Step |
Hyp |
Ref |
Expression |
1 |
|
cdlemg1c.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
2 |
|
cdlemg1c.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
3 |
|
cdlemg1c.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
4 |
|
cdlemg1c.t |
⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) |
5 |
|
eqidd |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) → ( 𝐹 ‘ 𝑃 ) = ( 𝐹 ‘ 𝑃 ) ) |
6 |
|
simp3 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) → 𝐹 ∈ 𝑇 ) |
7 |
1 2 3 4
|
ltrnel |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( ( 𝐹 ‘ 𝑃 ) ∈ 𝐴 ∧ ¬ ( 𝐹 ‘ 𝑃 ) ≤ 𝑊 ) ) |
8 |
7
|
3com23 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) → ( ( 𝐹 ‘ 𝑃 ) ∈ 𝐴 ∧ ¬ ( 𝐹 ‘ 𝑃 ) ≤ 𝑊 ) ) |
9 |
1 2 3 4
|
cdleme |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ∈ 𝐴 ∧ ¬ ( 𝐹 ‘ 𝑃 ) ≤ 𝑊 ) ) → ∃! 𝑓 ∈ 𝑇 ( 𝑓 ‘ 𝑃 ) = ( 𝐹 ‘ 𝑃 ) ) |
10 |
8 9
|
syld3an3 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) → ∃! 𝑓 ∈ 𝑇 ( 𝑓 ‘ 𝑃 ) = ( 𝐹 ‘ 𝑃 ) ) |
11 |
|
fveq1 |
⊢ ( 𝑓 = 𝐹 → ( 𝑓 ‘ 𝑃 ) = ( 𝐹 ‘ 𝑃 ) ) |
12 |
11
|
eqeq1d |
⊢ ( 𝑓 = 𝐹 → ( ( 𝑓 ‘ 𝑃 ) = ( 𝐹 ‘ 𝑃 ) ↔ ( 𝐹 ‘ 𝑃 ) = ( 𝐹 ‘ 𝑃 ) ) ) |
13 |
12
|
riota2 |
⊢ ( ( 𝐹 ∈ 𝑇 ∧ ∃! 𝑓 ∈ 𝑇 ( 𝑓 ‘ 𝑃 ) = ( 𝐹 ‘ 𝑃 ) ) → ( ( 𝐹 ‘ 𝑃 ) = ( 𝐹 ‘ 𝑃 ) ↔ ( ℩ 𝑓 ∈ 𝑇 ( 𝑓 ‘ 𝑃 ) = ( 𝐹 ‘ 𝑃 ) ) = 𝐹 ) ) |
14 |
6 10 13
|
syl2anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) → ( ( 𝐹 ‘ 𝑃 ) = ( 𝐹 ‘ 𝑃 ) ↔ ( ℩ 𝑓 ∈ 𝑇 ( 𝑓 ‘ 𝑃 ) = ( 𝐹 ‘ 𝑃 ) ) = 𝐹 ) ) |
15 |
5 14
|
mpbid |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) → ( ℩ 𝑓 ∈ 𝑇 ( 𝑓 ‘ 𝑃 ) = ( 𝐹 ‘ 𝑃 ) ) = 𝐹 ) |
16 |
15
|
eqcomd |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) → 𝐹 = ( ℩ 𝑓 ∈ 𝑇 ( 𝑓 ‘ 𝑃 ) = ( 𝐹 ‘ 𝑃 ) ) ) |