Step |
Hyp |
Ref |
Expression |
1 |
|
diam.m |
⊢ ∧ = ( meet ‘ 𝐾 ) |
2 |
|
diam.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
3 |
|
diam.i |
⊢ 𝐼 = ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 ) |
4 |
|
eqid |
⊢ ( glb ‘ 𝐾 ) = ( glb ‘ 𝐾 ) |
5 |
|
simpll |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼 ) ) → 𝐾 ∈ HL ) |
6 |
|
eqid |
⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 ) |
7 |
6 2 3
|
diadmclN |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ dom 𝐼 ) → 𝑋 ∈ ( Base ‘ 𝐾 ) ) |
8 |
7
|
adantrr |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼 ) ) → 𝑋 ∈ ( Base ‘ 𝐾 ) ) |
9 |
6 2 3
|
diadmclN |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑌 ∈ dom 𝐼 ) → 𝑌 ∈ ( Base ‘ 𝐾 ) ) |
10 |
9
|
adantrl |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼 ) ) → 𝑌 ∈ ( Base ‘ 𝐾 ) ) |
11 |
4 1 5 8 10
|
meetval |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼 ) ) → ( 𝑋 ∧ 𝑌 ) = ( ( glb ‘ 𝐾 ) ‘ { 𝑋 , 𝑌 } ) ) |
12 |
11
|
fveq2d |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼 ) ) → ( 𝐼 ‘ ( 𝑋 ∧ 𝑌 ) ) = ( 𝐼 ‘ ( ( glb ‘ 𝐾 ) ‘ { 𝑋 , 𝑌 } ) ) ) |
13 |
|
simpl |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼 ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
14 |
|
prssi |
⊢ ( ( 𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼 ) → { 𝑋 , 𝑌 } ⊆ dom 𝐼 ) |
15 |
14
|
adantl |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼 ) ) → { 𝑋 , 𝑌 } ⊆ dom 𝐼 ) |
16 |
|
prnzg |
⊢ ( 𝑋 ∈ dom 𝐼 → { 𝑋 , 𝑌 } ≠ ∅ ) |
17 |
16
|
ad2antrl |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼 ) ) → { 𝑋 , 𝑌 } ≠ ∅ ) |
18 |
4 2 3
|
diaglbN |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( { 𝑋 , 𝑌 } ⊆ dom 𝐼 ∧ { 𝑋 , 𝑌 } ≠ ∅ ) ) → ( 𝐼 ‘ ( ( glb ‘ 𝐾 ) ‘ { 𝑋 , 𝑌 } ) ) = ∩ 𝑥 ∈ { 𝑋 , 𝑌 } ( 𝐼 ‘ 𝑥 ) ) |
19 |
13 15 17 18
|
syl12anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼 ) ) → ( 𝐼 ‘ ( ( glb ‘ 𝐾 ) ‘ { 𝑋 , 𝑌 } ) ) = ∩ 𝑥 ∈ { 𝑋 , 𝑌 } ( 𝐼 ‘ 𝑥 ) ) |
20 |
|
fveq2 |
⊢ ( 𝑥 = 𝑋 → ( 𝐼 ‘ 𝑥 ) = ( 𝐼 ‘ 𝑋 ) ) |
21 |
|
fveq2 |
⊢ ( 𝑥 = 𝑌 → ( 𝐼 ‘ 𝑥 ) = ( 𝐼 ‘ 𝑌 ) ) |
22 |
20 21
|
iinxprg |
⊢ ( ( 𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼 ) → ∩ 𝑥 ∈ { 𝑋 , 𝑌 } ( 𝐼 ‘ 𝑥 ) = ( ( 𝐼 ‘ 𝑋 ) ∩ ( 𝐼 ‘ 𝑌 ) ) ) |
23 |
22
|
adantl |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼 ) ) → ∩ 𝑥 ∈ { 𝑋 , 𝑌 } ( 𝐼 ‘ 𝑥 ) = ( ( 𝐼 ‘ 𝑋 ) ∩ ( 𝐼 ‘ 𝑌 ) ) ) |
24 |
12 19 23
|
3eqtrd |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼 ) ) → ( 𝐼 ‘ ( 𝑋 ∧ 𝑌 ) ) = ( ( 𝐼 ‘ 𝑋 ) ∩ ( 𝐼 ‘ 𝑌 ) ) ) |