Step |
Hyp |
Ref |
Expression |
1 |
|
dochkrsm.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
2 |
|
dochkrsm.i |
⊢ 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) |
3 |
|
dochkrsm.o |
⊢ ⊥ = ( ( ocH ‘ 𝐾 ) ‘ 𝑊 ) |
4 |
|
dochkrsm.u |
⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) |
5 |
|
dochkrsm.p |
⊢ ⊕ = ( LSSum ‘ 𝑈 ) |
6 |
|
dochkrsm.f |
⊢ 𝐹 = ( LFnl ‘ 𝑈 ) |
7 |
|
dochkrsm.l |
⊢ 𝐿 = ( LKer ‘ 𝑈 ) |
8 |
|
dochkrsm.k |
⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
9 |
|
dochkrsm.x |
⊢ ( 𝜑 → 𝑋 ∈ ran 𝐼 ) |
10 |
|
dochkrsm.g |
⊢ ( 𝜑 → 𝐺 ∈ 𝐹 ) |
11 |
|
eqid |
⊢ ( LSAtoms ‘ 𝑈 ) = ( LSAtoms ‘ 𝑈 ) |
12 |
8
|
adantr |
⊢ ( ( 𝜑 ∧ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ∈ ( LSAtoms ‘ 𝑈 ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
13 |
9
|
adantr |
⊢ ( ( 𝜑 ∧ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ∈ ( LSAtoms ‘ 𝑈 ) ) → 𝑋 ∈ ran 𝐼 ) |
14 |
|
simpr |
⊢ ( ( 𝜑 ∧ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ∈ ( LSAtoms ‘ 𝑈 ) ) → ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ∈ ( LSAtoms ‘ 𝑈 ) ) |
15 |
1 2 4 5 11 12 13 14
|
dihsmatrn |
⊢ ( ( 𝜑 ∧ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ∈ ( LSAtoms ‘ 𝑈 ) ) → ( 𝑋 ⊕ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ) ∈ ran 𝐼 ) |
16 |
|
oveq2 |
⊢ ( ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) = { ( 0g ‘ 𝑈 ) } → ( 𝑋 ⊕ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ) = ( 𝑋 ⊕ { ( 0g ‘ 𝑈 ) } ) ) |
17 |
1 4 8
|
dvhlmod |
⊢ ( 𝜑 → 𝑈 ∈ LMod ) |
18 |
|
eqid |
⊢ ( LSubSp ‘ 𝑈 ) = ( LSubSp ‘ 𝑈 ) |
19 |
1 4 2 18
|
dihrnlss |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ ran 𝐼 ) → 𝑋 ∈ ( LSubSp ‘ 𝑈 ) ) |
20 |
8 9 19
|
syl2anc |
⊢ ( 𝜑 → 𝑋 ∈ ( LSubSp ‘ 𝑈 ) ) |
21 |
18
|
lsssubg |
⊢ ( ( 𝑈 ∈ LMod ∧ 𝑋 ∈ ( LSubSp ‘ 𝑈 ) ) → 𝑋 ∈ ( SubGrp ‘ 𝑈 ) ) |
22 |
17 20 21
|
syl2anc |
⊢ ( 𝜑 → 𝑋 ∈ ( SubGrp ‘ 𝑈 ) ) |
23 |
|
eqid |
⊢ ( 0g ‘ 𝑈 ) = ( 0g ‘ 𝑈 ) |
24 |
23 5
|
lsm01 |
⊢ ( 𝑋 ∈ ( SubGrp ‘ 𝑈 ) → ( 𝑋 ⊕ { ( 0g ‘ 𝑈 ) } ) = 𝑋 ) |
25 |
22 24
|
syl |
⊢ ( 𝜑 → ( 𝑋 ⊕ { ( 0g ‘ 𝑈 ) } ) = 𝑋 ) |
26 |
16 25
|
sylan9eqr |
⊢ ( ( 𝜑 ∧ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) = { ( 0g ‘ 𝑈 ) } ) → ( 𝑋 ⊕ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ) = 𝑋 ) |
27 |
9
|
adantr |
⊢ ( ( 𝜑 ∧ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) = { ( 0g ‘ 𝑈 ) } ) → 𝑋 ∈ ran 𝐼 ) |
28 |
26 27
|
eqeltrd |
⊢ ( ( 𝜑 ∧ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) = { ( 0g ‘ 𝑈 ) } ) → ( 𝑋 ⊕ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ) ∈ ran 𝐼 ) |
29 |
1 3 4 23 11 6 7 8 10
|
dochsat0 |
⊢ ( 𝜑 → ( ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ∈ ( LSAtoms ‘ 𝑈 ) ∨ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) = { ( 0g ‘ 𝑈 ) } ) ) |
30 |
15 28 29
|
mpjaodan |
⊢ ( 𝜑 → ( 𝑋 ⊕ ( ⊥ ‘ ( 𝐿 ‘ 𝐺 ) ) ) ∈ ran 𝐼 ) |