Metamath Proof Explorer


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 → ( ( ( 𝐴C𝐵C ) ∧ 𝐴 ⊆ ( ⊥ ‘ 𝐵 ) ) → ( 𝑆 ‘ ( 𝐴 𝐵 ) ) = ( ( 𝑆𝐴 ) + ( 𝑆𝐵 ) ) ) )

Proof

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