Step |
Hyp |
Ref |
Expression |
1 |
|
quslmod.n |
⊢ 𝑁 = ( 𝑀 /s ( 𝑀 ~QG 𝐺 ) ) |
2 |
|
quslmod.v |
⊢ 𝑉 = ( Base ‘ 𝑀 ) |
3 |
|
quslmod.1 |
⊢ ( 𝜑 → 𝑀 ∈ LMod ) |
4 |
|
quslmod.2 |
⊢ ( 𝜑 → 𝐺 ∈ ( LSubSp ‘ 𝑀 ) ) |
5 |
1
|
a1i |
⊢ ( 𝜑 → 𝑁 = ( 𝑀 /s ( 𝑀 ~QG 𝐺 ) ) ) |
6 |
2
|
a1i |
⊢ ( 𝜑 → 𝑉 = ( Base ‘ 𝑀 ) ) |
7 |
|
eqid |
⊢ ( 𝑥 ∈ 𝑉 ↦ [ 𝑥 ] ( 𝑀 ~QG 𝐺 ) ) = ( 𝑥 ∈ 𝑉 ↦ [ 𝑥 ] ( 𝑀 ~QG 𝐺 ) ) |
8 |
|
ovexd |
⊢ ( 𝜑 → ( 𝑀 ~QG 𝐺 ) ∈ V ) |
9 |
5 6 7 8 3
|
qusval |
⊢ ( 𝜑 → 𝑁 = ( ( 𝑥 ∈ 𝑉 ↦ [ 𝑥 ] ( 𝑀 ~QG 𝐺 ) ) “s 𝑀 ) ) |
10 |
|
eqid |
⊢ ( Base ‘ ( Scalar ‘ 𝑀 ) ) = ( Base ‘ ( Scalar ‘ 𝑀 ) ) |
11 |
|
eqid |
⊢ ( +g ‘ 𝑀 ) = ( +g ‘ 𝑀 ) |
12 |
|
eqid |
⊢ ( ·𝑠 ‘ 𝑀 ) = ( ·𝑠 ‘ 𝑀 ) |
13 |
|
eqid |
⊢ ( 0g ‘ 𝑀 ) = ( 0g ‘ 𝑀 ) |
14 |
5 6 7 8 3
|
quslem |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝑉 ↦ [ 𝑥 ] ( 𝑀 ~QG 𝐺 ) ) : 𝑉 –onto→ ( 𝑉 / ( 𝑀 ~QG 𝐺 ) ) ) |
15 |
|
eqid |
⊢ ( LSubSp ‘ 𝑀 ) = ( LSubSp ‘ 𝑀 ) |
16 |
15
|
lsssubg |
⊢ ( ( 𝑀 ∈ LMod ∧ 𝐺 ∈ ( LSubSp ‘ 𝑀 ) ) → 𝐺 ∈ ( SubGrp ‘ 𝑀 ) ) |
17 |
3 4 16
|
syl2anc |
⊢ ( 𝜑 → 𝐺 ∈ ( SubGrp ‘ 𝑀 ) ) |
18 |
|
eqid |
⊢ ( 𝑀 ~QG 𝐺 ) = ( 𝑀 ~QG 𝐺 ) |
19 |
2 18
|
eqger |
⊢ ( 𝐺 ∈ ( SubGrp ‘ 𝑀 ) → ( 𝑀 ~QG 𝐺 ) Er 𝑉 ) |
20 |
17 19
|
syl |
⊢ ( 𝜑 → ( 𝑀 ~QG 𝐺 ) Er 𝑉 ) |
21 |
2
|
fvexi |
⊢ 𝑉 ∈ V |
22 |
21
|
a1i |
⊢ ( 𝜑 → 𝑉 ∈ V ) |
23 |
|
lmodgrp |
⊢ ( 𝑀 ∈ LMod → 𝑀 ∈ Grp ) |
24 |
3 23
|
syl |
⊢ ( 𝜑 → 𝑀 ∈ Grp ) |
25 |
24
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉 ) ) → 𝑀 ∈ Grp ) |
26 |
|
simprl |
⊢ ( ( 𝜑 ∧ ( 𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉 ) ) → 𝑝 ∈ 𝑉 ) |
27 |
|
simprr |
⊢ ( ( 𝜑 ∧ ( 𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉 ) ) → 𝑞 ∈ 𝑉 ) |
28 |
2 11
|
grpcl |
⊢ ( ( 𝑀 ∈ Grp ∧ 𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉 ) → ( 𝑝 ( +g ‘ 𝑀 ) 𝑞 ) ∈ 𝑉 ) |
29 |
25 26 27 28
|
syl3anc |
⊢ ( ( 𝜑 ∧ ( 𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉 ) ) → ( 𝑝 ( +g ‘ 𝑀 ) 𝑞 ) ∈ 𝑉 ) |
30 |
|
lmodabl |
⊢ ( 𝑀 ∈ LMod → 𝑀 ∈ Abel ) |
31 |
|
ablnsg |
⊢ ( 𝑀 ∈ Abel → ( NrmSGrp ‘ 𝑀 ) = ( SubGrp ‘ 𝑀 ) ) |
32 |
3 30 31
|
3syl |
⊢ ( 𝜑 → ( NrmSGrp ‘ 𝑀 ) = ( SubGrp ‘ 𝑀 ) ) |
33 |
17 32
|
eleqtrrd |
⊢ ( 𝜑 → 𝐺 ∈ ( NrmSGrp ‘ 𝑀 ) ) |
34 |
2 18 11
|
eqgcpbl |
⊢ ( 𝐺 ∈ ( NrmSGrp ‘ 𝑀 ) → ( ( 𝑎 ( 𝑀 ~QG 𝐺 ) 𝑝 ∧ 𝑏 ( 𝑀 ~QG 𝐺 ) 𝑞 ) → ( 𝑎 ( +g ‘ 𝑀 ) 𝑏 ) ( 𝑀 ~QG 𝐺 ) ( 𝑝 ( +g ‘ 𝑀 ) 𝑞 ) ) ) |
35 |
33 34
|
syl |
⊢ ( 𝜑 → ( ( 𝑎 ( 𝑀 ~QG 𝐺 ) 𝑝 ∧ 𝑏 ( 𝑀 ~QG 𝐺 ) 𝑞 ) → ( 𝑎 ( +g ‘ 𝑀 ) 𝑏 ) ( 𝑀 ~QG 𝐺 ) ( 𝑝 ( +g ‘ 𝑀 ) 𝑞 ) ) ) |
36 |
20 22 7 29 35
|
ercpbl |
⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ∧ ( 𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉 ) ) → ( ( ( ( 𝑥 ∈ 𝑉 ↦ [ 𝑥 ] ( 𝑀 ~QG 𝐺 ) ) ‘ 𝑎 ) = ( ( 𝑥 ∈ 𝑉 ↦ [ 𝑥 ] ( 𝑀 ~QG 𝐺 ) ) ‘ 𝑝 ) ∧ ( ( 𝑥 ∈ 𝑉 ↦ [ 𝑥 ] ( 𝑀 ~QG 𝐺 ) ) ‘ 𝑏 ) = ( ( 𝑥 ∈ 𝑉 ↦ [ 𝑥 ] ( 𝑀 ~QG 𝐺 ) ) ‘ 𝑞 ) ) → ( ( 𝑥 ∈ 𝑉 ↦ [ 𝑥 ] ( 𝑀 ~QG 𝐺 ) ) ‘ ( 𝑎 ( +g ‘ 𝑀 ) 𝑏 ) ) = ( ( 𝑥 ∈ 𝑉 ↦ [ 𝑥 ] ( 𝑀 ~QG 𝐺 ) ) ‘ ( 𝑝 ( +g ‘ 𝑀 ) 𝑞 ) ) ) ) |
37 |
3
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑘 ∈ ( Base ‘ ( Scalar ‘ 𝑀 ) ) ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ) → 𝑀 ∈ LMod ) |
38 |
4
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑘 ∈ ( Base ‘ ( Scalar ‘ 𝑀 ) ) ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ) → 𝐺 ∈ ( LSubSp ‘ 𝑀 ) ) |
39 |
|
simpr1 |
⊢ ( ( 𝜑 ∧ ( 𝑘 ∈ ( Base ‘ ( Scalar ‘ 𝑀 ) ) ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ) → 𝑘 ∈ ( Base ‘ ( Scalar ‘ 𝑀 ) ) ) |
40 |
|
eqid |
⊢ ( ·𝑠 ‘ 𝑁 ) = ( ·𝑠 ‘ 𝑁 ) |
41 |
|
simpr2 |
⊢ ( ( 𝜑 ∧ ( 𝑘 ∈ ( Base ‘ ( Scalar ‘ 𝑀 ) ) ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ) → 𝑎 ∈ 𝑉 ) |
42 |
|
simpr3 |
⊢ ( ( 𝜑 ∧ ( 𝑘 ∈ ( Base ‘ ( Scalar ‘ 𝑀 ) ) ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ) → 𝑏 ∈ 𝑉 ) |
43 |
2 18 10 12 37 38 39 1 40 7 41 42
|
qusvscpbl |
⊢ ( ( 𝜑 ∧ ( 𝑘 ∈ ( Base ‘ ( Scalar ‘ 𝑀 ) ) ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ) → ( ( ( 𝑥 ∈ 𝑉 ↦ [ 𝑥 ] ( 𝑀 ~QG 𝐺 ) ) ‘ 𝑎 ) = ( ( 𝑥 ∈ 𝑉 ↦ [ 𝑥 ] ( 𝑀 ~QG 𝐺 ) ) ‘ 𝑏 ) → ( ( 𝑥 ∈ 𝑉 ↦ [ 𝑥 ] ( 𝑀 ~QG 𝐺 ) ) ‘ ( 𝑘 ( ·𝑠 ‘ 𝑀 ) 𝑎 ) ) = ( ( 𝑥 ∈ 𝑉 ↦ [ 𝑥 ] ( 𝑀 ~QG 𝐺 ) ) ‘ ( 𝑘 ( ·𝑠 ‘ 𝑀 ) 𝑏 ) ) ) ) |
44 |
9 2 10 11 12 13 14 36 43 3
|
imaslmod |
⊢ ( 𝜑 → 𝑁 ∈ LMod ) |