Step |
Hyp |
Ref |
Expression |
1 |
|
hgmapadd.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
2 |
|
hgmapadd.u |
⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) |
3 |
|
hgmapadd.r |
⊢ 𝑅 = ( Scalar ‘ 𝑈 ) |
4 |
|
hgmapadd.b |
⊢ 𝐵 = ( Base ‘ 𝑅 ) |
5 |
|
hgmapadd.p |
⊢ + = ( +g ‘ 𝑅 ) |
6 |
|
hgmapadd.g |
⊢ 𝐺 = ( ( HGMap ‘ 𝐾 ) ‘ 𝑊 ) |
7 |
|
hgmapadd.k |
⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
8 |
|
hgmapadd.x |
⊢ ( 𝜑 → 𝑋 ∈ 𝐵 ) |
9 |
|
hgmapadd.y |
⊢ ( 𝜑 → 𝑌 ∈ 𝐵 ) |
10 |
|
eqid |
⊢ ( Base ‘ 𝑈 ) = ( Base ‘ 𝑈 ) |
11 |
|
eqid |
⊢ ( 0g ‘ 𝑈 ) = ( 0g ‘ 𝑈 ) |
12 |
1 2 10 11 7
|
dvh1dim |
⊢ ( 𝜑 → ∃ 𝑡 ∈ ( Base ‘ 𝑈 ) 𝑡 ≠ ( 0g ‘ 𝑈 ) ) |
13 |
|
eqid |
⊢ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) |
14 |
1 13 7
|
lcdlmod |
⊢ ( 𝜑 → ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ∈ LMod ) |
15 |
14
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( Base ‘ 𝑈 ) ∧ 𝑡 ≠ ( 0g ‘ 𝑈 ) ) → ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ∈ LMod ) |
16 |
|
eqid |
⊢ ( Scalar ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) = ( Scalar ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) |
17 |
|
eqid |
⊢ ( Base ‘ ( Scalar ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ) = ( Base ‘ ( Scalar ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ) |
18 |
7
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( Base ‘ 𝑈 ) ∧ 𝑡 ≠ ( 0g ‘ 𝑈 ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
19 |
8
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( Base ‘ 𝑈 ) ∧ 𝑡 ≠ ( 0g ‘ 𝑈 ) ) → 𝑋 ∈ 𝐵 ) |
20 |
1 2 3 4 13 16 17 6 18 19
|
hgmapdcl |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( Base ‘ 𝑈 ) ∧ 𝑡 ≠ ( 0g ‘ 𝑈 ) ) → ( 𝐺 ‘ 𝑋 ) ∈ ( Base ‘ ( Scalar ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ) ) |
21 |
1 2 3 4 13 16 17 6 7 9
|
hgmapdcl |
⊢ ( 𝜑 → ( 𝐺 ‘ 𝑌 ) ∈ ( Base ‘ ( Scalar ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ) ) |
22 |
21
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( Base ‘ 𝑈 ) ∧ 𝑡 ≠ ( 0g ‘ 𝑈 ) ) → ( 𝐺 ‘ 𝑌 ) ∈ ( Base ‘ ( Scalar ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ) ) |
23 |
|
eqid |
⊢ ( Base ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) = ( Base ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) |
24 |
|
eqid |
⊢ ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) = ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) |
25 |
|
simp2 |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( Base ‘ 𝑈 ) ∧ 𝑡 ≠ ( 0g ‘ 𝑈 ) ) → 𝑡 ∈ ( Base ‘ 𝑈 ) ) |
26 |
1 2 10 13 23 24 18 25
|
hdmapcl |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( Base ‘ 𝑈 ) ∧ 𝑡 ≠ ( 0g ‘ 𝑈 ) ) → ( ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑡 ) ∈ ( Base ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ) |
27 |
|
eqid |
⊢ ( +g ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) = ( +g ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) |
28 |
|
eqid |
⊢ ( ·𝑠 ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) = ( ·𝑠 ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) |
29 |
|
eqid |
⊢ ( +g ‘ ( Scalar ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ) = ( +g ‘ ( Scalar ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ) |
30 |
23 27 16 28 17 29
|
lmodvsdir |
⊢ ( ( ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ∈ LMod ∧ ( ( 𝐺 ‘ 𝑋 ) ∈ ( Base ‘ ( Scalar ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ) ∧ ( 𝐺 ‘ 𝑌 ) ∈ ( Base ‘ ( Scalar ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ) ∧ ( ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑡 ) ∈ ( Base ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ) ) → ( ( ( 𝐺 ‘ 𝑋 ) ( +g ‘ ( Scalar ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ) ( 𝐺 ‘ 𝑌 ) ) ( ·𝑠 ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ( ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑡 ) ) = ( ( ( 𝐺 ‘ 𝑋 ) ( ·𝑠 ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ( ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑡 ) ) ( +g ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ( ( 𝐺 ‘ 𝑌 ) ( ·𝑠 ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ( ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑡 ) ) ) ) |
31 |
15 20 22 26 30
|
syl13anc |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( Base ‘ 𝑈 ) ∧ 𝑡 ≠ ( 0g ‘ 𝑈 ) ) → ( ( ( 𝐺 ‘ 𝑋 ) ( +g ‘ ( Scalar ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ) ( 𝐺 ‘ 𝑌 ) ) ( ·𝑠 ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ( ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑡 ) ) = ( ( ( 𝐺 ‘ 𝑋 ) ( ·𝑠 ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ( ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑡 ) ) ( +g ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ( ( 𝐺 ‘ 𝑌 ) ( ·𝑠 ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ( ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑡 ) ) ) ) |
32 |
1 2 7
|
dvhlmod |
⊢ ( 𝜑 → 𝑈 ∈ LMod ) |
33 |
32
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( Base ‘ 𝑈 ) ∧ 𝑡 ≠ ( 0g ‘ 𝑈 ) ) → 𝑈 ∈ LMod ) |
34 |
9
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( Base ‘ 𝑈 ) ∧ 𝑡 ≠ ( 0g ‘ 𝑈 ) ) → 𝑌 ∈ 𝐵 ) |
35 |
|
eqid |
⊢ ( +g ‘ 𝑈 ) = ( +g ‘ 𝑈 ) |
36 |
|
eqid |
⊢ ( ·𝑠 ‘ 𝑈 ) = ( ·𝑠 ‘ 𝑈 ) |
37 |
10 35 3 36 4 5
|
lmodvsdir |
⊢ ( ( 𝑈 ∈ LMod ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑡 ∈ ( Base ‘ 𝑈 ) ) ) → ( ( 𝑋 + 𝑌 ) ( ·𝑠 ‘ 𝑈 ) 𝑡 ) = ( ( 𝑋 ( ·𝑠 ‘ 𝑈 ) 𝑡 ) ( +g ‘ 𝑈 ) ( 𝑌 ( ·𝑠 ‘ 𝑈 ) 𝑡 ) ) ) |
38 |
33 19 34 25 37
|
syl13anc |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( Base ‘ 𝑈 ) ∧ 𝑡 ≠ ( 0g ‘ 𝑈 ) ) → ( ( 𝑋 + 𝑌 ) ( ·𝑠 ‘ 𝑈 ) 𝑡 ) = ( ( 𝑋 ( ·𝑠 ‘ 𝑈 ) 𝑡 ) ( +g ‘ 𝑈 ) ( 𝑌 ( ·𝑠 ‘ 𝑈 ) 𝑡 ) ) ) |
39 |
38
|
fveq2d |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( Base ‘ 𝑈 ) ∧ 𝑡 ≠ ( 0g ‘ 𝑈 ) ) → ( ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( ( 𝑋 + 𝑌 ) ( ·𝑠 ‘ 𝑈 ) 𝑡 ) ) = ( ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( ( 𝑋 ( ·𝑠 ‘ 𝑈 ) 𝑡 ) ( +g ‘ 𝑈 ) ( 𝑌 ( ·𝑠 ‘ 𝑈 ) 𝑡 ) ) ) ) |
40 |
10 3 36 4
|
lmodvscl |
⊢ ( ( 𝑈 ∈ LMod ∧ 𝑋 ∈ 𝐵 ∧ 𝑡 ∈ ( Base ‘ 𝑈 ) ) → ( 𝑋 ( ·𝑠 ‘ 𝑈 ) 𝑡 ) ∈ ( Base ‘ 𝑈 ) ) |
41 |
33 19 25 40
|
syl3anc |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( Base ‘ 𝑈 ) ∧ 𝑡 ≠ ( 0g ‘ 𝑈 ) ) → ( 𝑋 ( ·𝑠 ‘ 𝑈 ) 𝑡 ) ∈ ( Base ‘ 𝑈 ) ) |
42 |
10 3 36 4
|
lmodvscl |
⊢ ( ( 𝑈 ∈ LMod ∧ 𝑌 ∈ 𝐵 ∧ 𝑡 ∈ ( Base ‘ 𝑈 ) ) → ( 𝑌 ( ·𝑠 ‘ 𝑈 ) 𝑡 ) ∈ ( Base ‘ 𝑈 ) ) |
43 |
33 34 25 42
|
syl3anc |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( Base ‘ 𝑈 ) ∧ 𝑡 ≠ ( 0g ‘ 𝑈 ) ) → ( 𝑌 ( ·𝑠 ‘ 𝑈 ) 𝑡 ) ∈ ( Base ‘ 𝑈 ) ) |
44 |
1 2 10 35 13 27 24 18 41 43
|
hdmapadd |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( Base ‘ 𝑈 ) ∧ 𝑡 ≠ ( 0g ‘ 𝑈 ) ) → ( ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( ( 𝑋 ( ·𝑠 ‘ 𝑈 ) 𝑡 ) ( +g ‘ 𝑈 ) ( 𝑌 ( ·𝑠 ‘ 𝑈 ) 𝑡 ) ) ) = ( ( ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑋 ( ·𝑠 ‘ 𝑈 ) 𝑡 ) ) ( +g ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ( ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑌 ( ·𝑠 ‘ 𝑈 ) 𝑡 ) ) ) ) |
45 |
1 2 10 36 3 4 13 28 24 6 18 25 19
|
hgmapvs |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( Base ‘ 𝑈 ) ∧ 𝑡 ≠ ( 0g ‘ 𝑈 ) ) → ( ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑋 ( ·𝑠 ‘ 𝑈 ) 𝑡 ) ) = ( ( 𝐺 ‘ 𝑋 ) ( ·𝑠 ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ( ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑡 ) ) ) |
46 |
1 2 10 36 3 4 13 28 24 6 18 25 34
|
hgmapvs |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( Base ‘ 𝑈 ) ∧ 𝑡 ≠ ( 0g ‘ 𝑈 ) ) → ( ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑌 ( ·𝑠 ‘ 𝑈 ) 𝑡 ) ) = ( ( 𝐺 ‘ 𝑌 ) ( ·𝑠 ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ( ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑡 ) ) ) |
47 |
45 46
|
oveq12d |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( Base ‘ 𝑈 ) ∧ 𝑡 ≠ ( 0g ‘ 𝑈 ) ) → ( ( ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑋 ( ·𝑠 ‘ 𝑈 ) 𝑡 ) ) ( +g ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ( ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑌 ( ·𝑠 ‘ 𝑈 ) 𝑡 ) ) ) = ( ( ( 𝐺 ‘ 𝑋 ) ( ·𝑠 ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ( ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑡 ) ) ( +g ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ( ( 𝐺 ‘ 𝑌 ) ( ·𝑠 ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ( ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑡 ) ) ) ) |
48 |
39 44 47
|
3eqtrrd |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( Base ‘ 𝑈 ) ∧ 𝑡 ≠ ( 0g ‘ 𝑈 ) ) → ( ( ( 𝐺 ‘ 𝑋 ) ( ·𝑠 ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ( ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑡 ) ) ( +g ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ( ( 𝐺 ‘ 𝑌 ) ( ·𝑠 ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ( ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑡 ) ) ) = ( ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( ( 𝑋 + 𝑌 ) ( ·𝑠 ‘ 𝑈 ) 𝑡 ) ) ) |
49 |
3 4 5
|
lmodacl |
⊢ ( ( 𝑈 ∈ LMod ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 + 𝑌 ) ∈ 𝐵 ) |
50 |
32 8 9 49
|
syl3anc |
⊢ ( 𝜑 → ( 𝑋 + 𝑌 ) ∈ 𝐵 ) |
51 |
50
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( Base ‘ 𝑈 ) ∧ 𝑡 ≠ ( 0g ‘ 𝑈 ) ) → ( 𝑋 + 𝑌 ) ∈ 𝐵 ) |
52 |
1 2 10 36 3 4 13 28 24 6 18 25 51
|
hgmapvs |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( Base ‘ 𝑈 ) ∧ 𝑡 ≠ ( 0g ‘ 𝑈 ) ) → ( ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( ( 𝑋 + 𝑌 ) ( ·𝑠 ‘ 𝑈 ) 𝑡 ) ) = ( ( 𝐺 ‘ ( 𝑋 + 𝑌 ) ) ( ·𝑠 ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ( ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑡 ) ) ) |
53 |
31 48 52
|
3eqtrrd |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( Base ‘ 𝑈 ) ∧ 𝑡 ≠ ( 0g ‘ 𝑈 ) ) → ( ( 𝐺 ‘ ( 𝑋 + 𝑌 ) ) ( ·𝑠 ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ( ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑡 ) ) = ( ( ( 𝐺 ‘ 𝑋 ) ( +g ‘ ( Scalar ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ) ( 𝐺 ‘ 𝑌 ) ) ( ·𝑠 ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ( ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑡 ) ) ) |
54 |
|
eqid |
⊢ ( 0g ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) = ( 0g ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) |
55 |
1 13 7
|
lcdlvec |
⊢ ( 𝜑 → ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ∈ LVec ) |
56 |
55
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( Base ‘ 𝑈 ) ∧ 𝑡 ≠ ( 0g ‘ 𝑈 ) ) → ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ∈ LVec ) |
57 |
1 2 3 4 13 16 17 6 7 50
|
hgmapdcl |
⊢ ( 𝜑 → ( 𝐺 ‘ ( 𝑋 + 𝑌 ) ) ∈ ( Base ‘ ( Scalar ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ) ) |
58 |
57
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( Base ‘ 𝑈 ) ∧ 𝑡 ≠ ( 0g ‘ 𝑈 ) ) → ( 𝐺 ‘ ( 𝑋 + 𝑌 ) ) ∈ ( Base ‘ ( Scalar ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ) ) |
59 |
1 2 3 4 13 16 17 6 7 8
|
hgmapdcl |
⊢ ( 𝜑 → ( 𝐺 ‘ 𝑋 ) ∈ ( Base ‘ ( Scalar ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ) ) |
60 |
16 17 29
|
lmodacl |
⊢ ( ( ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ∈ LMod ∧ ( 𝐺 ‘ 𝑋 ) ∈ ( Base ‘ ( Scalar ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ) ∧ ( 𝐺 ‘ 𝑌 ) ∈ ( Base ‘ ( Scalar ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ) ) → ( ( 𝐺 ‘ 𝑋 ) ( +g ‘ ( Scalar ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ) ( 𝐺 ‘ 𝑌 ) ) ∈ ( Base ‘ ( Scalar ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ) ) |
61 |
14 59 21 60
|
syl3anc |
⊢ ( 𝜑 → ( ( 𝐺 ‘ 𝑋 ) ( +g ‘ ( Scalar ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ) ( 𝐺 ‘ 𝑌 ) ) ∈ ( Base ‘ ( Scalar ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ) ) |
62 |
61
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( Base ‘ 𝑈 ) ∧ 𝑡 ≠ ( 0g ‘ 𝑈 ) ) → ( ( 𝐺 ‘ 𝑋 ) ( +g ‘ ( Scalar ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ) ( 𝐺 ‘ 𝑌 ) ) ∈ ( Base ‘ ( Scalar ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ) ) |
63 |
|
simp3 |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( Base ‘ 𝑈 ) ∧ 𝑡 ≠ ( 0g ‘ 𝑈 ) ) → 𝑡 ≠ ( 0g ‘ 𝑈 ) ) |
64 |
1 2 10 11 13 54 24 18 25
|
hdmapeq0 |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( Base ‘ 𝑈 ) ∧ 𝑡 ≠ ( 0g ‘ 𝑈 ) ) → ( ( ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑡 ) = ( 0g ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ↔ 𝑡 = ( 0g ‘ 𝑈 ) ) ) |
65 |
64
|
necon3bid |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( Base ‘ 𝑈 ) ∧ 𝑡 ≠ ( 0g ‘ 𝑈 ) ) → ( ( ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑡 ) ≠ ( 0g ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ↔ 𝑡 ≠ ( 0g ‘ 𝑈 ) ) ) |
66 |
63 65
|
mpbird |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( Base ‘ 𝑈 ) ∧ 𝑡 ≠ ( 0g ‘ 𝑈 ) ) → ( ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑡 ) ≠ ( 0g ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ) |
67 |
23 28 16 17 54 56 58 62 26 66
|
lvecvscan2 |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( Base ‘ 𝑈 ) ∧ 𝑡 ≠ ( 0g ‘ 𝑈 ) ) → ( ( ( 𝐺 ‘ ( 𝑋 + 𝑌 ) ) ( ·𝑠 ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ( ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑡 ) ) = ( ( ( 𝐺 ‘ 𝑋 ) ( +g ‘ ( Scalar ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ) ( 𝐺 ‘ 𝑌 ) ) ( ·𝑠 ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ( ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑡 ) ) ↔ ( 𝐺 ‘ ( 𝑋 + 𝑌 ) ) = ( ( 𝐺 ‘ 𝑋 ) ( +g ‘ ( Scalar ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ) ( 𝐺 ‘ 𝑌 ) ) ) ) |
68 |
53 67
|
mpbid |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( Base ‘ 𝑈 ) ∧ 𝑡 ≠ ( 0g ‘ 𝑈 ) ) → ( 𝐺 ‘ ( 𝑋 + 𝑌 ) ) = ( ( 𝐺 ‘ 𝑋 ) ( +g ‘ ( Scalar ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ) ( 𝐺 ‘ 𝑌 ) ) ) |
69 |
68
|
rexlimdv3a |
⊢ ( 𝜑 → ( ∃ 𝑡 ∈ ( Base ‘ 𝑈 ) 𝑡 ≠ ( 0g ‘ 𝑈 ) → ( 𝐺 ‘ ( 𝑋 + 𝑌 ) ) = ( ( 𝐺 ‘ 𝑋 ) ( +g ‘ ( Scalar ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ) ( 𝐺 ‘ 𝑌 ) ) ) ) |
70 |
12 69
|
mpd |
⊢ ( 𝜑 → ( 𝐺 ‘ ( 𝑋 + 𝑌 ) ) = ( ( 𝐺 ‘ 𝑋 ) ( +g ‘ ( Scalar ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ) ( 𝐺 ‘ 𝑌 ) ) ) |
71 |
1 2 3 5 13 16 29 7
|
lcdsadd |
⊢ ( 𝜑 → ( +g ‘ ( Scalar ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ) = + ) |
72 |
71
|
oveqd |
⊢ ( 𝜑 → ( ( 𝐺 ‘ 𝑋 ) ( +g ‘ ( Scalar ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ) ( 𝐺 ‘ 𝑌 ) ) = ( ( 𝐺 ‘ 𝑋 ) + ( 𝐺 ‘ 𝑌 ) ) ) |
73 |
70 72
|
eqtrd |
⊢ ( 𝜑 → ( 𝐺 ‘ ( 𝑋 + 𝑌 ) ) = ( ( 𝐺 ‘ 𝑋 ) + ( 𝐺 ‘ 𝑌 ) ) ) |