Step |
Hyp |
Ref |
Expression |
1 |
|
ltrn2eq.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
2 |
|
ltrn2eq.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
3 |
|
ltrn2eq.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
4 |
|
ltrn2eq.t |
⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) |
5 |
|
simp3r |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ∧ ( 𝐹 ‘ 𝑄 ) = 𝑄 ) ) → ( 𝐹 ‘ 𝑄 ) = 𝑄 ) |
6 |
|
simpl1 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ∧ ( 𝐹 ‘ 𝑄 ) = 𝑄 ) ) ∧ ¬ 𝑄 ≤ 𝑊 ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
7 |
|
simpl21 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ∧ ( 𝐹 ‘ 𝑄 ) = 𝑄 ) ) ∧ ¬ 𝑄 ≤ 𝑊 ) → 𝐹 ∈ 𝑇 ) |
8 |
|
simpl22 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ∧ ( 𝐹 ‘ 𝑄 ) = 𝑄 ) ) ∧ ¬ 𝑄 ≤ 𝑊 ) → ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) |
9 |
|
simpl23 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ∧ ( 𝐹 ‘ 𝑄 ) = 𝑄 ) ) ∧ ¬ 𝑄 ≤ 𝑊 ) → 𝑄 ∈ 𝐴 ) |
10 |
|
simpr |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ∧ ( 𝐹 ‘ 𝑄 ) = 𝑄 ) ) ∧ ¬ 𝑄 ≤ 𝑊 ) → ¬ 𝑄 ≤ 𝑊 ) |
11 |
9 10
|
jca |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ∧ ( 𝐹 ‘ 𝑄 ) = 𝑄 ) ) ∧ ¬ 𝑄 ≤ 𝑊 ) → ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) |
12 |
|
simpl3l |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ∧ ( 𝐹 ‘ 𝑄 ) = 𝑄 ) ) ∧ ¬ 𝑄 ≤ 𝑊 ) → ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ) |
13 |
1 2 3 4
|
ltrnatneq |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ) → ( 𝐹 ‘ 𝑄 ) ≠ 𝑄 ) |
14 |
6 7 8 11 12 13
|
syl131anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ∧ ( 𝐹 ‘ 𝑄 ) = 𝑄 ) ) ∧ ¬ 𝑄 ≤ 𝑊 ) → ( 𝐹 ‘ 𝑄 ) ≠ 𝑄 ) |
15 |
14
|
ex |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ∧ ( 𝐹 ‘ 𝑄 ) = 𝑄 ) ) → ( ¬ 𝑄 ≤ 𝑊 → ( 𝐹 ‘ 𝑄 ) ≠ 𝑄 ) ) |
16 |
15
|
necon4bd |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ∧ ( 𝐹 ‘ 𝑄 ) = 𝑄 ) ) → ( ( 𝐹 ‘ 𝑄 ) = 𝑄 → 𝑄 ≤ 𝑊 ) ) |
17 |
5 16
|
mpd |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ∧ ( 𝐹 ‘ 𝑄 ) = 𝑄 ) ) → 𝑄 ≤ 𝑊 ) |