Step |
Hyp |
Ref |
Expression |
1 |
|
dihglblem.b |
⊢ 𝐵 = ( Base ‘ 𝐾 ) |
2 |
|
dihglblem.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
3 |
|
dihglblem.m |
⊢ ∧ = ( meet ‘ 𝐾 ) |
4 |
|
dihglblem.g |
⊢ 𝐺 = ( glb ‘ 𝐾 ) |
5 |
|
dihglblem.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
6 |
|
dihglblem.t |
⊢ 𝑇 = { 𝑢 ∈ 𝐵 ∣ ∃ 𝑣 ∈ 𝑆 𝑢 = ( 𝑣 ∧ 𝑊 ) } |
7 |
|
simpl1l |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ⊆ 𝐵 ∧ ( 𝐺 ‘ 𝑆 ) ≤ 𝑊 ) ∧ 𝑥 ∈ 𝑆 ) → 𝐾 ∈ HL ) |
8 |
7
|
hllatd |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ⊆ 𝐵 ∧ ( 𝐺 ‘ 𝑆 ) ≤ 𝑊 ) ∧ 𝑥 ∈ 𝑆 ) → 𝐾 ∈ Lat ) |
9 |
|
simp1l |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ⊆ 𝐵 ∧ ( 𝐺 ‘ 𝑆 ) ≤ 𝑊 ) → 𝐾 ∈ HL ) |
10 |
|
hlclat |
⊢ ( 𝐾 ∈ HL → 𝐾 ∈ CLat ) |
11 |
9 10
|
syl |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ⊆ 𝐵 ∧ ( 𝐺 ‘ 𝑆 ) ≤ 𝑊 ) → 𝐾 ∈ CLat ) |
12 |
|
ssrab2 |
⊢ { 𝑢 ∈ 𝐵 ∣ ∃ 𝑣 ∈ 𝑆 𝑢 = ( 𝑣 ∧ 𝑊 ) } ⊆ 𝐵 |
13 |
6 12
|
eqsstri |
⊢ 𝑇 ⊆ 𝐵 |
14 |
1 4
|
clatglbcl |
⊢ ( ( 𝐾 ∈ CLat ∧ 𝑇 ⊆ 𝐵 ) → ( 𝐺 ‘ 𝑇 ) ∈ 𝐵 ) |
15 |
11 13 14
|
sylancl |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ⊆ 𝐵 ∧ ( 𝐺 ‘ 𝑆 ) ≤ 𝑊 ) → ( 𝐺 ‘ 𝑇 ) ∈ 𝐵 ) |
16 |
15
|
adantr |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ⊆ 𝐵 ∧ ( 𝐺 ‘ 𝑆 ) ≤ 𝑊 ) ∧ 𝑥 ∈ 𝑆 ) → ( 𝐺 ‘ 𝑇 ) ∈ 𝐵 ) |
17 |
|
simpl2 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ⊆ 𝐵 ∧ ( 𝐺 ‘ 𝑆 ) ≤ 𝑊 ) ∧ 𝑥 ∈ 𝑆 ) → 𝑆 ⊆ 𝐵 ) |
18 |
|
simpr |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ⊆ 𝐵 ∧ ( 𝐺 ‘ 𝑆 ) ≤ 𝑊 ) ∧ 𝑥 ∈ 𝑆 ) → 𝑥 ∈ 𝑆 ) |
19 |
17 18
|
sseldd |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ⊆ 𝐵 ∧ ( 𝐺 ‘ 𝑆 ) ≤ 𝑊 ) ∧ 𝑥 ∈ 𝑆 ) → 𝑥 ∈ 𝐵 ) |
20 |
|
simpl1r |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ⊆ 𝐵 ∧ ( 𝐺 ‘ 𝑆 ) ≤ 𝑊 ) ∧ 𝑥 ∈ 𝑆 ) → 𝑊 ∈ 𝐻 ) |
21 |
1 5
|
lhpbase |
⊢ ( 𝑊 ∈ 𝐻 → 𝑊 ∈ 𝐵 ) |
22 |
20 21
|
syl |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ⊆ 𝐵 ∧ ( 𝐺 ‘ 𝑆 ) ≤ 𝑊 ) ∧ 𝑥 ∈ 𝑆 ) → 𝑊 ∈ 𝐵 ) |
23 |
1 3
|
latmcl |
⊢ ( ( 𝐾 ∈ Lat ∧ 𝑥 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ) → ( 𝑥 ∧ 𝑊 ) ∈ 𝐵 ) |
24 |
8 19 22 23
|
syl3anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ⊆ 𝐵 ∧ ( 𝐺 ‘ 𝑆 ) ≤ 𝑊 ) ∧ 𝑥 ∈ 𝑆 ) → ( 𝑥 ∧ 𝑊 ) ∈ 𝐵 ) |
25 |
7 10
|
syl |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ⊆ 𝐵 ∧ ( 𝐺 ‘ 𝑆 ) ≤ 𝑊 ) ∧ 𝑥 ∈ 𝑆 ) → 𝐾 ∈ CLat ) |
26 |
|
eqidd |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ⊆ 𝐵 ∧ ( 𝐺 ‘ 𝑆 ) ≤ 𝑊 ) ∧ 𝑥 ∈ 𝑆 ) → ( 𝑥 ∧ 𝑊 ) = ( 𝑥 ∧ 𝑊 ) ) |
27 |
|
oveq1 |
⊢ ( 𝑣 = 𝑥 → ( 𝑣 ∧ 𝑊 ) = ( 𝑥 ∧ 𝑊 ) ) |
28 |
27
|
rspceeqv |
⊢ ( ( 𝑥 ∈ 𝑆 ∧ ( 𝑥 ∧ 𝑊 ) = ( 𝑥 ∧ 𝑊 ) ) → ∃ 𝑣 ∈ 𝑆 ( 𝑥 ∧ 𝑊 ) = ( 𝑣 ∧ 𝑊 ) ) |
29 |
18 26 28
|
syl2anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ⊆ 𝐵 ∧ ( 𝐺 ‘ 𝑆 ) ≤ 𝑊 ) ∧ 𝑥 ∈ 𝑆 ) → ∃ 𝑣 ∈ 𝑆 ( 𝑥 ∧ 𝑊 ) = ( 𝑣 ∧ 𝑊 ) ) |
30 |
|
eqeq1 |
⊢ ( 𝑢 = ( 𝑥 ∧ 𝑊 ) → ( 𝑢 = ( 𝑣 ∧ 𝑊 ) ↔ ( 𝑥 ∧ 𝑊 ) = ( 𝑣 ∧ 𝑊 ) ) ) |
31 |
30
|
rexbidv |
⊢ ( 𝑢 = ( 𝑥 ∧ 𝑊 ) → ( ∃ 𝑣 ∈ 𝑆 𝑢 = ( 𝑣 ∧ 𝑊 ) ↔ ∃ 𝑣 ∈ 𝑆 ( 𝑥 ∧ 𝑊 ) = ( 𝑣 ∧ 𝑊 ) ) ) |
32 |
31
|
elrab |
⊢ ( ( 𝑥 ∧ 𝑊 ) ∈ { 𝑢 ∈ 𝐵 ∣ ∃ 𝑣 ∈ 𝑆 𝑢 = ( 𝑣 ∧ 𝑊 ) } ↔ ( ( 𝑥 ∧ 𝑊 ) ∈ 𝐵 ∧ ∃ 𝑣 ∈ 𝑆 ( 𝑥 ∧ 𝑊 ) = ( 𝑣 ∧ 𝑊 ) ) ) |
33 |
24 29 32
|
sylanbrc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ⊆ 𝐵 ∧ ( 𝐺 ‘ 𝑆 ) ≤ 𝑊 ) ∧ 𝑥 ∈ 𝑆 ) → ( 𝑥 ∧ 𝑊 ) ∈ { 𝑢 ∈ 𝐵 ∣ ∃ 𝑣 ∈ 𝑆 𝑢 = ( 𝑣 ∧ 𝑊 ) } ) |
34 |
33 6
|
eleqtrrdi |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ⊆ 𝐵 ∧ ( 𝐺 ‘ 𝑆 ) ≤ 𝑊 ) ∧ 𝑥 ∈ 𝑆 ) → ( 𝑥 ∧ 𝑊 ) ∈ 𝑇 ) |
35 |
1 2 4
|
clatglble |
⊢ ( ( 𝐾 ∈ CLat ∧ 𝑇 ⊆ 𝐵 ∧ ( 𝑥 ∧ 𝑊 ) ∈ 𝑇 ) → ( 𝐺 ‘ 𝑇 ) ≤ ( 𝑥 ∧ 𝑊 ) ) |
36 |
13 35
|
mp3an2 |
⊢ ( ( 𝐾 ∈ CLat ∧ ( 𝑥 ∧ 𝑊 ) ∈ 𝑇 ) → ( 𝐺 ‘ 𝑇 ) ≤ ( 𝑥 ∧ 𝑊 ) ) |
37 |
25 34 36
|
syl2anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ⊆ 𝐵 ∧ ( 𝐺 ‘ 𝑆 ) ≤ 𝑊 ) ∧ 𝑥 ∈ 𝑆 ) → ( 𝐺 ‘ 𝑇 ) ≤ ( 𝑥 ∧ 𝑊 ) ) |
38 |
1 2 3
|
latmle1 |
⊢ ( ( 𝐾 ∈ Lat ∧ 𝑥 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ) → ( 𝑥 ∧ 𝑊 ) ≤ 𝑥 ) |
39 |
8 19 22 38
|
syl3anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ⊆ 𝐵 ∧ ( 𝐺 ‘ 𝑆 ) ≤ 𝑊 ) ∧ 𝑥 ∈ 𝑆 ) → ( 𝑥 ∧ 𝑊 ) ≤ 𝑥 ) |
40 |
1 2 8 16 24 19 37 39
|
lattrd |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ⊆ 𝐵 ∧ ( 𝐺 ‘ 𝑆 ) ≤ 𝑊 ) ∧ 𝑥 ∈ 𝑆 ) → ( 𝐺 ‘ 𝑇 ) ≤ 𝑥 ) |
41 |
|
eqeq1 |
⊢ ( 𝑢 = 𝑤 → ( 𝑢 = ( 𝑣 ∧ 𝑊 ) ↔ 𝑤 = ( 𝑣 ∧ 𝑊 ) ) ) |
42 |
41
|
rexbidv |
⊢ ( 𝑢 = 𝑤 → ( ∃ 𝑣 ∈ 𝑆 𝑢 = ( 𝑣 ∧ 𝑊 ) ↔ ∃ 𝑣 ∈ 𝑆 𝑤 = ( 𝑣 ∧ 𝑊 ) ) ) |
43 |
|
oveq1 |
⊢ ( 𝑣 = 𝑦 → ( 𝑣 ∧ 𝑊 ) = ( 𝑦 ∧ 𝑊 ) ) |
44 |
43
|
eqeq2d |
⊢ ( 𝑣 = 𝑦 → ( 𝑤 = ( 𝑣 ∧ 𝑊 ) ↔ 𝑤 = ( 𝑦 ∧ 𝑊 ) ) ) |
45 |
44
|
cbvrexvw |
⊢ ( ∃ 𝑣 ∈ 𝑆 𝑤 = ( 𝑣 ∧ 𝑊 ) ↔ ∃ 𝑦 ∈ 𝑆 𝑤 = ( 𝑦 ∧ 𝑊 ) ) |
46 |
42 45
|
bitrdi |
⊢ ( 𝑢 = 𝑤 → ( ∃ 𝑣 ∈ 𝑆 𝑢 = ( 𝑣 ∧ 𝑊 ) ↔ ∃ 𝑦 ∈ 𝑆 𝑤 = ( 𝑦 ∧ 𝑊 ) ) ) |
47 |
46 6
|
elrab2 |
⊢ ( 𝑤 ∈ 𝑇 ↔ ( 𝑤 ∈ 𝐵 ∧ ∃ 𝑦 ∈ 𝑆 𝑤 = ( 𝑦 ∧ 𝑊 ) ) ) |
48 |
|
simp3 |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ⊆ 𝐵 ∧ ( 𝐺 ‘ 𝑆 ) ≤ 𝑊 ) ∧ 𝑧 ∈ 𝐵 ∧ ∀ 𝑥 ∈ 𝑆 𝑧 ≤ 𝑥 ) ∧ 𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝑆 ) → 𝑦 ∈ 𝑆 ) |
49 |
|
simp13 |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ⊆ 𝐵 ∧ ( 𝐺 ‘ 𝑆 ) ≤ 𝑊 ) ∧ 𝑧 ∈ 𝐵 ∧ ∀ 𝑥 ∈ 𝑆 𝑧 ≤ 𝑥 ) ∧ 𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝑆 ) → ∀ 𝑥 ∈ 𝑆 𝑧 ≤ 𝑥 ) |
50 |
|
breq2 |
⊢ ( 𝑥 = 𝑦 → ( 𝑧 ≤ 𝑥 ↔ 𝑧 ≤ 𝑦 ) ) |
51 |
50
|
rspcva |
⊢ ( ( 𝑦 ∈ 𝑆 ∧ ∀ 𝑥 ∈ 𝑆 𝑧 ≤ 𝑥 ) → 𝑧 ≤ 𝑦 ) |
52 |
48 49 51
|
syl2anc |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ⊆ 𝐵 ∧ ( 𝐺 ‘ 𝑆 ) ≤ 𝑊 ) ∧ 𝑧 ∈ 𝐵 ∧ ∀ 𝑥 ∈ 𝑆 𝑧 ≤ 𝑥 ) ∧ 𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝑆 ) → 𝑧 ≤ 𝑦 ) |
53 |
|
simp11l |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ⊆ 𝐵 ∧ ( 𝐺 ‘ 𝑆 ) ≤ 𝑊 ) ∧ 𝑧 ∈ 𝐵 ∧ ∀ 𝑥 ∈ 𝑆 𝑧 ≤ 𝑥 ) → 𝐾 ∈ HL ) |
54 |
53
|
3ad2ant1 |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ⊆ 𝐵 ∧ ( 𝐺 ‘ 𝑆 ) ≤ 𝑊 ) ∧ 𝑧 ∈ 𝐵 ∧ ∀ 𝑥 ∈ 𝑆 𝑧 ≤ 𝑥 ) ∧ 𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝑆 ) → 𝐾 ∈ HL ) |
55 |
54
|
hllatd |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ⊆ 𝐵 ∧ ( 𝐺 ‘ 𝑆 ) ≤ 𝑊 ) ∧ 𝑧 ∈ 𝐵 ∧ ∀ 𝑥 ∈ 𝑆 𝑧 ≤ 𝑥 ) ∧ 𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝑆 ) → 𝐾 ∈ Lat ) |
56 |
|
simp12 |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ⊆ 𝐵 ∧ ( 𝐺 ‘ 𝑆 ) ≤ 𝑊 ) ∧ 𝑧 ∈ 𝐵 ∧ ∀ 𝑥 ∈ 𝑆 𝑧 ≤ 𝑥 ) ∧ 𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝑆 ) → 𝑧 ∈ 𝐵 ) |
57 |
54 10
|
syl |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ⊆ 𝐵 ∧ ( 𝐺 ‘ 𝑆 ) ≤ 𝑊 ) ∧ 𝑧 ∈ 𝐵 ∧ ∀ 𝑥 ∈ 𝑆 𝑧 ≤ 𝑥 ) ∧ 𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝑆 ) → 𝐾 ∈ CLat ) |
58 |
|
simp112 |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ⊆ 𝐵 ∧ ( 𝐺 ‘ 𝑆 ) ≤ 𝑊 ) ∧ 𝑧 ∈ 𝐵 ∧ ∀ 𝑥 ∈ 𝑆 𝑧 ≤ 𝑥 ) ∧ 𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝑆 ) → 𝑆 ⊆ 𝐵 ) |
59 |
1 4
|
clatglbcl |
⊢ ( ( 𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵 ) → ( 𝐺 ‘ 𝑆 ) ∈ 𝐵 ) |
60 |
57 58 59
|
syl2anc |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ⊆ 𝐵 ∧ ( 𝐺 ‘ 𝑆 ) ≤ 𝑊 ) ∧ 𝑧 ∈ 𝐵 ∧ ∀ 𝑥 ∈ 𝑆 𝑧 ≤ 𝑥 ) ∧ 𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝑆 ) → ( 𝐺 ‘ 𝑆 ) ∈ 𝐵 ) |
61 |
|
simp11r |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ⊆ 𝐵 ∧ ( 𝐺 ‘ 𝑆 ) ≤ 𝑊 ) ∧ 𝑧 ∈ 𝐵 ∧ ∀ 𝑥 ∈ 𝑆 𝑧 ≤ 𝑥 ) → 𝑊 ∈ 𝐻 ) |
62 |
61
|
3ad2ant1 |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ⊆ 𝐵 ∧ ( 𝐺 ‘ 𝑆 ) ≤ 𝑊 ) ∧ 𝑧 ∈ 𝐵 ∧ ∀ 𝑥 ∈ 𝑆 𝑧 ≤ 𝑥 ) ∧ 𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝑆 ) → 𝑊 ∈ 𝐻 ) |
63 |
62 21
|
syl |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ⊆ 𝐵 ∧ ( 𝐺 ‘ 𝑆 ) ≤ 𝑊 ) ∧ 𝑧 ∈ 𝐵 ∧ ∀ 𝑥 ∈ 𝑆 𝑧 ≤ 𝑥 ) ∧ 𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝑆 ) → 𝑊 ∈ 𝐵 ) |
64 |
1 2 4
|
clatleglb |
⊢ ( ( 𝐾 ∈ CLat ∧ 𝑧 ∈ 𝐵 ∧ 𝑆 ⊆ 𝐵 ) → ( 𝑧 ≤ ( 𝐺 ‘ 𝑆 ) ↔ ∀ 𝑥 ∈ 𝑆 𝑧 ≤ 𝑥 ) ) |
65 |
57 56 58 64
|
syl3anc |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ⊆ 𝐵 ∧ ( 𝐺 ‘ 𝑆 ) ≤ 𝑊 ) ∧ 𝑧 ∈ 𝐵 ∧ ∀ 𝑥 ∈ 𝑆 𝑧 ≤ 𝑥 ) ∧ 𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝑆 ) → ( 𝑧 ≤ ( 𝐺 ‘ 𝑆 ) ↔ ∀ 𝑥 ∈ 𝑆 𝑧 ≤ 𝑥 ) ) |
66 |
49 65
|
mpbird |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ⊆ 𝐵 ∧ ( 𝐺 ‘ 𝑆 ) ≤ 𝑊 ) ∧ 𝑧 ∈ 𝐵 ∧ ∀ 𝑥 ∈ 𝑆 𝑧 ≤ 𝑥 ) ∧ 𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝑆 ) → 𝑧 ≤ ( 𝐺 ‘ 𝑆 ) ) |
67 |
|
simp113 |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ⊆ 𝐵 ∧ ( 𝐺 ‘ 𝑆 ) ≤ 𝑊 ) ∧ 𝑧 ∈ 𝐵 ∧ ∀ 𝑥 ∈ 𝑆 𝑧 ≤ 𝑥 ) ∧ 𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝑆 ) → ( 𝐺 ‘ 𝑆 ) ≤ 𝑊 ) |
68 |
1 2 55 56 60 63 66 67
|
lattrd |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ⊆ 𝐵 ∧ ( 𝐺 ‘ 𝑆 ) ≤ 𝑊 ) ∧ 𝑧 ∈ 𝐵 ∧ ∀ 𝑥 ∈ 𝑆 𝑧 ≤ 𝑥 ) ∧ 𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝑆 ) → 𝑧 ≤ 𝑊 ) |
69 |
58 48
|
sseldd |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ⊆ 𝐵 ∧ ( 𝐺 ‘ 𝑆 ) ≤ 𝑊 ) ∧ 𝑧 ∈ 𝐵 ∧ ∀ 𝑥 ∈ 𝑆 𝑧 ≤ 𝑥 ) ∧ 𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝑆 ) → 𝑦 ∈ 𝐵 ) |
70 |
1 2 3
|
latlem12 |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑧 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ) ) → ( ( 𝑧 ≤ 𝑦 ∧ 𝑧 ≤ 𝑊 ) ↔ 𝑧 ≤ ( 𝑦 ∧ 𝑊 ) ) ) |
71 |
55 56 69 63 70
|
syl13anc |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ⊆ 𝐵 ∧ ( 𝐺 ‘ 𝑆 ) ≤ 𝑊 ) ∧ 𝑧 ∈ 𝐵 ∧ ∀ 𝑥 ∈ 𝑆 𝑧 ≤ 𝑥 ) ∧ 𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝑆 ) → ( ( 𝑧 ≤ 𝑦 ∧ 𝑧 ≤ 𝑊 ) ↔ 𝑧 ≤ ( 𝑦 ∧ 𝑊 ) ) ) |
72 |
52 68 71
|
mpbi2and |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ⊆ 𝐵 ∧ ( 𝐺 ‘ 𝑆 ) ≤ 𝑊 ) ∧ 𝑧 ∈ 𝐵 ∧ ∀ 𝑥 ∈ 𝑆 𝑧 ≤ 𝑥 ) ∧ 𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝑆 ) → 𝑧 ≤ ( 𝑦 ∧ 𝑊 ) ) |
73 |
72
|
3expia |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ⊆ 𝐵 ∧ ( 𝐺 ‘ 𝑆 ) ≤ 𝑊 ) ∧ 𝑧 ∈ 𝐵 ∧ ∀ 𝑥 ∈ 𝑆 𝑧 ≤ 𝑥 ) ∧ 𝑤 ∈ 𝐵 ) → ( 𝑦 ∈ 𝑆 → 𝑧 ≤ ( 𝑦 ∧ 𝑊 ) ) ) |
74 |
|
breq2 |
⊢ ( 𝑤 = ( 𝑦 ∧ 𝑊 ) → ( 𝑧 ≤ 𝑤 ↔ 𝑧 ≤ ( 𝑦 ∧ 𝑊 ) ) ) |
75 |
74
|
biimprcd |
⊢ ( 𝑧 ≤ ( 𝑦 ∧ 𝑊 ) → ( 𝑤 = ( 𝑦 ∧ 𝑊 ) → 𝑧 ≤ 𝑤 ) ) |
76 |
73 75
|
syl6 |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ⊆ 𝐵 ∧ ( 𝐺 ‘ 𝑆 ) ≤ 𝑊 ) ∧ 𝑧 ∈ 𝐵 ∧ ∀ 𝑥 ∈ 𝑆 𝑧 ≤ 𝑥 ) ∧ 𝑤 ∈ 𝐵 ) → ( 𝑦 ∈ 𝑆 → ( 𝑤 = ( 𝑦 ∧ 𝑊 ) → 𝑧 ≤ 𝑤 ) ) ) |
77 |
76
|
rexlimdv |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ⊆ 𝐵 ∧ ( 𝐺 ‘ 𝑆 ) ≤ 𝑊 ) ∧ 𝑧 ∈ 𝐵 ∧ ∀ 𝑥 ∈ 𝑆 𝑧 ≤ 𝑥 ) ∧ 𝑤 ∈ 𝐵 ) → ( ∃ 𝑦 ∈ 𝑆 𝑤 = ( 𝑦 ∧ 𝑊 ) → 𝑧 ≤ 𝑤 ) ) |
78 |
77
|
expimpd |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ⊆ 𝐵 ∧ ( 𝐺 ‘ 𝑆 ) ≤ 𝑊 ) ∧ 𝑧 ∈ 𝐵 ∧ ∀ 𝑥 ∈ 𝑆 𝑧 ≤ 𝑥 ) → ( ( 𝑤 ∈ 𝐵 ∧ ∃ 𝑦 ∈ 𝑆 𝑤 = ( 𝑦 ∧ 𝑊 ) ) → 𝑧 ≤ 𝑤 ) ) |
79 |
47 78
|
syl5bi |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ⊆ 𝐵 ∧ ( 𝐺 ‘ 𝑆 ) ≤ 𝑊 ) ∧ 𝑧 ∈ 𝐵 ∧ ∀ 𝑥 ∈ 𝑆 𝑧 ≤ 𝑥 ) → ( 𝑤 ∈ 𝑇 → 𝑧 ≤ 𝑤 ) ) |
80 |
79
|
ralrimiv |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ⊆ 𝐵 ∧ ( 𝐺 ‘ 𝑆 ) ≤ 𝑊 ) ∧ 𝑧 ∈ 𝐵 ∧ ∀ 𝑥 ∈ 𝑆 𝑧 ≤ 𝑥 ) → ∀ 𝑤 ∈ 𝑇 𝑧 ≤ 𝑤 ) |
81 |
53 10
|
syl |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ⊆ 𝐵 ∧ ( 𝐺 ‘ 𝑆 ) ≤ 𝑊 ) ∧ 𝑧 ∈ 𝐵 ∧ ∀ 𝑥 ∈ 𝑆 𝑧 ≤ 𝑥 ) → 𝐾 ∈ CLat ) |
82 |
|
simp2 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ⊆ 𝐵 ∧ ( 𝐺 ‘ 𝑆 ) ≤ 𝑊 ) ∧ 𝑧 ∈ 𝐵 ∧ ∀ 𝑥 ∈ 𝑆 𝑧 ≤ 𝑥 ) → 𝑧 ∈ 𝐵 ) |
83 |
1 2 4
|
clatleglb |
⊢ ( ( 𝐾 ∈ CLat ∧ 𝑧 ∈ 𝐵 ∧ 𝑇 ⊆ 𝐵 ) → ( 𝑧 ≤ ( 𝐺 ‘ 𝑇 ) ↔ ∀ 𝑤 ∈ 𝑇 𝑧 ≤ 𝑤 ) ) |
84 |
13 83
|
mp3an3 |
⊢ ( ( 𝐾 ∈ CLat ∧ 𝑧 ∈ 𝐵 ) → ( 𝑧 ≤ ( 𝐺 ‘ 𝑇 ) ↔ ∀ 𝑤 ∈ 𝑇 𝑧 ≤ 𝑤 ) ) |
85 |
81 82 84
|
syl2anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ⊆ 𝐵 ∧ ( 𝐺 ‘ 𝑆 ) ≤ 𝑊 ) ∧ 𝑧 ∈ 𝐵 ∧ ∀ 𝑥 ∈ 𝑆 𝑧 ≤ 𝑥 ) → ( 𝑧 ≤ ( 𝐺 ‘ 𝑇 ) ↔ ∀ 𝑤 ∈ 𝑇 𝑧 ≤ 𝑤 ) ) |
86 |
80 85
|
mpbird |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ⊆ 𝐵 ∧ ( 𝐺 ‘ 𝑆 ) ≤ 𝑊 ) ∧ 𝑧 ∈ 𝐵 ∧ ∀ 𝑥 ∈ 𝑆 𝑧 ≤ 𝑥 ) → 𝑧 ≤ ( 𝐺 ‘ 𝑇 ) ) |
87 |
|
simp2 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ⊆ 𝐵 ∧ ( 𝐺 ‘ 𝑆 ) ≤ 𝑊 ) → 𝑆 ⊆ 𝐵 ) |
88 |
1 2 4 40 86 11 87 15
|
isglbd |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ⊆ 𝐵 ∧ ( 𝐺 ‘ 𝑆 ) ≤ 𝑊 ) → ( 𝐺 ‘ 𝑆 ) = ( 𝐺 ‘ 𝑇 ) ) |