Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  smflimsupmpt Structured version   Visualization version   GIF version

Theorem smflimsupmpt 41699
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 477 . . . . . . . . . . 11 ((𝜑𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴) → 𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴)
7 smflimsupmpt.n . . . . . . . . . . . . 13 𝑛𝜑
8 smflimsupmpt.p . . . . . . . . . . . . . . 15 𝑚𝜑
9 nfv 2009 . . . . . . . . . . . . . . 15 𝑚 𝑛𝑍
108, 9nfan 1998 . . . . . . . . . . . . . 14 𝑚(𝜑𝑛𝑍)
11 simpll 783 . . . . . . . . . . . . . . 15 (((𝜑𝑛𝑍) ∧ 𝑚 ∈ (ℤ𝑛)) → 𝜑)
12 smflimsupmpt.z . . . . . . . . . . . . . . . . 17 𝑍 = (ℤ𝑀)
1312uztrn2 11909 . . . . . . . . . . . . . . . 16 ((𝑛𝑍𝑚 ∈ (ℤ𝑛)) → 𝑚𝑍)
1413adantll 705 . . . . . . . . . . . . . . 15 (((𝜑𝑛𝑍) ∧ 𝑚 ∈ (ℤ𝑛)) → 𝑚𝑍)
15 simpr 477 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑚𝑍) → 𝑚𝑍)
16 smflimsupmpt.f . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑚𝑍) → (𝑥𝐴𝐵) ∈ (SMblFn‘𝑆))
1716elexd 3367 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑚𝑍) → (𝑥𝐴𝐵) ∈ V)
18 eqid 2765 . . . . . . . . . . . . . . . . . . 19 (𝑚𝑍 ↦ (𝑥𝐴𝐵)) = (𝑚𝑍 ↦ (𝑥𝐴𝐵))
1918fvmpt2 6484 . . . . . . . . . . . . . . . . . 18 ((𝑚𝑍 ∧ (𝑥𝐴𝐵) ∈ V) → ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚) = (𝑥𝐴𝐵))
2015, 17, 19syl2anc 579 . . . . . . . . . . . . . . . . 17 ((𝜑𝑚𝑍) → ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚) = (𝑥𝐴𝐵))
2120dmeqd 5496 . . . . . . . . . . . . . . . 16 ((𝜑𝑚𝑍) → dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚) = dom (𝑥𝐴𝐵))
22 nfv 2009 . . . . . . . . . . . . . . . . . 18 𝑥 𝑚𝑍
233, 22nfan 1998 . . . . . . . . . . . . . . . . 17 𝑥(𝜑𝑚𝑍)
24 eqid 2765 . . . . . . . . . . . . . . . . 17 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
25 smflimsupmpt.b . . . . . . . . . . . . . . . . . 18 ((𝜑𝑚𝑍𝑥𝐴) → 𝐵𝑊)
26253expa 1147 . . . . . . . . . . . . . . . . 17 (((𝜑𝑚𝑍) ∧ 𝑥𝐴) → 𝐵𝑊)
2723, 24, 26dmmptdf 40086 . . . . . . . . . . . . . . . 16 ((𝜑𝑚𝑍) → dom (𝑥𝐴𝐵) = 𝐴)
2821, 27eqtr2d 2800 . . . . . . . . . . . . . . 15 ((𝜑𝑚𝑍) → 𝐴 = dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚))
2911, 14, 28syl2anc 579 . . . . . . . . . . . . . 14 (((𝜑𝑛𝑍) ∧ 𝑚 ∈ (ℤ𝑛)) → 𝐴 = dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚))
3010, 29iineq2d 4699 . . . . . . . . . . . . 13 ((𝜑𝑛𝑍) → 𝑚 ∈ (ℤ𝑛)𝐴 = 𝑚 ∈ (ℤ𝑛)dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚))
317, 30iuneq2df 39887 . . . . . . . . . . . 12 (𝜑 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴 = 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚))
3231adantr 472 . . . . . . . . . . 11 ((𝜑𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴) → 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴 = 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚))
336, 32eleqtrd 2846 . . . . . . . . . 10 ((𝜑𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴) → 𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚))
3433adantrr 708 . . . . . . . . 9 ((𝜑 ∧ (𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴 ∧ (lim sup‘(𝑚𝑍𝐵)) ∈ ℝ)) → 𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚))
35 eliun 4682 . . . . . . . . . . . . . 14 (𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴 ↔ ∃𝑛𝑍 𝑥 𝑚 ∈ (ℤ𝑛)𝐴)
3635biimpi 207 . . . . . . . . . . . . 13 (𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴 → ∃𝑛𝑍 𝑥 𝑚 ∈ (ℤ𝑛)𝐴)
3736adantl 473 . . . . . . . . . . . 12 ((𝜑𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴) → ∃𝑛𝑍 𝑥 𝑚 ∈ (ℤ𝑛)𝐴)
38 nfv 2009 . . . . . . . . . . . . . 14 𝑛(lim sup‘(𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥))) = (lim sup‘(𝑚𝑍𝐵))
39 nfcv 2907 . . . . . . . . . . . . . . . . . . . 20 𝑚𝑥
40 nfii1 4709 . . . . . . . . . . . . . . . . . . . 20 𝑚 𝑚 ∈ (ℤ𝑛)𝐴
4139, 40nfel 2920 . . . . . . . . . . . . . . . . . . 19 𝑚 𝑥 𝑚 ∈ (ℤ𝑛)𝐴
428, 9, 41nf3an 2000 . . . . . . . . . . . . . . . . . 18 𝑚(𝜑𝑛𝑍𝑥 𝑚 ∈ (ℤ𝑛)𝐴)
4320fveq1d 6381 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑚𝑍) → (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥) = ((𝑥𝐴𝐵)‘𝑥))
4411, 14, 43syl2anc 579 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑛𝑍) ∧ 𝑚 ∈ (ℤ𝑛)) → (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥) = ((𝑥𝐴𝐵)‘𝑥))
45443adantl3 1209 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑛𝑍𝑥 𝑚 ∈ (ℤ𝑛)𝐴) ∧ 𝑚 ∈ (ℤ𝑛)) → (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥) = ((𝑥𝐴𝐵)‘𝑥))
46 eliinid 39968 . . . . . . . . . . . . . . . . . . . . 21 ((𝑥 𝑚 ∈ (ℤ𝑛)𝐴𝑚 ∈ (ℤ𝑛)) → 𝑥𝐴)
47463ad2antl3 1238 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑛𝑍𝑥 𝑚 ∈ (ℤ𝑛)𝐴) ∧ 𝑚 ∈ (ℤ𝑛)) → 𝑥𝐴)
48 simpl1 1242 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑛𝑍𝑥 𝑚 ∈ (ℤ𝑛)𝐴) ∧ 𝑚 ∈ (ℤ𝑛)) → 𝜑)
49143adantl3 1209 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑛𝑍𝑥 𝑚 ∈ (ℤ𝑛)𝐴) ∧ 𝑚 ∈ (ℤ𝑛)) → 𝑚𝑍)
5048, 49, 47, 25syl3anc 1490 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑛𝑍𝑥 𝑚 ∈ (ℤ𝑛)𝐴) ∧ 𝑚 ∈ (ℤ𝑛)) → 𝐵𝑊)
5124fvmpt2 6484 . . . . . . . . . . . . . . . . . . . 20 ((𝑥𝐴𝐵𝑊) → ((𝑥𝐴𝐵)‘𝑥) = 𝐵)
5247, 50, 51syl2anc 579 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑛𝑍𝑥 𝑚 ∈ (ℤ𝑛)𝐴) ∧ 𝑚 ∈ (ℤ𝑛)) → ((𝑥𝐴𝐵)‘𝑥) = 𝐵)
5345, 52eqtrd 2799 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑛𝑍𝑥 𝑚 ∈ (ℤ𝑛)𝐴) ∧ 𝑚 ∈ (ℤ𝑛)) → (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥) = 𝐵)
5442, 53mpteq2da 4904 . . . . . . . . . . . . . . . . 17 ((𝜑𝑛𝑍𝑥 𝑚 ∈ (ℤ𝑛)𝐴) → (𝑚 ∈ (ℤ𝑛) ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥)) = (𝑚 ∈ (ℤ𝑛) ↦ 𝐵))
5554fveq2d 6383 . . . . . . . . . . . . . . . 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 1163 . . . . . . . . . . . . . . . . 17 ((𝜑𝑛𝑍𝑥 𝑚 ∈ (ℤ𝑛)𝐴) → 𝑀 ∈ ℤ)
5812eluzelz2 40288 . . . . . . . . . . . . . . . . . 18 (𝑛𝑍𝑛 ∈ ℤ)
59583ad2ant2 1164 . . . . . . . . . . . . . . . . 17 ((𝜑𝑛𝑍𝑥 𝑚 ∈ (ℤ𝑛)𝐴) → 𝑛 ∈ ℤ)
60 eqid 2765 . . . . . . . . . . . . . . . . 17 (ℤ𝑛) = (ℤ𝑛)
61 fvexd 6394 . . . . . . . . . . . . . . . . 17 (((𝜑𝑛𝑍𝑥 𝑚 ∈ (ℤ𝑛)𝐴) ∧ 𝑚𝑍) → (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥) ∈ V)
6249, 61syldan 585 . . . . . . . . . . . . . . . . 17 (((𝜑𝑛𝑍𝑥 𝑚 ∈ (ℤ𝑛)𝐴) ∧ 𝑚 ∈ (ℤ𝑛)) → (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥) ∈ V)
6342, 57, 59, 12, 60, 61, 62limsupequzmpt 40623 . . . . . . . . . . . . . . . 16 ((𝜑𝑛𝑍𝑥 𝑚 ∈ (ℤ𝑛)𝐴) → (lim sup‘(𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥))) = (lim sup‘(𝑚 ∈ (ℤ𝑛) ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥))))
649nfci 2897 . . . . . . . . . . . . . . . . 17 𝑚𝑍
65 nfcv 2907 . . . . . . . . . . . . . . . . 17 𝑚(ℤ𝑛)
66 simp2 1167 . . . . . . . . . . . . . . . . 17 ((𝜑𝑛𝑍𝑥 𝑚 ∈ (ℤ𝑛)𝐴) → 𝑛𝑍)
6759uzidd 40292 . . . . . . . . . . . . . . . . 17 ((𝜑𝑛𝑍𝑥 𝑚 ∈ (ℤ𝑛)𝐴) → 𝑛 ∈ (ℤ𝑛))
6842, 64, 65, 12, 60, 66, 67, 50limsupequzmpt2 40612 . . . . . . . . . . . . . . . 16 ((𝜑𝑛𝑍𝑥 𝑚 ∈ (ℤ𝑛)𝐴) → (lim sup‘(𝑚𝑍𝐵)) = (lim sup‘(𝑚 ∈ (ℤ𝑛) ↦ 𝐵)))
6955, 63, 683eqtr4d 2809 . . . . . . . . . . . . . . 15 ((𝜑𝑛𝑍𝑥 𝑚 ∈ (ℤ𝑛)𝐴) → (lim sup‘(𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥))) = (lim sup‘(𝑚𝑍𝐵)))
70693exp 1148 . . . . . . . . . . . . . 14 (𝜑 → (𝑛𝑍 → (𝑥 𝑚 ∈ (ℤ𝑛)𝐴 → (lim sup‘(𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥))) = (lim sup‘(𝑚𝑍𝐵)))))
717, 38, 70rexlimd 3173 . . . . . . . . . . . . 13 (𝜑 → (∃𝑛𝑍 𝑥 𝑚 ∈ (ℤ𝑛)𝐴 → (lim sup‘(𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥))) = (lim sup‘(𝑚𝑍𝐵))))
7271adantr 472 . . . . . . . . . . . 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 708 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴 ∧ (lim sup‘(𝑚𝑍𝐵)) ∈ ℝ)) → (lim sup‘(𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥))) = (lim sup‘(𝑚𝑍𝐵)))
75 simprr 789 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴 ∧ (lim sup‘(𝑚𝑍𝐵)) ∈ ℝ)) → (lim sup‘(𝑚𝑍𝐵)) ∈ ℝ)
7674, 75eqeltrd 2844 . . . . . . . . 9 ((𝜑 ∧ (𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴 ∧ (lim sup‘(𝑚𝑍𝐵)) ∈ ℝ)) → (lim sup‘(𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥))) ∈ ℝ)
7734, 76jca 507 . . . . . . . 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 401 . . . . . . 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 474 . . . . . . . . 9 ((𝜑 ∧ (𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚) ∧ (lim sup‘(𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥))) ∈ ℝ)) → 𝜑)
80 simpr 477 . . . . . . . . . . 11 ((𝜑𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)) → 𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚))
8131eqcomd 2771 . . . . . . . . . . . 12 (𝜑 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚) = 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴)
8281adantr 472 . . . . . . . . . . 11 ((𝜑𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)) → 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚) = 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴)
8380, 82eleqtrd 2846 . . . . . . . . . 10 ((𝜑𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)) → 𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴)
8483adantrr 708 . . . . . . . . 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 789 . . . . . . . . 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 1167 . . . . . . . . . 10 ((𝜑𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴 ∧ (lim sup‘(𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥))) ∈ ℝ) → 𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴)
8773eqcomd 2771 . . . . . . . . . . . 12 ((𝜑𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴) → (lim sup‘(𝑚𝑍𝐵)) = (lim sup‘(𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥))))
88873adant3 1162 . . . . . . . . . . 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 1168 . . . . . . . . . . 11 ((𝜑𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴 ∧ (lim sup‘(𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥))) ∈ ℝ) → (lim sup‘(𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥))) ∈ ℝ)
9088, 89eqeltrd 2844 . . . . . . . . . 10 ((𝜑𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴 ∧ (lim sup‘(𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥))) ∈ ℝ) → (lim sup‘(𝑚𝑍𝐵)) ∈ ℝ)
9186, 90jca 507 . . . . . . . . 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 1490 . . . . . . . 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 401 . . . . . . 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 203 . . . . . 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 39993 . . . . 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 2799 . . . 4 (𝜑𝐷 = {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚) ∣ (lim sup‘(𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥))) ∈ ℝ})
974eleq2i 2836 . . . . . . 7 (𝑥𝐷𝑥 ∈ {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴 ∣ (lim sup‘(𝑚𝑍𝐵)) ∈ ℝ})
9897biimpi 207 . . . . . 6 (𝑥𝐷𝑥 ∈ {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴 ∣ (lim sup‘(𝑚𝑍𝐵)) ∈ ℝ})
99 rabidim1 3265 . . . . . 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 586 . . . 4 ((𝜑𝑥𝐷) → (lim sup‘(𝑚𝑍𝐵)) = (lim sup‘(𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥))))
1023, 96, 101mpteq12da 40118 . . 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 2799 . 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 4908 . . 3 𝑚(𝑚𝑍 ↦ (𝑥𝐴𝐵))
105 nfcv 2907 . . . 4 𝑥𝑍
106 nfmpt1 4908 . . . 4 𝑥(𝑥𝐴𝐵)
107105, 106nfmpt 4907 . . 3 𝑥(𝑚𝑍 ↦ (𝑥𝐴𝐵))
108 smflimsupmpt.s . . 3 (𝜑𝑆 ∈ SAlg)
1098, 16fmptd2f 40108 . . 3 (𝜑 → (𝑚𝑍 ↦ (𝑥𝐴𝐵)):𝑍⟶(SMblFn‘𝑆))
110 eqid 2765 . . 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 2765 . . 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 41698 . 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 2844 1 (𝜑𝐺 ∈ (SMblFn‘𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1107   = wceq 1652  wnf 1878  wcel 2155  wrex 3056  {crab 3059  Vcvv 3350   ciun 4678   ciin 4679  cmpt 4890  dom cdm 5279  cfv 6070  cr 10192  cz 11628  cuz 11891  lim supclsp 14500  SAlgcsalg 41189  SMblFncsmblfn 41573
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4932  ax-sep 4943  ax-nul 4951  ax-pow 5003  ax-pr 5064  ax-un 7151  ax-inf2 8757  ax-cc 9514  ax-ac2 9542  ax-cnex 10249  ax-resscn 10250  ax-1cn 10251  ax-icn 10252  ax-addcl 10253  ax-addrcl 10254  ax-mulcl 10255  ax-mulrcl 10256  ax-mulcom 10257  ax-addass 10258  ax-mulass 10259  ax-distr 10260  ax-i2m1 10261  ax-1ne0 10262  ax-1rid 10263  ax-rnegex 10264  ax-rrecex 10265  ax-cnre 10266  ax-pre-lttri 10267  ax-pre-lttrn 10268  ax-pre-ltadd 10269  ax-pre-mulgt0 10270  ax-pre-sup 10271
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3599  df-csb 3694  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-pss 3750  df-nul 4082  df-if 4246  df-pw 4319  df-sn 4337  df-pr 4339  df-tp 4341  df-op 4343  df-uni 4597  df-int 4636  df-iun 4680  df-iin 4681  df-br 4812  df-opab 4874  df-mpt 4891  df-tr 4914  df-id 5187  df-eprel 5192  df-po 5200  df-so 5201  df-fr 5238  df-se 5239  df-we 5240  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-res 5291  df-ima 5292  df-pred 5867  df-ord 5913  df-on 5914  df-lim 5915  df-suc 5916  df-iota 6033  df-fun 6072  df-fn 6073  df-f 6074  df-f1 6075  df-fo 6076  df-f1o 6077  df-fv 6078  df-isom 6079  df-riota 6807  df-ov 6849  df-oprab 6850  df-mpt2 6851  df-om 7268  df-1st 7370  df-2nd 7371  df-wrecs 7614  df-recs 7676  df-rdg 7714  df-1o 7768  df-oadd 7772  df-omul 7773  df-er 7951  df-map 8066  df-pm 8067  df-en 8165  df-dom 8166  df-sdom 8167  df-fin 8168  df-sup 8559  df-inf 8560  df-oi 8626  df-card 9020  df-acn 9023  df-ac 9194  df-pnf 10334  df-mnf 10335  df-xr 10336  df-ltxr 10337  df-le 10338  df-sub 10526  df-neg 10527  df-div 10943  df-nn 11279  df-2 11339  df-3 11340  df-n0 11543  df-z 11629  df-uz 11892  df-q 11995  df-rp 12034  df-ioo 12386  df-ioc 12387  df-ico 12388  df-fz 12539  df-fl 12806  df-ceil 12807  df-seq 13014  df-exp 13073  df-cj 14138  df-re 14139  df-im 14140  df-sqrt 14274  df-abs 14275  df-limsup 14501  df-clim 14518  df-rlim 14519  df-rest 16363  df-topgen 16384  df-top 20992  df-bases 21044  df-salg 41190  df-salgen 41194  df-smblfn 41574
This theorem is referenced by:  smfliminflem  41700
  Copyright terms: Public domain W3C validator