| Step |
Hyp |
Ref |
Expression |
| 1 |
|
diaintcl.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
| 2 |
|
diaintcl.i |
⊢ 𝐼 = ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 ) |
| 3 |
1 2
|
diaf11N |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → 𝐼 : dom 𝐼 –1-1-onto→ ran 𝐼 ) |
| 4 |
3
|
adantr |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅ ) ) → 𝐼 : dom 𝐼 –1-1-onto→ ran 𝐼 ) |
| 5 |
|
f1ofn |
⊢ ( 𝐼 : dom 𝐼 –1-1-onto→ ran 𝐼 → 𝐼 Fn dom 𝐼 ) |
| 6 |
4 5
|
syl |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅ ) ) → 𝐼 Fn dom 𝐼 ) |
| 7 |
|
cnvimass |
⊢ ( ◡ 𝐼 “ 𝑆 ) ⊆ dom 𝐼 |
| 8 |
|
fnssres |
⊢ ( ( 𝐼 Fn dom 𝐼 ∧ ( ◡ 𝐼 “ 𝑆 ) ⊆ dom 𝐼 ) → ( 𝐼 ↾ ( ◡ 𝐼 “ 𝑆 ) ) Fn ( ◡ 𝐼 “ 𝑆 ) ) |
| 9 |
6 7 8
|
sylancl |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅ ) ) → ( 𝐼 ↾ ( ◡ 𝐼 “ 𝑆 ) ) Fn ( ◡ 𝐼 “ 𝑆 ) ) |
| 10 |
|
fniinfv |
⊢ ( ( 𝐼 ↾ ( ◡ 𝐼 “ 𝑆 ) ) Fn ( ◡ 𝐼 “ 𝑆 ) → ∩ 𝑦 ∈ ( ◡ 𝐼 “ 𝑆 ) ( ( 𝐼 ↾ ( ◡ 𝐼 “ 𝑆 ) ) ‘ 𝑦 ) = ∩ ran ( 𝐼 ↾ ( ◡ 𝐼 “ 𝑆 ) ) ) |
| 11 |
9 10
|
syl |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅ ) ) → ∩ 𝑦 ∈ ( ◡ 𝐼 “ 𝑆 ) ( ( 𝐼 ↾ ( ◡ 𝐼 “ 𝑆 ) ) ‘ 𝑦 ) = ∩ ran ( 𝐼 ↾ ( ◡ 𝐼 “ 𝑆 ) ) ) |
| 12 |
|
df-ima |
⊢ ( 𝐼 “ ( ◡ 𝐼 “ 𝑆 ) ) = ran ( 𝐼 ↾ ( ◡ 𝐼 “ 𝑆 ) ) |
| 13 |
|
f1ofo |
⊢ ( 𝐼 : dom 𝐼 –1-1-onto→ ran 𝐼 → 𝐼 : dom 𝐼 –onto→ ran 𝐼 ) |
| 14 |
3 13
|
syl |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → 𝐼 : dom 𝐼 –onto→ ran 𝐼 ) |
| 15 |
14
|
adantr |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅ ) ) → 𝐼 : dom 𝐼 –onto→ ran 𝐼 ) |
| 16 |
|
simprl |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅ ) ) → 𝑆 ⊆ ran 𝐼 ) |
| 17 |
|
foimacnv |
⊢ ( ( 𝐼 : dom 𝐼 –onto→ ran 𝐼 ∧ 𝑆 ⊆ ran 𝐼 ) → ( 𝐼 “ ( ◡ 𝐼 “ 𝑆 ) ) = 𝑆 ) |
| 18 |
15 16 17
|
syl2anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅ ) ) → ( 𝐼 “ ( ◡ 𝐼 “ 𝑆 ) ) = 𝑆 ) |
| 19 |
12 18
|
eqtr3id |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅ ) ) → ran ( 𝐼 ↾ ( ◡ 𝐼 “ 𝑆 ) ) = 𝑆 ) |
| 20 |
19
|
inteqd |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅ ) ) → ∩ ran ( 𝐼 ↾ ( ◡ 𝐼 “ 𝑆 ) ) = ∩ 𝑆 ) |
| 21 |
11 20
|
eqtrd |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅ ) ) → ∩ 𝑦 ∈ ( ◡ 𝐼 “ 𝑆 ) ( ( 𝐼 ↾ ( ◡ 𝐼 “ 𝑆 ) ) ‘ 𝑦 ) = ∩ 𝑆 ) |
| 22 |
|
simpl |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅ ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
| 23 |
7
|
a1i |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅ ) ) → ( ◡ 𝐼 “ 𝑆 ) ⊆ dom 𝐼 ) |
| 24 |
|
simprr |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅ ) ) → 𝑆 ≠ ∅ ) |
| 25 |
|
n0 |
⊢ ( 𝑆 ≠ ∅ ↔ ∃ 𝑦 𝑦 ∈ 𝑆 ) |
| 26 |
24 25
|
sylib |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅ ) ) → ∃ 𝑦 𝑦 ∈ 𝑆 ) |
| 27 |
16
|
sselda |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅ ) ) ∧ 𝑦 ∈ 𝑆 ) → 𝑦 ∈ ran 𝐼 ) |
| 28 |
3
|
ad2antrr |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅ ) ) ∧ 𝑦 ∈ 𝑆 ) → 𝐼 : dom 𝐼 –1-1-onto→ ran 𝐼 ) |
| 29 |
28 5
|
syl |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅ ) ) ∧ 𝑦 ∈ 𝑆 ) → 𝐼 Fn dom 𝐼 ) |
| 30 |
|
fvelrnb |
⊢ ( 𝐼 Fn dom 𝐼 → ( 𝑦 ∈ ran 𝐼 ↔ ∃ 𝑥 ∈ dom 𝐼 ( 𝐼 ‘ 𝑥 ) = 𝑦 ) ) |
| 31 |
29 30
|
syl |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅ ) ) ∧ 𝑦 ∈ 𝑆 ) → ( 𝑦 ∈ ran 𝐼 ↔ ∃ 𝑥 ∈ dom 𝐼 ( 𝐼 ‘ 𝑥 ) = 𝑦 ) ) |
| 32 |
27 31
|
mpbid |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅ ) ) ∧ 𝑦 ∈ 𝑆 ) → ∃ 𝑥 ∈ dom 𝐼 ( 𝐼 ‘ 𝑥 ) = 𝑦 ) |
| 33 |
|
f1ofun |
⊢ ( 𝐼 : dom 𝐼 –1-1-onto→ ran 𝐼 → Fun 𝐼 ) |
| 34 |
3 33
|
syl |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → Fun 𝐼 ) |
| 35 |
34
|
adantr |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅ ) ) → Fun 𝐼 ) |
| 36 |
|
fvimacnv |
⊢ ( ( Fun 𝐼 ∧ 𝑥 ∈ dom 𝐼 ) → ( ( 𝐼 ‘ 𝑥 ) ∈ 𝑆 ↔ 𝑥 ∈ ( ◡ 𝐼 “ 𝑆 ) ) ) |
| 37 |
35 36
|
sylan |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅ ) ) ∧ 𝑥 ∈ dom 𝐼 ) → ( ( 𝐼 ‘ 𝑥 ) ∈ 𝑆 ↔ 𝑥 ∈ ( ◡ 𝐼 “ 𝑆 ) ) ) |
| 38 |
|
ne0i |
⊢ ( 𝑥 ∈ ( ◡ 𝐼 “ 𝑆 ) → ( ◡ 𝐼 “ 𝑆 ) ≠ ∅ ) |
| 39 |
37 38
|
syl6bi |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅ ) ) ∧ 𝑥 ∈ dom 𝐼 ) → ( ( 𝐼 ‘ 𝑥 ) ∈ 𝑆 → ( ◡ 𝐼 “ 𝑆 ) ≠ ∅ ) ) |
| 40 |
39
|
ex |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅ ) ) → ( 𝑥 ∈ dom 𝐼 → ( ( 𝐼 ‘ 𝑥 ) ∈ 𝑆 → ( ◡ 𝐼 “ 𝑆 ) ≠ ∅ ) ) ) |
| 41 |
|
eleq1 |
⊢ ( ( 𝐼 ‘ 𝑥 ) = 𝑦 → ( ( 𝐼 ‘ 𝑥 ) ∈ 𝑆 ↔ 𝑦 ∈ 𝑆 ) ) |
| 42 |
41
|
biimprd |
⊢ ( ( 𝐼 ‘ 𝑥 ) = 𝑦 → ( 𝑦 ∈ 𝑆 → ( 𝐼 ‘ 𝑥 ) ∈ 𝑆 ) ) |
| 43 |
42
|
imim1d |
⊢ ( ( 𝐼 ‘ 𝑥 ) = 𝑦 → ( ( ( 𝐼 ‘ 𝑥 ) ∈ 𝑆 → ( ◡ 𝐼 “ 𝑆 ) ≠ ∅ ) → ( 𝑦 ∈ 𝑆 → ( ◡ 𝐼 “ 𝑆 ) ≠ ∅ ) ) ) |
| 44 |
40 43
|
syl9 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅ ) ) → ( ( 𝐼 ‘ 𝑥 ) = 𝑦 → ( 𝑥 ∈ dom 𝐼 → ( 𝑦 ∈ 𝑆 → ( ◡ 𝐼 “ 𝑆 ) ≠ ∅ ) ) ) ) |
| 45 |
44
|
com24 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅ ) ) → ( 𝑦 ∈ 𝑆 → ( 𝑥 ∈ dom 𝐼 → ( ( 𝐼 ‘ 𝑥 ) = 𝑦 → ( ◡ 𝐼 “ 𝑆 ) ≠ ∅ ) ) ) ) |
| 46 |
45
|
imp |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅ ) ) ∧ 𝑦 ∈ 𝑆 ) → ( 𝑥 ∈ dom 𝐼 → ( ( 𝐼 ‘ 𝑥 ) = 𝑦 → ( ◡ 𝐼 “ 𝑆 ) ≠ ∅ ) ) ) |
| 47 |
46
|
rexlimdv |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅ ) ) ∧ 𝑦 ∈ 𝑆 ) → ( ∃ 𝑥 ∈ dom 𝐼 ( 𝐼 ‘ 𝑥 ) = 𝑦 → ( ◡ 𝐼 “ 𝑆 ) ≠ ∅ ) ) |
| 48 |
32 47
|
mpd |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅ ) ) ∧ 𝑦 ∈ 𝑆 ) → ( ◡ 𝐼 “ 𝑆 ) ≠ ∅ ) |
| 49 |
26 48
|
exlimddv |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅ ) ) → ( ◡ 𝐼 “ 𝑆 ) ≠ ∅ ) |
| 50 |
|
eqid |
⊢ ( glb ‘ 𝐾 ) = ( glb ‘ 𝐾 ) |
| 51 |
50 1 2
|
diaglbN |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( ◡ 𝐼 “ 𝑆 ) ⊆ dom 𝐼 ∧ ( ◡ 𝐼 “ 𝑆 ) ≠ ∅ ) ) → ( 𝐼 ‘ ( ( glb ‘ 𝐾 ) ‘ ( ◡ 𝐼 “ 𝑆 ) ) ) = ∩ 𝑦 ∈ ( ◡ 𝐼 “ 𝑆 ) ( 𝐼 ‘ 𝑦 ) ) |
| 52 |
22 23 49 51
|
syl12anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅ ) ) → ( 𝐼 ‘ ( ( glb ‘ 𝐾 ) ‘ ( ◡ 𝐼 “ 𝑆 ) ) ) = ∩ 𝑦 ∈ ( ◡ 𝐼 “ 𝑆 ) ( 𝐼 ‘ 𝑦 ) ) |
| 53 |
|
fvres |
⊢ ( 𝑦 ∈ ( ◡ 𝐼 “ 𝑆 ) → ( ( 𝐼 ↾ ( ◡ 𝐼 “ 𝑆 ) ) ‘ 𝑦 ) = ( 𝐼 ‘ 𝑦 ) ) |
| 54 |
53
|
iineq2i |
⊢ ∩ 𝑦 ∈ ( ◡ 𝐼 “ 𝑆 ) ( ( 𝐼 ↾ ( ◡ 𝐼 “ 𝑆 ) ) ‘ 𝑦 ) = ∩ 𝑦 ∈ ( ◡ 𝐼 “ 𝑆 ) ( 𝐼 ‘ 𝑦 ) |
| 55 |
52 54
|
eqtr4di |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅ ) ) → ( 𝐼 ‘ ( ( glb ‘ 𝐾 ) ‘ ( ◡ 𝐼 “ 𝑆 ) ) ) = ∩ 𝑦 ∈ ( ◡ 𝐼 “ 𝑆 ) ( ( 𝐼 ↾ ( ◡ 𝐼 “ 𝑆 ) ) ‘ 𝑦 ) ) |
| 56 |
|
hlclat |
⊢ ( 𝐾 ∈ HL → 𝐾 ∈ CLat ) |
| 57 |
56
|
ad2antrr |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅ ) ) → 𝐾 ∈ CLat ) |
| 58 |
|
eqid |
⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 ) |
| 59 |
|
eqid |
⊢ ( le ‘ 𝐾 ) = ( le ‘ 𝐾 ) |
| 60 |
58 59 1 2
|
diadm |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → dom 𝐼 = { 𝑥 ∈ ( Base ‘ 𝐾 ) ∣ 𝑥 ( le ‘ 𝐾 ) 𝑊 } ) |
| 61 |
|
ssrab2 |
⊢ { 𝑥 ∈ ( Base ‘ 𝐾 ) ∣ 𝑥 ( le ‘ 𝐾 ) 𝑊 } ⊆ ( Base ‘ 𝐾 ) |
| 62 |
60 61
|
eqsstrdi |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → dom 𝐼 ⊆ ( Base ‘ 𝐾 ) ) |
| 63 |
62
|
adantr |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅ ) ) → dom 𝐼 ⊆ ( Base ‘ 𝐾 ) ) |
| 64 |
7 63
|
sstrid |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅ ) ) → ( ◡ 𝐼 “ 𝑆 ) ⊆ ( Base ‘ 𝐾 ) ) |
| 65 |
58 50
|
clatglbcl |
⊢ ( ( 𝐾 ∈ CLat ∧ ( ◡ 𝐼 “ 𝑆 ) ⊆ ( Base ‘ 𝐾 ) ) → ( ( glb ‘ 𝐾 ) ‘ ( ◡ 𝐼 “ 𝑆 ) ) ∈ ( Base ‘ 𝐾 ) ) |
| 66 |
57 64 65
|
syl2anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅ ) ) → ( ( glb ‘ 𝐾 ) ‘ ( ◡ 𝐼 “ 𝑆 ) ) ∈ ( Base ‘ 𝐾 ) ) |
| 67 |
|
n0 |
⊢ ( ( ◡ 𝐼 “ 𝑆 ) ≠ ∅ ↔ ∃ 𝑦 𝑦 ∈ ( ◡ 𝐼 “ 𝑆 ) ) |
| 68 |
49 67
|
sylib |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅ ) ) → ∃ 𝑦 𝑦 ∈ ( ◡ 𝐼 “ 𝑆 ) ) |
| 69 |
|
hllat |
⊢ ( 𝐾 ∈ HL → 𝐾 ∈ Lat ) |
| 70 |
69
|
ad3antrrr |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅ ) ) ∧ 𝑦 ∈ ( ◡ 𝐼 “ 𝑆 ) ) → 𝐾 ∈ Lat ) |
| 71 |
66
|
adantr |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅ ) ) ∧ 𝑦 ∈ ( ◡ 𝐼 “ 𝑆 ) ) → ( ( glb ‘ 𝐾 ) ‘ ( ◡ 𝐼 “ 𝑆 ) ) ∈ ( Base ‘ 𝐾 ) ) |
| 72 |
64
|
sselda |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅ ) ) ∧ 𝑦 ∈ ( ◡ 𝐼 “ 𝑆 ) ) → 𝑦 ∈ ( Base ‘ 𝐾 ) ) |
| 73 |
58 1
|
lhpbase |
⊢ ( 𝑊 ∈ 𝐻 → 𝑊 ∈ ( Base ‘ 𝐾 ) ) |
| 74 |
73
|
ad3antlr |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅ ) ) ∧ 𝑦 ∈ ( ◡ 𝐼 “ 𝑆 ) ) → 𝑊 ∈ ( Base ‘ 𝐾 ) ) |
| 75 |
56
|
ad3antrrr |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅ ) ) ∧ 𝑦 ∈ ( ◡ 𝐼 “ 𝑆 ) ) → 𝐾 ∈ CLat ) |
| 76 |
60
|
adantr |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅ ) ) → dom 𝐼 = { 𝑥 ∈ ( Base ‘ 𝐾 ) ∣ 𝑥 ( le ‘ 𝐾 ) 𝑊 } ) |
| 77 |
7 76
|
sseqtrid |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅ ) ) → ( ◡ 𝐼 “ 𝑆 ) ⊆ { 𝑥 ∈ ( Base ‘ 𝐾 ) ∣ 𝑥 ( le ‘ 𝐾 ) 𝑊 } ) |
| 78 |
77 61
|
sstrdi |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅ ) ) → ( ◡ 𝐼 “ 𝑆 ) ⊆ ( Base ‘ 𝐾 ) ) |
| 79 |
78
|
adantr |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅ ) ) ∧ 𝑦 ∈ ( ◡ 𝐼 “ 𝑆 ) ) → ( ◡ 𝐼 “ 𝑆 ) ⊆ ( Base ‘ 𝐾 ) ) |
| 80 |
|
simpr |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅ ) ) ∧ 𝑦 ∈ ( ◡ 𝐼 “ 𝑆 ) ) → 𝑦 ∈ ( ◡ 𝐼 “ 𝑆 ) ) |
| 81 |
58 59 50
|
clatglble |
⊢ ( ( 𝐾 ∈ CLat ∧ ( ◡ 𝐼 “ 𝑆 ) ⊆ ( Base ‘ 𝐾 ) ∧ 𝑦 ∈ ( ◡ 𝐼 “ 𝑆 ) ) → ( ( glb ‘ 𝐾 ) ‘ ( ◡ 𝐼 “ 𝑆 ) ) ( le ‘ 𝐾 ) 𝑦 ) |
| 82 |
75 79 80 81
|
syl3anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅ ) ) ∧ 𝑦 ∈ ( ◡ 𝐼 “ 𝑆 ) ) → ( ( glb ‘ 𝐾 ) ‘ ( ◡ 𝐼 “ 𝑆 ) ) ( le ‘ 𝐾 ) 𝑦 ) |
| 83 |
7
|
sseli |
⊢ ( 𝑦 ∈ ( ◡ 𝐼 “ 𝑆 ) → 𝑦 ∈ dom 𝐼 ) |
| 84 |
83
|
adantl |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅ ) ) ∧ 𝑦 ∈ ( ◡ 𝐼 “ 𝑆 ) ) → 𝑦 ∈ dom 𝐼 ) |
| 85 |
58 59 1 2
|
diaeldm |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( 𝑦 ∈ dom 𝐼 ↔ ( 𝑦 ∈ ( Base ‘ 𝐾 ) ∧ 𝑦 ( le ‘ 𝐾 ) 𝑊 ) ) ) |
| 86 |
85
|
ad2antrr |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅ ) ) ∧ 𝑦 ∈ ( ◡ 𝐼 “ 𝑆 ) ) → ( 𝑦 ∈ dom 𝐼 ↔ ( 𝑦 ∈ ( Base ‘ 𝐾 ) ∧ 𝑦 ( le ‘ 𝐾 ) 𝑊 ) ) ) |
| 87 |
84 86
|
mpbid |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅ ) ) ∧ 𝑦 ∈ ( ◡ 𝐼 “ 𝑆 ) ) → ( 𝑦 ∈ ( Base ‘ 𝐾 ) ∧ 𝑦 ( le ‘ 𝐾 ) 𝑊 ) ) |
| 88 |
87
|
simprd |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅ ) ) ∧ 𝑦 ∈ ( ◡ 𝐼 “ 𝑆 ) ) → 𝑦 ( le ‘ 𝐾 ) 𝑊 ) |
| 89 |
58 59 70 71 72 74 82 88
|
lattrd |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅ ) ) ∧ 𝑦 ∈ ( ◡ 𝐼 “ 𝑆 ) ) → ( ( glb ‘ 𝐾 ) ‘ ( ◡ 𝐼 “ 𝑆 ) ) ( le ‘ 𝐾 ) 𝑊 ) |
| 90 |
68 89
|
exlimddv |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅ ) ) → ( ( glb ‘ 𝐾 ) ‘ ( ◡ 𝐼 “ 𝑆 ) ) ( le ‘ 𝐾 ) 𝑊 ) |
| 91 |
58 59 1 2
|
diaeldm |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( ( ( glb ‘ 𝐾 ) ‘ ( ◡ 𝐼 “ 𝑆 ) ) ∈ dom 𝐼 ↔ ( ( ( glb ‘ 𝐾 ) ‘ ( ◡ 𝐼 “ 𝑆 ) ) ∈ ( Base ‘ 𝐾 ) ∧ ( ( glb ‘ 𝐾 ) ‘ ( ◡ 𝐼 “ 𝑆 ) ) ( le ‘ 𝐾 ) 𝑊 ) ) ) |
| 92 |
91
|
adantr |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅ ) ) → ( ( ( glb ‘ 𝐾 ) ‘ ( ◡ 𝐼 “ 𝑆 ) ) ∈ dom 𝐼 ↔ ( ( ( glb ‘ 𝐾 ) ‘ ( ◡ 𝐼 “ 𝑆 ) ) ∈ ( Base ‘ 𝐾 ) ∧ ( ( glb ‘ 𝐾 ) ‘ ( ◡ 𝐼 “ 𝑆 ) ) ( le ‘ 𝐾 ) 𝑊 ) ) ) |
| 93 |
66 90 92
|
mpbir2and |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅ ) ) → ( ( glb ‘ 𝐾 ) ‘ ( ◡ 𝐼 “ 𝑆 ) ) ∈ dom 𝐼 ) |
| 94 |
1 2
|
diaclN |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( glb ‘ 𝐾 ) ‘ ( ◡ 𝐼 “ 𝑆 ) ) ∈ dom 𝐼 ) → ( 𝐼 ‘ ( ( glb ‘ 𝐾 ) ‘ ( ◡ 𝐼 “ 𝑆 ) ) ) ∈ ran 𝐼 ) |
| 95 |
93 94
|
syldan |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅ ) ) → ( 𝐼 ‘ ( ( glb ‘ 𝐾 ) ‘ ( ◡ 𝐼 “ 𝑆 ) ) ) ∈ ran 𝐼 ) |
| 96 |
55 95
|
eqeltrrd |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅ ) ) → ∩ 𝑦 ∈ ( ◡ 𝐼 “ 𝑆 ) ( ( 𝐼 ↾ ( ◡ 𝐼 “ 𝑆 ) ) ‘ 𝑦 ) ∈ ran 𝐼 ) |
| 97 |
21 96
|
eqeltrrd |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅ ) ) → ∩ 𝑆 ∈ ran 𝐼 ) |