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Theorem smfliminfmpt 46358
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 483 . . . . . . . . . 10 ((𝜑𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴) → 𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴)
7 smfliminfmpt.n . . . . . . . . . . . 12 𝑛𝜑
8 smfliminfmpt.p . . . . . . . . . . . . . 14 𝑚𝜑
9 nfv 1909 . . . . . . . . . . . . . 14 𝑚 𝑛𝑍
108, 9nfan 1894 . . . . . . . . . . . . 13 𝑚(𝜑𝑛𝑍)
11 simpll 765 . . . . . . . . . . . . . 14 (((𝜑𝑛𝑍) ∧ 𝑚 ∈ (ℤ𝑛)) → 𝜑)
12 smfliminfmpt.z . . . . . . . . . . . . . . . 16 𝑍 = (ℤ𝑀)
1312uztrn2 12874 . . . . . . . . . . . . . . 15 ((𝑛𝑍𝑚 ∈ (ℤ𝑛)) → 𝑚𝑍)
1413adantll 712 . . . . . . . . . . . . . 14 (((𝜑𝑛𝑍) ∧ 𝑚 ∈ (ℤ𝑛)) → 𝑚𝑍)
15 simpr 483 . . . . . . . . . . . . . . . . 17 ((𝜑𝑚𝑍) → 𝑚𝑍)
16 smfliminfmpt.f . . . . . . . . . . . . . . . . . 18 ((𝜑𝑚𝑍) → (𝑥𝐴𝐵) ∈ (SMblFn‘𝑆))
1716elexd 3483 . . . . . . . . . . . . . . . . 17 ((𝜑𝑚𝑍) → (𝑥𝐴𝐵) ∈ V)
18 eqid 2725 . . . . . . . . . . . . . . . . . 18 (𝑚𝑍 ↦ (𝑥𝐴𝐵)) = (𝑚𝑍 ↦ (𝑥𝐴𝐵))
1918fvmpt2 7015 . . . . . . . . . . . . . . . . 17 ((𝑚𝑍 ∧ (𝑥𝐴𝐵) ∈ V) → ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚) = (𝑥𝐴𝐵))
2015, 17, 19syl2anc 582 . . . . . . . . . . . . . . . 16 ((𝜑𝑚𝑍) → ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚) = (𝑥𝐴𝐵))
2120dmeqd 5908 . . . . . . . . . . . . . . 15 ((𝜑𝑚𝑍) → dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚) = dom (𝑥𝐴𝐵))
22 nfv 1909 . . . . . . . . . . . . . . . . 17 𝑥 𝑚𝑍
233, 22nfan 1894 . . . . . . . . . . . . . . . 16 𝑥(𝜑𝑚𝑍)
24 eqid 2725 . . . . . . . . . . . . . . . 16 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
25 smfliminfmpt.b . . . . . . . . . . . . . . . . 17 ((𝜑𝑚𝑍𝑥𝐴) → 𝐵𝑉)
26253expa 1115 . . . . . . . . . . . . . . . 16 (((𝜑𝑚𝑍) ∧ 𝑥𝐴) → 𝐵𝑉)
2723, 24, 26dmmptdf 44736 . . . . . . . . . . . . . . 15 ((𝜑𝑚𝑍) → dom (𝑥𝐴𝐵) = 𝐴)
2821, 27eqtr2d 2766 . . . . . . . . . . . . . 14 ((𝜑𝑚𝑍) → 𝐴 = dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚))
2911, 14, 28syl2anc 582 . . . . . . . . . . . . 13 (((𝜑𝑛𝑍) ∧ 𝑚 ∈ (ℤ𝑛)) → 𝐴 = dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚))
3010, 29iineq2d 5020 . . . . . . . . . . . 12 ((𝜑𝑛𝑍) → 𝑚 ∈ (ℤ𝑛)𝐴 = 𝑚 ∈ (ℤ𝑛)dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚))
317, 30iuneq2df 44552 . . . . . . . . . . 11 (𝜑 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴 = 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚))
3231adantr 479 . . . . . . . . . 10 ((𝜑𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴) → 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴 = 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚))
336, 32eleqtrd 2827 . . . . . . . . 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 5001 . . . . . . . . . . . . 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 480 . . . . . . . . . . 11 ((𝜑𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴) → ∃𝑛𝑍 𝑥 𝑚 ∈ (ℤ𝑛)𝐴)
38 nfv 1909 . . . . . . . . . . . . 13 𝑛(lim inf‘(𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥))) = (lim inf‘(𝑚𝑍𝐵))
39 nfcv 2891 . . . . . . . . . . . . . . . . . . 19 𝑚𝑥
40 nfii1 5033 . . . . . . . . . . . . . . . . . . 19 𝑚 𝑚 ∈ (ℤ𝑛)𝐴
4139, 40nfel 2906 . . . . . . . . . . . . . . . . . 18 𝑚 𝑥 𝑚 ∈ (ℤ𝑛)𝐴
428, 9, 41nf3an 1896 . . . . . . . . . . . . . . . . 17 𝑚(𝜑𝑛𝑍𝑥 𝑚 ∈ (ℤ𝑛)𝐴)
4320fveq1d 6898 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑚𝑍) → (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥) = ((𝑥𝐴𝐵)‘𝑥))
4411, 14, 43syl2anc 582 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑛𝑍) ∧ 𝑚 ∈ (ℤ𝑛)) → (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥) = ((𝑥𝐴𝐵)‘𝑥))
45443adantl3 1165 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑛𝑍𝑥 𝑚 ∈ (ℤ𝑛)𝐴) ∧ 𝑚 ∈ (ℤ𝑛)) → (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥) = ((𝑥𝐴𝐵)‘𝑥))
46 eliinid 44617 . . . . . . . . . . . . . . . . . . . 20 ((𝑥 𝑚 ∈ (ℤ𝑛)𝐴𝑚 ∈ (ℤ𝑛)) → 𝑥𝐴)
47463ad2antl3 1184 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑛𝑍𝑥 𝑚 ∈ (ℤ𝑛)𝐴) ∧ 𝑚 ∈ (ℤ𝑛)) → 𝑥𝐴)
48 simpl1 1188 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑛𝑍𝑥 𝑚 ∈ (ℤ𝑛)𝐴) ∧ 𝑚 ∈ (ℤ𝑛)) → 𝜑)
49 simp2 1134 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑛𝑍𝑥 𝑚 ∈ (ℤ𝑛)𝐴) → 𝑛𝑍)
5049, 13sylan 578 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑛𝑍𝑥 𝑚 ∈ (ℤ𝑛)𝐴) ∧ 𝑚 ∈ (ℤ𝑛)) → 𝑚𝑍)
5148, 50, 47, 25syl3anc 1368 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑛𝑍𝑥 𝑚 ∈ (ℤ𝑛)𝐴) ∧ 𝑚 ∈ (ℤ𝑛)) → 𝐵𝑉)
5224fvmpt2 7015 . . . . . . . . . . . . . . . . . . 19 ((𝑥𝐴𝐵𝑉) → ((𝑥𝐴𝐵)‘𝑥) = 𝐵)
5347, 51, 52syl2anc 582 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑛𝑍𝑥 𝑚 ∈ (ℤ𝑛)𝐴) ∧ 𝑚 ∈ (ℤ𝑛)) → ((𝑥𝐴𝐵)‘𝑥) = 𝐵)
5445, 53eqtrd 2765 . . . . . . . . . . . . . . . . 17 (((𝜑𝑛𝑍𝑥 𝑚 ∈ (ℤ𝑛)𝐴) ∧ 𝑚 ∈ (ℤ𝑛)) → (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥) = 𝐵)
5542, 54mpteq2da 5247 . . . . . . . . . . . . . . . 16 ((𝜑𝑛𝑍𝑥 𝑚 ∈ (ℤ𝑛)𝐴) → (𝑚 ∈ (ℤ𝑛) ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥)) = (𝑚 ∈ (ℤ𝑛) ↦ 𝐵))
5655fveq2d 6900 . . . . . . . . . . . . . . 15 ((𝜑𝑛𝑍𝑥 𝑚 ∈ (ℤ𝑛)𝐴) → (lim inf‘(𝑚 ∈ (ℤ𝑛) ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥))) = (lim inf‘(𝑚 ∈ (ℤ𝑛) ↦ 𝐵)))
57 nfcv 2891 . . . . . . . . . . . . . . . 16 𝑚𝑍
58 nfcv 2891 . . . . . . . . . . . . . . . 16 𝑚(ℤ𝑛)
59 eqid 2725 . . . . . . . . . . . . . . . 16 (ℤ𝑛) = (ℤ𝑛)
6012eluzelz2 44923 . . . . . . . . . . . . . . . . . 18 (𝑛𝑍𝑛 ∈ ℤ)
6160uzidd 12871 . . . . . . . . . . . . . . . . 17 (𝑛𝑍𝑛 ∈ (ℤ𝑛))
62613ad2ant2 1131 . . . . . . . . . . . . . . . 16 ((𝜑𝑛𝑍𝑥 𝑚 ∈ (ℤ𝑛)𝐴) → 𝑛 ∈ (ℤ𝑛))
63 fvexd 6911 . . . . . . . . . . . . . . . 16 (((𝜑𝑛𝑍𝑥 𝑚 ∈ (ℤ𝑛)𝐴) ∧ 𝑚 ∈ (ℤ𝑛)) → (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥) ∈ V)
6442, 57, 58, 12, 59, 49, 62, 63liminfequzmpt2 45317 . . . . . . . . . . . . . . 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 45317 . . . . . . . . . . . . . . 15 ((𝜑𝑛𝑍𝑥 𝑚 ∈ (ℤ𝑛)𝐴) → (lim inf‘(𝑚𝑍𝐵)) = (lim inf‘(𝑚 ∈ (ℤ𝑛) ↦ 𝐵)))
6656, 64, 653eqtr4d 2775 . . . . . . . . . . . . . 14 ((𝜑𝑛𝑍𝑥 𝑚 ∈ (ℤ𝑛)𝐴) → (lim inf‘(𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥))) = (lim inf‘(𝑚𝑍𝐵)))
67663exp 1116 . . . . . . . . . . . . 13 (𝜑 → (𝑛𝑍 → (𝑥 𝑚 ∈ (ℤ𝑛)𝐴 → (lim inf‘(𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥))) = (lim inf‘(𝑚𝑍𝐵)))))
687, 38, 67rexlimd 3253 . . . . . . . . . . . 12 (𝜑 → (∃𝑛𝑍 𝑥 𝑚 ∈ (ℤ𝑛)𝐴 → (lim inf‘(𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥))) = (lim inf‘(𝑚𝑍𝐵))))
6968adantr 479 . . . . . . . . . . 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 2825 . . . . . . . 8 ((𝜑 ∧ (𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴 ∧ (lim inf‘(𝑚𝑍𝐵)) ∈ ℝ)) → (lim inf‘(𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥))) ∈ ℝ)
7434, 73jca 510 . . . . . . 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 481 . . . . . . . 8 ((𝜑 ∧ (𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚) ∧ (lim inf‘(𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥))) ∈ ℝ)) → 𝜑)
76 simpr 483 . . . . . . . . . 10 ((𝜑𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)) → 𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚))
7731eqcomd 2731 . . . . . . . . . . 11 (𝜑 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚) = 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴)
7877adantr 479 . . . . . . . . . 10 ((𝜑𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)) → 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚) = 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴)
7976, 78eleqtrd 2827 . . . . . . . . 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 1134 . . . . . . . . 9 ((𝜑𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴 ∧ (lim inf‘(𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥))) ∈ ℝ) → 𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴)
8370eqcomd 2731 . . . . . . . . . . 11 ((𝜑𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴) → (lim inf‘(𝑚𝑍𝐵)) = (lim inf‘(𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥))))
84833adant3 1129 . . . . . . . . . 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 1135 . . . . . . . . . 10 ((𝜑𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴 ∧ (lim inf‘(𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥))) ∈ ℝ) → (lim inf‘(𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥))) ∈ ℝ)
8684, 85eqeltrd 2825 . . . . . . . . 9 ((𝜑𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴 ∧ (lim inf‘(𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥))) ∈ ℝ) → (lim inf‘(𝑚𝑍𝐵)) ∈ ℝ)
8782, 86jca 510 . . . . . . . 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 1368 . . . . . . 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 44641 . . . . 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 2765 . . . 4 (𝜑𝐷 = {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚) ∣ (lim inf‘(𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥))) ∈ ℝ})
924eleq2i 2817 . . . . . . 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 3440 . . . . . 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 591 . . . 4 ((𝜑𝑥𝐷) → (lim inf‘(𝑚𝑍𝐵)) = (lim inf‘(𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥))))
973, 91, 96mpteq12da 5234 . . 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 2765 . 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 5257 . . 3 𝑚(𝑚𝑍 ↦ (𝑥𝐴𝐵))
100 nfcv 2891 . . . 4 𝑥𝑍
101 nfmpt1 5257 . . . 4 𝑥(𝑥𝐴𝐵)
102100, 101nfmpt 5256 . . 3 𝑥(𝑚𝑍 ↦ (𝑥𝐴𝐵))
103 smfliminfmpt.m . . 3 (𝜑𝑀 ∈ ℤ)
104 smfliminfmpt.s . . 3 (𝜑𝑆 ∈ SAlg)
1058, 16fmptd2f 44747 . . 3 (𝜑 → (𝑚𝑍 ↦ (𝑥𝐴𝐵)):𝑍⟶(SMblFn‘𝑆))
106 eqid 2725 . . 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 2725 . . 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 46357 . 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 2825 1 (𝜑𝐺 ∈ (SMblFn‘𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  w3a 1084   = wceq 1533  wnf 1777  wcel 2098  wrex 3059  {crab 3418  Vcvv 3461   ciun 4997   ciin 4998  cmpt 5232  dom cdm 5678  cfv 6549  cr 11139  cz 12591  cuz 12855  lim infclsi 45277  SAlgcsalg 45834  SMblFncsmblfn 46221
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741  ax-inf2 9666  ax-cc 10460  ax-ac2 10488  ax-cnex 11196  ax-resscn 11197  ax-1cn 11198  ax-icn 11199  ax-addcl 11200  ax-addrcl 11201  ax-mulcl 11202  ax-mulrcl 11203  ax-mulcom 11204  ax-addass 11205  ax-mulass 11206  ax-distr 11207  ax-i2m1 11208  ax-1ne0 11209  ax-1rid 11210  ax-rnegex 11211  ax-rrecex 11212  ax-cnre 11213  ax-pre-lttri 11214  ax-pre-lttrn 11215  ax-pre-ltadd 11216  ax-pre-mulgt0 11217  ax-pre-sup 11218
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-nel 3036  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-tp 4635  df-op 4637  df-uni 4910  df-int 4951  df-iun 4999  df-iin 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-se 5634  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6307  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-isom 6558  df-riota 7375  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7872  df-1st 7994  df-2nd 7995  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-1o 8487  df-oadd 8491  df-omul 8492  df-er 8725  df-map 8847  df-pm 8848  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-sup 9467  df-inf 9468  df-oi 9535  df-card 9964  df-acn 9967  df-ac 10141  df-pnf 11282  df-mnf 11283  df-xr 11284  df-ltxr 11285  df-le 11286  df-sub 11478  df-neg 11479  df-div 11904  df-nn 12246  df-2 12308  df-3 12309  df-4 12310  df-n0 12506  df-z 12592  df-uz 12856  df-q 12966  df-rp 13010  df-xneg 13127  df-ioo 13363  df-ioc 13364  df-ico 13365  df-icc 13366  df-fz 13520  df-fzo 13663  df-fl 13793  df-ceil 13794  df-seq 14003  df-exp 14063  df-hash 14326  df-word 14501  df-concat 14557  df-s1 14582  df-s2 14835  df-s3 14836  df-s4 14837  df-cj 15082  df-re 15083  df-im 15084  df-sqrt 15218  df-abs 15219  df-limsup 15451  df-clim 15468  df-rlim 15469  df-rest 17407  df-topgen 17428  df-top 22840  df-bases 22893  df-liminf 45278  df-salg 45835  df-salgen 45839  df-smblfn 46222
This theorem is referenced by: (None)
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