Step |
Hyp |
Ref |
Expression |
1 |
|
cdlemk6.b |
⊢ 𝐵 = ( Base ‘ 𝐾 ) |
2 |
|
cdlemk6.j |
⊢ ∨ = ( join ‘ 𝐾 ) |
3 |
|
cdlemk6.m |
⊢ ∧ = ( meet ‘ 𝐾 ) |
4 |
|
cdlemk6.o |
⊢ ⊥ = ( oc ‘ 𝐾 ) |
5 |
|
cdlemk6.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
6 |
|
cdlemk6.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
7 |
|
cdlemk6.t |
⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) |
8 |
|
cdlemk6.r |
⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) |
9 |
|
cdlemk6.p |
⊢ 𝑃 = ( ⊥ ‘ 𝑊 ) |
10 |
|
cdlemk6.z |
⊢ 𝑍 = ( ( 𝑃 ∨ ( 𝑅 ‘ 𝑏 ) ) ∧ ( ( 𝑁 ‘ 𝑃 ) ∨ ( 𝑅 ‘ ( 𝑏 ∘ ◡ 𝐹 ) ) ) ) |
11 |
|
cdlemk6.y |
⊢ 𝑌 = ( ( 𝑃 ∨ ( 𝑅 ‘ 𝑔 ) ) ∧ ( 𝑍 ∨ ( 𝑅 ‘ ( 𝑔 ∘ ◡ 𝑏 ) ) ) ) |
12 |
|
cdlemk6.x |
⊢ 𝑋 = ( ℩ 𝑧 ∈ 𝑇 ∀ 𝑏 ∈ 𝑇 ( ( 𝑏 ≠ ( I ↾ 𝐵 ) ∧ ( 𝑅 ‘ 𝑏 ) ≠ ( 𝑅 ‘ 𝐹 ) ∧ ( 𝑅 ‘ 𝑏 ) ≠ ( 𝑅 ‘ 𝑔 ) ) → ( 𝑧 ‘ 𝑃 ) = 𝑌 ) ) |
13 |
|
cdlemk6.u |
⊢ 𝑈 = ( 𝑔 ∈ 𝑇 ↦ if ( 𝐹 = 𝑁 , 𝑔 , 𝑋 ) ) |
14 |
|
cdlemk6.e |
⊢ 𝐸 = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) |
15 |
|
simp1 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇 ) ∧ ( 𝑅 ‘ 𝐹 ) = ( 𝑅 ‘ 𝑁 ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
16 |
|
simp2l |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇 ) ∧ ( 𝑅 ‘ 𝐹 ) = ( 𝑅 ‘ 𝑁 ) ) → 𝐹 ∈ 𝑇 ) |
17 |
|
simp2r |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇 ) ∧ ( 𝑅 ‘ 𝐹 ) = ( 𝑅 ‘ 𝑁 ) ) → 𝑁 ∈ 𝑇 ) |
18 |
|
simp3 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇 ) ∧ ( 𝑅 ‘ 𝐹 ) = ( 𝑅 ‘ 𝑁 ) ) → ( 𝑅 ‘ 𝐹 ) = ( 𝑅 ‘ 𝑁 ) ) |
19 |
|
eqid |
⊢ ( le ‘ 𝐾 ) = ( le ‘ 𝐾 ) |
20 |
4
|
fveq1i |
⊢ ( ⊥ ‘ 𝑊 ) = ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) |
21 |
9 20
|
eqtri |
⊢ 𝑃 = ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) |
22 |
19 5 6 21
|
lhpocnel2 |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ( le ‘ 𝐾 ) 𝑊 ) ) |
23 |
22
|
3ad2ant1 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇 ) ∧ ( 𝑅 ‘ 𝐹 ) = ( 𝑅 ‘ 𝑁 ) ) → ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ( le ‘ 𝐾 ) 𝑊 ) ) |
24 |
1 19 2 3 5 6 7 8 10 11 12 13 14
|
cdlemk56 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇 ) ∧ ( 𝑅 ‘ 𝐹 ) = ( 𝑅 ‘ 𝑁 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ( le ‘ 𝐾 ) 𝑊 ) ) → 𝑈 ∈ 𝐸 ) |
25 |
15 16 17 18 23 24
|
syl311anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇 ) ∧ ( 𝑅 ‘ 𝐹 ) = ( 𝑅 ‘ 𝑁 ) ) → 𝑈 ∈ 𝐸 ) |
26 |
1 2 3 4 5 6 7 8 9 10 11 12 13
|
cdlemk19w |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇 ) ∧ ( 𝑅 ‘ 𝐹 ) = ( 𝑅 ‘ 𝑁 ) ) → ( 𝑈 ‘ 𝐹 ) = 𝑁 ) |
27 |
25 26
|
jca |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇 ) ∧ ( 𝑅 ‘ 𝐹 ) = ( 𝑅 ‘ 𝑁 ) ) → ( 𝑈 ∈ 𝐸 ∧ ( 𝑈 ‘ 𝐹 ) = 𝑁 ) ) |