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Theorem mbflimsup 23352
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 = (ℤ𝑀)
mbflimsup.2 𝐺 = (𝑥𝐴 ↦ (lim sup‘(𝑛𝑍𝐵)))
mbflimsup.h 𝐻 = (𝑚 ∈ ℝ ↦ sup((((𝑛𝑍𝐵) “ (𝑚[,)+∞)) ∩ ℝ*), ℝ*, < ))
mbflimsup.3 (𝜑𝑀 ∈ ℤ)
mbflimsup.4 ((𝜑𝑥𝐴) → (lim sup‘(𝑛𝑍𝐵)) ∈ ℝ)
mbflimsup.5 ((𝜑𝑛𝑍) → (𝑥𝐴𝐵) ∈ MblFn)
mbflimsup.6 ((𝜑 ∧ (𝑛𝑍𝑥𝐴)) → 𝐵 ∈ ℝ)
Assertion
Ref Expression
mbflimsup (𝜑𝐺 ∈ 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.
StepHypRef Expression
1 mbflimsup.2 . . 3 𝐺 = (𝑥𝐴 ↦ (lim sup‘(𝑛𝑍𝐵)))
2 mbflimsup.h . . . . . 6 𝐻 = (𝑚 ∈ ℝ ↦ sup((((𝑛𝑍𝐵) “ (𝑚[,)+∞)) ∩ ℝ*), ℝ*, < ))
3 mbflimsup.1 . . . . . . . . 9 𝑍 = (ℤ𝑀)
4 fvex 6163 . . . . . . . . 9 (ℤ𝑀) ∈ V
53, 4eqeltri 2694 . . . . . . . 8 𝑍 ∈ V
65mptex 6446 . . . . . . 7 (𝑛𝑍𝐵) ∈ V
76a1i 11 . . . . . 6 ((𝜑𝑥𝐴) → (𝑛𝑍𝐵) ∈ V)
8 uzssz 11658 . . . . . . . . 9 (ℤ𝑀) ⊆ ℤ
93, 8eqsstri 3619 . . . . . . . 8 𝑍 ⊆ ℤ
10 zssre 11335 . . . . . . . 8 ℤ ⊆ ℝ
119, 10sstri 3596 . . . . . . 7 𝑍 ⊆ ℝ
1211a1i 11 . . . . . 6 ((𝜑𝑥𝐴) → 𝑍 ⊆ ℝ)
13 mbflimsup.3 . . . . . . . 8 (𝜑𝑀 ∈ ℤ)
143uzsup 12609 . . . . . . . 8 (𝑀 ∈ ℤ → sup(𝑍, ℝ*, < ) = +∞)
1513, 14syl 17 . . . . . . 7 (𝜑 → sup(𝑍, ℝ*, < ) = +∞)
1615adantr 481 . . . . . 6 ((𝜑𝑥𝐴) → sup(𝑍, ℝ*, < ) = +∞)
172, 7, 12, 16limsupval2 14152 . . . . 5 ((𝜑𝑥𝐴) → (lim sup‘(𝑛𝑍𝐵)) = inf((𝐻𝑍), ℝ*, < ))
18 imassrn 5441 . . . . . . 7 (𝐻𝑍) ⊆ ran 𝐻
1913adantr 481 . . . . . . . . 9 ((𝜑𝑥𝐴) → 𝑀 ∈ ℤ)
20 mbflimsup.6 . . . . . . . . . . 11 ((𝜑 ∧ (𝑛𝑍𝑥𝐴)) → 𝐵 ∈ ℝ)
2120anass1rs 848 . . . . . . . . . 10 (((𝜑𝑥𝐴) ∧ 𝑛𝑍) → 𝐵 ∈ ℝ)
22 eqid 2621 . . . . . . . . . 10 (𝑛𝑍𝐵) = (𝑛𝑍𝐵)
2321, 22fmptd 6346 . . . . . . . . 9 ((𝜑𝑥𝐴) → (𝑛𝑍𝐵):𝑍⟶ℝ)
24 mbflimsup.4 . . . . . . . . . 10 ((𝜑𝑥𝐴) → (lim sup‘(𝑛𝑍𝐵)) ∈ ℝ)
2524ltpnfd 11906 . . . . . . . . 9 ((𝜑𝑥𝐴) → (lim sup‘(𝑛𝑍𝐵)) < +∞)
262, 3limsupgre 14153 . . . . . . . . 9 ((𝑀 ∈ ℤ ∧ (𝑛𝑍𝐵):𝑍⟶ℝ ∧ (lim sup‘(𝑛𝑍𝐵)) < +∞) → 𝐻:ℝ⟶ℝ)
2719, 23, 25, 26syl3anc 1323 . . . . . . . 8 ((𝜑𝑥𝐴) → 𝐻:ℝ⟶ℝ)
28 frn 6015 . . . . . . . 8 (𝐻:ℝ⟶ℝ → ran 𝐻 ⊆ ℝ)
2927, 28syl 17 . . . . . . 7 ((𝜑𝑥𝐴) → ran 𝐻 ⊆ ℝ)
3018, 29syl5ss 3598 . . . . . 6 ((𝜑𝑥𝐴) → (𝐻𝑍) ⊆ ℝ)
31 fdm 6013 . . . . . . . . . . 11 (𝐻:ℝ⟶ℝ → dom 𝐻 = ℝ)
3227, 31syl 17 . . . . . . . . . 10 ((𝜑𝑥𝐴) → dom 𝐻 = ℝ)
3332ineq1d 3796 . . . . . . . . 9 ((𝜑𝑥𝐴) → (dom 𝐻𝑍) = (ℝ ∩ 𝑍))
34 sseqin2 3800 . . . . . . . . . 10 (𝑍 ⊆ ℝ ↔ (ℝ ∩ 𝑍) = 𝑍)
3511, 34mpbi 220 . . . . . . . . 9 (ℝ ∩ 𝑍) = 𝑍
3633, 35syl6eq 2671 . . . . . . . 8 ((𝜑𝑥𝐴) → (dom 𝐻𝑍) = 𝑍)
37 uzid 11653 . . . . . . . . . . . 12 (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ𝑀))
3813, 37syl 17 . . . . . . . . . . 11 (𝜑𝑀 ∈ (ℤ𝑀))
3938, 3syl6eleqr 2709 . . . . . . . . . 10 (𝜑𝑀𝑍)
4039adantr 481 . . . . . . . . 9 ((𝜑𝑥𝐴) → 𝑀𝑍)
41 ne0i 3902 . . . . . . . . 9 (𝑀𝑍𝑍 ≠ ∅)
4240, 41syl 17 . . . . . . . 8 ((𝜑𝑥𝐴) → 𝑍 ≠ ∅)
4336, 42eqnetrd 2857 . . . . . . 7 ((𝜑𝑥𝐴) → (dom 𝐻𝑍) ≠ ∅)
44 imadisj 5448 . . . . . . . 8 ((𝐻𝑍) = ∅ ↔ (dom 𝐻𝑍) = ∅)
4544necon3bii 2842 . . . . . . 7 ((𝐻𝑍) ≠ ∅ ↔ (dom 𝐻𝑍) ≠ ∅)
4643, 45sylibr 224 . . . . . 6 ((𝜑𝑥𝐴) → (𝐻𝑍) ≠ ∅)
4724leidd 10545 . . . . . . . . . 10 ((𝜑𝑥𝐴) → (lim sup‘(𝑛𝑍𝐵)) ≤ (lim sup‘(𝑛𝑍𝐵)))
4821rexrd 10040 . . . . . . . . . . . 12 (((𝜑𝑥𝐴) ∧ 𝑛𝑍) → 𝐵 ∈ ℝ*)
4948, 22fmptd 6346 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → (𝑛𝑍𝐵):𝑍⟶ℝ*)
5024rexrd 10040 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → (lim sup‘(𝑛𝑍𝐵)) ∈ ℝ*)
512limsuple 14150 . . . . . . . . . . 11 ((𝑍 ⊆ ℝ ∧ (𝑛𝑍𝐵):𝑍⟶ℝ* ∧ (lim sup‘(𝑛𝑍𝐵)) ∈ ℝ*) → ((lim sup‘(𝑛𝑍𝐵)) ≤ (lim sup‘(𝑛𝑍𝐵)) ↔ ∀𝑦 ∈ ℝ (lim sup‘(𝑛𝑍𝐵)) ≤ (𝐻𝑦)))
5212, 49, 50, 51syl3anc 1323 . . . . . . . . . 10 ((𝜑𝑥𝐴) → ((lim sup‘(𝑛𝑍𝐵)) ≤ (lim sup‘(𝑛𝑍𝐵)) ↔ ∀𝑦 ∈ ℝ (lim sup‘(𝑛𝑍𝐵)) ≤ (𝐻𝑦)))
5347, 52mpbid 222 . . . . . . . . 9 ((𝜑𝑥𝐴) → ∀𝑦 ∈ ℝ (lim sup‘(𝑛𝑍𝐵)) ≤ (𝐻𝑦))
54 ssralv 3650 . . . . . . . . 9 (𝑍 ⊆ ℝ → (∀𝑦 ∈ ℝ (lim sup‘(𝑛𝑍𝐵)) ≤ (𝐻𝑦) → ∀𝑦𝑍 (lim sup‘(𝑛𝑍𝐵)) ≤ (𝐻𝑦)))
5511, 53, 54mpsyl 68 . . . . . . . 8 ((𝜑𝑥𝐴) → ∀𝑦𝑍 (lim sup‘(𝑛𝑍𝐵)) ≤ (𝐻𝑦))
562limsupgf 14147 . . . . . . . . . 10 𝐻:ℝ⟶ℝ*
57 ffn 6007 . . . . . . . . . 10 (𝐻:ℝ⟶ℝ*𝐻 Fn ℝ)
5856, 57ax-mp 5 . . . . . . . . 9 𝐻 Fn ℝ
59 breq2 4622 . . . . . . . . . 10 (𝑧 = (𝐻𝑦) → ((lim sup‘(𝑛𝑍𝐵)) ≤ 𝑧 ↔ (lim sup‘(𝑛𝑍𝐵)) ≤ (𝐻𝑦)))
6059ralima 6458 . . . . . . . . 9 ((𝐻 Fn ℝ ∧ 𝑍 ⊆ ℝ) → (∀𝑧 ∈ (𝐻𝑍)(lim sup‘(𝑛𝑍𝐵)) ≤ 𝑧 ↔ ∀𝑦𝑍 (lim sup‘(𝑛𝑍𝐵)) ≤ (𝐻𝑦)))
6158, 12, 60sylancr 694 . . . . . . . 8 ((𝜑𝑥𝐴) → (∀𝑧 ∈ (𝐻𝑍)(lim sup‘(𝑛𝑍𝐵)) ≤ 𝑧 ↔ ∀𝑦𝑍 (lim sup‘(𝑛𝑍𝐵)) ≤ (𝐻𝑦)))
6255, 61mpbird 247 . . . . . . 7 ((𝜑𝑥𝐴) → ∀𝑧 ∈ (𝐻𝑍)(lim sup‘(𝑛𝑍𝐵)) ≤ 𝑧)
63 breq1 4621 . . . . . . . . 9 (𝑦 = (lim sup‘(𝑛𝑍𝐵)) → (𝑦𝑧 ↔ (lim sup‘(𝑛𝑍𝐵)) ≤ 𝑧))
6463ralbidv 2981 . . . . . . . 8 (𝑦 = (lim sup‘(𝑛𝑍𝐵)) → (∀𝑧 ∈ (𝐻𝑍)𝑦𝑧 ↔ ∀𝑧 ∈ (𝐻𝑍)(lim sup‘(𝑛𝑍𝐵)) ≤ 𝑧))
6564rspcev 3298 . . . . . . 7 (((lim sup‘(𝑛𝑍𝐵)) ∈ ℝ ∧ ∀𝑧 ∈ (𝐻𝑍)(lim sup‘(𝑛𝑍𝐵)) ≤ 𝑧) → ∃𝑦 ∈ ℝ ∀𝑧 ∈ (𝐻𝑍)𝑦𝑧)
6624, 62, 65syl2anc 692 . . . . . 6 ((𝜑𝑥𝐴) → ∃𝑦 ∈ ℝ ∀𝑧 ∈ (𝐻𝑍)𝑦𝑧)
67 infxrre 12116 . . . . . 6 (((𝐻𝑍) ⊆ ℝ ∧ (𝐻𝑍) ≠ ∅ ∧ ∃𝑦 ∈ ℝ ∀𝑧 ∈ (𝐻𝑍)𝑦𝑧) → inf((𝐻𝑍), ℝ*, < ) = inf((𝐻𝑍), ℝ, < ))
6830, 46, 66, 67syl3anc 1323 . . . . 5 ((𝜑𝑥𝐴) → inf((𝐻𝑍), ℝ*, < ) = inf((𝐻𝑍), ℝ, < ))
69 df-ima 5092 . . . . . . 7 (𝐻𝑍) = ran (𝐻𝑍)
7027feqmptd 6211 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → 𝐻 = (𝑖 ∈ ℝ ↦ (𝐻𝑖)))
7170reseq1d 5360 . . . . . . . . . 10 ((𝜑𝑥𝐴) → (𝐻𝑍) = ((𝑖 ∈ ℝ ↦ (𝐻𝑖)) ↾ 𝑍))
72 resmpt 5413 . . . . . . . . . . 11 (𝑍 ⊆ ℝ → ((𝑖 ∈ ℝ ↦ (𝐻𝑖)) ↾ 𝑍) = (𝑖𝑍 ↦ (𝐻𝑖)))
7311, 72ax-mp 5 . . . . . . . . . 10 ((𝑖 ∈ ℝ ↦ (𝐻𝑖)) ↾ 𝑍) = (𝑖𝑍 ↦ (𝐻𝑖))
7471, 73syl6eq 2671 . . . . . . . . 9 ((𝜑𝑥𝐴) → (𝐻𝑍) = (𝑖𝑍 ↦ (𝐻𝑖)))
7511sseli 3583 . . . . . . . . . . . . 13 (𝑖𝑍𝑖 ∈ ℝ)
76 ffvelrn 6318 . . . . . . . . . . . . 13 ((𝐻:ℝ⟶ℝ ∧ 𝑖 ∈ ℝ) → (𝐻𝑖) ∈ ℝ)
7727, 75, 76syl2an 494 . . . . . . . . . . . 12 (((𝜑𝑥𝐴) ∧ 𝑖𝑍) → (𝐻𝑖) ∈ ℝ)
7877rexrd 10040 . . . . . . . . . . 11 (((𝜑𝑥𝐴) ∧ 𝑖𝑍) → (𝐻𝑖) ∈ ℝ*)
79 simplll 797 . . . . . . . . . . . . . . . 16 ((((𝜑𝑥𝐴) ∧ 𝑖𝑍) ∧ 𝑛 ∈ (ℤ𝑖)) → 𝜑)
803uztrn2 11656 . . . . . . . . . . . . . . . . 17 ((𝑖𝑍𝑛 ∈ (ℤ𝑖)) → 𝑛𝑍)
8180adantll 749 . . . . . . . . . . . . . . . 16 ((((𝜑𝑥𝐴) ∧ 𝑖𝑍) ∧ 𝑛 ∈ (ℤ𝑖)) → 𝑛𝑍)
82 simpllr 798 . . . . . . . . . . . . . . . 16 ((((𝜑𝑥𝐴) ∧ 𝑖𝑍) ∧ 𝑛 ∈ (ℤ𝑖)) → 𝑥𝐴)
8379, 81, 82, 20syl12anc 1321 . . . . . . . . . . . . . . 15 ((((𝜑𝑥𝐴) ∧ 𝑖𝑍) ∧ 𝑛 ∈ (ℤ𝑖)) → 𝐵 ∈ ℝ)
84 eqid 2621 . . . . . . . . . . . . . . 15 (𝑛 ∈ (ℤ𝑖) ↦ 𝐵) = (𝑛 ∈ (ℤ𝑖) ↦ 𝐵)
8583, 84fmptd 6346 . . . . . . . . . . . . . 14 (((𝜑𝑥𝐴) ∧ 𝑖𝑍) → (𝑛 ∈ (ℤ𝑖) ↦ 𝐵):(ℤ𝑖)⟶ℝ)
86 frn 6015 . . . . . . . . . . . . . 14 ((𝑛 ∈ (ℤ𝑖) ↦ 𝐵):(ℤ𝑖)⟶ℝ → ran (𝑛 ∈ (ℤ𝑖) ↦ 𝐵) ⊆ ℝ)
8785, 86syl 17 . . . . . . . . . . . . 13 (((𝜑𝑥𝐴) ∧ 𝑖𝑍) → ran (𝑛 ∈ (ℤ𝑖) ↦ 𝐵) ⊆ ℝ)
8884, 83dmmptd 5986 . . . . . . . . . . . . . . 15 (((𝜑𝑥𝐴) ∧ 𝑖𝑍) → dom (𝑛 ∈ (ℤ𝑖) ↦ 𝐵) = (ℤ𝑖))
89 simpr 477 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑖𝑍) → 𝑖𝑍)
9089, 3syl6eleq 2708 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑖𝑍) → 𝑖 ∈ (ℤ𝑀))
91 eluzelz 11648 . . . . . . . . . . . . . . . . . 18 (𝑖 ∈ (ℤ𝑀) → 𝑖 ∈ ℤ)
9290, 91syl 17 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖𝑍) → 𝑖 ∈ ℤ)
9392adantlr 750 . . . . . . . . . . . . . . . 16 (((𝜑𝑥𝐴) ∧ 𝑖𝑍) → 𝑖 ∈ ℤ)
94 uzid 11653 . . . . . . . . . . . . . . . 16 (𝑖 ∈ ℤ → 𝑖 ∈ (ℤ𝑖))
95 ne0i 3902 . . . . . . . . . . . . . . . 16 (𝑖 ∈ (ℤ𝑖) → (ℤ𝑖) ≠ ∅)
9693, 94, 953syl 18 . . . . . . . . . . . . . . 15 (((𝜑𝑥𝐴) ∧ 𝑖𝑍) → (ℤ𝑖) ≠ ∅)
9788, 96eqnetrd 2857 . . . . . . . . . . . . . 14 (((𝜑𝑥𝐴) ∧ 𝑖𝑍) → dom (𝑛 ∈ (ℤ𝑖) ↦ 𝐵) ≠ ∅)
98 dm0rn0 5307 . . . . . . . . . . . . . . 15 (dom (𝑛 ∈ (ℤ𝑖) ↦ 𝐵) = ∅ ↔ ran (𝑛 ∈ (ℤ𝑖) ↦ 𝐵) = ∅)
9998necon3bii 2842 . . . . . . . . . . . . . 14 (dom (𝑛 ∈ (ℤ𝑖) ↦ 𝐵) ≠ ∅ ↔ ran (𝑛 ∈ (ℤ𝑖) ↦ 𝐵) ≠ ∅)
10097, 99sylib 208 . . . . . . . . . . . . 13 (((𝜑𝑥𝐴) ∧ 𝑖𝑍) → ran (𝑛 ∈ (ℤ𝑖) ↦ 𝐵) ≠ ∅)
10190adantlr 750 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑥𝐴) ∧ 𝑖𝑍) → 𝑖 ∈ (ℤ𝑀))
102 uzss 11659 . . . . . . . . . . . . . . . . . . 19 (𝑖 ∈ (ℤ𝑀) → (ℤ𝑖) ⊆ (ℤ𝑀))
103101, 102syl 17 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑥𝐴) ∧ 𝑖𝑍) → (ℤ𝑖) ⊆ (ℤ𝑀))
104103, 3syl6sseqr 3636 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥𝐴) ∧ 𝑖𝑍) → (ℤ𝑖) ⊆ 𝑍)
10577leidd 10545 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑥𝐴) ∧ 𝑖𝑍) → (𝐻𝑖) ≤ (𝐻𝑖))
10611a1i 11 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑥𝐴) ∧ 𝑖𝑍) → 𝑍 ⊆ ℝ)
10749adantr 481 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑥𝐴) ∧ 𝑖𝑍) → (𝑛𝑍𝐵):𝑍⟶ℝ*)
108 simpr 477 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑥𝐴) ∧ 𝑖𝑍) → 𝑖𝑍)
10911, 108sseldi 3585 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑥𝐴) ∧ 𝑖𝑍) → 𝑖 ∈ ℝ)
1102limsupgle 14149 . . . . . . . . . . . . . . . . . . 19 (((𝑍 ⊆ ℝ ∧ (𝑛𝑍𝐵):𝑍⟶ℝ*) ∧ 𝑖 ∈ ℝ ∧ (𝐻𝑖) ∈ ℝ*) → ((𝐻𝑖) ≤ (𝐻𝑖) ↔ ∀𝑘𝑍 (𝑖𝑘 → ((𝑛𝑍𝐵)‘𝑘) ≤ (𝐻𝑖))))
111106, 107, 109, 78, 110syl211anc 1329 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑥𝐴) ∧ 𝑖𝑍) → ((𝐻𝑖) ≤ (𝐻𝑖) ↔ ∀𝑘𝑍 (𝑖𝑘 → ((𝑛𝑍𝐵)‘𝑘) ≤ (𝐻𝑖))))
112105, 111mpbid 222 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥𝐴) ∧ 𝑖𝑍) → ∀𝑘𝑍 (𝑖𝑘 → ((𝑛𝑍𝐵)‘𝑘) ≤ (𝐻𝑖)))
113 ssralv 3650 . . . . . . . . . . . . . . . . 17 ((ℤ𝑖) ⊆ 𝑍 → (∀𝑘𝑍 (𝑖𝑘 → ((𝑛𝑍𝐵)‘𝑘) ≤ (𝐻𝑖)) → ∀𝑘 ∈ (ℤ𝑖)(𝑖𝑘 → ((𝑛𝑍𝐵)‘𝑘) ≤ (𝐻𝑖))))
114104, 112, 113sylc 65 . . . . . . . . . . . . . . . 16 (((𝜑𝑥𝐴) ∧ 𝑖𝑍) → ∀𝑘 ∈ (ℤ𝑖)(𝑖𝑘 → ((𝑛𝑍𝐵)‘𝑘) ≤ (𝐻𝑖)))
115104adantr 481 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑥𝐴) ∧ 𝑖𝑍) ∧ 𝑘 ∈ (ℤ𝑖)) → (ℤ𝑖) ⊆ 𝑍)
116115resmptd 5416 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑥𝐴) ∧ 𝑖𝑍) ∧ 𝑘 ∈ (ℤ𝑖)) → ((𝑛𝑍𝐵) ↾ (ℤ𝑖)) = (𝑛 ∈ (ℤ𝑖) ↦ 𝐵))
117116fveq1d 6155 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑥𝐴) ∧ 𝑖𝑍) ∧ 𝑘 ∈ (ℤ𝑖)) → (((𝑛𝑍𝐵) ↾ (ℤ𝑖))‘𝑘) = ((𝑛 ∈ (ℤ𝑖) ↦ 𝐵)‘𝑘))
118 fvres 6169 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 ∈ (ℤ𝑖) → (((𝑛𝑍𝐵) ↾ (ℤ𝑖))‘𝑘) = ((𝑛𝑍𝐵)‘𝑘))
119118adantl 482 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑥𝐴) ∧ 𝑖𝑍) ∧ 𝑘 ∈ (ℤ𝑖)) → (((𝑛𝑍𝐵) ↾ (ℤ𝑖))‘𝑘) = ((𝑛𝑍𝐵)‘𝑘))
120117, 119eqtr3d 2657 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑥𝐴) ∧ 𝑖𝑍) ∧ 𝑘 ∈ (ℤ𝑖)) → ((𝑛 ∈ (ℤ𝑖) ↦ 𝐵)‘𝑘) = ((𝑛𝑍𝐵)‘𝑘))
121120breq1d 4628 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑥𝐴) ∧ 𝑖𝑍) ∧ 𝑘 ∈ (ℤ𝑖)) → (((𝑛 ∈ (ℤ𝑖) ↦ 𝐵)‘𝑘) ≤ (𝐻𝑖) ↔ ((𝑛𝑍𝐵)‘𝑘) ≤ (𝐻𝑖)))
122 eluzle 11651 . . . . . . . . . . . . . . . . . . . 20 (𝑘 ∈ (ℤ𝑖) → 𝑖𝑘)
123122adantl 482 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑥𝐴) ∧ 𝑖𝑍) ∧ 𝑘 ∈ (ℤ𝑖)) → 𝑖𝑘)
124 biimt 350 . . . . . . . . . . . . . . . . . . 19 (𝑖𝑘 → (((𝑛𝑍𝐵)‘𝑘) ≤ (𝐻𝑖) ↔ (𝑖𝑘 → ((𝑛𝑍𝐵)‘𝑘) ≤ (𝐻𝑖))))
125123, 124syl 17 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑥𝐴) ∧ 𝑖𝑍) ∧ 𝑘 ∈ (ℤ𝑖)) → (((𝑛𝑍𝐵)‘𝑘) ≤ (𝐻𝑖) ↔ (𝑖𝑘 → ((𝑛𝑍𝐵)‘𝑘) ≤ (𝐻𝑖))))
126121, 125bitrd 268 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑥𝐴) ∧ 𝑖𝑍) ∧ 𝑘 ∈ (ℤ𝑖)) → (((𝑛 ∈ (ℤ𝑖) ↦ 𝐵)‘𝑘) ≤ (𝐻𝑖) ↔ (𝑖𝑘 → ((𝑛𝑍𝐵)‘𝑘) ≤ (𝐻𝑖))))
127126ralbidva 2980 . . . . . . . . . . . . . . . 16 (((𝜑𝑥𝐴) ∧ 𝑖𝑍) → (∀𝑘 ∈ (ℤ𝑖)((𝑛 ∈ (ℤ𝑖) ↦ 𝐵)‘𝑘) ≤ (𝐻𝑖) ↔ ∀𝑘 ∈ (ℤ𝑖)(𝑖𝑘 → ((𝑛𝑍𝐵)‘𝑘) ≤ (𝐻𝑖))))
128114, 127mpbird 247 . . . . . . . . . . . . . . 15 (((𝜑𝑥𝐴) ∧ 𝑖𝑍) → ∀𝑘 ∈ (ℤ𝑖)((𝑛 ∈ (ℤ𝑖) ↦ 𝐵)‘𝑘) ≤ (𝐻𝑖))
129 ffn 6007 . . . . . . . . . . . . . . . 16 ((𝑛 ∈ (ℤ𝑖) ↦ 𝐵):(ℤ𝑖)⟶ℝ → (𝑛 ∈ (ℤ𝑖) ↦ 𝐵) Fn (ℤ𝑖))
130 breq1 4621 . . . . . . . . . . . . . . . . 17 (𝑧 = ((𝑛 ∈ (ℤ𝑖) ↦ 𝐵)‘𝑘) → (𝑧 ≤ (𝐻𝑖) ↔ ((𝑛 ∈ (ℤ𝑖) ↦ 𝐵)‘𝑘) ≤ (𝐻𝑖)))
131130ralrn 6323 . . . . . . . . . . . . . . . 16 ((𝑛 ∈ (ℤ𝑖) ↦ 𝐵) Fn (ℤ𝑖) → (∀𝑧 ∈ ran (𝑛 ∈ (ℤ𝑖) ↦ 𝐵)𝑧 ≤ (𝐻𝑖) ↔ ∀𝑘 ∈ (ℤ𝑖)((𝑛 ∈ (ℤ𝑖) ↦ 𝐵)‘𝑘) ≤ (𝐻𝑖)))
13285, 129, 1313syl 18 . . . . . . . . . . . . . . 15 (((𝜑𝑥𝐴) ∧ 𝑖𝑍) → (∀𝑧 ∈ ran (𝑛 ∈ (ℤ𝑖) ↦ 𝐵)𝑧 ≤ (𝐻𝑖) ↔ ∀𝑘 ∈ (ℤ𝑖)((𝑛 ∈ (ℤ𝑖) ↦ 𝐵)‘𝑘) ≤ (𝐻𝑖)))
133128, 132mpbird 247 . . . . . . . . . . . . . 14 (((𝜑𝑥𝐴) ∧ 𝑖𝑍) → ∀𝑧 ∈ ran (𝑛 ∈ (ℤ𝑖) ↦ 𝐵)𝑧 ≤ (𝐻𝑖))
134 breq2 4622 . . . . . . . . . . . . . . . 16 (𝑦 = (𝐻𝑖) → (𝑧𝑦𝑧 ≤ (𝐻𝑖)))
135134ralbidv 2981 . . . . . . . . . . . . . . 15 (𝑦 = (𝐻𝑖) → (∀𝑧 ∈ ran (𝑛 ∈ (ℤ𝑖) ↦ 𝐵)𝑧𝑦 ↔ ∀𝑧 ∈ ran (𝑛 ∈ (ℤ𝑖) ↦ 𝐵)𝑧 ≤ (𝐻𝑖)))
136135rspcev 3298 . . . . . . . . . . . . . 14 (((𝐻𝑖) ∈ ℝ ∧ ∀𝑧 ∈ ran (𝑛 ∈ (ℤ𝑖) ↦ 𝐵)𝑧 ≤ (𝐻𝑖)) → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑛 ∈ (ℤ𝑖) ↦ 𝐵)𝑧𝑦)
13777, 133, 136syl2anc 692 . . . . . . . . . . . . 13 (((𝜑𝑥𝐴) ∧ 𝑖𝑍) → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑛 ∈ (ℤ𝑖) ↦ 𝐵)𝑧𝑦)
138 suprcl 10934 . . . . . . . . . . . . 13 ((ran (𝑛 ∈ (ℤ𝑖) ↦ 𝐵) ⊆ ℝ ∧ ran (𝑛 ∈ (ℤ𝑖) ↦ 𝐵) ≠ ∅ ∧ ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑛 ∈ (ℤ𝑖) ↦ 𝐵)𝑧𝑦) → sup(ran (𝑛 ∈ (ℤ𝑖) ↦ 𝐵), ℝ, < ) ∈ ℝ)
13987, 100, 137, 138syl3anc 1323 . . . . . . . . . . . 12 (((𝜑𝑥𝐴) ∧ 𝑖𝑍) → sup(ran (𝑛 ∈ (ℤ𝑖) ↦ 𝐵), ℝ, < ) ∈ ℝ)
140139rexrd 10040 . . . . . . . . . . 11 (((𝜑𝑥𝐴) ∧ 𝑖𝑍) → sup(ran (𝑛 ∈ (ℤ𝑖) ↦ 𝐵), ℝ, < ) ∈ ℝ*)
14187adantr 481 . . . . . . . . . . . . . . 15 ((((𝜑𝑥𝐴) ∧ 𝑖𝑍) ∧ (𝑘𝑍𝑖𝑘)) → ran (𝑛 ∈ (ℤ𝑖) ↦ 𝐵) ⊆ ℝ)
142100adantr 481 . . . . . . . . . . . . . . 15 ((((𝜑𝑥𝐴) ∧ 𝑖𝑍) ∧ (𝑘𝑍𝑖𝑘)) → ran (𝑛 ∈ (ℤ𝑖) ↦ 𝐵) ≠ ∅)
143137adantr 481 . . . . . . . . . . . . . . 15 ((((𝜑𝑥𝐴) ∧ 𝑖𝑍) ∧ (𝑘𝑍𝑖𝑘)) → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑛 ∈ (ℤ𝑖) ↦ 𝐵)𝑧𝑦)
1449sseli 3583 . . . . . . . . . . . . . . . . . . . 20 (𝑘𝑍𝑘 ∈ ℤ)
145 eluz 11652 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 ∈ ℤ ∧ 𝑘 ∈ ℤ) → (𝑘 ∈ (ℤ𝑖) ↔ 𝑖𝑘))
14693, 144, 145syl2an 494 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑥𝐴) ∧ 𝑖𝑍) ∧ 𝑘𝑍) → (𝑘 ∈ (ℤ𝑖) ↔ 𝑖𝑘))
147146biimprd 238 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑥𝐴) ∧ 𝑖𝑍) ∧ 𝑘𝑍) → (𝑖𝑘𝑘 ∈ (ℤ𝑖)))
148147impr 648 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑥𝐴) ∧ 𝑖𝑍) ∧ (𝑘𝑍𝑖𝑘)) → 𝑘 ∈ (ℤ𝑖))
149148, 120syldan 487 . . . . . . . . . . . . . . . 16 ((((𝜑𝑥𝐴) ∧ 𝑖𝑍) ∧ (𝑘𝑍𝑖𝑘)) → ((𝑛 ∈ (ℤ𝑖) ↦ 𝐵)‘𝑘) = ((𝑛𝑍𝐵)‘𝑘))
15085adantr 481 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑥𝐴) ∧ 𝑖𝑍) ∧ (𝑘𝑍𝑖𝑘)) → (𝑛 ∈ (ℤ𝑖) ↦ 𝐵):(ℤ𝑖)⟶ℝ)
151150, 129syl 17 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑥𝐴) ∧ 𝑖𝑍) ∧ (𝑘𝑍𝑖𝑘)) → (𝑛 ∈ (ℤ𝑖) ↦ 𝐵) Fn (ℤ𝑖))
152 fnfvelrn 6317 . . . . . . . . . . . . . . . . 17 (((𝑛 ∈ (ℤ𝑖) ↦ 𝐵) Fn (ℤ𝑖) ∧ 𝑘 ∈ (ℤ𝑖)) → ((𝑛 ∈ (ℤ𝑖) ↦ 𝐵)‘𝑘) ∈ ran (𝑛 ∈ (ℤ𝑖) ↦ 𝐵))
153151, 148, 152syl2anc 692 . . . . . . . . . . . . . . . 16 ((((𝜑𝑥𝐴) ∧ 𝑖𝑍) ∧ (𝑘𝑍𝑖𝑘)) → ((𝑛 ∈ (ℤ𝑖) ↦ 𝐵)‘𝑘) ∈ ran (𝑛 ∈ (ℤ𝑖) ↦ 𝐵))
154149, 153eqeltrrd 2699 . . . . . . . . . . . . . . 15 ((((𝜑𝑥𝐴) ∧ 𝑖𝑍) ∧ (𝑘𝑍𝑖𝑘)) → ((𝑛𝑍𝐵)‘𝑘) ∈ ran (𝑛 ∈ (ℤ𝑖) ↦ 𝐵))
155 suprub 10935 . . . . . . . . . . . . . . 15 (((ran (𝑛 ∈ (ℤ𝑖) ↦ 𝐵) ⊆ ℝ ∧ ran (𝑛 ∈ (ℤ𝑖) ↦ 𝐵) ≠ ∅ ∧ ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑛 ∈ (ℤ𝑖) ↦ 𝐵)𝑧𝑦) ∧ ((𝑛𝑍𝐵)‘𝑘) ∈ ran (𝑛 ∈ (ℤ𝑖) ↦ 𝐵)) → ((𝑛𝑍𝐵)‘𝑘) ≤ sup(ran (𝑛 ∈ (ℤ𝑖) ↦ 𝐵), ℝ, < ))
156141, 142, 143, 154, 155syl31anc 1326 . . . . . . . . . . . . . 14 ((((𝜑𝑥𝐴) ∧ 𝑖𝑍) ∧ (𝑘𝑍𝑖𝑘)) → ((𝑛𝑍𝐵)‘𝑘) ≤ sup(ran (𝑛 ∈ (ℤ𝑖) ↦ 𝐵), ℝ, < ))
157156expr 642 . . . . . . . . . . . . 13 ((((𝜑𝑥𝐴) ∧ 𝑖𝑍) ∧ 𝑘𝑍) → (𝑖𝑘 → ((𝑛𝑍𝐵)‘𝑘) ≤ sup(ran (𝑛 ∈ (ℤ𝑖) ↦ 𝐵), ℝ, < )))
158157ralrimiva 2961 . . . . . . . . . . . 12 (((𝜑𝑥𝐴) ∧ 𝑖𝑍) → ∀𝑘𝑍 (𝑖𝑘 → ((𝑛𝑍𝐵)‘𝑘) ≤ sup(ran (𝑛 ∈ (ℤ𝑖) ↦ 𝐵), ℝ, < )))
1592limsupgle 14149 . . . . . . . . . . . . 13 (((𝑍 ⊆ ℝ ∧ (𝑛𝑍𝐵):𝑍⟶ℝ*) ∧ 𝑖 ∈ ℝ ∧ sup(ran (𝑛 ∈ (ℤ𝑖) ↦ 𝐵), ℝ, < ) ∈ ℝ*) → ((𝐻𝑖) ≤ sup(ran (𝑛 ∈ (ℤ𝑖) ↦ 𝐵), ℝ, < ) ↔ ∀𝑘𝑍 (𝑖𝑘 → ((𝑛𝑍𝐵)‘𝑘) ≤ sup(ran (𝑛 ∈ (ℤ𝑖) ↦ 𝐵), ℝ, < ))))
160106, 107, 109, 140, 159syl211anc 1329 . . . . . . . . . . . 12 (((𝜑𝑥𝐴) ∧ 𝑖𝑍) → ((𝐻𝑖) ≤ sup(ran (𝑛 ∈ (ℤ𝑖) ↦ 𝐵), ℝ, < ) ↔ ∀𝑘𝑍 (𝑖𝑘 → ((𝑛𝑍𝐵)‘𝑘) ≤ sup(ran (𝑛 ∈ (ℤ𝑖) ↦ 𝐵), ℝ, < ))))
161158, 160mpbird 247 . . . . . . . . . . 11 (((𝜑𝑥𝐴) ∧ 𝑖𝑍) → (𝐻𝑖) ≤ sup(ran (𝑛 ∈ (ℤ𝑖) ↦ 𝐵), ℝ, < ))
162 suprleub 10940 . . . . . . . . . . . . 13 (((ran (𝑛 ∈ (ℤ𝑖) ↦ 𝐵) ⊆ ℝ ∧ ran (𝑛 ∈ (ℤ𝑖) ↦ 𝐵) ≠ ∅ ∧ ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑛 ∈ (ℤ𝑖) ↦ 𝐵)𝑧𝑦) ∧ (𝐻𝑖) ∈ ℝ) → (sup(ran (𝑛 ∈ (ℤ𝑖) ↦ 𝐵), ℝ, < ) ≤ (𝐻𝑖) ↔ ∀𝑧 ∈ ran (𝑛 ∈ (ℤ𝑖) ↦ 𝐵)𝑧 ≤ (𝐻𝑖)))
16387, 100, 137, 77, 162syl31anc 1326 . . . . . . . . . . . 12 (((𝜑𝑥𝐴) ∧ 𝑖𝑍) → (sup(ran (𝑛 ∈ (ℤ𝑖) ↦ 𝐵), ℝ, < ) ≤ (𝐻𝑖) ↔ ∀𝑧 ∈ ran (𝑛 ∈ (ℤ𝑖) ↦ 𝐵)𝑧 ≤ (𝐻𝑖)))
164133, 163mpbird 247 . . . . . . . . . . 11 (((𝜑𝑥𝐴) ∧ 𝑖𝑍) → sup(ran (𝑛 ∈ (ℤ𝑖) ↦ 𝐵), ℝ, < ) ≤ (𝐻𝑖))
16578, 140, 161, 164xrletrid 11937 . . . . . . . . . 10 (((𝜑𝑥𝐴) ∧ 𝑖𝑍) → (𝐻𝑖) = sup(ran (𝑛 ∈ (ℤ𝑖) ↦ 𝐵), ℝ, < ))
166165mpteq2dva 4709 . . . . . . . . 9 ((𝜑𝑥𝐴) → (𝑖𝑍 ↦ (𝐻𝑖)) = (𝑖𝑍 ↦ sup(ran (𝑛 ∈ (ℤ𝑖) ↦ 𝐵), ℝ, < )))
16774, 166eqtrd 2655 . . . . . . . 8 ((𝜑𝑥𝐴) → (𝐻𝑍) = (𝑖𝑍 ↦ sup(ran (𝑛 ∈ (ℤ𝑖) ↦ 𝐵), ℝ, < )))
168167rneqd 5318 . . . . . . 7 ((𝜑𝑥𝐴) → ran (𝐻𝑍) = ran (𝑖𝑍 ↦ sup(ran (𝑛 ∈ (ℤ𝑖) ↦ 𝐵), ℝ, < )))
16969, 168syl5eq 2667 . . . . . 6 ((𝜑𝑥𝐴) → (𝐻𝑍) = ran (𝑖𝑍 ↦ sup(ran (𝑛 ∈ (ℤ𝑖) ↦ 𝐵), ℝ, < )))
170169infeq1d 8334 . . . . 5 ((𝜑𝑥𝐴) → inf((𝐻𝑍), ℝ, < ) = inf(ran (𝑖𝑍 ↦ sup(ran (𝑛 ∈ (ℤ𝑖) ↦ 𝐵), ℝ, < )), ℝ, < ))
17117, 68, 1703eqtrd 2659 . . . 4 ((𝜑𝑥𝐴) → (lim sup‘(𝑛𝑍𝐵)) = inf(ran (𝑖𝑍 ↦ sup(ran (𝑛 ∈ (ℤ𝑖) ↦ 𝐵), ℝ, < )), ℝ, < ))
172171mpteq2dva 4709 . . 3 (𝜑 → (𝑥𝐴 ↦ (lim sup‘(𝑛𝑍𝐵))) = (𝑥𝐴 ↦ inf(ran (𝑖𝑍 ↦ sup(ran (𝑛 ∈ (ℤ𝑖) ↦ 𝐵), ℝ, < )), ℝ, < )))
1731, 172syl5eq 2667 . 2 (𝜑𝐺 = (𝑥𝐴 ↦ inf(ran (𝑖𝑍 ↦ sup(ran (𝑛 ∈ (ℤ𝑖) ↦ 𝐵), ℝ, < )), ℝ, < )))
174 eqid 2621 . . 3 (𝑥𝐴 ↦ inf(ran (𝑖𝑍 ↦ sup(ran (𝑛 ∈ (ℤ𝑖) ↦ 𝐵), ℝ, < )), ℝ, < )) = (𝑥𝐴 ↦ inf(ran (𝑖𝑍 ↦ sup(ran (𝑛 ∈ (ℤ𝑖) ↦ 𝐵), ℝ, < )), ℝ, < ))
175 eqid 2621 . . . 4 (ℤ𝑖) = (ℤ𝑖)
176 eqid 2621 . . . 4 (𝑥𝐴 ↦ sup(ran (𝑛 ∈ (ℤ𝑖) ↦ 𝐵), ℝ, < )) = (𝑥𝐴 ↦ sup(ran (𝑛 ∈ (ℤ𝑖) ↦ 𝐵), ℝ, < ))
177 simpll 789 . . . . 5 (((𝜑𝑖𝑍) ∧ 𝑛 ∈ (ℤ𝑖)) → 𝜑)
17880adantll 749 . . . . 5 (((𝜑𝑖𝑍) ∧ 𝑛 ∈ (ℤ𝑖)) → 𝑛𝑍)
179 mbflimsup.5 . . . . 5 ((𝜑𝑛𝑍) → (𝑥𝐴𝐵) ∈ MblFn)
180177, 178, 179syl2anc 692 . . . 4 (((𝜑𝑖𝑍) ∧ 𝑛 ∈ (ℤ𝑖)) → (𝑥𝐴𝐵) ∈ MblFn)
181 simpll 789 . . . . 5 (((𝜑𝑖𝑍) ∧ (𝑛 ∈ (ℤ𝑖) ∧ 𝑥𝐴)) → 𝜑)
18280ad2ant2lr 783 . . . . 5 (((𝜑𝑖𝑍) ∧ (𝑛 ∈ (ℤ𝑖) ∧ 𝑥𝐴)) → 𝑛𝑍)
183 simprr 795 . . . . 5 (((𝜑𝑖𝑍) ∧ (𝑛 ∈ (ℤ𝑖) ∧ 𝑥𝐴)) → 𝑥𝐴)
184181, 182, 183, 20syl12anc 1321 . . . 4 (((𝜑𝑖𝑍) ∧ (𝑛 ∈ (ℤ𝑖) ∧ 𝑥𝐴)) → 𝐵 ∈ ℝ)
18583ralrimiva 2961 . . . . . . . 8 (((𝜑𝑥𝐴) ∧ 𝑖𝑍) → ∀𝑛 ∈ (ℤ𝑖)𝐵 ∈ ℝ)
186 breq1 4621 . . . . . . . . 9 (𝑧 = 𝐵 → (𝑧𝑦𝐵𝑦))
18784, 186ralrnmpt 6329 . . . . . . . 8 (∀𝑛 ∈ (ℤ𝑖)𝐵 ∈ ℝ → (∀𝑧 ∈ ran (𝑛 ∈ (ℤ𝑖) ↦ 𝐵)𝑧𝑦 ↔ ∀𝑛 ∈ (ℤ𝑖)𝐵𝑦))
188185, 187syl 17 . . . . . . 7 (((𝜑𝑥𝐴) ∧ 𝑖𝑍) → (∀𝑧 ∈ ran (𝑛 ∈ (ℤ𝑖) ↦ 𝐵)𝑧𝑦 ↔ ∀𝑛 ∈ (ℤ𝑖)𝐵𝑦))
189188rexbidv 3046 . . . . . 6 (((𝜑𝑥𝐴) ∧ 𝑖𝑍) → (∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑛 ∈ (ℤ𝑖) ↦ 𝐵)𝑧𝑦 ↔ ∃𝑦 ∈ ℝ ∀𝑛 ∈ (ℤ𝑖)𝐵𝑦))
190137, 189mpbid 222 . . . . 5 (((𝜑𝑥𝐴) ∧ 𝑖𝑍) → ∃𝑦 ∈ ℝ ∀𝑛 ∈ (ℤ𝑖)𝐵𝑦)
191190an32s 845 . . . 4 (((𝜑𝑖𝑍) ∧ 𝑥𝐴) → ∃𝑦 ∈ ℝ ∀𝑛 ∈ (ℤ𝑖)𝐵𝑦)
192175, 176, 92, 180, 184, 191mbfsup 23350 . . 3 ((𝜑𝑖𝑍) → (𝑥𝐴 ↦ sup(ran (𝑛 ∈ (ℤ𝑖) ↦ 𝐵), ℝ, < )) ∈ MblFn)
193139an32s 845 . . . 4 (((𝜑𝑖𝑍) ∧ 𝑥𝐴) → sup(ran (𝑛 ∈ (ℤ𝑖) ↦ 𝐵), ℝ, < ) ∈ ℝ)
194193anasss 678 . . 3 ((𝜑 ∧ (𝑖𝑍𝑥𝐴)) → sup(ran (𝑛 ∈ (ℤ𝑖) ↦ 𝐵), ℝ, < ) ∈ ℝ)
1952limsuple 14150 . . . . . . . 8 ((𝑍 ⊆ ℝ ∧ (𝑛𝑍𝐵):𝑍⟶ℝ* ∧ (lim sup‘(𝑛𝑍𝐵)) ∈ ℝ*) → ((lim sup‘(𝑛𝑍𝐵)) ≤ (lim sup‘(𝑛𝑍𝐵)) ↔ ∀𝑖 ∈ ℝ (lim sup‘(𝑛𝑍𝐵)) ≤ (𝐻𝑖)))
19612, 49, 50, 195syl3anc 1323 . . . . . . 7 ((𝜑𝑥𝐴) → ((lim sup‘(𝑛𝑍𝐵)) ≤ (lim sup‘(𝑛𝑍𝐵)) ↔ ∀𝑖 ∈ ℝ (lim sup‘(𝑛𝑍𝐵)) ≤ (𝐻𝑖)))
19747, 196mpbid 222 . . . . . 6 ((𝜑𝑥𝐴) → ∀𝑖 ∈ ℝ (lim sup‘(𝑛𝑍𝐵)) ≤ (𝐻𝑖))
198 ssralv 3650 . . . . . 6 (𝑍 ⊆ ℝ → (∀𝑖 ∈ ℝ (lim sup‘(𝑛𝑍𝐵)) ≤ (𝐻𝑖) → ∀𝑖𝑍 (lim sup‘(𝑛𝑍𝐵)) ≤ (𝐻𝑖)))
19911, 197, 198mpsyl 68 . . . . 5 ((𝜑𝑥𝐴) → ∀𝑖𝑍 (lim sup‘(𝑛𝑍𝐵)) ≤ (𝐻𝑖))
200165breq2d 4630 . . . . . 6 (((𝜑𝑥𝐴) ∧ 𝑖𝑍) → ((lim sup‘(𝑛𝑍𝐵)) ≤ (𝐻𝑖) ↔ (lim sup‘(𝑛𝑍𝐵)) ≤ sup(ran (𝑛 ∈ (ℤ𝑖) ↦ 𝐵), ℝ, < )))
201200ralbidva 2980 . . . . 5 ((𝜑𝑥𝐴) → (∀𝑖𝑍 (lim sup‘(𝑛𝑍𝐵)) ≤ (𝐻𝑖) ↔ ∀𝑖𝑍 (lim sup‘(𝑛𝑍𝐵)) ≤ sup(ran (𝑛 ∈ (ℤ𝑖) ↦ 𝐵), ℝ, < )))
202199, 201mpbid 222 . . . 4 ((𝜑𝑥𝐴) → ∀𝑖𝑍 (lim sup‘(𝑛𝑍𝐵)) ≤ sup(ran (𝑛 ∈ (ℤ𝑖) ↦ 𝐵), ℝ, < ))
203 breq1 4621 . . . . . 6 (𝑦 = (lim sup‘(𝑛𝑍𝐵)) → (𝑦 ≤ sup(ran (𝑛 ∈ (ℤ𝑖) ↦ 𝐵), ℝ, < ) ↔ (lim sup‘(𝑛𝑍𝐵)) ≤ sup(ran (𝑛 ∈ (ℤ𝑖) ↦ 𝐵), ℝ, < )))
204203ralbidv 2981 . . . . 5 (𝑦 = (lim sup‘(𝑛𝑍𝐵)) → (∀𝑖𝑍 𝑦 ≤ sup(ran (𝑛 ∈ (ℤ𝑖) ↦ 𝐵), ℝ, < ) ↔ ∀𝑖𝑍 (lim sup‘(𝑛𝑍𝐵)) ≤ sup(ran (𝑛 ∈ (ℤ𝑖) ↦ 𝐵), ℝ, < )))
205204rspcev 3298 . . . 4 (((lim sup‘(𝑛𝑍𝐵)) ∈ ℝ ∧ ∀𝑖𝑍 (lim sup‘(𝑛𝑍𝐵)) ≤ sup(ran (𝑛 ∈ (ℤ𝑖) ↦ 𝐵), ℝ, < )) → ∃𝑦 ∈ ℝ ∀𝑖𝑍 𝑦 ≤ sup(ran (𝑛 ∈ (ℤ𝑖) ↦ 𝐵), ℝ, < ))
20624, 202, 205syl2anc 692 . . 3 ((𝜑𝑥𝐴) → ∃𝑦 ∈ ℝ ∀𝑖𝑍 𝑦 ≤ sup(ran (𝑛 ∈ (ℤ𝑖) ↦ 𝐵), ℝ, < ))
2073, 174, 13, 192, 194, 206mbfinf 23351 . 2 (𝜑 → (𝑥𝐴 ↦ inf(ran (𝑖𝑍 ↦ sup(ran (𝑛 ∈ (ℤ𝑖) ↦ 𝐵), ℝ, < )), ℝ, < )) ∈ MblFn)
208173, 207eqeltrd 2698 1 (𝜑𝐺 ∈ MblFn)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wne 2790  wral 2907  wrex 2908  Vcvv 3189  cin 3558  wss 3559  c0 3896   class class class wbr 4618  cmpt 4678  dom cdm 5079  ran crn 5080  cres 5081  cima 5082   Fn wfn 5847  wf 5848  cfv 5852  (class class class)co 6610  supcsup 8297  infcinf 8298  cr 9886  +∞cpnf 10022  *cxr 10024   < clt 10025  cle 10026  cz 11328  cuz 11638  [,)cico 12126  lim supclsp 14142  MblFncmbf 23302
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-inf2 8489  ax-cc 9208  ax-cnex 9943  ax-resscn 9944  ax-1cn 9945  ax-icn 9946  ax-addcl 9947  ax-addrcl 9948  ax-mulcl 9949  ax-mulrcl 9950  ax-mulcom 9951  ax-addass 9952  ax-mulass 9953  ax-distr 9954  ax-i2m1 9955  ax-1ne0 9956  ax-1rid 9957  ax-rnegex 9958  ax-rrecex 9959  ax-cnre 9960  ax-pre-lttri 9961  ax-pre-lttrn 9962  ax-pre-ltadd 9963  ax-pre-mulgt0 9964  ax-pre-sup 9965
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-disj 4589  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-se 5039  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-isom 5861  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-of 6857  df-om 7020  df-1st 7120  df-2nd 7121  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-1o 7512  df-2o 7513  df-oadd 7516  df-omul 7517  df-er 7694  df-map 7811  df-pm 7812  df-en 7907  df-dom 7908  df-sdom 7909  df-fin 7910  df-sup 8299  df-inf 8300  df-oi 8366  df-card 8716  df-acn 8719  df-cda 8941  df-pnf 10027  df-mnf 10028  df-xr 10029  df-ltxr 10030  df-le 10031  df-sub 10219  df-neg 10220  df-div 10636  df-nn 10972  df-2 11030  df-3 11031  df-n0 11244  df-z 11329  df-uz 11639  df-q 11740  df-rp 11784  df-xadd 11898  df-ioo 12128  df-ioc 12129  df-ico 12130  df-icc 12131  df-fz 12276  df-fzo 12414  df-fl 12540  df-seq 12749  df-exp 12808  df-hash 13065  df-cj 13780  df-re 13781  df-im 13782  df-sqrt 13916  df-abs 13917  df-limsup 14143  df-clim 14160  df-rlim 14161  df-sum 14358  df-xmet 19667  df-met 19668  df-ovol 23152  df-vol 23153  df-mbf 23307
This theorem is referenced by:  mbflimlem  23353
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