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Theorem smfliminfmpt 45063
Description: The inferior limit of a countable set of sigma-measurable functions is sigma-measurable. Proposition 121F (e) 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, 2-Jan-2022.)
Hypotheses
Ref Expression
smfliminfmpt.p 𝑚𝜑
smfliminfmpt.x 𝑥𝜑
smfliminfmpt.n 𝑛𝜑
smfliminfmpt.m (𝜑𝑀 ∈ ℤ)
smfliminfmpt.z 𝑍 = (ℤ𝑀)
smfliminfmpt.s (𝜑𝑆 ∈ SAlg)
smfliminfmpt.b ((𝜑𝑚𝑍𝑥𝐴) → 𝐵𝑉)
smfliminfmpt.f ((𝜑𝑚𝑍) → (𝑥𝐴𝐵) ∈ (SMblFn‘𝑆))
smfliminfmpt.d 𝐷 = {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴 ∣ (lim inf‘(𝑚𝑍𝐵)) ∈ ℝ}
smfliminfmpt.g 𝐺 = (𝑥𝐷 ↦ (lim inf‘(𝑚𝑍𝐵)))
Assertion
Ref Expression
smfliminfmpt (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)   𝑉(𝑥,𝑚,𝑛)

Proof of Theorem smfliminfmpt
StepHypRef Expression
1 smfliminfmpt.g . . . 4 𝐺 = (𝑥𝐷 ↦ (lim inf‘(𝑚𝑍𝐵)))
21a1i 11 . . 3 (𝜑𝐺 = (𝑥𝐷 ↦ (lim inf‘(𝑚𝑍𝐵))))
3 smfliminfmpt.x . . . 4 𝑥𝜑
4 smfliminfmpt.d . . . . . 6 𝐷 = {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴 ∣ (lim inf‘(𝑚𝑍𝐵)) ∈ ℝ}
54a1i 11 . . . . 5 (𝜑𝐷 = {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴 ∣ (lim inf‘(𝑚𝑍𝐵)) ∈ ℝ})
6 simpr 485 . . . . . . . . . 10 ((𝜑𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴) → 𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴)
7 smfliminfmpt.n . . . . . . . . . . . 12 𝑛𝜑
8 smfliminfmpt.p . . . . . . . . . . . . . 14 𝑚𝜑
9 nfv 1917 . . . . . . . . . . . . . 14 𝑚 𝑛𝑍
108, 9nfan 1902 . . . . . . . . . . . . 13 𝑚(𝜑𝑛𝑍)
11 simpll 765 . . . . . . . . . . . . . 14 (((𝜑𝑛𝑍) ∧ 𝑚 ∈ (ℤ𝑛)) → 𝜑)
12 smfliminfmpt.z . . . . . . . . . . . . . . . 16 𝑍 = (ℤ𝑀)
1312uztrn2 12782 . . . . . . . . . . . . . . 15 ((𝑛𝑍𝑚 ∈ (ℤ𝑛)) → 𝑚𝑍)
1413adantll 712 . . . . . . . . . . . . . 14 (((𝜑𝑛𝑍) ∧ 𝑚 ∈ (ℤ𝑛)) → 𝑚𝑍)
15 simpr 485 . . . . . . . . . . . . . . . . 17 ((𝜑𝑚𝑍) → 𝑚𝑍)
16 smfliminfmpt.f . . . . . . . . . . . . . . . . . 18 ((𝜑𝑚𝑍) → (𝑥𝐴𝐵) ∈ (SMblFn‘𝑆))
1716elexd 3465 . . . . . . . . . . . . . . . . 17 ((𝜑𝑚𝑍) → (𝑥𝐴𝐵) ∈ V)
18 eqid 2736 . . . . . . . . . . . . . . . . . 18 (𝑚𝑍 ↦ (𝑥𝐴𝐵)) = (𝑚𝑍 ↦ (𝑥𝐴𝐵))
1918fvmpt2 6959 . . . . . . . . . . . . . . . . 17 ((𝑚𝑍 ∧ (𝑥𝐴𝐵) ∈ V) → ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚) = (𝑥𝐴𝐵))
2015, 17, 19syl2anc 584 . . . . . . . . . . . . . . . 16 ((𝜑𝑚𝑍) → ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚) = (𝑥𝐴𝐵))
2120dmeqd 5861 . . . . . . . . . . . . . . 15 ((𝜑𝑚𝑍) → dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚) = dom (𝑥𝐴𝐵))
22 nfv 1917 . . . . . . . . . . . . . . . . 17 𝑥 𝑚𝑍
233, 22nfan 1902 . . . . . . . . . . . . . . . 16 𝑥(𝜑𝑚𝑍)
24 eqid 2736 . . . . . . . . . . . . . . . 16 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
25 smfliminfmpt.b . . . . . . . . . . . . . . . . 17 ((𝜑𝑚𝑍𝑥𝐴) → 𝐵𝑉)
26253expa 1118 . . . . . . . . . . . . . . . 16 (((𝜑𝑚𝑍) ∧ 𝑥𝐴) → 𝐵𝑉)
2723, 24, 26dmmptdf 43435 . . . . . . . . . . . . . . 15 ((𝜑𝑚𝑍) → dom (𝑥𝐴𝐵) = 𝐴)
2821, 27eqtr2d 2777 . . . . . . . . . . . . . 14 ((𝜑𝑚𝑍) → 𝐴 = dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚))
2911, 14, 28syl2anc 584 . . . . . . . . . . . . 13 (((𝜑𝑛𝑍) ∧ 𝑚 ∈ (ℤ𝑛)) → 𝐴 = dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚))
3010, 29iineq2d 4977 . . . . . . . . . . . 12 ((𝜑𝑛𝑍) → 𝑚 ∈ (ℤ𝑛)𝐴 = 𝑚 ∈ (ℤ𝑛)dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚))
317, 30iuneq2df 43244 . . . . . . . . . . 11 (𝜑 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴 = 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚))
3231adantr 481 . . . . . . . . . 10 ((𝜑𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴) → 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴 = 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚))
336, 32eleqtrd 2840 . . . . . . . . 9 ((𝜑𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴) → 𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚))
3433adantrr 715 . . . . . . . 8 ((𝜑 ∧ (𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴 ∧ (lim inf‘(𝑚𝑍𝐵)) ∈ ℝ)) → 𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚))
35 eliun 4958 . . . . . . . . . . . . 13 (𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴 ↔ ∃𝑛𝑍 𝑥 𝑚 ∈ (ℤ𝑛)𝐴)
3635biimpi 215 . . . . . . . . . . . 12 (𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴 → ∃𝑛𝑍 𝑥 𝑚 ∈ (ℤ𝑛)𝐴)
3736adantl 482 . . . . . . . . . . 11 ((𝜑𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴) → ∃𝑛𝑍 𝑥 𝑚 ∈ (ℤ𝑛)𝐴)
38 nfv 1917 . . . . . . . . . . . . 13 𝑛(lim inf‘(𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥))) = (lim inf‘(𝑚𝑍𝐵))
39 nfcv 2907 . . . . . . . . . . . . . . . . . . 19 𝑚𝑥
40 nfii1 4989 . . . . . . . . . . . . . . . . . . 19 𝑚 𝑚 ∈ (ℤ𝑛)𝐴
4139, 40nfel 2921 . . . . . . . . . . . . . . . . . 18 𝑚 𝑥 𝑚 ∈ (ℤ𝑛)𝐴
428, 9, 41nf3an 1904 . . . . . . . . . . . . . . . . 17 𝑚(𝜑𝑛𝑍𝑥 𝑚 ∈ (ℤ𝑛)𝐴)
4320fveq1d 6844 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑚𝑍) → (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥) = ((𝑥𝐴𝐵)‘𝑥))
4411, 14, 43syl2anc 584 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑛𝑍) ∧ 𝑚 ∈ (ℤ𝑛)) → (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥) = ((𝑥𝐴𝐵)‘𝑥))
45443adantl3 1168 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑛𝑍𝑥 𝑚 ∈ (ℤ𝑛)𝐴) ∧ 𝑚 ∈ (ℤ𝑛)) → (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥) = ((𝑥𝐴𝐵)‘𝑥))
46 eliinid 43311 . . . . . . . . . . . . . . . . . . . 20 ((𝑥 𝑚 ∈ (ℤ𝑛)𝐴𝑚 ∈ (ℤ𝑛)) → 𝑥𝐴)
47463ad2antl3 1187 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑛𝑍𝑥 𝑚 ∈ (ℤ𝑛)𝐴) ∧ 𝑚 ∈ (ℤ𝑛)) → 𝑥𝐴)
48 simpl1 1191 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑛𝑍𝑥 𝑚 ∈ (ℤ𝑛)𝐴) ∧ 𝑚 ∈ (ℤ𝑛)) → 𝜑)
49 simp2 1137 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑛𝑍𝑥 𝑚 ∈ (ℤ𝑛)𝐴) → 𝑛𝑍)
5049, 13sylan 580 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑛𝑍𝑥 𝑚 ∈ (ℤ𝑛)𝐴) ∧ 𝑚 ∈ (ℤ𝑛)) → 𝑚𝑍)
5148, 50, 47, 25syl3anc 1371 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑛𝑍𝑥 𝑚 ∈ (ℤ𝑛)𝐴) ∧ 𝑚 ∈ (ℤ𝑛)) → 𝐵𝑉)
5224fvmpt2 6959 . . . . . . . . . . . . . . . . . . 19 ((𝑥𝐴𝐵𝑉) → ((𝑥𝐴𝐵)‘𝑥) = 𝐵)
5347, 51, 52syl2anc 584 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑛𝑍𝑥 𝑚 ∈ (ℤ𝑛)𝐴) ∧ 𝑚 ∈ (ℤ𝑛)) → ((𝑥𝐴𝐵)‘𝑥) = 𝐵)
5445, 53eqtrd 2776 . . . . . . . . . . . . . . . . 17 (((𝜑𝑛𝑍𝑥 𝑚 ∈ (ℤ𝑛)𝐴) ∧ 𝑚 ∈ (ℤ𝑛)) → (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥) = 𝐵)
5542, 54mpteq2da 5203 . . . . . . . . . . . . . . . 16 ((𝜑𝑛𝑍𝑥 𝑚 ∈ (ℤ𝑛)𝐴) → (𝑚 ∈ (ℤ𝑛) ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥)) = (𝑚 ∈ (ℤ𝑛) ↦ 𝐵))
5655fveq2d 6846 . . . . . . . . . . . . . . 15 ((𝜑𝑛𝑍𝑥 𝑚 ∈ (ℤ𝑛)𝐴) → (lim inf‘(𝑚 ∈ (ℤ𝑛) ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥))) = (lim inf‘(𝑚 ∈ (ℤ𝑛) ↦ 𝐵)))
57 nfcv 2907 . . . . . . . . . . . . . . . 16 𝑚𝑍
58 nfcv 2907 . . . . . . . . . . . . . . . 16 𝑚(ℤ𝑛)
59 eqid 2736 . . . . . . . . . . . . . . . 16 (ℤ𝑛) = (ℤ𝑛)
6012eluzelz2 43628 . . . . . . . . . . . . . . . . . 18 (𝑛𝑍𝑛 ∈ ℤ)
6160uzidd 12779 . . . . . . . . . . . . . . . . 17 (𝑛𝑍𝑛 ∈ (ℤ𝑛))
62613ad2ant2 1134 . . . . . . . . . . . . . . . 16 ((𝜑𝑛𝑍𝑥 𝑚 ∈ (ℤ𝑛)𝐴) → 𝑛 ∈ (ℤ𝑛))
63 fvexd 6857 . . . . . . . . . . . . . . . 16 (((𝜑𝑛𝑍𝑥 𝑚 ∈ (ℤ𝑛)𝐴) ∧ 𝑚 ∈ (ℤ𝑛)) → (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥) ∈ V)
6442, 57, 58, 12, 59, 49, 62, 63liminfequzmpt2 44022 . . . . . . . . . . . . . . 15 ((𝜑𝑛𝑍𝑥 𝑚 ∈ (ℤ𝑛)𝐴) → (lim inf‘(𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥))) = (lim inf‘(𝑚 ∈ (ℤ𝑛) ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥))))
6542, 57, 58, 12, 59, 49, 62, 51liminfequzmpt2 44022 . . . . . . . . . . . . . . 15 ((𝜑𝑛𝑍𝑥 𝑚 ∈ (ℤ𝑛)𝐴) → (lim inf‘(𝑚𝑍𝐵)) = (lim inf‘(𝑚 ∈ (ℤ𝑛) ↦ 𝐵)))
6656, 64, 653eqtr4d 2786 . . . . . . . . . . . . . 14 ((𝜑𝑛𝑍𝑥 𝑚 ∈ (ℤ𝑛)𝐴) → (lim inf‘(𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥))) = (lim inf‘(𝑚𝑍𝐵)))
67663exp 1119 . . . . . . . . . . . . 13 (𝜑 → (𝑛𝑍 → (𝑥 𝑚 ∈ (ℤ𝑛)𝐴 → (lim inf‘(𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥))) = (lim inf‘(𝑚𝑍𝐵)))))
687, 38, 67rexlimd 3249 . . . . . . . . . . . 12 (𝜑 → (∃𝑛𝑍 𝑥 𝑚 ∈ (ℤ𝑛)𝐴 → (lim inf‘(𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥))) = (lim inf‘(𝑚𝑍𝐵))))
6968adantr 481 . . . . . . . . . . 11 ((𝜑𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴) → (∃𝑛𝑍 𝑥 𝑚 ∈ (ℤ𝑛)𝐴 → (lim inf‘(𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥))) = (lim inf‘(𝑚𝑍𝐵))))
7037, 69mpd 15 . . . . . . . . . 10 ((𝜑𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴) → (lim inf‘(𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥))) = (lim inf‘(𝑚𝑍𝐵)))
7170adantrr 715 . . . . . . . . 9 ((𝜑 ∧ (𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴 ∧ (lim inf‘(𝑚𝑍𝐵)) ∈ ℝ)) → (lim inf‘(𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥))) = (lim inf‘(𝑚𝑍𝐵)))
72 simprr 771 . . . . . . . . 9 ((𝜑 ∧ (𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴 ∧ (lim inf‘(𝑚𝑍𝐵)) ∈ ℝ)) → (lim inf‘(𝑚𝑍𝐵)) ∈ ℝ)
7371, 72eqeltrd 2838 . . . . . . . 8 ((𝜑 ∧ (𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴 ∧ (lim inf‘(𝑚𝑍𝐵)) ∈ ℝ)) → (lim inf‘(𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥))) ∈ ℝ)
7434, 73jca 512 . . . . . . 7 ((𝜑 ∧ (𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴 ∧ (lim inf‘(𝑚𝑍𝐵)) ∈ ℝ)) → (𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚) ∧ (lim inf‘(𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥))) ∈ ℝ))
75 simpl 483 . . . . . . . 8 ((𝜑 ∧ (𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚) ∧ (lim inf‘(𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥))) ∈ ℝ)) → 𝜑)
76 simpr 485 . . . . . . . . . 10 ((𝜑𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)) → 𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚))
7731eqcomd 2742 . . . . . . . . . . 11 (𝜑 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚) = 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴)
7877adantr 481 . . . . . . . . . 10 ((𝜑𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)) → 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚) = 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴)
7976, 78eleqtrd 2840 . . . . . . . . 9 ((𝜑𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)) → 𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴)
8079adantrr 715 . . . . . . . 8 ((𝜑 ∧ (𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚) ∧ (lim inf‘(𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥))) ∈ ℝ)) → 𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴)
81 simprr 771 . . . . . . . 8 ((𝜑 ∧ (𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚) ∧ (lim inf‘(𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥))) ∈ ℝ)) → (lim inf‘(𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥))) ∈ ℝ)
82 simp2 1137 . . . . . . . . 9 ((𝜑𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴 ∧ (lim inf‘(𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥))) ∈ ℝ) → 𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴)
8370eqcomd 2742 . . . . . . . . . . 11 ((𝜑𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴) → (lim inf‘(𝑚𝑍𝐵)) = (lim inf‘(𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥))))
84833adant3 1132 . . . . . . . . . 10 ((𝜑𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴 ∧ (lim inf‘(𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥))) ∈ ℝ) → (lim inf‘(𝑚𝑍𝐵)) = (lim inf‘(𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥))))
85 simp3 1138 . . . . . . . . . 10 ((𝜑𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴 ∧ (lim inf‘(𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥))) ∈ ℝ) → (lim inf‘(𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥))) ∈ ℝ)
8684, 85eqeltrd 2838 . . . . . . . . 9 ((𝜑𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴 ∧ (lim inf‘(𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥))) ∈ ℝ) → (lim inf‘(𝑚𝑍𝐵)) ∈ ℝ)
8782, 86jca 512 . . . . . . . 8 ((𝜑𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴 ∧ (lim inf‘(𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥))) ∈ ℝ) → (𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴 ∧ (lim inf‘(𝑚𝑍𝐵)) ∈ ℝ))
8875, 80, 81, 87syl3anc 1371 . . . . . . 7 ((𝜑 ∧ (𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚) ∧ (lim inf‘(𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥))) ∈ ℝ)) → (𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴 ∧ (lim inf‘(𝑚𝑍𝐵)) ∈ ℝ))
8974, 88impbida 799 . . . . . 6 (𝜑 → ((𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴 ∧ (lim inf‘(𝑚𝑍𝐵)) ∈ ℝ) ↔ (𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚) ∧ (lim inf‘(𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥))) ∈ ℝ)))
903, 89rabbida3 43335 . . . . 5 (𝜑 → {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴 ∣ (lim inf‘(𝑚𝑍𝐵)) ∈ ℝ} = {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚) ∣ (lim inf‘(𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥))) ∈ ℝ})
915, 90eqtrd 2776 . . . 4 (𝜑𝐷 = {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚) ∣ (lim inf‘(𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥))) ∈ ℝ})
924eleq2i 2829 . . . . . . 7 (𝑥𝐷𝑥 ∈ {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴 ∣ (lim inf‘(𝑚𝑍𝐵)) ∈ ℝ})
9392biimpi 215 . . . . . 6 (𝑥𝐷𝑥 ∈ {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴 ∣ (lim inf‘(𝑚𝑍𝐵)) ∈ ℝ})
94 rabidim1 3428 . . . . . 6 (𝑥 ∈ {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴 ∣ (lim inf‘(𝑚𝑍𝐵)) ∈ ℝ} → 𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴)
9593, 94syl 17 . . . . 5 (𝑥𝐷𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴)
9695, 83sylan2 593 . . . 4 ((𝜑𝑥𝐷) → (lim inf‘(𝑚𝑍𝐵)) = (lim inf‘(𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥))))
973, 91, 96mpteq12da 5190 . . 3 (𝜑 → (𝑥𝐷 ↦ (lim inf‘(𝑚𝑍𝐵))) = (𝑥 ∈ {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚) ∣ (lim inf‘(𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥))) ∈ ℝ} ↦ (lim inf‘(𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥)))))
982, 97eqtrd 2776 . 2 (𝜑𝐺 = (𝑥 ∈ {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚) ∣ (lim inf‘(𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥))) ∈ ℝ} ↦ (lim inf‘(𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥)))))
99 nfmpt1 5213 . . 3 𝑚(𝑚𝑍 ↦ (𝑥𝐴𝐵))
100 nfcv 2907 . . . 4 𝑥𝑍
101 nfmpt1 5213 . . . 4 𝑥(𝑥𝐴𝐵)
102100, 101nfmpt 5212 . . 3 𝑥(𝑚𝑍 ↦ (𝑥𝐴𝐵))
103 smfliminfmpt.m . . 3 (𝜑𝑀 ∈ ℤ)
104 smfliminfmpt.s . . 3 (𝜑𝑆 ∈ SAlg)
1058, 16fmptd2f 43450 . . 3 (𝜑 → (𝑚𝑍 ↦ (𝑥𝐴𝐵)):𝑍⟶(SMblFn‘𝑆))
106 eqid 2736 . . 3 {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚) ∣ (lim inf‘(𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥))) ∈ ℝ} = {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚) ∣ (lim inf‘(𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥))) ∈ ℝ}
107 eqid 2736 . . 3 (𝑥 ∈ {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚) ∣ (lim inf‘(𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥))) ∈ ℝ} ↦ (lim inf‘(𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥)))) = (𝑥 ∈ {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚) ∣ (lim inf‘(𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥))) ∈ ℝ} ↦ (lim inf‘(𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥))))
10899, 102, 103, 12, 104, 105, 106, 107smfliminf 45062 . 2 (𝜑 → (𝑥 ∈ {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚) ∣ (lim inf‘(𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥))) ∈ ℝ} ↦ (lim inf‘(𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥)))) ∈ (SMblFn‘𝑆))
10998, 108eqeltrd 2838 1 (𝜑𝐺 ∈ (SMblFn‘𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1087   = wceq 1541  wnf 1785  wcel 2106  wrex 3073  {crab 3407  Vcvv 3445   ciun 4954   ciin 4955  cmpt 5188  dom cdm 5633  cfv 6496  cr 11050  cz 12499  cuz 12763  lim infclsi 43982  SAlgcsalg 44539  SMblFncsmblfn 44926
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 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-inf2 9577  ax-cc 10371  ax-ac2 10399  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129
This theorem depends on definitions:  df-bi 206  df-an 397  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 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-iin 4957  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-oadd 8416  df-omul 8417  df-er 8648  df-map 8767  df-pm 8768  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-sup 9378  df-inf 9379  df-oi 9446  df-card 9875  df-acn 9878  df-ac 10052  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-3 12217  df-4 12218  df-n0 12414  df-z 12500  df-uz 12764  df-q 12874  df-rp 12916  df-xneg 13033  df-ioo 13268  df-ioc 13269  df-ico 13270  df-icc 13271  df-fz 13425  df-fzo 13568  df-fl 13697  df-ceil 13698  df-seq 13907  df-exp 13968  df-hash 14231  df-word 14403  df-concat 14459  df-s1 14484  df-s2 14737  df-s3 14738  df-s4 14739  df-cj 14984  df-re 14985  df-im 14986  df-sqrt 15120  df-abs 15121  df-limsup 15353  df-clim 15370  df-rlim 15371  df-rest 17304  df-topgen 17325  df-top 22243  df-bases 22296  df-liminf 43983  df-salg 44540  df-salgen 44544  df-smblfn 44927
This theorem is referenced by: (None)
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