Step |
Hyp |
Ref |
Expression |
1 |
|
dihjust.b |
⊢ 𝐵 = ( Base ‘ 𝐾 ) |
2 |
|
dihjust.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
3 |
|
dihjust.j |
⊢ ∨ = ( join ‘ 𝐾 ) |
4 |
|
dihjust.m |
⊢ ∧ = ( meet ‘ 𝐾 ) |
5 |
|
dihjust.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
6 |
|
dihjust.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
7 |
|
dihjust.i |
⊢ 𝐼 = ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) |
8 |
|
dihjust.J |
⊢ 𝐽 = ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 ) |
9 |
|
dihjust.u |
⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) |
10 |
|
dihjust.s |
⊢ ⊕ = ( LSSum ‘ 𝑈 ) |
11 |
|
dihord2c.t |
⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) |
12 |
|
dihord2c.r |
⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) |
13 |
|
dihord2c.o |
⊢ 𝑂 = ( ℎ ∈ 𝑇 ↦ ( I ↾ 𝐵 ) ) |
14 |
|
dihord2.p |
⊢ 𝑃 = ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) |
15 |
|
dihord2.e |
⊢ 𝐸 = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) |
16 |
|
dihord2.d |
⊢ + = ( +g ‘ 𝑈 ) |
17 |
|
dihord2.g |
⊢ 𝐺 = ( ℩ ℎ ∈ 𝑇 ( ℎ ‘ 𝑃 ) = 𝑁 ) |
18 |
|
simpl1 |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ⊆ ( ( 𝐽 ‘ 𝑁 ) ⊕ ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) ) ) ∧ ( 𝑓 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑓 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ) → ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ) ) |
19 |
|
simpl2l |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ⊆ ( ( 𝐽 ‘ 𝑁 ) ⊕ ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) ) ) ∧ ( 𝑓 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑓 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ) → 𝑋 ∈ 𝐵 ) |
20 |
|
simpl2r |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ⊆ ( ( 𝐽 ‘ 𝑁 ) ⊕ ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) ) ) ∧ ( 𝑓 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑓 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ) → 𝑌 ∈ 𝐵 ) |
21 |
|
simpl3 |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ⊆ ( ( 𝐽 ‘ 𝑁 ) ⊕ ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) ) ) ∧ ( 𝑓 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑓 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ) → ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ⊆ ( ( 𝐽 ‘ 𝑁 ) ⊕ ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) ) ) |
22 |
|
simprl |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ⊆ ( ( 𝐽 ‘ 𝑁 ) ⊕ ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) ) ) ∧ ( 𝑓 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑓 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ) → 𝑓 ∈ 𝑇 ) |
23 |
|
simprr |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ⊆ ( ( 𝐽 ‘ 𝑁 ) ⊕ ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) ) ) ∧ ( 𝑓 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑓 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ) → ( 𝑅 ‘ 𝑓 ) ≤ ( 𝑋 ∧ 𝑊 ) ) |
24 |
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
|
dihord11c |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ⊆ ( ( 𝐽 ‘ 𝑁 ) ⊕ ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) ) ∧ 𝑓 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑓 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ) → ∃ 𝑦 ∈ ( 𝐽 ‘ 𝑁 ) ∃ 𝑧 ∈ ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) 〈 𝑓 , 𝑂 〉 = ( 𝑦 + 𝑧 ) ) |
25 |
18 19 20 21 22 23 24
|
syl123anc |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ⊆ ( ( 𝐽 ‘ 𝑁 ) ⊕ ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) ) ) ∧ ( 𝑓 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑓 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ) → ∃ 𝑦 ∈ ( 𝐽 ‘ 𝑁 ) ∃ 𝑧 ∈ ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) 〈 𝑓 , 𝑂 〉 = ( 𝑦 + 𝑧 ) ) |
26 |
|
simpl11 |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ⊆ ( ( 𝐽 ‘ 𝑁 ) ⊕ ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) ) ) ∧ ( 𝑓 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑓 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
27 |
|
simpl13 |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ⊆ ( ( 𝐽 ‘ 𝑁 ) ⊕ ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) ) ) ∧ ( 𝑓 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑓 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ) → ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ) |
28 |
2 5 6 14 11 15 8 17
|
dicelval3 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ) → ( 𝑦 ∈ ( 𝐽 ‘ 𝑁 ) ↔ ∃ 𝑠 ∈ 𝐸 𝑦 = 〈 ( 𝑠 ‘ 𝐺 ) , 𝑠 〉 ) ) |
29 |
26 27 28
|
syl2anc |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ⊆ ( ( 𝐽 ‘ 𝑁 ) ⊕ ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) ) ) ∧ ( 𝑓 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑓 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ) → ( 𝑦 ∈ ( 𝐽 ‘ 𝑁 ) ↔ ∃ 𝑠 ∈ 𝐸 𝑦 = 〈 ( 𝑠 ‘ 𝐺 ) , 𝑠 〉 ) ) |
30 |
|
simp11l |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ⊆ ( ( 𝐽 ‘ 𝑁 ) ⊕ ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) ) ) → 𝐾 ∈ HL ) |
31 |
30
|
adantr |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ⊆ ( ( 𝐽 ‘ 𝑁 ) ⊕ ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) ) ) ∧ ( 𝑓 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑓 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ) → 𝐾 ∈ HL ) |
32 |
31
|
hllatd |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ⊆ ( ( 𝐽 ‘ 𝑁 ) ⊕ ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) ) ) ∧ ( 𝑓 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑓 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ) → 𝐾 ∈ Lat ) |
33 |
|
simp11r |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ⊆ ( ( 𝐽 ‘ 𝑁 ) ⊕ ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) ) ) → 𝑊 ∈ 𝐻 ) |
34 |
33
|
adantr |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ⊆ ( ( 𝐽 ‘ 𝑁 ) ⊕ ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) ) ) ∧ ( 𝑓 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑓 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ) → 𝑊 ∈ 𝐻 ) |
35 |
1 6
|
lhpbase |
⊢ ( 𝑊 ∈ 𝐻 → 𝑊 ∈ 𝐵 ) |
36 |
34 35
|
syl |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ⊆ ( ( 𝐽 ‘ 𝑁 ) ⊕ ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) ) ) ∧ ( 𝑓 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑓 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ) → 𝑊 ∈ 𝐵 ) |
37 |
1 4
|
latmcl |
⊢ ( ( 𝐾 ∈ Lat ∧ 𝑌 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ) → ( 𝑌 ∧ 𝑊 ) ∈ 𝐵 ) |
38 |
32 20 36 37
|
syl3anc |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ⊆ ( ( 𝐽 ‘ 𝑁 ) ⊕ ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) ) ) ∧ ( 𝑓 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑓 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ) → ( 𝑌 ∧ 𝑊 ) ∈ 𝐵 ) |
39 |
1 2 4
|
latmle2 |
⊢ ( ( 𝐾 ∈ Lat ∧ 𝑌 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ) → ( 𝑌 ∧ 𝑊 ) ≤ 𝑊 ) |
40 |
32 20 36 39
|
syl3anc |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ⊆ ( ( 𝐽 ‘ 𝑁 ) ⊕ ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) ) ) ∧ ( 𝑓 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑓 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ) → ( 𝑌 ∧ 𝑊 ) ≤ 𝑊 ) |
41 |
1 2 6 11 12 13 7
|
dibelval3 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑌 ∧ 𝑊 ) ∈ 𝐵 ∧ ( 𝑌 ∧ 𝑊 ) ≤ 𝑊 ) ) → ( 𝑧 ∈ ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) ↔ ∃ 𝑔 ∈ 𝑇 ( 𝑧 = 〈 𝑔 , 𝑂 〉 ∧ ( 𝑅 ‘ 𝑔 ) ≤ ( 𝑌 ∧ 𝑊 ) ) ) ) |
42 |
26 38 40 41
|
syl12anc |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ⊆ ( ( 𝐽 ‘ 𝑁 ) ⊕ ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) ) ) ∧ ( 𝑓 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑓 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ) → ( 𝑧 ∈ ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) ↔ ∃ 𝑔 ∈ 𝑇 ( 𝑧 = 〈 𝑔 , 𝑂 〉 ∧ ( 𝑅 ‘ 𝑔 ) ≤ ( 𝑌 ∧ 𝑊 ) ) ) ) |
43 |
29 42
|
anbi12d |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ⊆ ( ( 𝐽 ‘ 𝑁 ) ⊕ ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) ) ) ∧ ( 𝑓 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑓 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ) → ( ( 𝑦 ∈ ( 𝐽 ‘ 𝑁 ) ∧ 𝑧 ∈ ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) ) ↔ ( ∃ 𝑠 ∈ 𝐸 𝑦 = 〈 ( 𝑠 ‘ 𝐺 ) , 𝑠 〉 ∧ ∃ 𝑔 ∈ 𝑇 ( 𝑧 = 〈 𝑔 , 𝑂 〉 ∧ ( 𝑅 ‘ 𝑔 ) ≤ ( 𝑌 ∧ 𝑊 ) ) ) ) ) |
44 |
|
reeanv |
⊢ ( ∃ 𝑠 ∈ 𝐸 ∃ 𝑔 ∈ 𝑇 ( 𝑦 = 〈 ( 𝑠 ‘ 𝐺 ) , 𝑠 〉 ∧ ( 𝑧 = 〈 𝑔 , 𝑂 〉 ∧ ( 𝑅 ‘ 𝑔 ) ≤ ( 𝑌 ∧ 𝑊 ) ) ) ↔ ( ∃ 𝑠 ∈ 𝐸 𝑦 = 〈 ( 𝑠 ‘ 𝐺 ) , 𝑠 〉 ∧ ∃ 𝑔 ∈ 𝑇 ( 𝑧 = 〈 𝑔 , 𝑂 〉 ∧ ( 𝑅 ‘ 𝑔 ) ≤ ( 𝑌 ∧ 𝑊 ) ) ) ) |
45 |
|
simpll1 |
⊢ ( ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ⊆ ( ( 𝐽 ‘ 𝑁 ) ⊕ ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) ) ) ∧ ( 𝑓 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑓 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ) ∧ ( ( 𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ) ∧ ( 𝑅 ‘ 𝑔 ) ≤ ( 𝑌 ∧ 𝑊 ) ∧ 〈 𝑓 , 𝑂 〉 = ( 〈 ( 𝑠 ‘ 𝐺 ) , 𝑠 〉 + 〈 𝑔 , 𝑂 〉 ) ) ) → ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ) ) |
46 |
|
simplr |
⊢ ( ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ⊆ ( ( 𝐽 ‘ 𝑁 ) ⊕ ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) ) ) ∧ ( 𝑓 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑓 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ) ∧ ( ( 𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ) ∧ ( 𝑅 ‘ 𝑔 ) ≤ ( 𝑌 ∧ 𝑊 ) ∧ 〈 𝑓 , 𝑂 〉 = ( 〈 ( 𝑠 ‘ 𝐺 ) , 𝑠 〉 + 〈 𝑔 , 𝑂 〉 ) ) ) → ( 𝑓 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑓 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ) |
47 |
|
simpr |
⊢ ( ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ⊆ ( ( 𝐽 ‘ 𝑁 ) ⊕ ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) ) ) ∧ ( 𝑓 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑓 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ) ∧ ( ( 𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ) ∧ ( 𝑅 ‘ 𝑔 ) ≤ ( 𝑌 ∧ 𝑊 ) ∧ 〈 𝑓 , 𝑂 〉 = ( 〈 ( 𝑠 ‘ 𝐺 ) , 𝑠 〉 + 〈 𝑔 , 𝑂 〉 ) ) ) → ( ( 𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ) ∧ ( 𝑅 ‘ 𝑔 ) ≤ ( 𝑌 ∧ 𝑊 ) ∧ 〈 𝑓 , 𝑂 〉 = ( 〈 ( 𝑠 ‘ 𝐺 ) , 𝑠 〉 + 〈 𝑔 , 𝑂 〉 ) ) ) |
48 |
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
|
dihord10 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ) ∧ ( 𝑓 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑓 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ∧ ( ( 𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ) ∧ ( 𝑅 ‘ 𝑔 ) ≤ ( 𝑌 ∧ 𝑊 ) ∧ 〈 𝑓 , 𝑂 〉 = ( 〈 ( 𝑠 ‘ 𝐺 ) , 𝑠 〉 + 〈 𝑔 , 𝑂 〉 ) ) ) → ( 𝑅 ‘ 𝑓 ) ≤ ( 𝑌 ∧ 𝑊 ) ) |
49 |
45 46 47 48
|
syl3anc |
⊢ ( ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ⊆ ( ( 𝐽 ‘ 𝑁 ) ⊕ ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) ) ) ∧ ( 𝑓 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑓 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ) ∧ ( ( 𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ) ∧ ( 𝑅 ‘ 𝑔 ) ≤ ( 𝑌 ∧ 𝑊 ) ∧ 〈 𝑓 , 𝑂 〉 = ( 〈 ( 𝑠 ‘ 𝐺 ) , 𝑠 〉 + 〈 𝑔 , 𝑂 〉 ) ) ) → ( 𝑅 ‘ 𝑓 ) ≤ ( 𝑌 ∧ 𝑊 ) ) |
50 |
49
|
3exp2 |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ⊆ ( ( 𝐽 ‘ 𝑁 ) ⊕ ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) ) ) ∧ ( 𝑓 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑓 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ) → ( ( 𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ) → ( ( 𝑅 ‘ 𝑔 ) ≤ ( 𝑌 ∧ 𝑊 ) → ( 〈 𝑓 , 𝑂 〉 = ( 〈 ( 𝑠 ‘ 𝐺 ) , 𝑠 〉 + 〈 𝑔 , 𝑂 〉 ) → ( 𝑅 ‘ 𝑓 ) ≤ ( 𝑌 ∧ 𝑊 ) ) ) ) ) |
51 |
|
oveq12 |
⊢ ( ( 𝑦 = 〈 ( 𝑠 ‘ 𝐺 ) , 𝑠 〉 ∧ 𝑧 = 〈 𝑔 , 𝑂 〉 ) → ( 𝑦 + 𝑧 ) = ( 〈 ( 𝑠 ‘ 𝐺 ) , 𝑠 〉 + 〈 𝑔 , 𝑂 〉 ) ) |
52 |
51
|
eqeq2d |
⊢ ( ( 𝑦 = 〈 ( 𝑠 ‘ 𝐺 ) , 𝑠 〉 ∧ 𝑧 = 〈 𝑔 , 𝑂 〉 ) → ( 〈 𝑓 , 𝑂 〉 = ( 𝑦 + 𝑧 ) ↔ 〈 𝑓 , 𝑂 〉 = ( 〈 ( 𝑠 ‘ 𝐺 ) , 𝑠 〉 + 〈 𝑔 , 𝑂 〉 ) ) ) |
53 |
52
|
imbi1d |
⊢ ( ( 𝑦 = 〈 ( 𝑠 ‘ 𝐺 ) , 𝑠 〉 ∧ 𝑧 = 〈 𝑔 , 𝑂 〉 ) → ( ( 〈 𝑓 , 𝑂 〉 = ( 𝑦 + 𝑧 ) → ( 𝑅 ‘ 𝑓 ) ≤ ( 𝑌 ∧ 𝑊 ) ) ↔ ( 〈 𝑓 , 𝑂 〉 = ( 〈 ( 𝑠 ‘ 𝐺 ) , 𝑠 〉 + 〈 𝑔 , 𝑂 〉 ) → ( 𝑅 ‘ 𝑓 ) ≤ ( 𝑌 ∧ 𝑊 ) ) ) ) |
54 |
53
|
imbi2d |
⊢ ( ( 𝑦 = 〈 ( 𝑠 ‘ 𝐺 ) , 𝑠 〉 ∧ 𝑧 = 〈 𝑔 , 𝑂 〉 ) → ( ( ( 𝑅 ‘ 𝑔 ) ≤ ( 𝑌 ∧ 𝑊 ) → ( 〈 𝑓 , 𝑂 〉 = ( 𝑦 + 𝑧 ) → ( 𝑅 ‘ 𝑓 ) ≤ ( 𝑌 ∧ 𝑊 ) ) ) ↔ ( ( 𝑅 ‘ 𝑔 ) ≤ ( 𝑌 ∧ 𝑊 ) → ( 〈 𝑓 , 𝑂 〉 = ( 〈 ( 𝑠 ‘ 𝐺 ) , 𝑠 〉 + 〈 𝑔 , 𝑂 〉 ) → ( 𝑅 ‘ 𝑓 ) ≤ ( 𝑌 ∧ 𝑊 ) ) ) ) ) |
55 |
54
|
biimprd |
⊢ ( ( 𝑦 = 〈 ( 𝑠 ‘ 𝐺 ) , 𝑠 〉 ∧ 𝑧 = 〈 𝑔 , 𝑂 〉 ) → ( ( ( 𝑅 ‘ 𝑔 ) ≤ ( 𝑌 ∧ 𝑊 ) → ( 〈 𝑓 , 𝑂 〉 = ( 〈 ( 𝑠 ‘ 𝐺 ) , 𝑠 〉 + 〈 𝑔 , 𝑂 〉 ) → ( 𝑅 ‘ 𝑓 ) ≤ ( 𝑌 ∧ 𝑊 ) ) ) → ( ( 𝑅 ‘ 𝑔 ) ≤ ( 𝑌 ∧ 𝑊 ) → ( 〈 𝑓 , 𝑂 〉 = ( 𝑦 + 𝑧 ) → ( 𝑅 ‘ 𝑓 ) ≤ ( 𝑌 ∧ 𝑊 ) ) ) ) ) |
56 |
55
|
com23 |
⊢ ( ( 𝑦 = 〈 ( 𝑠 ‘ 𝐺 ) , 𝑠 〉 ∧ 𝑧 = 〈 𝑔 , 𝑂 〉 ) → ( ( 𝑅 ‘ 𝑔 ) ≤ ( 𝑌 ∧ 𝑊 ) → ( ( ( 𝑅 ‘ 𝑔 ) ≤ ( 𝑌 ∧ 𝑊 ) → ( 〈 𝑓 , 𝑂 〉 = ( 〈 ( 𝑠 ‘ 𝐺 ) , 𝑠 〉 + 〈 𝑔 , 𝑂 〉 ) → ( 𝑅 ‘ 𝑓 ) ≤ ( 𝑌 ∧ 𝑊 ) ) ) → ( 〈 𝑓 , 𝑂 〉 = ( 𝑦 + 𝑧 ) → ( 𝑅 ‘ 𝑓 ) ≤ ( 𝑌 ∧ 𝑊 ) ) ) ) ) |
57 |
56
|
impr |
⊢ ( ( 𝑦 = 〈 ( 𝑠 ‘ 𝐺 ) , 𝑠 〉 ∧ ( 𝑧 = 〈 𝑔 , 𝑂 〉 ∧ ( 𝑅 ‘ 𝑔 ) ≤ ( 𝑌 ∧ 𝑊 ) ) ) → ( ( ( 𝑅 ‘ 𝑔 ) ≤ ( 𝑌 ∧ 𝑊 ) → ( 〈 𝑓 , 𝑂 〉 = ( 〈 ( 𝑠 ‘ 𝐺 ) , 𝑠 〉 + 〈 𝑔 , 𝑂 〉 ) → ( 𝑅 ‘ 𝑓 ) ≤ ( 𝑌 ∧ 𝑊 ) ) ) → ( 〈 𝑓 , 𝑂 〉 = ( 𝑦 + 𝑧 ) → ( 𝑅 ‘ 𝑓 ) ≤ ( 𝑌 ∧ 𝑊 ) ) ) ) |
58 |
57
|
com12 |
⊢ ( ( ( 𝑅 ‘ 𝑔 ) ≤ ( 𝑌 ∧ 𝑊 ) → ( 〈 𝑓 , 𝑂 〉 = ( 〈 ( 𝑠 ‘ 𝐺 ) , 𝑠 〉 + 〈 𝑔 , 𝑂 〉 ) → ( 𝑅 ‘ 𝑓 ) ≤ ( 𝑌 ∧ 𝑊 ) ) ) → ( ( 𝑦 = 〈 ( 𝑠 ‘ 𝐺 ) , 𝑠 〉 ∧ ( 𝑧 = 〈 𝑔 , 𝑂 〉 ∧ ( 𝑅 ‘ 𝑔 ) ≤ ( 𝑌 ∧ 𝑊 ) ) ) → ( 〈 𝑓 , 𝑂 〉 = ( 𝑦 + 𝑧 ) → ( 𝑅 ‘ 𝑓 ) ≤ ( 𝑌 ∧ 𝑊 ) ) ) ) |
59 |
50 58
|
syl6 |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ⊆ ( ( 𝐽 ‘ 𝑁 ) ⊕ ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) ) ) ∧ ( 𝑓 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑓 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ) → ( ( 𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ) → ( ( 𝑦 = 〈 ( 𝑠 ‘ 𝐺 ) , 𝑠 〉 ∧ ( 𝑧 = 〈 𝑔 , 𝑂 〉 ∧ ( 𝑅 ‘ 𝑔 ) ≤ ( 𝑌 ∧ 𝑊 ) ) ) → ( 〈 𝑓 , 𝑂 〉 = ( 𝑦 + 𝑧 ) → ( 𝑅 ‘ 𝑓 ) ≤ ( 𝑌 ∧ 𝑊 ) ) ) ) ) |
60 |
59
|
rexlimdvv |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ⊆ ( ( 𝐽 ‘ 𝑁 ) ⊕ ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) ) ) ∧ ( 𝑓 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑓 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ) → ( ∃ 𝑠 ∈ 𝐸 ∃ 𝑔 ∈ 𝑇 ( 𝑦 = 〈 ( 𝑠 ‘ 𝐺 ) , 𝑠 〉 ∧ ( 𝑧 = 〈 𝑔 , 𝑂 〉 ∧ ( 𝑅 ‘ 𝑔 ) ≤ ( 𝑌 ∧ 𝑊 ) ) ) → ( 〈 𝑓 , 𝑂 〉 = ( 𝑦 + 𝑧 ) → ( 𝑅 ‘ 𝑓 ) ≤ ( 𝑌 ∧ 𝑊 ) ) ) ) |
61 |
44 60
|
syl5bir |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ⊆ ( ( 𝐽 ‘ 𝑁 ) ⊕ ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) ) ) ∧ ( 𝑓 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑓 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ) → ( ( ∃ 𝑠 ∈ 𝐸 𝑦 = 〈 ( 𝑠 ‘ 𝐺 ) , 𝑠 〉 ∧ ∃ 𝑔 ∈ 𝑇 ( 𝑧 = 〈 𝑔 , 𝑂 〉 ∧ ( 𝑅 ‘ 𝑔 ) ≤ ( 𝑌 ∧ 𝑊 ) ) ) → ( 〈 𝑓 , 𝑂 〉 = ( 𝑦 + 𝑧 ) → ( 𝑅 ‘ 𝑓 ) ≤ ( 𝑌 ∧ 𝑊 ) ) ) ) |
62 |
43 61
|
sylbid |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ⊆ ( ( 𝐽 ‘ 𝑁 ) ⊕ ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) ) ) ∧ ( 𝑓 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑓 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ) → ( ( 𝑦 ∈ ( 𝐽 ‘ 𝑁 ) ∧ 𝑧 ∈ ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) ) → ( 〈 𝑓 , 𝑂 〉 = ( 𝑦 + 𝑧 ) → ( 𝑅 ‘ 𝑓 ) ≤ ( 𝑌 ∧ 𝑊 ) ) ) ) |
63 |
62
|
rexlimdvv |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ⊆ ( ( 𝐽 ‘ 𝑁 ) ⊕ ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) ) ) ∧ ( 𝑓 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑓 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ) → ( ∃ 𝑦 ∈ ( 𝐽 ‘ 𝑁 ) ∃ 𝑧 ∈ ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) 〈 𝑓 , 𝑂 〉 = ( 𝑦 + 𝑧 ) → ( 𝑅 ‘ 𝑓 ) ≤ ( 𝑌 ∧ 𝑊 ) ) ) |
64 |
25 63
|
mpd |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ⊆ ( ( 𝐽 ‘ 𝑁 ) ⊕ ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) ) ) ∧ ( 𝑓 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑓 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ) → ( 𝑅 ‘ 𝑓 ) ≤ ( 𝑌 ∧ 𝑊 ) ) |
65 |
64
|
exp32 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ⊆ ( ( 𝐽 ‘ 𝑁 ) ⊕ ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) ) ) → ( 𝑓 ∈ 𝑇 → ( ( 𝑅 ‘ 𝑓 ) ≤ ( 𝑋 ∧ 𝑊 ) → ( 𝑅 ‘ 𝑓 ) ≤ ( 𝑌 ∧ 𝑊 ) ) ) ) |
66 |
65
|
ralrimiv |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ⊆ ( ( 𝐽 ‘ 𝑁 ) ⊕ ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) ) ) → ∀ 𝑓 ∈ 𝑇 ( ( 𝑅 ‘ 𝑓 ) ≤ ( 𝑋 ∧ 𝑊 ) → ( 𝑅 ‘ 𝑓 ) ≤ ( 𝑌 ∧ 𝑊 ) ) ) |
67 |
|
simp11 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ⊆ ( ( 𝐽 ‘ 𝑁 ) ⊕ ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
68 |
30
|
hllatd |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ⊆ ( ( 𝐽 ‘ 𝑁 ) ⊕ ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) ) ) → 𝐾 ∈ Lat ) |
69 |
|
simp2l |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ⊆ ( ( 𝐽 ‘ 𝑁 ) ⊕ ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) ) ) → 𝑋 ∈ 𝐵 ) |
70 |
33 35
|
syl |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ⊆ ( ( 𝐽 ‘ 𝑁 ) ⊕ ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) ) ) → 𝑊 ∈ 𝐵 ) |
71 |
1 4
|
latmcl |
⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ) → ( 𝑋 ∧ 𝑊 ) ∈ 𝐵 ) |
72 |
68 69 70 71
|
syl3anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ⊆ ( ( 𝐽 ‘ 𝑁 ) ⊕ ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) ) ) → ( 𝑋 ∧ 𝑊 ) ∈ 𝐵 ) |
73 |
1 2 4
|
latmle2 |
⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ) → ( 𝑋 ∧ 𝑊 ) ≤ 𝑊 ) |
74 |
68 69 70 73
|
syl3anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ⊆ ( ( 𝐽 ‘ 𝑁 ) ⊕ ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) ) ) → ( 𝑋 ∧ 𝑊 ) ≤ 𝑊 ) |
75 |
|
simp2r |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ⊆ ( ( 𝐽 ‘ 𝑁 ) ⊕ ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) ) ) → 𝑌 ∈ 𝐵 ) |
76 |
68 75 70 37
|
syl3anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ⊆ ( ( 𝐽 ‘ 𝑁 ) ⊕ ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) ) ) → ( 𝑌 ∧ 𝑊 ) ∈ 𝐵 ) |
77 |
68 75 70 39
|
syl3anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ⊆ ( ( 𝐽 ‘ 𝑁 ) ⊕ ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) ) ) → ( 𝑌 ∧ 𝑊 ) ≤ 𝑊 ) |
78 |
1 2 5 6 11 12
|
trlord |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑋 ∧ 𝑊 ) ∈ 𝐵 ∧ ( 𝑋 ∧ 𝑊 ) ≤ 𝑊 ) ∧ ( ( 𝑌 ∧ 𝑊 ) ∈ 𝐵 ∧ ( 𝑌 ∧ 𝑊 ) ≤ 𝑊 ) ) → ( ( 𝑋 ∧ 𝑊 ) ≤ ( 𝑌 ∧ 𝑊 ) ↔ ∀ 𝑓 ∈ 𝑇 ( ( 𝑅 ‘ 𝑓 ) ≤ ( 𝑋 ∧ 𝑊 ) → ( 𝑅 ‘ 𝑓 ) ≤ ( 𝑌 ∧ 𝑊 ) ) ) ) |
79 |
67 72 74 76 77 78
|
syl122anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ⊆ ( ( 𝐽 ‘ 𝑁 ) ⊕ ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) ) ) → ( ( 𝑋 ∧ 𝑊 ) ≤ ( 𝑌 ∧ 𝑊 ) ↔ ∀ 𝑓 ∈ 𝑇 ( ( 𝑅 ‘ 𝑓 ) ≤ ( 𝑋 ∧ 𝑊 ) → ( 𝑅 ‘ 𝑓 ) ≤ ( 𝑌 ∧ 𝑊 ) ) ) ) |
80 |
66 79
|
mpbird |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ⊆ ( ( 𝐽 ‘ 𝑁 ) ⊕ ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) ) ) → ( 𝑋 ∧ 𝑊 ) ≤ ( 𝑌 ∧ 𝑊 ) ) |