Metamath Proof Explorer

Theorem measunl

Description: A measure is sub-additive with respect to union. (Contributed by Thierry Arnoux, 20-Oct-2017)

Ref Expression
Hypotheses measunl.1 ⊢ ( 𝜑 → 𝑀 ∈ ( measures ‘ 𝑆 ) )
measunl.2 ⊢ ( 𝜑 → 𝐴 ∈ 𝑆 )
measunl.3 ⊢ ( 𝜑 → 𝐵 ∈ 𝑆 )
Assertion measunl ( 𝜑 → ( 𝑀 ‘ ( 𝐴 ∪ 𝐵 ) ) ≤ ( ( 𝑀 ‘ 𝐴 ) +e ( 𝑀 ‘ 𝐵 ) ) )

Proof

Step Hyp Ref Expression
1 measunl.1 ⊢ ( 𝜑 → 𝑀 ∈ ( measures ‘ 𝑆 ) )
2 measunl.2 ⊢ ( 𝜑 → 𝐴 ∈ 𝑆 )
3 measunl.3 ⊢ ( 𝜑 → 𝐵 ∈ 𝑆 )
4 undif1 ⊢ ( ( 𝐴 ∖ 𝐵 ) ∪ 𝐵 ) = ( 𝐴 ∪ 𝐵 )
5 4 fveq2i ⊢ ( 𝑀 ‘ ( ( 𝐴 ∖ 𝐵 ) ∪ 𝐵 ) ) = ( 𝑀 ‘ ( 𝐴 ∪ 𝐵 ) )
6 measbase ⊢ ( 𝑀 ∈ ( measures ‘ 𝑆 ) → 𝑆 ∈ ∪ ran sigAlgebra )
7 1 6 syl ⊢ ( 𝜑 → 𝑆 ∈ ∪ ran sigAlgebra )
8 difelsiga ⊢ ( ( 𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ) → ( 𝐴 ∖ 𝐵 ) ∈ 𝑆 )
9 7 2 3 8 syl3anc ⊢ ( 𝜑 → ( 𝐴 ∖ 𝐵 ) ∈ 𝑆 )
10 disjdifr ⊢ ( ( 𝐴 ∖ 𝐵 ) ∩ 𝐵 ) = ∅
11 10 a1i ⊢ ( 𝜑 → ( ( 𝐴 ∖ 𝐵 ) ∩ 𝐵 ) = ∅ )
12 measun ⊢ ( ( 𝑀 ∈ ( measures ‘ 𝑆 ) ∧ ( ( 𝐴 ∖ 𝐵 ) ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ) ∧ ( ( 𝐴 ∖ 𝐵 ) ∩ 𝐵 ) = ∅ ) → ( 𝑀 ‘ ( ( 𝐴 ∖ 𝐵 ) ∪ 𝐵 ) ) = ( ( 𝑀 ‘ ( 𝐴 ∖ 𝐵 ) ) +e ( 𝑀 ‘ 𝐵 ) ) )
13 1 9 3 11 12 syl121anc ⊢ ( 𝜑 → ( 𝑀 ‘ ( ( 𝐴 ∖ 𝐵 ) ∪ 𝐵 ) ) = ( ( 𝑀 ‘ ( 𝐴 ∖ 𝐵 ) ) +e ( 𝑀 ‘ 𝐵 ) ) )
14 5 13 eqtr3id ⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝐴 ∪ 𝐵 ) ) = ( ( 𝑀 ‘ ( 𝐴 ∖ 𝐵 ) ) +e ( 𝑀 ‘ 𝐵 ) ) )
15 iccssxr ⊢ ( 0 [,] +∞ ) ⊆ ℝ*
16 measvxrge0 ⊢ ( ( 𝑀 ∈ ( measures ‘ 𝑆 ) ∧ ( 𝐴 ∖ 𝐵 ) ∈ 𝑆 ) → ( 𝑀 ‘ ( 𝐴 ∖ 𝐵 ) ) ∈ ( 0 [,] +∞ ) )
17 1 9 16 syl2anc ⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝐴 ∖ 𝐵 ) ) ∈ ( 0 [,] +∞ ) )
18 15 17 sselid ⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝐴 ∖ 𝐵 ) ) ∈ ℝ* )
19 measvxrge0 ⊢ ( ( 𝑀 ∈ ( measures ‘ 𝑆 ) ∧ 𝐴 ∈ 𝑆 ) → ( 𝑀 ‘ 𝐴 ) ∈ ( 0 [,] +∞ ) )
20 1 2 19 syl2anc ⊢ ( 𝜑 → ( 𝑀 ‘ 𝐴 ) ∈ ( 0 [,] +∞ ) )
21 15 20 sselid ⊢ ( 𝜑 → ( 𝑀 ‘ 𝐴 ) ∈ ℝ* )
22 measvxrge0 ⊢ ( ( 𝑀 ∈ ( measures ‘ 𝑆 ) ∧ 𝐵 ∈ 𝑆 ) → ( 𝑀 ‘ 𝐵 ) ∈ ( 0 [,] +∞ ) )
23 1 3 22 syl2anc ⊢ ( 𝜑 → ( 𝑀 ‘ 𝐵 ) ∈ ( 0 [,] +∞ ) )
24 15 23 sselid ⊢ ( 𝜑 → ( 𝑀 ‘ 𝐵 ) ∈ ℝ* )
25 inelsiga ⊢ ( ( 𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ) → ( 𝐴 ∩ 𝐵 ) ∈ 𝑆 )
26 7 2 3 25 syl3anc ⊢ ( 𝜑 → ( 𝐴 ∩ 𝐵 ) ∈ 𝑆 )
27 measvxrge0 ⊢ ( ( 𝑀 ∈ ( measures ‘ 𝑆 ) ∧ ( 𝐴 ∩ 𝐵 ) ∈ 𝑆 ) → ( 𝑀 ‘ ( 𝐴 ∩ 𝐵 ) ) ∈ ( 0 [,] +∞ ) )
28 1 26 27 syl2anc ⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝐴 ∩ 𝐵 ) ) ∈ ( 0 [,] +∞ ) )
29 elxrge0 ⊢ ( ( 𝑀 ‘ ( 𝐴 ∩ 𝐵 ) ) ∈ ( 0 [,] +∞ ) ↔ ( ( 𝑀 ‘ ( 𝐴 ∩ 𝐵 ) ) ∈ ℝ* ∧ 0 ≤ ( 𝑀 ‘ ( 𝐴 ∩ 𝐵 ) ) ) )
30 28 29 sylib ⊢ ( 𝜑 → ( ( 𝑀 ‘ ( 𝐴 ∩ 𝐵 ) ) ∈ ℝ* ∧ 0 ≤ ( 𝑀 ‘ ( 𝐴 ∩ 𝐵 ) ) ) )
31 30 simprd ⊢ ( 𝜑 → 0 ≤ ( 𝑀 ‘ ( 𝐴 ∩ 𝐵 ) ) )
32 15 28 sselid ⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝐴 ∩ 𝐵 ) ) ∈ ℝ* )
33 xraddge02 ⊢ ( ( ( 𝑀 ‘ ( 𝐴 ∖ 𝐵 ) ) ∈ ℝ* ∧ ( 𝑀 ‘ ( 𝐴 ∩ 𝐵 ) ) ∈ ℝ* ) → ( 0 ≤ ( 𝑀 ‘ ( 𝐴 ∩ 𝐵 ) ) → ( 𝑀 ‘ ( 𝐴 ∖ 𝐵 ) ) ≤ ( ( 𝑀 ‘ ( 𝐴 ∖ 𝐵 ) ) +e ( 𝑀 ‘ ( 𝐴 ∩ 𝐵 ) ) ) ) )
34 18 32 33 syl2anc ⊢ ( 𝜑 → ( 0 ≤ ( 𝑀 ‘ ( 𝐴 ∩ 𝐵 ) ) → ( 𝑀 ‘ ( 𝐴 ∖ 𝐵 ) ) ≤ ( ( 𝑀 ‘ ( 𝐴 ∖ 𝐵 ) ) +e ( 𝑀 ‘ ( 𝐴 ∩ 𝐵 ) ) ) ) )
35 31 34 mpd ⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝐴 ∖ 𝐵 ) ) ≤ ( ( 𝑀 ‘ ( 𝐴 ∖ 𝐵 ) ) +e ( 𝑀 ‘ ( 𝐴 ∩ 𝐵 ) ) ) )
36 uncom ⊢ ( ( 𝐴 ∩ 𝐵 ) ∪ ( 𝐴 ∖ 𝐵 ) ) = ( ( 𝐴 ∖ 𝐵 ) ∪ ( 𝐴 ∩ 𝐵 ) )
37 inundif ⊢ ( ( 𝐴 ∩ 𝐵 ) ∪ ( 𝐴 ∖ 𝐵 ) ) = 𝐴
38 36 37 eqtr3i ⊢ ( ( 𝐴 ∖ 𝐵 ) ∪ ( 𝐴 ∩ 𝐵 ) ) = 𝐴
39 38 fveq2i ⊢ ( 𝑀 ‘ ( ( 𝐴 ∖ 𝐵 ) ∪ ( 𝐴 ∩ 𝐵 ) ) ) = ( 𝑀 ‘ 𝐴 )
40 incom ⊢ ( ( 𝐴 ∩ 𝐵 ) ∩ ( 𝐴 ∖ 𝐵 ) ) = ( ( 𝐴 ∖ 𝐵 ) ∩ ( 𝐴 ∩ 𝐵 ) )
41 inindif ⊢ ( ( 𝐴 ∩ 𝐵 ) ∩ ( 𝐴 ∖ 𝐵 ) ) = ∅
42 40 41 eqtr3i ⊢ ( ( 𝐴 ∖ 𝐵 ) ∩ ( 𝐴 ∩ 𝐵 ) ) = ∅
43 42 a1i ⊢ ( 𝜑 → ( ( 𝐴 ∖ 𝐵 ) ∩ ( 𝐴 ∩ 𝐵 ) ) = ∅ )
44 measun ⊢ ( ( 𝑀 ∈ ( measures ‘ 𝑆 ) ∧ ( ( 𝐴 ∖ 𝐵 ) ∈ 𝑆 ∧ ( 𝐴 ∩ 𝐵 ) ∈ 𝑆 ) ∧ ( ( 𝐴 ∖ 𝐵 ) ∩ ( 𝐴 ∩ 𝐵 ) ) = ∅ ) → ( 𝑀 ‘ ( ( 𝐴 ∖ 𝐵 ) ∪ ( 𝐴 ∩ 𝐵 ) ) ) = ( ( 𝑀 ‘ ( 𝐴 ∖ 𝐵 ) ) +e ( 𝑀 ‘ ( 𝐴 ∩ 𝐵 ) ) ) )
45 1 9 26 43 44 syl121anc ⊢ ( 𝜑 → ( 𝑀 ‘ ( ( 𝐴 ∖ 𝐵 ) ∪ ( 𝐴 ∩ 𝐵 ) ) ) = ( ( 𝑀 ‘ ( 𝐴 ∖ 𝐵 ) ) +e ( 𝑀 ‘ ( 𝐴 ∩ 𝐵 ) ) ) )
46 39 45 eqtr3id ⊢ ( 𝜑 → ( 𝑀 ‘ 𝐴 ) = ( ( 𝑀 ‘ ( 𝐴 ∖ 𝐵 ) ) +e ( 𝑀 ‘ ( 𝐴 ∩ 𝐵 ) ) ) )
47 35 46 breqtrrd ⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝐴 ∖ 𝐵 ) ) ≤ ( 𝑀 ‘ 𝐴 ) )
48 xleadd1a ⊢ ( ( ( ( 𝑀 ‘ ( 𝐴 ∖ 𝐵 ) ) ∈ ℝ* ∧ ( 𝑀 ‘ 𝐴 ) ∈ ℝ* ∧ ( 𝑀 ‘ 𝐵 ) ∈ ℝ* ) ∧ ( 𝑀 ‘ ( 𝐴 ∖ 𝐵 ) ) ≤ ( 𝑀 ‘ 𝐴 ) ) → ( ( 𝑀 ‘ ( 𝐴 ∖ 𝐵 ) ) +e ( 𝑀 ‘ 𝐵 ) ) ≤ ( ( 𝑀 ‘ 𝐴 ) +e ( 𝑀 ‘ 𝐵 ) ) )
49 18 21 24 47 48 syl31anc ⊢ ( 𝜑 → ( ( 𝑀 ‘ ( 𝐴 ∖ 𝐵 ) ) +e ( 𝑀 ‘ 𝐵 ) ) ≤ ( ( 𝑀 ‘ 𝐴 ) +e ( 𝑀 ‘ 𝐵 ) ) )
50 14 49 eqbrtrd ⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝐴 ∪ 𝐵 ) ) ≤ ( ( 𝑀 ‘ 𝐴 ) +e ( 𝑀 ‘ 𝐵 ) ) )