Step |
Hyp |
Ref |
Expression |
1 |
|
eqgvscpbl.v |
⊢ 𝐵 = ( Base ‘ 𝑀 ) |
2 |
|
eqgvscpbl.e |
⊢ ∼ = ( 𝑀 ~QG 𝐺 ) |
3 |
|
eqgvscpbl.s |
⊢ 𝑆 = ( Base ‘ ( Scalar ‘ 𝑀 ) ) |
4 |
|
eqgvscpbl.p |
⊢ · = ( ·𝑠 ‘ 𝑀 ) |
5 |
|
eqgvscpbl.m |
⊢ ( 𝜑 → 𝑀 ∈ LMod ) |
6 |
|
eqgvscpbl.g |
⊢ ( 𝜑 → 𝐺 ∈ ( LSubSp ‘ 𝑀 ) ) |
7 |
|
eqgvscpbl.k |
⊢ ( 𝜑 → 𝐾 ∈ 𝑆 ) |
8 |
5
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ ( ( ( invg ‘ 𝑀 ) ‘ 𝑋 ) ( +g ‘ 𝑀 ) 𝑌 ) ∈ 𝐺 ) ) → 𝑀 ∈ LMod ) |
9 |
7
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ ( ( ( invg ‘ 𝑀 ) ‘ 𝑋 ) ( +g ‘ 𝑀 ) 𝑌 ) ∈ 𝐺 ) ) → 𝐾 ∈ 𝑆 ) |
10 |
|
simpr1 |
⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ ( ( ( invg ‘ 𝑀 ) ‘ 𝑋 ) ( +g ‘ 𝑀 ) 𝑌 ) ∈ 𝐺 ) ) → 𝑋 ∈ 𝐵 ) |
11 |
|
eqid |
⊢ ( Scalar ‘ 𝑀 ) = ( Scalar ‘ 𝑀 ) |
12 |
1 11 4 3
|
lmodvscl |
⊢ ( ( 𝑀 ∈ LMod ∧ 𝐾 ∈ 𝑆 ∧ 𝑋 ∈ 𝐵 ) → ( 𝐾 · 𝑋 ) ∈ 𝐵 ) |
13 |
8 9 10 12
|
syl3anc |
⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ ( ( ( invg ‘ 𝑀 ) ‘ 𝑋 ) ( +g ‘ 𝑀 ) 𝑌 ) ∈ 𝐺 ) ) → ( 𝐾 · 𝑋 ) ∈ 𝐵 ) |
14 |
|
simpr2 |
⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ ( ( ( invg ‘ 𝑀 ) ‘ 𝑋 ) ( +g ‘ 𝑀 ) 𝑌 ) ∈ 𝐺 ) ) → 𝑌 ∈ 𝐵 ) |
15 |
1 11 4 3
|
lmodvscl |
⊢ ( ( 𝑀 ∈ LMod ∧ 𝐾 ∈ 𝑆 ∧ 𝑌 ∈ 𝐵 ) → ( 𝐾 · 𝑌 ) ∈ 𝐵 ) |
16 |
8 9 14 15
|
syl3anc |
⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ ( ( ( invg ‘ 𝑀 ) ‘ 𝑋 ) ( +g ‘ 𝑀 ) 𝑌 ) ∈ 𝐺 ) ) → ( 𝐾 · 𝑌 ) ∈ 𝐵 ) |
17 |
5
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑋 ∈ 𝐵 ) ∧ 𝑌 ∈ 𝐵 ) → 𝑀 ∈ LMod ) |
18 |
7
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑋 ∈ 𝐵 ) ∧ 𝑌 ∈ 𝐵 ) → 𝐾 ∈ 𝑆 ) |
19 |
|
lmodgrp |
⊢ ( 𝑀 ∈ LMod → 𝑀 ∈ Grp ) |
20 |
17 19
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝑋 ∈ 𝐵 ) ∧ 𝑌 ∈ 𝐵 ) → 𝑀 ∈ Grp ) |
21 |
|
simplr |
⊢ ( ( ( 𝜑 ∧ 𝑋 ∈ 𝐵 ) ∧ 𝑌 ∈ 𝐵 ) → 𝑋 ∈ 𝐵 ) |
22 |
|
eqid |
⊢ ( invg ‘ 𝑀 ) = ( invg ‘ 𝑀 ) |
23 |
1 22
|
grpinvcl |
⊢ ( ( 𝑀 ∈ Grp ∧ 𝑋 ∈ 𝐵 ) → ( ( invg ‘ 𝑀 ) ‘ 𝑋 ) ∈ 𝐵 ) |
24 |
20 21 23
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑋 ∈ 𝐵 ) ∧ 𝑌 ∈ 𝐵 ) → ( ( invg ‘ 𝑀 ) ‘ 𝑋 ) ∈ 𝐵 ) |
25 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝑋 ∈ 𝐵 ) ∧ 𝑌 ∈ 𝐵 ) → 𝑌 ∈ 𝐵 ) |
26 |
|
eqid |
⊢ ( +g ‘ 𝑀 ) = ( +g ‘ 𝑀 ) |
27 |
1 26 11 4 3
|
lmodvsdi |
⊢ ( ( 𝑀 ∈ LMod ∧ ( 𝐾 ∈ 𝑆 ∧ ( ( invg ‘ 𝑀 ) ‘ 𝑋 ) ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( 𝐾 · ( ( ( invg ‘ 𝑀 ) ‘ 𝑋 ) ( +g ‘ 𝑀 ) 𝑌 ) ) = ( ( 𝐾 · ( ( invg ‘ 𝑀 ) ‘ 𝑋 ) ) ( +g ‘ 𝑀 ) ( 𝐾 · 𝑌 ) ) ) |
28 |
17 18 24 25 27
|
syl13anc |
⊢ ( ( ( 𝜑 ∧ 𝑋 ∈ 𝐵 ) ∧ 𝑌 ∈ 𝐵 ) → ( 𝐾 · ( ( ( invg ‘ 𝑀 ) ‘ 𝑋 ) ( +g ‘ 𝑀 ) 𝑌 ) ) = ( ( 𝐾 · ( ( invg ‘ 𝑀 ) ‘ 𝑋 ) ) ( +g ‘ 𝑀 ) ( 𝐾 · 𝑌 ) ) ) |
29 |
1 11 4 22 3
|
lmodvsinv2 |
⊢ ( ( 𝑀 ∈ LMod ∧ 𝐾 ∈ 𝑆 ∧ 𝑋 ∈ 𝐵 ) → ( 𝐾 · ( ( invg ‘ 𝑀 ) ‘ 𝑋 ) ) = ( ( invg ‘ 𝑀 ) ‘ ( 𝐾 · 𝑋 ) ) ) |
30 |
17 18 21 29
|
syl3anc |
⊢ ( ( ( 𝜑 ∧ 𝑋 ∈ 𝐵 ) ∧ 𝑌 ∈ 𝐵 ) → ( 𝐾 · ( ( invg ‘ 𝑀 ) ‘ 𝑋 ) ) = ( ( invg ‘ 𝑀 ) ‘ ( 𝐾 · 𝑋 ) ) ) |
31 |
30
|
oveq1d |
⊢ ( ( ( 𝜑 ∧ 𝑋 ∈ 𝐵 ) ∧ 𝑌 ∈ 𝐵 ) → ( ( 𝐾 · ( ( invg ‘ 𝑀 ) ‘ 𝑋 ) ) ( +g ‘ 𝑀 ) ( 𝐾 · 𝑌 ) ) = ( ( ( invg ‘ 𝑀 ) ‘ ( 𝐾 · 𝑋 ) ) ( +g ‘ 𝑀 ) ( 𝐾 · 𝑌 ) ) ) |
32 |
28 31
|
eqtrd |
⊢ ( ( ( 𝜑 ∧ 𝑋 ∈ 𝐵 ) ∧ 𝑌 ∈ 𝐵 ) → ( 𝐾 · ( ( ( invg ‘ 𝑀 ) ‘ 𝑋 ) ( +g ‘ 𝑀 ) 𝑌 ) ) = ( ( ( invg ‘ 𝑀 ) ‘ ( 𝐾 · 𝑋 ) ) ( +g ‘ 𝑀 ) ( 𝐾 · 𝑌 ) ) ) |
33 |
32
|
anasss |
⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( 𝐾 · ( ( ( invg ‘ 𝑀 ) ‘ 𝑋 ) ( +g ‘ 𝑀 ) 𝑌 ) ) = ( ( ( invg ‘ 𝑀 ) ‘ ( 𝐾 · 𝑋 ) ) ( +g ‘ 𝑀 ) ( 𝐾 · 𝑌 ) ) ) |
34 |
33
|
3adantr3 |
⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ ( ( ( invg ‘ 𝑀 ) ‘ 𝑋 ) ( +g ‘ 𝑀 ) 𝑌 ) ∈ 𝐺 ) ) → ( 𝐾 · ( ( ( invg ‘ 𝑀 ) ‘ 𝑋 ) ( +g ‘ 𝑀 ) 𝑌 ) ) = ( ( ( invg ‘ 𝑀 ) ‘ ( 𝐾 · 𝑋 ) ) ( +g ‘ 𝑀 ) ( 𝐾 · 𝑌 ) ) ) |
35 |
6
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ ( ( ( invg ‘ 𝑀 ) ‘ 𝑋 ) ( +g ‘ 𝑀 ) 𝑌 ) ∈ 𝐺 ) ) → 𝐺 ∈ ( LSubSp ‘ 𝑀 ) ) |
36 |
|
simpr3 |
⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ ( ( ( invg ‘ 𝑀 ) ‘ 𝑋 ) ( +g ‘ 𝑀 ) 𝑌 ) ∈ 𝐺 ) ) → ( ( ( invg ‘ 𝑀 ) ‘ 𝑋 ) ( +g ‘ 𝑀 ) 𝑌 ) ∈ 𝐺 ) |
37 |
|
eqid |
⊢ ( LSubSp ‘ 𝑀 ) = ( LSubSp ‘ 𝑀 ) |
38 |
11 4 3 37
|
lssvscl |
⊢ ( ( ( 𝑀 ∈ LMod ∧ 𝐺 ∈ ( LSubSp ‘ 𝑀 ) ) ∧ ( 𝐾 ∈ 𝑆 ∧ ( ( ( invg ‘ 𝑀 ) ‘ 𝑋 ) ( +g ‘ 𝑀 ) 𝑌 ) ∈ 𝐺 ) ) → ( 𝐾 · ( ( ( invg ‘ 𝑀 ) ‘ 𝑋 ) ( +g ‘ 𝑀 ) 𝑌 ) ) ∈ 𝐺 ) |
39 |
8 35 9 36 38
|
syl22anc |
⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ ( ( ( invg ‘ 𝑀 ) ‘ 𝑋 ) ( +g ‘ 𝑀 ) 𝑌 ) ∈ 𝐺 ) ) → ( 𝐾 · ( ( ( invg ‘ 𝑀 ) ‘ 𝑋 ) ( +g ‘ 𝑀 ) 𝑌 ) ) ∈ 𝐺 ) |
40 |
34 39
|
eqeltrrd |
⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ ( ( ( invg ‘ 𝑀 ) ‘ 𝑋 ) ( +g ‘ 𝑀 ) 𝑌 ) ∈ 𝐺 ) ) → ( ( ( invg ‘ 𝑀 ) ‘ ( 𝐾 · 𝑋 ) ) ( +g ‘ 𝑀 ) ( 𝐾 · 𝑌 ) ) ∈ 𝐺 ) |
41 |
13 16 40
|
3jca |
⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ ( ( ( invg ‘ 𝑀 ) ‘ 𝑋 ) ( +g ‘ 𝑀 ) 𝑌 ) ∈ 𝐺 ) ) → ( ( 𝐾 · 𝑋 ) ∈ 𝐵 ∧ ( 𝐾 · 𝑌 ) ∈ 𝐵 ∧ ( ( ( invg ‘ 𝑀 ) ‘ ( 𝐾 · 𝑋 ) ) ( +g ‘ 𝑀 ) ( 𝐾 · 𝑌 ) ) ∈ 𝐺 ) ) |
42 |
41
|
ex |
⊢ ( 𝜑 → ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ ( ( ( invg ‘ 𝑀 ) ‘ 𝑋 ) ( +g ‘ 𝑀 ) 𝑌 ) ∈ 𝐺 ) → ( ( 𝐾 · 𝑋 ) ∈ 𝐵 ∧ ( 𝐾 · 𝑌 ) ∈ 𝐵 ∧ ( ( ( invg ‘ 𝑀 ) ‘ ( 𝐾 · 𝑋 ) ) ( +g ‘ 𝑀 ) ( 𝐾 · 𝑌 ) ) ∈ 𝐺 ) ) ) |
43 |
5 19
|
syl |
⊢ ( 𝜑 → 𝑀 ∈ Grp ) |
44 |
37
|
lsssubg |
⊢ ( ( 𝑀 ∈ LMod ∧ 𝐺 ∈ ( LSubSp ‘ 𝑀 ) ) → 𝐺 ∈ ( SubGrp ‘ 𝑀 ) ) |
45 |
5 6 44
|
syl2anc |
⊢ ( 𝜑 → 𝐺 ∈ ( SubGrp ‘ 𝑀 ) ) |
46 |
1
|
subgss |
⊢ ( 𝐺 ∈ ( SubGrp ‘ 𝑀 ) → 𝐺 ⊆ 𝐵 ) |
47 |
45 46
|
syl |
⊢ ( 𝜑 → 𝐺 ⊆ 𝐵 ) |
48 |
1 22 26 2
|
eqgval |
⊢ ( ( 𝑀 ∈ Grp ∧ 𝐺 ⊆ 𝐵 ) → ( 𝑋 ∼ 𝑌 ↔ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ ( ( ( invg ‘ 𝑀 ) ‘ 𝑋 ) ( +g ‘ 𝑀 ) 𝑌 ) ∈ 𝐺 ) ) ) |
49 |
43 47 48
|
syl2anc |
⊢ ( 𝜑 → ( 𝑋 ∼ 𝑌 ↔ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ ( ( ( invg ‘ 𝑀 ) ‘ 𝑋 ) ( +g ‘ 𝑀 ) 𝑌 ) ∈ 𝐺 ) ) ) |
50 |
1 22 26 2
|
eqgval |
⊢ ( ( 𝑀 ∈ Grp ∧ 𝐺 ⊆ 𝐵 ) → ( ( 𝐾 · 𝑋 ) ∼ ( 𝐾 · 𝑌 ) ↔ ( ( 𝐾 · 𝑋 ) ∈ 𝐵 ∧ ( 𝐾 · 𝑌 ) ∈ 𝐵 ∧ ( ( ( invg ‘ 𝑀 ) ‘ ( 𝐾 · 𝑋 ) ) ( +g ‘ 𝑀 ) ( 𝐾 · 𝑌 ) ) ∈ 𝐺 ) ) ) |
51 |
43 47 50
|
syl2anc |
⊢ ( 𝜑 → ( ( 𝐾 · 𝑋 ) ∼ ( 𝐾 · 𝑌 ) ↔ ( ( 𝐾 · 𝑋 ) ∈ 𝐵 ∧ ( 𝐾 · 𝑌 ) ∈ 𝐵 ∧ ( ( ( invg ‘ 𝑀 ) ‘ ( 𝐾 · 𝑋 ) ) ( +g ‘ 𝑀 ) ( 𝐾 · 𝑌 ) ) ∈ 𝐺 ) ) ) |
52 |
42 49 51
|
3imtr4d |
⊢ ( 𝜑 → ( 𝑋 ∼ 𝑌 → ( 𝐾 · 𝑋 ) ∼ ( 𝐾 · 𝑌 ) ) ) |