Step |
Hyp |
Ref |
Expression |
1 |
|
pjhfval |
⊢ ( 𝐻 ∈ Cℋ → ( projℎ ‘ 𝐻 ) = ( 𝑧 ∈ ℋ ↦ ( ℩ 𝑥 ∈ 𝐻 ∃ 𝑦 ∈ ( ⊥ ‘ 𝐻 ) 𝑧 = ( 𝑥 +ℎ 𝑦 ) ) ) ) |
2 |
1
|
fveq1d |
⊢ ( 𝐻 ∈ Cℋ → ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) = ( ( 𝑧 ∈ ℋ ↦ ( ℩ 𝑥 ∈ 𝐻 ∃ 𝑦 ∈ ( ⊥ ‘ 𝐻 ) 𝑧 = ( 𝑥 +ℎ 𝑦 ) ) ) ‘ 𝐴 ) ) |
3 |
|
eqeq1 |
⊢ ( 𝑧 = 𝐴 → ( 𝑧 = ( 𝑥 +ℎ 𝑦 ) ↔ 𝐴 = ( 𝑥 +ℎ 𝑦 ) ) ) |
4 |
3
|
rexbidv |
⊢ ( 𝑧 = 𝐴 → ( ∃ 𝑦 ∈ ( ⊥ ‘ 𝐻 ) 𝑧 = ( 𝑥 +ℎ 𝑦 ) ↔ ∃ 𝑦 ∈ ( ⊥ ‘ 𝐻 ) 𝐴 = ( 𝑥 +ℎ 𝑦 ) ) ) |
5 |
4
|
riotabidv |
⊢ ( 𝑧 = 𝐴 → ( ℩ 𝑥 ∈ 𝐻 ∃ 𝑦 ∈ ( ⊥ ‘ 𝐻 ) 𝑧 = ( 𝑥 +ℎ 𝑦 ) ) = ( ℩ 𝑥 ∈ 𝐻 ∃ 𝑦 ∈ ( ⊥ ‘ 𝐻 ) 𝐴 = ( 𝑥 +ℎ 𝑦 ) ) ) |
6 |
|
eqid |
⊢ ( 𝑧 ∈ ℋ ↦ ( ℩ 𝑥 ∈ 𝐻 ∃ 𝑦 ∈ ( ⊥ ‘ 𝐻 ) 𝑧 = ( 𝑥 +ℎ 𝑦 ) ) ) = ( 𝑧 ∈ ℋ ↦ ( ℩ 𝑥 ∈ 𝐻 ∃ 𝑦 ∈ ( ⊥ ‘ 𝐻 ) 𝑧 = ( 𝑥 +ℎ 𝑦 ) ) ) |
7 |
|
riotaex |
⊢ ( ℩ 𝑥 ∈ 𝐻 ∃ 𝑦 ∈ ( ⊥ ‘ 𝐻 ) 𝐴 = ( 𝑥 +ℎ 𝑦 ) ) ∈ V |
8 |
5 6 7
|
fvmpt |
⊢ ( 𝐴 ∈ ℋ → ( ( 𝑧 ∈ ℋ ↦ ( ℩ 𝑥 ∈ 𝐻 ∃ 𝑦 ∈ ( ⊥ ‘ 𝐻 ) 𝑧 = ( 𝑥 +ℎ 𝑦 ) ) ) ‘ 𝐴 ) = ( ℩ 𝑥 ∈ 𝐻 ∃ 𝑦 ∈ ( ⊥ ‘ 𝐻 ) 𝐴 = ( 𝑥 +ℎ 𝑦 ) ) ) |
9 |
2 8
|
sylan9eq |
⊢ ( ( 𝐻 ∈ Cℋ ∧ 𝐴 ∈ ℋ ) → ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) = ( ℩ 𝑥 ∈ 𝐻 ∃ 𝑦 ∈ ( ⊥ ‘ 𝐻 ) 𝐴 = ( 𝑥 +ℎ 𝑦 ) ) ) |