Step |
Hyp |
Ref |
Expression |
1 |
|
lsmelvalm.m |
⊢ − = ( -g ‘ 𝐺 ) |
2 |
|
lsmelvalm.p |
⊢ ⊕ = ( LSSum ‘ 𝐺 ) |
3 |
|
lsmelvalm.t |
⊢ ( 𝜑 → 𝑇 ∈ ( SubGrp ‘ 𝐺 ) ) |
4 |
|
lsmelvalm.u |
⊢ ( 𝜑 → 𝑈 ∈ ( SubGrp ‘ 𝐺 ) ) |
5 |
|
eqid |
⊢ ( +g ‘ 𝐺 ) = ( +g ‘ 𝐺 ) |
6 |
5 2
|
lsmelval |
⊢ ( ( 𝑇 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑈 ∈ ( SubGrp ‘ 𝐺 ) ) → ( 𝑋 ∈ ( 𝑇 ⊕ 𝑈 ) ↔ ∃ 𝑦 ∈ 𝑇 ∃ 𝑥 ∈ 𝑈 𝑋 = ( 𝑦 ( +g ‘ 𝐺 ) 𝑥 ) ) ) |
7 |
3 4 6
|
syl2anc |
⊢ ( 𝜑 → ( 𝑋 ∈ ( 𝑇 ⊕ 𝑈 ) ↔ ∃ 𝑦 ∈ 𝑇 ∃ 𝑥 ∈ 𝑈 𝑋 = ( 𝑦 ( +g ‘ 𝐺 ) 𝑥 ) ) ) |
8 |
4
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝑇 ) → 𝑈 ∈ ( SubGrp ‘ 𝐺 ) ) |
9 |
|
eqid |
⊢ ( invg ‘ 𝐺 ) = ( invg ‘ 𝐺 ) |
10 |
9
|
subginvcl |
⊢ ( ( 𝑈 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑥 ∈ 𝑈 ) → ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) ∈ 𝑈 ) |
11 |
8 10
|
sylan |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝑇 ) ∧ 𝑥 ∈ 𝑈 ) → ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) ∈ 𝑈 ) |
12 |
|
eqid |
⊢ ( Base ‘ 𝐺 ) = ( Base ‘ 𝐺 ) |
13 |
|
subgrcl |
⊢ ( 𝑇 ∈ ( SubGrp ‘ 𝐺 ) → 𝐺 ∈ Grp ) |
14 |
3 13
|
syl |
⊢ ( 𝜑 → 𝐺 ∈ Grp ) |
15 |
14
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝑇 ) ∧ 𝑥 ∈ 𝑈 ) → 𝐺 ∈ Grp ) |
16 |
12
|
subgss |
⊢ ( 𝑇 ∈ ( SubGrp ‘ 𝐺 ) → 𝑇 ⊆ ( Base ‘ 𝐺 ) ) |
17 |
3 16
|
syl |
⊢ ( 𝜑 → 𝑇 ⊆ ( Base ‘ 𝐺 ) ) |
18 |
17
|
sselda |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝑇 ) → 𝑦 ∈ ( Base ‘ 𝐺 ) ) |
19 |
18
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝑇 ) ∧ 𝑥 ∈ 𝑈 ) → 𝑦 ∈ ( Base ‘ 𝐺 ) ) |
20 |
12
|
subgss |
⊢ ( 𝑈 ∈ ( SubGrp ‘ 𝐺 ) → 𝑈 ⊆ ( Base ‘ 𝐺 ) ) |
21 |
8 20
|
syl |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝑇 ) → 𝑈 ⊆ ( Base ‘ 𝐺 ) ) |
22 |
21
|
sselda |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝑇 ) ∧ 𝑥 ∈ 𝑈 ) → 𝑥 ∈ ( Base ‘ 𝐺 ) ) |
23 |
12 5 1 9 15 19 22
|
grpsubinv |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝑇 ) ∧ 𝑥 ∈ 𝑈 ) → ( 𝑦 − ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) ) = ( 𝑦 ( +g ‘ 𝐺 ) 𝑥 ) ) |
24 |
23
|
eqcomd |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝑇 ) ∧ 𝑥 ∈ 𝑈 ) → ( 𝑦 ( +g ‘ 𝐺 ) 𝑥 ) = ( 𝑦 − ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) ) ) |
25 |
|
oveq2 |
⊢ ( 𝑧 = ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) → ( 𝑦 − 𝑧 ) = ( 𝑦 − ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) ) ) |
26 |
25
|
rspceeqv |
⊢ ( ( ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) ∈ 𝑈 ∧ ( 𝑦 ( +g ‘ 𝐺 ) 𝑥 ) = ( 𝑦 − ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) ) ) → ∃ 𝑧 ∈ 𝑈 ( 𝑦 ( +g ‘ 𝐺 ) 𝑥 ) = ( 𝑦 − 𝑧 ) ) |
27 |
11 24 26
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝑇 ) ∧ 𝑥 ∈ 𝑈 ) → ∃ 𝑧 ∈ 𝑈 ( 𝑦 ( +g ‘ 𝐺 ) 𝑥 ) = ( 𝑦 − 𝑧 ) ) |
28 |
|
eqeq1 |
⊢ ( 𝑋 = ( 𝑦 ( +g ‘ 𝐺 ) 𝑥 ) → ( 𝑋 = ( 𝑦 − 𝑧 ) ↔ ( 𝑦 ( +g ‘ 𝐺 ) 𝑥 ) = ( 𝑦 − 𝑧 ) ) ) |
29 |
28
|
rexbidv |
⊢ ( 𝑋 = ( 𝑦 ( +g ‘ 𝐺 ) 𝑥 ) → ( ∃ 𝑧 ∈ 𝑈 𝑋 = ( 𝑦 − 𝑧 ) ↔ ∃ 𝑧 ∈ 𝑈 ( 𝑦 ( +g ‘ 𝐺 ) 𝑥 ) = ( 𝑦 − 𝑧 ) ) ) |
30 |
27 29
|
syl5ibrcom |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝑇 ) ∧ 𝑥 ∈ 𝑈 ) → ( 𝑋 = ( 𝑦 ( +g ‘ 𝐺 ) 𝑥 ) → ∃ 𝑧 ∈ 𝑈 𝑋 = ( 𝑦 − 𝑧 ) ) ) |
31 |
30
|
rexlimdva |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝑇 ) → ( ∃ 𝑥 ∈ 𝑈 𝑋 = ( 𝑦 ( +g ‘ 𝐺 ) 𝑥 ) → ∃ 𝑧 ∈ 𝑈 𝑋 = ( 𝑦 − 𝑧 ) ) ) |
32 |
9
|
subginvcl |
⊢ ( ( 𝑈 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑧 ∈ 𝑈 ) → ( ( invg ‘ 𝐺 ) ‘ 𝑧 ) ∈ 𝑈 ) |
33 |
8 32
|
sylan |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝑇 ) ∧ 𝑧 ∈ 𝑈 ) → ( ( invg ‘ 𝐺 ) ‘ 𝑧 ) ∈ 𝑈 ) |
34 |
18
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝑇 ) ∧ 𝑧 ∈ 𝑈 ) → 𝑦 ∈ ( Base ‘ 𝐺 ) ) |
35 |
21
|
sselda |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝑇 ) ∧ 𝑧 ∈ 𝑈 ) → 𝑧 ∈ ( Base ‘ 𝐺 ) ) |
36 |
12 5 9 1
|
grpsubval |
⊢ ( ( 𝑦 ∈ ( Base ‘ 𝐺 ) ∧ 𝑧 ∈ ( Base ‘ 𝐺 ) ) → ( 𝑦 − 𝑧 ) = ( 𝑦 ( +g ‘ 𝐺 ) ( ( invg ‘ 𝐺 ) ‘ 𝑧 ) ) ) |
37 |
34 35 36
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝑇 ) ∧ 𝑧 ∈ 𝑈 ) → ( 𝑦 − 𝑧 ) = ( 𝑦 ( +g ‘ 𝐺 ) ( ( invg ‘ 𝐺 ) ‘ 𝑧 ) ) ) |
38 |
|
oveq2 |
⊢ ( 𝑥 = ( ( invg ‘ 𝐺 ) ‘ 𝑧 ) → ( 𝑦 ( +g ‘ 𝐺 ) 𝑥 ) = ( 𝑦 ( +g ‘ 𝐺 ) ( ( invg ‘ 𝐺 ) ‘ 𝑧 ) ) ) |
39 |
38
|
rspceeqv |
⊢ ( ( ( ( invg ‘ 𝐺 ) ‘ 𝑧 ) ∈ 𝑈 ∧ ( 𝑦 − 𝑧 ) = ( 𝑦 ( +g ‘ 𝐺 ) ( ( invg ‘ 𝐺 ) ‘ 𝑧 ) ) ) → ∃ 𝑥 ∈ 𝑈 ( 𝑦 − 𝑧 ) = ( 𝑦 ( +g ‘ 𝐺 ) 𝑥 ) ) |
40 |
33 37 39
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝑇 ) ∧ 𝑧 ∈ 𝑈 ) → ∃ 𝑥 ∈ 𝑈 ( 𝑦 − 𝑧 ) = ( 𝑦 ( +g ‘ 𝐺 ) 𝑥 ) ) |
41 |
|
eqeq1 |
⊢ ( 𝑋 = ( 𝑦 − 𝑧 ) → ( 𝑋 = ( 𝑦 ( +g ‘ 𝐺 ) 𝑥 ) ↔ ( 𝑦 − 𝑧 ) = ( 𝑦 ( +g ‘ 𝐺 ) 𝑥 ) ) ) |
42 |
41
|
rexbidv |
⊢ ( 𝑋 = ( 𝑦 − 𝑧 ) → ( ∃ 𝑥 ∈ 𝑈 𝑋 = ( 𝑦 ( +g ‘ 𝐺 ) 𝑥 ) ↔ ∃ 𝑥 ∈ 𝑈 ( 𝑦 − 𝑧 ) = ( 𝑦 ( +g ‘ 𝐺 ) 𝑥 ) ) ) |
43 |
40 42
|
syl5ibrcom |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝑇 ) ∧ 𝑧 ∈ 𝑈 ) → ( 𝑋 = ( 𝑦 − 𝑧 ) → ∃ 𝑥 ∈ 𝑈 𝑋 = ( 𝑦 ( +g ‘ 𝐺 ) 𝑥 ) ) ) |
44 |
43
|
rexlimdva |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝑇 ) → ( ∃ 𝑧 ∈ 𝑈 𝑋 = ( 𝑦 − 𝑧 ) → ∃ 𝑥 ∈ 𝑈 𝑋 = ( 𝑦 ( +g ‘ 𝐺 ) 𝑥 ) ) ) |
45 |
31 44
|
impbid |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝑇 ) → ( ∃ 𝑥 ∈ 𝑈 𝑋 = ( 𝑦 ( +g ‘ 𝐺 ) 𝑥 ) ↔ ∃ 𝑧 ∈ 𝑈 𝑋 = ( 𝑦 − 𝑧 ) ) ) |
46 |
45
|
rexbidva |
⊢ ( 𝜑 → ( ∃ 𝑦 ∈ 𝑇 ∃ 𝑥 ∈ 𝑈 𝑋 = ( 𝑦 ( +g ‘ 𝐺 ) 𝑥 ) ↔ ∃ 𝑦 ∈ 𝑇 ∃ 𝑧 ∈ 𝑈 𝑋 = ( 𝑦 − 𝑧 ) ) ) |
47 |
7 46
|
bitrd |
⊢ ( 𝜑 → ( 𝑋 ∈ ( 𝑇 ⊕ 𝑈 ) ↔ ∃ 𝑦 ∈ 𝑇 ∃ 𝑧 ∈ 𝑈 𝑋 = ( 𝑦 − 𝑧 ) ) ) |