Step | Hyp | Ref
| Expression |
1 | | reex 9906 |
. . . . 5
⊢ ℝ
∈ V |
2 | 1 | mptex 6390 |
. . . 4
⊢ (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (-𝑚[,]𝑚), if((𝑚𝐽𝑥) ≤ 𝑚, (𝑚𝐽𝑥), 𝑚), 0)) ∈ V |
3 | | mbfi1fseq.4 |
. . . 4
⊢ 𝐺 = (𝑚 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (-𝑚[,]𝑚), if((𝑚𝐽𝑥) ≤ 𝑚, (𝑚𝐽𝑥), 𝑚), 0))) |
4 | 2, 3 | fnmpti 5935 |
. . 3
⊢ 𝐺 Fn ℕ |
5 | 4 | a1i 11 |
. 2
⊢ (𝜑 → 𝐺 Fn ℕ) |
6 | | mbfi1fseq.1 |
. . . . . 6
⊢ (𝜑 → 𝐹 ∈ MblFn) |
7 | | mbfi1fseq.2 |
. . . . . 6
⊢ (𝜑 → 𝐹:ℝ⟶(0[,)+∞)) |
8 | | mbfi1fseq.3 |
. . . . . 6
⊢ 𝐽 = (𝑚 ∈ ℕ, 𝑦 ∈ ℝ ↦
((⌊‘((𝐹‘𝑦) · (2↑𝑚))) / (2↑𝑚))) |
9 | 6, 7, 8, 3 | mbfi1fseqlem3 23290 |
. . . . 5
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐺‘𝑛):ℝ⟶ran (𝑚 ∈ (0...(𝑛 · (2↑𝑛))) ↦ (𝑚 / (2↑𝑛)))) |
10 | | elfznn0 12302 |
. . . . . . . . 9
⊢ (𝑚 ∈ (0...(𝑛 · (2↑𝑛))) → 𝑚 ∈ ℕ0) |
11 | 10 | nn0red 11229 |
. . . . . . . 8
⊢ (𝑚 ∈ (0...(𝑛 · (2↑𝑛))) → 𝑚 ∈ ℝ) |
12 | | 2nn 11062 |
. . . . . . . . . 10
⊢ 2 ∈
ℕ |
13 | | nnnn0 11176 |
. . . . . . . . . 10
⊢ (𝑛 ∈ ℕ → 𝑛 ∈
ℕ0) |
14 | | nnexpcl 12735 |
. . . . . . . . . 10
⊢ ((2
∈ ℕ ∧ 𝑛
∈ ℕ0) → (2↑𝑛) ∈ ℕ) |
15 | 12, 13, 14 | sylancr 694 |
. . . . . . . . 9
⊢ (𝑛 ∈ ℕ →
(2↑𝑛) ∈
ℕ) |
16 | 15 | adantl 481 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2↑𝑛) ∈
ℕ) |
17 | | nndivre 10933 |
. . . . . . . 8
⊢ ((𝑚 ∈ ℝ ∧
(2↑𝑛) ∈ ℕ)
→ (𝑚 / (2↑𝑛)) ∈
ℝ) |
18 | 11, 16, 17 | syl2anr 494 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑚 ∈ (0...(𝑛 · (2↑𝑛)))) → (𝑚 / (2↑𝑛)) ∈ ℝ) |
19 | | eqid 2610 |
. . . . . . 7
⊢ (𝑚 ∈ (0...(𝑛 · (2↑𝑛))) ↦ (𝑚 / (2↑𝑛))) = (𝑚 ∈ (0...(𝑛 · (2↑𝑛))) ↦ (𝑚 / (2↑𝑛))) |
20 | 18, 19 | fmptd 6292 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑚 ∈ (0...(𝑛 · (2↑𝑛))) ↦ (𝑚 / (2↑𝑛))):(0...(𝑛 · (2↑𝑛)))⟶ℝ) |
21 | | frn 5966 |
. . . . . 6
⊢ ((𝑚 ∈ (0...(𝑛 · (2↑𝑛))) ↦ (𝑚 / (2↑𝑛))):(0...(𝑛 · (2↑𝑛)))⟶ℝ → ran (𝑚 ∈ (0...(𝑛 · (2↑𝑛))) ↦ (𝑚 / (2↑𝑛))) ⊆ ℝ) |
22 | 20, 21 | syl 17 |
. . . . 5
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ran (𝑚 ∈ (0...(𝑛 · (2↑𝑛))) ↦ (𝑚 / (2↑𝑛))) ⊆ ℝ) |
23 | 9, 22 | fssd 5970 |
. . . 4
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐺‘𝑛):ℝ⟶ℝ) |
24 | | fzfid 12634 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (0...(𝑛 · (2↑𝑛))) ∈ Fin) |
25 | | ffn 5958 |
. . . . . . . 8
⊢ ((𝑚 ∈ (0...(𝑛 · (2↑𝑛))) ↦ (𝑚 / (2↑𝑛))):(0...(𝑛 · (2↑𝑛)))⟶ℝ → (𝑚 ∈ (0...(𝑛 · (2↑𝑛))) ↦ (𝑚 / (2↑𝑛))) Fn (0...(𝑛 · (2↑𝑛)))) |
26 | 20, 25 | syl 17 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑚 ∈ (0...(𝑛 · (2↑𝑛))) ↦ (𝑚 / (2↑𝑛))) Fn (0...(𝑛 · (2↑𝑛)))) |
27 | | dffn4 6034 |
. . . . . . 7
⊢ ((𝑚 ∈ (0...(𝑛 · (2↑𝑛))) ↦ (𝑚 / (2↑𝑛))) Fn (0...(𝑛 · (2↑𝑛))) ↔ (𝑚 ∈ (0...(𝑛 · (2↑𝑛))) ↦ (𝑚 / (2↑𝑛))):(0...(𝑛 · (2↑𝑛)))–onto→ran (𝑚 ∈ (0...(𝑛 · (2↑𝑛))) ↦ (𝑚 / (2↑𝑛)))) |
28 | 26, 27 | sylib 207 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑚 ∈ (0...(𝑛 · (2↑𝑛))) ↦ (𝑚 / (2↑𝑛))):(0...(𝑛 · (2↑𝑛)))–onto→ran (𝑚 ∈ (0...(𝑛 · (2↑𝑛))) ↦ (𝑚 / (2↑𝑛)))) |
29 | | fofi 8135 |
. . . . . 6
⊢
(((0...(𝑛 ·
(2↑𝑛))) ∈ Fin
∧ (𝑚 ∈ (0...(𝑛 · (2↑𝑛))) ↦ (𝑚 / (2↑𝑛))):(0...(𝑛 · (2↑𝑛)))–onto→ran (𝑚 ∈ (0...(𝑛 · (2↑𝑛))) ↦ (𝑚 / (2↑𝑛)))) → ran (𝑚 ∈ (0...(𝑛 · (2↑𝑛))) ↦ (𝑚 / (2↑𝑛))) ∈ Fin) |
30 | 24, 28, 29 | syl2anc 691 |
. . . . 5
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ran (𝑚 ∈ (0...(𝑛 · (2↑𝑛))) ↦ (𝑚 / (2↑𝑛))) ∈ Fin) |
31 | | frn 5966 |
. . . . . 6
⊢ ((𝐺‘𝑛):ℝ⟶ran (𝑚 ∈ (0...(𝑛 · (2↑𝑛))) ↦ (𝑚 / (2↑𝑛))) → ran (𝐺‘𝑛) ⊆ ran (𝑚 ∈ (0...(𝑛 · (2↑𝑛))) ↦ (𝑚 / (2↑𝑛)))) |
32 | 9, 31 | syl 17 |
. . . . 5
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ran (𝐺‘𝑛) ⊆ ran (𝑚 ∈ (0...(𝑛 · (2↑𝑛))) ↦ (𝑚 / (2↑𝑛)))) |
33 | | ssfi 8065 |
. . . . 5
⊢ ((ran
(𝑚 ∈ (0...(𝑛 · (2↑𝑛))) ↦ (𝑚 / (2↑𝑛))) ∈ Fin ∧ ran (𝐺‘𝑛) ⊆ ran (𝑚 ∈ (0...(𝑛 · (2↑𝑛))) ↦ (𝑚 / (2↑𝑛)))) → ran (𝐺‘𝑛) ∈ Fin) |
34 | 30, 32, 33 | syl2anc 691 |
. . . 4
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ran (𝐺‘𝑛) ∈ Fin) |
35 | 6, 7, 8, 3 | mbfi1fseqlem2 23289 |
. . . . . . . . . . . . . 14
⊢ (𝑛 ∈ ℕ → (𝐺‘𝑛) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (-𝑛[,]𝑛), if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛), 0))) |
36 | 35 | fveq1d 6105 |
. . . . . . . . . . . . 13
⊢ (𝑛 ∈ ℕ → ((𝐺‘𝑛)‘𝑥) = ((𝑥 ∈ ℝ ↦ if(𝑥 ∈ (-𝑛[,]𝑛), if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛), 0))‘𝑥)) |
37 | 36 | ad2antlr 759 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝐺‘𝑛)‘𝑥) = ((𝑥 ∈ ℝ ↦ if(𝑥 ∈ (-𝑛[,]𝑛), if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛), 0))‘𝑥)) |
38 | | simpr 476 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → 𝑥 ∈ ℝ) |
39 | | ovex 6577 |
. . . . . . . . . . . . . . 15
⊢ (𝑛𝐽𝑥) ∈ V |
40 | | vex 3176 |
. . . . . . . . . . . . . . 15
⊢ 𝑛 ∈ V |
41 | 39, 40 | ifex 4106 |
. . . . . . . . . . . . . 14
⊢ if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛) ∈ V |
42 | | c0ex 9913 |
. . . . . . . . . . . . . 14
⊢ 0 ∈
V |
43 | 41, 42 | ifex 4106 |
. . . . . . . . . . . . 13
⊢ if(𝑥 ∈ (-𝑛[,]𝑛), if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛), 0) ∈ V |
44 | | eqid 2610 |
. . . . . . . . . . . . . 14
⊢ (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (-𝑛[,]𝑛), if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛), 0)) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (-𝑛[,]𝑛), if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛), 0)) |
45 | 44 | fvmpt2 6200 |
. . . . . . . . . . . . 13
⊢ ((𝑥 ∈ ℝ ∧ if(𝑥 ∈ (-𝑛[,]𝑛), if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛), 0) ∈ V) → ((𝑥 ∈ ℝ ↦ if(𝑥 ∈ (-𝑛[,]𝑛), if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛), 0))‘𝑥) = if(𝑥 ∈ (-𝑛[,]𝑛), if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛), 0)) |
46 | 38, 43, 45 | sylancl 693 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝑥 ∈ ℝ ↦ if(𝑥 ∈ (-𝑛[,]𝑛), if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛), 0))‘𝑥) = if(𝑥 ∈ (-𝑛[,]𝑛), if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛), 0)) |
47 | 37, 46 | eqtrd 2644 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝐺‘𝑛)‘𝑥) = if(𝑥 ∈ (-𝑛[,]𝑛), if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛), 0)) |
48 | 47 | adantlr 747 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺‘𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → ((𝐺‘𝑛)‘𝑥) = if(𝑥 ∈ (-𝑛[,]𝑛), if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛), 0)) |
49 | 48 | eqeq1d 2612 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺‘𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → (((𝐺‘𝑛)‘𝑥) = 𝑘 ↔ if(𝑥 ∈ (-𝑛[,]𝑛), if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛), 0) = 𝑘)) |
50 | | eldifsni 4261 |
. . . . . . . . . . . . 13
⊢ (𝑘 ∈ (ran (𝐺‘𝑛) ∖ {0}) → 𝑘 ≠ 0) |
51 | 50 | ad2antlr 759 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺‘𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → 𝑘 ≠ 0) |
52 | | neeq1 2844 |
. . . . . . . . . . . 12
⊢ (if(𝑥 ∈ (-𝑛[,]𝑛), if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛), 0) = 𝑘 → (if(𝑥 ∈ (-𝑛[,]𝑛), if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛), 0) ≠ 0 ↔ 𝑘 ≠ 0)) |
53 | 51, 52 | syl5ibrcom 236 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺‘𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → (if(𝑥 ∈ (-𝑛[,]𝑛), if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛), 0) = 𝑘 → if(𝑥 ∈ (-𝑛[,]𝑛), if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛), 0) ≠ 0)) |
54 | | iffalse 4045 |
. . . . . . . . . . . 12
⊢ (¬
𝑥 ∈ (-𝑛[,]𝑛) → if(𝑥 ∈ (-𝑛[,]𝑛), if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛), 0) = 0) |
55 | 54 | necon1ai 2809 |
. . . . . . . . . . 11
⊢ (if(𝑥 ∈ (-𝑛[,]𝑛), if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛), 0) ≠ 0 → 𝑥 ∈ (-𝑛[,]𝑛)) |
56 | 53, 55 | syl6 34 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺‘𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → (if(𝑥 ∈ (-𝑛[,]𝑛), if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛), 0) = 𝑘 → 𝑥 ∈ (-𝑛[,]𝑛))) |
57 | 56 | pm4.71rd 665 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺‘𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → (if(𝑥 ∈ (-𝑛[,]𝑛), if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛), 0) = 𝑘 ↔ (𝑥 ∈ (-𝑛[,]𝑛) ∧ if(𝑥 ∈ (-𝑛[,]𝑛), if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛), 0) = 𝑘))) |
58 | | iftrue 4042 |
. . . . . . . . . . . 12
⊢ (𝑥 ∈ (-𝑛[,]𝑛) → if(𝑥 ∈ (-𝑛[,]𝑛), if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛), 0) = if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛)) |
59 | 58 | eqeq1d 2612 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ (-𝑛[,]𝑛) → (if(𝑥 ∈ (-𝑛[,]𝑛), if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛), 0) = 𝑘 ↔ if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛) = 𝑘)) |
60 | | simpllr 795 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺‘𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → 𝑛 ∈ ℕ) |
61 | 60 | nnred 10912 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺‘𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → 𝑛 ∈ ℝ) |
62 | 61 | adantr 480 |
. . . . . . . . . . . . . . . 16
⊢
(((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺‘𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) ∧ 𝑘 = 𝑛) → 𝑛 ∈ ℝ) |
63 | | rge0ssre 12151 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢
(0[,)+∞) ⊆ ℝ |
64 | | simpr 476 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝑚 ∈ ℕ ∧ 𝑦 ∈ ℝ) → 𝑦 ∈
ℝ) |
65 | | ffvelrn 6265 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝐹:ℝ⟶(0[,)+∞)
∧ 𝑦 ∈ ℝ)
→ (𝐹‘𝑦) ∈
(0[,)+∞)) |
66 | 7, 64, 65 | syl2an 493 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑦 ∈ ℝ)) → (𝐹‘𝑦) ∈ (0[,)+∞)) |
67 | 63, 66 | sseldi 3566 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑦 ∈ ℝ)) → (𝐹‘𝑦) ∈ ℝ) |
68 | | nnnn0 11176 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑚 ∈ ℕ → 𝑚 ∈
ℕ0) |
69 | | nnexpcl 12735 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((2
∈ ℕ ∧ 𝑚
∈ ℕ0) → (2↑𝑚) ∈ ℕ) |
70 | 12, 68, 69 | sylancr 694 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑚 ∈ ℕ →
(2↑𝑚) ∈
ℕ) |
71 | 70 | ad2antrl 760 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑦 ∈ ℝ)) → (2↑𝑚) ∈
ℕ) |
72 | 71 | nnred 10912 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑦 ∈ ℝ)) → (2↑𝑚) ∈
ℝ) |
73 | 67, 72 | remulcld 9949 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑦 ∈ ℝ)) → ((𝐹‘𝑦) · (2↑𝑚)) ∈ ℝ) |
74 | | reflcl 12459 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (((𝐹‘𝑦) · (2↑𝑚)) ∈ ℝ →
(⌊‘((𝐹‘𝑦) · (2↑𝑚))) ∈ ℝ) |
75 | 73, 74 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑦 ∈ ℝ)) →
(⌊‘((𝐹‘𝑦) · (2↑𝑚))) ∈ ℝ) |
76 | 75, 71 | nndivred 10946 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑦 ∈ ℝ)) →
((⌊‘((𝐹‘𝑦) · (2↑𝑚))) / (2↑𝑚)) ∈ ℝ) |
77 | 76 | ralrimivva 2954 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → ∀𝑚 ∈ ℕ ∀𝑦 ∈ ℝ ((⌊‘((𝐹‘𝑦) · (2↑𝑚))) / (2↑𝑚)) ∈ ℝ) |
78 | 8 | fmpt2 7126 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(∀𝑚 ∈
ℕ ∀𝑦 ∈
ℝ ((⌊‘((𝐹‘𝑦) · (2↑𝑚))) / (2↑𝑚)) ∈ ℝ ↔ 𝐽:(ℕ ×
ℝ)⟶ℝ) |
79 | 77, 78 | sylib 207 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → 𝐽:(ℕ ×
ℝ)⟶ℝ) |
80 | | fovrn 6702 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝐽:(ℕ ×
ℝ)⟶ℝ ∧ 𝑛 ∈ ℕ ∧ 𝑥 ∈ ℝ) → (𝑛𝐽𝑥) ∈ ℝ) |
81 | 79, 80 | syl3an1 1351 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑥 ∈ ℝ) → (𝑛𝐽𝑥) ∈ ℝ) |
82 | 81 | 3expa 1257 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝑛𝐽𝑥) ∈ ℝ) |
83 | 82 | adantlr 747 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺‘𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → (𝑛𝐽𝑥) ∈ ℝ) |
84 | 83 | adantr 480 |
. . . . . . . . . . . . . . . 16
⊢
(((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺‘𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) ∧ 𝑘 = 𝑛) → (𝑛𝐽𝑥) ∈ ℝ) |
85 | | lemin 11897 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑛 ∈ ℝ ∧ (𝑛𝐽𝑥) ∈ ℝ ∧ 𝑛 ∈ ℝ) → (𝑛 ≤ if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛) ↔ (𝑛 ≤ (𝑛𝐽𝑥) ∧ 𝑛 ≤ 𝑛))) |
86 | 62, 84, 62, 85 | syl3anc 1318 |
. . . . . . . . . . . . . . 15
⊢
(((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺‘𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) ∧ 𝑘 = 𝑛) → (𝑛 ≤ if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛) ↔ (𝑛 ≤ (𝑛𝐽𝑥) ∧ 𝑛 ≤ 𝑛))) |
87 | 84, 62 | ifcld 4081 |
. . . . . . . . . . . . . . . . 17
⊢
(((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺‘𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) ∧ 𝑘 = 𝑛) → if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛) ∈ ℝ) |
88 | 87, 62 | letri3d 10058 |
. . . . . . . . . . . . . . . 16
⊢
(((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺‘𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) ∧ 𝑘 = 𝑛) → (if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛) = 𝑛 ↔ (if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛) ≤ 𝑛 ∧ 𝑛 ≤ if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛)))) |
89 | | simpr 476 |
. . . . . . . . . . . . . . . . 17
⊢
(((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺‘𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) ∧ 𝑘 = 𝑛) → 𝑘 = 𝑛) |
90 | 89 | eqeq2d 2620 |
. . . . . . . . . . . . . . . 16
⊢
(((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺‘𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) ∧ 𝑘 = 𝑛) → (if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛) = 𝑘 ↔ if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛) = 𝑛)) |
91 | | min2 11895 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝑛𝐽𝑥) ∈ ℝ ∧ 𝑛 ∈ ℝ) → if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛) ≤ 𝑛) |
92 | 84, 62, 91 | syl2anc 691 |
. . . . . . . . . . . . . . . . 17
⊢
(((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺‘𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) ∧ 𝑘 = 𝑛) → if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛) ≤ 𝑛) |
93 | 92 | biantrurd 528 |
. . . . . . . . . . . . . . . 16
⊢
(((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺‘𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) ∧ 𝑘 = 𝑛) → (𝑛 ≤ if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛) ↔ (if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛) ≤ 𝑛 ∧ 𝑛 ≤ if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛)))) |
94 | 88, 90, 93 | 3bitr4d 299 |
. . . . . . . . . . . . . . 15
⊢
(((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺‘𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) ∧ 𝑘 = 𝑛) → (if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛) = 𝑘 ↔ 𝑛 ≤ if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛))) |
95 | 62 | leidd 10473 |
. . . . . . . . . . . . . . . 16
⊢
(((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺‘𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) ∧ 𝑘 = 𝑛) → 𝑛 ≤ 𝑛) |
96 | 95 | biantrud 527 |
. . . . . . . . . . . . . . 15
⊢
(((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺‘𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) ∧ 𝑘 = 𝑛) → (𝑛 ≤ (𝑛𝐽𝑥) ↔ (𝑛 ≤ (𝑛𝐽𝑥) ∧ 𝑛 ≤ 𝑛))) |
97 | 86, 94, 96 | 3bitr4d 299 |
. . . . . . . . . . . . . 14
⊢
(((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺‘𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) ∧ 𝑘 = 𝑛) → (if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛) = 𝑘 ↔ 𝑛 ≤ (𝑛𝐽𝑥))) |
98 | | breq1 4586 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 = 𝑛 → (𝑘 ≤ (𝐹‘𝑥) ↔ 𝑛 ≤ (𝐹‘𝑥))) |
99 | 7 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐹:ℝ⟶(0[,)+∞)) |
100 | 99 | ffvelrnda 6267 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝐹‘𝑥) ∈ (0[,)+∞)) |
101 | | elrege0 12149 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝐹‘𝑥) ∈ (0[,)+∞) ↔ ((𝐹‘𝑥) ∈ ℝ ∧ 0 ≤ (𝐹‘𝑥))) |
102 | 100, 101 | sylib 207 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝐹‘𝑥) ∈ ℝ ∧ 0 ≤ (𝐹‘𝑥))) |
103 | 102 | simpld 474 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝐹‘𝑥) ∈ ℝ) |
104 | 103 | adantlr 747 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺‘𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → (𝐹‘𝑥) ∈ ℝ) |
105 | 60, 15 | syl 17 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺‘𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → (2↑𝑛) ∈
ℕ) |
106 | 105 | nnred 10912 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺‘𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → (2↑𝑛) ∈
ℝ) |
107 | 104, 106 | remulcld 9949 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺‘𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → ((𝐹‘𝑥) · (2↑𝑛)) ∈ ℝ) |
108 | | reflcl 12459 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝐹‘𝑥) · (2↑𝑛)) ∈ ℝ →
(⌊‘((𝐹‘𝑥) · (2↑𝑛))) ∈ ℝ) |
109 | 107, 108 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺‘𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) →
(⌊‘((𝐹‘𝑥) · (2↑𝑛))) ∈ ℝ) |
110 | 105 | nngt0d 10941 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺‘𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → 0 < (2↑𝑛)) |
111 | | lemuldiv 10782 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑛 ∈ ℝ ∧
(⌊‘((𝐹‘𝑥) · (2↑𝑛))) ∈ ℝ ∧ ((2↑𝑛) ∈ ℝ ∧ 0 <
(2↑𝑛))) → ((𝑛 · (2↑𝑛)) ≤ (⌊‘((𝐹‘𝑥) · (2↑𝑛))) ↔ 𝑛 ≤ ((⌊‘((𝐹‘𝑥) · (2↑𝑛))) / (2↑𝑛)))) |
112 | 61, 109, 106, 110, 111 | syl112anc 1322 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺‘𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → ((𝑛 · (2↑𝑛)) ≤ (⌊‘((𝐹‘𝑥) · (2↑𝑛))) ↔ 𝑛 ≤ ((⌊‘((𝐹‘𝑥) · (2↑𝑛))) / (2↑𝑛)))) |
113 | | lemul1 10754 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑛 ∈ ℝ ∧ (𝐹‘𝑥) ∈ ℝ ∧ ((2↑𝑛) ∈ ℝ ∧ 0 <
(2↑𝑛))) → (𝑛 ≤ (𝐹‘𝑥) ↔ (𝑛 · (2↑𝑛)) ≤ ((𝐹‘𝑥) · (2↑𝑛)))) |
114 | 61, 104, 106, 110, 113 | syl112anc 1322 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺‘𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → (𝑛 ≤ (𝐹‘𝑥) ↔ (𝑛 · (2↑𝑛)) ≤ ((𝐹‘𝑥) · (2↑𝑛)))) |
115 | | nnmulcl 10920 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑛 ∈ ℕ ∧
(2↑𝑛) ∈ ℕ)
→ (𝑛 ·
(2↑𝑛)) ∈
ℕ) |
116 | 15, 115 | mpdan 699 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑛 ∈ ℕ → (𝑛 · (2↑𝑛)) ∈
ℕ) |
117 | 60, 116 | syl 17 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺‘𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → (𝑛 · (2↑𝑛)) ∈ ℕ) |
118 | 117 | nnzd 11357 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺‘𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → (𝑛 · (2↑𝑛)) ∈ ℤ) |
119 | | flge 12468 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝐹‘𝑥) · (2↑𝑛)) ∈ ℝ ∧ (𝑛 · (2↑𝑛)) ∈ ℤ) → ((𝑛 · (2↑𝑛)) ≤ ((𝐹‘𝑥) · (2↑𝑛)) ↔ (𝑛 · (2↑𝑛)) ≤ (⌊‘((𝐹‘𝑥) · (2↑𝑛))))) |
120 | 107, 118,
119 | syl2anc 691 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺‘𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → ((𝑛 · (2↑𝑛)) ≤ ((𝐹‘𝑥) · (2↑𝑛)) ↔ (𝑛 · (2↑𝑛)) ≤ (⌊‘((𝐹‘𝑥) · (2↑𝑛))))) |
121 | 114, 120 | bitrd 267 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺‘𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → (𝑛 ≤ (𝐹‘𝑥) ↔ (𝑛 · (2↑𝑛)) ≤ (⌊‘((𝐹‘𝑥) · (2↑𝑛))))) |
122 | | simpr 476 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺‘𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → 𝑥 ∈ ℝ) |
123 | | simpr 476 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑚 = 𝑛 ∧ 𝑦 = 𝑥) → 𝑦 = 𝑥) |
124 | 123 | fveq2d 6107 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑚 = 𝑛 ∧ 𝑦 = 𝑥) → (𝐹‘𝑦) = (𝐹‘𝑥)) |
125 | | simpl 472 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑚 = 𝑛 ∧ 𝑦 = 𝑥) → 𝑚 = 𝑛) |
126 | 125 | oveq2d 6565 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑚 = 𝑛 ∧ 𝑦 = 𝑥) → (2↑𝑚) = (2↑𝑛)) |
127 | 124, 126 | oveq12d 6567 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑚 = 𝑛 ∧ 𝑦 = 𝑥) → ((𝐹‘𝑦) · (2↑𝑚)) = ((𝐹‘𝑥) · (2↑𝑛))) |
128 | 127 | fveq2d 6107 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑚 = 𝑛 ∧ 𝑦 = 𝑥) → (⌊‘((𝐹‘𝑦) · (2↑𝑚))) = (⌊‘((𝐹‘𝑥) · (2↑𝑛)))) |
129 | 128, 126 | oveq12d 6567 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑚 = 𝑛 ∧ 𝑦 = 𝑥) → ((⌊‘((𝐹‘𝑦) · (2↑𝑚))) / (2↑𝑚)) = ((⌊‘((𝐹‘𝑥) · (2↑𝑛))) / (2↑𝑛))) |
130 | | ovex 6577 |
. . . . . . . . . . . . . . . . . . 19
⊢
((⌊‘((𝐹‘𝑥) · (2↑𝑛))) / (2↑𝑛)) ∈ V |
131 | 129, 8, 130 | ovmpt2a 6689 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑛 ∈ ℕ ∧ 𝑥 ∈ ℝ) → (𝑛𝐽𝑥) = ((⌊‘((𝐹‘𝑥) · (2↑𝑛))) / (2↑𝑛))) |
132 | 60, 122, 131 | syl2anc 691 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺‘𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → (𝑛𝐽𝑥) = ((⌊‘((𝐹‘𝑥) · (2↑𝑛))) / (2↑𝑛))) |
133 | 132 | breq2d 4595 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺‘𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → (𝑛 ≤ (𝑛𝐽𝑥) ↔ 𝑛 ≤ ((⌊‘((𝐹‘𝑥) · (2↑𝑛))) / (2↑𝑛)))) |
134 | 112, 121,
133 | 3bitr4d 299 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺‘𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → (𝑛 ≤ (𝐹‘𝑥) ↔ 𝑛 ≤ (𝑛𝐽𝑥))) |
135 | 98, 134 | sylan9bbr 733 |
. . . . . . . . . . . . . 14
⊢
(((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺‘𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) ∧ 𝑘 = 𝑛) → (𝑘 ≤ (𝐹‘𝑥) ↔ 𝑛 ≤ (𝑛𝐽𝑥))) |
136 | 122 | adantr 480 |
. . . . . . . . . . . . . . . 16
⊢
(((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺‘𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) ∧ 𝑘 = 𝑛) → 𝑥 ∈ ℝ) |
137 | | iftrue 4042 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑘 = 𝑛 → if(𝑘 = 𝑛, ℝ, (◡𝐹 “ (-∞(,)(𝑘 + (1 / (2↑𝑛)))))) = ℝ) |
138 | 137 | adantl 481 |
. . . . . . . . . . . . . . . 16
⊢
(((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺‘𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) ∧ 𝑘 = 𝑛) → if(𝑘 = 𝑛, ℝ, (◡𝐹 “ (-∞(,)(𝑘 + (1 / (2↑𝑛)))))) = ℝ) |
139 | 136, 138 | eleqtrrd 2691 |
. . . . . . . . . . . . . . 15
⊢
(((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺‘𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) ∧ 𝑘 = 𝑛) → 𝑥 ∈ if(𝑘 = 𝑛, ℝ, (◡𝐹 “ (-∞(,)(𝑘 + (1 / (2↑𝑛))))))) |
140 | 139 | biantrurd 528 |
. . . . . . . . . . . . . 14
⊢
(((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺‘𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) ∧ 𝑘 = 𝑛) → (𝑘 ≤ (𝐹‘𝑥) ↔ (𝑥 ∈ if(𝑘 = 𝑛, ℝ, (◡𝐹 “ (-∞(,)(𝑘 + (1 / (2↑𝑛)))))) ∧ 𝑘 ≤ (𝐹‘𝑥)))) |
141 | 97, 135, 140 | 3bitr2d 295 |
. . . . . . . . . . . . 13
⊢
(((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺‘𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) ∧ 𝑘 = 𝑛) → (if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛) = 𝑘 ↔ (𝑥 ∈ if(𝑘 = 𝑛, ℝ, (◡𝐹 “ (-∞(,)(𝑘 + (1 / (2↑𝑛)))))) ∧ 𝑘 ≤ (𝐹‘𝑥)))) |
142 | 32 | ssdifssd 3710 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (ran (𝐺‘𝑛) ∖ {0}) ⊆ ran (𝑚 ∈ (0...(𝑛 · (2↑𝑛))) ↦ (𝑚 / (2↑𝑛)))) |
143 | 142 | sselda 3568 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺‘𝑛) ∖ {0})) → 𝑘 ∈ ran (𝑚 ∈ (0...(𝑛 · (2↑𝑛))) ↦ (𝑚 / (2↑𝑛)))) |
144 | 19 | rnmpt 5292 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ran
(𝑚 ∈ (0...(𝑛 · (2↑𝑛))) ↦ (𝑚 / (2↑𝑛))) = {𝑘 ∣ ∃𝑚 ∈ (0...(𝑛 · (2↑𝑛)))𝑘 = (𝑚 / (2↑𝑛))} |
145 | 144 | abeq2i 2722 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑘 ∈ ran (𝑚 ∈ (0...(𝑛 · (2↑𝑛))) ↦ (𝑚 / (2↑𝑛))) ↔ ∃𝑚 ∈ (0...(𝑛 · (2↑𝑛)))𝑘 = (𝑚 / (2↑𝑛))) |
146 | | elfzelz 12213 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑚 ∈ (0...(𝑛 · (2↑𝑛))) → 𝑚 ∈ ℤ) |
147 | 146 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑚 ∈ (0...(𝑛 · (2↑𝑛)))) → 𝑚 ∈ ℤ) |
148 | 147 | zcnd 11359 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑚 ∈ (0...(𝑛 · (2↑𝑛)))) → 𝑚 ∈ ℂ) |
149 | 15 | ad2antlr 759 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑚 ∈ (0...(𝑛 · (2↑𝑛)))) → (2↑𝑛) ∈ ℕ) |
150 | 149 | nncnd 10913 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑚 ∈ (0...(𝑛 · (2↑𝑛)))) → (2↑𝑛) ∈ ℂ) |
151 | 149 | nnne0d 10942 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑚 ∈ (0...(𝑛 · (2↑𝑛)))) → (2↑𝑛) ≠ 0) |
152 | 148, 150,
151 | divcan1d 10681 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑚 ∈ (0...(𝑛 · (2↑𝑛)))) → ((𝑚 / (2↑𝑛)) · (2↑𝑛)) = 𝑚) |
153 | 152, 147 | eqeltrd 2688 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑚 ∈ (0...(𝑛 · (2↑𝑛)))) → ((𝑚 / (2↑𝑛)) · (2↑𝑛)) ∈ ℤ) |
154 | | oveq1 6556 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑘 = (𝑚 / (2↑𝑛)) → (𝑘 · (2↑𝑛)) = ((𝑚 / (2↑𝑛)) · (2↑𝑛))) |
155 | 154 | eleq1d 2672 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑘 = (𝑚 / (2↑𝑛)) → ((𝑘 · (2↑𝑛)) ∈ ℤ ↔ ((𝑚 / (2↑𝑛)) · (2↑𝑛)) ∈ ℤ)) |
156 | 153, 155 | syl5ibrcom 236 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑚 ∈ (0...(𝑛 · (2↑𝑛)))) → (𝑘 = (𝑚 / (2↑𝑛)) → (𝑘 · (2↑𝑛)) ∈ ℤ)) |
157 | 156 | rexlimdva 3013 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (∃𝑚 ∈ (0...(𝑛 · (2↑𝑛)))𝑘 = (𝑚 / (2↑𝑛)) → (𝑘 · (2↑𝑛)) ∈ ℤ)) |
158 | 145, 157 | syl5bi 231 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑘 ∈ ran (𝑚 ∈ (0...(𝑛 · (2↑𝑛))) ↦ (𝑚 / (2↑𝑛))) → (𝑘 · (2↑𝑛)) ∈ ℤ)) |
159 | 158 | imp 444 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ran (𝑚 ∈ (0...(𝑛 · (2↑𝑛))) ↦ (𝑚 / (2↑𝑛)))) → (𝑘 · (2↑𝑛)) ∈ ℤ) |
160 | 143, 159 | syldan 486 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺‘𝑛) ∖ {0})) → (𝑘 · (2↑𝑛)) ∈ ℤ) |
161 | 160 | adantr 480 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺‘𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → (𝑘 · (2↑𝑛)) ∈ ℤ) |
162 | | flbi 12479 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝐹‘𝑥) · (2↑𝑛)) ∈ ℝ ∧ (𝑘 · (2↑𝑛)) ∈ ℤ) →
((⌊‘((𝐹‘𝑥) · (2↑𝑛))) = (𝑘 · (2↑𝑛)) ↔ ((𝑘 · (2↑𝑛)) ≤ ((𝐹‘𝑥) · (2↑𝑛)) ∧ ((𝐹‘𝑥) · (2↑𝑛)) < ((𝑘 · (2↑𝑛)) + 1)))) |
163 | 107, 161,
162 | syl2anc 691 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺‘𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) →
((⌊‘((𝐹‘𝑥) · (2↑𝑛))) = (𝑘 · (2↑𝑛)) ↔ ((𝑘 · (2↑𝑛)) ≤ ((𝐹‘𝑥) · (2↑𝑛)) ∧ ((𝐹‘𝑥) · (2↑𝑛)) < ((𝑘 · (2↑𝑛)) + 1)))) |
164 | 163 | adantr 480 |
. . . . . . . . . . . . . 14
⊢
(((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺‘𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ≠ 𝑛) → ((⌊‘((𝐹‘𝑥) · (2↑𝑛))) = (𝑘 · (2↑𝑛)) ↔ ((𝑘 · (2↑𝑛)) ≤ ((𝐹‘𝑥) · (2↑𝑛)) ∧ ((𝐹‘𝑥) · (2↑𝑛)) < ((𝑘 · (2↑𝑛)) + 1)))) |
165 | | neeq1 2844 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
(if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛) = 𝑘 → (if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛) ≠ 𝑛 ↔ 𝑘 ≠ 𝑛)) |
166 | 165 | biimparc 503 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑘 ≠ 𝑛 ∧ if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛) = 𝑘) → if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛) ≠ 𝑛) |
167 | | iffalse 4045 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (¬
(𝑛𝐽𝑥) ≤ 𝑛 → if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛) = 𝑛) |
168 | 167 | necon1ai 2809 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
(if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛) ≠ 𝑛 → (𝑛𝐽𝑥) ≤ 𝑛) |
169 | 166, 168 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑘 ≠ 𝑛 ∧ if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛) = 𝑘) → (𝑛𝐽𝑥) ≤ 𝑛) |
170 | 169 | iftrued 4044 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑘 ≠ 𝑛 ∧ if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛) = 𝑘) → if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛) = (𝑛𝐽𝑥)) |
171 | | simpr 476 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑘 ≠ 𝑛 ∧ if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛) = 𝑘) → if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛) = 𝑘) |
172 | 170, 171 | eqtr3d 2646 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑘 ≠ 𝑛 ∧ if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛) = 𝑘) → (𝑛𝐽𝑥) = 𝑘) |
173 | 172, 169 | eqbrtrrd 4607 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑘 ≠ 𝑛 ∧ if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛) = 𝑘) → 𝑘 ≤ 𝑛) |
174 | 173, 172 | jca 553 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑘 ≠ 𝑛 ∧ if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛) = 𝑘) → (𝑘 ≤ 𝑛 ∧ (𝑛𝐽𝑥) = 𝑘)) |
175 | 174 | ex 449 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑘 ≠ 𝑛 → (if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛) = 𝑘 → (𝑘 ≤ 𝑛 ∧ (𝑛𝐽𝑥) = 𝑘))) |
176 | | breq1 4586 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑛𝐽𝑥) = 𝑘 → ((𝑛𝐽𝑥) ≤ 𝑛 ↔ 𝑘 ≤ 𝑛)) |
177 | 176 | biimparc 503 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑘 ≤ 𝑛 ∧ (𝑛𝐽𝑥) = 𝑘) → (𝑛𝐽𝑥) ≤ 𝑛) |
178 | 177 | iftrued 4044 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑘 ≤ 𝑛 ∧ (𝑛𝐽𝑥) = 𝑘) → if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛) = (𝑛𝐽𝑥)) |
179 | | simpr 476 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑘 ≤ 𝑛 ∧ (𝑛𝐽𝑥) = 𝑘) → (𝑛𝐽𝑥) = 𝑘) |
180 | 178, 179 | eqtrd 2644 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑘 ≤ 𝑛 ∧ (𝑛𝐽𝑥) = 𝑘) → if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛) = 𝑘) |
181 | 175, 180 | impbid1 214 |
. . . . . . . . . . . . . . . 16
⊢ (𝑘 ≠ 𝑛 → (if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛) = 𝑘 ↔ (𝑘 ≤ 𝑛 ∧ (𝑛𝐽𝑥) = 𝑘))) |
182 | 181 | adantl 481 |
. . . . . . . . . . . . . . 15
⊢
(((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺‘𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ≠ 𝑛) → (if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛) = 𝑘 ↔ (𝑘 ≤ 𝑛 ∧ (𝑛𝐽𝑥) = 𝑘))) |
183 | | eldifi 3694 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑘 ∈ (ran (𝐺‘𝑛) ∖ {0}) → 𝑘 ∈ ran (𝐺‘𝑛)) |
184 | | nnre 10904 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑛 ∈ ℕ → 𝑛 ∈
ℝ) |
185 | 184 | ad2antlr 759 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → 𝑛 ∈ ℝ) |
186 | 82, 185, 91 | syl2anc 691 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛) ≤ 𝑛) |
187 | 13 | ad2antlr 759 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → 𝑛 ∈ ℕ0) |
188 | 187 | nn0ge0d 11231 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → 0 ≤ 𝑛) |
189 | | breq1 4586 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
(if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛) = if(𝑥 ∈ (-𝑛[,]𝑛), if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛), 0) → (if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛) ≤ 𝑛 ↔ if(𝑥 ∈ (-𝑛[,]𝑛), if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛), 0) ≤ 𝑛)) |
190 | | breq1 4586 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (0 =
if(𝑥 ∈ (-𝑛[,]𝑛), if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛), 0) → (0 ≤ 𝑛 ↔ if(𝑥 ∈ (-𝑛[,]𝑛), if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛), 0) ≤ 𝑛)) |
191 | 189, 190 | ifboth 4074 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
((if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛) ≤ 𝑛 ∧ 0 ≤ 𝑛) → if(𝑥 ∈ (-𝑛[,]𝑛), if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛), 0) ≤ 𝑛) |
192 | 186, 188,
191 | syl2anc 691 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → if(𝑥 ∈ (-𝑛[,]𝑛), if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛), 0) ≤ 𝑛) |
193 | 47, 192 | eqbrtrd 4605 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝐺‘𝑛)‘𝑥) ≤ 𝑛) |
194 | 193 | ralrimiva 2949 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∀𝑥 ∈ ℝ ((𝐺‘𝑛)‘𝑥) ≤ 𝑛) |
195 | | ffn 5958 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝐺‘𝑛):ℝ⟶ℝ → (𝐺‘𝑛) Fn ℝ) |
196 | 23, 195 | syl 17 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐺‘𝑛) Fn ℝ) |
197 | | breq1 4586 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑘 = ((𝐺‘𝑛)‘𝑥) → (𝑘 ≤ 𝑛 ↔ ((𝐺‘𝑛)‘𝑥) ≤ 𝑛)) |
198 | 197 | ralrn 6270 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝐺‘𝑛) Fn ℝ → (∀𝑘 ∈ ran (𝐺‘𝑛)𝑘 ≤ 𝑛 ↔ ∀𝑥 ∈ ℝ ((𝐺‘𝑛)‘𝑥) ≤ 𝑛)) |
199 | 196, 198 | syl 17 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (∀𝑘 ∈ ran (𝐺‘𝑛)𝑘 ≤ 𝑛 ↔ ∀𝑥 ∈ ℝ ((𝐺‘𝑛)‘𝑥) ≤ 𝑛)) |
200 | 194, 199 | mpbird 246 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∀𝑘 ∈ ran (𝐺‘𝑛)𝑘 ≤ 𝑛) |
201 | 200 | r19.21bi 2916 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ran (𝐺‘𝑛)) → 𝑘 ≤ 𝑛) |
202 | 183, 201 | sylan2 490 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺‘𝑛) ∖ {0})) → 𝑘 ≤ 𝑛) |
203 | 202 | ad2antrr 758 |
. . . . . . . . . . . . . . . 16
⊢
(((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺‘𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ≠ 𝑛) → 𝑘 ≤ 𝑛) |
204 | 203 | biantrurd 528 |
. . . . . . . . . . . . . . 15
⊢
(((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺‘𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ≠ 𝑛) → ((𝑛𝐽𝑥) = 𝑘 ↔ (𝑘 ≤ 𝑛 ∧ (𝑛𝐽𝑥) = 𝑘))) |
205 | 132 | eqeq1d 2612 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺‘𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → ((𝑛𝐽𝑥) = 𝑘 ↔ ((⌊‘((𝐹‘𝑥) · (2↑𝑛))) / (2↑𝑛)) = 𝑘)) |
206 | 109 | recnd 9947 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺‘𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) →
(⌊‘((𝐹‘𝑥) · (2↑𝑛))) ∈ ℂ) |
207 | 32, 22 | sstrd 3578 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ran (𝐺‘𝑛) ⊆ ℝ) |
208 | 207 | ssdifssd 3710 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (ran (𝐺‘𝑛) ∖ {0}) ⊆
ℝ) |
209 | 208 | sselda 3568 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺‘𝑛) ∖ {0})) → 𝑘 ∈ ℝ) |
210 | 209 | adantr 480 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺‘𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → 𝑘 ∈ ℝ) |
211 | 210 | recnd 9947 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺‘𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → 𝑘 ∈ ℂ) |
212 | 105 | nncnd 10913 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺‘𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → (2↑𝑛) ∈
ℂ) |
213 | 105 | nnne0d 10942 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺‘𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → (2↑𝑛) ≠ 0) |
214 | 206, 211,
212, 213 | divmul3d 10714 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺‘𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) →
(((⌊‘((𝐹‘𝑥) · (2↑𝑛))) / (2↑𝑛)) = 𝑘 ↔ (⌊‘((𝐹‘𝑥) · (2↑𝑛))) = (𝑘 · (2↑𝑛)))) |
215 | 205, 214 | bitrd 267 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺‘𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → ((𝑛𝐽𝑥) = 𝑘 ↔ (⌊‘((𝐹‘𝑥) · (2↑𝑛))) = (𝑘 · (2↑𝑛)))) |
216 | 215 | adantr 480 |
. . . . . . . . . . . . . . 15
⊢
(((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺‘𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ≠ 𝑛) → ((𝑛𝐽𝑥) = 𝑘 ↔ (⌊‘((𝐹‘𝑥) · (2↑𝑛))) = (𝑘 · (2↑𝑛)))) |
217 | 182, 204,
216 | 3bitr2d 295 |
. . . . . . . . . . . . . 14
⊢
(((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺‘𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ≠ 𝑛) → (if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛) = 𝑘 ↔ (⌊‘((𝐹‘𝑥) · (2↑𝑛))) = (𝑘 · (2↑𝑛)))) |
218 | | ifnefalse 4048 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑘 ≠ 𝑛 → if(𝑘 = 𝑛, ℝ, (◡𝐹 “ (-∞(,)(𝑘 + (1 / (2↑𝑛)))))) = (◡𝐹 “ (-∞(,)(𝑘 + (1 / (2↑𝑛)))))) |
219 | 218 | eleq2d 2673 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑘 ≠ 𝑛 → (𝑥 ∈ if(𝑘 = 𝑛, ℝ, (◡𝐹 “ (-∞(,)(𝑘 + (1 / (2↑𝑛)))))) ↔ 𝑥 ∈ (◡𝐹 “ (-∞(,)(𝑘 + (1 / (2↑𝑛))))))) |
220 | 105 | nnrecred 10943 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺‘𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → (1 / (2↑𝑛)) ∈
ℝ) |
221 | 210, 220 | readdcld 9948 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺‘𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → (𝑘 + (1 / (2↑𝑛))) ∈ ℝ) |
222 | 221 | rexrd 9968 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺‘𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → (𝑘 + (1 / (2↑𝑛))) ∈
ℝ*) |
223 | | elioomnf 12139 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑘 + (1 / (2↑𝑛))) ∈ ℝ*
→ ((𝐹‘𝑥) ∈ (-∞(,)(𝑘 + (1 / (2↑𝑛)))) ↔ ((𝐹‘𝑥) ∈ ℝ ∧ (𝐹‘𝑥) < (𝑘 + (1 / (2↑𝑛)))))) |
224 | 222, 223 | syl 17 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺‘𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → ((𝐹‘𝑥) ∈ (-∞(,)(𝑘 + (1 / (2↑𝑛)))) ↔ ((𝐹‘𝑥) ∈ ℝ ∧ (𝐹‘𝑥) < (𝑘 + (1 / (2↑𝑛)))))) |
225 | 99 | ad2antrr 758 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺‘𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → 𝐹:ℝ⟶(0[,)+∞)) |
226 | | ffn 5958 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝐹:ℝ⟶(0[,)+∞)
→ 𝐹 Fn
ℝ) |
227 | 225, 226 | syl 17 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺‘𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → 𝐹 Fn ℝ) |
228 | | elpreima 6245 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝐹 Fn ℝ → (𝑥 ∈ (◡𝐹 “ (-∞(,)(𝑘 + (1 / (2↑𝑛))))) ↔ (𝑥 ∈ ℝ ∧ (𝐹‘𝑥) ∈ (-∞(,)(𝑘 + (1 / (2↑𝑛))))))) |
229 | 227, 228 | syl 17 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺‘𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → (𝑥 ∈ (◡𝐹 “ (-∞(,)(𝑘 + (1 / (2↑𝑛))))) ↔ (𝑥 ∈ ℝ ∧ (𝐹‘𝑥) ∈ (-∞(,)(𝑘 + (1 / (2↑𝑛))))))) |
230 | 122 | biantrurd 528 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺‘𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → ((𝐹‘𝑥) ∈ (-∞(,)(𝑘 + (1 / (2↑𝑛)))) ↔ (𝑥 ∈ ℝ ∧ (𝐹‘𝑥) ∈ (-∞(,)(𝑘 + (1 / (2↑𝑛))))))) |
231 | 229, 230 | bitr4d 270 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺‘𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → (𝑥 ∈ (◡𝐹 “ (-∞(,)(𝑘 + (1 / (2↑𝑛))))) ↔ (𝐹‘𝑥) ∈ (-∞(,)(𝑘 + (1 / (2↑𝑛)))))) |
232 | 104 | biantrurd 528 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺‘𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → ((𝐹‘𝑥) < (𝑘 + (1 / (2↑𝑛))) ↔ ((𝐹‘𝑥) ∈ ℝ ∧ (𝐹‘𝑥) < (𝑘 + (1 / (2↑𝑛)))))) |
233 | 224, 231,
232 | 3bitr4d 299 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺‘𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → (𝑥 ∈ (◡𝐹 “ (-∞(,)(𝑘 + (1 / (2↑𝑛))))) ↔ (𝐹‘𝑥) < (𝑘 + (1 / (2↑𝑛))))) |
234 | | ltmul1 10752 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝐹‘𝑥) ∈ ℝ ∧ (𝑘 + (1 / (2↑𝑛))) ∈ ℝ ∧ ((2↑𝑛) ∈ ℝ ∧ 0 <
(2↑𝑛))) → ((𝐹‘𝑥) < (𝑘 + (1 / (2↑𝑛))) ↔ ((𝐹‘𝑥) · (2↑𝑛)) < ((𝑘 + (1 / (2↑𝑛))) · (2↑𝑛)))) |
235 | 104, 221,
106, 110, 234 | syl112anc 1322 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺‘𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → ((𝐹‘𝑥) < (𝑘 + (1 / (2↑𝑛))) ↔ ((𝐹‘𝑥) · (2↑𝑛)) < ((𝑘 + (1 / (2↑𝑛))) · (2↑𝑛)))) |
236 | 220 | recnd 9947 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺‘𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → (1 / (2↑𝑛)) ∈
ℂ) |
237 | 211, 236,
212 | adddird 9944 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺‘𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → ((𝑘 + (1 / (2↑𝑛))) · (2↑𝑛)) = ((𝑘 · (2↑𝑛)) + ((1 / (2↑𝑛)) · (2↑𝑛)))) |
238 | 212, 213 | recid2d 10676 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺‘𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → ((1 / (2↑𝑛)) · (2↑𝑛)) = 1) |
239 | 238 | oveq2d 6565 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺‘𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → ((𝑘 · (2↑𝑛)) + ((1 / (2↑𝑛)) · (2↑𝑛))) = ((𝑘 · (2↑𝑛)) + 1)) |
240 | 237, 239 | eqtrd 2644 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺‘𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → ((𝑘 + (1 / (2↑𝑛))) · (2↑𝑛)) = ((𝑘 · (2↑𝑛)) + 1)) |
241 | 240 | breq2d 4595 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺‘𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → (((𝐹‘𝑥) · (2↑𝑛)) < ((𝑘 + (1 / (2↑𝑛))) · (2↑𝑛)) ↔ ((𝐹‘𝑥) · (2↑𝑛)) < ((𝑘 · (2↑𝑛)) + 1))) |
242 | 233, 235,
241 | 3bitrd 293 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺‘𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → (𝑥 ∈ (◡𝐹 “ (-∞(,)(𝑘 + (1 / (2↑𝑛))))) ↔ ((𝐹‘𝑥) · (2↑𝑛)) < ((𝑘 · (2↑𝑛)) + 1))) |
243 | 219, 242 | sylan9bbr 733 |
. . . . . . . . . . . . . . . 16
⊢
(((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺‘𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ≠ 𝑛) → (𝑥 ∈ if(𝑘 = 𝑛, ℝ, (◡𝐹 “ (-∞(,)(𝑘 + (1 / (2↑𝑛)))))) ↔ ((𝐹‘𝑥) · (2↑𝑛)) < ((𝑘 · (2↑𝑛)) + 1))) |
244 | | lemul1 10754 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑘 ∈ ℝ ∧ (𝐹‘𝑥) ∈ ℝ ∧ ((2↑𝑛) ∈ ℝ ∧ 0 <
(2↑𝑛))) → (𝑘 ≤ (𝐹‘𝑥) ↔ (𝑘 · (2↑𝑛)) ≤ ((𝐹‘𝑥) · (2↑𝑛)))) |
245 | 210, 104,
106, 110, 244 | syl112anc 1322 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺‘𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → (𝑘 ≤ (𝐹‘𝑥) ↔ (𝑘 · (2↑𝑛)) ≤ ((𝐹‘𝑥) · (2↑𝑛)))) |
246 | 245 | adantr 480 |
. . . . . . . . . . . . . . . 16
⊢
(((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺‘𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ≠ 𝑛) → (𝑘 ≤ (𝐹‘𝑥) ↔ (𝑘 · (2↑𝑛)) ≤ ((𝐹‘𝑥) · (2↑𝑛)))) |
247 | 243, 246 | anbi12d 743 |
. . . . . . . . . . . . . . 15
⊢
(((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺‘𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ≠ 𝑛) → ((𝑥 ∈ if(𝑘 = 𝑛, ℝ, (◡𝐹 “ (-∞(,)(𝑘 + (1 / (2↑𝑛)))))) ∧ 𝑘 ≤ (𝐹‘𝑥)) ↔ (((𝐹‘𝑥) · (2↑𝑛)) < ((𝑘 · (2↑𝑛)) + 1) ∧ (𝑘 · (2↑𝑛)) ≤ ((𝐹‘𝑥) · (2↑𝑛))))) |
248 | | ancom 465 |
. . . . . . . . . . . . . . 15
⊢ ((((𝐹‘𝑥) · (2↑𝑛)) < ((𝑘 · (2↑𝑛)) + 1) ∧ (𝑘 · (2↑𝑛)) ≤ ((𝐹‘𝑥) · (2↑𝑛))) ↔ ((𝑘 · (2↑𝑛)) ≤ ((𝐹‘𝑥) · (2↑𝑛)) ∧ ((𝐹‘𝑥) · (2↑𝑛)) < ((𝑘 · (2↑𝑛)) + 1))) |
249 | 247, 248 | syl6bb 275 |
. . . . . . . . . . . . . 14
⊢
(((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺‘𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ≠ 𝑛) → ((𝑥 ∈ if(𝑘 = 𝑛, ℝ, (◡𝐹 “ (-∞(,)(𝑘 + (1 / (2↑𝑛)))))) ∧ 𝑘 ≤ (𝐹‘𝑥)) ↔ ((𝑘 · (2↑𝑛)) ≤ ((𝐹‘𝑥) · (2↑𝑛)) ∧ ((𝐹‘𝑥) · (2↑𝑛)) < ((𝑘 · (2↑𝑛)) + 1)))) |
250 | 164, 217,
249 | 3bitr4d 299 |
. . . . . . . . . . . . 13
⊢
(((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺‘𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ≠ 𝑛) → (if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛) = 𝑘 ↔ (𝑥 ∈ if(𝑘 = 𝑛, ℝ, (◡𝐹 “ (-∞(,)(𝑘 + (1 / (2↑𝑛)))))) ∧ 𝑘 ≤ (𝐹‘𝑥)))) |
251 | 141, 250 | pm2.61dane 2869 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺‘𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → (if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛) = 𝑘 ↔ (𝑥 ∈ if(𝑘 = 𝑛, ℝ, (◡𝐹 “ (-∞(,)(𝑘 + (1 / (2↑𝑛)))))) ∧ 𝑘 ≤ (𝐹‘𝑥)))) |
252 | | eldif 3550 |
. . . . . . . . . . . . 13
⊢ (𝑥 ∈ (if(𝑘 = 𝑛, ℝ, (◡𝐹 “ (-∞(,)(𝑘 + (1 / (2↑𝑛)))))) ∖ (◡𝐹 “ (-∞(,)𝑘))) ↔ (𝑥 ∈ if(𝑘 = 𝑛, ℝ, (◡𝐹 “ (-∞(,)(𝑘 + (1 / (2↑𝑛)))))) ∧ ¬ 𝑥 ∈ (◡𝐹 “ (-∞(,)𝑘)))) |
253 | 210 | rexrd 9968 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺‘𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → 𝑘 ∈ ℝ*) |
254 | | elioomnf 12139 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑘 ∈ ℝ*
→ ((𝐹‘𝑥) ∈ (-∞(,)𝑘) ↔ ((𝐹‘𝑥) ∈ ℝ ∧ (𝐹‘𝑥) < 𝑘))) |
255 | 253, 254 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺‘𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → ((𝐹‘𝑥) ∈ (-∞(,)𝑘) ↔ ((𝐹‘𝑥) ∈ ℝ ∧ (𝐹‘𝑥) < 𝑘))) |
256 | | elpreima 6245 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝐹 Fn ℝ → (𝑥 ∈ (◡𝐹 “ (-∞(,)𝑘)) ↔ (𝑥 ∈ ℝ ∧ (𝐹‘𝑥) ∈ (-∞(,)𝑘)))) |
257 | 227, 256 | syl 17 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺‘𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → (𝑥 ∈ (◡𝐹 “ (-∞(,)𝑘)) ↔ (𝑥 ∈ ℝ ∧ (𝐹‘𝑥) ∈ (-∞(,)𝑘)))) |
258 | 122 | biantrurd 528 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺‘𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → ((𝐹‘𝑥) ∈ (-∞(,)𝑘) ↔ (𝑥 ∈ ℝ ∧ (𝐹‘𝑥) ∈ (-∞(,)𝑘)))) |
259 | 257, 258 | bitr4d 270 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺‘𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → (𝑥 ∈ (◡𝐹 “ (-∞(,)𝑘)) ↔ (𝐹‘𝑥) ∈ (-∞(,)𝑘))) |
260 | 104 | biantrurd 528 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺‘𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → ((𝐹‘𝑥) < 𝑘 ↔ ((𝐹‘𝑥) ∈ ℝ ∧ (𝐹‘𝑥) < 𝑘))) |
261 | 255, 259,
260 | 3bitr4d 299 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺‘𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → (𝑥 ∈ (◡𝐹 “ (-∞(,)𝑘)) ↔ (𝐹‘𝑥) < 𝑘)) |
262 | 261 | notbid 307 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺‘𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → (¬ 𝑥 ∈ (◡𝐹 “ (-∞(,)𝑘)) ↔ ¬ (𝐹‘𝑥) < 𝑘)) |
263 | 210, 104 | lenltd 10062 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺‘𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → (𝑘 ≤ (𝐹‘𝑥) ↔ ¬ (𝐹‘𝑥) < 𝑘)) |
264 | 262, 263 | bitr4d 270 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺‘𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → (¬ 𝑥 ∈ (◡𝐹 “ (-∞(,)𝑘)) ↔ 𝑘 ≤ (𝐹‘𝑥))) |
265 | 264 | anbi2d 736 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺‘𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → ((𝑥 ∈ if(𝑘 = 𝑛, ℝ, (◡𝐹 “ (-∞(,)(𝑘 + (1 / (2↑𝑛)))))) ∧ ¬ 𝑥 ∈ (◡𝐹 “ (-∞(,)𝑘))) ↔ (𝑥 ∈ if(𝑘 = 𝑛, ℝ, (◡𝐹 “ (-∞(,)(𝑘 + (1 / (2↑𝑛)))))) ∧ 𝑘 ≤ (𝐹‘𝑥)))) |
266 | 252, 265 | syl5bb 271 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺‘𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → (𝑥 ∈ (if(𝑘 = 𝑛, ℝ, (◡𝐹 “ (-∞(,)(𝑘 + (1 / (2↑𝑛)))))) ∖ (◡𝐹 “ (-∞(,)𝑘))) ↔ (𝑥 ∈ if(𝑘 = 𝑛, ℝ, (◡𝐹 “ (-∞(,)(𝑘 + (1 / (2↑𝑛)))))) ∧ 𝑘 ≤ (𝐹‘𝑥)))) |
267 | 251, 266 | bitr4d 270 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺‘𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → (if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛) = 𝑘 ↔ 𝑥 ∈ (if(𝑘 = 𝑛, ℝ, (◡𝐹 “ (-∞(,)(𝑘 + (1 / (2↑𝑛)))))) ∖ (◡𝐹 “ (-∞(,)𝑘))))) |
268 | 59, 267 | sylan9bbr 733 |
. . . . . . . . . 10
⊢
(((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺‘𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 ∈ (-𝑛[,]𝑛)) → (if(𝑥 ∈ (-𝑛[,]𝑛), if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛), 0) = 𝑘 ↔ 𝑥 ∈ (if(𝑘 = 𝑛, ℝ, (◡𝐹 “ (-∞(,)(𝑘 + (1 / (2↑𝑛)))))) ∖ (◡𝐹 “ (-∞(,)𝑘))))) |
269 | 268 | pm5.32da 671 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺‘𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → ((𝑥 ∈ (-𝑛[,]𝑛) ∧ if(𝑥 ∈ (-𝑛[,]𝑛), if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛), 0) = 𝑘) ↔ (𝑥 ∈ (-𝑛[,]𝑛) ∧ 𝑥 ∈ (if(𝑘 = 𝑛, ℝ, (◡𝐹 “ (-∞(,)(𝑘 + (1 / (2↑𝑛)))))) ∖ (◡𝐹 “ (-∞(,)𝑘)))))) |
270 | 49, 57, 269 | 3bitrd 293 |
. . . . . . . 8
⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺‘𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → (((𝐺‘𝑛)‘𝑥) = 𝑘 ↔ (𝑥 ∈ (-𝑛[,]𝑛) ∧ 𝑥 ∈ (if(𝑘 = 𝑛, ℝ, (◡𝐹 “ (-∞(,)(𝑘 + (1 / (2↑𝑛)))))) ∖ (◡𝐹 “ (-∞(,)𝑘)))))) |
271 | 270 | pm5.32da 671 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺‘𝑛) ∖ {0})) → ((𝑥 ∈ ℝ ∧ ((𝐺‘𝑛)‘𝑥) = 𝑘) ↔ (𝑥 ∈ ℝ ∧ (𝑥 ∈ (-𝑛[,]𝑛) ∧ 𝑥 ∈ (if(𝑘 = 𝑛, ℝ, (◡𝐹 “ (-∞(,)(𝑘 + (1 / (2↑𝑛)))))) ∖ (◡𝐹 “ (-∞(,)𝑘))))))) |
272 | 23 | adantr 480 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺‘𝑛) ∖ {0})) → (𝐺‘𝑛):ℝ⟶ℝ) |
273 | 272, 195 | syl 17 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺‘𝑛) ∖ {0})) → (𝐺‘𝑛) Fn ℝ) |
274 | | fniniseg 6246 |
. . . . . . . 8
⊢ ((𝐺‘𝑛) Fn ℝ → (𝑥 ∈ (◡(𝐺‘𝑛) “ {𝑘}) ↔ (𝑥 ∈ ℝ ∧ ((𝐺‘𝑛)‘𝑥) = 𝑘))) |
275 | 273, 274 | syl 17 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺‘𝑛) ∖ {0})) → (𝑥 ∈ (◡(𝐺‘𝑛) “ {𝑘}) ↔ (𝑥 ∈ ℝ ∧ ((𝐺‘𝑛)‘𝑥) = 𝑘))) |
276 | | elin 3758 |
. . . . . . . 8
⊢ (𝑥 ∈ ((-𝑛[,]𝑛) ∩ (if(𝑘 = 𝑛, ℝ, (◡𝐹 “ (-∞(,)(𝑘 + (1 / (2↑𝑛)))))) ∖ (◡𝐹 “ (-∞(,)𝑘)))) ↔ (𝑥 ∈ (-𝑛[,]𝑛) ∧ 𝑥 ∈ (if(𝑘 = 𝑛, ℝ, (◡𝐹 “ (-∞(,)(𝑘 + (1 / (2↑𝑛)))))) ∖ (◡𝐹 “ (-∞(,)𝑘))))) |
277 | 184 | ad2antlr 759 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺‘𝑛) ∖ {0})) → 𝑛 ∈ ℝ) |
278 | 277 | renegcld 10336 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺‘𝑛) ∖ {0})) → -𝑛 ∈ ℝ) |
279 | | iccmbl 23141 |
. . . . . . . . . . . . 13
⊢ ((-𝑛 ∈ ℝ ∧ 𝑛 ∈ ℝ) → (-𝑛[,]𝑛) ∈ dom vol) |
280 | 278, 277,
279 | syl2anc 691 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺‘𝑛) ∖ {0})) → (-𝑛[,]𝑛) ∈ dom vol) |
281 | | mblss 23106 |
. . . . . . . . . . . 12
⊢ ((-𝑛[,]𝑛) ∈ dom vol → (-𝑛[,]𝑛) ⊆ ℝ) |
282 | 280, 281 | syl 17 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺‘𝑛) ∖ {0})) → (-𝑛[,]𝑛) ⊆ ℝ) |
283 | 282 | sseld 3567 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺‘𝑛) ∖ {0})) → (𝑥 ∈ (-𝑛[,]𝑛) → 𝑥 ∈ ℝ)) |
284 | 283 | adantrd 483 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺‘𝑛) ∖ {0})) → ((𝑥 ∈ (-𝑛[,]𝑛) ∧ 𝑥 ∈ (if(𝑘 = 𝑛, ℝ, (◡𝐹 “ (-∞(,)(𝑘 + (1 / (2↑𝑛)))))) ∖ (◡𝐹 “ (-∞(,)𝑘)))) → 𝑥 ∈ ℝ)) |
285 | 284 | pm4.71rd 665 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺‘𝑛) ∖ {0})) → ((𝑥 ∈ (-𝑛[,]𝑛) ∧ 𝑥 ∈ (if(𝑘 = 𝑛, ℝ, (◡𝐹 “ (-∞(,)(𝑘 + (1 / (2↑𝑛)))))) ∖ (◡𝐹 “ (-∞(,)𝑘)))) ↔ (𝑥 ∈ ℝ ∧ (𝑥 ∈ (-𝑛[,]𝑛) ∧ 𝑥 ∈ (if(𝑘 = 𝑛, ℝ, (◡𝐹 “ (-∞(,)(𝑘 + (1 / (2↑𝑛)))))) ∖ (◡𝐹 “ (-∞(,)𝑘))))))) |
286 | 276, 285 | syl5bb 271 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺‘𝑛) ∖ {0})) → (𝑥 ∈ ((-𝑛[,]𝑛) ∩ (if(𝑘 = 𝑛, ℝ, (◡𝐹 “ (-∞(,)(𝑘 + (1 / (2↑𝑛)))))) ∖ (◡𝐹 “ (-∞(,)𝑘)))) ↔ (𝑥 ∈ ℝ ∧ (𝑥 ∈ (-𝑛[,]𝑛) ∧ 𝑥 ∈ (if(𝑘 = 𝑛, ℝ, (◡𝐹 “ (-∞(,)(𝑘 + (1 / (2↑𝑛)))))) ∖ (◡𝐹 “ (-∞(,)𝑘))))))) |
287 | 271, 275,
286 | 3bitr4d 299 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺‘𝑛) ∖ {0})) → (𝑥 ∈ (◡(𝐺‘𝑛) “ {𝑘}) ↔ 𝑥 ∈ ((-𝑛[,]𝑛) ∩ (if(𝑘 = 𝑛, ℝ, (◡𝐹 “ (-∞(,)(𝑘 + (1 / (2↑𝑛)))))) ∖ (◡𝐹 “ (-∞(,)𝑘)))))) |
288 | 287 | eqrdv 2608 |
. . . . 5
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺‘𝑛) ∖ {0})) → (◡(𝐺‘𝑛) “ {𝑘}) = ((-𝑛[,]𝑛) ∩ (if(𝑘 = 𝑛, ℝ, (◡𝐹 “ (-∞(,)(𝑘 + (1 / (2↑𝑛)))))) ∖ (◡𝐹 “ (-∞(,)𝑘))))) |
289 | | rembl 23115 |
. . . . . . . . 9
⊢ ℝ
∈ dom vol |
290 | | fss 5969 |
. . . . . . . . . . 11
⊢ ((𝐹:ℝ⟶(0[,)+∞)
∧ (0[,)+∞) ⊆ ℝ) → 𝐹:ℝ⟶ℝ) |
291 | 7, 63, 290 | sylancl 693 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐹:ℝ⟶ℝ) |
292 | | mbfima 23205 |
. . . . . . . . . 10
⊢ ((𝐹 ∈ MblFn ∧ 𝐹:ℝ⟶ℝ) →
(◡𝐹 “ (-∞(,)(𝑘 + (1 / (2↑𝑛))))) ∈ dom vol) |
293 | 6, 291, 292 | syl2anc 691 |
. . . . . . . . 9
⊢ (𝜑 → (◡𝐹 “ (-∞(,)(𝑘 + (1 / (2↑𝑛))))) ∈ dom vol) |
294 | | ifcl 4080 |
. . . . . . . . 9
⊢ ((ℝ
∈ dom vol ∧ (◡𝐹 “ (-∞(,)(𝑘 + (1 / (2↑𝑛))))) ∈ dom vol) →
if(𝑘 = 𝑛, ℝ, (◡𝐹 “ (-∞(,)(𝑘 + (1 / (2↑𝑛)))))) ∈ dom vol) |
295 | 289, 293,
294 | sylancr 694 |
. . . . . . . 8
⊢ (𝜑 → if(𝑘 = 𝑛, ℝ, (◡𝐹 “ (-∞(,)(𝑘 + (1 / (2↑𝑛)))))) ∈ dom vol) |
296 | | mbfima 23205 |
. . . . . . . . 9
⊢ ((𝐹 ∈ MblFn ∧ 𝐹:ℝ⟶ℝ) →
(◡𝐹 “ (-∞(,)𝑘)) ∈ dom vol) |
297 | 6, 291, 296 | syl2anc 691 |
. . . . . . . 8
⊢ (𝜑 → (◡𝐹 “ (-∞(,)𝑘)) ∈ dom vol) |
298 | | difmbl 23118 |
. . . . . . . 8
⊢
((if(𝑘 = 𝑛, ℝ, (◡𝐹 “ (-∞(,)(𝑘 + (1 / (2↑𝑛)))))) ∈ dom vol ∧ (◡𝐹 “ (-∞(,)𝑘)) ∈ dom vol) → (if(𝑘 = 𝑛, ℝ, (◡𝐹 “ (-∞(,)(𝑘 + (1 / (2↑𝑛)))))) ∖ (◡𝐹 “ (-∞(,)𝑘))) ∈ dom vol) |
299 | 295, 297,
298 | syl2anc 691 |
. . . . . . 7
⊢ (𝜑 → (if(𝑘 = 𝑛, ℝ, (◡𝐹 “ (-∞(,)(𝑘 + (1 / (2↑𝑛)))))) ∖ (◡𝐹 “ (-∞(,)𝑘))) ∈ dom vol) |
300 | 299 | ad2antrr 758 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺‘𝑛) ∖ {0})) → (if(𝑘 = 𝑛, ℝ, (◡𝐹 “ (-∞(,)(𝑘 + (1 / (2↑𝑛)))))) ∖ (◡𝐹 “ (-∞(,)𝑘))) ∈ dom vol) |
301 | | inmbl 23117 |
. . . . . 6
⊢ (((-𝑛[,]𝑛) ∈ dom vol ∧ (if(𝑘 = 𝑛, ℝ, (◡𝐹 “ (-∞(,)(𝑘 + (1 / (2↑𝑛)))))) ∖ (◡𝐹 “ (-∞(,)𝑘))) ∈ dom vol) → ((-𝑛[,]𝑛) ∩ (if(𝑘 = 𝑛, ℝ, (◡𝐹 “ (-∞(,)(𝑘 + (1 / (2↑𝑛)))))) ∖ (◡𝐹 “ (-∞(,)𝑘)))) ∈ dom vol) |
302 | 280, 300,
301 | syl2anc 691 |
. . . . 5
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺‘𝑛) ∖ {0})) → ((-𝑛[,]𝑛) ∩ (if(𝑘 = 𝑛, ℝ, (◡𝐹 “ (-∞(,)(𝑘 + (1 / (2↑𝑛)))))) ∖ (◡𝐹 “ (-∞(,)𝑘)))) ∈ dom vol) |
303 | 288, 302 | eqeltrd 2688 |
. . . 4
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺‘𝑛) ∖ {0})) → (◡(𝐺‘𝑛) “ {𝑘}) ∈ dom vol) |
304 | | mblvol 23105 |
. . . . . 6
⊢ ((◡(𝐺‘𝑛) “ {𝑘}) ∈ dom vol → (vol‘(◡(𝐺‘𝑛) “ {𝑘})) = (vol*‘(◡(𝐺‘𝑛) “ {𝑘}))) |
305 | 303, 304 | syl 17 |
. . . . 5
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺‘𝑛) ∖ {0})) → (vol‘(◡(𝐺‘𝑛) “ {𝑘})) = (vol*‘(◡(𝐺‘𝑛) “ {𝑘}))) |
306 | 196 | adantr 480 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺‘𝑛) ∖ {0})) → (𝐺‘𝑛) Fn ℝ) |
307 | 306, 274 | syl 17 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺‘𝑛) ∖ {0})) → (𝑥 ∈ (◡(𝐺‘𝑛) “ {𝑘}) ↔ (𝑥 ∈ ℝ ∧ ((𝐺‘𝑛)‘𝑥) = 𝑘))) |
308 | 82, 185 | ifcld 4081 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛) ∈ ℝ) |
309 | | 0re 9919 |
. . . . . . . . . . . . . . 15
⊢ 0 ∈
ℝ |
310 | | ifcl 4080 |
. . . . . . . . . . . . . . 15
⊢
((if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛) ∈ ℝ ∧ 0 ∈ ℝ)
→ if(𝑥 ∈ (-𝑛[,]𝑛), if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛), 0) ∈ ℝ) |
311 | 308, 309,
310 | sylancl 693 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → if(𝑥 ∈ (-𝑛[,]𝑛), if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛), 0) ∈ ℝ) |
312 | 44 | fvmpt2 6200 |
. . . . . . . . . . . . . 14
⊢ ((𝑥 ∈ ℝ ∧ if(𝑥 ∈ (-𝑛[,]𝑛), if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛), 0) ∈ ℝ) → ((𝑥 ∈ ℝ ↦ if(𝑥 ∈ (-𝑛[,]𝑛), if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛), 0))‘𝑥) = if(𝑥 ∈ (-𝑛[,]𝑛), if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛), 0)) |
313 | 38, 311, 312 | syl2anc 691 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝑥 ∈ ℝ ↦ if(𝑥 ∈ (-𝑛[,]𝑛), if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛), 0))‘𝑥) = if(𝑥 ∈ (-𝑛[,]𝑛), if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛), 0)) |
314 | 37, 313 | eqtrd 2644 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝐺‘𝑛)‘𝑥) = if(𝑥 ∈ (-𝑛[,]𝑛), if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛), 0)) |
315 | 314 | adantlr 747 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺‘𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → ((𝐺‘𝑛)‘𝑥) = if(𝑥 ∈ (-𝑛[,]𝑛), if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛), 0)) |
316 | 315 | eqeq1d 2612 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺‘𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → (((𝐺‘𝑛)‘𝑥) = 𝑘 ↔ if(𝑥 ∈ (-𝑛[,]𝑛), if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛), 0) = 𝑘)) |
317 | 316, 56 | sylbid 229 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺‘𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → (((𝐺‘𝑛)‘𝑥) = 𝑘 → 𝑥 ∈ (-𝑛[,]𝑛))) |
318 | 317 | expimpd 627 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺‘𝑛) ∖ {0})) → ((𝑥 ∈ ℝ ∧ ((𝐺‘𝑛)‘𝑥) = 𝑘) → 𝑥 ∈ (-𝑛[,]𝑛))) |
319 | 307, 318 | sylbid 229 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺‘𝑛) ∖ {0})) → (𝑥 ∈ (◡(𝐺‘𝑛) “ {𝑘}) → 𝑥 ∈ (-𝑛[,]𝑛))) |
320 | 319 | ssrdv 3574 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺‘𝑛) ∖ {0})) → (◡(𝐺‘𝑛) “ {𝑘}) ⊆ (-𝑛[,]𝑛)) |
321 | | iccssre 12126 |
. . . . . . 7
⊢ ((-𝑛 ∈ ℝ ∧ 𝑛 ∈ ℝ) → (-𝑛[,]𝑛) ⊆ ℝ) |
322 | 278, 277,
321 | syl2anc 691 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺‘𝑛) ∖ {0})) → (-𝑛[,]𝑛) ⊆ ℝ) |
323 | | mblvol 23105 |
. . . . . . . 8
⊢ ((-𝑛[,]𝑛) ∈ dom vol → (vol‘(-𝑛[,]𝑛)) = (vol*‘(-𝑛[,]𝑛))) |
324 | 280, 323 | syl 17 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺‘𝑛) ∖ {0})) → (vol‘(-𝑛[,]𝑛)) = (vol*‘(-𝑛[,]𝑛))) |
325 | | iccvolcl 23142 |
. . . . . . . 8
⊢ ((-𝑛 ∈ ℝ ∧ 𝑛 ∈ ℝ) →
(vol‘(-𝑛[,]𝑛)) ∈
ℝ) |
326 | 278, 277,
325 | syl2anc 691 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺‘𝑛) ∖ {0})) → (vol‘(-𝑛[,]𝑛)) ∈ ℝ) |
327 | 324, 326 | eqeltrrd 2689 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺‘𝑛) ∖ {0})) → (vol*‘(-𝑛[,]𝑛)) ∈ ℝ) |
328 | | ovolsscl 23061 |
. . . . . 6
⊢ (((◡(𝐺‘𝑛) “ {𝑘}) ⊆ (-𝑛[,]𝑛) ∧ (-𝑛[,]𝑛) ⊆ ℝ ∧ (vol*‘(-𝑛[,]𝑛)) ∈ ℝ) → (vol*‘(◡(𝐺‘𝑛) “ {𝑘})) ∈ ℝ) |
329 | 320, 322,
327, 328 | syl3anc 1318 |
. . . . 5
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺‘𝑛) ∖ {0})) → (vol*‘(◡(𝐺‘𝑛) “ {𝑘})) ∈ ℝ) |
330 | 305, 329 | eqeltrd 2688 |
. . . 4
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺‘𝑛) ∖ {0})) → (vol‘(◡(𝐺‘𝑛) “ {𝑘})) ∈ ℝ) |
331 | 23, 34, 303, 330 | i1fd 23254 |
. . 3
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐺‘𝑛) ∈ dom
∫1) |
332 | 331 | ralrimiva 2949 |
. 2
⊢ (𝜑 → ∀𝑛 ∈ ℕ (𝐺‘𝑛) ∈ dom
∫1) |
333 | | ffnfv 6295 |
. 2
⊢ (𝐺:ℕ⟶dom
∫1 ↔ (𝐺
Fn ℕ ∧ ∀𝑛
∈ ℕ (𝐺‘𝑛) ∈ dom
∫1)) |
334 | 5, 332, 333 | sylanbrc 695 |
1
⊢ (𝜑 → 𝐺:ℕ⟶dom
∫1) |