| Step |
Hyp |
Ref |
Expression |
| 1 |
|
lsmmod.p |
⊢ ⊕ = ( LSSum ‘ 𝐺 ) |
| 2 |
|
simpl1 |
⊢ ( ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑇 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑈 ∈ ( SubGrp ‘ 𝐺 ) ) ∧ 𝑆 ⊆ 𝑈 ) → 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ) |
| 3 |
|
simpl2 |
⊢ ( ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑇 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑈 ∈ ( SubGrp ‘ 𝐺 ) ) ∧ 𝑆 ⊆ 𝑈 ) → 𝑇 ∈ ( SubGrp ‘ 𝐺 ) ) |
| 4 |
|
inss1 |
⊢ ( 𝑇 ∩ 𝑈 ) ⊆ 𝑇 |
| 5 |
4
|
a1i |
⊢ ( ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑇 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑈 ∈ ( SubGrp ‘ 𝐺 ) ) ∧ 𝑆 ⊆ 𝑈 ) → ( 𝑇 ∩ 𝑈 ) ⊆ 𝑇 ) |
| 6 |
1
|
lsmless2 |
⊢ ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑇 ∈ ( SubGrp ‘ 𝐺 ) ∧ ( 𝑇 ∩ 𝑈 ) ⊆ 𝑇 ) → ( 𝑆 ⊕ ( 𝑇 ∩ 𝑈 ) ) ⊆ ( 𝑆 ⊕ 𝑇 ) ) |
| 7 |
2 3 5 6
|
syl3anc |
⊢ ( ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑇 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑈 ∈ ( SubGrp ‘ 𝐺 ) ) ∧ 𝑆 ⊆ 𝑈 ) → ( 𝑆 ⊕ ( 𝑇 ∩ 𝑈 ) ) ⊆ ( 𝑆 ⊕ 𝑇 ) ) |
| 8 |
|
simpr |
⊢ ( ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑇 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑈 ∈ ( SubGrp ‘ 𝐺 ) ) ∧ 𝑆 ⊆ 𝑈 ) → 𝑆 ⊆ 𝑈 ) |
| 9 |
|
inss2 |
⊢ ( 𝑇 ∩ 𝑈 ) ⊆ 𝑈 |
| 10 |
9
|
a1i |
⊢ ( ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑇 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑈 ∈ ( SubGrp ‘ 𝐺 ) ) ∧ 𝑆 ⊆ 𝑈 ) → ( 𝑇 ∩ 𝑈 ) ⊆ 𝑈 ) |
| 11 |
|
subgrcl |
⊢ ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) → 𝐺 ∈ Grp ) |
| 12 |
|
eqid |
⊢ ( Base ‘ 𝐺 ) = ( Base ‘ 𝐺 ) |
| 13 |
12
|
subgacs |
⊢ ( 𝐺 ∈ Grp → ( SubGrp ‘ 𝐺 ) ∈ ( ACS ‘ ( Base ‘ 𝐺 ) ) ) |
| 14 |
|
acsmre |
⊢ ( ( SubGrp ‘ 𝐺 ) ∈ ( ACS ‘ ( Base ‘ 𝐺 ) ) → ( SubGrp ‘ 𝐺 ) ∈ ( Moore ‘ ( Base ‘ 𝐺 ) ) ) |
| 15 |
2 11 13 14
|
4syl |
⊢ ( ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑇 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑈 ∈ ( SubGrp ‘ 𝐺 ) ) ∧ 𝑆 ⊆ 𝑈 ) → ( SubGrp ‘ 𝐺 ) ∈ ( Moore ‘ ( Base ‘ 𝐺 ) ) ) |
| 16 |
|
simpl3 |
⊢ ( ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑇 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑈 ∈ ( SubGrp ‘ 𝐺 ) ) ∧ 𝑆 ⊆ 𝑈 ) → 𝑈 ∈ ( SubGrp ‘ 𝐺 ) ) |
| 17 |
|
mreincl |
⊢ ( ( ( SubGrp ‘ 𝐺 ) ∈ ( Moore ‘ ( Base ‘ 𝐺 ) ) ∧ 𝑇 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑈 ∈ ( SubGrp ‘ 𝐺 ) ) → ( 𝑇 ∩ 𝑈 ) ∈ ( SubGrp ‘ 𝐺 ) ) |
| 18 |
15 3 16 17
|
syl3anc |
⊢ ( ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑇 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑈 ∈ ( SubGrp ‘ 𝐺 ) ) ∧ 𝑆 ⊆ 𝑈 ) → ( 𝑇 ∩ 𝑈 ) ∈ ( SubGrp ‘ 𝐺 ) ) |
| 19 |
1
|
lsmlub |
⊢ ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ ( 𝑇 ∩ 𝑈 ) ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑈 ∈ ( SubGrp ‘ 𝐺 ) ) → ( ( 𝑆 ⊆ 𝑈 ∧ ( 𝑇 ∩ 𝑈 ) ⊆ 𝑈 ) ↔ ( 𝑆 ⊕ ( 𝑇 ∩ 𝑈 ) ) ⊆ 𝑈 ) ) |
| 20 |
2 18 16 19
|
syl3anc |
⊢ ( ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑇 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑈 ∈ ( SubGrp ‘ 𝐺 ) ) ∧ 𝑆 ⊆ 𝑈 ) → ( ( 𝑆 ⊆ 𝑈 ∧ ( 𝑇 ∩ 𝑈 ) ⊆ 𝑈 ) ↔ ( 𝑆 ⊕ ( 𝑇 ∩ 𝑈 ) ) ⊆ 𝑈 ) ) |
| 21 |
8 10 20
|
mpbi2and |
⊢ ( ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑇 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑈 ∈ ( SubGrp ‘ 𝐺 ) ) ∧ 𝑆 ⊆ 𝑈 ) → ( 𝑆 ⊕ ( 𝑇 ∩ 𝑈 ) ) ⊆ 𝑈 ) |
| 22 |
7 21
|
ssind |
⊢ ( ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑇 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑈 ∈ ( SubGrp ‘ 𝐺 ) ) ∧ 𝑆 ⊆ 𝑈 ) → ( 𝑆 ⊕ ( 𝑇 ∩ 𝑈 ) ) ⊆ ( ( 𝑆 ⊕ 𝑇 ) ∩ 𝑈 ) ) |
| 23 |
|
elin |
⊢ ( 𝑥 ∈ ( ( 𝑆 ⊕ 𝑇 ) ∩ 𝑈 ) ↔ ( 𝑥 ∈ ( 𝑆 ⊕ 𝑇 ) ∧ 𝑥 ∈ 𝑈 ) ) |
| 24 |
|
eqid |
⊢ ( +g ‘ 𝐺 ) = ( +g ‘ 𝐺 ) |
| 25 |
24 1
|
lsmelval |
⊢ ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑇 ∈ ( SubGrp ‘ 𝐺 ) ) → ( 𝑥 ∈ ( 𝑆 ⊕ 𝑇 ) ↔ ∃ 𝑦 ∈ 𝑆 ∃ 𝑧 ∈ 𝑇 𝑥 = ( 𝑦 ( +g ‘ 𝐺 ) 𝑧 ) ) ) |
| 26 |
2 3 25
|
syl2anc |
⊢ ( ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑇 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑈 ∈ ( SubGrp ‘ 𝐺 ) ) ∧ 𝑆 ⊆ 𝑈 ) → ( 𝑥 ∈ ( 𝑆 ⊕ 𝑇 ) ↔ ∃ 𝑦 ∈ 𝑆 ∃ 𝑧 ∈ 𝑇 𝑥 = ( 𝑦 ( +g ‘ 𝐺 ) 𝑧 ) ) ) |
| 27 |
2
|
adantr |
⊢ ( ( ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑇 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑈 ∈ ( SubGrp ‘ 𝐺 ) ) ∧ 𝑆 ⊆ 𝑈 ) ∧ ( ( 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑇 ) ∧ ( 𝑦 ( +g ‘ 𝐺 ) 𝑧 ) ∈ 𝑈 ) ) → 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ) |
| 28 |
18
|
adantr |
⊢ ( ( ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑇 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑈 ∈ ( SubGrp ‘ 𝐺 ) ) ∧ 𝑆 ⊆ 𝑈 ) ∧ ( ( 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑇 ) ∧ ( 𝑦 ( +g ‘ 𝐺 ) 𝑧 ) ∈ 𝑈 ) ) → ( 𝑇 ∩ 𝑈 ) ∈ ( SubGrp ‘ 𝐺 ) ) |
| 29 |
|
simprll |
⊢ ( ( ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑇 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑈 ∈ ( SubGrp ‘ 𝐺 ) ) ∧ 𝑆 ⊆ 𝑈 ) ∧ ( ( 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑇 ) ∧ ( 𝑦 ( +g ‘ 𝐺 ) 𝑧 ) ∈ 𝑈 ) ) → 𝑦 ∈ 𝑆 ) |
| 30 |
|
simprlr |
⊢ ( ( ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑇 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑈 ∈ ( SubGrp ‘ 𝐺 ) ) ∧ 𝑆 ⊆ 𝑈 ) ∧ ( ( 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑇 ) ∧ ( 𝑦 ( +g ‘ 𝐺 ) 𝑧 ) ∈ 𝑈 ) ) → 𝑧 ∈ 𝑇 ) |
| 31 |
27 11
|
syl |
⊢ ( ( ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑇 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑈 ∈ ( SubGrp ‘ 𝐺 ) ) ∧ 𝑆 ⊆ 𝑈 ) ∧ ( ( 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑇 ) ∧ ( 𝑦 ( +g ‘ 𝐺 ) 𝑧 ) ∈ 𝑈 ) ) → 𝐺 ∈ Grp ) |
| 32 |
16
|
adantr |
⊢ ( ( ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑇 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑈 ∈ ( SubGrp ‘ 𝐺 ) ) ∧ 𝑆 ⊆ 𝑈 ) ∧ ( ( 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑇 ) ∧ ( 𝑦 ( +g ‘ 𝐺 ) 𝑧 ) ∈ 𝑈 ) ) → 𝑈 ∈ ( SubGrp ‘ 𝐺 ) ) |
| 33 |
12
|
subgss |
⊢ ( 𝑈 ∈ ( SubGrp ‘ 𝐺 ) → 𝑈 ⊆ ( Base ‘ 𝐺 ) ) |
| 34 |
32 33
|
syl |
⊢ ( ( ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑇 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑈 ∈ ( SubGrp ‘ 𝐺 ) ) ∧ 𝑆 ⊆ 𝑈 ) ∧ ( ( 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑇 ) ∧ ( 𝑦 ( +g ‘ 𝐺 ) 𝑧 ) ∈ 𝑈 ) ) → 𝑈 ⊆ ( Base ‘ 𝐺 ) ) |
| 35 |
8
|
adantr |
⊢ ( ( ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑇 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑈 ∈ ( SubGrp ‘ 𝐺 ) ) ∧ 𝑆 ⊆ 𝑈 ) ∧ ( ( 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑇 ) ∧ ( 𝑦 ( +g ‘ 𝐺 ) 𝑧 ) ∈ 𝑈 ) ) → 𝑆 ⊆ 𝑈 ) |
| 36 |
35 29
|
sseldd |
⊢ ( ( ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑇 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑈 ∈ ( SubGrp ‘ 𝐺 ) ) ∧ 𝑆 ⊆ 𝑈 ) ∧ ( ( 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑇 ) ∧ ( 𝑦 ( +g ‘ 𝐺 ) 𝑧 ) ∈ 𝑈 ) ) → 𝑦 ∈ 𝑈 ) |
| 37 |
34 36
|
sseldd |
⊢ ( ( ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑇 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑈 ∈ ( SubGrp ‘ 𝐺 ) ) ∧ 𝑆 ⊆ 𝑈 ) ∧ ( ( 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑇 ) ∧ ( 𝑦 ( +g ‘ 𝐺 ) 𝑧 ) ∈ 𝑈 ) ) → 𝑦 ∈ ( Base ‘ 𝐺 ) ) |
| 38 |
|
eqid |
⊢ ( 0g ‘ 𝐺 ) = ( 0g ‘ 𝐺 ) |
| 39 |
|
eqid |
⊢ ( invg ‘ 𝐺 ) = ( invg ‘ 𝐺 ) |
| 40 |
12 24 38 39
|
grplinv |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑦 ∈ ( Base ‘ 𝐺 ) ) → ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑦 ) = ( 0g ‘ 𝐺 ) ) |
| 41 |
31 37 40
|
syl2anc |
⊢ ( ( ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑇 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑈 ∈ ( SubGrp ‘ 𝐺 ) ) ∧ 𝑆 ⊆ 𝑈 ) ∧ ( ( 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑇 ) ∧ ( 𝑦 ( +g ‘ 𝐺 ) 𝑧 ) ∈ 𝑈 ) ) → ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑦 ) = ( 0g ‘ 𝐺 ) ) |
| 42 |
41
|
oveq1d |
⊢ ( ( ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑇 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑈 ∈ ( SubGrp ‘ 𝐺 ) ) ∧ 𝑆 ⊆ 𝑈 ) ∧ ( ( 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑇 ) ∧ ( 𝑦 ( +g ‘ 𝐺 ) 𝑧 ) ∈ 𝑈 ) ) → ( ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑦 ) ( +g ‘ 𝐺 ) 𝑧 ) = ( ( 0g ‘ 𝐺 ) ( +g ‘ 𝐺 ) 𝑧 ) ) |
| 43 |
39
|
subginvcl |
⊢ ( ( 𝑈 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑦 ∈ 𝑈 ) → ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ∈ 𝑈 ) |
| 44 |
32 36 43
|
syl2anc |
⊢ ( ( ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑇 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑈 ∈ ( SubGrp ‘ 𝐺 ) ) ∧ 𝑆 ⊆ 𝑈 ) ∧ ( ( 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑇 ) ∧ ( 𝑦 ( +g ‘ 𝐺 ) 𝑧 ) ∈ 𝑈 ) ) → ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ∈ 𝑈 ) |
| 45 |
34 44
|
sseldd |
⊢ ( ( ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑇 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑈 ∈ ( SubGrp ‘ 𝐺 ) ) ∧ 𝑆 ⊆ 𝑈 ) ∧ ( ( 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑇 ) ∧ ( 𝑦 ( +g ‘ 𝐺 ) 𝑧 ) ∈ 𝑈 ) ) → ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ∈ ( Base ‘ 𝐺 ) ) |
| 46 |
|
simpll2 |
⊢ ( ( ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑇 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑈 ∈ ( SubGrp ‘ 𝐺 ) ) ∧ 𝑆 ⊆ 𝑈 ) ∧ ( ( 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑇 ) ∧ ( 𝑦 ( +g ‘ 𝐺 ) 𝑧 ) ∈ 𝑈 ) ) → 𝑇 ∈ ( SubGrp ‘ 𝐺 ) ) |
| 47 |
12
|
subgss |
⊢ ( 𝑇 ∈ ( SubGrp ‘ 𝐺 ) → 𝑇 ⊆ ( Base ‘ 𝐺 ) ) |
| 48 |
46 47
|
syl |
⊢ ( ( ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑇 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑈 ∈ ( SubGrp ‘ 𝐺 ) ) ∧ 𝑆 ⊆ 𝑈 ) ∧ ( ( 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑇 ) ∧ ( 𝑦 ( +g ‘ 𝐺 ) 𝑧 ) ∈ 𝑈 ) ) → 𝑇 ⊆ ( Base ‘ 𝐺 ) ) |
| 49 |
48 30
|
sseldd |
⊢ ( ( ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑇 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑈 ∈ ( SubGrp ‘ 𝐺 ) ) ∧ 𝑆 ⊆ 𝑈 ) ∧ ( ( 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑇 ) ∧ ( 𝑦 ( +g ‘ 𝐺 ) 𝑧 ) ∈ 𝑈 ) ) → 𝑧 ∈ ( Base ‘ 𝐺 ) ) |
| 50 |
12 24
|
grpass |
⊢ ( ( 𝐺 ∈ Grp ∧ ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ∈ ( Base ‘ 𝐺 ) ∧ 𝑦 ∈ ( Base ‘ 𝐺 ) ∧ 𝑧 ∈ ( Base ‘ 𝐺 ) ) ) → ( ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑦 ) ( +g ‘ 𝐺 ) 𝑧 ) = ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) ( 𝑦 ( +g ‘ 𝐺 ) 𝑧 ) ) ) |
| 51 |
31 45 37 49 50
|
syl13anc |
⊢ ( ( ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑇 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑈 ∈ ( SubGrp ‘ 𝐺 ) ) ∧ 𝑆 ⊆ 𝑈 ) ∧ ( ( 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑇 ) ∧ ( 𝑦 ( +g ‘ 𝐺 ) 𝑧 ) ∈ 𝑈 ) ) → ( ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑦 ) ( +g ‘ 𝐺 ) 𝑧 ) = ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) ( 𝑦 ( +g ‘ 𝐺 ) 𝑧 ) ) ) |
| 52 |
12 24 38
|
grplid |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑧 ∈ ( Base ‘ 𝐺 ) ) → ( ( 0g ‘ 𝐺 ) ( +g ‘ 𝐺 ) 𝑧 ) = 𝑧 ) |
| 53 |
31 49 52
|
syl2anc |
⊢ ( ( ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑇 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑈 ∈ ( SubGrp ‘ 𝐺 ) ) ∧ 𝑆 ⊆ 𝑈 ) ∧ ( ( 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑇 ) ∧ ( 𝑦 ( +g ‘ 𝐺 ) 𝑧 ) ∈ 𝑈 ) ) → ( ( 0g ‘ 𝐺 ) ( +g ‘ 𝐺 ) 𝑧 ) = 𝑧 ) |
| 54 |
42 51 53
|
3eqtr3d |
⊢ ( ( ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑇 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑈 ∈ ( SubGrp ‘ 𝐺 ) ) ∧ 𝑆 ⊆ 𝑈 ) ∧ ( ( 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑇 ) ∧ ( 𝑦 ( +g ‘ 𝐺 ) 𝑧 ) ∈ 𝑈 ) ) → ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) ( 𝑦 ( +g ‘ 𝐺 ) 𝑧 ) ) = 𝑧 ) |
| 55 |
|
simprr |
⊢ ( ( ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑇 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑈 ∈ ( SubGrp ‘ 𝐺 ) ) ∧ 𝑆 ⊆ 𝑈 ) ∧ ( ( 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑇 ) ∧ ( 𝑦 ( +g ‘ 𝐺 ) 𝑧 ) ∈ 𝑈 ) ) → ( 𝑦 ( +g ‘ 𝐺 ) 𝑧 ) ∈ 𝑈 ) |
| 56 |
24
|
subgcl |
⊢ ( ( 𝑈 ∈ ( SubGrp ‘ 𝐺 ) ∧ ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ∈ 𝑈 ∧ ( 𝑦 ( +g ‘ 𝐺 ) 𝑧 ) ∈ 𝑈 ) → ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) ( 𝑦 ( +g ‘ 𝐺 ) 𝑧 ) ) ∈ 𝑈 ) |
| 57 |
32 44 55 56
|
syl3anc |
⊢ ( ( ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑇 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑈 ∈ ( SubGrp ‘ 𝐺 ) ) ∧ 𝑆 ⊆ 𝑈 ) ∧ ( ( 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑇 ) ∧ ( 𝑦 ( +g ‘ 𝐺 ) 𝑧 ) ∈ 𝑈 ) ) → ( ( ( invg ‘ 𝐺 ) ‘ 𝑦 ) ( +g ‘ 𝐺 ) ( 𝑦 ( +g ‘ 𝐺 ) 𝑧 ) ) ∈ 𝑈 ) |
| 58 |
54 57
|
eqeltrrd |
⊢ ( ( ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑇 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑈 ∈ ( SubGrp ‘ 𝐺 ) ) ∧ 𝑆 ⊆ 𝑈 ) ∧ ( ( 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑇 ) ∧ ( 𝑦 ( +g ‘ 𝐺 ) 𝑧 ) ∈ 𝑈 ) ) → 𝑧 ∈ 𝑈 ) |
| 59 |
30 58
|
elind |
⊢ ( ( ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑇 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑈 ∈ ( SubGrp ‘ 𝐺 ) ) ∧ 𝑆 ⊆ 𝑈 ) ∧ ( ( 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑇 ) ∧ ( 𝑦 ( +g ‘ 𝐺 ) 𝑧 ) ∈ 𝑈 ) ) → 𝑧 ∈ ( 𝑇 ∩ 𝑈 ) ) |
| 60 |
24 1
|
lsmelvali |
⊢ ( ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ ( 𝑇 ∩ 𝑈 ) ∈ ( SubGrp ‘ 𝐺 ) ) ∧ ( 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ ( 𝑇 ∩ 𝑈 ) ) ) → ( 𝑦 ( +g ‘ 𝐺 ) 𝑧 ) ∈ ( 𝑆 ⊕ ( 𝑇 ∩ 𝑈 ) ) ) |
| 61 |
27 28 29 59 60
|
syl22anc |
⊢ ( ( ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑇 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑈 ∈ ( SubGrp ‘ 𝐺 ) ) ∧ 𝑆 ⊆ 𝑈 ) ∧ ( ( 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑇 ) ∧ ( 𝑦 ( +g ‘ 𝐺 ) 𝑧 ) ∈ 𝑈 ) ) → ( 𝑦 ( +g ‘ 𝐺 ) 𝑧 ) ∈ ( 𝑆 ⊕ ( 𝑇 ∩ 𝑈 ) ) ) |
| 62 |
61
|
expr |
⊢ ( ( ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑇 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑈 ∈ ( SubGrp ‘ 𝐺 ) ) ∧ 𝑆 ⊆ 𝑈 ) ∧ ( 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑇 ) ) → ( ( 𝑦 ( +g ‘ 𝐺 ) 𝑧 ) ∈ 𝑈 → ( 𝑦 ( +g ‘ 𝐺 ) 𝑧 ) ∈ ( 𝑆 ⊕ ( 𝑇 ∩ 𝑈 ) ) ) ) |
| 63 |
|
eleq1 |
⊢ ( 𝑥 = ( 𝑦 ( +g ‘ 𝐺 ) 𝑧 ) → ( 𝑥 ∈ 𝑈 ↔ ( 𝑦 ( +g ‘ 𝐺 ) 𝑧 ) ∈ 𝑈 ) ) |
| 64 |
|
eleq1 |
⊢ ( 𝑥 = ( 𝑦 ( +g ‘ 𝐺 ) 𝑧 ) → ( 𝑥 ∈ ( 𝑆 ⊕ ( 𝑇 ∩ 𝑈 ) ) ↔ ( 𝑦 ( +g ‘ 𝐺 ) 𝑧 ) ∈ ( 𝑆 ⊕ ( 𝑇 ∩ 𝑈 ) ) ) ) |
| 65 |
63 64
|
imbi12d |
⊢ ( 𝑥 = ( 𝑦 ( +g ‘ 𝐺 ) 𝑧 ) → ( ( 𝑥 ∈ 𝑈 → 𝑥 ∈ ( 𝑆 ⊕ ( 𝑇 ∩ 𝑈 ) ) ) ↔ ( ( 𝑦 ( +g ‘ 𝐺 ) 𝑧 ) ∈ 𝑈 → ( 𝑦 ( +g ‘ 𝐺 ) 𝑧 ) ∈ ( 𝑆 ⊕ ( 𝑇 ∩ 𝑈 ) ) ) ) ) |
| 66 |
62 65
|
syl5ibrcom |
⊢ ( ( ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑇 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑈 ∈ ( SubGrp ‘ 𝐺 ) ) ∧ 𝑆 ⊆ 𝑈 ) ∧ ( 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑇 ) ) → ( 𝑥 = ( 𝑦 ( +g ‘ 𝐺 ) 𝑧 ) → ( 𝑥 ∈ 𝑈 → 𝑥 ∈ ( 𝑆 ⊕ ( 𝑇 ∩ 𝑈 ) ) ) ) ) |
| 67 |
66
|
rexlimdvva |
⊢ ( ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑇 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑈 ∈ ( SubGrp ‘ 𝐺 ) ) ∧ 𝑆 ⊆ 𝑈 ) → ( ∃ 𝑦 ∈ 𝑆 ∃ 𝑧 ∈ 𝑇 𝑥 = ( 𝑦 ( +g ‘ 𝐺 ) 𝑧 ) → ( 𝑥 ∈ 𝑈 → 𝑥 ∈ ( 𝑆 ⊕ ( 𝑇 ∩ 𝑈 ) ) ) ) ) |
| 68 |
26 67
|
sylbid |
⊢ ( ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑇 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑈 ∈ ( SubGrp ‘ 𝐺 ) ) ∧ 𝑆 ⊆ 𝑈 ) → ( 𝑥 ∈ ( 𝑆 ⊕ 𝑇 ) → ( 𝑥 ∈ 𝑈 → 𝑥 ∈ ( 𝑆 ⊕ ( 𝑇 ∩ 𝑈 ) ) ) ) ) |
| 69 |
68
|
impd |
⊢ ( ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑇 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑈 ∈ ( SubGrp ‘ 𝐺 ) ) ∧ 𝑆 ⊆ 𝑈 ) → ( ( 𝑥 ∈ ( 𝑆 ⊕ 𝑇 ) ∧ 𝑥 ∈ 𝑈 ) → 𝑥 ∈ ( 𝑆 ⊕ ( 𝑇 ∩ 𝑈 ) ) ) ) |
| 70 |
23 69
|
syl5bi |
⊢ ( ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑇 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑈 ∈ ( SubGrp ‘ 𝐺 ) ) ∧ 𝑆 ⊆ 𝑈 ) → ( 𝑥 ∈ ( ( 𝑆 ⊕ 𝑇 ) ∩ 𝑈 ) → 𝑥 ∈ ( 𝑆 ⊕ ( 𝑇 ∩ 𝑈 ) ) ) ) |
| 71 |
70
|
ssrdv |
⊢ ( ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑇 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑈 ∈ ( SubGrp ‘ 𝐺 ) ) ∧ 𝑆 ⊆ 𝑈 ) → ( ( 𝑆 ⊕ 𝑇 ) ∩ 𝑈 ) ⊆ ( 𝑆 ⊕ ( 𝑇 ∩ 𝑈 ) ) ) |
| 72 |
22 71
|
eqssd |
⊢ ( ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑇 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑈 ∈ ( SubGrp ‘ 𝐺 ) ) ∧ 𝑆 ⊆ 𝑈 ) → ( 𝑆 ⊕ ( 𝑇 ∩ 𝑈 ) ) = ( ( 𝑆 ⊕ 𝑇 ) ∩ 𝑈 ) ) |