Metamath Proof Explorer


Theorem smfadd

Description: The sum of two sigma-measurable functions is measurable. Proposition 121E (b) of Fremlin1 p. 37 . (Contributed by Glauco Siliprandi, 26-Jun-2021)

Ref Expression
Hypotheses smfadd.x 𝑥 𝜑
smfadd.s ( 𝜑𝑆 ∈ SAlg )
smfadd.a ( 𝜑𝐴𝑉 )
smfadd.b ( ( 𝜑𝑥𝐴 ) → 𝐵 ∈ ℝ )
smfadd.d ( ( 𝜑𝑥𝐶 ) → 𝐷 ∈ ℝ )
smfadd.m ( 𝜑 → ( 𝑥𝐴𝐵 ) ∈ ( SMblFn ‘ 𝑆 ) )
smfadd.n ( 𝜑 → ( 𝑥𝐶𝐷 ) ∈ ( SMblFn ‘ 𝑆 ) )
Assertion smfadd ( 𝜑 → ( 𝑥 ∈ ( 𝐴𝐶 ) ↦ ( 𝐵 + 𝐷 ) ) ∈ ( SMblFn ‘ 𝑆 ) )

Proof

Step Hyp Ref Expression
1 smfadd.x 𝑥 𝜑
2 smfadd.s ( 𝜑𝑆 ∈ SAlg )
3 smfadd.a ( 𝜑𝐴𝑉 )
4 smfadd.b ( ( 𝜑𝑥𝐴 ) → 𝐵 ∈ ℝ )
5 smfadd.d ( ( 𝜑𝑥𝐶 ) → 𝐷 ∈ ℝ )
6 smfadd.m ( 𝜑 → ( 𝑥𝐴𝐵 ) ∈ ( SMblFn ‘ 𝑆 ) )
7 smfadd.n ( 𝜑 → ( 𝑥𝐶𝐷 ) ∈ ( SMblFn ‘ 𝑆 ) )
8 nfv 𝑎 𝜑
9 elinel1 ( 𝑥 ∈ ( 𝐴𝐶 ) → 𝑥𝐴 )
10 9 adantl ( ( 𝜑𝑥 ∈ ( 𝐴𝐶 ) ) → 𝑥𝐴 )
11 1 10 ssdf ( 𝜑 → ( 𝐴𝐶 ) ⊆ 𝐴 )
12 eqid ( 𝑥𝐴𝐵 ) = ( 𝑥𝐴𝐵 )
13 1 12 4 dmmptdf ( 𝜑 → dom ( 𝑥𝐴𝐵 ) = 𝐴 )
14 13 eqcomd ( 𝜑𝐴 = dom ( 𝑥𝐴𝐵 ) )
15 eqid dom ( 𝑥𝐴𝐵 ) = dom ( 𝑥𝐴𝐵 )
16 2 6 15 smfdmss ( 𝜑 → dom ( 𝑥𝐴𝐵 ) ⊆ 𝑆 )
17 14 16 eqsstrd ( 𝜑𝐴 𝑆 )
18 11 17 sstrd ( 𝜑 → ( 𝐴𝐶 ) ⊆ 𝑆 )
19 10 4 syldan ( ( 𝜑𝑥 ∈ ( 𝐴𝐶 ) ) → 𝐵 ∈ ℝ )
20 elinel2 ( 𝑥 ∈ ( 𝐴𝐶 ) → 𝑥𝐶 )
21 20 adantl ( ( 𝜑𝑥 ∈ ( 𝐴𝐶 ) ) → 𝑥𝐶 )
22 21 5 syldan ( ( 𝜑𝑥 ∈ ( 𝐴𝐶 ) ) → 𝐷 ∈ ℝ )
23 19 22 readdcld ( ( 𝜑𝑥 ∈ ( 𝐴𝐶 ) ) → ( 𝐵 + 𝐷 ) ∈ ℝ )
24 eqid ( 𝑥 ∈ ( 𝐴𝐶 ) ↦ ( 𝐵 + 𝐷 ) ) = ( 𝑥 ∈ ( 𝐴𝐶 ) ↦ ( 𝐵 + 𝐷 ) )
25 1 23 24 fmptdf ( 𝜑 → ( 𝑥 ∈ ( 𝐴𝐶 ) ↦ ( 𝐵 + 𝐷 ) ) : ( 𝐴𝐶 ) ⟶ ℝ )
26 25 fvmptelrn ( ( 𝜑𝑥 ∈ ( 𝐴𝐶 ) ) → ( 𝐵 + 𝐷 ) ∈ ℝ )
27 nfv 𝑥 𝑎 ∈ ℝ
28 1 27 nfan 𝑥 ( 𝜑𝑎 ∈ ℝ )
29 2 adantr ( ( 𝜑𝑎 ∈ ℝ ) → 𝑆 ∈ SAlg )
30 3 adantr ( ( 𝜑𝑎 ∈ ℝ ) → 𝐴𝑉 )
31 4 adantlr ( ( ( 𝜑𝑎 ∈ ℝ ) ∧ 𝑥𝐴 ) → 𝐵 ∈ ℝ )
32 5 adantlr ( ( ( 𝜑𝑎 ∈ ℝ ) ∧ 𝑥𝐶 ) → 𝐷 ∈ ℝ )
33 6 adantr ( ( 𝜑𝑎 ∈ ℝ ) → ( 𝑥𝐴𝐵 ) ∈ ( SMblFn ‘ 𝑆 ) )
34 7 adantr ( ( 𝜑𝑎 ∈ ℝ ) → ( 𝑥𝐶𝐷 ) ∈ ( SMblFn ‘ 𝑆 ) )
35 simpr ( ( 𝜑𝑎 ∈ ℝ ) → 𝑎 ∈ ℝ )
36 oveq2 ( 𝑟 = 𝑞 → ( 𝑝 + 𝑟 ) = ( 𝑝 + 𝑞 ) )
37 36 breq1d ( 𝑟 = 𝑞 → ( ( 𝑝 + 𝑟 ) < 𝑎 ↔ ( 𝑝 + 𝑞 ) < 𝑎 ) )
38 37 cbvrabv { 𝑟 ∈ ℚ ∣ ( 𝑝 + 𝑟 ) < 𝑎 } = { 𝑞 ∈ ℚ ∣ ( 𝑝 + 𝑞 ) < 𝑎 }
39 38 mpteq2i ( 𝑝 ∈ ℚ ↦ { 𝑟 ∈ ℚ ∣ ( 𝑝 + 𝑟 ) < 𝑎 } ) = ( 𝑝 ∈ ℚ ↦ { 𝑞 ∈ ℚ ∣ ( 𝑝 + 𝑞 ) < 𝑎 } )
40 28 29 30 31 32 33 34 35 39 smfaddlem2 ( ( 𝜑𝑎 ∈ ℝ ) → { 𝑥 ∈ ( 𝐴𝐶 ) ∣ ( 𝐵 + 𝐷 ) < 𝑎 } ∈ ( 𝑆t ( 𝐴𝐶 ) ) )
41 1 8 2 18 26 40 issmfdmpt ( 𝜑 → ( 𝑥 ∈ ( 𝐴𝐶 ) ↦ ( 𝐵 + 𝐷 ) ) ∈ ( SMblFn ‘ 𝑆 ) )