Step |
Hyp |
Ref |
Expression |
1 |
|
diaoc.j |
⊢ ∨ = ( join ‘ 𝐾 ) |
2 |
|
diaoc.m |
⊢ ∧ = ( meet ‘ 𝐾 ) |
3 |
|
diaoc.o |
⊢ ⊥ = ( oc ‘ 𝐾 ) |
4 |
|
diaoc.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
5 |
|
diaoc.t |
⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) |
6 |
|
diaoc.i |
⊢ 𝐼 = ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 ) |
7 |
|
diaoc.n |
⊢ 𝑁 = ( ( ocA ‘ 𝐾 ) ‘ 𝑊 ) |
8 |
|
simpl |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ dom 𝐼 ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
9 |
|
eqid |
⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 ) |
10 |
9 4 6
|
diadmclN |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ dom 𝐼 ) → 𝑋 ∈ ( Base ‘ 𝐾 ) ) |
11 |
|
eqid |
⊢ ( le ‘ 𝐾 ) = ( le ‘ 𝐾 ) |
12 |
11 4 6
|
diadmleN |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ dom 𝐼 ) → 𝑋 ( le ‘ 𝐾 ) 𝑊 ) |
13 |
9 11 4 5 6
|
diass |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ ( Base ‘ 𝐾 ) ∧ 𝑋 ( le ‘ 𝐾 ) 𝑊 ) ) → ( 𝐼 ‘ 𝑋 ) ⊆ 𝑇 ) |
14 |
8 10 12 13
|
syl12anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ dom 𝐼 ) → ( 𝐼 ‘ 𝑋 ) ⊆ 𝑇 ) |
15 |
1 2 3 4 5 6 7
|
docavalN |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐼 ‘ 𝑋 ) ⊆ 𝑇 ) → ( 𝑁 ‘ ( 𝐼 ‘ 𝑋 ) ) = ( 𝐼 ‘ ( ( ( ⊥ ‘ ( ◡ 𝐼 ‘ ∩ { 𝑧 ∈ ran 𝐼 ∣ ( 𝐼 ‘ 𝑋 ) ⊆ 𝑧 } ) ) ∨ ( ⊥ ‘ 𝑊 ) ) ∧ 𝑊 ) ) ) |
16 |
14 15
|
syldan |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ dom 𝐼 ) → ( 𝑁 ‘ ( 𝐼 ‘ 𝑋 ) ) = ( 𝐼 ‘ ( ( ( ⊥ ‘ ( ◡ 𝐼 ‘ ∩ { 𝑧 ∈ ran 𝐼 ∣ ( 𝐼 ‘ 𝑋 ) ⊆ 𝑧 } ) ) ∨ ( ⊥ ‘ 𝑊 ) ) ∧ 𝑊 ) ) ) |
17 |
4 6
|
diaclN |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ dom 𝐼 ) → ( 𝐼 ‘ 𝑋 ) ∈ ran 𝐼 ) |
18 |
|
intmin |
⊢ ( ( 𝐼 ‘ 𝑋 ) ∈ ran 𝐼 → ∩ { 𝑧 ∈ ran 𝐼 ∣ ( 𝐼 ‘ 𝑋 ) ⊆ 𝑧 } = ( 𝐼 ‘ 𝑋 ) ) |
19 |
17 18
|
syl |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ dom 𝐼 ) → ∩ { 𝑧 ∈ ran 𝐼 ∣ ( 𝐼 ‘ 𝑋 ) ⊆ 𝑧 } = ( 𝐼 ‘ 𝑋 ) ) |
20 |
19
|
fveq2d |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ dom 𝐼 ) → ( ◡ 𝐼 ‘ ∩ { 𝑧 ∈ ran 𝐼 ∣ ( 𝐼 ‘ 𝑋 ) ⊆ 𝑧 } ) = ( ◡ 𝐼 ‘ ( 𝐼 ‘ 𝑋 ) ) ) |
21 |
4 6
|
diaf11N |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → 𝐼 : dom 𝐼 –1-1-onto→ ran 𝐼 ) |
22 |
|
f1ocnvfv1 |
⊢ ( ( 𝐼 : dom 𝐼 –1-1-onto→ ran 𝐼 ∧ 𝑋 ∈ dom 𝐼 ) → ( ◡ 𝐼 ‘ ( 𝐼 ‘ 𝑋 ) ) = 𝑋 ) |
23 |
21 22
|
sylan |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ dom 𝐼 ) → ( ◡ 𝐼 ‘ ( 𝐼 ‘ 𝑋 ) ) = 𝑋 ) |
24 |
20 23
|
eqtrd |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ dom 𝐼 ) → ( ◡ 𝐼 ‘ ∩ { 𝑧 ∈ ran 𝐼 ∣ ( 𝐼 ‘ 𝑋 ) ⊆ 𝑧 } ) = 𝑋 ) |
25 |
24
|
fveq2d |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ dom 𝐼 ) → ( ⊥ ‘ ( ◡ 𝐼 ‘ ∩ { 𝑧 ∈ ran 𝐼 ∣ ( 𝐼 ‘ 𝑋 ) ⊆ 𝑧 } ) ) = ( ⊥ ‘ 𝑋 ) ) |
26 |
25
|
oveq1d |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ dom 𝐼 ) → ( ( ⊥ ‘ ( ◡ 𝐼 ‘ ∩ { 𝑧 ∈ ran 𝐼 ∣ ( 𝐼 ‘ 𝑋 ) ⊆ 𝑧 } ) ) ∨ ( ⊥ ‘ 𝑊 ) ) = ( ( ⊥ ‘ 𝑋 ) ∨ ( ⊥ ‘ 𝑊 ) ) ) |
27 |
26
|
fvoveq1d |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ dom 𝐼 ) → ( 𝐼 ‘ ( ( ( ⊥ ‘ ( ◡ 𝐼 ‘ ∩ { 𝑧 ∈ ran 𝐼 ∣ ( 𝐼 ‘ 𝑋 ) ⊆ 𝑧 } ) ) ∨ ( ⊥ ‘ 𝑊 ) ) ∧ 𝑊 ) ) = ( 𝐼 ‘ ( ( ( ⊥ ‘ 𝑋 ) ∨ ( ⊥ ‘ 𝑊 ) ) ∧ 𝑊 ) ) ) |
28 |
16 27
|
eqtr2d |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ dom 𝐼 ) → ( 𝐼 ‘ ( ( ( ⊥ ‘ 𝑋 ) ∨ ( ⊥ ‘ 𝑊 ) ) ∧ 𝑊 ) ) = ( 𝑁 ‘ ( 𝐼 ‘ 𝑋 ) ) ) |