑍 ∧ 𝑖 ≤ 𝑘)) → 𝑘 ∈ (ℤ_≥‘𝑖))
140	139, 114	syldan 593	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑖 ∈ 𝑍) ∧ (𝑘 ∈ 𝑍 ∧ 𝑖 ≤ 𝑘)) → ((𝑛 ∈ (ℤ_≥‘𝑖) ↦ 𝐵)‘𝑘) = ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑘))
141	79	adantr 483	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑖 ∈ 𝑍) ∧ (𝑘 ∈ 𝑍 ∧ 𝑖 ≤ 𝑘)) → (𝑛 ∈ (ℤ_≥‘𝑖) ↦ 𝐵):(ℤ_≥‘𝑖)⟶ℝ)
142	141, 123	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑖 ∈ 𝑍) ∧ (𝑘 ∈ 𝑍 ∧ 𝑖 ≤ 𝑘)) → (𝑛 ∈ (ℤ_≥‘𝑖) ↦ 𝐵) Fn (ℤ_≥‘𝑖))
143		fnfvelrn 6842	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑛 ∈ (ℤ_≥‘𝑖) ↦ 𝐵) Fn (ℤ_≥‘𝑖) ∧ 𝑘 ∈ (ℤ_≥‘𝑖)) → ((𝑛 ∈ (ℤ_≥‘𝑖) ↦ 𝐵)‘𝑘) ∈ ran (𝑛 ∈ (ℤ_≥‘𝑖) ↦ 𝐵))
144	142, 139, 143	syl2anc 586	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑖 ∈ 𝑍) ∧ (𝑘 ∈ 𝑍 ∧ 𝑖 ≤ 𝑘)) → ((𝑛 ∈ (ℤ_≥‘𝑖) ↦ 𝐵)‘𝑘) ∈ ran (𝑛 ∈ (ℤ_≥‘𝑖) ↦ 𝐵))
145	140, 144	eqeltrrd 2914	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑖 ∈ 𝑍) ∧ (𝑘 ∈ 𝑍 ∧ 𝑖 ≤ 𝑘)) → ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑘) ∈ ran (𝑛 ∈ (ℤ_≥‘𝑖) ↦ 𝐵))
146	132, 133, 134, 145	suprubd 11597	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑖 ∈ 𝑍) ∧ (𝑘 ∈ 𝑍 ∧ 𝑖 ≤ 𝑘)) → ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑘) ≤ sup(ran (𝑛 ∈ (ℤ_≥‘𝑖) ↦ 𝐵), ℝ, < ))
147	146	expr 459	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑖 ∈ 𝑍) ∧ 𝑘 ∈ 𝑍) → (𝑖 ≤ 𝑘 → ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑘) ≤ sup(ran (𝑛 ∈ (ℤ_≥‘𝑖) ↦ 𝐵), ℝ, < )))
148	147	ralrimiva 3182	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑖 ∈ 𝑍) → ∀𝑘 ∈ 𝑍 (𝑖 ≤ 𝑘 → ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑘) ≤ sup(ran (𝑛 ∈ (ℤ_≥‘𝑖) ↦ 𝐵), ℝ, < )))
149	2	limsupgle 14828	. . . . . . . . . . . . 13 ⊢ (((𝑍 ⊆ ℝ ∧ (𝑛 ∈ 𝑍 ↦ 𝐵):𝑍⟶ℝ^) ∧ 𝑖 ∈ ℝ ∧ sup(ran (𝑛 ∈ (ℤ_≥‘𝑖) ↦ 𝐵), ℝ, < ) ∈ ℝ^) → ((𝐻‘𝑖) ≤ sup(ran (𝑛 ∈ (ℤ_≥‘𝑖) ↦ 𝐵), ℝ, < ) ↔ ∀𝑘 ∈ 𝑍 (𝑖 ≤ 𝑘 → ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑘) ≤ sup(ran (𝑛 ∈ (ℤ_≥‘𝑖) ↦ 𝐵), ℝ, < ))))
150	100, 101, 103, 131, 149	syl211anc 1372	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑖 ∈ 𝑍) → ((𝐻‘𝑖) ≤ sup(ran (𝑛 ∈ (ℤ_≥‘𝑖) ↦ 𝐵), ℝ, < ) ↔ ∀𝑘 ∈ 𝑍 (𝑖 ≤ 𝑘 → ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑘) ≤ sup(ran (𝑛 ∈ (ℤ_≥‘𝑖) ↦ 𝐵), ℝ, < ))))
151	148, 150	mpbird 259	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑖 ∈ 𝑍) → (𝐻‘𝑖) ≤ sup(ran (𝑛 ∈ (ℤ_≥‘𝑖) ↦ 𝐵), ℝ, < ))
152		suprleub 11601	. . . . . . . . . . . . 13 ⊢ (((ran (𝑛 ∈ (ℤ_≥‘𝑖) ↦ 𝐵) ⊆ ℝ ∧ ran (𝑛 ∈ (ℤ_≥‘𝑖) ↦ 𝐵) ≠ ∅ ∧ ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑛 ∈ (ℤ_≥‘𝑖) ↦ 𝐵)𝑧 ≤ 𝑦) ∧ (𝐻‘𝑖) ∈ ℝ) → (sup(ran (𝑛 ∈ (ℤ_≥‘𝑖) ↦ 𝐵), ℝ, < ) ≤ (𝐻‘𝑖) ↔ ∀𝑧 ∈ ran (𝑛 ∈ (ℤ_≥‘𝑖) ↦ 𝐵)𝑧 ≤ (𝐻‘𝑖)))
153	80, 94, 129, 72, 152	syl31anc 1369	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑖 ∈ 𝑍) → (sup(ran (𝑛 ∈ (ℤ_≥‘𝑖) ↦ 𝐵), ℝ, < ) ≤ (𝐻‘𝑖) ↔ ∀𝑧 ∈ ran (𝑛 ∈ (ℤ_≥‘𝑖) ↦ 𝐵)𝑧 ≤ (𝐻‘𝑖)))
154	127, 153	mpbird 259	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑖 ∈ 𝑍) → sup(ran (𝑛 ∈ (ℤ_≥‘𝑖) ↦ 𝐵), ℝ, < ) ≤ (𝐻‘𝑖))
155	73, 131, 151, 154	xrletrid 12542	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑖 ∈ 𝑍) → (𝐻‘𝑖) = sup(ran (𝑛 ∈ (ℤ_≥‘𝑖) ↦ 𝐵), ℝ, < ))
156	155	mpteq2dva 5153	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑖 ∈ 𝑍 ↦ (𝐻‘𝑖)) = (𝑖 ∈ 𝑍 ↦ sup(ran (𝑛 ∈ (ℤ_≥‘𝑖) ↦ 𝐵), ℝ, < )))
157	69, 156	eqtrd 2856	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐻 ↾ 𝑍) = (𝑖 ∈ 𝑍 ↦ sup(ran (𝑛 ∈ (ℤ_≥‘𝑖) ↦ 𝐵), ℝ, < )))
158	157	rneqd 5802	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ran (𝐻 ↾ 𝑍) = ran (𝑖 ∈ 𝑍 ↦ sup(ran (𝑛 ∈ (ℤ_≥‘𝑖) ↦ 𝐵), ℝ, < )))
159	64, 158	syl5eq 2868	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐻 “ 𝑍) = ran (𝑖 ∈ 𝑍 ↦ sup(ran (𝑛 ∈ (ℤ_≥‘𝑖) ↦ 𝐵), ℝ, < )))
160	159	infeq1d 8935	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → inf((𝐻 “ 𝑍), ℝ, < ) = inf(ran (𝑖 ∈ 𝑍 ↦ sup(ran (𝑛 ∈ (ℤ_≥‘𝑖) ↦ 𝐵), ℝ, < )), ℝ, < ))
161	16, 63, 160	3eqtrd 2860	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (lim sup‘(𝑛 ∈ 𝑍 ↦ 𝐵)) = inf(ran (𝑖 ∈ 𝑍 ↦ sup(ran (𝑛 ∈ (ℤ_≥‘𝑖) ↦ 𝐵), ℝ, < )), ℝ, < ))
162	161	mpteq2dva 5153	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (lim sup‘(𝑛 ∈ 𝑍 ↦ 𝐵))) = (𝑥 ∈ 𝐴 ↦ inf(ran (𝑖 ∈ 𝑍 ↦ sup(ran (𝑛 ∈ (ℤ_≥‘𝑖) ↦ 𝐵), ℝ, < )), ℝ, < )))
163	1, 162	syl5eq 2868	. 2 ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐴 ↦ inf(ran (𝑖 ∈ 𝑍 ↦ sup(ran (𝑛 ∈ (ℤ_≥‘𝑖) ↦ 𝐵), ℝ, < )), ℝ, < )))
164		eqid 2821	. . 3 ⊢ (𝑥 ∈ 𝐴 ↦ inf(ran (𝑖 ∈ 𝑍 ↦ sup(ran (𝑛 ∈ (ℤ_≥‘𝑖) ↦ 𝐵), ℝ, < )), ℝ, < )) = (𝑥 ∈ 𝐴 ↦ inf(ran (𝑖 ∈ 𝑍 ↦ sup(ran (𝑛 ∈ (ℤ_≥‘𝑖) ↦ 𝐵), ℝ, < )), ℝ, < ))
165		eqid 2821	. . . 4 ⊢ (ℤ_≥‘𝑖) = (ℤ_≥‘𝑖)
166		eqid 2821	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ sup(ran (𝑛 ∈ (ℤ_≥‘𝑖) ↦ 𝐵), ℝ, < )) = (𝑥 ∈ 𝐴 ↦ sup(ran (𝑛 ∈ (ℤ_≥‘𝑖) ↦ 𝐵), ℝ, < ))
167		simpll 765	. . . . 5 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑍) ∧ 𝑛 ∈ (ℤ_≥‘𝑖)) → 𝜑)
168	75	adantll 712	. . . . 5 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑍) ∧ 𝑛 ∈ (ℤ_≥‘𝑖)) → 𝑛 ∈ 𝑍)
169		mbflimsup.5	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn)
170	167, 168, 169	syl2anc 586	. . . 4 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑍) ∧ 𝑛 ∈ (ℤ_≥‘𝑖)) → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn)
171		simpll 765	. . . . 5 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑍) ∧ (𝑛 ∈ (ℤ_≥‘𝑖) ∧ 𝑥 ∈ 𝐴)) → 𝜑)
172	75	ad2ant2lr 746	. . . . 5 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑍) ∧ (𝑛 ∈ (ℤ_≥‘𝑖) ∧ 𝑥 ∈ 𝐴)) → 𝑛 ∈ 𝑍)
173		simprr 771	. . . . 5 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑍) ∧ (𝑛 ∈ (ℤ_≥‘𝑖) ∧ 𝑥 ∈ 𝐴)) → 𝑥 ∈ 𝐴)
174	171, 172, 173, 19	syl12anc 834	. . . 4 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑍) ∧ (𝑛 ∈ (ℤ_≥‘𝑖) ∧ 𝑥 ∈ 𝐴)) → 𝐵 ∈ ℝ)
175	78	ralrimiva 3182	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑖 ∈ 𝑍) → ∀𝑛 ∈ (ℤ_≥‘𝑖)𝐵 ∈ ℝ)
176		breq1 5061	. . . . . . . . 9 ⊢ (𝑧 = 𝐵 → (𝑧 ≤ 𝑦 ↔ 𝐵 ≤ 𝑦))
177	81, 176	ralrnmptw 6854	. . . . . . . 8 ⊢ (∀𝑛 ∈ (ℤ_≥‘𝑖)𝐵 ∈ ℝ → (∀𝑧 ∈ ran (𝑛 ∈ (ℤ_≥‘𝑖) ↦ 𝐵)𝑧 ≤ 𝑦 ↔ ∀𝑛 ∈ (ℤ_≥‘𝑖)𝐵 ≤ 𝑦))
178	175, 177	syl 17	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑖 ∈ 𝑍) → (∀𝑧 ∈ ran (𝑛 ∈ (ℤ_≥‘𝑖) ↦ 𝐵)𝑧 ≤ 𝑦 ↔ ∀𝑛 ∈ (ℤ_≥‘𝑖)𝐵 ≤ 𝑦))
179	178	rexbidv 3297	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑖 ∈ 𝑍) → (∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑛 ∈ (ℤ_≥‘𝑖) ↦ 𝐵)𝑧 ≤ 𝑦 ↔ ∃𝑦 ∈ ℝ ∀𝑛 ∈ (ℤ_≥‘𝑖)𝐵 ≤ 𝑦))
180	129, 179	mpbid 234	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑖 ∈ 𝑍) → ∃𝑦 ∈ ℝ ∀𝑛 ∈ (ℤ_≥‘𝑖)𝐵 ≤ 𝑦)
181	180	an32s 650	. . . 4 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑍) ∧ 𝑥 ∈ 𝐴) → ∃𝑦 ∈ ℝ ∀𝑛 ∈ (ℤ_≥‘𝑖)𝐵 ≤ 𝑦)
182	165, 166, 86, 170, 174, 181	mbfsup 24259	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → (𝑥 ∈ 𝐴 ↦ sup(ran (𝑛 ∈ (ℤ_≥‘𝑖) ↦ 𝐵), ℝ, < )) ∈ MblFn)
183	130	an32s 650	. . . 4 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑍) ∧ 𝑥 ∈ 𝐴) → sup(ran (𝑛 ∈ (ℤ_≥‘𝑖) ↦ 𝐵), ℝ, < ) ∈ ℝ)
184	183	anasss 469	. . 3 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝑍 ∧ 𝑥 ∈ 𝐴)) → sup(ran (𝑛 ∈ (ℤ_≥‘𝑖) ↦ 𝐵), ℝ, < ) ∈ ℝ)
185	2	limsuple 14829	. . . . . . . 8 ⊢ ((𝑍 ⊆ ℝ ∧ (𝑛 ∈ 𝑍 ↦ 𝐵):𝑍⟶ℝ^* ∧ (lim sup‘(𝑛 ∈ 𝑍 ↦ 𝐵)) ∈ ℝ^*) → ((lim sup‘(𝑛 ∈ 𝑍 ↦ 𝐵)) ≤ (lim sup‘(𝑛 ∈ 𝑍 ↦ 𝐵)) ↔ ∀𝑖 ∈ ℝ (lim sup‘(𝑛 ∈ 𝑍 ↦ 𝐵)) ≤ (𝐻‘𝑖)))
186	11, 44, 45, 185	syl3anc 1367	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((lim sup‘(𝑛 ∈ 𝑍 ↦ 𝐵)) ≤ (lim sup‘(𝑛 ∈ 𝑍 ↦ 𝐵)) ↔ ∀𝑖 ∈ ℝ (lim sup‘(𝑛 ∈ 𝑍 ↦ 𝐵)) ≤ (𝐻‘𝑖)))
187	42, 186	mpbid 234	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∀𝑖 ∈ ℝ (lim sup‘(𝑛 ∈ 𝑍 ↦ 𝐵)) ≤ (𝐻‘𝑖))
188		ssralv 4032	. . . . . 6 ⊢ (𝑍 ⊆ ℝ → (∀𝑖 ∈ ℝ (lim sup‘(𝑛 ∈ 𝑍 ↦ 𝐵)) ≤ (𝐻‘𝑖) → ∀𝑖 ∈ 𝑍 (lim sup‘(𝑛 ∈ 𝑍 ↦ 𝐵)) ≤ (𝐻‘𝑖)))
189	10, 187, 188	mpsyl 68	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∀𝑖 ∈ 𝑍 (lim sup‘(𝑛 ∈ 𝑍 ↦ 𝐵)) ≤ (𝐻‘𝑖))
190	155	breq2d 5070	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑖 ∈ 𝑍) → ((lim sup‘(𝑛 ∈ 𝑍 ↦ 𝐵)) ≤ (𝐻‘𝑖) ↔ (lim sup‘(𝑛 ∈ 𝑍 ↦ 𝐵)) ≤ sup(ran (𝑛 ∈ (ℤ_≥‘𝑖) ↦ 𝐵), ℝ, < )))
191	190	ralbidva 3196	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (∀𝑖 ∈ 𝑍 (lim sup‘(𝑛 ∈ 𝑍 ↦ 𝐵)) ≤ (𝐻‘𝑖) ↔ ∀𝑖 ∈ 𝑍 (lim sup‘(𝑛 ∈ 𝑍 ↦ 𝐵)) ≤ sup(ran (𝑛 ∈ (ℤ_≥‘𝑖) ↦ 𝐵), ℝ, < )))
192	189, 191	mpbid 234	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∀𝑖 ∈ 𝑍 (lim sup‘(𝑛 ∈ 𝑍 ↦ 𝐵)) ≤ sup(ran (𝑛 ∈ (ℤ_≥‘𝑖) ↦ 𝐵), ℝ, < ))
193		breq1 5061	. . . . . 6 ⊢ (𝑦 = (lim sup‘(𝑛 ∈ 𝑍 ↦ 𝐵)) → (𝑦 ≤ sup(ran (𝑛 ∈ (ℤ_≥‘𝑖) ↦ 𝐵), ℝ, < ) ↔ (lim sup‘(𝑛 ∈ 𝑍 ↦ 𝐵)) ≤ sup(ran (𝑛 ∈ (ℤ_≥‘𝑖) ↦ 𝐵), ℝ, < )))
194	193	ralbidv 3197	. . . . 5 ⊢ (𝑦 = (lim sup‘(𝑛 ∈ 𝑍 ↦ 𝐵)) → (∀𝑖 ∈ 𝑍 𝑦 ≤ sup(ran (𝑛 ∈ (ℤ_≥‘𝑖) ↦ 𝐵), ℝ, < ) ↔ ∀𝑖 ∈ 𝑍 (lim sup‘(𝑛 ∈ 𝑍 ↦ 𝐵)) ≤ sup(ran (𝑛 ∈ (ℤ_≥‘𝑖) ↦ 𝐵), ℝ, < )))
195	194	rspcev 3622	. . . 4 ⊢ (((lim sup‘(𝑛 ∈ 𝑍 ↦ 𝐵)) ∈ ℝ ∧ ∀𝑖 ∈ 𝑍 (lim sup‘(𝑛 ∈ 𝑍 ↦ 𝐵)) ≤ sup(ran (𝑛 ∈ (ℤ_≥‘𝑖) ↦ 𝐵), ℝ, < )) → ∃𝑦 ∈ ℝ ∀𝑖 ∈ 𝑍 𝑦 ≤ sup(ran (𝑛 ∈ (ℤ_≥‘𝑖) ↦ 𝐵), ℝ, < ))
196	22, 192, 195	syl2anc 586	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∃𝑦 ∈ ℝ ∀𝑖 ∈ 𝑍 𝑦 ≤ sup(ran (𝑛 ∈ (ℤ_≥‘𝑖) ↦ 𝐵), ℝ, < ))
197	3, 164, 12, 182, 184, 196	mbfinf 24260	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ inf(ran (𝑖 ∈ 𝑍 ↦ sup(ran (𝑛 ∈ (ℤ_≥‘𝑖) ↦ 𝐵), ℝ, < )), ℝ, < )) ∈ MblFn)
198	163, 197	eqeltrd 2913	1 ⊢ (𝜑 → 𝐺 ∈ MblFn)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 ∀wral 3138 ∃wrex 3139 Vcvv 3494 ∩ cin 3934 ⊆ wss 3935 ∅c0 4290 class class class wbr 5058 ↦ cmpt 5138 dom cdm 5549 ran crn 5550 ↾ cres 5551 “ cima 5552 Fn wfn 6344 ⟶wf 6345 ‘cfv 6349 (class class class)co 7150 supcsup 8898 infcinf 8899 ℝcr 10530 +∞cpnf 10666 ℝ^*cxr 10668 < clt 10669 ≤ cle 10670 ℤcz 11975 ℤ_≥cuz 12237 [,)cico 12734 lim supclsp 14821 MblFncmbf 24209

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-inf2 9098 ax-cc 9851 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-disj 5024 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-se 5509 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-isom 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-of 7403 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-2o 8097 df-oadd 8100 df-omul 8101 df-er 8283 df-map 8402 df-pm 8403 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-sup 8900 df-inf 8901 df-oi 8968 df-dju 9324 df-card 9362 df-acn 9365 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-2 11694 df-3 11695 df-n0 11892 df-z 11976 df-uz 12238 df-q 12343 df-rp 12384 df-xadd 12502 df-ioo 12736 df-ioc 12737 df-ico 12738 df-icc 12739 df-fz 12887 df-fzo 13028 df-fl 13156 df-seq 13364 df-exp 13424 df-hash 13685 df-cj 14452 df-re 14453 df-im 14454 df-sqrt 14588 df-abs 14589 df-limsup 14822 df-clim 14839 df-rlim 14840 df-sum 15037 df-xmet 20532 df-met 20533 df-ovol 24059 df-vol 24060 df-mbf 24214

This theorem is referenced by: mbflimlem 24262

W3C validator