Step |
Hyp |
Ref |
Expression |
1 |
|
hgmapmul.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
2 |
|
hgmapmul.u |
⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) |
3 |
|
hgmapmul.r |
⊢ 𝑅 = ( Scalar ‘ 𝑈 ) |
4 |
|
hgmapmul.b |
⊢ 𝐵 = ( Base ‘ 𝑅 ) |
5 |
|
hgmapmul.t |
⊢ · = ( .r ‘ 𝑅 ) |
6 |
|
hgmapmul.g |
⊢ 𝐺 = ( ( HGMap ‘ 𝐾 ) ‘ 𝑊 ) |
7 |
|
hgmapmul.k |
⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
8 |
|
hgmapmul.x |
⊢ ( 𝜑 → 𝑋 ∈ 𝐵 ) |
9 |
|
hgmapmul.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 |
1 2 3 4 13 16 17 6 7 8
|
hgmapdcl |
⊢ ( 𝜑 → ( 𝐺 ‘ 𝑋 ) ∈ ( Base ‘ ( Scalar ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ) ) |
19 |
18
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( Base ‘ 𝑈 ) ∧ 𝑡 ≠ ( 0g ‘ 𝑈 ) ) → ( 𝐺 ‘ 𝑋 ) ∈ ( Base ‘ ( Scalar ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ) ) |
20 |
1 2 3 4 13 16 17 6 7 9
|
hgmapdcl |
⊢ ( 𝜑 → ( 𝐺 ‘ 𝑌 ) ∈ ( Base ‘ ( Scalar ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ) ) |
21 |
20
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( Base ‘ 𝑈 ) ∧ 𝑡 ≠ ( 0g ‘ 𝑈 ) ) → ( 𝐺 ‘ 𝑌 ) ∈ ( Base ‘ ( Scalar ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ) ) |
22 |
|
eqid |
⊢ ( Base ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) = ( Base ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) |
23 |
|
eqid |
⊢ ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) = ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) |
24 |
7
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( Base ‘ 𝑈 ) ∧ 𝑡 ≠ ( 0g ‘ 𝑈 ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
25 |
|
simp2 |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( Base ‘ 𝑈 ) ∧ 𝑡 ≠ ( 0g ‘ 𝑈 ) ) → 𝑡 ∈ ( Base ‘ 𝑈 ) ) |
26 |
1 2 10 13 22 23 24 25
|
hdmapcl |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( Base ‘ 𝑈 ) ∧ 𝑡 ≠ ( 0g ‘ 𝑈 ) ) → ( ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑡 ) ∈ ( Base ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ) |
27 |
|
eqid |
⊢ ( ·𝑠 ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) = ( ·𝑠 ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) |
28 |
|
eqid |
⊢ ( .r ‘ ( Scalar ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ) = ( .r ‘ ( Scalar ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ) |
29 |
22 16 27 17 28
|
lmodvsass |
⊢ ( ( ( ( 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 ) ( .r ‘ ( Scalar ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ) ( 𝐺 ‘ 𝑌 ) ) ( ·𝑠 ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ( ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑡 ) ) = ( ( 𝐺 ‘ 𝑋 ) ( ·𝑠 ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ( ( 𝐺 ‘ 𝑌 ) ( ·𝑠 ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ( ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑡 ) ) ) ) |
30 |
15 19 21 26 29
|
syl13anc |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( Base ‘ 𝑈 ) ∧ 𝑡 ≠ ( 0g ‘ 𝑈 ) ) → ( ( ( 𝐺 ‘ 𝑋 ) ( .r ‘ ( Scalar ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ) ( 𝐺 ‘ 𝑌 ) ) ( ·𝑠 ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ( ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑡 ) ) = ( ( 𝐺 ‘ 𝑋 ) ( ·𝑠 ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ( ( 𝐺 ‘ 𝑌 ) ( ·𝑠 ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ( ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑡 ) ) ) ) |
31 |
1 2 7
|
dvhlmod |
⊢ ( 𝜑 → 𝑈 ∈ LMod ) |
32 |
31
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( Base ‘ 𝑈 ) ∧ 𝑡 ≠ ( 0g ‘ 𝑈 ) ) → 𝑈 ∈ LMod ) |
33 |
8
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( Base ‘ 𝑈 ) ∧ 𝑡 ≠ ( 0g ‘ 𝑈 ) ) → 𝑋 ∈ 𝐵 ) |
34 |
9
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( Base ‘ 𝑈 ) ∧ 𝑡 ≠ ( 0g ‘ 𝑈 ) ) → 𝑌 ∈ 𝐵 ) |
35 |
|
eqid |
⊢ ( ·𝑠 ‘ 𝑈 ) = ( ·𝑠 ‘ 𝑈 ) |
36 |
10 3 35 4 5
|
lmodvsass |
⊢ ( ( 𝑈 ∈ LMod ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑡 ∈ ( Base ‘ 𝑈 ) ) ) → ( ( 𝑋 · 𝑌 ) ( ·𝑠 ‘ 𝑈 ) 𝑡 ) = ( 𝑋 ( ·𝑠 ‘ 𝑈 ) ( 𝑌 ( ·𝑠 ‘ 𝑈 ) 𝑡 ) ) ) |
37 |
32 33 34 25 36
|
syl13anc |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( Base ‘ 𝑈 ) ∧ 𝑡 ≠ ( 0g ‘ 𝑈 ) ) → ( ( 𝑋 · 𝑌 ) ( ·𝑠 ‘ 𝑈 ) 𝑡 ) = ( 𝑋 ( ·𝑠 ‘ 𝑈 ) ( 𝑌 ( ·𝑠 ‘ 𝑈 ) 𝑡 ) ) ) |
38 |
37
|
fveq2d |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( Base ‘ 𝑈 ) ∧ 𝑡 ≠ ( 0g ‘ 𝑈 ) ) → ( ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( ( 𝑋 · 𝑌 ) ( ·𝑠 ‘ 𝑈 ) 𝑡 ) ) = ( ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑋 ( ·𝑠 ‘ 𝑈 ) ( 𝑌 ( ·𝑠 ‘ 𝑈 ) 𝑡 ) ) ) ) |
39 |
10 3 35 4
|
lmodvscl |
⊢ ( ( 𝑈 ∈ LMod ∧ 𝑌 ∈ 𝐵 ∧ 𝑡 ∈ ( Base ‘ 𝑈 ) ) → ( 𝑌 ( ·𝑠 ‘ 𝑈 ) 𝑡 ) ∈ ( Base ‘ 𝑈 ) ) |
40 |
32 34 25 39
|
syl3anc |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( Base ‘ 𝑈 ) ∧ 𝑡 ≠ ( 0g ‘ 𝑈 ) ) → ( 𝑌 ( ·𝑠 ‘ 𝑈 ) 𝑡 ) ∈ ( Base ‘ 𝑈 ) ) |
41 |
1 2 10 35 3 4 13 27 23 6 24 40 33
|
hgmapvs |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( Base ‘ 𝑈 ) ∧ 𝑡 ≠ ( 0g ‘ 𝑈 ) ) → ( ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑋 ( ·𝑠 ‘ 𝑈 ) ( 𝑌 ( ·𝑠 ‘ 𝑈 ) 𝑡 ) ) ) = ( ( 𝐺 ‘ 𝑋 ) ( ·𝑠 ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ( ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑌 ( ·𝑠 ‘ 𝑈 ) 𝑡 ) ) ) ) |
42 |
1 2 10 35 3 4 13 27 23 6 24 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 ) ) ) |
43 |
42
|
oveq2d |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( Base ‘ 𝑈 ) ∧ 𝑡 ≠ ( 0g ‘ 𝑈 ) ) → ( ( 𝐺 ‘ 𝑋 ) ( ·𝑠 ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ( ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑌 ( ·𝑠 ‘ 𝑈 ) 𝑡 ) ) ) = ( ( 𝐺 ‘ 𝑋 ) ( ·𝑠 ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ( ( 𝐺 ‘ 𝑌 ) ( ·𝑠 ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ( ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑡 ) ) ) ) |
44 |
38 41 43
|
3eqtrd |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( Base ‘ 𝑈 ) ∧ 𝑡 ≠ ( 0g ‘ 𝑈 ) ) → ( ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( ( 𝑋 · 𝑌 ) ( ·𝑠 ‘ 𝑈 ) 𝑡 ) ) = ( ( 𝐺 ‘ 𝑋 ) ( ·𝑠 ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ( ( 𝐺 ‘ 𝑌 ) ( ·𝑠 ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ( ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑡 ) ) ) ) |
45 |
3 4 5
|
lmodmcl |
⊢ ( ( 𝑈 ∈ LMod ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 · 𝑌 ) ∈ 𝐵 ) |
46 |
31 8 9 45
|
syl3anc |
⊢ ( 𝜑 → ( 𝑋 · 𝑌 ) ∈ 𝐵 ) |
47 |
46
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( Base ‘ 𝑈 ) ∧ 𝑡 ≠ ( 0g ‘ 𝑈 ) ) → ( 𝑋 · 𝑌 ) ∈ 𝐵 ) |
48 |
1 2 10 35 3 4 13 27 23 6 24 25 47
|
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 ) ) ) |
49 |
30 44 48
|
3eqtr2rd |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( Base ‘ 𝑈 ) ∧ 𝑡 ≠ ( 0g ‘ 𝑈 ) ) → ( ( 𝐺 ‘ ( 𝑋 · 𝑌 ) ) ( ·𝑠 ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ( ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑡 ) ) = ( ( ( 𝐺 ‘ 𝑋 ) ( .r ‘ ( Scalar ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ) ( 𝐺 ‘ 𝑌 ) ) ( ·𝑠 ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ( ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑡 ) ) ) |
50 |
|
eqid |
⊢ ( 0g ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) = ( 0g ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) |
51 |
1 13 7
|
lcdlvec |
⊢ ( 𝜑 → ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ∈ LVec ) |
52 |
51
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( Base ‘ 𝑈 ) ∧ 𝑡 ≠ ( 0g ‘ 𝑈 ) ) → ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ∈ LVec ) |
53 |
1 2 3 4 13 16 17 6 7 46
|
hgmapdcl |
⊢ ( 𝜑 → ( 𝐺 ‘ ( 𝑋 · 𝑌 ) ) ∈ ( Base ‘ ( Scalar ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ) ) |
54 |
53
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( Base ‘ 𝑈 ) ∧ 𝑡 ≠ ( 0g ‘ 𝑈 ) ) → ( 𝐺 ‘ ( 𝑋 · 𝑌 ) ) ∈ ( Base ‘ ( Scalar ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ) ) |
55 |
16 17 28
|
lmodmcl |
⊢ ( ( ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ∈ LMod ∧ ( 𝐺 ‘ 𝑋 ) ∈ ( Base ‘ ( Scalar ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ) ∧ ( 𝐺 ‘ 𝑌 ) ∈ ( Base ‘ ( Scalar ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ) ) → ( ( 𝐺 ‘ 𝑋 ) ( .r ‘ ( Scalar ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ) ( 𝐺 ‘ 𝑌 ) ) ∈ ( Base ‘ ( Scalar ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ) ) |
56 |
14 18 20 55
|
syl3anc |
⊢ ( 𝜑 → ( ( 𝐺 ‘ 𝑋 ) ( .r ‘ ( Scalar ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ) ( 𝐺 ‘ 𝑌 ) ) ∈ ( Base ‘ ( Scalar ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ) ) |
57 |
56
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( Base ‘ 𝑈 ) ∧ 𝑡 ≠ ( 0g ‘ 𝑈 ) ) → ( ( 𝐺 ‘ 𝑋 ) ( .r ‘ ( Scalar ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ) ( 𝐺 ‘ 𝑌 ) ) ∈ ( Base ‘ ( Scalar ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ) ) |
58 |
|
simp3 |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( Base ‘ 𝑈 ) ∧ 𝑡 ≠ ( 0g ‘ 𝑈 ) ) → 𝑡 ≠ ( 0g ‘ 𝑈 ) ) |
59 |
1 2 10 11 13 50 23 24 25
|
hdmapeq0 |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( Base ‘ 𝑈 ) ∧ 𝑡 ≠ ( 0g ‘ 𝑈 ) ) → ( ( ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑡 ) = ( 0g ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ↔ 𝑡 = ( 0g ‘ 𝑈 ) ) ) |
60 |
59
|
necon3bid |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( Base ‘ 𝑈 ) ∧ 𝑡 ≠ ( 0g ‘ 𝑈 ) ) → ( ( ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑡 ) ≠ ( 0g ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ↔ 𝑡 ≠ ( 0g ‘ 𝑈 ) ) ) |
61 |
58 60
|
mpbird |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( Base ‘ 𝑈 ) ∧ 𝑡 ≠ ( 0g ‘ 𝑈 ) ) → ( ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑡 ) ≠ ( 0g ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ) |
62 |
22 27 16 17 50 52 54 57 26 61
|
lvecvscan2 |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( Base ‘ 𝑈 ) ∧ 𝑡 ≠ ( 0g ‘ 𝑈 ) ) → ( ( ( 𝐺 ‘ ( 𝑋 · 𝑌 ) ) ( ·𝑠 ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ( ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑡 ) ) = ( ( ( 𝐺 ‘ 𝑋 ) ( .r ‘ ( Scalar ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ) ( 𝐺 ‘ 𝑌 ) ) ( ·𝑠 ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ( ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑡 ) ) ↔ ( 𝐺 ‘ ( 𝑋 · 𝑌 ) ) = ( ( 𝐺 ‘ 𝑋 ) ( .r ‘ ( Scalar ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ) ( 𝐺 ‘ 𝑌 ) ) ) ) |
63 |
49 62
|
mpbid |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( Base ‘ 𝑈 ) ∧ 𝑡 ≠ ( 0g ‘ 𝑈 ) ) → ( 𝐺 ‘ ( 𝑋 · 𝑌 ) ) = ( ( 𝐺 ‘ 𝑋 ) ( .r ‘ ( Scalar ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ) ( 𝐺 ‘ 𝑌 ) ) ) |
64 |
63
|
rexlimdv3a |
⊢ ( 𝜑 → ( ∃ 𝑡 ∈ ( Base ‘ 𝑈 ) 𝑡 ≠ ( 0g ‘ 𝑈 ) → ( 𝐺 ‘ ( 𝑋 · 𝑌 ) ) = ( ( 𝐺 ‘ 𝑋 ) ( .r ‘ ( Scalar ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ) ( 𝐺 ‘ 𝑌 ) ) ) ) |
65 |
12 64
|
mpd |
⊢ ( 𝜑 → ( 𝐺 ‘ ( 𝑋 · 𝑌 ) ) = ( ( 𝐺 ‘ 𝑋 ) ( .r ‘ ( Scalar ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ) ( 𝐺 ‘ 𝑌 ) ) ) |
66 |
1 2 3 4 6 7 8
|
hgmapcl |
⊢ ( 𝜑 → ( 𝐺 ‘ 𝑋 ) ∈ 𝐵 ) |
67 |
1 2 3 4 6 7 9
|
hgmapcl |
⊢ ( 𝜑 → ( 𝐺 ‘ 𝑌 ) ∈ 𝐵 ) |
68 |
1 2 3 4 5 13 16 28 7 66 67
|
lcdsmul |
⊢ ( 𝜑 → ( ( 𝐺 ‘ 𝑋 ) ( .r ‘ ( Scalar ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ) ( 𝐺 ‘ 𝑌 ) ) = ( ( 𝐺 ‘ 𝑌 ) · ( 𝐺 ‘ 𝑋 ) ) ) |
69 |
65 68
|
eqtrd |
⊢ ( 𝜑 → ( 𝐺 ‘ ( 𝑋 · 𝑌 ) ) = ( ( 𝐺 ‘ 𝑌 ) · ( 𝐺 ‘ 𝑋 ) ) ) |