| 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 767 |
. . 3
⊢ (((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ (𝑥 ∈ 𝑋 ∧ (𝑂‘𝑥) = sup(ran 𝑂, ℝ, < ))) → 𝐺 ∈ Abel) |
| 5 | | simplr 769 |
. . 3
⊢ (((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ (𝑥 ∈ 𝑋 ∧ (𝑂‘𝑥) = sup(ran 𝑂, ℝ, < ))) → 𝐸 ∈ ℕ) |
| 6 | | simprl 771 |
. . 3
⊢ (((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ (𝑥 ∈ 𝑋 ∧ (𝑂‘𝑥) = sup(ran 𝑂, ℝ, < ))) → 𝑥 ∈ 𝑋) |
| 7 | 1, 3 | odf 19555 |
. . . . . . 7
⊢ 𝑂:𝑋⟶ℕ0 |
| 8 | | frn 6743 |
. . . . . . 7
⊢ (𝑂:𝑋⟶ℕ0 → ran 𝑂 ⊆
ℕ0) |
| 9 | 7, 8 | ax-mp 5 |
. . . . . 6
⊢ ran 𝑂 ⊆
ℕ0 |
| 10 | | nn0ssz 12636 |
. . . . . 6
⊢
ℕ0 ⊆ ℤ |
| 11 | 9, 10 | sstri 3993 |
. . . . 5
⊢ ran 𝑂 ⊆
ℤ |
| 12 | | nnz 12634 |
. . . . . . . 8
⊢ (𝐸 ∈ ℕ → 𝐸 ∈
ℤ) |
| 13 | 12 | adantl 481 |
. . . . . . 7
⊢ ((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) → 𝐸 ∈
ℤ) |
| 14 | | ablgrp 19803 |
. . . . . . . . . . . 12
⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp) |
| 15 | 14 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) → 𝐺 ∈ Grp) |
| 16 | 1, 2, 3 | gexod 19604 |
. . . . . . . . . . 11
⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝑋) → (𝑂‘𝑥) ∥ 𝐸) |
| 17 | 15, 16 | sylan 580 |
. . . . . . . . . 10
⊢ (((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ 𝑥 ∈ 𝑋) → (𝑂‘𝑥) ∥ 𝐸) |
| 18 | 1, 3 | odcl 19554 |
. . . . . . . . . . . . 13
⊢ (𝑥 ∈ 𝑋 → (𝑂‘𝑥) ∈
ℕ0) |
| 19 | 18 | adantl 481 |
. . . . . . . . . . . 12
⊢ (((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ 𝑥 ∈ 𝑋) → (𝑂‘𝑥) ∈
ℕ0) |
| 20 | 19 | nn0zd 12639 |
. . . . . . . . . . 11
⊢ (((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ 𝑥 ∈ 𝑋) → (𝑂‘𝑥) ∈ ℤ) |
| 21 | | simplr 769 |
. . . . . . . . . . 11
⊢ (((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ 𝑥 ∈ 𝑋) → 𝐸 ∈ ℕ) |
| 22 | | dvdsle 16347 |
. . . . . . . . . . 11
⊢ (((𝑂‘𝑥) ∈ ℤ ∧ 𝐸 ∈ ℕ) → ((𝑂‘𝑥) ∥ 𝐸 → (𝑂‘𝑥) ≤ 𝐸)) |
| 23 | 20, 21, 22 | syl2anc 584 |
. . . . . . . . . 10
⊢ (((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ 𝑥 ∈ 𝑋) → ((𝑂‘𝑥) ∥ 𝐸 → (𝑂‘𝑥) ≤ 𝐸)) |
| 24 | 17, 23 | mpd 15 |
. . . . . . . . 9
⊢ (((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ 𝑥 ∈ 𝑋) → (𝑂‘𝑥) ≤ 𝐸) |
| 25 | 24 | ralrimiva 3146 |
. . . . . . . 8
⊢ ((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) →
∀𝑥 ∈ 𝑋 (𝑂‘𝑥) ≤ 𝐸) |
| 26 | | ffn 6736 |
. . . . . . . . . 10
⊢ (𝑂:𝑋⟶ℕ0 → 𝑂 Fn 𝑋) |
| 27 | 7, 26 | ax-mp 5 |
. . . . . . . . 9
⊢ 𝑂 Fn 𝑋 |
| 28 | | breq1 5146 |
. . . . . . . . . 10
⊢ (𝑦 = (𝑂‘𝑥) → (𝑦 ≤ 𝐸 ↔ (𝑂‘𝑥) ≤ 𝐸)) |
| 29 | 28 | ralrn 7108 |
. . . . . . . . 9
⊢ (𝑂 Fn 𝑋 → (∀𝑦 ∈ ran 𝑂 𝑦 ≤ 𝐸 ↔ ∀𝑥 ∈ 𝑋 (𝑂‘𝑥) ≤ 𝐸)) |
| 30 | 27, 29 | ax-mp 5 |
. . . . . . . 8
⊢
(∀𝑦 ∈
ran 𝑂 𝑦 ≤ 𝐸 ↔ ∀𝑥 ∈ 𝑋 (𝑂‘𝑥) ≤ 𝐸) |
| 31 | 25, 30 | sylibr 234 |
. . . . . . 7
⊢ ((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) →
∀𝑦 ∈ ran 𝑂 𝑦 ≤ 𝐸) |
| 32 | | brralrspcev 5203 |
. . . . . . 7
⊢ ((𝐸 ∈ ℤ ∧
∀𝑦 ∈ ran 𝑂 𝑦 ≤ 𝐸) → ∃𝑛 ∈ ℤ ∀𝑦 ∈ ran 𝑂 𝑦 ≤ 𝑛) |
| 33 | 13, 31, 32 | syl2anc 584 |
. . . . . 6
⊢ ((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) →
∃𝑛 ∈ ℤ
∀𝑦 ∈ ran 𝑂 𝑦 ≤ 𝑛) |
| 34 | 33 | ad2antrr 726 |
. . . . 5
⊢ ((((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ (𝑥 ∈ 𝑋 ∧ (𝑂‘𝑥) = sup(ran 𝑂, ℝ, < ))) ∧ 𝑦 ∈ 𝑋) → ∃𝑛 ∈ ℤ ∀𝑦 ∈ ran 𝑂 𝑦 ≤ 𝑛) |
| 35 | 27 | a1i 11 |
. . . . . 6
⊢ (((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ (𝑥 ∈ 𝑋 ∧ (𝑂‘𝑥) = sup(ran 𝑂, ℝ, < ))) → 𝑂 Fn 𝑋) |
| 36 | | fnfvelrn 7100 |
. . . . . 6
⊢ ((𝑂 Fn 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑂‘𝑦) ∈ ran 𝑂) |
| 37 | 35, 36 | sylan 580 |
. . . . 5
⊢ ((((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ (𝑥 ∈ 𝑋 ∧ (𝑂‘𝑥) = sup(ran 𝑂, ℝ, < ))) ∧ 𝑦 ∈ 𝑋) → (𝑂‘𝑦) ∈ ran 𝑂) |
| 38 | | suprzub 12981 |
. . . . 5
⊢ ((ran
𝑂 ⊆ ℤ ∧
∃𝑛 ∈ ℤ
∀𝑦 ∈ ran 𝑂 𝑦 ≤ 𝑛 ∧ (𝑂‘𝑦) ∈ ran 𝑂) → (𝑂‘𝑦) ≤ sup(ran 𝑂, ℝ, < )) |
| 39 | 11, 34, 37, 38 | mp3an2i 1468 |
. . . 4
⊢ ((((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ (𝑥 ∈ 𝑋 ∧ (𝑂‘𝑥) = sup(ran 𝑂, ℝ, < ))) ∧ 𝑦 ∈ 𝑋) → (𝑂‘𝑦) ≤ sup(ran 𝑂, ℝ, < )) |
| 40 | | simplrr 778 |
. . . 4
⊢ ((((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ (𝑥 ∈ 𝑋 ∧ (𝑂‘𝑥) = sup(ran 𝑂, ℝ, < ))) ∧ 𝑦 ∈ 𝑋) → (𝑂‘𝑥) = sup(ran 𝑂, ℝ, < )) |
| 41 | 39, 40 | breqtrrd 5171 |
. . 3
⊢ ((((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ (𝑥 ∈ 𝑋 ∧ (𝑂‘𝑥) = sup(ran 𝑂, ℝ, < ))) ∧ 𝑦 ∈ 𝑋) → (𝑂‘𝑦) ≤ (𝑂‘𝑥)) |
| 42 | 1, 2, 3, 4, 5, 6, 41 | gexexlem 19870 |
. 2
⊢ (((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) ∧ (𝑥 ∈ 𝑋 ∧ (𝑂‘𝑥) = sup(ran 𝑂, ℝ, < ))) → (𝑂‘𝑥) = 𝐸) |
| 43 | 1 | grpbn0 18984 |
. . . . . 6
⊢ (𝐺 ∈ Grp → 𝑋 ≠ ∅) |
| 44 | 15, 43 | syl 17 |
. . . . 5
⊢ ((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) → 𝑋 ≠ ∅) |
| 45 | 7 | fdmi 6747 |
. . . . . . . 8
⊢ dom 𝑂 = 𝑋 |
| 46 | 45 | eqeq1i 2742 |
. . . . . . 7
⊢ (dom
𝑂 = ∅ ↔ 𝑋 = ∅) |
| 47 | | dm0rn0 5935 |
. . . . . . 7
⊢ (dom
𝑂 = ∅ ↔ ran
𝑂 =
∅) |
| 48 | 46, 47 | bitr3i 277 |
. . . . . 6
⊢ (𝑋 = ∅ ↔ ran 𝑂 = ∅) |
| 49 | 48 | necon3bii 2993 |
. . . . 5
⊢ (𝑋 ≠ ∅ ↔ ran 𝑂 ≠ ∅) |
| 50 | 44, 49 | sylib 218 |
. . . 4
⊢ ((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) → ran
𝑂 ≠
∅) |
| 51 | | suprzcl2 12980 |
. . . 4
⊢ ((ran
𝑂 ⊆ ℤ ∧ ran
𝑂 ≠ ∅ ∧
∃𝑛 ∈ ℤ
∀𝑦 ∈ ran 𝑂 𝑦 ≤ 𝑛) → sup(ran 𝑂, ℝ, < ) ∈ ran 𝑂) |
| 52 | 11, 50, 33, 51 | mp3an2i 1468 |
. . 3
⊢ ((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) → sup(ran
𝑂, ℝ, < ) ∈
ran 𝑂) |
| 53 | | fvelrnb 6969 |
. . . 4
⊢ (𝑂 Fn 𝑋 → (sup(ran 𝑂, ℝ, < ) ∈ ran 𝑂 ↔ ∃𝑥 ∈ 𝑋 (𝑂‘𝑥) = sup(ran 𝑂, ℝ, < ))) |
| 54 | 27, 53 | ax-mp 5 |
. . 3
⊢ (sup(ran
𝑂, ℝ, < ) ∈
ran 𝑂 ↔ ∃𝑥 ∈ 𝑋 (𝑂‘𝑥) = sup(ran 𝑂, ℝ, < )) |
| 55 | 52, 54 | sylib 218 |
. 2
⊢ ((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) →
∃𝑥 ∈ 𝑋 (𝑂‘𝑥) = sup(ran 𝑂, ℝ, < )) |
| 56 | 42, 55 | reximddv 3171 |
1
⊢ ((𝐺 ∈ Abel ∧ 𝐸 ∈ ℕ) →
∃𝑥 ∈ 𝑋 (𝑂‘𝑥) = 𝐸) |