Step | Hyp | Ref
| Expression |
1 | | 2lgslem1b.f |
. . . 4
⊢ 𝐹 = (𝑗 ∈ 𝐼 ↦ (𝑗 · 2)) |
2 | | eqeq1 2742 |
. . . . . 6
⊢ (𝑥 = (𝑗 · 2) → (𝑥 = (𝑖 · 2) ↔ (𝑗 · 2) = (𝑖 · 2))) |
3 | 2 | rexbidv 3206 |
. . . . 5
⊢ (𝑥 = (𝑗 · 2) → (∃𝑖 ∈ 𝐼 𝑥 = (𝑖 · 2) ↔ ∃𝑖 ∈ 𝐼 (𝑗 · 2) = (𝑖 · 2))) |
4 | | elfzelz 12991 |
. . . . . . 7
⊢ (𝑗 ∈ (𝐴...𝐵) → 𝑗 ∈ ℤ) |
5 | | 2lgslem1b.i |
. . . . . . 7
⊢ 𝐼 = (𝐴...𝐵) |
6 | 4, 5 | eleq2s 2851 |
. . . . . 6
⊢ (𝑗 ∈ 𝐼 → 𝑗 ∈ ℤ) |
7 | | 2z 12088 |
. . . . . . 7
⊢ 2 ∈
ℤ |
8 | 7 | a1i 11 |
. . . . . 6
⊢ (𝑗 ∈ 𝐼 → 2 ∈ ℤ) |
9 | 6, 8 | zmulcld 12167 |
. . . . 5
⊢ (𝑗 ∈ 𝐼 → (𝑗 · 2) ∈ ℤ) |
10 | | id 22 |
. . . . . 6
⊢ (𝑗 ∈ 𝐼 → 𝑗 ∈ 𝐼) |
11 | | oveq1 7171 |
. . . . . . . 8
⊢ (𝑖 = 𝑗 → (𝑖 · 2) = (𝑗 · 2)) |
12 | 11 | eqeq2d 2749 |
. . . . . . 7
⊢ (𝑖 = 𝑗 → ((𝑗 · 2) = (𝑖 · 2) ↔ (𝑗 · 2) = (𝑗 · 2))) |
13 | 12 | adantl 485 |
. . . . . 6
⊢ ((𝑗 ∈ 𝐼 ∧ 𝑖 = 𝑗) → ((𝑗 · 2) = (𝑖 · 2) ↔ (𝑗 · 2) = (𝑗 · 2))) |
14 | | eqidd 2739 |
. . . . . 6
⊢ (𝑗 ∈ 𝐼 → (𝑗 · 2) = (𝑗 · 2)) |
15 | 10, 13, 14 | rspcedvd 3527 |
. . . . 5
⊢ (𝑗 ∈ 𝐼 → ∃𝑖 ∈ 𝐼 (𝑗 · 2) = (𝑖 · 2)) |
16 | 3, 9, 15 | elrabd 3587 |
. . . 4
⊢ (𝑗 ∈ 𝐼 → (𝑗 · 2) ∈ {𝑥 ∈ ℤ ∣ ∃𝑖 ∈ 𝐼 𝑥 = (𝑖 · 2)}) |
17 | 1, 16 | fmpti 6880 |
. . 3
⊢ 𝐹:𝐼⟶{𝑥 ∈ ℤ ∣ ∃𝑖 ∈ 𝐼 𝑥 = (𝑖 · 2)} |
18 | | oveq1 7171 |
. . . . . . 7
⊢ (𝑗 = 𝑦 → (𝑗 · 2) = (𝑦 · 2)) |
19 | | simpl 486 |
. . . . . . 7
⊢ ((𝑦 ∈ 𝐼 ∧ 𝑧 ∈ 𝐼) → 𝑦 ∈ 𝐼) |
20 | | ovexd 7199 |
. . . . . . 7
⊢ ((𝑦 ∈ 𝐼 ∧ 𝑧 ∈ 𝐼) → (𝑦 · 2) ∈ V) |
21 | 1, 18, 19, 20 | fvmptd3 6792 |
. . . . . 6
⊢ ((𝑦 ∈ 𝐼 ∧ 𝑧 ∈ 𝐼) → (𝐹‘𝑦) = (𝑦 · 2)) |
22 | | oveq1 7171 |
. . . . . . 7
⊢ (𝑗 = 𝑧 → (𝑗 · 2) = (𝑧 · 2)) |
23 | | simpr 488 |
. . . . . . 7
⊢ ((𝑦 ∈ 𝐼 ∧ 𝑧 ∈ 𝐼) → 𝑧 ∈ 𝐼) |
24 | | ovexd 7199 |
. . . . . . 7
⊢ ((𝑦 ∈ 𝐼 ∧ 𝑧 ∈ 𝐼) → (𝑧 · 2) ∈ V) |
25 | 1, 22, 23, 24 | fvmptd3 6792 |
. . . . . 6
⊢ ((𝑦 ∈ 𝐼 ∧ 𝑧 ∈ 𝐼) → (𝐹‘𝑧) = (𝑧 · 2)) |
26 | 21, 25 | eqeq12d 2754 |
. . . . 5
⊢ ((𝑦 ∈ 𝐼 ∧ 𝑧 ∈ 𝐼) → ((𝐹‘𝑦) = (𝐹‘𝑧) ↔ (𝑦 · 2) = (𝑧 · 2))) |
27 | | elfzelz 12991 |
. . . . . . . . . 10
⊢ (𝑦 ∈ (𝐴...𝐵) → 𝑦 ∈ ℤ) |
28 | 27, 5 | eleq2s 2851 |
. . . . . . . . 9
⊢ (𝑦 ∈ 𝐼 → 𝑦 ∈ ℤ) |
29 | 28 | zcnd 12162 |
. . . . . . . 8
⊢ (𝑦 ∈ 𝐼 → 𝑦 ∈ ℂ) |
30 | 29 | adantr 484 |
. . . . . . 7
⊢ ((𝑦 ∈ 𝐼 ∧ 𝑧 ∈ 𝐼) → 𝑦 ∈ ℂ) |
31 | | elfzelz 12991 |
. . . . . . . . . 10
⊢ (𝑧 ∈ (𝐴...𝐵) → 𝑧 ∈ ℤ) |
32 | 31, 5 | eleq2s 2851 |
. . . . . . . . 9
⊢ (𝑧 ∈ 𝐼 → 𝑧 ∈ ℤ) |
33 | 32 | zcnd 12162 |
. . . . . . . 8
⊢ (𝑧 ∈ 𝐼 → 𝑧 ∈ ℂ) |
34 | 33 | adantl 485 |
. . . . . . 7
⊢ ((𝑦 ∈ 𝐼 ∧ 𝑧 ∈ 𝐼) → 𝑧 ∈ ℂ) |
35 | | 2cnd 11787 |
. . . . . . 7
⊢ ((𝑦 ∈ 𝐼 ∧ 𝑧 ∈ 𝐼) → 2 ∈ ℂ) |
36 | | 2ne0 11813 |
. . . . . . . 8
⊢ 2 ≠
0 |
37 | 36 | a1i 11 |
. . . . . . 7
⊢ ((𝑦 ∈ 𝐼 ∧ 𝑧 ∈ 𝐼) → 2 ≠ 0) |
38 | 30, 34, 35, 37 | mulcan2d 11345 |
. . . . . 6
⊢ ((𝑦 ∈ 𝐼 ∧ 𝑧 ∈ 𝐼) → ((𝑦 · 2) = (𝑧 · 2) ↔ 𝑦 = 𝑧)) |
39 | 38 | biimpd 232 |
. . . . 5
⊢ ((𝑦 ∈ 𝐼 ∧ 𝑧 ∈ 𝐼) → ((𝑦 · 2) = (𝑧 · 2) → 𝑦 = 𝑧)) |
40 | 26, 39 | sylbid 243 |
. . . 4
⊢ ((𝑦 ∈ 𝐼 ∧ 𝑧 ∈ 𝐼) → ((𝐹‘𝑦) = (𝐹‘𝑧) → 𝑦 = 𝑧)) |
41 | 40 | rgen2 3115 |
. . 3
⊢
∀𝑦 ∈
𝐼 ∀𝑧 ∈ 𝐼 ((𝐹‘𝑦) = (𝐹‘𝑧) → 𝑦 = 𝑧) |
42 | | dff13 7018 |
. . 3
⊢ (𝐹:𝐼–1-1→{𝑥 ∈ ℤ ∣ ∃𝑖 ∈ 𝐼 𝑥 = (𝑖 · 2)} ↔ (𝐹:𝐼⟶{𝑥 ∈ ℤ ∣ ∃𝑖 ∈ 𝐼 𝑥 = (𝑖 · 2)} ∧ ∀𝑦 ∈ 𝐼 ∀𝑧 ∈ 𝐼 ((𝐹‘𝑦) = (𝐹‘𝑧) → 𝑦 = 𝑧))) |
43 | 17, 41, 42 | mpbir2an 711 |
. 2
⊢ 𝐹:𝐼–1-1→{𝑥 ∈ ℤ ∣ ∃𝑖 ∈ 𝐼 𝑥 = (𝑖 · 2)} |
44 | | oveq1 7171 |
. . . . . . 7
⊢ (𝑗 = 𝑖 → (𝑗 · 2) = (𝑖 · 2)) |
45 | 44 | eqeq2d 2749 |
. . . . . 6
⊢ (𝑗 = 𝑖 → (𝑥 = (𝑗 · 2) ↔ 𝑥 = (𝑖 · 2))) |
46 | 45 | cbvrexvw 3349 |
. . . . 5
⊢
(∃𝑗 ∈
𝐼 𝑥 = (𝑗 · 2) ↔ ∃𝑖 ∈ 𝐼 𝑥 = (𝑖 · 2)) |
47 | | elfzelz 12991 |
. . . . . . . . . 10
⊢ (𝑖 ∈ (𝐴...𝐵) → 𝑖 ∈ ℤ) |
48 | 7 | a1i 11 |
. . . . . . . . . 10
⊢ (𝑖 ∈ (𝐴...𝐵) → 2 ∈ ℤ) |
49 | 47, 48 | zmulcld 12167 |
. . . . . . . . 9
⊢ (𝑖 ∈ (𝐴...𝐵) → (𝑖 · 2) ∈ ℤ) |
50 | 49, 5 | eleq2s 2851 |
. . . . . . . 8
⊢ (𝑖 ∈ 𝐼 → (𝑖 · 2) ∈ ℤ) |
51 | | eleq1 2820 |
. . . . . . . 8
⊢ (𝑥 = (𝑖 · 2) → (𝑥 ∈ ℤ ↔ (𝑖 · 2) ∈
ℤ)) |
52 | 50, 51 | syl5ibrcom 250 |
. . . . . . 7
⊢ (𝑖 ∈ 𝐼 → (𝑥 = (𝑖 · 2) → 𝑥 ∈ ℤ)) |
53 | 52 | rexlimiv 3189 |
. . . . . 6
⊢
(∃𝑖 ∈
𝐼 𝑥 = (𝑖 · 2) → 𝑥 ∈ ℤ) |
54 | 53 | pm4.71ri 564 |
. . . . 5
⊢
(∃𝑖 ∈
𝐼 𝑥 = (𝑖 · 2) ↔ (𝑥 ∈ ℤ ∧ ∃𝑖 ∈ 𝐼 𝑥 = (𝑖 · 2))) |
55 | 46, 54 | bitri 278 |
. . . 4
⊢
(∃𝑗 ∈
𝐼 𝑥 = (𝑗 · 2) ↔ (𝑥 ∈ ℤ ∧ ∃𝑖 ∈ 𝐼 𝑥 = (𝑖 · 2))) |
56 | 55 | abbii 2803 |
. . 3
⊢ {𝑥 ∣ ∃𝑗 ∈ 𝐼 𝑥 = (𝑗 · 2)} = {𝑥 ∣ (𝑥 ∈ ℤ ∧ ∃𝑖 ∈ 𝐼 𝑥 = (𝑖 · 2))} |
57 | 1 | rnmpt 5792 |
. . 3
⊢ ran 𝐹 = {𝑥 ∣ ∃𝑗 ∈ 𝐼 𝑥 = (𝑗 · 2)} |
58 | | df-rab 3062 |
. . 3
⊢ {𝑥 ∈ ℤ ∣
∃𝑖 ∈ 𝐼 𝑥 = (𝑖 · 2)} = {𝑥 ∣ (𝑥 ∈ ℤ ∧ ∃𝑖 ∈ 𝐼 𝑥 = (𝑖 · 2))} |
59 | 56, 57, 58 | 3eqtr4i 2771 |
. 2
⊢ ran 𝐹 = {𝑥 ∈ ℤ ∣ ∃𝑖 ∈ 𝐼 𝑥 = (𝑖 · 2)} |
60 | | dff1o5 6621 |
. 2
⊢ (𝐹:𝐼–1-1-onto→{𝑥 ∈ ℤ ∣ ∃𝑖 ∈ 𝐼 𝑥 = (𝑖 · 2)} ↔ (𝐹:𝐼–1-1→{𝑥 ∈ ℤ ∣ ∃𝑖 ∈ 𝐼 𝑥 = (𝑖 · 2)} ∧ ran 𝐹 = {𝑥 ∈ ℤ ∣ ∃𝑖 ∈ 𝐼 𝑥 = (𝑖 · 2)})) |
61 | 43, 59, 60 | mpbir2an 711 |
1
⊢ 𝐹:𝐼–1-1-onto→{𝑥 ∈ ℤ ∣ ∃𝑖 ∈ 𝐼 𝑥 = (𝑖 · 2)} |