Metamath Proof Explorer


Theorem pjocini

Description: Membership of projection in orthocomplement of intersection. (Contributed by NM, 21-Apr-2001) (New usage is discouraged.)

Ref Expression
Hypotheses pjocin.1 𝐺C
pjocin.2 𝐻C
Assertion pjocini ( 𝐴 ∈ ( ⊥ ‘ ( 𝐺𝐻 ) ) → ( ( proj𝐺 ) ‘ 𝐴 ) ∈ ( ⊥ ‘ ( 𝐺𝐻 ) ) )

Proof

Step Hyp Ref Expression
1 pjocin.1 𝐺C
2 pjocin.2 𝐻C
3 1 2 chincli ( 𝐺𝐻 ) ∈ C
4 3 choccli ( ⊥ ‘ ( 𝐺𝐻 ) ) ∈ C
5 4 cheli ( 𝐴 ∈ ( ⊥ ‘ ( 𝐺𝐻 ) ) → 𝐴 ∈ ℋ )
6 pjpo ( ( 𝐺C𝐴 ∈ ℋ ) → ( ( proj𝐺 ) ‘ 𝐴 ) = ( 𝐴 ( ( proj ‘ ( ⊥ ‘ 𝐺 ) ) ‘ 𝐴 ) ) )
7 1 5 6 sylancr ( 𝐴 ∈ ( ⊥ ‘ ( 𝐺𝐻 ) ) → ( ( proj𝐺 ) ‘ 𝐴 ) = ( 𝐴 ( ( proj ‘ ( ⊥ ‘ 𝐺 ) ) ‘ 𝐴 ) ) )
8 inss1 ( 𝐺𝐻 ) ⊆ 𝐺
9 3 1 chsscon3i ( ( 𝐺𝐻 ) ⊆ 𝐺 ↔ ( ⊥ ‘ 𝐺 ) ⊆ ( ⊥ ‘ ( 𝐺𝐻 ) ) )
10 8 9 mpbi ( ⊥ ‘ 𝐺 ) ⊆ ( ⊥ ‘ ( 𝐺𝐻 ) )
11 1 choccli ( ⊥ ‘ 𝐺 ) ∈ C
12 11 pjcli ( 𝐴 ∈ ℋ → ( ( proj ‘ ( ⊥ ‘ 𝐺 ) ) ‘ 𝐴 ) ∈ ( ⊥ ‘ 𝐺 ) )
13 5 12 syl ( 𝐴 ∈ ( ⊥ ‘ ( 𝐺𝐻 ) ) → ( ( proj ‘ ( ⊥ ‘ 𝐺 ) ) ‘ 𝐴 ) ∈ ( ⊥ ‘ 𝐺 ) )
14 10 13 sselid ( 𝐴 ∈ ( ⊥ ‘ ( 𝐺𝐻 ) ) → ( ( proj ‘ ( ⊥ ‘ 𝐺 ) ) ‘ 𝐴 ) ∈ ( ⊥ ‘ ( 𝐺𝐻 ) ) )
15 4 chshii ( ⊥ ‘ ( 𝐺𝐻 ) ) ∈ S
16 shsubcl ( ( ( ⊥ ‘ ( 𝐺𝐻 ) ) ∈ S𝐴 ∈ ( ⊥ ‘ ( 𝐺𝐻 ) ) ∧ ( ( proj ‘ ( ⊥ ‘ 𝐺 ) ) ‘ 𝐴 ) ∈ ( ⊥ ‘ ( 𝐺𝐻 ) ) ) → ( 𝐴 ( ( proj ‘ ( ⊥ ‘ 𝐺 ) ) ‘ 𝐴 ) ) ∈ ( ⊥ ‘ ( 𝐺𝐻 ) ) )
17 15 16 mp3an1 ( ( 𝐴 ∈ ( ⊥ ‘ ( 𝐺𝐻 ) ) ∧ ( ( proj ‘ ( ⊥ ‘ 𝐺 ) ) ‘ 𝐴 ) ∈ ( ⊥ ‘ ( 𝐺𝐻 ) ) ) → ( 𝐴 ( ( proj ‘ ( ⊥ ‘ 𝐺 ) ) ‘ 𝐴 ) ) ∈ ( ⊥ ‘ ( 𝐺𝐻 ) ) )
18 14 17 mpdan ( 𝐴 ∈ ( ⊥ ‘ ( 𝐺𝐻 ) ) → ( 𝐴 ( ( proj ‘ ( ⊥ ‘ 𝐺 ) ) ‘ 𝐴 ) ) ∈ ( ⊥ ‘ ( 𝐺𝐻 ) ) )
19 7 18 eqeltrd ( 𝐴 ∈ ( ⊥ ‘ ( 𝐺𝐻 ) ) → ( ( proj𝐺 ) ‘ 𝐴 ) ∈ ( ⊥ ‘ ( 𝐺𝐻 ) ) )