Step |
Hyp |
Ref |
Expression |
1 |
|
dihjat2.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
2 |
|
dihjat2.i |
⊢ 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) |
3 |
|
dihjat2.j |
⊢ ∨ = ( ( joinH ‘ 𝐾 ) ‘ 𝑊 ) |
4 |
|
dihjat2.u |
⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) |
5 |
|
dihjat2.p |
⊢ ⊕ = ( LSSum ‘ 𝑈 ) |
6 |
|
dihjat2.a |
⊢ 𝐴 = ( LSAtoms ‘ 𝑈 ) |
7 |
|
dihjat2.k |
⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
8 |
|
dihjat2.x |
⊢ ( 𝜑 → 𝑋 ∈ ran 𝐼 ) |
9 |
|
dihjat2.q |
⊢ ( 𝜑 → 𝑄 ∈ 𝐴 ) |
10 |
|
eqid |
⊢ ( Base ‘ 𝑈 ) = ( Base ‘ 𝑈 ) |
11 |
|
eqid |
⊢ ( LSpan ‘ 𝑈 ) = ( LSpan ‘ 𝑈 ) |
12 |
7
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑣 ∈ ( Base ‘ 𝑈 ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
13 |
8
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑣 ∈ ( Base ‘ 𝑈 ) ) → 𝑋 ∈ ran 𝐼 ) |
14 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑣 ∈ ( Base ‘ 𝑈 ) ) → 𝑣 ∈ ( Base ‘ 𝑈 ) ) |
15 |
1 4 10 5 11 2 3 12 13 14
|
dihjat1 |
⊢ ( ( 𝜑 ∧ 𝑣 ∈ ( Base ‘ 𝑈 ) ) → ( 𝑋 ∨ ( ( LSpan ‘ 𝑈 ) ‘ { 𝑣 } ) ) = ( 𝑋 ⊕ ( ( LSpan ‘ 𝑈 ) ‘ { 𝑣 } ) ) ) |
16 |
15
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑣 ∈ ( Base ‘ 𝑈 ) ) ∧ 𝑄 = ( ( LSpan ‘ 𝑈 ) ‘ { 𝑣 } ) ) → ( 𝑋 ∨ ( ( LSpan ‘ 𝑈 ) ‘ { 𝑣 } ) ) = ( 𝑋 ⊕ ( ( LSpan ‘ 𝑈 ) ‘ { 𝑣 } ) ) ) |
17 |
|
oveq2 |
⊢ ( 𝑄 = ( ( LSpan ‘ 𝑈 ) ‘ { 𝑣 } ) → ( 𝑋 ∨ 𝑄 ) = ( 𝑋 ∨ ( ( LSpan ‘ 𝑈 ) ‘ { 𝑣 } ) ) ) |
18 |
17
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑣 ∈ ( Base ‘ 𝑈 ) ) ∧ 𝑄 = ( ( LSpan ‘ 𝑈 ) ‘ { 𝑣 } ) ) → ( 𝑋 ∨ 𝑄 ) = ( 𝑋 ∨ ( ( LSpan ‘ 𝑈 ) ‘ { 𝑣 } ) ) ) |
19 |
|
oveq2 |
⊢ ( 𝑄 = ( ( LSpan ‘ 𝑈 ) ‘ { 𝑣 } ) → ( 𝑋 ⊕ 𝑄 ) = ( 𝑋 ⊕ ( ( LSpan ‘ 𝑈 ) ‘ { 𝑣 } ) ) ) |
20 |
19
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑣 ∈ ( Base ‘ 𝑈 ) ) ∧ 𝑄 = ( ( LSpan ‘ 𝑈 ) ‘ { 𝑣 } ) ) → ( 𝑋 ⊕ 𝑄 ) = ( 𝑋 ⊕ ( ( LSpan ‘ 𝑈 ) ‘ { 𝑣 } ) ) ) |
21 |
16 18 20
|
3eqtr4d |
⊢ ( ( ( 𝜑 ∧ 𝑣 ∈ ( Base ‘ 𝑈 ) ) ∧ 𝑄 = ( ( LSpan ‘ 𝑈 ) ‘ { 𝑣 } ) ) → ( 𝑋 ∨ 𝑄 ) = ( 𝑋 ⊕ 𝑄 ) ) |
22 |
1 4 7
|
dvhlmod |
⊢ ( 𝜑 → 𝑈 ∈ LMod ) |
23 |
10 11 6
|
islsati |
⊢ ( ( 𝑈 ∈ LMod ∧ 𝑄 ∈ 𝐴 ) → ∃ 𝑣 ∈ ( Base ‘ 𝑈 ) 𝑄 = ( ( LSpan ‘ 𝑈 ) ‘ { 𝑣 } ) ) |
24 |
22 9 23
|
syl2anc |
⊢ ( 𝜑 → ∃ 𝑣 ∈ ( Base ‘ 𝑈 ) 𝑄 = ( ( LSpan ‘ 𝑈 ) ‘ { 𝑣 } ) ) |
25 |
21 24
|
r19.29a |
⊢ ( 𝜑 → ( 𝑋 ∨ 𝑄 ) = ( 𝑋 ⊕ 𝑄 ) ) |