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Theorem smfliminfmpt 41558
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 indexes 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 471 . . . . . . . . . 10 ((𝜑𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴) → 𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴)
7 smfliminfmpt.n . . . . . . . . . . . 12 𝑛𝜑
8 smfliminfmpt.p . . . . . . . . . . . . . 14 𝑚𝜑
9 nfv 1995 . . . . . . . . . . . . . 14 𝑚 𝑛𝑍
108, 9nfan 1980 . . . . . . . . . . . . 13 𝑚(𝜑𝑛𝑍)
11 simpll 750 . . . . . . . . . . . . . 14 (((𝜑𝑛𝑍) ∧ 𝑚 ∈ (ℤ𝑛)) → 𝜑)
12 smfliminfmpt.z . . . . . . . . . . . . . . . 16 𝑍 = (ℤ𝑀)
1312uztrn2 11906 . . . . . . . . . . . . . . 15 ((𝑛𝑍𝑚 ∈ (ℤ𝑛)) → 𝑚𝑍)
1413adantll 693 . . . . . . . . . . . . . 14 (((𝜑𝑛𝑍) ∧ 𝑚 ∈ (ℤ𝑛)) → 𝑚𝑍)
15 simpr 471 . . . . . . . . . . . . . . . . 17 ((𝜑𝑚𝑍) → 𝑚𝑍)
16 smfliminfmpt.f . . . . . . . . . . . . . . . . . 18 ((𝜑𝑚𝑍) → (𝑥𝐴𝐵) ∈ (SMblFn‘𝑆))
1716elexd 3366 . . . . . . . . . . . . . . . . 17 ((𝜑𝑚𝑍) → (𝑥𝐴𝐵) ∈ V)
18 eqid 2771 . . . . . . . . . . . . . . . . . 18 (𝑚𝑍 ↦ (𝑥𝐴𝐵)) = (𝑚𝑍 ↦ (𝑥𝐴𝐵))
1918fvmpt2 6433 . . . . . . . . . . . . . . . . 17 ((𝑚𝑍 ∧ (𝑥𝐴𝐵) ∈ V) → ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚) = (𝑥𝐴𝐵))
2015, 17, 19syl2anc 573 . . . . . . . . . . . . . . . 16 ((𝜑𝑚𝑍) → ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚) = (𝑥𝐴𝐵))
2120dmeqd 5464 . . . . . . . . . . . . . . 15 ((𝜑𝑚𝑍) → dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚) = dom (𝑥𝐴𝐵))
22 nfv 1995 . . . . . . . . . . . . . . . . 17 𝑥 𝑚𝑍
233, 22nfan 1980 . . . . . . . . . . . . . . . 16 𝑥(𝜑𝑚𝑍)
24 eqid 2771 . . . . . . . . . . . . . . . 16 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
25 smfliminfmpt.b . . . . . . . . . . . . . . . . 17 ((𝜑𝑚𝑍𝑥𝐴) → 𝐵𝑉)
26253expa 1111 . . . . . . . . . . . . . . . 16 (((𝜑𝑚𝑍) ∧ 𝑥𝐴) → 𝐵𝑉)
2723, 24, 26dmmptdf 39935 . . . . . . . . . . . . . . 15 ((𝜑𝑚𝑍) → dom (𝑥𝐴𝐵) = 𝐴)
2821, 27eqtr2d 2806 . . . . . . . . . . . . . 14 ((𝜑𝑚𝑍) → 𝐴 = dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚))
2911, 14, 28syl2anc 573 . . . . . . . . . . . . 13 (((𝜑𝑛𝑍) ∧ 𝑚 ∈ (ℤ𝑛)) → 𝐴 = dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚))
3010, 29iineq2d 4675 . . . . . . . . . . . 12 ((𝜑𝑛𝑍) → 𝑚 ∈ (ℤ𝑛)𝐴 = 𝑚 ∈ (ℤ𝑛)dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚))
317, 30iuneq2df 39733 . . . . . . . . . . 11 (𝜑 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴 = 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚))
3231adantr 466 . . . . . . . . . 10 ((𝜑𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴) → 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴 = 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚))
336, 32eleqtrd 2852 . . . . . . . . 9 ((𝜑𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴) → 𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚))
3433adantrr 696 . . . . . . . 8 ((𝜑 ∧ (𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴 ∧ (lim inf‘(𝑚𝑍𝐵)) ∈ ℝ)) → 𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚))
35 eliun 4658 . . . . . . . . . . . . 13 (𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴 ↔ ∃𝑛𝑍 𝑥 𝑚 ∈ (ℤ𝑛)𝐴)
3635biimpi 206 . . . . . . . . . . . 12 (𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴 → ∃𝑛𝑍 𝑥 𝑚 ∈ (ℤ𝑛)𝐴)
3736adantl 467 . . . . . . . . . . 11 ((𝜑𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴) → ∃𝑛𝑍 𝑥 𝑚 ∈ (ℤ𝑛)𝐴)
38 nfv 1995 . . . . . . . . . . . . 13 𝑛(lim inf‘(𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥))) = (lim inf‘(𝑚𝑍𝐵))
39 nfcv 2913 . . . . . . . . . . . . . . . . . . 19 𝑚𝑥
40 nfii1 4685 . . . . . . . . . . . . . . . . . . 19 𝑚 𝑚 ∈ (ℤ𝑛)𝐴
4139, 40nfel 2926 . . . . . . . . . . . . . . . . . 18 𝑚 𝑥 𝑚 ∈ (ℤ𝑛)𝐴
428, 9, 41nf3an 1983 . . . . . . . . . . . . . . . . 17 𝑚(𝜑𝑛𝑍𝑥 𝑚 ∈ (ℤ𝑛)𝐴)
4320fveq1d 6334 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑚𝑍) → (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥) = ((𝑥𝐴𝐵)‘𝑥))
4411, 14, 43syl2anc 573 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑛𝑍) ∧ 𝑚 ∈ (ℤ𝑛)) → (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥) = ((𝑥𝐴𝐵)‘𝑥))
45443adantl3 1173 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑛𝑍𝑥 𝑚 ∈ (ℤ𝑛)𝐴) ∧ 𝑚 ∈ (ℤ𝑛)) → (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥) = ((𝑥𝐴𝐵)‘𝑥))
46 eliinid 39815 . . . . . . . . . . . . . . . . . . . 20 ((𝑥 𝑚 ∈ (ℤ𝑛)𝐴𝑚 ∈ (ℤ𝑛)) → 𝑥𝐴)
47463ad2antl3 1202 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑛𝑍𝑥 𝑚 ∈ (ℤ𝑛)𝐴) ∧ 𝑚 ∈ (ℤ𝑛)) → 𝑥𝐴)
48 simpl1 1227 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑛𝑍𝑥 𝑚 ∈ (ℤ𝑛)𝐴) ∧ 𝑚 ∈ (ℤ𝑛)) → 𝜑)
49 simp2 1131 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑛𝑍𝑥 𝑚 ∈ (ℤ𝑛)𝐴) → 𝑛𝑍)
5049, 13sylan 569 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑛𝑍𝑥 𝑚 ∈ (ℤ𝑛)𝐴) ∧ 𝑚 ∈ (ℤ𝑛)) → 𝑚𝑍)
5148, 50, 47, 25syl3anc 1476 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑛𝑍𝑥 𝑚 ∈ (ℤ𝑛)𝐴) ∧ 𝑚 ∈ (ℤ𝑛)) → 𝐵𝑉)
5224fvmpt2 6433 . . . . . . . . . . . . . . . . . . 19 ((𝑥𝐴𝐵𝑉) → ((𝑥𝐴𝐵)‘𝑥) = 𝐵)
5347, 51, 52syl2anc 573 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑛𝑍𝑥 𝑚 ∈ (ℤ𝑛)𝐴) ∧ 𝑚 ∈ (ℤ𝑛)) → ((𝑥𝐴𝐵)‘𝑥) = 𝐵)
5445, 53eqtrd 2805 . . . . . . . . . . . . . . . . 17 (((𝜑𝑛𝑍𝑥 𝑚 ∈ (ℤ𝑛)𝐴) ∧ 𝑚 ∈ (ℤ𝑛)) → (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥) = 𝐵)
5542, 54mpteq2da 4877 . . . . . . . . . . . . . . . 16 ((𝜑𝑛𝑍𝑥 𝑚 ∈ (ℤ𝑛)𝐴) → (𝑚 ∈ (ℤ𝑛) ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥)) = (𝑚 ∈ (ℤ𝑛) ↦ 𝐵))
5655fveq2d 6336 . . . . . . . . . . . . . . 15 ((𝜑𝑛𝑍𝑥 𝑚 ∈ (ℤ𝑛)𝐴) → (lim inf‘(𝑚 ∈ (ℤ𝑛) ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥))) = (lim inf‘(𝑚 ∈ (ℤ𝑛) ↦ 𝐵)))
57 nfcv 2913 . . . . . . . . . . . . . . . 16 𝑚𝑍
58 nfcv 2913 . . . . . . . . . . . . . . . 16 𝑚(ℤ𝑛)
59 eqid 2771 . . . . . . . . . . . . . . . 16 (ℤ𝑛) = (ℤ𝑛)
6012eluzelz2 40143 . . . . . . . . . . . . . . . . . 18 (𝑛𝑍𝑛 ∈ ℤ)
6160uzidd 40147 . . . . . . . . . . . . . . . . 17 (𝑛𝑍𝑛 ∈ (ℤ𝑛))
62613ad2ant2 1128 . . . . . . . . . . . . . . . 16 ((𝜑𝑛𝑍𝑥 𝑚 ∈ (ℤ𝑛)𝐴) → 𝑛 ∈ (ℤ𝑛))
63 fvexd 6344 . . . . . . . . . . . . . . . 16 (((𝜑𝑛𝑍𝑥 𝑚 ∈ (ℤ𝑛)𝐴) ∧ 𝑚 ∈ (ℤ𝑛)) → (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥) ∈ V)
6442, 57, 58, 12, 59, 49, 62, 63liminfequzmpt2 40541 . . . . . . . . . . . . . . 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 40541 . . . . . . . . . . . . . . 15 ((𝜑𝑛𝑍𝑥 𝑚 ∈ (ℤ𝑛)𝐴) → (lim inf‘(𝑚𝑍𝐵)) = (lim inf‘(𝑚 ∈ (ℤ𝑛) ↦ 𝐵)))
6656, 64, 653eqtr4d 2815 . . . . . . . . . . . . . 14 ((𝜑𝑛𝑍𝑥 𝑚 ∈ (ℤ𝑛)𝐴) → (lim inf‘(𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥))) = (lim inf‘(𝑚𝑍𝐵)))
67663exp 1112 . . . . . . . . . . . . 13 (𝜑 → (𝑛𝑍 → (𝑥 𝑚 ∈ (ℤ𝑛)𝐴 → (lim inf‘(𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥))) = (lim inf‘(𝑚𝑍𝐵)))))
687, 38, 67rexlimd 3174 . . . . . . . . . . . 12 (𝜑 → (∃𝑛𝑍 𝑥 𝑚 ∈ (ℤ𝑛)𝐴 → (lim inf‘(𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥))) = (lim inf‘(𝑚𝑍𝐵))))
6968adantr 466 . . . . . . . . . . 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 696 . . . . . . . . 9 ((𝜑 ∧ (𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴 ∧ (lim inf‘(𝑚𝑍𝐵)) ∈ ℝ)) → (lim inf‘(𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥))) = (lim inf‘(𝑚𝑍𝐵)))
72 simprr 756 . . . . . . . . 9 ((𝜑 ∧ (𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴 ∧ (lim inf‘(𝑚𝑍𝐵)) ∈ ℝ)) → (lim inf‘(𝑚𝑍𝐵)) ∈ ℝ)
7371, 72eqeltrd 2850 . . . . . . . 8 ((𝜑 ∧ (𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴 ∧ (lim inf‘(𝑚𝑍𝐵)) ∈ ℝ)) → (lim inf‘(𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥))) ∈ ℝ)
7434, 73jca 501 . . . . . . 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 468 . . . . . . . 8 ((𝜑 ∧ (𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚) ∧ (lim inf‘(𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥))) ∈ ℝ)) → 𝜑)
76 simpr 471 . . . . . . . . . 10 ((𝜑𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)) → 𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚))
7731eqcomd 2777 . . . . . . . . . . 11 (𝜑 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚) = 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴)
7877adantr 466 . . . . . . . . . 10 ((𝜑𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)) → 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚) = 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴)
7976, 78eleqtrd 2852 . . . . . . . . 9 ((𝜑𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)) → 𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴)
8079adantrr 696 . . . . . . . 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 756 . . . . . . . 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 1131 . . . . . . . . 9 ((𝜑𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴 ∧ (lim inf‘(𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥))) ∈ ℝ) → 𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴)
8370eqcomd 2777 . . . . . . . . . . 11 ((𝜑𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴) → (lim inf‘(𝑚𝑍𝐵)) = (lim inf‘(𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥))))
84833adant3 1126 . . . . . . . . . 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 1132 . . . . . . . . . 10 ((𝜑𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴 ∧ (lim inf‘(𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥))) ∈ ℝ) → (lim inf‘(𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥))) ∈ ℝ)
8684, 85eqeltrd 2850 . . . . . . . . 9 ((𝜑𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴 ∧ (lim inf‘(𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥))) ∈ ℝ) → (lim inf‘(𝑚𝑍𝐵)) ∈ ℝ)
8782, 86jca 501 . . . . . . . 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 1476 . . . . . . 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 802 . . . . . 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 39841 . . . . 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 2805 . . . 4 (𝜑𝐷 = {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚) ∣ (lim inf‘(𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥))) ∈ ℝ})
924eleq2i 2842 . . . . . . 7 (𝑥𝐷𝑥 ∈ {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴 ∣ (lim inf‘(𝑚𝑍𝐵)) ∈ ℝ})
9392biimpi 206 . . . . . 6 (𝑥𝐷𝑥 ∈ {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴 ∣ (lim inf‘(𝑚𝑍𝐵)) ∈ ℝ})
94 rabidim1 3266 . . . . . 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 580 . . . 4 ((𝜑𝑥𝐷) → (lim inf‘(𝑚𝑍𝐵)) = (lim inf‘(𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥))))
973, 91, 96mpteq12da 39970 . . 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 2805 . 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 4881 . . 3 𝑚(𝑚𝑍 ↦ (𝑥𝐴𝐵))
100 nfcv 2913 . . . 4 𝑥𝑍
101 nfmpt1 4881 . . . 4 𝑥(𝑥𝐴𝐵)
102100, 101nfmpt 4880 . . 3 𝑥(𝑚𝑍 ↦ (𝑥𝐴𝐵))
103 smfliminfmpt.m . . 3 (𝜑𝑀 ∈ ℤ)
104 smfliminfmpt.s . . 3 (𝜑𝑆 ∈ SAlg)
1058, 16fmptd2f 39960 . . 3 (𝜑 → (𝑚𝑍 ↦ (𝑥𝐴𝐵)):𝑍⟶(SMblFn‘𝑆))
106 eqid 2771 . . 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 2771 . . 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 41557 . 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 2850 1 (𝜑𝐺 ∈ (SMblFn‘𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3a 1071   = wceq 1631  wnf 1856  wcel 2145  wrex 3062  {crab 3065  Vcvv 3351   ciun 4654   ciin 4655  cmpt 4863  dom cdm 5249  cfv 6031  cr 10137  cz 11579  cuz 11888  lim infclsi 40501  SAlgcsalg 41045  SMblFncsmblfn 41429
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-inf2 8702  ax-cc 9459  ax-ac2 9487  ax-cnex 10194  ax-resscn 10195  ax-1cn 10196  ax-icn 10197  ax-addcl 10198  ax-addrcl 10199  ax-mulcl 10200  ax-mulrcl 10201  ax-mulcom 10202  ax-addass 10203  ax-mulass 10204  ax-distr 10205  ax-i2m1 10206  ax-1ne0 10207  ax-1rid 10208  ax-rnegex 10209  ax-rrecex 10210  ax-cnre 10211  ax-pre-lttri 10212  ax-pre-lttrn 10213  ax-pre-ltadd 10214  ax-pre-mulgt0 10215  ax-pre-sup 10216
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-iin 4657  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-se 5209  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-isom 6040  df-riota 6754  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-om 7213  df-1st 7315  df-2nd 7316  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-1o 7713  df-oadd 7717  df-omul 7718  df-er 7896  df-map 8011  df-pm 8012  df-en 8110  df-dom 8111  df-sdom 8112  df-fin 8113  df-sup 8504  df-inf 8505  df-oi 8571  df-card 8965  df-acn 8968  df-ac 9139  df-pnf 10278  df-mnf 10279  df-xr 10280  df-ltxr 10281  df-le 10282  df-sub 10470  df-neg 10471  df-div 10887  df-nn 11223  df-2 11281  df-3 11282  df-4 11283  df-n0 11495  df-z 11580  df-uz 11889  df-q 11992  df-rp 12036  df-xneg 12151  df-ioo 12384  df-ioc 12385  df-ico 12386  df-icc 12387  df-fz 12534  df-fzo 12674  df-fl 12801  df-ceil 12802  df-seq 13009  df-exp 13068  df-hash 13322  df-word 13495  df-concat 13497  df-s1 13498  df-s2 13802  df-s3 13803  df-s4 13804  df-cj 14047  df-re 14048  df-im 14049  df-sqrt 14183  df-abs 14184  df-limsup 14410  df-clim 14427  df-rlim 14428  df-rest 16291  df-topgen 16312  df-top 20919  df-bases 20971  df-liminf 40502  df-salg 41046  df-salgen 41050  df-smblfn 41430
This theorem is referenced by: (None)
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