Theorem pjssge0ii

Description: Theorem 4.5(iv)->(v) of Beran p. 112. (Contributed by NM, 13-Aug-2000) (New usage is discouraged.)

	Ref	Expression
Hypotheses	pjidm.1	⊢ 𝐻 ∈ Cℋ
	pjidm.2	⊢ 𝐴 ∈ ℋ
	pjsslem.1	⊢ 𝐺 ∈ Cℋ
Assertion	pjssge0ii	⊢ ( ( ( ( projℎ ‘ 𝐺 ) ‘ 𝐴 ) −ℎ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ) = ( ( projℎ ‘ ( 𝐺 ∩ ( ⊥ ‘ 𝐻 ) ) ) ‘ 𝐴 ) → 0 ≤ ( ( ( ( projℎ ‘ 𝐺 ) ‘ 𝐴 ) −ℎ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ) ·ih 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		pjidm.1	⊢ 𝐻 ∈ Cℋ
2		pjidm.2	⊢ 𝐴 ∈ ℋ
3		pjsslem.1	⊢ 𝐺 ∈ Cℋ
4	1	choccli	⊢ ( ⊥ ‘ 𝐻 ) ∈ Cℋ
5	3 4	chincli	⊢ ( 𝐺 ∩ ( ⊥ ‘ 𝐻 ) ) ∈ Cℋ
6	5 2	pjhclii	⊢ ( ( projℎ ‘ ( 𝐺 ∩ ( ⊥ ‘ 𝐻 ) ) ) ‘ 𝐴 ) ∈ ℋ
7	6	normcli	⊢ ( normℎ ‘ ( ( projℎ ‘ ( 𝐺 ∩ ( ⊥ ‘ 𝐻 ) ) ) ‘ 𝐴 ) ) ∈ ℝ
8	7	sqge0i	⊢ 0 ≤ ( ( normℎ ‘ ( ( projℎ ‘ ( 𝐺 ∩ ( ⊥ ‘ 𝐻 ) ) ) ‘ 𝐴 ) ) ↑ 2 )
9		oveq1	⊢ ( ( ( ( projℎ ‘ 𝐺 ) ‘ 𝐴 ) −ℎ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ) = ( ( projℎ ‘ ( 𝐺 ∩ ( ⊥ ‘ 𝐻 ) ) ) ‘ 𝐴 ) → ( ( ( ( projℎ ‘ 𝐺 ) ‘ 𝐴 ) −ℎ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ) ·ih 𝐴 ) = ( ( ( projℎ ‘ ( 𝐺 ∩ ( ⊥ ‘ 𝐻 ) ) ) ‘ 𝐴 ) ·ih 𝐴 ) )
10	5 2	pjinormii	⊢ ( ( ( projℎ ‘ ( 𝐺 ∩ ( ⊥ ‘ 𝐻 ) ) ) ‘ 𝐴 ) ·ih 𝐴 ) = ( ( normℎ ‘ ( ( projℎ ‘ ( 𝐺 ∩ ( ⊥ ‘ 𝐻 ) ) ) ‘ 𝐴 ) ) ↑ 2 )
11	9 10	eqtrdi	⊢ ( ( ( ( projℎ ‘ 𝐺 ) ‘ 𝐴 ) −ℎ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ) = ( ( projℎ ‘ ( 𝐺 ∩ ( ⊥ ‘ 𝐻 ) ) ) ‘ 𝐴 ) → ( ( ( ( projℎ ‘ 𝐺 ) ‘ 𝐴 ) −ℎ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ) ·ih 𝐴 ) = ( ( normℎ ‘ ( ( projℎ ‘ ( 𝐺 ∩ ( ⊥ ‘ 𝐻 ) ) ) ‘ 𝐴 ) ) ↑ 2 ) )
12	8 11	breqtrrid	⊢ ( ( ( ( projℎ ‘ 𝐺 ) ‘ 𝐴 ) −ℎ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ) = ( ( projℎ ‘ ( 𝐺 ∩ ( ⊥ ‘ 𝐻 ) ) ) ‘ 𝐴 ) → 0 ≤ ( ( ( ( projℎ ‘ 𝐺 ) ‘ 𝐴 ) −ℎ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ) ·ih 𝐴 ) )