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	sseldi	⊢ ( 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 ) ) )