Proof of Theorem smfliminflem
| Step | Hyp | Ref
| Expression |
| 1 | | smfliminflem.g |
. . . 4
⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ (lim inf‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)))) |
| 2 | 1 | a1i 11 |
. . 3
⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐷 ↦ (lim inf‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))))) |
| 3 | | smfliminflem.d |
. . . . . . . . . 10
⊢ 𝐷 = {𝑥 ∈ ∪
𝑛 ∈ 𝑍 ∩ 𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ (lim inf‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ} |
| 4 | | ssrab2 4080 |
. . . . . . . . . 10
⊢ {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ (lim inf‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ} ⊆ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚) |
| 5 | 3, 4 | eqsstri 4030 |
. . . . . . . . 9
⊢ 𝐷 ⊆ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚) |
| 6 | | id 22 |
. . . . . . . . 9
⊢ (𝑥 ∈ 𝐷 → 𝑥 ∈ 𝐷) |
| 7 | 5, 6 | sselid 3981 |
. . . . . . . 8
⊢ (𝑥 ∈ 𝐷 → 𝑥 ∈ ∪
𝑛 ∈ 𝑍 ∩ 𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚)) |
| 8 | | smfliminflem.z |
. . . . . . . . 9
⊢ 𝑍 =
(ℤ≥‘𝑀) |
| 9 | | eqid 2737 |
. . . . . . . . 9
⊢ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚) = ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚) |
| 10 | 8, 9 | allbutfi 45404 |
. . . . . . . 8
⊢ (𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚) ↔ ∃𝑛 ∈ 𝑍 ∀𝑚 ∈ (ℤ≥‘𝑛)𝑥 ∈ dom (𝐹‘𝑚)) |
| 11 | 7, 10 | sylib 218 |
. . . . . . 7
⊢ (𝑥 ∈ 𝐷 → ∃𝑛 ∈ 𝑍 ∀𝑚 ∈ (ℤ≥‘𝑛)𝑥 ∈ dom (𝐹‘𝑚)) |
| 12 | 11 | adantl 481 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ∃𝑛 ∈ 𝑍 ∀𝑚 ∈ (ℤ≥‘𝑛)𝑥 ∈ dom (𝐹‘𝑚)) |
| 13 | | nfv 1914 |
. . . . . . . . . 10
⊢
Ⅎ𝑚(𝜑 ∧ 𝑛 ∈ 𝑍) |
| 14 | | nfra1 3284 |
. . . . . . . . . 10
⊢
Ⅎ𝑚∀𝑚 ∈ (ℤ≥‘𝑛)𝑥 ∈ dom (𝐹‘𝑚) |
| 15 | 13, 14 | nfan 1899 |
. . . . . . . . 9
⊢
Ⅎ𝑚((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ ∀𝑚 ∈ (ℤ≥‘𝑛)𝑥 ∈ dom (𝐹‘𝑚)) |
| 16 | 8 | fvexi 6920 |
. . . . . . . . . 10
⊢ 𝑍 ∈ V |
| 17 | 16 | a1i 11 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ ∀𝑚 ∈ (ℤ≥‘𝑛)𝑥 ∈ dom (𝐹‘𝑚)) → 𝑍 ∈ V) |
| 18 | 8 | eluzelz2 45414 |
. . . . . . . . . . 11
⊢ (𝑛 ∈ 𝑍 → 𝑛 ∈ ℤ) |
| 19 | 18 | zred 12722 |
. . . . . . . . . 10
⊢ (𝑛 ∈ 𝑍 → 𝑛 ∈ ℝ) |
| 20 | 19 | ad2antlr 727 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ ∀𝑚 ∈ (ℤ≥‘𝑛)𝑥 ∈ dom (𝐹‘𝑚)) → 𝑛 ∈ ℝ) |
| 21 | | simpll 767 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ ∀𝑚 ∈ (ℤ≥‘𝑛)𝑥 ∈ dom (𝐹‘𝑚)) → 𝜑) |
| 22 | | elinel1 4201 |
. . . . . . . . . . 11
⊢ (𝑚 ∈ (𝑍 ∩ (𝑛[,)+∞)) → 𝑚 ∈ 𝑍) |
| 23 | | smfliminflem.s |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝑆 ∈ SAlg) |
| 24 | 23 | adantr 480 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → 𝑆 ∈ SAlg) |
| 25 | | smfliminflem.f |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆)) |
| 26 | 25 | ffvelcdmda 7104 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (𝐹‘𝑚) ∈ (SMblFn‘𝑆)) |
| 27 | | eqid 2737 |
. . . . . . . . . . . 12
⊢ dom
(𝐹‘𝑚) = dom (𝐹‘𝑚) |
| 28 | 24, 26, 27 | smff 46747 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (𝐹‘𝑚):dom (𝐹‘𝑚)⟶ℝ) |
| 29 | 21, 22, 28 | syl2an 596 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ ∀𝑚 ∈ (ℤ≥‘𝑛)𝑥 ∈ dom (𝐹‘𝑚)) ∧ 𝑚 ∈ (𝑍 ∩ (𝑛[,)+∞))) → (𝐹‘𝑚):dom (𝐹‘𝑚)⟶ℝ) |
| 30 | | simplr 769 |
. . . . . . . . . . . 12
⊢ (((𝑛 ∈ 𝑍 ∧ ∀𝑚 ∈ (ℤ≥‘𝑛)𝑥 ∈ dom (𝐹‘𝑚)) ∧ 𝑚 ∈ (𝑍 ∩ (𝑛[,)+∞))) → ∀𝑚 ∈
(ℤ≥‘𝑛)𝑥 ∈ dom (𝐹‘𝑚)) |
| 31 | | eqid 2737 |
. . . . . . . . . . . . . 14
⊢
(ℤ≥‘𝑛) = (ℤ≥‘𝑛) |
| 32 | 18 | adantr 480 |
. . . . . . . . . . . . . 14
⊢ ((𝑛 ∈ 𝑍 ∧ 𝑚 ∈ (𝑍 ∩ (𝑛[,)+∞))) → 𝑛 ∈ ℤ) |
| 33 | 8, 22 | eluzelz2d 45424 |
. . . . . . . . . . . . . . 15
⊢ (𝑚 ∈ (𝑍 ∩ (𝑛[,)+∞)) → 𝑚 ∈ ℤ) |
| 34 | 33 | adantl 481 |
. . . . . . . . . . . . . 14
⊢ ((𝑛 ∈ 𝑍 ∧ 𝑚 ∈ (𝑍 ∩ (𝑛[,)+∞))) → 𝑚 ∈ ℤ) |
| 35 | 19 | rexrd 11311 |
. . . . . . . . . . . . . . . 16
⊢ (𝑛 ∈ 𝑍 → 𝑛 ∈ ℝ*) |
| 36 | 35 | adantr 480 |
. . . . . . . . . . . . . . 15
⊢ ((𝑛 ∈ 𝑍 ∧ 𝑚 ∈ (𝑍 ∩ (𝑛[,)+∞))) → 𝑛 ∈ ℝ*) |
| 37 | | pnfxr 11315 |
. . . . . . . . . . . . . . . 16
⊢ +∞
∈ ℝ* |
| 38 | 37 | a1i 11 |
. . . . . . . . . . . . . . 15
⊢ ((𝑛 ∈ 𝑍 ∧ 𝑚 ∈ (𝑍 ∩ (𝑛[,)+∞))) → +∞ ∈
ℝ*) |
| 39 | | elinel2 4202 |
. . . . . . . . . . . . . . . 16
⊢ (𝑚 ∈ (𝑍 ∩ (𝑛[,)+∞)) → 𝑚 ∈ (𝑛[,)+∞)) |
| 40 | 39 | adantl 481 |
. . . . . . . . . . . . . . 15
⊢ ((𝑛 ∈ 𝑍 ∧ 𝑚 ∈ (𝑍 ∩ (𝑛[,)+∞))) → 𝑚 ∈ (𝑛[,)+∞)) |
| 41 | 36, 38, 40 | icogelbd 45571 |
. . . . . . . . . . . . . 14
⊢ ((𝑛 ∈ 𝑍 ∧ 𝑚 ∈ (𝑍 ∩ (𝑛[,)+∞))) → 𝑛 ≤ 𝑚) |
| 42 | 31, 32, 34, 41 | eluzd 45420 |
. . . . . . . . . . . . 13
⊢ ((𝑛 ∈ 𝑍 ∧ 𝑚 ∈ (𝑍 ∩ (𝑛[,)+∞))) → 𝑚 ∈ (ℤ≥‘𝑛)) |
| 43 | 42 | adantlr 715 |
. . . . . . . . . . . 12
⊢ (((𝑛 ∈ 𝑍 ∧ ∀𝑚 ∈ (ℤ≥‘𝑛)𝑥 ∈ dom (𝐹‘𝑚)) ∧ 𝑚 ∈ (𝑍 ∩ (𝑛[,)+∞))) → 𝑚 ∈ (ℤ≥‘𝑛)) |
| 44 | | rspa 3248 |
. . . . . . . . . . . 12
⊢
((∀𝑚 ∈
(ℤ≥‘𝑛)𝑥 ∈ dom (𝐹‘𝑚) ∧ 𝑚 ∈ (ℤ≥‘𝑛)) → 𝑥 ∈ dom (𝐹‘𝑚)) |
| 45 | 30, 43, 44 | syl2anc 584 |
. . . . . . . . . . 11
⊢ (((𝑛 ∈ 𝑍 ∧ ∀𝑚 ∈ (ℤ≥‘𝑛)𝑥 ∈ dom (𝐹‘𝑚)) ∧ 𝑚 ∈ (𝑍 ∩ (𝑛[,)+∞))) → 𝑥 ∈ dom (𝐹‘𝑚)) |
| 46 | 45 | adantlll 718 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ ∀𝑚 ∈ (ℤ≥‘𝑛)𝑥 ∈ dom (𝐹‘𝑚)) ∧ 𝑚 ∈ (𝑍 ∩ (𝑛[,)+∞))) → 𝑥 ∈ dom (𝐹‘𝑚)) |
| 47 | 29, 46 | ffvelcdmd 7105 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ ∀𝑚 ∈ (ℤ≥‘𝑛)𝑥 ∈ dom (𝐹‘𝑚)) ∧ 𝑚 ∈ (𝑍 ∩ (𝑛[,)+∞))) → ((𝐹‘𝑚)‘𝑥) ∈ ℝ) |
| 48 | 15, 17, 20, 47 | liminfval4 45804 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ ∀𝑚 ∈ (ℤ≥‘𝑛)𝑥 ∈ dom (𝐹‘𝑚)) → (lim inf‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) = -𝑒(lim
sup‘(𝑚 ∈ 𝑍 ↦ -((𝐹‘𝑚)‘𝑥)))) |
| 49 | 48 | rexlimdva2 3157 |
. . . . . . 7
⊢ (𝜑 → (∃𝑛 ∈ 𝑍 ∀𝑚 ∈ (ℤ≥‘𝑛)𝑥 ∈ dom (𝐹‘𝑚) → (lim inf‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) = -𝑒(lim
sup‘(𝑚 ∈ 𝑍 ↦ -((𝐹‘𝑚)‘𝑥))))) |
| 50 | 49 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (∃𝑛 ∈ 𝑍 ∀𝑚 ∈ (ℤ≥‘𝑛)𝑥 ∈ dom (𝐹‘𝑚) → (lim inf‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) = -𝑒(lim
sup‘(𝑚 ∈ 𝑍 ↦ -((𝐹‘𝑚)‘𝑥))))) |
| 51 | 12, 50 | mpd 15 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (lim inf‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) = -𝑒(lim
sup‘(𝑚 ∈ 𝑍 ↦ -((𝐹‘𝑚)‘𝑥)))) |
| 52 | 51 | xnegeqd 45448 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → -𝑒(lim
inf‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) =
-𝑒-𝑒(lim sup‘(𝑚 ∈ 𝑍 ↦ -((𝐹‘𝑚)‘𝑥)))) |
| 53 | 16 | mptex 7243 |
. . . . . . . . . . . 12
⊢ (𝑚 ∈ 𝑍 ↦ -((𝐹‘𝑚)‘𝑥)) ∈ V |
| 54 | 53 | limsupcli 45772 |
. . . . . . . . . . 11
⊢ (lim
sup‘(𝑚 ∈ 𝑍 ↦ -((𝐹‘𝑚)‘𝑥))) ∈
ℝ* |
| 55 | 54 | xnegnegi 45450 |
. . . . . . . . . 10
⊢
-𝑒-𝑒(lim sup‘(𝑚 ∈ 𝑍 ↦ -((𝐹‘𝑚)‘𝑥))) = (lim sup‘(𝑚 ∈ 𝑍 ↦ -((𝐹‘𝑚)‘𝑥))) |
| 56 | 55 | a1i 11 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) →
-𝑒-𝑒(lim sup‘(𝑚 ∈ 𝑍 ↦ -((𝐹‘𝑚)‘𝑥))) = (lim sup‘(𝑚 ∈ 𝑍 ↦ -((𝐹‘𝑚)‘𝑥)))) |
| 57 | 52, 56 | eqtr2d 2778 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (lim sup‘(𝑚 ∈ 𝑍 ↦ -((𝐹‘𝑚)‘𝑥))) = -𝑒(lim
inf‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)))) |
| 58 | 3 | reqabi 3460 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ 𝐷 ↔ (𝑥 ∈ ∪
𝑛 ∈ 𝑍 ∩ 𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚) ∧ (lim inf‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ)) |
| 59 | 58 | simprbi 496 |
. . . . . . . . . 10
⊢ (𝑥 ∈ 𝐷 → (lim inf‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ) |
| 60 | 59 | adantl 481 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (lim inf‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ) |
| 61 | 60 | rexnegd 45148 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → -𝑒(lim
inf‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) = -(lim inf‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)))) |
| 62 | 57, 61 | eqtr2d 2778 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → -(lim inf‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) = (lim sup‘(𝑚 ∈ 𝑍 ↦ -((𝐹‘𝑚)‘𝑥)))) |
| 63 | 60 | renegcld 11690 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → -(lim inf‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ) |
| 64 | 62, 63 | eqeltrrd 2842 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (lim sup‘(𝑚 ∈ 𝑍 ↦ -((𝐹‘𝑚)‘𝑥))) ∈ ℝ) |
| 65 | 64 | rexnegd 45148 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → -𝑒(lim
sup‘(𝑚 ∈ 𝑍 ↦ -((𝐹‘𝑚)‘𝑥))) = -(lim sup‘(𝑚 ∈ 𝑍 ↦ -((𝐹‘𝑚)‘𝑥)))) |
| 66 | 51, 65 | eqtrd 2777 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (lim inf‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) = -(lim sup‘(𝑚 ∈ 𝑍 ↦ -((𝐹‘𝑚)‘𝑥)))) |
| 67 | 66 | mpteq2dva 5242 |
. . 3
⊢ (𝜑 → (𝑥 ∈ 𝐷 ↦ (lim inf‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)))) = (𝑥 ∈ 𝐷 ↦ -(lim sup‘(𝑚 ∈ 𝑍 ↦ -((𝐹‘𝑚)‘𝑥))))) |
| 68 | 2, 67 | eqtrd 2777 |
. 2
⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐷 ↦ -(lim sup‘(𝑚 ∈ 𝑍 ↦ -((𝐹‘𝑚)‘𝑥))))) |
| 69 | | nfv 1914 |
. . 3
⊢
Ⅎ𝑥𝜑 |
| 70 | 18, 31 | uzn0d 45436 |
. . . . . . . 8
⊢ (𝑛 ∈ 𝑍 → (ℤ≥‘𝑛) ≠ ∅) |
| 71 | | fvex 6919 |
. . . . . . . . . . 11
⊢ (𝐹‘𝑚) ∈ V |
| 72 | 71 | dmex 7931 |
. . . . . . . . . 10
⊢ dom
(𝐹‘𝑚) ∈ V |
| 73 | 72 | rgenw 3065 |
. . . . . . . . 9
⊢
∀𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚) ∈ V |
| 74 | 73 | a1i 11 |
. . . . . . . 8
⊢ (𝑛 ∈ 𝑍 → ∀𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∈ V) |
| 75 | | iinexg 5348 |
. . . . . . . 8
⊢
(((ℤ≥‘𝑛) ≠ ∅ ∧ ∀𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚) ∈ V) → ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∈ V) |
| 76 | 70, 74, 75 | syl2anc 584 |
. . . . . . 7
⊢ (𝑛 ∈ 𝑍 → ∩
𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚) ∈ V) |
| 77 | 76 | rgen 3063 |
. . . . . 6
⊢
∀𝑛 ∈
𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∈ V |
| 78 | | iunexg 7988 |
. . . . . 6
⊢ ((𝑍 ∈ V ∧ ∀𝑛 ∈ 𝑍 ∩ 𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚) ∈ V) → ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚) ∈ V) |
| 79 | 16, 77, 78 | mp2an 692 |
. . . . 5
⊢ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚) ∈ V |
| 80 | 79, 5 | ssexi 5322 |
. . . 4
⊢ 𝐷 ∈ V |
| 81 | 80 | a1i 11 |
. . 3
⊢ (𝜑 → 𝐷 ∈ V) |
| 82 | 3 | a1i 11 |
. . . . . 6
⊢ (𝜑 → 𝐷 = {𝑥 ∈ ∪
𝑛 ∈ 𝑍 ∩ 𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ (lim inf‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ}) |
| 83 | 10 | biimpi 216 |
. . . . . . . . 9
⊢ (𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚) → ∃𝑛 ∈ 𝑍 ∀𝑚 ∈ (ℤ≥‘𝑛)𝑥 ∈ dom (𝐹‘𝑚)) |
| 84 | 49 | imp 406 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ∃𝑛 ∈ 𝑍 ∀𝑚 ∈ (ℤ≥‘𝑛)𝑥 ∈ dom (𝐹‘𝑚)) → (lim inf‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) = -𝑒(lim
sup‘(𝑚 ∈ 𝑍 ↦ -((𝐹‘𝑚)‘𝑥)))) |
| 85 | 83, 84 | sylan2 593 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ ∪
𝑛 ∈ 𝑍 ∩ 𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚)) → (lim inf‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) = -𝑒(lim
sup‘(𝑚 ∈ 𝑍 ↦ -((𝐹‘𝑚)‘𝑥)))) |
| 86 | 54 | a1i 11 |
. . . . . . . . . 10
⊢ (((lim
inf‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) = -𝑒(lim
sup‘(𝑚 ∈ 𝑍 ↦ -((𝐹‘𝑚)‘𝑥))) ∧ (lim inf‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ) → (lim
sup‘(𝑚 ∈ 𝑍 ↦ -((𝐹‘𝑚)‘𝑥))) ∈
ℝ*) |
| 87 | | simpl 482 |
. . . . . . . . . . 11
⊢ (((lim
inf‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) = -𝑒(lim
sup‘(𝑚 ∈ 𝑍 ↦ -((𝐹‘𝑚)‘𝑥))) ∧ (lim inf‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ) → (lim
inf‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) = -𝑒(lim
sup‘(𝑚 ∈ 𝑍 ↦ -((𝐹‘𝑚)‘𝑥)))) |
| 88 | | simpr 484 |
. . . . . . . . . . 11
⊢ (((lim
inf‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) = -𝑒(lim
sup‘(𝑚 ∈ 𝑍 ↦ -((𝐹‘𝑚)‘𝑥))) ∧ (lim inf‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ) → (lim
inf‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ) |
| 89 | 87, 88 | eqeltrrd 2842 |
. . . . . . . . . 10
⊢ (((lim
inf‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) = -𝑒(lim
sup‘(𝑚 ∈ 𝑍 ↦ -((𝐹‘𝑚)‘𝑥))) ∧ (lim inf‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ) →
-𝑒(lim sup‘(𝑚 ∈ 𝑍 ↦ -((𝐹‘𝑚)‘𝑥))) ∈ ℝ) |
| 90 | | xnegrecl2 45471 |
. . . . . . . . . 10
⊢ (((lim
sup‘(𝑚 ∈ 𝑍 ↦ -((𝐹‘𝑚)‘𝑥))) ∈ ℝ* ∧
-𝑒(lim sup‘(𝑚 ∈ 𝑍 ↦ -((𝐹‘𝑚)‘𝑥))) ∈ ℝ) → (lim
sup‘(𝑚 ∈ 𝑍 ↦ -((𝐹‘𝑚)‘𝑥))) ∈ ℝ) |
| 91 | 86, 89, 90 | syl2anc 584 |
. . . . . . . . 9
⊢ (((lim
inf‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) = -𝑒(lim
sup‘(𝑚 ∈ 𝑍 ↦ -((𝐹‘𝑚)‘𝑥))) ∧ (lim inf‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ) → (lim
sup‘(𝑚 ∈ 𝑍 ↦ -((𝐹‘𝑚)‘𝑥))) ∈ ℝ) |
| 92 | | simpl 482 |
. . . . . . . . . 10
⊢ (((lim
inf‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) = -𝑒(lim
sup‘(𝑚 ∈ 𝑍 ↦ -((𝐹‘𝑚)‘𝑥))) ∧ (lim sup‘(𝑚 ∈ 𝑍 ↦ -((𝐹‘𝑚)‘𝑥))) ∈ ℝ) → (lim
inf‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) = -𝑒(lim
sup‘(𝑚 ∈ 𝑍 ↦ -((𝐹‘𝑚)‘𝑥)))) |
| 93 | | xnegrecl 45449 |
. . . . . . . . . . 11
⊢ ((lim
sup‘(𝑚 ∈ 𝑍 ↦ -((𝐹‘𝑚)‘𝑥))) ∈ ℝ →
-𝑒(lim sup‘(𝑚 ∈ 𝑍 ↦ -((𝐹‘𝑚)‘𝑥))) ∈ ℝ) |
| 94 | 93 | adantl 481 |
. . . . . . . . . 10
⊢ (((lim
inf‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) = -𝑒(lim
sup‘(𝑚 ∈ 𝑍 ↦ -((𝐹‘𝑚)‘𝑥))) ∧ (lim sup‘(𝑚 ∈ 𝑍 ↦ -((𝐹‘𝑚)‘𝑥))) ∈ ℝ) →
-𝑒(lim sup‘(𝑚 ∈ 𝑍 ↦ -((𝐹‘𝑚)‘𝑥))) ∈ ℝ) |
| 95 | 92, 94 | eqeltrd 2841 |
. . . . . . . . 9
⊢ (((lim
inf‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) = -𝑒(lim
sup‘(𝑚 ∈ 𝑍 ↦ -((𝐹‘𝑚)‘𝑥))) ∧ (lim sup‘(𝑚 ∈ 𝑍 ↦ -((𝐹‘𝑚)‘𝑥))) ∈ ℝ) → (lim
inf‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ) |
| 96 | 91, 95 | impbida 801 |
. . . . . . . 8
⊢ ((lim
inf‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) = -𝑒(lim
sup‘(𝑚 ∈ 𝑍 ↦ -((𝐹‘𝑚)‘𝑥))) → ((lim inf‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ ↔ (lim
sup‘(𝑚 ∈ 𝑍 ↦ -((𝐹‘𝑚)‘𝑥))) ∈ ℝ)) |
| 97 | 85, 96 | syl 17 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ ∪
𝑛 ∈ 𝑍 ∩ 𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚)) → ((lim inf‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ ↔ (lim
sup‘(𝑚 ∈ 𝑍 ↦ -((𝐹‘𝑚)‘𝑥))) ∈ ℝ)) |
| 98 | 97 | rabbidva 3443 |
. . . . . 6
⊢ (𝜑 → {𝑥 ∈ ∪
𝑛 ∈ 𝑍 ∩ 𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ (lim inf‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ} = {𝑥 ∈ ∪
𝑛 ∈ 𝑍 ∩ 𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ (lim sup‘(𝑚 ∈ 𝑍 ↦ -((𝐹‘𝑚)‘𝑥))) ∈ ℝ}) |
| 99 | 82, 98 | eqtrd 2777 |
. . . . 5
⊢ (𝜑 → 𝐷 = {𝑥 ∈ ∪
𝑛 ∈ 𝑍 ∩ 𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ (lim sup‘(𝑚 ∈ 𝑍 ↦ -((𝐹‘𝑚)‘𝑥))) ∈ ℝ}) |
| 100 | 69, 99 | mpteq1df 45241 |
. . . 4
⊢ (𝜑 → (𝑥 ∈ 𝐷 ↦ (lim sup‘(𝑚 ∈ 𝑍 ↦ -((𝐹‘𝑚)‘𝑥)))) = (𝑥 ∈ {𝑥 ∈ ∪
𝑛 ∈ 𝑍 ∩ 𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ (lim sup‘(𝑚 ∈ 𝑍 ↦ -((𝐹‘𝑚)‘𝑥))) ∈ ℝ} ↦ (lim
sup‘(𝑚 ∈ 𝑍 ↦ -((𝐹‘𝑚)‘𝑥))))) |
| 101 | | nfv 1914 |
. . . . 5
⊢
Ⅎ𝑚𝜑 |
| 102 | | nfv 1914 |
. . . . 5
⊢
Ⅎ𝑛𝜑 |
| 103 | | smfliminflem.m |
. . . . 5
⊢ (𝜑 → 𝑀 ∈ ℤ) |
| 104 | | negex 11506 |
. . . . . 6
⊢ -((𝐹‘𝑚)‘𝑥) ∈ V |
| 105 | 104 | a1i 11 |
. . . . 5
⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍 ∧ 𝑥 ∈ dom (𝐹‘𝑚)) → -((𝐹‘𝑚)‘𝑥) ∈ V) |
| 106 | | nfv 1914 |
. . . . . 6
⊢
Ⅎ𝑥(𝜑 ∧ 𝑚 ∈ 𝑍) |
| 107 | 72 | a1i 11 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → dom (𝐹‘𝑚) ∈ V) |
| 108 | 28 | ffvelcdmda 7104 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑚 ∈ 𝑍) ∧ 𝑥 ∈ dom (𝐹‘𝑚)) → ((𝐹‘𝑚)‘𝑥) ∈ ℝ) |
| 109 | 28 | feqmptd 6977 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (𝐹‘𝑚) = (𝑥 ∈ dom (𝐹‘𝑚) ↦ ((𝐹‘𝑚)‘𝑥))) |
| 110 | 109, 26 | eqeltrrd 2842 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (𝑥 ∈ dom (𝐹‘𝑚) ↦ ((𝐹‘𝑚)‘𝑥)) ∈ (SMblFn‘𝑆)) |
| 111 | 106, 24, 107, 108, 110 | smfneg 46818 |
. . . . 5
⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (𝑥 ∈ dom (𝐹‘𝑚) ↦ -((𝐹‘𝑚)‘𝑥)) ∈ (SMblFn‘𝑆)) |
| 112 | | eqid 2737 |
. . . . 5
⊢ {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ (lim sup‘(𝑚 ∈ 𝑍 ↦ -((𝐹‘𝑚)‘𝑥))) ∈ ℝ} = {𝑥 ∈ ∪
𝑛 ∈ 𝑍 ∩ 𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ (lim sup‘(𝑚 ∈ 𝑍 ↦ -((𝐹‘𝑚)‘𝑥))) ∈ ℝ} |
| 113 | | eqid 2737 |
. . . . 5
⊢ (𝑥 ∈ {𝑥 ∈ ∪
𝑛 ∈ 𝑍 ∩ 𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ (lim sup‘(𝑚 ∈ 𝑍 ↦ -((𝐹‘𝑚)‘𝑥))) ∈ ℝ} ↦ (lim
sup‘(𝑚 ∈ 𝑍 ↦ -((𝐹‘𝑚)‘𝑥)))) = (𝑥 ∈ {𝑥 ∈ ∪
𝑛 ∈ 𝑍 ∩ 𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ (lim sup‘(𝑚 ∈ 𝑍 ↦ -((𝐹‘𝑚)‘𝑥))) ∈ ℝ} ↦ (lim
sup‘(𝑚 ∈ 𝑍 ↦ -((𝐹‘𝑚)‘𝑥)))) |
| 114 | 101, 69, 102, 103, 8, 23, 105, 111, 112, 113 | smflimsupmpt 46844 |
. . . 4
⊢ (𝜑 → (𝑥 ∈ {𝑥 ∈ ∪
𝑛 ∈ 𝑍 ∩ 𝑚 ∈
(ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ (lim sup‘(𝑚 ∈ 𝑍 ↦ -((𝐹‘𝑚)‘𝑥))) ∈ ℝ} ↦ (lim
sup‘(𝑚 ∈ 𝑍 ↦ -((𝐹‘𝑚)‘𝑥)))) ∈ (SMblFn‘𝑆)) |
| 115 | 100, 114 | eqeltrd 2841 |
. . 3
⊢ (𝜑 → (𝑥 ∈ 𝐷 ↦ (lim sup‘(𝑚 ∈ 𝑍 ↦ -((𝐹‘𝑚)‘𝑥)))) ∈ (SMblFn‘𝑆)) |
| 116 | 69, 23, 81, 64, 115 | smfneg 46818 |
. 2
⊢ (𝜑 → (𝑥 ∈ 𝐷 ↦ -(lim sup‘(𝑚 ∈ 𝑍 ↦ -((𝐹‘𝑚)‘𝑥)))) ∈ (SMblFn‘𝑆)) |
| 117 | 68, 116 | eqeltrd 2841 |
1
⊢ (𝜑 → 𝐺 ∈ (SMblFn‘𝑆)) |