Step |
Hyp |
Ref |
Expression |
1 |
|
ltrnel.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
2 |
|
ltrnel.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
3 |
|
ltrnel.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
4 |
|
ltrnel.t |
⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) |
5 |
1 2 3 4
|
ltrncnvat |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝑃 ∈ 𝐴 ) → ( ◡ 𝐹 ‘ 𝑃 ) ∈ 𝐴 ) |
6 |
5
|
3adant3r |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( ◡ 𝐹 ‘ 𝑃 ) ∈ 𝐴 ) |
7 |
|
simp3r |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ¬ 𝑃 ≤ 𝑊 ) |
8 |
|
simp1 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
9 |
|
simp2 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → 𝐹 ∈ 𝑇 ) |
10 |
|
eqid |
⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 ) |
11 |
10 2
|
atbase |
⊢ ( ( ◡ 𝐹 ‘ 𝑃 ) ∈ 𝐴 → ( ◡ 𝐹 ‘ 𝑃 ) ∈ ( Base ‘ 𝐾 ) ) |
12 |
6 11
|
syl |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( ◡ 𝐹 ‘ 𝑃 ) ∈ ( Base ‘ 𝐾 ) ) |
13 |
|
simp1r |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → 𝑊 ∈ 𝐻 ) |
14 |
10 3
|
lhpbase |
⊢ ( 𝑊 ∈ 𝐻 → 𝑊 ∈ ( Base ‘ 𝐾 ) ) |
15 |
13 14
|
syl |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → 𝑊 ∈ ( Base ‘ 𝐾 ) ) |
16 |
10 1 3 4
|
ltrnle |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( ( ◡ 𝐹 ‘ 𝑃 ) ∈ ( Base ‘ 𝐾 ) ∧ 𝑊 ∈ ( Base ‘ 𝐾 ) ) ) → ( ( ◡ 𝐹 ‘ 𝑃 ) ≤ 𝑊 ↔ ( 𝐹 ‘ ( ◡ 𝐹 ‘ 𝑃 ) ) ≤ ( 𝐹 ‘ 𝑊 ) ) ) |
17 |
8 9 12 15 16
|
syl112anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( ( ◡ 𝐹 ‘ 𝑃 ) ≤ 𝑊 ↔ ( 𝐹 ‘ ( ◡ 𝐹 ‘ 𝑃 ) ) ≤ ( 𝐹 ‘ 𝑊 ) ) ) |
18 |
10 3 4
|
ltrn1o |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → 𝐹 : ( Base ‘ 𝐾 ) –1-1-onto→ ( Base ‘ 𝐾 ) ) |
19 |
18
|
3adant3 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → 𝐹 : ( Base ‘ 𝐾 ) –1-1-onto→ ( Base ‘ 𝐾 ) ) |
20 |
|
simp3l |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → 𝑃 ∈ 𝐴 ) |
21 |
10 2
|
atbase |
⊢ ( 𝑃 ∈ 𝐴 → 𝑃 ∈ ( Base ‘ 𝐾 ) ) |
22 |
20 21
|
syl |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → 𝑃 ∈ ( Base ‘ 𝐾 ) ) |
23 |
|
f1ocnvfv2 |
⊢ ( ( 𝐹 : ( Base ‘ 𝐾 ) –1-1-onto→ ( Base ‘ 𝐾 ) ∧ 𝑃 ∈ ( Base ‘ 𝐾 ) ) → ( 𝐹 ‘ ( ◡ 𝐹 ‘ 𝑃 ) ) = 𝑃 ) |
24 |
19 22 23
|
syl2anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( 𝐹 ‘ ( ◡ 𝐹 ‘ 𝑃 ) ) = 𝑃 ) |
25 |
|
simp1l |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → 𝐾 ∈ HL ) |
26 |
25
|
hllatd |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → 𝐾 ∈ Lat ) |
27 |
10 1
|
latref |
⊢ ( ( 𝐾 ∈ Lat ∧ 𝑊 ∈ ( Base ‘ 𝐾 ) ) → 𝑊 ≤ 𝑊 ) |
28 |
26 15 27
|
syl2anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → 𝑊 ≤ 𝑊 ) |
29 |
10 1 3 4
|
ltrnval1 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑊 ∈ ( Base ‘ 𝐾 ) ∧ 𝑊 ≤ 𝑊 ) ) → ( 𝐹 ‘ 𝑊 ) = 𝑊 ) |
30 |
8 9 15 28 29
|
syl112anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( 𝐹 ‘ 𝑊 ) = 𝑊 ) |
31 |
24 30
|
breq12d |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( ( 𝐹 ‘ ( ◡ 𝐹 ‘ 𝑃 ) ) ≤ ( 𝐹 ‘ 𝑊 ) ↔ 𝑃 ≤ 𝑊 ) ) |
32 |
17 31
|
bitrd |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( ( ◡ 𝐹 ‘ 𝑃 ) ≤ 𝑊 ↔ 𝑃 ≤ 𝑊 ) ) |
33 |
7 32
|
mtbird |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ¬ ( ◡ 𝐹 ‘ 𝑃 ) ≤ 𝑊 ) |
34 |
6 33
|
jca |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( ( ◡ 𝐹 ‘ 𝑃 ) ∈ 𝐴 ∧ ¬ ( ◡ 𝐹 ‘ 𝑃 ) ≤ 𝑊 ) ) |