Step |
Hyp |
Ref |
Expression |
1 |
|
docacl.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
2 |
|
docacl.t |
⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) |
3 |
|
docacl.i |
⊢ 𝐼 = ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 ) |
4 |
|
docacl.n |
⊢ ⊥ = ( ( ocA ‘ 𝐾 ) ‘ 𝑊 ) |
5 |
|
eqid |
⊢ ( join ‘ 𝐾 ) = ( join ‘ 𝐾 ) |
6 |
|
eqid |
⊢ ( meet ‘ 𝐾 ) = ( meet ‘ 𝐾 ) |
7 |
|
eqid |
⊢ ( oc ‘ 𝐾 ) = ( oc ‘ 𝐾 ) |
8 |
5 6 7 1 2 3 4
|
docavalN |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ⊆ 𝑇 ) → ( ⊥ ‘ 𝑋 ) = ( 𝐼 ‘ ( ( ( ( oc ‘ 𝐾 ) ‘ ( ◡ 𝐼 ‘ ∩ { 𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧 } ) ) ( join ‘ 𝐾 ) ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ) ) |
9 |
1 3
|
diaf11N |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → 𝐼 : dom 𝐼 –1-1-onto→ ran 𝐼 ) |
10 |
|
f1ofun |
⊢ ( 𝐼 : dom 𝐼 –1-1-onto→ ran 𝐼 → Fun 𝐼 ) |
11 |
9 10
|
syl |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → Fun 𝐼 ) |
12 |
11
|
adantr |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ⊆ 𝑇 ) → Fun 𝐼 ) |
13 |
|
hllat |
⊢ ( 𝐾 ∈ HL → 𝐾 ∈ Lat ) |
14 |
13
|
ad2antrr |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ⊆ 𝑇 ) → 𝐾 ∈ Lat ) |
15 |
|
hlop |
⊢ ( 𝐾 ∈ HL → 𝐾 ∈ OP ) |
16 |
15
|
ad2antrr |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ⊆ 𝑇 ) → 𝐾 ∈ OP ) |
17 |
|
simpl |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ⊆ 𝑇 ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
18 |
|
ssrab2 |
⊢ { 𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧 } ⊆ ran 𝐼 |
19 |
18
|
a1i |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ⊆ 𝑇 ) → { 𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧 } ⊆ ran 𝐼 ) |
20 |
1 2 3
|
dia1elN |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → 𝑇 ∈ ran 𝐼 ) |
21 |
20
|
anim1i |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ⊆ 𝑇 ) → ( 𝑇 ∈ ran 𝐼 ∧ 𝑋 ⊆ 𝑇 ) ) |
22 |
|
sseq2 |
⊢ ( 𝑧 = 𝑇 → ( 𝑋 ⊆ 𝑧 ↔ 𝑋 ⊆ 𝑇 ) ) |
23 |
22
|
elrab |
⊢ ( 𝑇 ∈ { 𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧 } ↔ ( 𝑇 ∈ ran 𝐼 ∧ 𝑋 ⊆ 𝑇 ) ) |
24 |
21 23
|
sylibr |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ⊆ 𝑇 ) → 𝑇 ∈ { 𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧 } ) |
25 |
24
|
ne0d |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ⊆ 𝑇 ) → { 𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧 } ≠ ∅ ) |
26 |
1 3
|
diaintclN |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( { 𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧 } ⊆ ran 𝐼 ∧ { 𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧 } ≠ ∅ ) ) → ∩ { 𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧 } ∈ ran 𝐼 ) |
27 |
17 19 25 26
|
syl12anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ⊆ 𝑇 ) → ∩ { 𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧 } ∈ ran 𝐼 ) |
28 |
1 3
|
diacnvclN |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ∩ { 𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧 } ∈ ran 𝐼 ) → ( ◡ 𝐼 ‘ ∩ { 𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧 } ) ∈ dom 𝐼 ) |
29 |
27 28
|
syldan |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ⊆ 𝑇 ) → ( ◡ 𝐼 ‘ ∩ { 𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧 } ) ∈ dom 𝐼 ) |
30 |
|
eqid |
⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 ) |
31 |
30 1 3
|
diadmclN |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ◡ 𝐼 ‘ ∩ { 𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧 } ) ∈ dom 𝐼 ) → ( ◡ 𝐼 ‘ ∩ { 𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧 } ) ∈ ( Base ‘ 𝐾 ) ) |
32 |
29 31
|
syldan |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ⊆ 𝑇 ) → ( ◡ 𝐼 ‘ ∩ { 𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧 } ) ∈ ( Base ‘ 𝐾 ) ) |
33 |
30 7
|
opoccl |
⊢ ( ( 𝐾 ∈ OP ∧ ( ◡ 𝐼 ‘ ∩ { 𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧 } ) ∈ ( Base ‘ 𝐾 ) ) → ( ( oc ‘ 𝐾 ) ‘ ( ◡ 𝐼 ‘ ∩ { 𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧 } ) ) ∈ ( Base ‘ 𝐾 ) ) |
34 |
16 32 33
|
syl2anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ⊆ 𝑇 ) → ( ( oc ‘ 𝐾 ) ‘ ( ◡ 𝐼 ‘ ∩ { 𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧 } ) ) ∈ ( Base ‘ 𝐾 ) ) |
35 |
30 1
|
lhpbase |
⊢ ( 𝑊 ∈ 𝐻 → 𝑊 ∈ ( Base ‘ 𝐾 ) ) |
36 |
35
|
ad2antlr |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ⊆ 𝑇 ) → 𝑊 ∈ ( Base ‘ 𝐾 ) ) |
37 |
30 7
|
opoccl |
⊢ ( ( 𝐾 ∈ OP ∧ 𝑊 ∈ ( Base ‘ 𝐾 ) ) → ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ∈ ( Base ‘ 𝐾 ) ) |
38 |
16 36 37
|
syl2anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ⊆ 𝑇 ) → ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ∈ ( Base ‘ 𝐾 ) ) |
39 |
30 5
|
latjcl |
⊢ ( ( 𝐾 ∈ Lat ∧ ( ( oc ‘ 𝐾 ) ‘ ( ◡ 𝐼 ‘ ∩ { 𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧 } ) ) ∈ ( Base ‘ 𝐾 ) ∧ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ∈ ( Base ‘ 𝐾 ) ) → ( ( ( oc ‘ 𝐾 ) ‘ ( ◡ 𝐼 ‘ ∩ { 𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧 } ) ) ( join ‘ 𝐾 ) ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ∈ ( Base ‘ 𝐾 ) ) |
40 |
14 34 38 39
|
syl3anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ⊆ 𝑇 ) → ( ( ( oc ‘ 𝐾 ) ‘ ( ◡ 𝐼 ‘ ∩ { 𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧 } ) ) ( join ‘ 𝐾 ) ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ∈ ( Base ‘ 𝐾 ) ) |
41 |
30 6
|
latmcl |
⊢ ( ( 𝐾 ∈ Lat ∧ ( ( ( oc ‘ 𝐾 ) ‘ ( ◡ 𝐼 ‘ ∩ { 𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧 } ) ) ( join ‘ 𝐾 ) ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ∈ ( Base ‘ 𝐾 ) ∧ 𝑊 ∈ ( Base ‘ 𝐾 ) ) → ( ( ( ( oc ‘ 𝐾 ) ‘ ( ◡ 𝐼 ‘ ∩ { 𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧 } ) ) ( join ‘ 𝐾 ) ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ∈ ( Base ‘ 𝐾 ) ) |
42 |
14 40 36 41
|
syl3anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ⊆ 𝑇 ) → ( ( ( ( oc ‘ 𝐾 ) ‘ ( ◡ 𝐼 ‘ ∩ { 𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧 } ) ) ( join ‘ 𝐾 ) ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ∈ ( Base ‘ 𝐾 ) ) |
43 |
|
eqid |
⊢ ( le ‘ 𝐾 ) = ( le ‘ 𝐾 ) |
44 |
30 43 6
|
latmle2 |
⊢ ( ( 𝐾 ∈ Lat ∧ ( ( ( oc ‘ 𝐾 ) ‘ ( ◡ 𝐼 ‘ ∩ { 𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧 } ) ) ( join ‘ 𝐾 ) ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ∈ ( Base ‘ 𝐾 ) ∧ 𝑊 ∈ ( Base ‘ 𝐾 ) ) → ( ( ( ( oc ‘ 𝐾 ) ‘ ( ◡ 𝐼 ‘ ∩ { 𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧 } ) ) ( join ‘ 𝐾 ) ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ( le ‘ 𝐾 ) 𝑊 ) |
45 |
14 40 36 44
|
syl3anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ⊆ 𝑇 ) → ( ( ( ( oc ‘ 𝐾 ) ‘ ( ◡ 𝐼 ‘ ∩ { 𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧 } ) ) ( join ‘ 𝐾 ) ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ( le ‘ 𝐾 ) 𝑊 ) |
46 |
30 43 1 3
|
diaeldm |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( ( ( ( ( oc ‘ 𝐾 ) ‘ ( ◡ 𝐼 ‘ ∩ { 𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧 } ) ) ( join ‘ 𝐾 ) ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ∈ dom 𝐼 ↔ ( ( ( ( ( oc ‘ 𝐾 ) ‘ ( ◡ 𝐼 ‘ ∩ { 𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧 } ) ) ( join ‘ 𝐾 ) ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ∈ ( Base ‘ 𝐾 ) ∧ ( ( ( ( oc ‘ 𝐾 ) ‘ ( ◡ 𝐼 ‘ ∩ { 𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧 } ) ) ( join ‘ 𝐾 ) ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ( le ‘ 𝐾 ) 𝑊 ) ) ) |
47 |
46
|
adantr |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ⊆ 𝑇 ) → ( ( ( ( ( oc ‘ 𝐾 ) ‘ ( ◡ 𝐼 ‘ ∩ { 𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧 } ) ) ( join ‘ 𝐾 ) ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ∈ dom 𝐼 ↔ ( ( ( ( ( oc ‘ 𝐾 ) ‘ ( ◡ 𝐼 ‘ ∩ { 𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧 } ) ) ( join ‘ 𝐾 ) ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ∈ ( Base ‘ 𝐾 ) ∧ ( ( ( ( oc ‘ 𝐾 ) ‘ ( ◡ 𝐼 ‘ ∩ { 𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧 } ) ) ( join ‘ 𝐾 ) ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ( le ‘ 𝐾 ) 𝑊 ) ) ) |
48 |
42 45 47
|
mpbir2and |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ⊆ 𝑇 ) → ( ( ( ( oc ‘ 𝐾 ) ‘ ( ◡ 𝐼 ‘ ∩ { 𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧 } ) ) ( join ‘ 𝐾 ) ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ∈ dom 𝐼 ) |
49 |
|
fvelrn |
⊢ ( ( Fun 𝐼 ∧ ( ( ( ( oc ‘ 𝐾 ) ‘ ( ◡ 𝐼 ‘ ∩ { 𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧 } ) ) ( join ‘ 𝐾 ) ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ∈ dom 𝐼 ) → ( 𝐼 ‘ ( ( ( ( oc ‘ 𝐾 ) ‘ ( ◡ 𝐼 ‘ ∩ { 𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧 } ) ) ( join ‘ 𝐾 ) ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ) ∈ ran 𝐼 ) |
50 |
12 48 49
|
syl2anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ⊆ 𝑇 ) → ( 𝐼 ‘ ( ( ( ( oc ‘ 𝐾 ) ‘ ( ◡ 𝐼 ‘ ∩ { 𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧 } ) ) ( join ‘ 𝐾 ) ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ) ∈ ran 𝐼 ) |
51 |
8 50
|
eqeltrd |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ⊆ 𝑇 ) → ( ⊥ ‘ 𝑋 ) ∈ ran 𝐼 ) |