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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 ⊢ 𝐺 ∈ Cℋ
pjocin.2 ⊢ 𝐻 ∈ Cℋ
Assertion pjocini ( 𝐴 ∈ ( ⊥ ‘ ( 𝐺 ∩ 𝐻 ) ) → ( ( projℎ ‘ 𝐺 ) ‘ 𝐴 ) ∈ ( ⊥ ‘ ( 𝐺 ∩ 𝐻 ) ) )

Proof

Step Hyp Ref Expression
1 pjocin.1 ⊢ 𝐺 ∈ Cℋ
2 pjocin.2 ⊢ 𝐻 ∈ Cℋ
3 1 2 chincli ⊢ ( 𝐺 ∩ 𝐻 ) ∈ Cℋ
4 3 choccli ⊢ ( ⊥ ‘ ( 𝐺 ∩ 𝐻 ) ) ∈ Cℋ
5 4 cheli ⊢ ( 𝐴 ∈ ( ⊥ ‘ ( 𝐺 ∩ 𝐻 ) ) → 𝐴 ∈ ℋ )
6 pjpo ⊢ ( ( 𝐺 ∈ Cℋ ∧ 𝐴 ∈ ℋ ) → ( ( projℎ ‘ 𝐺 ) ‘ 𝐴 ) = ( 𝐴 −ℎ ( ( projℎ ‘ ( ⊥ ‘ 𝐺 ) ) ‘ 𝐴 ) ) )
7 1 5 6 sylancr ⊢ ( 𝐴 ∈ ( ⊥ ‘ ( 𝐺 ∩ 𝐻 ) ) → ( ( projℎ ‘ 𝐺 ) ‘ 𝐴 ) = ( 𝐴 −ℎ ( ( projℎ ‘ ( ⊥ ‘ 𝐺 ) ) ‘ 𝐴 ) ) )
8 inss1 ⊢ ( 𝐺 ∩ 𝐻 ) ⊆ 𝐺
9 3 1 chsscon3i ⊢ ( ( 𝐺 ∩ 𝐻 ) ⊆ 𝐺 ↔ ( ⊥ ‘ 𝐺 ) ⊆ ( ⊥ ‘ ( 𝐺 ∩ 𝐻 ) ) )
10 8 9 mpbi ⊢ ( ⊥ ‘ 𝐺 ) ⊆ ( ⊥ ‘ ( 𝐺 ∩ 𝐻 ) )
11 1 choccli ⊢ ( ⊥ ‘ 𝐺 ) ∈ Cℋ
12 11 pjcli ⊢ ( 𝐴 ∈ ℋ → ( ( projℎ ‘ ( ⊥ ‘ 𝐺 ) ) ‘ 𝐴 ) ∈ ( ⊥ ‘ 𝐺 ) )
13 5 12 syl ⊢ ( 𝐴 ∈ ( ⊥ ‘ ( 𝐺 ∩ 𝐻 ) ) → ( ( projℎ ‘ ( ⊥ ‘ 𝐺 ) ) ‘ 𝐴 ) ∈ ( ⊥ ‘ 𝐺 ) )
14 10 13 sselid ⊢ ( 𝐴 ∈ ( ⊥ ‘ ( 𝐺 ∩ 𝐻 ) ) → ( ( projℎ ‘ ( ⊥ ‘ 𝐺 ) ) ‘ 𝐴 ) ∈ ( ⊥ ‘ ( 𝐺 ∩ 𝐻 ) ) )
15 4 chshii ⊢ ( ⊥ ‘ ( 𝐺 ∩ 𝐻 ) ) ∈ Sℋ
16 shsubcl ⊢ ( ( ( ⊥ ‘ ( 𝐺 ∩ 𝐻 ) ) ∈ Sℋ ∧ 𝐴 ∈ ( ⊥ ‘ ( 𝐺 ∩ 𝐻 ) ) ∧ ( ( projℎ ‘ ( ⊥ ‘ 𝐺 ) ) ‘ 𝐴 ) ∈ ( ⊥ ‘ ( 𝐺 ∩ 𝐻 ) ) ) → ( 𝐴 −ℎ ( ( projℎ ‘ ( ⊥ ‘ 𝐺 ) ) ‘ 𝐴 ) ) ∈ ( ⊥ ‘ ( 𝐺 ∩ 𝐻 ) ) )
17 15 16 mp3an1 ⊢ ( ( 𝐴 ∈ ( ⊥ ‘ ( 𝐺 ∩ 𝐻 ) ) ∧ ( ( projℎ ‘ ( ⊥ ‘ 𝐺 ) ) ‘ 𝐴 ) ∈ ( ⊥ ‘ ( 𝐺 ∩ 𝐻 ) ) ) → ( 𝐴 −ℎ ( ( projℎ ‘ ( ⊥ ‘ 𝐺 ) ) ‘ 𝐴 ) ) ∈ ( ⊥ ‘ ( 𝐺 ∩ 𝐻 ) ) )
18 14 17 mpdan ⊢ ( 𝐴 ∈ ( ⊥ ‘ ( 𝐺 ∩ 𝐻 ) ) → ( 𝐴 −ℎ ( ( projℎ ‘ ( ⊥ ‘ 𝐺 ) ) ‘ 𝐴 ) ) ∈ ( ⊥ ‘ ( 𝐺 ∩ 𝐻 ) ) )
19 7 18 eqeltrd ⊢ ( 𝐴 ∈ ( ⊥ ‘ ( 𝐺 ∩ 𝐻 ) ) → ( ( projℎ ‘ 𝐺 ) ‘ 𝐴 ) ∈ ( ⊥ ‘ ( 𝐺 ∩ 𝐻 ) ) )