Step |
Hyp |
Ref |
Expression |
1 |
|
dihglblem5a.b |
⊢ 𝐵 = ( Base ‘ 𝐾 ) |
2 |
|
dihglblem5a.m |
⊢ ∧ = ( meet ‘ 𝐾 ) |
3 |
|
dihglblem5a.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
4 |
|
dihglblem5a.i |
⊢ 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) |
5 |
|
simpr |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ) ∧ 𝑋 ( le ‘ 𝐾 ) 𝑊 ) → 𝑋 ( le ‘ 𝐾 ) 𝑊 ) |
6 |
|
hllat |
⊢ ( 𝐾 ∈ HL → 𝐾 ∈ Lat ) |
7 |
6
|
ad3antrrr |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ) ∧ 𝑋 ( le ‘ 𝐾 ) 𝑊 ) → 𝐾 ∈ Lat ) |
8 |
|
simplr |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ) ∧ 𝑋 ( le ‘ 𝐾 ) 𝑊 ) → 𝑋 ∈ 𝐵 ) |
9 |
1 3
|
lhpbase |
⊢ ( 𝑊 ∈ 𝐻 → 𝑊 ∈ 𝐵 ) |
10 |
9
|
ad3antlr |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ) ∧ 𝑋 ( le ‘ 𝐾 ) 𝑊 ) → 𝑊 ∈ 𝐵 ) |
11 |
|
eqid |
⊢ ( le ‘ 𝐾 ) = ( le ‘ 𝐾 ) |
12 |
1 11 2
|
latleeqm1 |
⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ) → ( 𝑋 ( le ‘ 𝐾 ) 𝑊 ↔ ( 𝑋 ∧ 𝑊 ) = 𝑋 ) ) |
13 |
7 8 10 12
|
syl3anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ) ∧ 𝑋 ( le ‘ 𝐾 ) 𝑊 ) → ( 𝑋 ( le ‘ 𝐾 ) 𝑊 ↔ ( 𝑋 ∧ 𝑊 ) = 𝑋 ) ) |
14 |
5 13
|
mpbid |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ) ∧ 𝑋 ( le ‘ 𝐾 ) 𝑊 ) → ( 𝑋 ∧ 𝑊 ) = 𝑋 ) |
15 |
14
|
fveq2d |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ) ∧ 𝑋 ( le ‘ 𝐾 ) 𝑊 ) → ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) = ( 𝐼 ‘ 𝑋 ) ) |
16 |
|
simpll |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ) ∧ 𝑋 ( le ‘ 𝐾 ) 𝑊 ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
17 |
1 11 3 4
|
dihord |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ) → ( ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑊 ) ↔ 𝑋 ( le ‘ 𝐾 ) 𝑊 ) ) |
18 |
16 8 10 17
|
syl3anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ) ∧ 𝑋 ( le ‘ 𝐾 ) 𝑊 ) → ( ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑊 ) ↔ 𝑋 ( le ‘ 𝐾 ) 𝑊 ) ) |
19 |
5 18
|
mpbird |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ) ∧ 𝑋 ( le ‘ 𝐾 ) 𝑊 ) → ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑊 ) ) |
20 |
|
df-ss |
⊢ ( ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑊 ) ↔ ( ( 𝐼 ‘ 𝑋 ) ∩ ( 𝐼 ‘ 𝑊 ) ) = ( 𝐼 ‘ 𝑋 ) ) |
21 |
19 20
|
sylib |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ) ∧ 𝑋 ( le ‘ 𝐾 ) 𝑊 ) → ( ( 𝐼 ‘ 𝑋 ) ∩ ( 𝐼 ‘ 𝑊 ) ) = ( 𝐼 ‘ 𝑋 ) ) |
22 |
15 21
|
eqtr4d |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ) ∧ 𝑋 ( le ‘ 𝐾 ) 𝑊 ) → ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) = ( ( 𝐼 ‘ 𝑋 ) ∩ ( 𝐼 ‘ 𝑊 ) ) ) |
23 |
|
eqid |
⊢ ( join ‘ 𝐾 ) = ( join ‘ 𝐾 ) |
24 |
|
eqid |
⊢ ( Atoms ‘ 𝐾 ) = ( Atoms ‘ 𝐾 ) |
25 |
|
eqid |
⊢ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) = ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) |
26 |
|
eqid |
⊢ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) |
27 |
|
eqid |
⊢ ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) = ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) |
28 |
|
eqid |
⊢ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) |
29 |
|
eqid |
⊢ ( ℩ ℎ ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ( ℎ ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) = 𝑞 ) = ( ℩ ℎ ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ( ℎ ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) = 𝑞 ) |
30 |
|
eqid |
⊢ ( ℎ ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( I ↾ 𝐵 ) ) = ( ℎ ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( I ↾ 𝐵 ) ) |
31 |
1 2 3 4 11 23 24 25 26 27 28 29 30
|
dihglblem5apreN |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ( le ‘ 𝐾 ) 𝑊 ) ) → ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) = ( ( 𝐼 ‘ 𝑋 ) ∩ ( 𝐼 ‘ 𝑊 ) ) ) |
32 |
31
|
anassrs |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ) ∧ ¬ 𝑋 ( le ‘ 𝐾 ) 𝑊 ) → ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) = ( ( 𝐼 ‘ 𝑋 ) ∩ ( 𝐼 ‘ 𝑊 ) ) ) |
33 |
22 32
|
pm2.61dan |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ) → ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) = ( ( 𝐼 ‘ 𝑋 ) ∩ ( 𝐼 ‘ 𝑊 ) ) ) |