Step |
Hyp |
Ref |
Expression |
1 |
|
lkrin.f |
⊢ 𝐹 = ( LFnl ‘ 𝑊 ) |
2 |
|
lkrin.k |
⊢ 𝐾 = ( LKer ‘ 𝑊 ) |
3 |
|
lkrin.d |
⊢ 𝐷 = ( LDual ‘ 𝑊 ) |
4 |
|
lkrin.p |
⊢ + = ( +g ‘ 𝐷 ) |
5 |
|
lkrin.w |
⊢ ( 𝜑 → 𝑊 ∈ LMod ) |
6 |
|
lkrin.e |
⊢ ( 𝜑 → 𝐺 ∈ 𝐹 ) |
7 |
|
lkrin.g |
⊢ ( 𝜑 → 𝐻 ∈ 𝐹 ) |
8 |
|
elin |
⊢ ( 𝑣 ∈ ( ( 𝐾 ‘ 𝐺 ) ∩ ( 𝐾 ‘ 𝐻 ) ) ↔ ( 𝑣 ∈ ( 𝐾 ‘ 𝐺 ) ∧ 𝑣 ∈ ( 𝐾 ‘ 𝐻 ) ) ) |
9 |
5
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑣 ∈ ( 𝐾 ‘ 𝐺 ) ∧ 𝑣 ∈ ( 𝐾 ‘ 𝐻 ) ) ) → 𝑊 ∈ LMod ) |
10 |
6
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑣 ∈ ( 𝐾 ‘ 𝐺 ) ∧ 𝑣 ∈ ( 𝐾 ‘ 𝐻 ) ) ) → 𝐺 ∈ 𝐹 ) |
11 |
|
simprl |
⊢ ( ( 𝜑 ∧ ( 𝑣 ∈ ( 𝐾 ‘ 𝐺 ) ∧ 𝑣 ∈ ( 𝐾 ‘ 𝐻 ) ) ) → 𝑣 ∈ ( 𝐾 ‘ 𝐺 ) ) |
12 |
|
eqid |
⊢ ( Base ‘ 𝑊 ) = ( Base ‘ 𝑊 ) |
13 |
12 1 2
|
lkrcl |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹 ∧ 𝑣 ∈ ( 𝐾 ‘ 𝐺 ) ) → 𝑣 ∈ ( Base ‘ 𝑊 ) ) |
14 |
9 10 11 13
|
syl3anc |
⊢ ( ( 𝜑 ∧ ( 𝑣 ∈ ( 𝐾 ‘ 𝐺 ) ∧ 𝑣 ∈ ( 𝐾 ‘ 𝐻 ) ) ) → 𝑣 ∈ ( Base ‘ 𝑊 ) ) |
15 |
|
eqid |
⊢ ( Scalar ‘ 𝑊 ) = ( Scalar ‘ 𝑊 ) |
16 |
|
eqid |
⊢ ( +g ‘ ( Scalar ‘ 𝑊 ) ) = ( +g ‘ ( Scalar ‘ 𝑊 ) ) |
17 |
7
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑣 ∈ ( 𝐾 ‘ 𝐺 ) ∧ 𝑣 ∈ ( 𝐾 ‘ 𝐻 ) ) ) → 𝐻 ∈ 𝐹 ) |
18 |
12 15 16 1 3 4 9 10 17 14
|
ldualvaddval |
⊢ ( ( 𝜑 ∧ ( 𝑣 ∈ ( 𝐾 ‘ 𝐺 ) ∧ 𝑣 ∈ ( 𝐾 ‘ 𝐻 ) ) ) → ( ( 𝐺 + 𝐻 ) ‘ 𝑣 ) = ( ( 𝐺 ‘ 𝑣 ) ( +g ‘ ( Scalar ‘ 𝑊 ) ) ( 𝐻 ‘ 𝑣 ) ) ) |
19 |
|
eqid |
⊢ ( 0g ‘ ( Scalar ‘ 𝑊 ) ) = ( 0g ‘ ( Scalar ‘ 𝑊 ) ) |
20 |
15 19 1 2
|
lkrf0 |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹 ∧ 𝑣 ∈ ( 𝐾 ‘ 𝐺 ) ) → ( 𝐺 ‘ 𝑣 ) = ( 0g ‘ ( Scalar ‘ 𝑊 ) ) ) |
21 |
9 10 11 20
|
syl3anc |
⊢ ( ( 𝜑 ∧ ( 𝑣 ∈ ( 𝐾 ‘ 𝐺 ) ∧ 𝑣 ∈ ( 𝐾 ‘ 𝐻 ) ) ) → ( 𝐺 ‘ 𝑣 ) = ( 0g ‘ ( Scalar ‘ 𝑊 ) ) ) |
22 |
|
simprr |
⊢ ( ( 𝜑 ∧ ( 𝑣 ∈ ( 𝐾 ‘ 𝐺 ) ∧ 𝑣 ∈ ( 𝐾 ‘ 𝐻 ) ) ) → 𝑣 ∈ ( 𝐾 ‘ 𝐻 ) ) |
23 |
15 19 1 2
|
lkrf0 |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝐻 ∈ 𝐹 ∧ 𝑣 ∈ ( 𝐾 ‘ 𝐻 ) ) → ( 𝐻 ‘ 𝑣 ) = ( 0g ‘ ( Scalar ‘ 𝑊 ) ) ) |
24 |
9 17 22 23
|
syl3anc |
⊢ ( ( 𝜑 ∧ ( 𝑣 ∈ ( 𝐾 ‘ 𝐺 ) ∧ 𝑣 ∈ ( 𝐾 ‘ 𝐻 ) ) ) → ( 𝐻 ‘ 𝑣 ) = ( 0g ‘ ( Scalar ‘ 𝑊 ) ) ) |
25 |
21 24
|
oveq12d |
⊢ ( ( 𝜑 ∧ ( 𝑣 ∈ ( 𝐾 ‘ 𝐺 ) ∧ 𝑣 ∈ ( 𝐾 ‘ 𝐻 ) ) ) → ( ( 𝐺 ‘ 𝑣 ) ( +g ‘ ( Scalar ‘ 𝑊 ) ) ( 𝐻 ‘ 𝑣 ) ) = ( ( 0g ‘ ( Scalar ‘ 𝑊 ) ) ( +g ‘ ( Scalar ‘ 𝑊 ) ) ( 0g ‘ ( Scalar ‘ 𝑊 ) ) ) ) |
26 |
15
|
lmodring |
⊢ ( 𝑊 ∈ LMod → ( Scalar ‘ 𝑊 ) ∈ Ring ) |
27 |
5 26
|
syl |
⊢ ( 𝜑 → ( Scalar ‘ 𝑊 ) ∈ Ring ) |
28 |
|
ringgrp |
⊢ ( ( Scalar ‘ 𝑊 ) ∈ Ring → ( Scalar ‘ 𝑊 ) ∈ Grp ) |
29 |
27 28
|
syl |
⊢ ( 𝜑 → ( Scalar ‘ 𝑊 ) ∈ Grp ) |
30 |
|
eqid |
⊢ ( Base ‘ ( Scalar ‘ 𝑊 ) ) = ( Base ‘ ( Scalar ‘ 𝑊 ) ) |
31 |
30 19
|
grpidcl |
⊢ ( ( Scalar ‘ 𝑊 ) ∈ Grp → ( 0g ‘ ( Scalar ‘ 𝑊 ) ) ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ) |
32 |
30 16 19
|
grplid |
⊢ ( ( ( Scalar ‘ 𝑊 ) ∈ Grp ∧ ( 0g ‘ ( Scalar ‘ 𝑊 ) ) ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ) → ( ( 0g ‘ ( Scalar ‘ 𝑊 ) ) ( +g ‘ ( Scalar ‘ 𝑊 ) ) ( 0g ‘ ( Scalar ‘ 𝑊 ) ) ) = ( 0g ‘ ( Scalar ‘ 𝑊 ) ) ) |
33 |
29 31 32
|
syl2anc2 |
⊢ ( 𝜑 → ( ( 0g ‘ ( Scalar ‘ 𝑊 ) ) ( +g ‘ ( Scalar ‘ 𝑊 ) ) ( 0g ‘ ( Scalar ‘ 𝑊 ) ) ) = ( 0g ‘ ( Scalar ‘ 𝑊 ) ) ) |
34 |
33
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑣 ∈ ( 𝐾 ‘ 𝐺 ) ∧ 𝑣 ∈ ( 𝐾 ‘ 𝐻 ) ) ) → ( ( 0g ‘ ( Scalar ‘ 𝑊 ) ) ( +g ‘ ( Scalar ‘ 𝑊 ) ) ( 0g ‘ ( Scalar ‘ 𝑊 ) ) ) = ( 0g ‘ ( Scalar ‘ 𝑊 ) ) ) |
35 |
18 25 34
|
3eqtrd |
⊢ ( ( 𝜑 ∧ ( 𝑣 ∈ ( 𝐾 ‘ 𝐺 ) ∧ 𝑣 ∈ ( 𝐾 ‘ 𝐻 ) ) ) → ( ( 𝐺 + 𝐻 ) ‘ 𝑣 ) = ( 0g ‘ ( Scalar ‘ 𝑊 ) ) ) |
36 |
1 3 4 5 6 7
|
ldualvaddcl |
⊢ ( 𝜑 → ( 𝐺 + 𝐻 ) ∈ 𝐹 ) |
37 |
36
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑣 ∈ ( 𝐾 ‘ 𝐺 ) ∧ 𝑣 ∈ ( 𝐾 ‘ 𝐻 ) ) ) → ( 𝐺 + 𝐻 ) ∈ 𝐹 ) |
38 |
12 15 19 1 2
|
ellkr |
⊢ ( ( 𝑊 ∈ LMod ∧ ( 𝐺 + 𝐻 ) ∈ 𝐹 ) → ( 𝑣 ∈ ( 𝐾 ‘ ( 𝐺 + 𝐻 ) ) ↔ ( 𝑣 ∈ ( Base ‘ 𝑊 ) ∧ ( ( 𝐺 + 𝐻 ) ‘ 𝑣 ) = ( 0g ‘ ( Scalar ‘ 𝑊 ) ) ) ) ) |
39 |
9 37 38
|
syl2anc |
⊢ ( ( 𝜑 ∧ ( 𝑣 ∈ ( 𝐾 ‘ 𝐺 ) ∧ 𝑣 ∈ ( 𝐾 ‘ 𝐻 ) ) ) → ( 𝑣 ∈ ( 𝐾 ‘ ( 𝐺 + 𝐻 ) ) ↔ ( 𝑣 ∈ ( Base ‘ 𝑊 ) ∧ ( ( 𝐺 + 𝐻 ) ‘ 𝑣 ) = ( 0g ‘ ( Scalar ‘ 𝑊 ) ) ) ) ) |
40 |
14 35 39
|
mpbir2and |
⊢ ( ( 𝜑 ∧ ( 𝑣 ∈ ( 𝐾 ‘ 𝐺 ) ∧ 𝑣 ∈ ( 𝐾 ‘ 𝐻 ) ) ) → 𝑣 ∈ ( 𝐾 ‘ ( 𝐺 + 𝐻 ) ) ) |
41 |
40
|
ex |
⊢ ( 𝜑 → ( ( 𝑣 ∈ ( 𝐾 ‘ 𝐺 ) ∧ 𝑣 ∈ ( 𝐾 ‘ 𝐻 ) ) → 𝑣 ∈ ( 𝐾 ‘ ( 𝐺 + 𝐻 ) ) ) ) |
42 |
8 41
|
syl5bi |
⊢ ( 𝜑 → ( 𝑣 ∈ ( ( 𝐾 ‘ 𝐺 ) ∩ ( 𝐾 ‘ 𝐻 ) ) → 𝑣 ∈ ( 𝐾 ‘ ( 𝐺 + 𝐻 ) ) ) ) |
43 |
42
|
ssrdv |
⊢ ( 𝜑 → ( ( 𝐾 ‘ 𝐺 ) ∩ ( 𝐾 ‘ 𝐻 ) ) ⊆ ( 𝐾 ‘ ( 𝐺 + 𝐻 ) ) ) |