Metamath Proof Explorer


Theorem meassre

Description: If the measure of a measurable set is real, then the measure of any of its measurable subsets is real. (Contributed by Glauco Siliprandi, 8-Apr-2021)

Ref Expression
Hypotheses meassre.m ( 𝜑𝑀 ∈ Meas )
meassre.a ( 𝜑𝐴 ∈ dom 𝑀 )
meassre.r ( 𝜑 → ( 𝑀𝐴 ) ∈ ℝ )
meassre.s ( 𝜑𝐵𝐴 )
meassre.b ( 𝜑𝐵 ∈ dom 𝑀 )
Assertion meassre ( 𝜑 → ( 𝑀𝐵 ) ∈ ℝ )

Proof

Step Hyp Ref Expression
1 meassre.m ( 𝜑𝑀 ∈ Meas )
2 meassre.a ( 𝜑𝐴 ∈ dom 𝑀 )
3 meassre.r ( 𝜑 → ( 𝑀𝐴 ) ∈ ℝ )
4 meassre.s ( 𝜑𝐵𝐴 )
5 meassre.b ( 𝜑𝐵 ∈ dom 𝑀 )
6 rge0ssre ( 0 [,) +∞ ) ⊆ ℝ
7 0xr 0 ∈ ℝ*
8 7 a1i ( 𝜑 → 0 ∈ ℝ* )
9 pnfxr +∞ ∈ ℝ*
10 9 a1i ( 𝜑 → +∞ ∈ ℝ* )
11 eqid dom 𝑀 = dom 𝑀
12 1 11 5 meaxrcl ( 𝜑 → ( 𝑀𝐵 ) ∈ ℝ* )
13 1 5 meage0 ( 𝜑 → 0 ≤ ( 𝑀𝐵 ) )
14 3 rexrd ( 𝜑 → ( 𝑀𝐴 ) ∈ ℝ* )
15 1 11 5 2 4 meassle ( 𝜑 → ( 𝑀𝐵 ) ≤ ( 𝑀𝐴 ) )
16 3 ltpnfd ( 𝜑 → ( 𝑀𝐴 ) < +∞ )
17 12 14 10 15 16 xrlelttrd ( 𝜑 → ( 𝑀𝐵 ) < +∞ )
18 8 10 12 13 17 elicod ( 𝜑 → ( 𝑀𝐵 ) ∈ ( 0 [,) +∞ ) )
19 6 18 sseldi ( 𝜑 → ( 𝑀𝐵 ) ∈ ℝ )