Step |
Hyp |
Ref |
Expression |
1 |
|
shocel |
⊢ ( 𝐴 ∈ Sℋ → ( 𝑥 ∈ ( ⊥ ‘ 𝐴 ) ↔ ( 𝑥 ∈ ℋ ∧ ∀ 𝑦 ∈ 𝐴 ( 𝑥 ·ih 𝑦 ) = 0 ) ) ) |
2 |
|
oveq2 |
⊢ ( 𝑦 = 𝑥 → ( 𝑥 ·ih 𝑦 ) = ( 𝑥 ·ih 𝑥 ) ) |
3 |
2
|
eqeq1d |
⊢ ( 𝑦 = 𝑥 → ( ( 𝑥 ·ih 𝑦 ) = 0 ↔ ( 𝑥 ·ih 𝑥 ) = 0 ) ) |
4 |
3
|
rspccv |
⊢ ( ∀ 𝑦 ∈ 𝐴 ( 𝑥 ·ih 𝑦 ) = 0 → ( 𝑥 ∈ 𝐴 → ( 𝑥 ·ih 𝑥 ) = 0 ) ) |
5 |
|
his6 |
⊢ ( 𝑥 ∈ ℋ → ( ( 𝑥 ·ih 𝑥 ) = 0 ↔ 𝑥 = 0ℎ ) ) |
6 |
5
|
biimpd |
⊢ ( 𝑥 ∈ ℋ → ( ( 𝑥 ·ih 𝑥 ) = 0 → 𝑥 = 0ℎ ) ) |
7 |
4 6
|
sylan9r |
⊢ ( ( 𝑥 ∈ ℋ ∧ ∀ 𝑦 ∈ 𝐴 ( 𝑥 ·ih 𝑦 ) = 0 ) → ( 𝑥 ∈ 𝐴 → 𝑥 = 0ℎ ) ) |
8 |
1 7
|
syl6bi |
⊢ ( 𝐴 ∈ Sℋ → ( 𝑥 ∈ ( ⊥ ‘ 𝐴 ) → ( 𝑥 ∈ 𝐴 → 𝑥 = 0ℎ ) ) ) |
9 |
8
|
com23 |
⊢ ( 𝐴 ∈ Sℋ → ( 𝑥 ∈ 𝐴 → ( 𝑥 ∈ ( ⊥ ‘ 𝐴 ) → 𝑥 = 0ℎ ) ) ) |
10 |
9
|
impd |
⊢ ( 𝐴 ∈ Sℋ → ( ( 𝑥 ∈ 𝐴 ∧ 𝑥 ∈ ( ⊥ ‘ 𝐴 ) ) → 𝑥 = 0ℎ ) ) |
11 |
|
sh0 |
⊢ ( 𝐴 ∈ Sℋ → 0ℎ ∈ 𝐴 ) |
12 |
|
oc0 |
⊢ ( 𝐴 ∈ Sℋ → 0ℎ ∈ ( ⊥ ‘ 𝐴 ) ) |
13 |
11 12
|
jca |
⊢ ( 𝐴 ∈ Sℋ → ( 0ℎ ∈ 𝐴 ∧ 0ℎ ∈ ( ⊥ ‘ 𝐴 ) ) ) |
14 |
|
eleq1 |
⊢ ( 𝑥 = 0ℎ → ( 𝑥 ∈ 𝐴 ↔ 0ℎ ∈ 𝐴 ) ) |
15 |
|
eleq1 |
⊢ ( 𝑥 = 0ℎ → ( 𝑥 ∈ ( ⊥ ‘ 𝐴 ) ↔ 0ℎ ∈ ( ⊥ ‘ 𝐴 ) ) ) |
16 |
14 15
|
anbi12d |
⊢ ( 𝑥 = 0ℎ → ( ( 𝑥 ∈ 𝐴 ∧ 𝑥 ∈ ( ⊥ ‘ 𝐴 ) ) ↔ ( 0ℎ ∈ 𝐴 ∧ 0ℎ ∈ ( ⊥ ‘ 𝐴 ) ) ) ) |
17 |
13 16
|
syl5ibrcom |
⊢ ( 𝐴 ∈ Sℋ → ( 𝑥 = 0ℎ → ( 𝑥 ∈ 𝐴 ∧ 𝑥 ∈ ( ⊥ ‘ 𝐴 ) ) ) ) |
18 |
10 17
|
impbid |
⊢ ( 𝐴 ∈ Sℋ → ( ( 𝑥 ∈ 𝐴 ∧ 𝑥 ∈ ( ⊥ ‘ 𝐴 ) ) ↔ 𝑥 = 0ℎ ) ) |
19 |
|
elin |
⊢ ( 𝑥 ∈ ( 𝐴 ∩ ( ⊥ ‘ 𝐴 ) ) ↔ ( 𝑥 ∈ 𝐴 ∧ 𝑥 ∈ ( ⊥ ‘ 𝐴 ) ) ) |
20 |
|
elch0 |
⊢ ( 𝑥 ∈ 0ℋ ↔ 𝑥 = 0ℎ ) |
21 |
18 19 20
|
3bitr4g |
⊢ ( 𝐴 ∈ Sℋ → ( 𝑥 ∈ ( 𝐴 ∩ ( ⊥ ‘ 𝐴 ) ) ↔ 𝑥 ∈ 0ℋ ) ) |
22 |
21
|
eqrdv |
⊢ ( 𝐴 ∈ Sℋ → ( 𝐴 ∩ ( ⊥ ‘ 𝐴 ) ) = 0ℋ ) |