Metamath Proof Explorer


Theorem stge1i

Description: If a state is greater than or equal to 1, it is 1. (Contributed by NM, 11-Nov-1999) (New usage is discouraged.)

Ref Expression
Hypothesis sto1.1 𝐴C
Assertion stge1i ( 𝑆 ∈ States → ( 1 ≤ ( 𝑆𝐴 ) ↔ ( 𝑆𝐴 ) = 1 ) )

Proof

Step Hyp Ref Expression
1 sto1.1 𝐴C
2 stle1 ( 𝑆 ∈ States → ( 𝐴C → ( 𝑆𝐴 ) ≤ 1 ) )
3 1 2 mpi ( 𝑆 ∈ States → ( 𝑆𝐴 ) ≤ 1 )
4 3 anim1i ( ( 𝑆 ∈ States ∧ 1 ≤ ( 𝑆𝐴 ) ) → ( ( 𝑆𝐴 ) ≤ 1 ∧ 1 ≤ ( 𝑆𝐴 ) ) )
5 4 ex ( 𝑆 ∈ States → ( 1 ≤ ( 𝑆𝐴 ) → ( ( 𝑆𝐴 ) ≤ 1 ∧ 1 ≤ ( 𝑆𝐴 ) ) ) )
6 stcl ( 𝑆 ∈ States → ( 𝐴C → ( 𝑆𝐴 ) ∈ ℝ ) )
7 1 6 mpi ( 𝑆 ∈ States → ( 𝑆𝐴 ) ∈ ℝ )
8 1re 1 ∈ ℝ
9 letri3 ( ( ( 𝑆𝐴 ) ∈ ℝ ∧ 1 ∈ ℝ ) → ( ( 𝑆𝐴 ) = 1 ↔ ( ( 𝑆𝐴 ) ≤ 1 ∧ 1 ≤ ( 𝑆𝐴 ) ) ) )
10 7 8 9 sylancl ( 𝑆 ∈ States → ( ( 𝑆𝐴 ) = 1 ↔ ( ( 𝑆𝐴 ) ≤ 1 ∧ 1 ≤ ( 𝑆𝐴 ) ) ) )
11 5 10 sylibrd ( 𝑆 ∈ States → ( 1 ≤ ( 𝑆𝐴 ) → ( 𝑆𝐴 ) = 1 ) )
12 1le1 1 ≤ 1
13 breq2 ( ( 𝑆𝐴 ) = 1 → ( 1 ≤ ( 𝑆𝐴 ) ↔ 1 ≤ 1 ) )
14 12 13 mpbiri ( ( 𝑆𝐴 ) = 1 → 1 ≤ ( 𝑆𝐴 ) )
15 11 14 impbid1 ( 𝑆 ∈ States → ( 1 ≤ ( 𝑆𝐴 ) ↔ ( 𝑆𝐴 ) = 1 ) )