Metamath Proof Explorer


Theorem sge0lessmpt

Description: A shorter sum of nonnegative extended reals is smaller than a longer one. (Contributed by Glauco Siliprandi, 17-Aug-2020)

Ref Expression
Hypotheses sge0lessmpt.a ( 𝜑𝐴𝑉 )
sge0lessmpt.b ( ( 𝜑𝑥𝐴 ) → 𝐵 ∈ ( 0 [,] +∞ ) )
sge0lessmpt.c ( 𝜑𝐶𝐴 )
Assertion sge0lessmpt ( 𝜑 → ( Σ^ ‘ ( 𝑥𝐶𝐵 ) ) ≤ ( Σ^ ‘ ( 𝑥𝐴𝐵 ) ) )

Proof

Step Hyp Ref Expression
1 sge0lessmpt.a ( 𝜑𝐴𝑉 )
2 sge0lessmpt.b ( ( 𝜑𝑥𝐴 ) → 𝐵 ∈ ( 0 [,] +∞ ) )
3 sge0lessmpt.c ( 𝜑𝐶𝐴 )
4 id ( 𝜑𝜑 )
5 3 resmptd ( 𝜑 → ( ( 𝑥𝐴𝐵 ) ↾ 𝐶 ) = ( 𝑥𝐶𝐵 ) )
6 5 eqcomd ( 𝜑 → ( 𝑥𝐶𝐵 ) = ( ( 𝑥𝐴𝐵 ) ↾ 𝐶 ) )
7 4 6 syl ( 𝜑 → ( 𝑥𝐶𝐵 ) = ( ( 𝑥𝐴𝐵 ) ↾ 𝐶 ) )
8 7 fveq2d ( 𝜑 → ( Σ^ ‘ ( 𝑥𝐶𝐵 ) ) = ( Σ^ ‘ ( ( 𝑥𝐴𝐵 ) ↾ 𝐶 ) ) )
9 eqid ( 𝑥𝐴𝐵 ) = ( 𝑥𝐴𝐵 )
10 2 9 fmptd ( 𝜑 → ( 𝑥𝐴𝐵 ) : 𝐴 ⟶ ( 0 [,] +∞ ) )
11 1 10 sge0less ( 𝜑 → ( Σ^ ‘ ( ( 𝑥𝐴𝐵 ) ↾ 𝐶 ) ) ≤ ( Σ^ ‘ ( 𝑥𝐴𝐵 ) ) )
12 8 11 eqbrtrd ( 𝜑 → ( Σ^ ‘ ( 𝑥𝐶𝐵 ) ) ≤ ( Σ^ ‘ ( 𝑥𝐴𝐵 ) ) )