Step | Hyp | Ref
| Expression |
1 | | gexex.1 |
. . 3
⊢ 𝑋 = (Base‘𝐺) |
2 | | gexex.2 |
. . 3
⊢ 𝐸 = (gEx‘𝐺) |
3 | | gexex.3 |
. . 3
⊢ 𝑂 = (od‘𝐺) |
4 | | simpll 790 |
. . 3
⊢ (((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ (𝑥 ∈ 𝑋 ∧ (𝑂‘𝑥) = sup(ran 𝑂, ℝ, < ))) → 𝐺 ∈ Abel) |
5 | | simplr 792 |
. . 3
⊢ (((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ (𝑥 ∈ 𝑋 ∧ (𝑂‘𝑥) = sup(ran 𝑂, ℝ, < ))) → 𝐸 ∈ ℕ) |
6 | | simprl 794 |
. . 3
⊢ (((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ (𝑥 ∈ 𝑋 ∧ (𝑂‘𝑥) = sup(ran 𝑂, ℝ, < ))) → 𝑥 ∈ 𝑋) |
7 | 1, 3 | odf 17956 |
. . . . . . . 8
⊢ 𝑂:𝑋⟶ℕ0 |
8 | | frn 6053 |
. . . . . . . 8
⊢ (𝑂:𝑋⟶ℕ0 → ran 𝑂 ⊆
ℕ0) |
9 | 7, 8 | ax-mp 5 |
. . . . . . 7
⊢ ran 𝑂 ⊆
ℕ0 |
10 | | nn0ssz 11398 |
. . . . . . 7
⊢
ℕ0 ⊆ ℤ |
11 | 9, 10 | sstri 3612 |
. . . . . 6
⊢ ran 𝑂 ⊆
ℤ |
12 | 11 | a1i 11 |
. . . . 5
⊢ ((((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ (𝑥 ∈ 𝑋 ∧ (𝑂‘𝑥) = sup(ran 𝑂, ℝ, < ))) ∧ 𝑦 ∈ 𝑋) → ran 𝑂 ⊆ ℤ) |
13 | | nnz 11399 |
. . . . . . . 8
⊢ (𝐸 ∈ ℕ → 𝐸 ∈
ℤ) |
14 | 13 | adantl 482 |
. . . . . . 7
⊢ ((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) → 𝐸 ∈
ℤ) |
15 | | ablgrp 18198 |
. . . . . . . . . . . 12
⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp) |
16 | 15 | adantr 481 |
. . . . . . . . . . 11
⊢ ((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) → 𝐺 ∈ Grp) |
17 | 1, 2, 3 | gexod 18001 |
. . . . . . . . . . 11
⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝑋) → (𝑂‘𝑥) ∥ 𝐸) |
18 | 16, 17 | sylan 488 |
. . . . . . . . . 10
⊢ (((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ 𝑥 ∈ 𝑋) → (𝑂‘𝑥) ∥ 𝐸) |
19 | 1, 3 | odcl 17955 |
. . . . . . . . . . . . 13
⊢ (𝑥 ∈ 𝑋 → (𝑂‘𝑥) ∈
ℕ0) |
20 | 19 | adantl 482 |
. . . . . . . . . . . 12
⊢ (((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ 𝑥 ∈ 𝑋) → (𝑂‘𝑥) ∈
ℕ0) |
21 | 20 | nn0zd 11480 |
. . . . . . . . . . 11
⊢ (((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ 𝑥 ∈ 𝑋) → (𝑂‘𝑥) ∈ ℤ) |
22 | | simplr 792 |
. . . . . . . . . . 11
⊢ (((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ 𝑥 ∈ 𝑋) → 𝐸 ∈ ℕ) |
23 | | dvdsle 15032 |
. . . . . . . . . . 11
⊢ (((𝑂‘𝑥) ∈ ℤ ∧ 𝐸 ∈ ℕ) → ((𝑂‘𝑥) ∥ 𝐸 → (𝑂‘𝑥) ≤ 𝐸)) |
24 | 21, 22, 23 | syl2anc 693 |
. . . . . . . . . 10
⊢ (((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ 𝑥 ∈ 𝑋) → ((𝑂‘𝑥) ∥ 𝐸 → (𝑂‘𝑥) ≤ 𝐸)) |
25 | 18, 24 | mpd 15 |
. . . . . . . . 9
⊢ (((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ 𝑥 ∈ 𝑋) → (𝑂‘𝑥) ≤ 𝐸) |
26 | 25 | ralrimiva 2966 |
. . . . . . . 8
⊢ ((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) →
∀𝑥 ∈ 𝑋 (𝑂‘𝑥) ≤ 𝐸) |
27 | | ffn 6045 |
. . . . . . . . . 10
⊢ (𝑂:𝑋⟶ℕ0 → 𝑂 Fn 𝑋) |
28 | 7, 27 | ax-mp 5 |
. . . . . . . . 9
⊢ 𝑂 Fn 𝑋 |
29 | | breq1 4656 |
. . . . . . . . . 10
⊢ (𝑦 = (𝑂‘𝑥) → (𝑦 ≤ 𝐸 ↔ (𝑂‘𝑥) ≤ 𝐸)) |
30 | 29 | ralrn 6362 |
. . . . . . . . 9
⊢ (𝑂 Fn 𝑋 → (∀𝑦 ∈ ran 𝑂 𝑦 ≤ 𝐸 ↔ ∀𝑥 ∈ 𝑋 (𝑂‘𝑥) ≤ 𝐸)) |
31 | 28, 30 | ax-mp 5 |
. . . . . . . 8
⊢
(∀𝑦 ∈
ran 𝑂 𝑦 ≤ 𝐸 ↔ ∀𝑥 ∈ 𝑋 (𝑂‘𝑥) ≤ 𝐸) |
32 | 26, 31 | sylibr 224 |
. . . . . . 7
⊢ ((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) →
∀𝑦 ∈ ran 𝑂 𝑦 ≤ 𝐸) |
33 | | breq2 4657 |
. . . . . . . . 9
⊢ (𝑛 = 𝐸 → (𝑦 ≤ 𝑛 ↔ 𝑦 ≤ 𝐸)) |
34 | 33 | ralbidv 2986 |
. . . . . . . 8
⊢ (𝑛 = 𝐸 → (∀𝑦 ∈ ran 𝑂 𝑦 ≤ 𝑛 ↔ ∀𝑦 ∈ ran 𝑂 𝑦 ≤ 𝐸)) |
35 | 34 | rspcev 3309 |
. . . . . . 7
⊢ ((𝐸 ∈ ℤ ∧
∀𝑦 ∈ ran 𝑂 𝑦 ≤ 𝐸) → ∃𝑛 ∈ ℤ ∀𝑦 ∈ ran 𝑂 𝑦 ≤ 𝑛) |
36 | 14, 32, 35 | syl2anc 693 |
. . . . . 6
⊢ ((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) →
∃𝑛 ∈ ℤ
∀𝑦 ∈ ran 𝑂 𝑦 ≤ 𝑛) |
37 | 36 | ad2antrr 762 |
. . . . 5
⊢ ((((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ (𝑥 ∈ 𝑋 ∧ (𝑂‘𝑥) = sup(ran 𝑂, ℝ, < ))) ∧ 𝑦 ∈ 𝑋) → ∃𝑛 ∈ ℤ ∀𝑦 ∈ ran 𝑂 𝑦 ≤ 𝑛) |
38 | 28 | a1i 11 |
. . . . . 6
⊢ (((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ (𝑥 ∈ 𝑋 ∧ (𝑂‘𝑥) = sup(ran 𝑂, ℝ, < ))) → 𝑂 Fn 𝑋) |
39 | | fnfvelrn 6356 |
. . . . . 6
⊢ ((𝑂 Fn 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑂‘𝑦) ∈ ran 𝑂) |
40 | 38, 39 | sylan 488 |
. . . . 5
⊢ ((((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ (𝑥 ∈ 𝑋 ∧ (𝑂‘𝑥) = sup(ran 𝑂, ℝ, < ))) ∧ 𝑦 ∈ 𝑋) → (𝑂‘𝑦) ∈ ran 𝑂) |
41 | | suprzub 11779 |
. . . . 5
⊢ ((ran
𝑂 ⊆ ℤ ∧
∃𝑛 ∈ ℤ
∀𝑦 ∈ ran 𝑂 𝑦 ≤ 𝑛 ∧ (𝑂‘𝑦) ∈ ran 𝑂) → (𝑂‘𝑦) ≤ sup(ran 𝑂, ℝ, < )) |
42 | 12, 37, 40, 41 | syl3anc 1326 |
. . . 4
⊢ ((((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ (𝑥 ∈ 𝑋 ∧ (𝑂‘𝑥) = sup(ran 𝑂, ℝ, < ))) ∧ 𝑦 ∈ 𝑋) → (𝑂‘𝑦) ≤ sup(ran 𝑂, ℝ, < )) |
43 | | simplrr 801 |
. . . 4
⊢ ((((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ (𝑥 ∈ 𝑋 ∧ (𝑂‘𝑥) = sup(ran 𝑂, ℝ, < ))) ∧ 𝑦 ∈ 𝑋) → (𝑂‘𝑥) = sup(ran 𝑂, ℝ, < )) |
44 | 42, 43 | breqtrrd 4681 |
. . 3
⊢ ((((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ (𝑥 ∈ 𝑋 ∧ (𝑂‘𝑥) = sup(ran 𝑂, ℝ, < ))) ∧ 𝑦 ∈ 𝑋) → (𝑂‘𝑦) ≤ (𝑂‘𝑥)) |
45 | 1, 2, 3, 4, 5, 6, 44 | gexexlem 18255 |
. 2
⊢ (((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ (𝑥 ∈ 𝑋 ∧ (𝑂‘𝑥) = sup(ran 𝑂, ℝ, < ))) → (𝑂‘𝑥) = 𝐸) |
46 | 11 | a1i 11 |
. . . 4
⊢ ((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) → ran
𝑂 ⊆
ℤ) |
47 | 1 | grpbn0 17451 |
. . . . . 6
⊢ (𝐺 ∈ Grp → 𝑋 ≠ ∅) |
48 | 16, 47 | syl 17 |
. . . . 5
⊢ ((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) → 𝑋 ≠ ∅) |
49 | 7 | fdmi 6052 |
. . . . . . . 8
⊢ dom 𝑂 = 𝑋 |
50 | 49 | eqeq1i 2627 |
. . . . . . 7
⊢ (dom
𝑂 = ∅ ↔ 𝑋 = ∅) |
51 | | dm0rn0 5342 |
. . . . . . 7
⊢ (dom
𝑂 = ∅ ↔ ran
𝑂 =
∅) |
52 | 50, 51 | bitr3i 266 |
. . . . . 6
⊢ (𝑋 = ∅ ↔ ran 𝑂 = ∅) |
53 | 52 | necon3bii 2846 |
. . . . 5
⊢ (𝑋 ≠ ∅ ↔ ran 𝑂 ≠ ∅) |
54 | 48, 53 | sylib 208 |
. . . 4
⊢ ((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) → ran
𝑂 ≠
∅) |
55 | | suprzcl2 11778 |
. . . 4
⊢ ((ran
𝑂 ⊆ ℤ ∧ ran
𝑂 ≠ ∅ ∧
∃𝑛 ∈ ℤ
∀𝑦 ∈ ran 𝑂 𝑦 ≤ 𝑛) → sup(ran 𝑂, ℝ, < ) ∈ ran 𝑂) |
56 | 46, 54, 36, 55 | syl3anc 1326 |
. . 3
⊢ ((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) → sup(ran
𝑂, ℝ, < ) ∈
ran 𝑂) |
57 | | fvelrnb 6243 |
. . . 4
⊢ (𝑂 Fn 𝑋 → (sup(ran 𝑂, ℝ, < ) ∈ ran 𝑂 ↔ ∃𝑥 ∈ 𝑋 (𝑂‘𝑥) = sup(ran 𝑂, ℝ, < ))) |
58 | 28, 57 | ax-mp 5 |
. . 3
⊢ (sup(ran
𝑂, ℝ, < ) ∈
ran 𝑂 ↔ ∃𝑥 ∈ 𝑋 (𝑂‘𝑥) = sup(ran 𝑂, ℝ, < )) |
59 | 56, 58 | sylib 208 |
. 2
⊢ ((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) →
∃𝑥 ∈ 𝑋 (𝑂‘𝑥) = sup(ran 𝑂, ℝ, < )) |
60 | 45, 59 | reximddv 3018 |
1
⊢ ((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) →
∃𝑥 ∈ 𝑋 (𝑂‘𝑥) = 𝐸) |