Theorem stj

Description: The value of a state on a join. (Contributed by NM, 23-Oct-1999) (New usage is discouraged.)

	Ref	Expression
Assertion	stj	⊢ ( 𝑆 ∈ States → ( ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) ∧ 𝐴 ⊆ ( ⊥ ‘ 𝐵 ) ) → ( 𝑆 ‘ ( 𝐴 ∨ℋ 𝐵 ) ) = ( ( 𝑆 ‘ 𝐴 ) + ( 𝑆 ‘ 𝐵 ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		isst	⊢ ( 𝑆 ∈ States ↔ ( 𝑆 : Cℋ ⟶ ( 0 [,] 1 ) ∧ ( 𝑆 ‘ ℋ ) = 1 ∧ ∀ 𝑥 ∈ Cℋ ∀ 𝑦 ∈ Cℋ ( 𝑥 ⊆ ( ⊥ ‘ 𝑦 ) → ( 𝑆 ‘ ( 𝑥 ∨ℋ 𝑦 ) ) = ( ( 𝑆 ‘ 𝑥 ) + ( 𝑆 ‘ 𝑦 ) ) ) ) )
2	1	simp3bi	⊢ ( 𝑆 ∈ States → ∀ 𝑥 ∈ Cℋ ∀ 𝑦 ∈ Cℋ ( 𝑥 ⊆ ( ⊥ ‘ 𝑦 ) → ( 𝑆 ‘ ( 𝑥 ∨ℋ 𝑦 ) ) = ( ( 𝑆 ‘ 𝑥 ) + ( 𝑆 ‘ 𝑦 ) ) ) )
3		sseq1	⊢ ( 𝑥 = 𝐴 → ( 𝑥 ⊆ ( ⊥ ‘ 𝑦 ) ↔ 𝐴 ⊆ ( ⊥ ‘ 𝑦 ) ) )
4		fvoveq1	⊢ ( 𝑥 = 𝐴 → ( 𝑆 ‘ ( 𝑥 ∨ℋ 𝑦 ) ) = ( 𝑆 ‘ ( 𝐴 ∨ℋ 𝑦 ) ) )
5		fveq2	⊢ ( 𝑥 = 𝐴 → ( 𝑆 ‘ 𝑥 ) = ( 𝑆 ‘ 𝐴 ) )
6	5	oveq1d	⊢ ( 𝑥 = 𝐴 → ( ( 𝑆 ‘ 𝑥 ) + ( 𝑆 ‘ 𝑦 ) ) = ( ( 𝑆 ‘ 𝐴 ) + ( 𝑆 ‘ 𝑦 ) ) )
7	4 6	eqeq12d	⊢ ( 𝑥 = 𝐴 → ( ( 𝑆 ‘ ( 𝑥 ∨ℋ 𝑦 ) ) = ( ( 𝑆 ‘ 𝑥 ) + ( 𝑆 ‘ 𝑦 ) ) ↔ ( 𝑆 ‘ ( 𝐴 ∨ℋ 𝑦 ) ) = ( ( 𝑆 ‘ 𝐴 ) + ( 𝑆 ‘ 𝑦 ) ) ) )
8	3 7	imbi12d	⊢ ( 𝑥 = 𝐴 → ( ( 𝑥 ⊆ ( ⊥ ‘ 𝑦 ) → ( 𝑆 ‘ ( 𝑥 ∨ℋ 𝑦 ) ) = ( ( 𝑆 ‘ 𝑥 ) + ( 𝑆 ‘ 𝑦 ) ) ) ↔ ( 𝐴 ⊆ ( ⊥ ‘ 𝑦 ) → ( 𝑆 ‘ ( 𝐴 ∨ℋ 𝑦 ) ) = ( ( 𝑆 ‘ 𝐴 ) + ( 𝑆 ‘ 𝑦 ) ) ) ) )
9		fveq2	⊢ ( 𝑦 = 𝐵 → ( ⊥ ‘ 𝑦 ) = ( ⊥ ‘ 𝐵 ) )
10	9	sseq2d	⊢ ( 𝑦 = 𝐵 → ( 𝐴 ⊆ ( ⊥ ‘ 𝑦 ) ↔ 𝐴 ⊆ ( ⊥ ‘ 𝐵 ) ) )
11		oveq2	⊢ ( 𝑦 = 𝐵 → ( 𝐴 ∨ℋ 𝑦 ) = ( 𝐴 ∨ℋ 𝐵 ) )
12	11	fveq2d	⊢ ( 𝑦 = 𝐵 → ( 𝑆 ‘ ( 𝐴 ∨ℋ 𝑦 ) ) = ( 𝑆 ‘ ( 𝐴 ∨ℋ 𝐵 ) ) )
13		fveq2	⊢ ( 𝑦 = 𝐵 → ( 𝑆 ‘ 𝑦 ) = ( 𝑆 ‘ 𝐵 ) )
14	13	oveq2d	⊢ ( 𝑦 = 𝐵 → ( ( 𝑆 ‘ 𝐴 ) + ( 𝑆 ‘ 𝑦 ) ) = ( ( 𝑆 ‘ 𝐴 ) + ( 𝑆 ‘ 𝐵 ) ) )
15	12 14	eqeq12d	⊢ ( 𝑦 = 𝐵 → ( ( 𝑆 ‘ ( 𝐴 ∨ℋ 𝑦 ) ) = ( ( 𝑆 ‘ 𝐴 ) + ( 𝑆 ‘ 𝑦 ) ) ↔ ( 𝑆 ‘ ( 𝐴 ∨ℋ 𝐵 ) ) = ( ( 𝑆 ‘ 𝐴 ) + ( 𝑆 ‘ 𝐵 ) ) ) )
16	10 15	imbi12d	⊢ ( 𝑦 = 𝐵 → ( ( 𝐴 ⊆ ( ⊥ ‘ 𝑦 ) → ( 𝑆 ‘ ( 𝐴 ∨ℋ 𝑦 ) ) = ( ( 𝑆 ‘ 𝐴 ) + ( 𝑆 ‘ 𝑦 ) ) ) ↔ ( 𝐴 ⊆ ( ⊥ ‘ 𝐵 ) → ( 𝑆 ‘ ( 𝐴 ∨ℋ 𝐵 ) ) = ( ( 𝑆 ‘ 𝐴 ) + ( 𝑆 ‘ 𝐵 ) ) ) ) )
17	8 16	rspc2v	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( ∀ 𝑥 ∈ Cℋ ∀ 𝑦 ∈ Cℋ ( 𝑥 ⊆ ( ⊥ ‘ 𝑦 ) → ( 𝑆 ‘ ( 𝑥 ∨ℋ 𝑦 ) ) = ( ( 𝑆 ‘ 𝑥 ) + ( 𝑆 ‘ 𝑦 ) ) ) → ( 𝐴 ⊆ ( ⊥ ‘ 𝐵 ) → ( 𝑆 ‘ ( 𝐴 ∨ℋ 𝐵 ) ) = ( ( 𝑆 ‘ 𝐴 ) + ( 𝑆 ‘ 𝐵 ) ) ) ) )
18	2 17	syl5com	⊢ ( 𝑆 ∈ States → ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( 𝐴 ⊆ ( ⊥ ‘ 𝐵 ) → ( 𝑆 ‘ ( 𝐴 ∨ℋ 𝐵 ) ) = ( ( 𝑆 ‘ 𝐴 ) + ( 𝑆 ‘ 𝐵 ) ) ) ) )
19	18	impd	⊢ ( 𝑆 ∈ States → ( ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) ∧ 𝐴 ⊆ ( ⊥ ‘ 𝐵 ) ) → ( 𝑆 ‘ ( 𝐴 ∨ℋ 𝐵 ) ) = ( ( 𝑆 ‘ 𝐴 ) + ( 𝑆 ‘ 𝐵 ) ) ) )