Step |
Hyp |
Ref |
Expression |
1 |
|
dia2dimlem4.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
2 |
|
dia2dimlem4.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
3 |
|
dia2dimlem4.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
4 |
|
dia2dimlem4.t |
⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) |
5 |
|
dia2dimlem4.k |
⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
6 |
|
dia2dimlem4.p |
⊢ ( 𝜑 → ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) |
7 |
|
dia2dimlem4.f |
⊢ ( 𝜑 → 𝐹 ∈ 𝑇 ) |
8 |
|
dia2dimlem4.g |
⊢ ( 𝜑 → 𝐺 ∈ 𝑇 ) |
9 |
|
dia2dimlem4.gv |
⊢ ( 𝜑 → ( 𝐺 ‘ 𝑃 ) = 𝑄 ) |
10 |
|
dia2dimlem4.d |
⊢ ( 𝜑 → 𝐷 ∈ 𝑇 ) |
11 |
|
dia2dimlem4.dv |
⊢ ( 𝜑 → ( 𝐷 ‘ 𝑄 ) = ( 𝐹 ‘ 𝑃 ) ) |
12 |
3 4
|
ltrnco |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐷 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) → ( 𝐷 ∘ 𝐺 ) ∈ 𝑇 ) |
13 |
5 10 8 12
|
syl3anc |
⊢ ( 𝜑 → ( 𝐷 ∘ 𝐺 ) ∈ 𝑇 ) |
14 |
6
|
simpld |
⊢ ( 𝜑 → 𝑃 ∈ 𝐴 ) |
15 |
1 2 3 4
|
ltrncoval |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐷 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑃 ∈ 𝐴 ) → ( ( 𝐷 ∘ 𝐺 ) ‘ 𝑃 ) = ( 𝐷 ‘ ( 𝐺 ‘ 𝑃 ) ) ) |
16 |
5 10 8 14 15
|
syl121anc |
⊢ ( 𝜑 → ( ( 𝐷 ∘ 𝐺 ) ‘ 𝑃 ) = ( 𝐷 ‘ ( 𝐺 ‘ 𝑃 ) ) ) |
17 |
9
|
fveq2d |
⊢ ( 𝜑 → ( 𝐷 ‘ ( 𝐺 ‘ 𝑃 ) ) = ( 𝐷 ‘ 𝑄 ) ) |
18 |
16 17 11
|
3eqtrd |
⊢ ( 𝜑 → ( ( 𝐷 ∘ 𝐺 ) ‘ 𝑃 ) = ( 𝐹 ‘ 𝑃 ) ) |
19 |
1 2 3 4
|
cdlemd |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐷 ∘ 𝐺 ) ∈ 𝑇 ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( ( 𝐷 ∘ 𝐺 ) ‘ 𝑃 ) = ( 𝐹 ‘ 𝑃 ) ) → ( 𝐷 ∘ 𝐺 ) = 𝐹 ) |
20 |
5 13 7 6 18 19
|
syl311anc |
⊢ ( 𝜑 → ( 𝐷 ∘ 𝐺 ) = 𝐹 ) |