Proof of Theorem ltrneq
Step | Hyp | Ref
| Expression |
1 | | simp11 1202 |
. . . . . . . 8
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑝 ∈ 𝐴 ∧ 𝑝 ≤ 𝑊) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
2 | | simp12 1203 |
. . . . . . . 8
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑝 ∈ 𝐴 ∧ 𝑝 ≤ 𝑊) → 𝐹 ∈ 𝑇) |
3 | | eqid 2738 |
. . . . . . . . . 10
⊢
(Base‘𝐾) =
(Base‘𝐾) |
4 | | ltrne.a |
. . . . . . . . . 10
⊢ 𝐴 = (Atoms‘𝐾) |
5 | 3, 4 | atbase 37300 |
. . . . . . . . 9
⊢ (𝑝 ∈ 𝐴 → 𝑝 ∈ (Base‘𝐾)) |
6 | 5 | 3ad2ant2 1133 |
. . . . . . . 8
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑝 ∈ 𝐴 ∧ 𝑝 ≤ 𝑊) → 𝑝 ∈ (Base‘𝐾)) |
7 | | simp3 1137 |
. . . . . . . 8
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑝 ∈ 𝐴 ∧ 𝑝 ≤ 𝑊) → 𝑝 ≤ 𝑊) |
8 | | ltrne.l |
. . . . . . . . 9
⊢ ≤ =
(le‘𝐾) |
9 | | ltrne.h |
. . . . . . . . 9
⊢ 𝐻 = (LHyp‘𝐾) |
10 | | ltrne.t |
. . . . . . . . 9
⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) |
11 | 3, 8, 9, 10 | ltrnval1 38145 |
. . . . . . . 8
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑝 ∈ (Base‘𝐾) ∧ 𝑝 ≤ 𝑊)) → (𝐹‘𝑝) = 𝑝) |
12 | 1, 2, 6, 7, 11 | syl112anc 1373 |
. . . . . . 7
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑝 ∈ 𝐴 ∧ 𝑝 ≤ 𝑊) → (𝐹‘𝑝) = 𝑝) |
13 | | simp13 1204 |
. . . . . . . 8
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑝 ∈ 𝐴 ∧ 𝑝 ≤ 𝑊) → 𝐺 ∈ 𝑇) |
14 | 3, 8, 9, 10 | ltrnval1 38145 |
. . . . . . . 8
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇 ∧ (𝑝 ∈ (Base‘𝐾) ∧ 𝑝 ≤ 𝑊)) → (𝐺‘𝑝) = 𝑝) |
15 | 1, 13, 6, 7, 14 | syl112anc 1373 |
. . . . . . 7
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑝 ∈ 𝐴 ∧ 𝑝 ≤ 𝑊) → (𝐺‘𝑝) = 𝑝) |
16 | 12, 15 | eqtr4d 2781 |
. . . . . 6
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑝 ∈ 𝐴 ∧ 𝑝 ≤ 𝑊) → (𝐹‘𝑝) = (𝐺‘𝑝)) |
17 | 16 | 3expia 1120 |
. . . . 5
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑝 ∈ 𝐴) → (𝑝 ≤ 𝑊 → (𝐹‘𝑝) = (𝐺‘𝑝))) |
18 | | pm2.61 191 |
. . . . 5
⊢ ((𝑝 ≤ 𝑊 → (𝐹‘𝑝) = (𝐺‘𝑝)) → ((¬ 𝑝 ≤ 𝑊 → (𝐹‘𝑝) = (𝐺‘𝑝)) → (𝐹‘𝑝) = (𝐺‘𝑝))) |
19 | 17, 18 | syl 17 |
. . . 4
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑝 ∈ 𝐴) → ((¬ 𝑝 ≤ 𝑊 → (𝐹‘𝑝) = (𝐺‘𝑝)) → (𝐹‘𝑝) = (𝐺‘𝑝))) |
20 | | re1tbw2 1749 |
. . . 4
⊢ ((𝐹‘𝑝) = (𝐺‘𝑝) → (¬ 𝑝 ≤ 𝑊 → (𝐹‘𝑝) = (𝐺‘𝑝))) |
21 | 19, 20 | impbid1 224 |
. . 3
⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑝 ∈ 𝐴) → ((¬ 𝑝 ≤ 𝑊 → (𝐹‘𝑝) = (𝐺‘𝑝)) ↔ (𝐹‘𝑝) = (𝐺‘𝑝))) |
22 | 21 | ralbidva 3117 |
. 2
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → (∀𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 → (𝐹‘𝑝) = (𝐺‘𝑝)) ↔ ∀𝑝 ∈ 𝐴 (𝐹‘𝑝) = (𝐺‘𝑝))) |
23 | 4, 9, 10 | ltrneq2 38159 |
. 2
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → (∀𝑝 ∈ 𝐴 (𝐹‘𝑝) = (𝐺‘𝑝) ↔ 𝐹 = 𝐺)) |
24 | 22, 23 | bitrd 278 |
1
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → (∀𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 → (𝐹‘𝑝) = (𝐺‘𝑝)) ↔ 𝐹 = 𝐺)) |