Metamath Proof Explorer


Theorem pjhval

Description: Value of a projection. (Contributed by NM, 23-Oct-1999) (Revised by Mario Carneiro, 15-Dec-2013) (New usage is discouraged.)

Ref Expression
Assertion pjhval ( ( 𝐻C𝐴 ∈ ℋ ) → ( ( proj𝐻 ) ‘ 𝐴 ) = ( 𝑥𝐻𝑦 ∈ ( ⊥ ‘ 𝐻 ) 𝐴 = ( 𝑥 + 𝑦 ) ) )

Proof

Step Hyp Ref Expression
1 pjhfval ( 𝐻C → ( proj𝐻 ) = ( 𝑧 ∈ ℋ ↦ ( 𝑥𝐻𝑦 ∈ ( ⊥ ‘ 𝐻 ) 𝑧 = ( 𝑥 + 𝑦 ) ) ) )
2 1 fveq1d ( 𝐻C → ( ( proj𝐻 ) ‘ 𝐴 ) = ( ( 𝑧 ∈ ℋ ↦ ( 𝑥𝐻𝑦 ∈ ( ⊥ ‘ 𝐻 ) 𝑧 = ( 𝑥 + 𝑦 ) ) ) ‘ 𝐴 ) )
3 eqeq1 ( 𝑧 = 𝐴 → ( 𝑧 = ( 𝑥 + 𝑦 ) ↔ 𝐴 = ( 𝑥 + 𝑦 ) ) )
4 3 rexbidv ( 𝑧 = 𝐴 → ( ∃ 𝑦 ∈ ( ⊥ ‘ 𝐻 ) 𝑧 = ( 𝑥 + 𝑦 ) ↔ ∃ 𝑦 ∈ ( ⊥ ‘ 𝐻 ) 𝐴 = ( 𝑥 + 𝑦 ) ) )
5 4 riotabidv ( 𝑧 = 𝐴 → ( 𝑥𝐻𝑦 ∈ ( ⊥ ‘ 𝐻 ) 𝑧 = ( 𝑥 + 𝑦 ) ) = ( 𝑥𝐻𝑦 ∈ ( ⊥ ‘ 𝐻 ) 𝐴 = ( 𝑥 + 𝑦 ) ) )
6 eqid ( 𝑧 ∈ ℋ ↦ ( 𝑥𝐻𝑦 ∈ ( ⊥ ‘ 𝐻 ) 𝑧 = ( 𝑥 + 𝑦 ) ) ) = ( 𝑧 ∈ ℋ ↦ ( 𝑥𝐻𝑦 ∈ ( ⊥ ‘ 𝐻 ) 𝑧 = ( 𝑥 + 𝑦 ) ) )
7 riotaex ( 𝑥𝐻𝑦 ∈ ( ⊥ ‘ 𝐻 ) 𝐴 = ( 𝑥 + 𝑦 ) ) ∈ V
8 5 6 7 fvmpt ( 𝐴 ∈ ℋ → ( ( 𝑧 ∈ ℋ ↦ ( 𝑥𝐻𝑦 ∈ ( ⊥ ‘ 𝐻 ) 𝑧 = ( 𝑥 + 𝑦 ) ) ) ‘ 𝐴 ) = ( 𝑥𝐻𝑦 ∈ ( ⊥ ‘ 𝐻 ) 𝐴 = ( 𝑥 + 𝑦 ) ) )
9 2 8 sylan9eq ( ( 𝐻C𝐴 ∈ ℋ ) → ( ( proj𝐻 ) ‘ 𝐴 ) = ( 𝑥𝐻𝑦 ∈ ( ⊥ ‘ 𝐻 ) 𝐴 = ( 𝑥 + 𝑦 ) ) )