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Theorem mbflimsup 24563

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 6731	. . . . . . . 8 ⊢ 𝑍 ∈ V
5	4	mptex 7039	. . . . . . 7 ⊢ (𝑛 ∈ 𝑍 ↦ 𝐵) ∈ V
6	5	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑛 ∈ 𝑍 ↦ 𝐵) ∈ V)
7		uzssz 12459	. . . . . . . . 9 ⊢ (ℤ_≥‘𝑀) ⊆ ℤ
8	3, 7	eqsstri 3935	. . . . . . . 8 ⊢ 𝑍 ⊆ ℤ
9		zssre 12183	. . . . . . . 8 ⊢ ℤ ⊆ ℝ
10	8, 9	sstri 3910	. . . . . . 7 ⊢ 𝑍 ⊆ ℝ
11	10	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑍 ⊆ ℝ)
12		mbflimsup.3	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ℤ)
13	3	uzsup 13436	. . . . . . . 8 ⊢ (𝑀 ∈ ℤ → sup(𝑍, ℝ^*, < ) = +∞)
14	12, 13	syl 17	. . . . . . 7 ⊢ (𝜑 → sup(𝑍, ℝ^*, < ) = +∞)
15	14	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → sup(𝑍, ℝ^*, < ) = +∞)
16	2, 6, 11, 15	limsupval2 15041	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (lim sup‘(𝑛 ∈ 𝑍 ↦ 𝐵)) = inf((𝐻 “ 𝑍), ℝ^*, < ))
17		imassrn 5940	. . . . . . 7 ⊢ (𝐻 “ 𝑍) ⊆ ran 𝐻
18	12	adantr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑀 ∈ ℤ)
19		mbflimsup.6	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑛 ∈ 𝑍 ∧ 𝑥 ∈ 𝐴)) → 𝐵 ∈ ℝ)
20	19	anass1rs 655	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑛 ∈ 𝑍) → 𝐵 ∈ ℝ)
21	20	fmpttd 6932	. . . . . . . . 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 12713	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (lim sup‘(𝑛 ∈ 𝑍 ↦ 𝐵)) < +∞)
24	2, 3	limsupgre 15042	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℤ ∧ (𝑛 ∈ 𝑍 ↦ 𝐵):𝑍⟶ℝ ∧ (lim sup‘(𝑛 ∈ 𝑍 ↦ 𝐵)) < +∞) → 𝐻:ℝ⟶ℝ)
25	18, 21, 23, 24	syl3anc 1373	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐻:ℝ⟶ℝ)
26	25	frnd 6553	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ran 𝐻 ⊆ ℝ)
27	17, 26	sstrid 3912	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐻 “ 𝑍) ⊆ ℝ)
28	25	fdmd 6556	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → dom 𝐻 = ℝ)
29	28	ineq1d 4126	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (dom 𝐻 ∩ 𝑍) = (ℝ ∩ 𝑍))
30		sseqin2 4130	. . . . . . . . . 10 ⊢ (𝑍 ⊆ ℝ ↔ (ℝ ∩ 𝑍) = 𝑍)
31	10, 30	mpbi 233	. . . . . . . . 9 ⊢ (ℝ ∩ 𝑍) = 𝑍
32	29, 31	eqtrdi 2794	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (dom 𝐻 ∩ 𝑍) = 𝑍)
33		uzid 12453	. . . . . . . . . . . 12 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ_≥‘𝑀))
34	12, 33	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ∈ (ℤ_≥‘𝑀))
35	34, 3	eleqtrrdi 2849	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ 𝑍)
36	35	adantr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑀 ∈ 𝑍)
37	36	ne0d 4250	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑍 ≠ ∅)
38	32, 37	eqnetrd 3008	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (dom 𝐻 ∩ 𝑍) ≠ ∅)
39		imadisj 5948	. . . . . . . 8 ⊢ ((𝐻 “ 𝑍) = ∅ ↔ (dom 𝐻 ∩ 𝑍) = ∅)
40	39	necon3bii 2993	. . . . . . 7 ⊢ ((𝐻 “ 𝑍) ≠ ∅ ↔ (dom 𝐻 ∩ 𝑍) ≠ ∅)
41	38, 40	sylibr 237	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐻 “ 𝑍) ≠ ∅)
42	22	leidd 11398	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (lim sup‘(𝑛 ∈ 𝑍 ↦ 𝐵)) ≤ (lim sup‘(𝑛 ∈ 𝑍 ↦ 𝐵)))
43	20	rexrd 10883	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑛 ∈ 𝑍) → 𝐵 ∈ ℝ^*)
44	43	fmpttd 6932	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑛 ∈ 𝑍 ↦ 𝐵):𝑍⟶ℝ^*)
45	22	rexrd 10883	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (lim sup‘(𝑛 ∈ 𝑍 ↦ 𝐵)) ∈ ℝ^*)
46	2	limsuple 15039	. . . . . . . . . . 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 1373	. . . . . . . . . 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 235	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∀𝑦 ∈ ℝ (lim sup‘(𝑛 ∈ 𝑍 ↦ 𝐵)) ≤ (𝐻‘𝑦))
49		ssralv 3967	. . . . . . . . 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 15036	. . . . . . . . . 10 ⊢ 𝐻:ℝ⟶ℝ^*
52		ffn 6545	. . . . . . . . . 10 ⊢ (𝐻:ℝ⟶ℝ^* → 𝐻 Fn ℝ)
53	51, 52	ax-mp 5	. . . . . . . . 9 ⊢ 𝐻 Fn ℝ
54		breq2 5057	. . . . . . . . . 10 ⊢ (𝑧 = (𝐻‘𝑦) → ((lim sup‘(𝑛 ∈ 𝑍 ↦ 𝐵)) ≤ 𝑧 ↔ (lim sup‘(𝑛 ∈ 𝑍 ↦ 𝐵)) ≤ (𝐻‘𝑦)))
55	54	ralima 7054	. . . . . . . . 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 590	. . . . . . . 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 260	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∀𝑧 ∈ (𝐻 “ 𝑍)(lim sup‘(𝑛 ∈ 𝑍 ↦ 𝐵)) ≤ 𝑧)
58		breq1 5056	. . . . . . . . 9 ⊢ (𝑦 = (lim sup‘(𝑛 ∈ 𝑍 ↦ 𝐵)) → (𝑦 ≤ 𝑧 ↔ (lim sup‘(𝑛 ∈ 𝑍 ↦ 𝐵)) ≤ 𝑧))
59	58	ralbidv 3118	. . . . . . . 8 ⊢ (𝑦 = (lim sup‘(𝑛 ∈ 𝑍 ↦ 𝐵)) → (∀𝑧 ∈ (𝐻 “ 𝑍)𝑦 ≤ 𝑧 ↔ ∀𝑧 ∈ (𝐻 “ 𝑍)(lim sup‘(𝑛 ∈ 𝑍 ↦ 𝐵)) ≤ 𝑧))
60	59	rspcev 3537	. . . . . . 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 587	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∃𝑦 ∈ ℝ ∀𝑧 ∈ (𝐻 “ 𝑍)𝑦 ≤ 𝑧)
62		infxrre 12926	. . . . . 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 1373	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → inf((𝐻 “ 𝑍), ℝ^*, < ) = inf((𝐻 “ 𝑍), ℝ, < ))
64		df-ima 5564	. . . . . . 7 ⊢ (𝐻 “ 𝑍) = ran (𝐻 ↾ 𝑍)
65	25	feqmptd 6780	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐻 = (𝑖 ∈ ℝ ↦ (𝐻‘𝑖)))
66	65	reseq1d 5850	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐻 ↾ 𝑍) = ((𝑖 ∈ ℝ ↦ (𝐻‘𝑖)) ↾ 𝑍))
67		resmpt 5905	. . . . . . . . . . 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	eqtrdi 2794	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐻 ↾ 𝑍) = (𝑖 ∈ 𝑍 ↦ (𝐻‘𝑖)))
70	10	sseli 3896	. . . . . . . . . . . . 13 ⊢ (𝑖 ∈ 𝑍 → 𝑖 ∈ ℝ)
71		ffvelrn 6902	. . . . . . . . . . . . 13 ⊢ ((𝐻:ℝ⟶ℝ ∧ 𝑖 ∈ ℝ) → (𝐻‘𝑖) ∈ ℝ)
72	25, 70, 71	syl2an 599	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑖 ∈ 𝑍) → (𝐻‘𝑖) ∈ ℝ)
73	72	rexrd 10883	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑖 ∈ 𝑍) → (𝐻‘𝑖) ∈ ℝ^*)
74		simplll 775	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑖 ∈ 𝑍) ∧ 𝑛 ∈ (ℤ_≥‘𝑖)) → 𝜑)
75	3	uztrn2 12457	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑖 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑖)) → 𝑛 ∈ 𝑍)
76	75	adantll 714	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑖 ∈ 𝑍) ∧ 𝑛 ∈ (ℤ_≥‘𝑖)) → 𝑛 ∈ 𝑍)
77		simpllr 776	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑖 ∈ 𝑍) ∧ 𝑛 ∈ (ℤ_≥‘𝑖)) → 𝑥 ∈ 𝐴)
78	74, 76, 77, 19	syl12anc 837	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑖 ∈ 𝑍) ∧ 𝑛 ∈ (ℤ_≥‘𝑖)) → 𝐵 ∈ ℝ)
79	78	fmpttd 6932	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑖 ∈ 𝑍) → (𝑛 ∈ (ℤ_≥‘𝑖) ↦ 𝐵):(ℤ_≥‘𝑖)⟶ℝ)
80	79	frnd 6553	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑖 ∈ 𝑍) → ran (𝑛 ∈ (ℤ_≥‘𝑖) ↦ 𝐵) ⊆ ℝ)
81		eqid 2737	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ (ℤ_≥‘𝑖) ↦ 𝐵) = (𝑛 ∈ (ℤ_≥‘𝑖) ↦ 𝐵)
82	81, 78	dmmptd 6523	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑖 ∈ 𝑍) → dom (𝑛 ∈ (ℤ_≥‘𝑖) ↦ 𝐵) = (ℤ_≥‘𝑖))
83		simpr 488	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → 𝑖 ∈ 𝑍)
84	83, 3	eleqtrdi 2848	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → 𝑖 ∈ (ℤ_≥‘𝑀))
85		eluzelz 12448	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 ∈ (ℤ_≥‘𝑀) → 𝑖 ∈ ℤ)
86	84, 85	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → 𝑖 ∈ ℤ)
87	86	adantlr 715	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑖 ∈ 𝑍) → 𝑖 ∈ ℤ)
88		uzid 12453	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ ℤ → 𝑖 ∈ (ℤ_≥‘𝑖))
89		ne0i 4249	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ (ℤ_≥‘𝑖) → (ℤ_≥‘𝑖) ≠ ∅)
90	87, 88, 89	3syl 18	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑖 ∈ 𝑍) → (ℤ_≥‘𝑖) ≠ ∅)
91	82, 90	eqnetrd 3008	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑖 ∈ 𝑍) → dom (𝑛 ∈ (ℤ_≥‘𝑖) ↦ 𝐵) ≠ ∅)
92		dm0rn0 5794	. . . . . . . . . . . . . . 15 ⊢ (dom (𝑛 ∈ (ℤ_≥‘𝑖) ↦ 𝐵) = ∅ ↔ ran (𝑛 ∈ (ℤ_≥‘𝑖) ↦ 𝐵) = ∅)
93	92	necon3bii 2993	. . . . . . . . . . . . . 14 ⊢ (dom (𝑛 ∈ (ℤ_≥‘𝑖) ↦ 𝐵) ≠ ∅ ↔ ran (𝑛 ∈ (ℤ_≥‘𝑖) ↦ 𝐵) ≠ ∅)
94	91, 93	sylib 221	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑖 ∈ 𝑍) → ran (𝑛 ∈ (ℤ_≥‘𝑖) ↦ 𝐵) ≠ ∅)
95	84	adantlr 715	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑖 ∈ 𝑍) → 𝑖 ∈ (ℤ_≥‘𝑀))
96		uzss 12461	. . . . . . . . . . . . . . . . . . 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 3952	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑖 ∈ 𝑍) → (ℤ_≥‘𝑖) ⊆ 𝑍)
99	72	leidd 11398	. . . . . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑖 ∈ 𝑍) → (𝑛 ∈ 𝑍 ↦ 𝐵):𝑍⟶ℝ^*)
102		simpr 488	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑖 ∈ 𝑍) → 𝑖 ∈ 𝑍)
103	10, 102	sseldi 3899	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑖 ∈ 𝑍) → 𝑖 ∈ ℝ)
104	2	limsupgle 15038	. . . . . . . . . . . . . . . . . . 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 1378	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑖 ∈ 𝑍) → ((𝐻‘𝑖) ≤ (𝐻‘𝑖) ↔ ∀𝑘 ∈ 𝑍 (𝑖 ≤ 𝑘 → ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑘) ≤ (𝐻‘𝑖))))
106	99, 105	mpbid 235	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑖 ∈ 𝑍) → ∀𝑘 ∈ 𝑍 (𝑖 ≤ 𝑘 → ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑘) ≤ (𝐻‘𝑖)))
107		ssralv 3967	. . . . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑖 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑖)) → (ℤ_≥‘𝑖) ⊆ 𝑍)
110	109	resmptd 5908	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑖 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑖)) → ((𝑛 ∈ 𝑍 ↦ 𝐵) ↾ (ℤ_≥‘𝑖)) = (𝑛 ∈ (ℤ_≥‘𝑖) ↦ 𝐵))
111	110	fveq1d 6719	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑖 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑖)) → (((𝑛 ∈ 𝑍 ↦ 𝐵) ↾ (ℤ_≥‘𝑖))‘𝑘) = ((𝑛 ∈ (ℤ_≥‘𝑖) ↦ 𝐵)‘𝑘))
112		fvres 6736	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 ∈ (ℤ_≥‘𝑖) → (((𝑛 ∈ 𝑍 ↦ 𝐵) ↾ (ℤ_≥‘𝑖))‘𝑘) = ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑘))
113	112	adantl 485	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑖 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑖)) → (((𝑛 ∈ 𝑍 ↦ 𝐵) ↾ (ℤ_≥‘𝑖))‘𝑘) = ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑘))
114	111, 113	eqtr3d 2779	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑖 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑖)) → ((𝑛 ∈ (ℤ_≥‘𝑖) ↦ 𝐵)‘𝑘) = ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑘))
115	114	breq1d 5063	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑖 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑖)) → (((𝑛 ∈ (ℤ_≥‘𝑖) ↦ 𝐵)‘𝑘) ≤ (𝐻‘𝑖) ↔ ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑘) ≤ (𝐻‘𝑖)))
116		eluzle 12451	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 ∈ (ℤ_≥‘𝑖) → 𝑖 ≤ 𝑘)
117	116	adantl 485	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑖 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑖)) → 𝑖 ≤ 𝑘)
118		biimt 364	. . . . . . . . . . . . . . . . . . 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 282	. . . . . . . . . . . . . . . . 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 3117	. . . . . . . . . . . . . . . 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 260	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑖 ∈ 𝑍) → ∀𝑘 ∈ (ℤ_≥‘𝑖)((𝑛 ∈ (ℤ_≥‘𝑖) ↦ 𝐵)‘𝑘) ≤ (𝐻‘𝑖))
123		ffn 6545	. . . . . . . . . . . . . . . 16 ⊢ ((𝑛 ∈ (ℤ_≥‘𝑖) ↦ 𝐵):(ℤ_≥‘𝑖)⟶ℝ → (𝑛 ∈ (ℤ_≥‘𝑖) ↦ 𝐵) Fn (ℤ_≥‘𝑖))
124		breq1 5056	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = ((𝑛 ∈ (ℤ_≥‘𝑖) ↦ 𝐵)‘𝑘) → (𝑧 ≤ (𝐻‘𝑖) ↔ ((𝑛 ∈ (ℤ_≥‘𝑖) ↦ 𝐵)‘𝑘) ≤ (𝐻‘𝑖)))
125	124	ralrn 6907	. . . . . . . . . . . . . . . 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 260	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑖 ∈ 𝑍) → ∀𝑧 ∈ ran (𝑛 ∈ (ℤ_≥‘𝑖) ↦ 𝐵)𝑧 ≤ (𝐻‘𝑖))
128		brralrspcev 5113	. . . . . . . . . . . . . 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 587	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑖 ∈ 𝑍) → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑛 ∈ (ℤ_≥‘𝑖) ↦ 𝐵)𝑧 ≤ 𝑦)
130	80, 94, 129	suprcld 11795	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑖 ∈ 𝑍) → sup(ran (𝑛 ∈ (ℤ_≥‘𝑖) ↦ 𝐵), ℝ, < ) ∈ ℝ)
131	130	rexrd 10883	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑖 ∈ 𝑍) → sup(ran (𝑛 ∈ (ℤ_≥‘𝑖) ↦ 𝐵), ℝ, < ) ∈ ℝ^*)
132	80	adantr 484	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑖 ∈ 𝑍) ∧ (𝑘 ∈ 𝑍 ∧ 𝑖 ≤ 𝑘)) → ran (𝑛 ∈ (ℤ_≥‘𝑖) ↦ 𝐵) ⊆ ℝ)
133	94	adantr 484	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑖 ∈ 𝑍) ∧ (𝑘 ∈ 𝑍 ∧ 𝑖 ≤ 𝑘)) → ran (𝑛 ∈ (ℤ_≥‘𝑖) ↦ 𝐵) ≠ ∅)
134	129	adantr 484	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑖 ∈ 𝑍) ∧ (𝑘 ∈ 𝑍 ∧ 𝑖 ≤ 𝑘)) → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑛 ∈ (ℤ_≥‘𝑖) ↦ 𝐵)𝑧 ≤ 𝑦)
135	8	sseli 3896	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 ∈ 𝑍 → 𝑘 ∈ ℤ)
136		eluz 12452	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑖 ∈ ℤ ∧ 𝑘 ∈ ℤ) → (𝑘 ∈ (ℤ_≥‘𝑖) ↔ 𝑖 ≤ 𝑘))
137	87, 135, 136	syl2an 599	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑖 ∈ 𝑍) ∧ 𝑘 ∈ 𝑍) → (𝑘 ∈ (ℤ_≥‘𝑖) ↔ 𝑖 ≤ 𝑘))
138	137	biimprd 251	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑖 ∈ 𝑍) ∧ 𝑘 ∈ 𝑍) → (𝑖 ≤ 𝑘 → 𝑘 ∈ (ℤ_≥‘𝑖)))
139	138	impr 458	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑖 ∈ 𝑍) ∧ (𝑘 ∈ 𝑍 ∧ 𝑖 ≤ 𝑘)) → 𝑘 ∈ (ℤ_≥‘𝑖))
140	139, 114	syldan 594	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑖 ∈ 𝑍) ∧ (𝑘 ∈ 𝑍 ∧ 𝑖 ≤ 𝑘)) → ((𝑛 ∈ (ℤ_≥‘𝑖) ↦ 𝐵)‘𝑘) = ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑘))
141	79	adantr 484	. . . . . . . . . . . . . . . . . 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 6901	. . . . . . . . . . . . . . . . 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 587	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑖 ∈ 𝑍) ∧ (𝑘 ∈ 𝑍 ∧ 𝑖 ≤ 𝑘)) → ((𝑛 ∈ (ℤ_≥‘𝑖) ↦ 𝐵)‘𝑘) ∈ ran (𝑛 ∈ (ℤ_≥‘𝑖) ↦ 𝐵))
145	140, 144	eqeltrrd 2839	. . . . . . . . . . . . . . 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 11794	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑖 ∈ 𝑍) ∧ (𝑘 ∈ 𝑍 ∧ 𝑖 ≤ 𝑘)) → ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑘) ≤ sup(ran (𝑛 ∈ (ℤ_≥‘𝑖) ↦ 𝐵), ℝ, < ))
147	146	expr 460	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑖 ∈ 𝑍) ∧ 𝑘 ∈ 𝑍) → (𝑖 ≤ 𝑘 → ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑘) ≤ sup(ran (𝑛 ∈ (ℤ_≥‘𝑖) ↦ 𝐵), ℝ, < )))
148	147	ralrimiva 3105	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑖 ∈ 𝑍) → ∀𝑘 ∈ 𝑍 (𝑖 ≤ 𝑘 → ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑘) ≤ sup(ran (𝑛 ∈ (ℤ_≥‘𝑖) ↦ 𝐵), ℝ, < )))
149	2	limsupgle 15038	. . . . . . . . . . . . 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 1378	. . . . . . . . . . . 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 260	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑖 ∈ 𝑍) → (𝐻‘𝑖) ≤ sup(ran (𝑛 ∈ (ℤ_≥‘𝑖) ↦ 𝐵), ℝ, < ))
152		suprleub 11798	. . . . . . . . . . . . 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 1375	. . . . . . . . . . . 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 260	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑖 ∈ 𝑍) → sup(ran (𝑛 ∈ (ℤ_≥‘𝑖) ↦ 𝐵), ℝ, < ) ≤ (𝐻‘𝑖))
155	73, 131, 151, 154	xrletrid 12745	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑖 ∈ 𝑍) → (𝐻‘𝑖) = sup(ran (𝑛 ∈ (ℤ_≥‘𝑖) ↦ 𝐵), ℝ, < ))
156	155	mpteq2dva 5150	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑖 ∈ 𝑍 ↦ (𝐻‘𝑖)) = (𝑖 ∈ 𝑍 ↦ sup(ran (𝑛 ∈ (ℤ_≥‘𝑖) ↦ 𝐵), ℝ, < )))
157	69, 156	eqtrd 2777	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐻 ↾ 𝑍) = (𝑖 ∈ 𝑍 ↦ sup(ran (𝑛 ∈ (ℤ_≥‘𝑖) ↦ 𝐵), ℝ, < )))
158	157	rneqd 5807	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ran (𝐻 ↾ 𝑍) = ran (𝑖 ∈ 𝑍 ↦ sup(ran (𝑛 ∈ (ℤ_≥‘𝑖) ↦ 𝐵), ℝ, < )))
159	64, 158	syl5eq 2790	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐻 “ 𝑍) = ran (𝑖 ∈ 𝑍 ↦ sup(ran (𝑛 ∈ (ℤ_≥‘𝑖) ↦ 𝐵), ℝ, < )))
160	159	infeq1d 9093	. . . . 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 2781	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (lim sup‘(𝑛 ∈ 𝑍 ↦ 𝐵)) = inf(ran (𝑖 ∈ 𝑍 ↦ sup(ran (𝑛 ∈ (ℤ_≥‘𝑖) ↦ 𝐵), ℝ, < )), ℝ, < ))
162	161	mpteq2dva 5150	. . 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 2790	. 2 ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐴 ↦ inf(ran (𝑖 ∈ 𝑍 ↦ sup(ran (𝑛 ∈ (ℤ_≥‘𝑖) ↦ 𝐵), ℝ, < )), ℝ, < )))
164		eqid 2737	. . 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 2737	. . . 4 ⊢ (ℤ_≥‘𝑖) = (ℤ_≥‘𝑖)
166		eqid 2737	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ sup(ran (𝑛 ∈ (ℤ_≥‘𝑖) ↦ 𝐵), ℝ, < )) = (𝑥 ∈ 𝐴 ↦ sup(ran (𝑛 ∈ (ℤ_≥‘𝑖) ↦ 𝐵), ℝ, < ))
167		simpll 767	. . . . 5 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑍) ∧ 𝑛 ∈ (ℤ_≥‘𝑖)) → 𝜑)
168	75	adantll 714	. . . . 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 587	. . . 4 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑍) ∧ 𝑛 ∈ (ℤ_≥‘𝑖)) → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn)
171		simpll 767	. . . . 5 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑍) ∧ (𝑛 ∈ (ℤ_≥‘𝑖) ∧ 𝑥 ∈ 𝐴)) → 𝜑)
172	75	ad2ant2lr 748	. . . . 5 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑍) ∧ (𝑛 ∈ (ℤ_≥‘𝑖) ∧ 𝑥 ∈ 𝐴)) → 𝑛 ∈ 𝑍)
173		simprr 773	. . . . 5 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑍) ∧ (𝑛 ∈ (ℤ_≥‘𝑖) ∧ 𝑥 ∈ 𝐴)) → 𝑥 ∈ 𝐴)
174	171, 172, 173, 19	syl12anc 837	. . . 4 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑍) ∧ (𝑛 ∈ (ℤ_≥‘𝑖) ∧ 𝑥 ∈ 𝐴)) → 𝐵 ∈ ℝ)
175	78	ralrimiva 3105	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑖 ∈ 𝑍) → ∀𝑛 ∈ (ℤ_≥‘𝑖)𝐵 ∈ ℝ)
176		breq1 5056	. . . . . . . . 9 ⊢ (𝑧 = 𝐵 → (𝑧 ≤ 𝑦 ↔ 𝐵 ≤ 𝑦))
177	81, 176	ralrnmptw 6913	. . . . . . . 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 3216	. . . . . 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 235	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑖 ∈ 𝑍) → ∃𝑦 ∈ ℝ ∀𝑛 ∈ (ℤ_≥‘𝑖)𝐵 ≤ 𝑦)
181	180	an32s 652	. . . 4 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑍) ∧ 𝑥 ∈ 𝐴) → ∃𝑦 ∈ ℝ ∀𝑛 ∈ (ℤ_≥‘𝑖)𝐵 ≤ 𝑦)
182	165, 166, 86, 170, 174, 181	mbfsup 24561	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → (𝑥 ∈ 𝐴 ↦ sup(ran (𝑛 ∈ (ℤ_≥‘𝑖) ↦ 𝐵), ℝ, < )) ∈ MblFn)
183	130	an32s 652	. . . 4 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑍) ∧ 𝑥 ∈ 𝐴) → sup(ran (𝑛 ∈ (ℤ_≥‘𝑖) ↦ 𝐵), ℝ, < ) ∈ ℝ)
184	183	anasss 470	. . 3 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝑍 ∧ 𝑥 ∈ 𝐴)) → sup(ran (𝑛 ∈ (ℤ_≥‘𝑖) ↦ 𝐵), ℝ, < ) ∈ ℝ)
185	2	limsuple 15039	. . . . . . . 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 1373	. . . . . . 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 235	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∀𝑖 ∈ ℝ (lim sup‘(𝑛 ∈ 𝑍 ↦ 𝐵)) ≤ (𝐻‘𝑖))
188		ssralv 3967	. . . . . 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 5065	. . . . . 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 3117	. . . . 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 235	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∀𝑖 ∈ 𝑍 (lim sup‘(𝑛 ∈ 𝑍 ↦ 𝐵)) ≤ sup(ran (𝑛 ∈ (ℤ_≥‘𝑖) ↦ 𝐵), ℝ, < ))
193		breq1 5056	. . . . . 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 3118	. . . . 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 3537	. . . 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 587	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∃𝑦 ∈ ℝ ∀𝑖 ∈ 𝑍 𝑦 ≤ sup(ran (𝑛 ∈ (ℤ_≥‘𝑖) ↦ 𝐵), ℝ, < ))
197	3, 164, 12, 182, 184, 196	mbfinf 24562	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ inf(ran (𝑖 ∈ 𝑍 ↦ sup(ran (𝑛 ∈ (ℤ_≥‘𝑖) ↦ 𝐵), ℝ, < )), ℝ, < )) ∈ MblFn)
198	163, 197	eqeltrd 2838	1 ⊢ (𝜑 → 𝐺 ∈ MblFn)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1543 ∈ wcel 2110 ≠ wne 2940 ∀wral 3061 ∃wrex 3062 Vcvv 3408 ∩ cin 3865 ⊆ wss 3866 ∅c0 4237 class class class wbr 5053 ↦ cmpt 5135 dom cdm 5551 ran crn 5552 ↾ cres 5553 “ cima 5554 Fn wfn 6375 ⟶wf 6376 ‘cfv 6380 (class class class)co 7213 supcsup 9056 infcinf 9057 ℝcr 10728 +∞cpnf 10864 ℝ^*cxr 10866 < clt 10867 ≤ cle 10868 ℤcz 12176 ℤ_≥cuz 12438 [,)cico 12937 lim supclsp 15031 MblFncmbf 24511

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 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-inf2 9256 ax-cc 10049 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 ax-pre-sup 10807

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 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-int 4860 df-iun 4906 df-disj 5019 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-se 5510 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-isom 6389 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-of 7469 df-om 7645 df-1st 7761 df-2nd 7762 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-1o 8202 df-2o 8203 df-oadd 8206 df-omul 8207 df-er 8391 df-map 8510 df-pm 8511 df-en 8627 df-dom 8628 df-sdom 8629 df-fin 8630 df-sup 9058 df-inf 9059 df-oi 9126 df-dju 9517 df-card 9555 df-acn 9558 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-div 11490 df-nn 11831 df-2 11893 df-3 11894 df-n0 12091 df-z 12177 df-uz 12439 df-q 12545 df-rp 12587 df-xadd 12705 df-ioo 12939 df-ioc 12940 df-ico 12941 df-icc 12942 df-fz 13096 df-fzo 13239 df-fl 13367 df-seq 13575 df-exp 13636 df-hash 13897 df-cj 14662 df-re 14663 df-im 14664 df-sqrt 14798 df-abs 14799 df-limsup 15032 df-clim 15049 df-rlim 15050 df-sum 15250 df-xmet 20356 df-met 20357 df-ovol 24361 df-vol 24362 df-mbf 24516

This theorem is referenced by: mbflimlem 24564

W3C validator