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Theorem smflimsupmpt 46784
Description: The superior limit of a sequence of sigma-measurable functions is sigma-measurable. Proposition 121F (d) of [Fremlin1] p. 39 . 𝐴 can contain 𝑚 as a free variable, in other words it can be thought of as an indexed collection 𝐴(𝑚). 𝐵 can be thought of as a collection with two indices 𝐵(𝑚, 𝑥). (Contributed by Glauco Siliprandi, 23-Oct-2021.)
Hypotheses
Ref Expression
smflimsupmpt.p 𝑚𝜑
smflimsupmpt.x 𝑥𝜑
smflimsupmpt.n 𝑛𝜑
smflimsupmpt.m (𝜑𝑀 ∈ ℤ)
smflimsupmpt.z 𝑍 = (ℤ𝑀)
smflimsupmpt.s (𝜑𝑆 ∈ SAlg)
smflimsupmpt.b ((𝜑𝑚𝑍𝑥𝐴) → 𝐵𝑊)
smflimsupmpt.f ((𝜑𝑚𝑍) → (𝑥𝐴𝐵) ∈ (SMblFn‘𝑆))
smflimsupmpt.d 𝐷 = {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴 ∣ (lim sup‘(𝑚𝑍𝐵)) ∈ ℝ}
smflimsupmpt.g 𝐺 = (𝑥𝐷 ↦ (lim sup‘(𝑚𝑍𝐵)))
Assertion
Ref Expression
smflimsupmpt (𝜑𝐺 ∈ (SMblFn‘𝑆))
Distinct variable groups:   𝐴,𝑛,𝑥   𝐵,𝑛   𝑚,𝑀   𝑆,𝑚   𝑚,𝑍,𝑛,𝑥
Allowed substitution hints:   𝜑(𝑥,𝑚,𝑛)   𝐴(𝑚)   𝐵(𝑥,𝑚)   𝐷(𝑥,𝑚,𝑛)   𝑆(𝑥,𝑛)   𝐺(𝑥,𝑚,𝑛)   𝑀(𝑥,𝑛)   𝑊(𝑥,𝑚,𝑛)

Proof of Theorem smflimsupmpt
StepHypRef Expression
1 smflimsupmpt.g . . . 4 𝐺 = (𝑥𝐷 ↦ (lim sup‘(𝑚𝑍𝐵)))
21a1i 11 . . 3 (𝜑𝐺 = (𝑥𝐷 ↦ (lim sup‘(𝑚𝑍𝐵))))
3 smflimsupmpt.x . . . 4 𝑥𝜑
4 smflimsupmpt.d . . . . . 6 𝐷 = {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴 ∣ (lim sup‘(𝑚𝑍𝐵)) ∈ ℝ}
54a1i 11 . . . . 5 (𝜑𝐷 = {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴 ∣ (lim sup‘(𝑚𝑍𝐵)) ∈ ℝ})
6 simpr 484 . . . . . . . . . . 11 ((𝜑𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴) → 𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴)
7 smflimsupmpt.n . . . . . . . . . . . . 13 𝑛𝜑
8 smflimsupmpt.p . . . . . . . . . . . . . . 15 𝑚𝜑
9 nfv 1911 . . . . . . . . . . . . . . 15 𝑚 𝑛𝑍
108, 9nfan 1896 . . . . . . . . . . . . . 14 𝑚(𝜑𝑛𝑍)
11 simpll 767 . . . . . . . . . . . . . . 15 (((𝜑𝑛𝑍) ∧ 𝑚 ∈ (ℤ𝑛)) → 𝜑)
12 smflimsupmpt.z . . . . . . . . . . . . . . . . 17 𝑍 = (ℤ𝑀)
1312uztrn2 12894 . . . . . . . . . . . . . . . 16 ((𝑛𝑍𝑚 ∈ (ℤ𝑛)) → 𝑚𝑍)
1413adantll 714 . . . . . . . . . . . . . . 15 (((𝜑𝑛𝑍) ∧ 𝑚 ∈ (ℤ𝑛)) → 𝑚𝑍)
15 simpr 484 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑚𝑍) → 𝑚𝑍)
16 smflimsupmpt.f . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑚𝑍) → (𝑥𝐴𝐵) ∈ (SMblFn‘𝑆))
1716elexd 3501 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑚𝑍) → (𝑥𝐴𝐵) ∈ V)
18 eqid 2734 . . . . . . . . . . . . . . . . . . 19 (𝑚𝑍 ↦ (𝑥𝐴𝐵)) = (𝑚𝑍 ↦ (𝑥𝐴𝐵))
1918fvmpt2 7026 . . . . . . . . . . . . . . . . . 18 ((𝑚𝑍 ∧ (𝑥𝐴𝐵) ∈ V) → ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚) = (𝑥𝐴𝐵))
2015, 17, 19syl2anc 584 . . . . . . . . . . . . . . . . 17 ((𝜑𝑚𝑍) → ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚) = (𝑥𝐴𝐵))
2120dmeqd 5918 . . . . . . . . . . . . . . . 16 ((𝜑𝑚𝑍) → dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚) = dom (𝑥𝐴𝐵))
22 nfv 1911 . . . . . . . . . . . . . . . . . 18 𝑥 𝑚𝑍
233, 22nfan 1896 . . . . . . . . . . . . . . . . 17 𝑥(𝜑𝑚𝑍)
24 eqid 2734 . . . . . . . . . . . . . . . . 17 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
25 smflimsupmpt.b . . . . . . . . . . . . . . . . . 18 ((𝜑𝑚𝑍𝑥𝐴) → 𝐵𝑊)
26253expa 1117 . . . . . . . . . . . . . . . . 17 (((𝜑𝑚𝑍) ∧ 𝑥𝐴) → 𝐵𝑊)
2723, 24, 26dmmptdf 45166 . . . . . . . . . . . . . . . 16 ((𝜑𝑚𝑍) → dom (𝑥𝐴𝐵) = 𝐴)
2821, 27eqtr2d 2775 . . . . . . . . . . . . . . 15 ((𝜑𝑚𝑍) → 𝐴 = dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚))
2911, 14, 28syl2anc 584 . . . . . . . . . . . . . 14 (((𝜑𝑛𝑍) ∧ 𝑚 ∈ (ℤ𝑛)) → 𝐴 = dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚))
3010, 29iineq2d 5019 . . . . . . . . . . . . 13 ((𝜑𝑛𝑍) → 𝑚 ∈ (ℤ𝑛)𝐴 = 𝑚 ∈ (ℤ𝑛)dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚))
317, 30iuneq2df 44985 . . . . . . . . . . . 12 (𝜑 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴 = 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚))
3231adantr 480 . . . . . . . . . . 11 ((𝜑𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴) → 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴 = 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚))
336, 32eleqtrd 2840 . . . . . . . . . 10 ((𝜑𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴) → 𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚))
3433adantrr 717 . . . . . . . . 9 ((𝜑 ∧ (𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴 ∧ (lim sup‘(𝑚𝑍𝐵)) ∈ ℝ)) → 𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚))
35 eliun 4999 . . . . . . . . . . . . . 14 (𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴 ↔ ∃𝑛𝑍 𝑥 𝑚 ∈ (ℤ𝑛)𝐴)
3635biimpi 216 . . . . . . . . . . . . 13 (𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴 → ∃𝑛𝑍 𝑥 𝑚 ∈ (ℤ𝑛)𝐴)
3736adantl 481 . . . . . . . . . . . 12 ((𝜑𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴) → ∃𝑛𝑍 𝑥 𝑚 ∈ (ℤ𝑛)𝐴)
38 nfv 1911 . . . . . . . . . . . . . 14 𝑛(lim sup‘(𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥))) = (lim sup‘(𝑚𝑍𝐵))
39 nfcv 2902 . . . . . . . . . . . . . . . . . . . 20 𝑚𝑥
40 nfii1 5033 . . . . . . . . . . . . . . . . . . . 20 𝑚 𝑚 ∈ (ℤ𝑛)𝐴
4139, 40nfel 2917 . . . . . . . . . . . . . . . . . . 19 𝑚 𝑥 𝑚 ∈ (ℤ𝑛)𝐴
428, 9, 41nf3an 1898 . . . . . . . . . . . . . . . . . 18 𝑚(𝜑𝑛𝑍𝑥 𝑚 ∈ (ℤ𝑛)𝐴)
4320fveq1d 6908 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑚𝑍) → (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥) = ((𝑥𝐴𝐵)‘𝑥))
4411, 14, 43syl2anc 584 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑛𝑍) ∧ 𝑚 ∈ (ℤ𝑛)) → (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥) = ((𝑥𝐴𝐵)‘𝑥))
45443adantl3 1167 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑛𝑍𝑥 𝑚 ∈ (ℤ𝑛)𝐴) ∧ 𝑚 ∈ (ℤ𝑛)) → (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥) = ((𝑥𝐴𝐵)‘𝑥))
46 eliinid 45050 . . . . . . . . . . . . . . . . . . . . 21 ((𝑥 𝑚 ∈ (ℤ𝑛)𝐴𝑚 ∈ (ℤ𝑛)) → 𝑥𝐴)
47463ad2antl3 1186 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑛𝑍𝑥 𝑚 ∈ (ℤ𝑛)𝐴) ∧ 𝑚 ∈ (ℤ𝑛)) → 𝑥𝐴)
48 simpl1 1190 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑛𝑍𝑥 𝑚 ∈ (ℤ𝑛)𝐴) ∧ 𝑚 ∈ (ℤ𝑛)) → 𝜑)
49143adantl3 1167 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑛𝑍𝑥 𝑚 ∈ (ℤ𝑛)𝐴) ∧ 𝑚 ∈ (ℤ𝑛)) → 𝑚𝑍)
5048, 49, 47, 25syl3anc 1370 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑛𝑍𝑥 𝑚 ∈ (ℤ𝑛)𝐴) ∧ 𝑚 ∈ (ℤ𝑛)) → 𝐵𝑊)
5124fvmpt2 7026 . . . . . . . . . . . . . . . . . . . 20 ((𝑥𝐴𝐵𝑊) → ((𝑥𝐴𝐵)‘𝑥) = 𝐵)
5247, 50, 51syl2anc 584 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑛𝑍𝑥 𝑚 ∈ (ℤ𝑛)𝐴) ∧ 𝑚 ∈ (ℤ𝑛)) → ((𝑥𝐴𝐵)‘𝑥) = 𝐵)
5345, 52eqtrd 2774 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑛𝑍𝑥 𝑚 ∈ (ℤ𝑛)𝐴) ∧ 𝑚 ∈ (ℤ𝑛)) → (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥) = 𝐵)
5442, 53mpteq2da 5245 . . . . . . . . . . . . . . . . 17 ((𝜑𝑛𝑍𝑥 𝑚 ∈ (ℤ𝑛)𝐴) → (𝑚 ∈ (ℤ𝑛) ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥)) = (𝑚 ∈ (ℤ𝑛) ↦ 𝐵))
5554fveq2d 6910 . . . . . . . . . . . . . . . 16 ((𝜑𝑛𝑍𝑥 𝑚 ∈ (ℤ𝑛)𝐴) → (lim sup‘(𝑚 ∈ (ℤ𝑛) ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥))) = (lim sup‘(𝑚 ∈ (ℤ𝑛) ↦ 𝐵)))
56 smflimsupmpt.m . . . . . . . . . . . . . . . . . 18 (𝜑𝑀 ∈ ℤ)
57563ad2ant1 1132 . . . . . . . . . . . . . . . . 17 ((𝜑𝑛𝑍𝑥 𝑚 ∈ (ℤ𝑛)𝐴) → 𝑀 ∈ ℤ)
5812eluzelz2 45352 . . . . . . . . . . . . . . . . . 18 (𝑛𝑍𝑛 ∈ ℤ)
59583ad2ant2 1133 . . . . . . . . . . . . . . . . 17 ((𝜑𝑛𝑍𝑥 𝑚 ∈ (ℤ𝑛)𝐴) → 𝑛 ∈ ℤ)
60 eqid 2734 . . . . . . . . . . . . . . . . 17 (ℤ𝑛) = (ℤ𝑛)
61 fvexd 6921 . . . . . . . . . . . . . . . . 17 (((𝜑𝑛𝑍𝑥 𝑚 ∈ (ℤ𝑛)𝐴) ∧ 𝑚𝑍) → (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥) ∈ V)
6249, 61syldan 591 . . . . . . . . . . . . . . . . 17 (((𝜑𝑛𝑍𝑥 𝑚 ∈ (ℤ𝑛)𝐴) ∧ 𝑚 ∈ (ℤ𝑛)) → (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥) ∈ V)
6342, 57, 59, 12, 60, 61, 62limsupequzmpt 45684 . . . . . . . . . . . . . . . 16 ((𝜑𝑛𝑍𝑥 𝑚 ∈ (ℤ𝑛)𝐴) → (lim sup‘(𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥))) = (lim sup‘(𝑚 ∈ (ℤ𝑛) ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥))))
649nfci 2890 . . . . . . . . . . . . . . . . 17 𝑚𝑍
65 nfcv 2902 . . . . . . . . . . . . . . . . 17 𝑚(ℤ𝑛)
66 simp2 1136 . . . . . . . . . . . . . . . . 17 ((𝜑𝑛𝑍𝑥 𝑚 ∈ (ℤ𝑛)𝐴) → 𝑛𝑍)
6759uzidd 12891 . . . . . . . . . . . . . . . . 17 ((𝜑𝑛𝑍𝑥 𝑚 ∈ (ℤ𝑛)𝐴) → 𝑛 ∈ (ℤ𝑛))
6842, 64, 65, 12, 60, 66, 67, 50limsupequzmpt2 45673 . . . . . . . . . . . . . . . 16 ((𝜑𝑛𝑍𝑥 𝑚 ∈ (ℤ𝑛)𝐴) → (lim sup‘(𝑚𝑍𝐵)) = (lim sup‘(𝑚 ∈ (ℤ𝑛) ↦ 𝐵)))
6955, 63, 683eqtr4d 2784 . . . . . . . . . . . . . . 15 ((𝜑𝑛𝑍𝑥 𝑚 ∈ (ℤ𝑛)𝐴) → (lim sup‘(𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥))) = (lim sup‘(𝑚𝑍𝐵)))
70693exp 1118 . . . . . . . . . . . . . 14 (𝜑 → (𝑛𝑍 → (𝑥 𝑚 ∈ (ℤ𝑛)𝐴 → (lim sup‘(𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥))) = (lim sup‘(𝑚𝑍𝐵)))))
717, 38, 70rexlimd 3263 . . . . . . . . . . . . 13 (𝜑 → (∃𝑛𝑍 𝑥 𝑚 ∈ (ℤ𝑛)𝐴 → (lim sup‘(𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥))) = (lim sup‘(𝑚𝑍𝐵))))
7271adantr 480 . . . . . . . . . . . 12 ((𝜑𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴) → (∃𝑛𝑍 𝑥 𝑚 ∈ (ℤ𝑛)𝐴 → (lim sup‘(𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥))) = (lim sup‘(𝑚𝑍𝐵))))
7337, 72mpd 15 . . . . . . . . . . 11 ((𝜑𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴) → (lim sup‘(𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥))) = (lim sup‘(𝑚𝑍𝐵)))
7473adantrr 717 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴 ∧ (lim sup‘(𝑚𝑍𝐵)) ∈ ℝ)) → (lim sup‘(𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥))) = (lim sup‘(𝑚𝑍𝐵)))
75 simprr 773 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴 ∧ (lim sup‘(𝑚𝑍𝐵)) ∈ ℝ)) → (lim sup‘(𝑚𝑍𝐵)) ∈ ℝ)
7674, 75eqeltrd 2838 . . . . . . . . 9 ((𝜑 ∧ (𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴 ∧ (lim sup‘(𝑚𝑍𝐵)) ∈ ℝ)) → (lim sup‘(𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥))) ∈ ℝ)
7734, 76jca 511 . . . . . . . 8 ((𝜑 ∧ (𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴 ∧ (lim sup‘(𝑚𝑍𝐵)) ∈ ℝ)) → (𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚) ∧ (lim sup‘(𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥))) ∈ ℝ))
7877ex 412 . . . . . . 7 (𝜑 → ((𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴 ∧ (lim sup‘(𝑚𝑍𝐵)) ∈ ℝ) → (𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚) ∧ (lim sup‘(𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥))) ∈ ℝ)))
79 simpl 482 . . . . . . . . 9 ((𝜑 ∧ (𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚) ∧ (lim sup‘(𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥))) ∈ ℝ)) → 𝜑)
80 simpr 484 . . . . . . . . . . 11 ((𝜑𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)) → 𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚))
8131eqcomd 2740 . . . . . . . . . . . 12 (𝜑 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚) = 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴)
8281adantr 480 . . . . . . . . . . 11 ((𝜑𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)) → 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚) = 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴)
8380, 82eleqtrd 2840 . . . . . . . . . 10 ((𝜑𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)) → 𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴)
8483adantrr 717 . . . . . . . . 9 ((𝜑 ∧ (𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚) ∧ (lim sup‘(𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥))) ∈ ℝ)) → 𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴)
85 simprr 773 . . . . . . . . 9 ((𝜑 ∧ (𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚) ∧ (lim sup‘(𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥))) ∈ ℝ)) → (lim sup‘(𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥))) ∈ ℝ)
86 simp2 1136 . . . . . . . . . 10 ((𝜑𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴 ∧ (lim sup‘(𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥))) ∈ ℝ) → 𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴)
8773eqcomd 2740 . . . . . . . . . . . 12 ((𝜑𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴) → (lim sup‘(𝑚𝑍𝐵)) = (lim sup‘(𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥))))
88873adant3 1131 . . . . . . . . . . 11 ((𝜑𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴 ∧ (lim sup‘(𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥))) ∈ ℝ) → (lim sup‘(𝑚𝑍𝐵)) = (lim sup‘(𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥))))
89 simp3 1137 . . . . . . . . . . 11 ((𝜑𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴 ∧ (lim sup‘(𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥))) ∈ ℝ) → (lim sup‘(𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥))) ∈ ℝ)
9088, 89eqeltrd 2838 . . . . . . . . . 10 ((𝜑𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴 ∧ (lim sup‘(𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥))) ∈ ℝ) → (lim sup‘(𝑚𝑍𝐵)) ∈ ℝ)
9186, 90jca 511 . . . . . . . . 9 ((𝜑𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴 ∧ (lim sup‘(𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥))) ∈ ℝ) → (𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴 ∧ (lim sup‘(𝑚𝑍𝐵)) ∈ ℝ))
9279, 84, 85, 91syl3anc 1370 . . . . . . . 8 ((𝜑 ∧ (𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚) ∧ (lim sup‘(𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥))) ∈ ℝ)) → (𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴 ∧ (lim sup‘(𝑚𝑍𝐵)) ∈ ℝ))
9392ex 412 . . . . . . 7 (𝜑 → ((𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚) ∧ (lim sup‘(𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥))) ∈ ℝ) → (𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴 ∧ (lim sup‘(𝑚𝑍𝐵)) ∈ ℝ)))
9478, 93impbid 212 . . . . . 6 (𝜑 → ((𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴 ∧ (lim sup‘(𝑚𝑍𝐵)) ∈ ℝ) ↔ (𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚) ∧ (lim sup‘(𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥))) ∈ ℝ)))
953, 94rabbida3 45074 . . . . 5 (𝜑 → {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴 ∣ (lim sup‘(𝑚𝑍𝐵)) ∈ ℝ} = {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚) ∣ (lim sup‘(𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥))) ∈ ℝ})
965, 95eqtrd 2774 . . . 4 (𝜑𝐷 = {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚) ∣ (lim sup‘(𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥))) ∈ ℝ})
974eleq2i 2830 . . . . . . 7 (𝑥𝐷𝑥 ∈ {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴 ∣ (lim sup‘(𝑚𝑍𝐵)) ∈ ℝ})
9897biimpi 216 . . . . . 6 (𝑥𝐷𝑥 ∈ {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴 ∣ (lim sup‘(𝑚𝑍𝐵)) ∈ ℝ})
99 rabidim1 3455 . . . . . 6 (𝑥 ∈ {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴 ∣ (lim sup‘(𝑚𝑍𝐵)) ∈ ℝ} → 𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴)
10098, 99syl 17 . . . . 5 (𝑥𝐷𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴)
101100, 87sylan2 593 . . . 4 ((𝜑𝑥𝐷) → (lim sup‘(𝑚𝑍𝐵)) = (lim sup‘(𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥))))
1023, 96, 101mpteq12da 5232 . . 3 (𝜑 → (𝑥𝐷 ↦ (lim sup‘(𝑚𝑍𝐵))) = (𝑥 ∈ {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚) ∣ (lim sup‘(𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥))) ∈ ℝ} ↦ (lim sup‘(𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥)))))
1032, 102eqtrd 2774 . 2 (𝜑𝐺 = (𝑥 ∈ {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚) ∣ (lim sup‘(𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥))) ∈ ℝ} ↦ (lim sup‘(𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥)))))
104 nfmpt1 5255 . . 3 𝑚(𝑚𝑍 ↦ (𝑥𝐴𝐵))
105 nfcv 2902 . . . 4 𝑥𝑍
106 nfmpt1 5255 . . . 4 𝑥(𝑥𝐴𝐵)
107105, 106nfmpt 5254 . . 3 𝑥(𝑚𝑍 ↦ (𝑥𝐴𝐵))
108 smflimsupmpt.s . . 3 (𝜑𝑆 ∈ SAlg)
1098, 16fmptd2f 45177 . . 3 (𝜑 → (𝑚𝑍 ↦ (𝑥𝐴𝐵)):𝑍⟶(SMblFn‘𝑆))
110 eqid 2734 . . 3 {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚) ∣ (lim sup‘(𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥))) ∈ ℝ} = {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚) ∣ (lim sup‘(𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥))) ∈ ℝ}
111 eqid 2734 . . 3 (𝑥 ∈ {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚) ∣ (lim sup‘(𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥))) ∈ ℝ} ↦ (lim sup‘(𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥)))) = (𝑥 ∈ {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚) ∣ (lim sup‘(𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥))) ∈ ℝ} ↦ (lim sup‘(𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥))))
112104, 107, 56, 12, 108, 109, 110, 111smflimsup 46783 . 2 (𝜑 → (𝑥 ∈ {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚) ∣ (lim sup‘(𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥))) ∈ ℝ} ↦ (lim sup‘(𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥)))) ∈ (SMblFn‘𝑆))
113103, 112eqeltrd 2838 1 (𝜑𝐺 ∈ (SMblFn‘𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1536  wnf 1779  wcel 2105  wrex 3067  {crab 3432  Vcvv 3477   ciun 4995   ciin 4996  cmpt 5230  dom cdm 5688  cfv 6562  cr 11151  cz 12610  cuz 12875  lim supclsp 15502  SAlgcsalg 46263  SMblFncsmblfn 46650
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-inf2 9678  ax-cc 10472  ax-ac2 10500  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229  ax-pre-sup 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-tp 4635  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-iin 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-se 5641  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-isom 6571  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-1st 8012  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-2o 8505  df-oadd 8508  df-omul 8509  df-er 8743  df-map 8866  df-pm 8867  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-sup 9479  df-inf 9480  df-oi 9547  df-card 9976  df-acn 9979  df-ac 10153  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-div 11918  df-nn 12264  df-2 12326  df-3 12327  df-n0 12524  df-z 12611  df-uz 12876  df-q 12988  df-rp 13032  df-ioo 13387  df-ioc 13388  df-ico 13389  df-fz 13544  df-fl 13828  df-ceil 13829  df-seq 14039  df-exp 14099  df-cj 15134  df-re 15135  df-im 15136  df-sqrt 15270  df-abs 15271  df-limsup 15503  df-clim 15520  df-rlim 15521  df-rest 17468  df-topgen 17489  df-top 22915  df-bases 22968  df-salg 46264  df-salgen 46268  df-smblfn 46651
This theorem is referenced by:  smfliminflem  46785
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