Metamath Proof Explorer


Theorem hstrlem6

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 = ( 𝑥C ↦ ( ( proj𝑥 ) ‘ 𝑢 ) )
hstrlem3.2 ( 𝜑 ↔ ( 𝑢 ∈ ( 𝐴𝐵 ) ∧ ( norm𝑢 ) = 1 ) )
hstrlem3.3 𝐴C
hstrlem3.4 𝐵C
Assertion hstrlem6 ( 𝜑 → ¬ ( ( norm ‘ ( 𝑆𝐴 ) ) = 1 → ( norm ‘ ( 𝑆𝐵 ) ) = 1 ) )

Proof

Step Hyp Ref Expression
1 hstrlem3.1 𝑆 = ( 𝑥C ↦ ( ( proj𝑥 ) ‘ 𝑢 ) )
2 hstrlem3.2 ( 𝜑 ↔ ( 𝑢 ∈ ( 𝐴𝐵 ) ∧ ( norm𝑢 ) = 1 ) )
3 hstrlem3.3 𝐴C
4 hstrlem3.4 𝐵C
5 1 2 3 4 hstrlem4 ( 𝜑 → ( norm ‘ ( 𝑆𝐴 ) ) = 1 )
6 1 2 3 4 hstrlem3 ( 𝜑𝑆 ∈ CHStates )
7 hstcl ( ( 𝑆 ∈ CHStates ∧ 𝐵C ) → ( 𝑆𝐵 ) ∈ ℋ )
8 6 4 7 sylancl ( 𝜑 → ( 𝑆𝐵 ) ∈ ℋ )
9 normcl ( ( 𝑆𝐵 ) ∈ ℋ → ( norm ‘ ( 𝑆𝐵 ) ) ∈ ℝ )
10 8 9 syl ( 𝜑 → ( norm ‘ ( 𝑆𝐵 ) ) ∈ ℝ )
11 1 2 3 4 hstrlem5 ( 𝜑 → ( norm ‘ ( 𝑆𝐵 ) ) < 1 )
12 10 11 ltned ( 𝜑 → ( norm ‘ ( 𝑆𝐵 ) ) ≠ 1 )
13 12 neneqd ( 𝜑 → ¬ ( norm ‘ ( 𝑆𝐵 ) ) = 1 )
14 5 13 jcnd ( 𝜑 → ¬ ( ( norm ‘ ( 𝑆𝐴 ) ) = 1 → ( norm ‘ ( 𝑆𝐵 ) ) = 1 ) )