Step |
Hyp |
Ref |
Expression |
1 |
|
ltrnnidn.b |
⊢ 𝐵 = ( Base ‘ 𝐾 ) |
2 |
|
ltrnnidn.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
3 |
|
ltrnnidn.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
4 |
|
ltrnnidn.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
5 |
|
ltrnnidn.t |
⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) |
6 |
|
simpr |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) ∧ 𝐹 = ( I ↾ 𝐵 ) ) → 𝐹 = ( I ↾ 𝐵 ) ) |
7 |
6
|
fveq1d |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) ∧ 𝐹 = ( I ↾ 𝐵 ) ) → ( 𝐹 ‘ 𝑃 ) = ( ( I ↾ 𝐵 ) ‘ 𝑃 ) ) |
8 |
|
simpl3l |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) ∧ 𝐹 = ( I ↾ 𝐵 ) ) → 𝑃 ∈ 𝐴 ) |
9 |
1 3
|
atbase |
⊢ ( 𝑃 ∈ 𝐴 → 𝑃 ∈ 𝐵 ) |
10 |
|
fvresi |
⊢ ( 𝑃 ∈ 𝐵 → ( ( I ↾ 𝐵 ) ‘ 𝑃 ) = 𝑃 ) |
11 |
8 9 10
|
3syl |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) ∧ 𝐹 = ( I ↾ 𝐵 ) ) → ( ( I ↾ 𝐵 ) ‘ 𝑃 ) = 𝑃 ) |
12 |
7 11
|
eqtrd |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) ∧ 𝐹 = ( I ↾ 𝐵 ) ) → ( 𝐹 ‘ 𝑃 ) = 𝑃 ) |
13 |
12
|
ex |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( 𝐹 = ( I ↾ 𝐵 ) → ( 𝐹 ‘ 𝑃 ) = 𝑃 ) ) |
14 |
|
simpl1 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) ∧ 𝐹 ≠ ( I ↾ 𝐵 ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
15 |
|
simpl2 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) ∧ 𝐹 ≠ ( I ↾ 𝐵 ) ) → 𝐹 ∈ 𝑇 ) |
16 |
|
simpr |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) ∧ 𝐹 ≠ ( I ↾ 𝐵 ) ) → 𝐹 ≠ ( I ↾ 𝐵 ) ) |
17 |
|
simpl3 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) ∧ 𝐹 ≠ ( I ↾ 𝐵 ) ) → ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) |
18 |
1 2 3 4 5
|
ltrnnidn |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵 ) ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ) |
19 |
14 15 16 17 18
|
syl121anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) ∧ 𝐹 ≠ ( I ↾ 𝐵 ) ) → ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ) |
20 |
19
|
ex |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( 𝐹 ≠ ( I ↾ 𝐵 ) → ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ) ) |
21 |
20
|
necon4d |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( ( 𝐹 ‘ 𝑃 ) = 𝑃 → 𝐹 = ( I ↾ 𝐵 ) ) ) |
22 |
13 21
|
impbid |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( 𝐹 = ( I ↾ 𝐵 ) ↔ ( 𝐹 ‘ 𝑃 ) = 𝑃 ) ) |