Step |
Hyp |
Ref |
Expression |
1 |
|
ldualvsubval.v |
⊢ 𝑉 = ( Base ‘ 𝑊 ) |
2 |
|
ldualvsubval.r |
⊢ 𝑅 = ( Scalar ‘ 𝑊 ) |
3 |
|
ldualvsubval.s |
⊢ 𝑆 = ( -g ‘ 𝑅 ) |
4 |
|
ldualvsubval.f |
⊢ 𝐹 = ( LFnl ‘ 𝑊 ) |
5 |
|
ldualvsubval.d |
⊢ 𝐷 = ( LDual ‘ 𝑊 ) |
6 |
|
ldualvsubval.m |
⊢ − = ( -g ‘ 𝐷 ) |
7 |
|
ldualvsubval.w |
⊢ ( 𝜑 → 𝑊 ∈ LMod ) |
8 |
|
ldualvsubval.g |
⊢ ( 𝜑 → 𝐺 ∈ 𝐹 ) |
9 |
|
ldualvsubval.h |
⊢ ( 𝜑 → 𝐻 ∈ 𝐹 ) |
10 |
|
ldualvsubval.x |
⊢ ( 𝜑 → 𝑋 ∈ 𝑉 ) |
11 |
5 7
|
lduallmod |
⊢ ( 𝜑 → 𝐷 ∈ LMod ) |
12 |
|
eqid |
⊢ ( Base ‘ 𝐷 ) = ( Base ‘ 𝐷 ) |
13 |
4 5 12 7 8
|
ldualelvbase |
⊢ ( 𝜑 → 𝐺 ∈ ( Base ‘ 𝐷 ) ) |
14 |
4 5 12 7 9
|
ldualelvbase |
⊢ ( 𝜑 → 𝐻 ∈ ( Base ‘ 𝐷 ) ) |
15 |
|
eqid |
⊢ ( +g ‘ 𝐷 ) = ( +g ‘ 𝐷 ) |
16 |
|
eqid |
⊢ ( Scalar ‘ 𝐷 ) = ( Scalar ‘ 𝐷 ) |
17 |
|
eqid |
⊢ ( ·𝑠 ‘ 𝐷 ) = ( ·𝑠 ‘ 𝐷 ) |
18 |
|
eqid |
⊢ ( invg ‘ ( Scalar ‘ 𝐷 ) ) = ( invg ‘ ( Scalar ‘ 𝐷 ) ) |
19 |
|
eqid |
⊢ ( 1r ‘ ( Scalar ‘ 𝐷 ) ) = ( 1r ‘ ( Scalar ‘ 𝐷 ) ) |
20 |
12 15 6 16 17 18 19
|
lmodvsubval2 |
⊢ ( ( 𝐷 ∈ LMod ∧ 𝐺 ∈ ( Base ‘ 𝐷 ) ∧ 𝐻 ∈ ( Base ‘ 𝐷 ) ) → ( 𝐺 − 𝐻 ) = ( 𝐺 ( +g ‘ 𝐷 ) ( ( ( invg ‘ ( Scalar ‘ 𝐷 ) ) ‘ ( 1r ‘ ( Scalar ‘ 𝐷 ) ) ) ( ·𝑠 ‘ 𝐷 ) 𝐻 ) ) ) |
21 |
11 13 14 20
|
syl3anc |
⊢ ( 𝜑 → ( 𝐺 − 𝐻 ) = ( 𝐺 ( +g ‘ 𝐷 ) ( ( ( invg ‘ ( Scalar ‘ 𝐷 ) ) ‘ ( 1r ‘ ( Scalar ‘ 𝐷 ) ) ) ( ·𝑠 ‘ 𝐷 ) 𝐻 ) ) ) |
22 |
21
|
fveq1d |
⊢ ( 𝜑 → ( ( 𝐺 − 𝐻 ) ‘ 𝑋 ) = ( ( 𝐺 ( +g ‘ 𝐷 ) ( ( ( invg ‘ ( Scalar ‘ 𝐷 ) ) ‘ ( 1r ‘ ( Scalar ‘ 𝐷 ) ) ) ( ·𝑠 ‘ 𝐷 ) 𝐻 ) ) ‘ 𝑋 ) ) |
23 |
|
eqid |
⊢ ( +g ‘ 𝑅 ) = ( +g ‘ 𝑅 ) |
24 |
|
eqid |
⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 ) |
25 |
16
|
lmodfgrp |
⊢ ( 𝐷 ∈ LMod → ( Scalar ‘ 𝐷 ) ∈ Grp ) |
26 |
11 25
|
syl |
⊢ ( 𝜑 → ( Scalar ‘ 𝐷 ) ∈ Grp ) |
27 |
16
|
lmodring |
⊢ ( 𝐷 ∈ LMod → ( Scalar ‘ 𝐷 ) ∈ Ring ) |
28 |
11 27
|
syl |
⊢ ( 𝜑 → ( Scalar ‘ 𝐷 ) ∈ Ring ) |
29 |
|
eqid |
⊢ ( Base ‘ ( Scalar ‘ 𝐷 ) ) = ( Base ‘ ( Scalar ‘ 𝐷 ) ) |
30 |
29 19
|
ringidcl |
⊢ ( ( Scalar ‘ 𝐷 ) ∈ Ring → ( 1r ‘ ( Scalar ‘ 𝐷 ) ) ∈ ( Base ‘ ( Scalar ‘ 𝐷 ) ) ) |
31 |
28 30
|
syl |
⊢ ( 𝜑 → ( 1r ‘ ( Scalar ‘ 𝐷 ) ) ∈ ( Base ‘ ( Scalar ‘ 𝐷 ) ) ) |
32 |
29 18
|
grpinvcl |
⊢ ( ( ( Scalar ‘ 𝐷 ) ∈ Grp ∧ ( 1r ‘ ( Scalar ‘ 𝐷 ) ) ∈ ( Base ‘ ( Scalar ‘ 𝐷 ) ) ) → ( ( invg ‘ ( Scalar ‘ 𝐷 ) ) ‘ ( 1r ‘ ( Scalar ‘ 𝐷 ) ) ) ∈ ( Base ‘ ( Scalar ‘ 𝐷 ) ) ) |
33 |
26 31 32
|
syl2anc |
⊢ ( 𝜑 → ( ( invg ‘ ( Scalar ‘ 𝐷 ) ) ‘ ( 1r ‘ ( Scalar ‘ 𝐷 ) ) ) ∈ ( Base ‘ ( Scalar ‘ 𝐷 ) ) ) |
34 |
2 24 5 16 29 7
|
ldualsbase |
⊢ ( 𝜑 → ( Base ‘ ( Scalar ‘ 𝐷 ) ) = ( Base ‘ 𝑅 ) ) |
35 |
33 34
|
eleqtrd |
⊢ ( 𝜑 → ( ( invg ‘ ( Scalar ‘ 𝐷 ) ) ‘ ( 1r ‘ ( Scalar ‘ 𝐷 ) ) ) ∈ ( Base ‘ 𝑅 ) ) |
36 |
4 2 24 5 17 7 35 9
|
ldualvscl |
⊢ ( 𝜑 → ( ( ( invg ‘ ( Scalar ‘ 𝐷 ) ) ‘ ( 1r ‘ ( Scalar ‘ 𝐷 ) ) ) ( ·𝑠 ‘ 𝐷 ) 𝐻 ) ∈ 𝐹 ) |
37 |
1 2 23 4 5 15 7 8 36 10
|
ldualvaddval |
⊢ ( 𝜑 → ( ( 𝐺 ( +g ‘ 𝐷 ) ( ( ( invg ‘ ( Scalar ‘ 𝐷 ) ) ‘ ( 1r ‘ ( Scalar ‘ 𝐷 ) ) ) ( ·𝑠 ‘ 𝐷 ) 𝐻 ) ) ‘ 𝑋 ) = ( ( 𝐺 ‘ 𝑋 ) ( +g ‘ 𝑅 ) ( ( ( ( invg ‘ ( Scalar ‘ 𝐷 ) ) ‘ ( 1r ‘ ( Scalar ‘ 𝐷 ) ) ) ( ·𝑠 ‘ 𝐷 ) 𝐻 ) ‘ 𝑋 ) ) ) |
38 |
|
eqid |
⊢ ( invg ‘ 𝑅 ) = ( invg ‘ 𝑅 ) |
39 |
2 38 5 16 18 7
|
ldualneg |
⊢ ( 𝜑 → ( invg ‘ ( Scalar ‘ 𝐷 ) ) = ( invg ‘ 𝑅 ) ) |
40 |
|
eqid |
⊢ ( 1r ‘ 𝑅 ) = ( 1r ‘ 𝑅 ) |
41 |
2 40 5 16 19 7
|
ldual1 |
⊢ ( 𝜑 → ( 1r ‘ ( Scalar ‘ 𝐷 ) ) = ( 1r ‘ 𝑅 ) ) |
42 |
39 41
|
fveq12d |
⊢ ( 𝜑 → ( ( invg ‘ ( Scalar ‘ 𝐷 ) ) ‘ ( 1r ‘ ( Scalar ‘ 𝐷 ) ) ) = ( ( invg ‘ 𝑅 ) ‘ ( 1r ‘ 𝑅 ) ) ) |
43 |
42
|
oveq1d |
⊢ ( 𝜑 → ( ( ( invg ‘ ( Scalar ‘ 𝐷 ) ) ‘ ( 1r ‘ ( Scalar ‘ 𝐷 ) ) ) ( ·𝑠 ‘ 𝐷 ) 𝐻 ) = ( ( ( invg ‘ 𝑅 ) ‘ ( 1r ‘ 𝑅 ) ) ( ·𝑠 ‘ 𝐷 ) 𝐻 ) ) |
44 |
43
|
fveq1d |
⊢ ( 𝜑 → ( ( ( ( invg ‘ ( Scalar ‘ 𝐷 ) ) ‘ ( 1r ‘ ( Scalar ‘ 𝐷 ) ) ) ( ·𝑠 ‘ 𝐷 ) 𝐻 ) ‘ 𝑋 ) = ( ( ( ( invg ‘ 𝑅 ) ‘ ( 1r ‘ 𝑅 ) ) ( ·𝑠 ‘ 𝐷 ) 𝐻 ) ‘ 𝑋 ) ) |
45 |
|
eqid |
⊢ ( .r ‘ 𝑅 ) = ( .r ‘ 𝑅 ) |
46 |
2
|
lmodring |
⊢ ( 𝑊 ∈ LMod → 𝑅 ∈ Ring ) |
47 |
7 46
|
syl |
⊢ ( 𝜑 → 𝑅 ∈ Ring ) |
48 |
|
ringgrp |
⊢ ( 𝑅 ∈ Ring → 𝑅 ∈ Grp ) |
49 |
47 48
|
syl |
⊢ ( 𝜑 → 𝑅 ∈ Grp ) |
50 |
2 24 40
|
lmod1cl |
⊢ ( 𝑊 ∈ LMod → ( 1r ‘ 𝑅 ) ∈ ( Base ‘ 𝑅 ) ) |
51 |
7 50
|
syl |
⊢ ( 𝜑 → ( 1r ‘ 𝑅 ) ∈ ( Base ‘ 𝑅 ) ) |
52 |
24 38
|
grpinvcl |
⊢ ( ( 𝑅 ∈ Grp ∧ ( 1r ‘ 𝑅 ) ∈ ( Base ‘ 𝑅 ) ) → ( ( invg ‘ 𝑅 ) ‘ ( 1r ‘ 𝑅 ) ) ∈ ( Base ‘ 𝑅 ) ) |
53 |
49 51 52
|
syl2anc |
⊢ ( 𝜑 → ( ( invg ‘ 𝑅 ) ‘ ( 1r ‘ 𝑅 ) ) ∈ ( Base ‘ 𝑅 ) ) |
54 |
4 1 2 24 45 5 17 7 53 9 10
|
ldualvsval |
⊢ ( 𝜑 → ( ( ( ( invg ‘ 𝑅 ) ‘ ( 1r ‘ 𝑅 ) ) ( ·𝑠 ‘ 𝐷 ) 𝐻 ) ‘ 𝑋 ) = ( ( 𝐻 ‘ 𝑋 ) ( .r ‘ 𝑅 ) ( ( invg ‘ 𝑅 ) ‘ ( 1r ‘ 𝑅 ) ) ) ) |
55 |
2 24 1 4
|
lflcl |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝐻 ∈ 𝐹 ∧ 𝑋 ∈ 𝑉 ) → ( 𝐻 ‘ 𝑋 ) ∈ ( Base ‘ 𝑅 ) ) |
56 |
7 9 10 55
|
syl3anc |
⊢ ( 𝜑 → ( 𝐻 ‘ 𝑋 ) ∈ ( Base ‘ 𝑅 ) ) |
57 |
24 45 40 38 47 56
|
rngnegr |
⊢ ( 𝜑 → ( ( 𝐻 ‘ 𝑋 ) ( .r ‘ 𝑅 ) ( ( invg ‘ 𝑅 ) ‘ ( 1r ‘ 𝑅 ) ) ) = ( ( invg ‘ 𝑅 ) ‘ ( 𝐻 ‘ 𝑋 ) ) ) |
58 |
44 54 57
|
3eqtrd |
⊢ ( 𝜑 → ( ( ( ( invg ‘ ( Scalar ‘ 𝐷 ) ) ‘ ( 1r ‘ ( Scalar ‘ 𝐷 ) ) ) ( ·𝑠 ‘ 𝐷 ) 𝐻 ) ‘ 𝑋 ) = ( ( invg ‘ 𝑅 ) ‘ ( 𝐻 ‘ 𝑋 ) ) ) |
59 |
58
|
oveq2d |
⊢ ( 𝜑 → ( ( 𝐺 ‘ 𝑋 ) ( +g ‘ 𝑅 ) ( ( ( ( invg ‘ ( Scalar ‘ 𝐷 ) ) ‘ ( 1r ‘ ( Scalar ‘ 𝐷 ) ) ) ( ·𝑠 ‘ 𝐷 ) 𝐻 ) ‘ 𝑋 ) ) = ( ( 𝐺 ‘ 𝑋 ) ( +g ‘ 𝑅 ) ( ( invg ‘ 𝑅 ) ‘ ( 𝐻 ‘ 𝑋 ) ) ) ) |
60 |
2 24 1 4
|
lflcl |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹 ∧ 𝑋 ∈ 𝑉 ) → ( 𝐺 ‘ 𝑋 ) ∈ ( Base ‘ 𝑅 ) ) |
61 |
7 8 10 60
|
syl3anc |
⊢ ( 𝜑 → ( 𝐺 ‘ 𝑋 ) ∈ ( Base ‘ 𝑅 ) ) |
62 |
24 23 38 3
|
grpsubval |
⊢ ( ( ( 𝐺 ‘ 𝑋 ) ∈ ( Base ‘ 𝑅 ) ∧ ( 𝐻 ‘ 𝑋 ) ∈ ( Base ‘ 𝑅 ) ) → ( ( 𝐺 ‘ 𝑋 ) 𝑆 ( 𝐻 ‘ 𝑋 ) ) = ( ( 𝐺 ‘ 𝑋 ) ( +g ‘ 𝑅 ) ( ( invg ‘ 𝑅 ) ‘ ( 𝐻 ‘ 𝑋 ) ) ) ) |
63 |
61 56 62
|
syl2anc |
⊢ ( 𝜑 → ( ( 𝐺 ‘ 𝑋 ) 𝑆 ( 𝐻 ‘ 𝑋 ) ) = ( ( 𝐺 ‘ 𝑋 ) ( +g ‘ 𝑅 ) ( ( invg ‘ 𝑅 ) ‘ ( 𝐻 ‘ 𝑋 ) ) ) ) |
64 |
59 63
|
eqtr4d |
⊢ ( 𝜑 → ( ( 𝐺 ‘ 𝑋 ) ( +g ‘ 𝑅 ) ( ( ( ( invg ‘ ( Scalar ‘ 𝐷 ) ) ‘ ( 1r ‘ ( Scalar ‘ 𝐷 ) ) ) ( ·𝑠 ‘ 𝐷 ) 𝐻 ) ‘ 𝑋 ) ) = ( ( 𝐺 ‘ 𝑋 ) 𝑆 ( 𝐻 ‘ 𝑋 ) ) ) |
65 |
22 37 64
|
3eqtrd |
⊢ ( 𝜑 → ( ( 𝐺 − 𝐻 ) ‘ 𝑋 ) = ( ( 𝐺 ‘ 𝑋 ) 𝑆 ( 𝐻 ‘ 𝑋 ) ) ) |