Metamath Proof Explorer

Theorem hgmapadd

Description: Part 15 of Baer p. 50 line 13. (Contributed by NM, 6-Jun-2015)

Ref Expression
Hypotheses hgmapadd.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
hgmapadd.u ⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
hgmapadd.r ⊢ 𝑅 = ( Scalar ‘ 𝑈 )
hgmapadd.b ⊢ 𝐵 = ( Base ‘ 𝑅 )
hgmapadd.p ⊢ + = ( +g ‘ 𝑅 )
hgmapadd.g ⊢ 𝐺 = ( ( HGMap ‘ 𝐾 ) ‘ 𝑊 )
hgmapadd.k ⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
hgmapadd.x ⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
hgmapadd.y ⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
Assertion hgmapadd ( 𝜑 → ( 𝐺 ‘ ( 𝑋 + 𝑌 ) ) = ( ( 𝐺 ‘ 𝑋 ) + ( 𝐺 ‘ 𝑌 ) ) )

Proof

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 ) ) )