Step |
Hyp |
Ref |
Expression |
1 |
|
lflsub.d |
⊢ 𝐷 = ( Scalar ‘ 𝑊 ) |
2 |
|
lflsub.m |
⊢ 𝑀 = ( -g ‘ 𝐷 ) |
3 |
|
lflsub.v |
⊢ 𝑉 = ( Base ‘ 𝑊 ) |
4 |
|
lflsub.a |
⊢ − = ( -g ‘ 𝑊 ) |
5 |
|
lflsub.f |
⊢ 𝐹 = ( LFnl ‘ 𝑊 ) |
6 |
|
simp1 |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹 ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ) ) → 𝑊 ∈ LMod ) |
7 |
|
simp3l |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹 ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ) ) → 𝑋 ∈ 𝑉 ) |
8 |
1
|
lmodring |
⊢ ( 𝑊 ∈ LMod → 𝐷 ∈ Ring ) |
9 |
8
|
3ad2ant1 |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹 ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ) ) → 𝐷 ∈ Ring ) |
10 |
|
ringgrp |
⊢ ( 𝐷 ∈ Ring → 𝐷 ∈ Grp ) |
11 |
9 10
|
syl |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹 ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ) ) → 𝐷 ∈ Grp ) |
12 |
|
eqid |
⊢ ( Base ‘ 𝐷 ) = ( Base ‘ 𝐷 ) |
13 |
|
eqid |
⊢ ( 1r ‘ 𝐷 ) = ( 1r ‘ 𝐷 ) |
14 |
12 13
|
ringidcl |
⊢ ( 𝐷 ∈ Ring → ( 1r ‘ 𝐷 ) ∈ ( Base ‘ 𝐷 ) ) |
15 |
9 14
|
syl |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹 ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ) ) → ( 1r ‘ 𝐷 ) ∈ ( Base ‘ 𝐷 ) ) |
16 |
|
eqid |
⊢ ( invg ‘ 𝐷 ) = ( invg ‘ 𝐷 ) |
17 |
12 16
|
grpinvcl |
⊢ ( ( 𝐷 ∈ Grp ∧ ( 1r ‘ 𝐷 ) ∈ ( Base ‘ 𝐷 ) ) → ( ( invg ‘ 𝐷 ) ‘ ( 1r ‘ 𝐷 ) ) ∈ ( Base ‘ 𝐷 ) ) |
18 |
11 15 17
|
syl2anc |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹 ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ) ) → ( ( invg ‘ 𝐷 ) ‘ ( 1r ‘ 𝐷 ) ) ∈ ( Base ‘ 𝐷 ) ) |
19 |
|
simp3r |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹 ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ) ) → 𝑌 ∈ 𝑉 ) |
20 |
|
eqid |
⊢ ( ·𝑠 ‘ 𝑊 ) = ( ·𝑠 ‘ 𝑊 ) |
21 |
3 1 20 12
|
lmodvscl |
⊢ ( ( 𝑊 ∈ LMod ∧ ( ( invg ‘ 𝐷 ) ‘ ( 1r ‘ 𝐷 ) ) ∈ ( Base ‘ 𝐷 ) ∧ 𝑌 ∈ 𝑉 ) → ( ( ( invg ‘ 𝐷 ) ‘ ( 1r ‘ 𝐷 ) ) ( ·𝑠 ‘ 𝑊 ) 𝑌 ) ∈ 𝑉 ) |
22 |
6 18 19 21
|
syl3anc |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹 ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ) ) → ( ( ( invg ‘ 𝐷 ) ‘ ( 1r ‘ 𝐷 ) ) ( ·𝑠 ‘ 𝑊 ) 𝑌 ) ∈ 𝑉 ) |
23 |
|
eqid |
⊢ ( +g ‘ 𝑊 ) = ( +g ‘ 𝑊 ) |
24 |
3 23
|
lmodcom |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ ( ( ( invg ‘ 𝐷 ) ‘ ( 1r ‘ 𝐷 ) ) ( ·𝑠 ‘ 𝑊 ) 𝑌 ) ∈ 𝑉 ) → ( 𝑋 ( +g ‘ 𝑊 ) ( ( ( invg ‘ 𝐷 ) ‘ ( 1r ‘ 𝐷 ) ) ( ·𝑠 ‘ 𝑊 ) 𝑌 ) ) = ( ( ( ( invg ‘ 𝐷 ) ‘ ( 1r ‘ 𝐷 ) ) ( ·𝑠 ‘ 𝑊 ) 𝑌 ) ( +g ‘ 𝑊 ) 𝑋 ) ) |
25 |
6 7 22 24
|
syl3anc |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹 ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ) ) → ( 𝑋 ( +g ‘ 𝑊 ) ( ( ( invg ‘ 𝐷 ) ‘ ( 1r ‘ 𝐷 ) ) ( ·𝑠 ‘ 𝑊 ) 𝑌 ) ) = ( ( ( ( invg ‘ 𝐷 ) ‘ ( 1r ‘ 𝐷 ) ) ( ·𝑠 ‘ 𝑊 ) 𝑌 ) ( +g ‘ 𝑊 ) 𝑋 ) ) |
26 |
25
|
fveq2d |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹 ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ) ) → ( 𝐺 ‘ ( 𝑋 ( +g ‘ 𝑊 ) ( ( ( invg ‘ 𝐷 ) ‘ ( 1r ‘ 𝐷 ) ) ( ·𝑠 ‘ 𝑊 ) 𝑌 ) ) ) = ( 𝐺 ‘ ( ( ( ( invg ‘ 𝐷 ) ‘ ( 1r ‘ 𝐷 ) ) ( ·𝑠 ‘ 𝑊 ) 𝑌 ) ( +g ‘ 𝑊 ) 𝑋 ) ) ) |
27 |
|
simp2 |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹 ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ) ) → 𝐺 ∈ 𝐹 ) |
28 |
|
eqid |
⊢ ( +g ‘ 𝐷 ) = ( +g ‘ 𝐷 ) |
29 |
|
eqid |
⊢ ( .r ‘ 𝐷 ) = ( .r ‘ 𝐷 ) |
30 |
3 23 1 20 12 28 29 5
|
lfli |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹 ∧ ( ( ( invg ‘ 𝐷 ) ‘ ( 1r ‘ 𝐷 ) ) ∈ ( Base ‘ 𝐷 ) ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉 ) ) → ( 𝐺 ‘ ( ( ( ( invg ‘ 𝐷 ) ‘ ( 1r ‘ 𝐷 ) ) ( ·𝑠 ‘ 𝑊 ) 𝑌 ) ( +g ‘ 𝑊 ) 𝑋 ) ) = ( ( ( ( invg ‘ 𝐷 ) ‘ ( 1r ‘ 𝐷 ) ) ( .r ‘ 𝐷 ) ( 𝐺 ‘ 𝑌 ) ) ( +g ‘ 𝐷 ) ( 𝐺 ‘ 𝑋 ) ) ) |
31 |
6 27 18 19 7 30
|
syl113anc |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹 ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ) ) → ( 𝐺 ‘ ( ( ( ( invg ‘ 𝐷 ) ‘ ( 1r ‘ 𝐷 ) ) ( ·𝑠 ‘ 𝑊 ) 𝑌 ) ( +g ‘ 𝑊 ) 𝑋 ) ) = ( ( ( ( invg ‘ 𝐷 ) ‘ ( 1r ‘ 𝐷 ) ) ( .r ‘ 𝐷 ) ( 𝐺 ‘ 𝑌 ) ) ( +g ‘ 𝐷 ) ( 𝐺 ‘ 𝑋 ) ) ) |
32 |
1 12 3 5
|
lflcl |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹 ∧ 𝑌 ∈ 𝑉 ) → ( 𝐺 ‘ 𝑌 ) ∈ ( Base ‘ 𝐷 ) ) |
33 |
32
|
3adant3l |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹 ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ) ) → ( 𝐺 ‘ 𝑌 ) ∈ ( Base ‘ 𝐷 ) ) |
34 |
12 29 13 16 9 33
|
ringnegl |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹 ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ) ) → ( ( ( invg ‘ 𝐷 ) ‘ ( 1r ‘ 𝐷 ) ) ( .r ‘ 𝐷 ) ( 𝐺 ‘ 𝑌 ) ) = ( ( invg ‘ 𝐷 ) ‘ ( 𝐺 ‘ 𝑌 ) ) ) |
35 |
34
|
oveq1d |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹 ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ) ) → ( ( ( ( invg ‘ 𝐷 ) ‘ ( 1r ‘ 𝐷 ) ) ( .r ‘ 𝐷 ) ( 𝐺 ‘ 𝑌 ) ) ( +g ‘ 𝐷 ) ( 𝐺 ‘ 𝑋 ) ) = ( ( ( invg ‘ 𝐷 ) ‘ ( 𝐺 ‘ 𝑌 ) ) ( +g ‘ 𝐷 ) ( 𝐺 ‘ 𝑋 ) ) ) |
36 |
|
ringabl |
⊢ ( 𝐷 ∈ Ring → 𝐷 ∈ Abel ) |
37 |
9 36
|
syl |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹 ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ) ) → 𝐷 ∈ Abel ) |
38 |
12 16
|
grpinvcl |
⊢ ( ( 𝐷 ∈ Grp ∧ ( 𝐺 ‘ 𝑌 ) ∈ ( Base ‘ 𝐷 ) ) → ( ( invg ‘ 𝐷 ) ‘ ( 𝐺 ‘ 𝑌 ) ) ∈ ( Base ‘ 𝐷 ) ) |
39 |
11 33 38
|
syl2anc |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹 ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ) ) → ( ( invg ‘ 𝐷 ) ‘ ( 𝐺 ‘ 𝑌 ) ) ∈ ( Base ‘ 𝐷 ) ) |
40 |
1 12 3 5
|
lflcl |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹 ∧ 𝑋 ∈ 𝑉 ) → ( 𝐺 ‘ 𝑋 ) ∈ ( Base ‘ 𝐷 ) ) |
41 |
40
|
3adant3r |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹 ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ) ) → ( 𝐺 ‘ 𝑋 ) ∈ ( Base ‘ 𝐷 ) ) |
42 |
12 28
|
ablcom |
⊢ ( ( 𝐷 ∈ Abel ∧ ( ( invg ‘ 𝐷 ) ‘ ( 𝐺 ‘ 𝑌 ) ) ∈ ( Base ‘ 𝐷 ) ∧ ( 𝐺 ‘ 𝑋 ) ∈ ( Base ‘ 𝐷 ) ) → ( ( ( invg ‘ 𝐷 ) ‘ ( 𝐺 ‘ 𝑌 ) ) ( +g ‘ 𝐷 ) ( 𝐺 ‘ 𝑋 ) ) = ( ( 𝐺 ‘ 𝑋 ) ( +g ‘ 𝐷 ) ( ( invg ‘ 𝐷 ) ‘ ( 𝐺 ‘ 𝑌 ) ) ) ) |
43 |
37 39 41 42
|
syl3anc |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹 ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ) ) → ( ( ( invg ‘ 𝐷 ) ‘ ( 𝐺 ‘ 𝑌 ) ) ( +g ‘ 𝐷 ) ( 𝐺 ‘ 𝑋 ) ) = ( ( 𝐺 ‘ 𝑋 ) ( +g ‘ 𝐷 ) ( ( invg ‘ 𝐷 ) ‘ ( 𝐺 ‘ 𝑌 ) ) ) ) |
44 |
35 43
|
eqtrd |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹 ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ) ) → ( ( ( ( invg ‘ 𝐷 ) ‘ ( 1r ‘ 𝐷 ) ) ( .r ‘ 𝐷 ) ( 𝐺 ‘ 𝑌 ) ) ( +g ‘ 𝐷 ) ( 𝐺 ‘ 𝑋 ) ) = ( ( 𝐺 ‘ 𝑋 ) ( +g ‘ 𝐷 ) ( ( invg ‘ 𝐷 ) ‘ ( 𝐺 ‘ 𝑌 ) ) ) ) |
45 |
26 31 44
|
3eqtrd |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹 ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ) ) → ( 𝐺 ‘ ( 𝑋 ( +g ‘ 𝑊 ) ( ( ( invg ‘ 𝐷 ) ‘ ( 1r ‘ 𝐷 ) ) ( ·𝑠 ‘ 𝑊 ) 𝑌 ) ) ) = ( ( 𝐺 ‘ 𝑋 ) ( +g ‘ 𝐷 ) ( ( invg ‘ 𝐷 ) ‘ ( 𝐺 ‘ 𝑌 ) ) ) ) |
46 |
3 23 4 1 20 16 13
|
lmodvsubval2 |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ) → ( 𝑋 − 𝑌 ) = ( 𝑋 ( +g ‘ 𝑊 ) ( ( ( invg ‘ 𝐷 ) ‘ ( 1r ‘ 𝐷 ) ) ( ·𝑠 ‘ 𝑊 ) 𝑌 ) ) ) |
47 |
6 7 19 46
|
syl3anc |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹 ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ) ) → ( 𝑋 − 𝑌 ) = ( 𝑋 ( +g ‘ 𝑊 ) ( ( ( invg ‘ 𝐷 ) ‘ ( 1r ‘ 𝐷 ) ) ( ·𝑠 ‘ 𝑊 ) 𝑌 ) ) ) |
48 |
47
|
fveq2d |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹 ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ) ) → ( 𝐺 ‘ ( 𝑋 − 𝑌 ) ) = ( 𝐺 ‘ ( 𝑋 ( +g ‘ 𝑊 ) ( ( ( invg ‘ 𝐷 ) ‘ ( 1r ‘ 𝐷 ) ) ( ·𝑠 ‘ 𝑊 ) 𝑌 ) ) ) ) |
49 |
12 28 16 2
|
grpsubval |
⊢ ( ( ( 𝐺 ‘ 𝑋 ) ∈ ( Base ‘ 𝐷 ) ∧ ( 𝐺 ‘ 𝑌 ) ∈ ( Base ‘ 𝐷 ) ) → ( ( 𝐺 ‘ 𝑋 ) 𝑀 ( 𝐺 ‘ 𝑌 ) ) = ( ( 𝐺 ‘ 𝑋 ) ( +g ‘ 𝐷 ) ( ( invg ‘ 𝐷 ) ‘ ( 𝐺 ‘ 𝑌 ) ) ) ) |
50 |
41 33 49
|
syl2anc |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹 ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ) ) → ( ( 𝐺 ‘ 𝑋 ) 𝑀 ( 𝐺 ‘ 𝑌 ) ) = ( ( 𝐺 ‘ 𝑋 ) ( +g ‘ 𝐷 ) ( ( invg ‘ 𝐷 ) ‘ ( 𝐺 ‘ 𝑌 ) ) ) ) |
51 |
45 48 50
|
3eqtr4d |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹 ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ) ) → ( 𝐺 ‘ ( 𝑋 − 𝑌 ) ) = ( ( 𝐺 ‘ 𝑋 ) 𝑀 ( 𝐺 ‘ 𝑌 ) ) ) |