Step |
Hyp |
Ref |
Expression |
1 |
|
tendopl.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
2 |
|
tendopl.t |
⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) |
3 |
|
tendopl.e |
⊢ 𝐸 = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) |
4 |
|
tendopl.p |
⊢ 𝑃 = ( 𝑠 ∈ 𝐸 , 𝑡 ∈ 𝐸 ↦ ( 𝑓 ∈ 𝑇 ↦ ( ( 𝑠 ‘ 𝑓 ) ∘ ( 𝑡 ‘ 𝑓 ) ) ) ) |
5 |
|
tendopltp.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
6 |
|
tendopltp.r |
⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) |
7 |
|
eqid |
⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 ) |
8 |
|
simp1l |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ) ∧ 𝐹 ∈ 𝑇 ) → 𝐾 ∈ HL ) |
9 |
8
|
hllatd |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ) ∧ 𝐹 ∈ 𝑇 ) → 𝐾 ∈ Lat ) |
10 |
|
simp1 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ) ∧ 𝐹 ∈ 𝑇 ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
11 |
1 2 3 4
|
tendoplcl2 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ) ∧ 𝐹 ∈ 𝑇 ) → ( ( 𝑈 𝑃 𝑉 ) ‘ 𝐹 ) ∈ 𝑇 ) |
12 |
7 1 2 6
|
trlcl |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑈 𝑃 𝑉 ) ‘ 𝐹 ) ∈ 𝑇 ) → ( 𝑅 ‘ ( ( 𝑈 𝑃 𝑉 ) ‘ 𝐹 ) ) ∈ ( Base ‘ 𝐾 ) ) |
13 |
10 11 12
|
syl2anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ) ∧ 𝐹 ∈ 𝑇 ) → ( 𝑅 ‘ ( ( 𝑈 𝑃 𝑉 ) ‘ 𝐹 ) ) ∈ ( Base ‘ 𝐾 ) ) |
14 |
1 2 3
|
tendocl |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑈 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ) → ( 𝑈 ‘ 𝐹 ) ∈ 𝑇 ) |
15 |
14
|
3adant2r |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ) ∧ 𝐹 ∈ 𝑇 ) → ( 𝑈 ‘ 𝐹 ) ∈ 𝑇 ) |
16 |
7 1 2 6
|
trlcl |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ‘ 𝐹 ) ∈ 𝑇 ) → ( 𝑅 ‘ ( 𝑈 ‘ 𝐹 ) ) ∈ ( Base ‘ 𝐾 ) ) |
17 |
10 15 16
|
syl2anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ) ∧ 𝐹 ∈ 𝑇 ) → ( 𝑅 ‘ ( 𝑈 ‘ 𝐹 ) ) ∈ ( Base ‘ 𝐾 ) ) |
18 |
1 2 3
|
tendocl |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑉 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ) → ( 𝑉 ‘ 𝐹 ) ∈ 𝑇 ) |
19 |
18
|
3adant2l |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ) ∧ 𝐹 ∈ 𝑇 ) → ( 𝑉 ‘ 𝐹 ) ∈ 𝑇 ) |
20 |
7 1 2 6
|
trlcl |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑉 ‘ 𝐹 ) ∈ 𝑇 ) → ( 𝑅 ‘ ( 𝑉 ‘ 𝐹 ) ) ∈ ( Base ‘ 𝐾 ) ) |
21 |
10 19 20
|
syl2anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ) ∧ 𝐹 ∈ 𝑇 ) → ( 𝑅 ‘ ( 𝑉 ‘ 𝐹 ) ) ∈ ( Base ‘ 𝐾 ) ) |
22 |
|
eqid |
⊢ ( join ‘ 𝐾 ) = ( join ‘ 𝐾 ) |
23 |
7 22
|
latjcl |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑅 ‘ ( 𝑈 ‘ 𝐹 ) ) ∈ ( Base ‘ 𝐾 ) ∧ ( 𝑅 ‘ ( 𝑉 ‘ 𝐹 ) ) ∈ ( Base ‘ 𝐾 ) ) → ( ( 𝑅 ‘ ( 𝑈 ‘ 𝐹 ) ) ( join ‘ 𝐾 ) ( 𝑅 ‘ ( 𝑉 ‘ 𝐹 ) ) ) ∈ ( Base ‘ 𝐾 ) ) |
24 |
9 17 21 23
|
syl3anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ) ∧ 𝐹 ∈ 𝑇 ) → ( ( 𝑅 ‘ ( 𝑈 ‘ 𝐹 ) ) ( join ‘ 𝐾 ) ( 𝑅 ‘ ( 𝑉 ‘ 𝐹 ) ) ) ∈ ( Base ‘ 𝐾 ) ) |
25 |
|
simp3 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ) ∧ 𝐹 ∈ 𝑇 ) → 𝐹 ∈ 𝑇 ) |
26 |
7 1 2 6
|
trlcl |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → ( 𝑅 ‘ 𝐹 ) ∈ ( Base ‘ 𝐾 ) ) |
27 |
10 25 26
|
syl2anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ) ∧ 𝐹 ∈ 𝑇 ) → ( 𝑅 ‘ 𝐹 ) ∈ ( Base ‘ 𝐾 ) ) |
28 |
|
simp2l |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ) ∧ 𝐹 ∈ 𝑇 ) → 𝑈 ∈ 𝐸 ) |
29 |
|
simp2r |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ) ∧ 𝐹 ∈ 𝑇 ) → 𝑉 ∈ 𝐸 ) |
30 |
4 2
|
tendopl2 |
⊢ ( ( 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ) → ( ( 𝑈 𝑃 𝑉 ) ‘ 𝐹 ) = ( ( 𝑈 ‘ 𝐹 ) ∘ ( 𝑉 ‘ 𝐹 ) ) ) |
31 |
28 29 25 30
|
syl3anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ) ∧ 𝐹 ∈ 𝑇 ) → ( ( 𝑈 𝑃 𝑉 ) ‘ 𝐹 ) = ( ( 𝑈 ‘ 𝐹 ) ∘ ( 𝑉 ‘ 𝐹 ) ) ) |
32 |
31
|
fveq2d |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ) ∧ 𝐹 ∈ 𝑇 ) → ( 𝑅 ‘ ( ( 𝑈 𝑃 𝑉 ) ‘ 𝐹 ) ) = ( 𝑅 ‘ ( ( 𝑈 ‘ 𝐹 ) ∘ ( 𝑉 ‘ 𝐹 ) ) ) ) |
33 |
5 22 1 2 6
|
trlco |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ‘ 𝐹 ) ∈ 𝑇 ∧ ( 𝑉 ‘ 𝐹 ) ∈ 𝑇 ) → ( 𝑅 ‘ ( ( 𝑈 ‘ 𝐹 ) ∘ ( 𝑉 ‘ 𝐹 ) ) ) ≤ ( ( 𝑅 ‘ ( 𝑈 ‘ 𝐹 ) ) ( join ‘ 𝐾 ) ( 𝑅 ‘ ( 𝑉 ‘ 𝐹 ) ) ) ) |
34 |
10 15 19 33
|
syl3anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ) ∧ 𝐹 ∈ 𝑇 ) → ( 𝑅 ‘ ( ( 𝑈 ‘ 𝐹 ) ∘ ( 𝑉 ‘ 𝐹 ) ) ) ≤ ( ( 𝑅 ‘ ( 𝑈 ‘ 𝐹 ) ) ( join ‘ 𝐾 ) ( 𝑅 ‘ ( 𝑉 ‘ 𝐹 ) ) ) ) |
35 |
32 34
|
eqbrtrd |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ) ∧ 𝐹 ∈ 𝑇 ) → ( 𝑅 ‘ ( ( 𝑈 𝑃 𝑉 ) ‘ 𝐹 ) ) ≤ ( ( 𝑅 ‘ ( 𝑈 ‘ 𝐹 ) ) ( join ‘ 𝐾 ) ( 𝑅 ‘ ( 𝑉 ‘ 𝐹 ) ) ) ) |
36 |
5 1 2 6 3
|
tendotp |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑈 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ) → ( 𝑅 ‘ ( 𝑈 ‘ 𝐹 ) ) ≤ ( 𝑅 ‘ 𝐹 ) ) |
37 |
36
|
3adant2r |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ) ∧ 𝐹 ∈ 𝑇 ) → ( 𝑅 ‘ ( 𝑈 ‘ 𝐹 ) ) ≤ ( 𝑅 ‘ 𝐹 ) ) |
38 |
5 1 2 6 3
|
tendotp |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑉 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ) → ( 𝑅 ‘ ( 𝑉 ‘ 𝐹 ) ) ≤ ( 𝑅 ‘ 𝐹 ) ) |
39 |
38
|
3adant2l |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ) ∧ 𝐹 ∈ 𝑇 ) → ( 𝑅 ‘ ( 𝑉 ‘ 𝐹 ) ) ≤ ( 𝑅 ‘ 𝐹 ) ) |
40 |
7 5 22
|
latjle12 |
⊢ ( ( 𝐾 ∈ Lat ∧ ( ( 𝑅 ‘ ( 𝑈 ‘ 𝐹 ) ) ∈ ( Base ‘ 𝐾 ) ∧ ( 𝑅 ‘ ( 𝑉 ‘ 𝐹 ) ) ∈ ( Base ‘ 𝐾 ) ∧ ( 𝑅 ‘ 𝐹 ) ∈ ( Base ‘ 𝐾 ) ) ) → ( ( ( 𝑅 ‘ ( 𝑈 ‘ 𝐹 ) ) ≤ ( 𝑅 ‘ 𝐹 ) ∧ ( 𝑅 ‘ ( 𝑉 ‘ 𝐹 ) ) ≤ ( 𝑅 ‘ 𝐹 ) ) ↔ ( ( 𝑅 ‘ ( 𝑈 ‘ 𝐹 ) ) ( join ‘ 𝐾 ) ( 𝑅 ‘ ( 𝑉 ‘ 𝐹 ) ) ) ≤ ( 𝑅 ‘ 𝐹 ) ) ) |
41 |
9 17 21 27 40
|
syl13anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ) ∧ 𝐹 ∈ 𝑇 ) → ( ( ( 𝑅 ‘ ( 𝑈 ‘ 𝐹 ) ) ≤ ( 𝑅 ‘ 𝐹 ) ∧ ( 𝑅 ‘ ( 𝑉 ‘ 𝐹 ) ) ≤ ( 𝑅 ‘ 𝐹 ) ) ↔ ( ( 𝑅 ‘ ( 𝑈 ‘ 𝐹 ) ) ( join ‘ 𝐾 ) ( 𝑅 ‘ ( 𝑉 ‘ 𝐹 ) ) ) ≤ ( 𝑅 ‘ 𝐹 ) ) ) |
42 |
37 39 41
|
mpbi2and |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ) ∧ 𝐹 ∈ 𝑇 ) → ( ( 𝑅 ‘ ( 𝑈 ‘ 𝐹 ) ) ( join ‘ 𝐾 ) ( 𝑅 ‘ ( 𝑉 ‘ 𝐹 ) ) ) ≤ ( 𝑅 ‘ 𝐹 ) ) |
43 |
7 5 9 13 24 27 35 42
|
lattrd |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ) ∧ 𝐹 ∈ 𝑇 ) → ( 𝑅 ‘ ( ( 𝑈 𝑃 𝑉 ) ‘ 𝐹 ) ) ≤ ( 𝑅 ‘ 𝐹 ) ) |