Step |
Hyp |
Ref |
Expression |
1 |
|
ltrne.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
2 |
|
ltrne.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
3 |
|
ltrne.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
4 |
|
ltrne.t |
⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) |
5 |
|
simp11 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑝 ∈ 𝐴 ∧ 𝑝 ≤ 𝑊 ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
6 |
|
simp12 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑝 ∈ 𝐴 ∧ 𝑝 ≤ 𝑊 ) → 𝐹 ∈ 𝑇 ) |
7 |
|
eqid |
⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 ) |
8 |
7 2
|
atbase |
⊢ ( 𝑝 ∈ 𝐴 → 𝑝 ∈ ( Base ‘ 𝐾 ) ) |
9 |
8
|
3ad2ant2 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑝 ∈ 𝐴 ∧ 𝑝 ≤ 𝑊 ) → 𝑝 ∈ ( Base ‘ 𝐾 ) ) |
10 |
|
simp3 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑝 ∈ 𝐴 ∧ 𝑝 ≤ 𝑊 ) → 𝑝 ≤ 𝑊 ) |
11 |
7 1 3 4
|
ltrnval1 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑝 ∈ ( Base ‘ 𝐾 ) ∧ 𝑝 ≤ 𝑊 ) ) → ( 𝐹 ‘ 𝑝 ) = 𝑝 ) |
12 |
5 6 9 10 11
|
syl112anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑝 ∈ 𝐴 ∧ 𝑝 ≤ 𝑊 ) → ( 𝐹 ‘ 𝑝 ) = 𝑝 ) |
13 |
|
simp13 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑝 ∈ 𝐴 ∧ 𝑝 ≤ 𝑊 ) → 𝐺 ∈ 𝑇 ) |
14 |
7 1 3 4
|
ltrnval1 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐺 ∈ 𝑇 ∧ ( 𝑝 ∈ ( Base ‘ 𝐾 ) ∧ 𝑝 ≤ 𝑊 ) ) → ( 𝐺 ‘ 𝑝 ) = 𝑝 ) |
15 |
5 13 9 10 14
|
syl112anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑝 ∈ 𝐴 ∧ 𝑝 ≤ 𝑊 ) → ( 𝐺 ‘ 𝑝 ) = 𝑝 ) |
16 |
12 15
|
eqtr4d |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑝 ∈ 𝐴 ∧ 𝑝 ≤ 𝑊 ) → ( 𝐹 ‘ 𝑝 ) = ( 𝐺 ‘ 𝑝 ) ) |
17 |
16
|
3expia |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑝 ∈ 𝐴 ) → ( 𝑝 ≤ 𝑊 → ( 𝐹 ‘ 𝑝 ) = ( 𝐺 ‘ 𝑝 ) ) ) |
18 |
|
pm2.61 |
⊢ ( ( 𝑝 ≤ 𝑊 → ( 𝐹 ‘ 𝑝 ) = ( 𝐺 ‘ 𝑝 ) ) → ( ( ¬ 𝑝 ≤ 𝑊 → ( 𝐹 ‘ 𝑝 ) = ( 𝐺 ‘ 𝑝 ) ) → ( 𝐹 ‘ 𝑝 ) = ( 𝐺 ‘ 𝑝 ) ) ) |
19 |
17 18
|
syl |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑝 ∈ 𝐴 ) → ( ( ¬ 𝑝 ≤ 𝑊 → ( 𝐹 ‘ 𝑝 ) = ( 𝐺 ‘ 𝑝 ) ) → ( 𝐹 ‘ 𝑝 ) = ( 𝐺 ‘ 𝑝 ) ) ) |
20 |
|
re1tbw2 |
⊢ ( ( 𝐹 ‘ 𝑝 ) = ( 𝐺 ‘ 𝑝 ) → ( ¬ 𝑝 ≤ 𝑊 → ( 𝐹 ‘ 𝑝 ) = ( 𝐺 ‘ 𝑝 ) ) ) |
21 |
19 20
|
impbid1 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑝 ∈ 𝐴 ) → ( ( ¬ 𝑝 ≤ 𝑊 → ( 𝐹 ‘ 𝑝 ) = ( 𝐺 ‘ 𝑝 ) ) ↔ ( 𝐹 ‘ 𝑝 ) = ( 𝐺 ‘ 𝑝 ) ) ) |
22 |
21
|
ralbidva |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) → ( ∀ 𝑝 ∈ 𝐴 ( ¬ 𝑝 ≤ 𝑊 → ( 𝐹 ‘ 𝑝 ) = ( 𝐺 ‘ 𝑝 ) ) ↔ ∀ 𝑝 ∈ 𝐴 ( 𝐹 ‘ 𝑝 ) = ( 𝐺 ‘ 𝑝 ) ) ) |
23 |
2 3 4
|
ltrneq2 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) → ( ∀ 𝑝 ∈ 𝐴 ( 𝐹 ‘ 𝑝 ) = ( 𝐺 ‘ 𝑝 ) ↔ 𝐹 = 𝐺 ) ) |
24 |
22 23
|
bitrd |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) → ( ∀ 𝑝 ∈ 𝐴 ( ¬ 𝑝 ≤ 𝑊 → ( 𝐹 ‘ 𝑝 ) = ( 𝐺 ‘ 𝑝 ) ) ↔ 𝐹 = 𝐺 ) ) |