Metamath Proof Explorer

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 ∈ C_ℋ
		pjidm.2	⊢ 𝐴 ∈ ℋ
		pjsslem.1	⊢ 𝐺 ∈ C_ℋ
	Assertion	pjssge0ii	⊢ ( ( ( ( proj_ℎ ‘ 𝐺 ) ‘ 𝐴 ) −_ℎ ( ( proj_ℎ ‘ 𝐻 ) ‘ 𝐴 ) ) = ( ( proj_ℎ ‘ ( 𝐺 ∩ ( ⊥ ‘ 𝐻 ) ) ) ‘ 𝐴 ) → 0 ≤ ( ( ( ( proj_ℎ ‘ 𝐺 ) ‘ 𝐴 ) −_ℎ ( ( proj_ℎ ‘ 𝐻 ) ‘ 𝐴 ) ) ·_ih 𝐴 ) )

Proof

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