Step |
Hyp |
Ref |
Expression |
1 |
|
dicvaddcl.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
2 |
|
dicvaddcl.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
3 |
|
dicvaddcl.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
4 |
|
dicvaddcl.u |
⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) |
5 |
|
dicvaddcl.i |
⊢ 𝐼 = ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 ) |
6 |
|
dicvaddcl.p |
⊢ + = ( +g ‘ 𝑈 ) |
7 |
|
simp1 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑋 ∈ ( 𝐼 ‘ 𝑄 ) ∧ 𝑌 ∈ ( 𝐼 ‘ 𝑄 ) ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
8 |
|
eqid |
⊢ ( Base ‘ 𝑈 ) = ( Base ‘ 𝑈 ) |
9 |
1 2 3 5 4 8
|
dicssdvh |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) → ( 𝐼 ‘ 𝑄 ) ⊆ ( Base ‘ 𝑈 ) ) |
10 |
|
eqid |
⊢ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) |
11 |
|
eqid |
⊢ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) |
12 |
3 10 11 4 8
|
dvhvbase |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( Base ‘ 𝑈 ) = ( ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) × ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ) ) |
13 |
12
|
eqcomd |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) × ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ) = ( Base ‘ 𝑈 ) ) |
14 |
13
|
adantr |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) → ( ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) × ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ) = ( Base ‘ 𝑈 ) ) |
15 |
9 14
|
sseqtrrd |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) → ( 𝐼 ‘ 𝑄 ) ⊆ ( ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) × ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ) ) |
16 |
15
|
3adant3 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑋 ∈ ( 𝐼 ‘ 𝑄 ) ∧ 𝑌 ∈ ( 𝐼 ‘ 𝑄 ) ) ) → ( 𝐼 ‘ 𝑄 ) ⊆ ( ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) × ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ) ) |
17 |
|
simp3l |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑋 ∈ ( 𝐼 ‘ 𝑄 ) ∧ 𝑌 ∈ ( 𝐼 ‘ 𝑄 ) ) ) → 𝑋 ∈ ( 𝐼 ‘ 𝑄 ) ) |
18 |
16 17
|
sseldd |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑋 ∈ ( 𝐼 ‘ 𝑄 ) ∧ 𝑌 ∈ ( 𝐼 ‘ 𝑄 ) ) ) → 𝑋 ∈ ( ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) × ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ) ) |
19 |
|
simp3r |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑋 ∈ ( 𝐼 ‘ 𝑄 ) ∧ 𝑌 ∈ ( 𝐼 ‘ 𝑄 ) ) ) → 𝑌 ∈ ( 𝐼 ‘ 𝑄 ) ) |
20 |
16 19
|
sseldd |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑋 ∈ ( 𝐼 ‘ 𝑄 ) ∧ 𝑌 ∈ ( 𝐼 ‘ 𝑄 ) ) ) → 𝑌 ∈ ( ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) × ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ) ) |
21 |
|
eqid |
⊢ ( Scalar ‘ 𝑈 ) = ( Scalar ‘ 𝑈 ) |
22 |
|
eqid |
⊢ ( +g ‘ ( Scalar ‘ 𝑈 ) ) = ( +g ‘ ( Scalar ‘ 𝑈 ) ) |
23 |
3 10 11 4 21 6 22
|
dvhvadd |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ ( ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) × ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ) ∧ 𝑌 ∈ ( ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) × ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ) ) ) → ( 𝑋 + 𝑌 ) = 〈 ( ( 1st ‘ 𝑋 ) ∘ ( 1st ‘ 𝑌 ) ) , ( ( 2nd ‘ 𝑋 ) ( +g ‘ ( Scalar ‘ 𝑈 ) ) ( 2nd ‘ 𝑌 ) ) 〉 ) |
24 |
7 18 20 23
|
syl12anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑋 ∈ ( 𝐼 ‘ 𝑄 ) ∧ 𝑌 ∈ ( 𝐼 ‘ 𝑄 ) ) ) → ( 𝑋 + 𝑌 ) = 〈 ( ( 1st ‘ 𝑋 ) ∘ ( 1st ‘ 𝑌 ) ) , ( ( 2nd ‘ 𝑋 ) ( +g ‘ ( Scalar ‘ 𝑈 ) ) ( 2nd ‘ 𝑌 ) ) 〉 ) |
25 |
1 2 3 11 5
|
dicelval2nd |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑋 ∈ ( 𝐼 ‘ 𝑄 ) ) → ( 2nd ‘ 𝑋 ) ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ) |
26 |
25
|
3adant3r |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑋 ∈ ( 𝐼 ‘ 𝑄 ) ∧ 𝑌 ∈ ( 𝐼 ‘ 𝑄 ) ) ) → ( 2nd ‘ 𝑋 ) ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ) |
27 |
1 2 3 11 5
|
dicelval2nd |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑌 ∈ ( 𝐼 ‘ 𝑄 ) ) → ( 2nd ‘ 𝑌 ) ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ) |
28 |
27
|
3adant3l |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑋 ∈ ( 𝐼 ‘ 𝑄 ) ∧ 𝑌 ∈ ( 𝐼 ‘ 𝑄 ) ) ) → ( 2nd ‘ 𝑌 ) ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ) |
29 |
|
eqid |
⊢ ( oc ‘ 𝐾 ) = ( oc ‘ 𝐾 ) |
30 |
1 29 2 3
|
lhpocnel |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ∈ 𝐴 ∧ ¬ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ≤ 𝑊 ) ) |
31 |
30
|
3ad2ant1 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑋 ∈ ( 𝐼 ‘ 𝑄 ) ∧ 𝑌 ∈ ( 𝐼 ‘ 𝑄 ) ) ) → ( ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ∈ 𝐴 ∧ ¬ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ≤ 𝑊 ) ) |
32 |
|
simp2 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑋 ∈ ( 𝐼 ‘ 𝑄 ) ∧ 𝑌 ∈ ( 𝐼 ‘ 𝑄 ) ) ) → ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) |
33 |
|
eqid |
⊢ ( ℩ 𝑔 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ( 𝑔 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) = 𝑄 ) = ( ℩ 𝑔 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ( 𝑔 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) = 𝑄 ) |
34 |
1 2 3 10 33
|
ltrniotacl |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ∈ 𝐴 ∧ ¬ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) → ( ℩ 𝑔 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ( 𝑔 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) = 𝑄 ) ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) |
35 |
7 31 32 34
|
syl3anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑋 ∈ ( 𝐼 ‘ 𝑄 ) ∧ 𝑌 ∈ ( 𝐼 ‘ 𝑄 ) ) ) → ( ℩ 𝑔 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ( 𝑔 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) = 𝑄 ) ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) |
36 |
|
eqid |
⊢ ( 𝑠 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) , 𝑡 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( ℎ ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( ( 𝑠 ‘ ℎ ) ∘ ( 𝑡 ‘ ℎ ) ) ) ) = ( 𝑠 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) , 𝑡 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( ℎ ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( ( 𝑠 ‘ ℎ ) ∘ ( 𝑡 ‘ ℎ ) ) ) ) |
37 |
10 36
|
tendospdi2 |
⊢ ( ( ( 2nd ‘ 𝑋 ) ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ∧ ( 2nd ‘ 𝑌 ) ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ∧ ( ℩ 𝑔 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ( 𝑔 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) = 𝑄 ) ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) → ( ( ( 2nd ‘ 𝑋 ) ( 𝑠 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) , 𝑡 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( ℎ ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( ( 𝑠 ‘ ℎ ) ∘ ( 𝑡 ‘ ℎ ) ) ) ) ( 2nd ‘ 𝑌 ) ) ‘ ( ℩ 𝑔 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ( 𝑔 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) = 𝑄 ) ) = ( ( ( 2nd ‘ 𝑋 ) ‘ ( ℩ 𝑔 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ( 𝑔 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) = 𝑄 ) ) ∘ ( ( 2nd ‘ 𝑌 ) ‘ ( ℩ 𝑔 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ( 𝑔 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) = 𝑄 ) ) ) ) |
38 |
26 28 35 37
|
syl3anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑋 ∈ ( 𝐼 ‘ 𝑄 ) ∧ 𝑌 ∈ ( 𝐼 ‘ 𝑄 ) ) ) → ( ( ( 2nd ‘ 𝑋 ) ( 𝑠 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) , 𝑡 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( ℎ ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( ( 𝑠 ‘ ℎ ) ∘ ( 𝑡 ‘ ℎ ) ) ) ) ( 2nd ‘ 𝑌 ) ) ‘ ( ℩ 𝑔 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ( 𝑔 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) = 𝑄 ) ) = ( ( ( 2nd ‘ 𝑋 ) ‘ ( ℩ 𝑔 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ( 𝑔 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) = 𝑄 ) ) ∘ ( ( 2nd ‘ 𝑌 ) ‘ ( ℩ 𝑔 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ( 𝑔 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) = 𝑄 ) ) ) ) |
39 |
3 10 11 4 21 36 22
|
dvhfplusr |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( +g ‘ ( Scalar ‘ 𝑈 ) ) = ( 𝑠 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) , 𝑡 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( ℎ ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( ( 𝑠 ‘ ℎ ) ∘ ( 𝑡 ‘ ℎ ) ) ) ) ) |
40 |
39
|
3ad2ant1 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑋 ∈ ( 𝐼 ‘ 𝑄 ) ∧ 𝑌 ∈ ( 𝐼 ‘ 𝑄 ) ) ) → ( +g ‘ ( Scalar ‘ 𝑈 ) ) = ( 𝑠 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) , 𝑡 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( ℎ ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( ( 𝑠 ‘ ℎ ) ∘ ( 𝑡 ‘ ℎ ) ) ) ) ) |
41 |
40
|
oveqd |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑋 ∈ ( 𝐼 ‘ 𝑄 ) ∧ 𝑌 ∈ ( 𝐼 ‘ 𝑄 ) ) ) → ( ( 2nd ‘ 𝑋 ) ( +g ‘ ( Scalar ‘ 𝑈 ) ) ( 2nd ‘ 𝑌 ) ) = ( ( 2nd ‘ 𝑋 ) ( 𝑠 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) , 𝑡 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( ℎ ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( ( 𝑠 ‘ ℎ ) ∘ ( 𝑡 ‘ ℎ ) ) ) ) ( 2nd ‘ 𝑌 ) ) ) |
42 |
41
|
fveq1d |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑋 ∈ ( 𝐼 ‘ 𝑄 ) ∧ 𝑌 ∈ ( 𝐼 ‘ 𝑄 ) ) ) → ( ( ( 2nd ‘ 𝑋 ) ( +g ‘ ( Scalar ‘ 𝑈 ) ) ( 2nd ‘ 𝑌 ) ) ‘ ( ℩ 𝑔 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ( 𝑔 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) = 𝑄 ) ) = ( ( ( 2nd ‘ 𝑋 ) ( 𝑠 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) , 𝑡 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( ℎ ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( ( 𝑠 ‘ ℎ ) ∘ ( 𝑡 ‘ ℎ ) ) ) ) ( 2nd ‘ 𝑌 ) ) ‘ ( ℩ 𝑔 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ( 𝑔 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) = 𝑄 ) ) ) |
43 |
|
eqid |
⊢ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) = ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) |
44 |
1 2 3 43 10 5
|
dicelval1sta |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑋 ∈ ( 𝐼 ‘ 𝑄 ) ) → ( 1st ‘ 𝑋 ) = ( ( 2nd ‘ 𝑋 ) ‘ ( ℩ 𝑔 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ( 𝑔 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) = 𝑄 ) ) ) |
45 |
44
|
3adant3r |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑋 ∈ ( 𝐼 ‘ 𝑄 ) ∧ 𝑌 ∈ ( 𝐼 ‘ 𝑄 ) ) ) → ( 1st ‘ 𝑋 ) = ( ( 2nd ‘ 𝑋 ) ‘ ( ℩ 𝑔 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ( 𝑔 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) = 𝑄 ) ) ) |
46 |
1 2 3 43 10 5
|
dicelval1sta |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑌 ∈ ( 𝐼 ‘ 𝑄 ) ) → ( 1st ‘ 𝑌 ) = ( ( 2nd ‘ 𝑌 ) ‘ ( ℩ 𝑔 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ( 𝑔 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) = 𝑄 ) ) ) |
47 |
46
|
3adant3l |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑋 ∈ ( 𝐼 ‘ 𝑄 ) ∧ 𝑌 ∈ ( 𝐼 ‘ 𝑄 ) ) ) → ( 1st ‘ 𝑌 ) = ( ( 2nd ‘ 𝑌 ) ‘ ( ℩ 𝑔 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ( 𝑔 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) = 𝑄 ) ) ) |
48 |
45 47
|
coeq12d |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑋 ∈ ( 𝐼 ‘ 𝑄 ) ∧ 𝑌 ∈ ( 𝐼 ‘ 𝑄 ) ) ) → ( ( 1st ‘ 𝑋 ) ∘ ( 1st ‘ 𝑌 ) ) = ( ( ( 2nd ‘ 𝑋 ) ‘ ( ℩ 𝑔 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ( 𝑔 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) = 𝑄 ) ) ∘ ( ( 2nd ‘ 𝑌 ) ‘ ( ℩ 𝑔 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ( 𝑔 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) = 𝑄 ) ) ) ) |
49 |
38 42 48
|
3eqtr4rd |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑋 ∈ ( 𝐼 ‘ 𝑄 ) ∧ 𝑌 ∈ ( 𝐼 ‘ 𝑄 ) ) ) → ( ( 1st ‘ 𝑋 ) ∘ ( 1st ‘ 𝑌 ) ) = ( ( ( 2nd ‘ 𝑋 ) ( +g ‘ ( Scalar ‘ 𝑈 ) ) ( 2nd ‘ 𝑌 ) ) ‘ ( ℩ 𝑔 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ( 𝑔 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) = 𝑄 ) ) ) |
50 |
3 10 11 36
|
tendoplcl |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 2nd ‘ 𝑋 ) ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ∧ ( 2nd ‘ 𝑌 ) ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ) → ( ( 2nd ‘ 𝑋 ) ( 𝑠 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) , 𝑡 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( ℎ ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( ( 𝑠 ‘ ℎ ) ∘ ( 𝑡 ‘ ℎ ) ) ) ) ( 2nd ‘ 𝑌 ) ) ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ) |
51 |
7 26 28 50
|
syl3anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑋 ∈ ( 𝐼 ‘ 𝑄 ) ∧ 𝑌 ∈ ( 𝐼 ‘ 𝑄 ) ) ) → ( ( 2nd ‘ 𝑋 ) ( 𝑠 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) , 𝑡 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( ℎ ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( ( 𝑠 ‘ ℎ ) ∘ ( 𝑡 ‘ ℎ ) ) ) ) ( 2nd ‘ 𝑌 ) ) ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ) |
52 |
41 51
|
eqeltrd |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑋 ∈ ( 𝐼 ‘ 𝑄 ) ∧ 𝑌 ∈ ( 𝐼 ‘ 𝑄 ) ) ) → ( ( 2nd ‘ 𝑋 ) ( +g ‘ ( Scalar ‘ 𝑈 ) ) ( 2nd ‘ 𝑌 ) ) ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ) |
53 |
|
fvex |
⊢ ( 1st ‘ 𝑋 ) ∈ V |
54 |
|
fvex |
⊢ ( 1st ‘ 𝑌 ) ∈ V |
55 |
53 54
|
coex |
⊢ ( ( 1st ‘ 𝑋 ) ∘ ( 1st ‘ 𝑌 ) ) ∈ V |
56 |
|
ovex |
⊢ ( ( 2nd ‘ 𝑋 ) ( +g ‘ ( Scalar ‘ 𝑈 ) ) ( 2nd ‘ 𝑌 ) ) ∈ V |
57 |
1 2 3 43 10 11 5 55 56
|
dicopelval |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) → ( 〈 ( ( 1st ‘ 𝑋 ) ∘ ( 1st ‘ 𝑌 ) ) , ( ( 2nd ‘ 𝑋 ) ( +g ‘ ( Scalar ‘ 𝑈 ) ) ( 2nd ‘ 𝑌 ) ) 〉 ∈ ( 𝐼 ‘ 𝑄 ) ↔ ( ( ( 1st ‘ 𝑋 ) ∘ ( 1st ‘ 𝑌 ) ) = ( ( ( 2nd ‘ 𝑋 ) ( +g ‘ ( Scalar ‘ 𝑈 ) ) ( 2nd ‘ 𝑌 ) ) ‘ ( ℩ 𝑔 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ( 𝑔 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) = 𝑄 ) ) ∧ ( ( 2nd ‘ 𝑋 ) ( +g ‘ ( Scalar ‘ 𝑈 ) ) ( 2nd ‘ 𝑌 ) ) ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ) ) ) |
58 |
57
|
3adant3 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑋 ∈ ( 𝐼 ‘ 𝑄 ) ∧ 𝑌 ∈ ( 𝐼 ‘ 𝑄 ) ) ) → ( 〈 ( ( 1st ‘ 𝑋 ) ∘ ( 1st ‘ 𝑌 ) ) , ( ( 2nd ‘ 𝑋 ) ( +g ‘ ( Scalar ‘ 𝑈 ) ) ( 2nd ‘ 𝑌 ) ) 〉 ∈ ( 𝐼 ‘ 𝑄 ) ↔ ( ( ( 1st ‘ 𝑋 ) ∘ ( 1st ‘ 𝑌 ) ) = ( ( ( 2nd ‘ 𝑋 ) ( +g ‘ ( Scalar ‘ 𝑈 ) ) ( 2nd ‘ 𝑌 ) ) ‘ ( ℩ 𝑔 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ( 𝑔 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) = 𝑄 ) ) ∧ ( ( 2nd ‘ 𝑋 ) ( +g ‘ ( Scalar ‘ 𝑈 ) ) ( 2nd ‘ 𝑌 ) ) ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ) ) ) |
59 |
49 52 58
|
mpbir2and |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑋 ∈ ( 𝐼 ‘ 𝑄 ) ∧ 𝑌 ∈ ( 𝐼 ‘ 𝑄 ) ) ) → 〈 ( ( 1st ‘ 𝑋 ) ∘ ( 1st ‘ 𝑌 ) ) , ( ( 2nd ‘ 𝑋 ) ( +g ‘ ( Scalar ‘ 𝑈 ) ) ( 2nd ‘ 𝑌 ) ) 〉 ∈ ( 𝐼 ‘ 𝑄 ) ) |
60 |
24 59
|
eqeltrd |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑋 ∈ ( 𝐼 ‘ 𝑄 ) ∧ 𝑌 ∈ ( 𝐼 ‘ 𝑄 ) ) ) → ( 𝑋 + 𝑌 ) ∈ ( 𝐼 ‘ 𝑄 ) ) |