Theorem pjs14i

Description: Theorem S-14 of Watanabe, p. 486. (Contributed by NM, 26-Sep-2001) (New usage is discouraged.)

	Ref	Expression
Hypotheses	pjs14.1	⊢ 𝐺 ∈ Cℋ
	pjs14.2	⊢ 𝐻 ∈ Cℋ
Assertion	pjs14i	⊢ ( 𝐴 ∈ ℋ → ( normℎ ‘ ( ( ( projℎ ‘ 𝐻 ) ∘ ( projℎ ‘ 𝐺 ) ) ‘ 𝐴 ) ) ≤ ( normℎ ‘ ( ( projℎ ‘ 𝐺 ) ‘ 𝐴 ) ) )

Proof

Step	Hyp	Ref	Expression
1		pjs14.1	⊢ 𝐺 ∈ Cℋ
2		pjs14.2	⊢ 𝐻 ∈ Cℋ
3	2 1	pjcoi	⊢ ( 𝐴 ∈ ℋ → ( ( ( projℎ ‘ 𝐻 ) ∘ ( projℎ ‘ 𝐺 ) ) ‘ 𝐴 ) = ( ( projℎ ‘ 𝐻 ) ‘ ( ( projℎ ‘ 𝐺 ) ‘ 𝐴 ) ) )
4	3	fveq2d	⊢ ( 𝐴 ∈ ℋ → ( normℎ ‘ ( ( ( projℎ ‘ 𝐻 ) ∘ ( projℎ ‘ 𝐺 ) ) ‘ 𝐴 ) ) = ( normℎ ‘ ( ( projℎ ‘ 𝐻 ) ‘ ( ( projℎ ‘ 𝐺 ) ‘ 𝐴 ) ) ) )
5	1	pjhcli	⊢ ( 𝐴 ∈ ℋ → ( ( projℎ ‘ 𝐺 ) ‘ 𝐴 ) ∈ ℋ )
6		pjnorm	⊢ ( ( 𝐻 ∈ Cℋ ∧ ( ( projℎ ‘ 𝐺 ) ‘ 𝐴 ) ∈ ℋ ) → ( normℎ ‘ ( ( projℎ ‘ 𝐻 ) ‘ ( ( projℎ ‘ 𝐺 ) ‘ 𝐴 ) ) ) ≤ ( normℎ ‘ ( ( projℎ ‘ 𝐺 ) ‘ 𝐴 ) ) )
7	2 5 6	sylancr	⊢ ( 𝐴 ∈ ℋ → ( normℎ ‘ ( ( projℎ ‘ 𝐻 ) ‘ ( ( projℎ ‘ 𝐺 ) ‘ 𝐴 ) ) ) ≤ ( normℎ ‘ ( ( projℎ ‘ 𝐺 ) ‘ 𝐴 ) ) )
8	4 7	eqbrtrd	⊢ ( 𝐴 ∈ ℋ → ( normℎ ‘ ( ( ( projℎ ‘ 𝐻 ) ∘ ( projℎ ‘ 𝐺 ) ) ‘ 𝐴 ) ) ≤ ( normℎ ‘ ( ( projℎ ‘ 𝐺 ) ‘ 𝐴 ) ) )