Theorem hstrlem4

Description: Lemma for strong set of CH states theorem. (Contributed by NM, 30-Jun-2006) (New usage is discouraged.)

	Ref	Expression
Hypotheses	hstrlem3.1	⊢ 𝑆 = ( 𝑥 ∈ Cℋ ↦ ( ( projℎ ‘ 𝑥 ) ‘ 𝑢 ) )
	hstrlem3.2	⊢ ( 𝜑 ↔ ( 𝑢 ∈ ( 𝐴 ∖ 𝐵 ) ∧ ( normℎ ‘ 𝑢 ) = 1 ) )
	hstrlem3.3	⊢ 𝐴 ∈ Cℋ
	hstrlem3.4	⊢ 𝐵 ∈ Cℋ
Assertion	hstrlem4	⊢ ( 𝜑 → ( normℎ ‘ ( 𝑆 ‘ 𝐴 ) ) = 1 )

Proof

Step	Hyp	Ref	Expression
1		hstrlem3.1	⊢ 𝑆 = ( 𝑥 ∈ Cℋ ↦ ( ( projℎ ‘ 𝑥 ) ‘ 𝑢 ) )
2		hstrlem3.2	⊢ ( 𝜑 ↔ ( 𝑢 ∈ ( 𝐴 ∖ 𝐵 ) ∧ ( normℎ ‘ 𝑢 ) = 1 ) )
3		hstrlem3.3	⊢ 𝐴 ∈ Cℋ
4		hstrlem3.4	⊢ 𝐵 ∈ Cℋ
5	1	hstrlem2	⊢ ( 𝐴 ∈ Cℋ → ( 𝑆 ‘ 𝐴 ) = ( ( projℎ ‘ 𝐴 ) ‘ 𝑢 ) )
6	3 5	ax-mp	⊢ ( 𝑆 ‘ 𝐴 ) = ( ( projℎ ‘ 𝐴 ) ‘ 𝑢 )
7	6	fveq2i	⊢ ( normℎ ‘ ( 𝑆 ‘ 𝐴 ) ) = ( normℎ ‘ ( ( projℎ ‘ 𝐴 ) ‘ 𝑢 ) )
8		eldifi	⊢ ( 𝑢 ∈ ( 𝐴 ∖ 𝐵 ) → 𝑢 ∈ 𝐴 )
9		pjid	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝑢 ∈ 𝐴 ) → ( ( projℎ ‘ 𝐴 ) ‘ 𝑢 ) = 𝑢 )
10	3 9	mpan	⊢ ( 𝑢 ∈ 𝐴 → ( ( projℎ ‘ 𝐴 ) ‘ 𝑢 ) = 𝑢 )
11	10	fveq2d	⊢ ( 𝑢 ∈ 𝐴 → ( normℎ ‘ ( ( projℎ ‘ 𝐴 ) ‘ 𝑢 ) ) = ( normℎ ‘ 𝑢 ) )
12		eqeq2	⊢ ( ( normℎ ‘ 𝑢 ) = 1 → ( ( normℎ ‘ ( ( projℎ ‘ 𝐴 ) ‘ 𝑢 ) ) = ( normℎ ‘ 𝑢 ) ↔ ( normℎ ‘ ( ( projℎ ‘ 𝐴 ) ‘ 𝑢 ) ) = 1 ) )
13	11 12	imbitrid	⊢ ( ( normℎ ‘ 𝑢 ) = 1 → ( 𝑢 ∈ 𝐴 → ( normℎ ‘ ( ( projℎ ‘ 𝐴 ) ‘ 𝑢 ) ) = 1 ) )
14	8 13	mpan9	⊢ ( ( 𝑢 ∈ ( 𝐴 ∖ 𝐵 ) ∧ ( normℎ ‘ 𝑢 ) = 1 ) → ( normℎ ‘ ( ( projℎ ‘ 𝐴 ) ‘ 𝑢 ) ) = 1 )
15	2 14	sylbi	⊢ ( 𝜑 → ( normℎ ‘ ( ( projℎ ‘ 𝐴 ) ‘ 𝑢 ) ) = 1 )
16	7 15	eqtrid	⊢ ( 𝜑 → ( normℎ ‘ ( 𝑆 ‘ 𝐴 ) ) = 1 )