Step |
Hyp |
Ref |
Expression |
1 |
|
measssd.1 |
⊢ ( 𝜑 → 𝑀 ∈ ( measures ‘ 𝑆 ) ) |
2 |
|
measssd.2 |
⊢ ( 𝜑 → 𝐴 ∈ 𝑆 ) |
3 |
|
measssd.3 |
⊢ ( 𝜑 → 𝐵 ∈ 𝑆 ) |
4 |
|
measssd.4 |
⊢ ( 𝜑 → 𝐴 ⊆ 𝐵 ) |
5 |
|
measbase |
⊢ ( 𝑀 ∈ ( measures ‘ 𝑆 ) → 𝑆 ∈ ∪ ran sigAlgebra ) |
6 |
1 5
|
syl |
⊢ ( 𝜑 → 𝑆 ∈ ∪ ran sigAlgebra ) |
7 |
|
difelsiga |
⊢ ( ( 𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ∈ 𝑆 ) → ( 𝐵 ∖ 𝐴 ) ∈ 𝑆 ) |
8 |
6 3 2 7
|
syl3anc |
⊢ ( 𝜑 → ( 𝐵 ∖ 𝐴 ) ∈ 𝑆 ) |
9 |
|
measvxrge0 |
⊢ ( ( 𝑀 ∈ ( measures ‘ 𝑆 ) ∧ ( 𝐵 ∖ 𝐴 ) ∈ 𝑆 ) → ( 𝑀 ‘ ( 𝐵 ∖ 𝐴 ) ) ∈ ( 0 [,] +∞ ) ) |
10 |
1 8 9
|
syl2anc |
⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝐵 ∖ 𝐴 ) ) ∈ ( 0 [,] +∞ ) ) |
11 |
|
elxrge0 |
⊢ ( ( 𝑀 ‘ ( 𝐵 ∖ 𝐴 ) ) ∈ ( 0 [,] +∞ ) ↔ ( ( 𝑀 ‘ ( 𝐵 ∖ 𝐴 ) ) ∈ ℝ* ∧ 0 ≤ ( 𝑀 ‘ ( 𝐵 ∖ 𝐴 ) ) ) ) |
12 |
11
|
simprbi |
⊢ ( ( 𝑀 ‘ ( 𝐵 ∖ 𝐴 ) ) ∈ ( 0 [,] +∞ ) → 0 ≤ ( 𝑀 ‘ ( 𝐵 ∖ 𝐴 ) ) ) |
13 |
10 12
|
syl |
⊢ ( 𝜑 → 0 ≤ ( 𝑀 ‘ ( 𝐵 ∖ 𝐴 ) ) ) |
14 |
|
measvxrge0 |
⊢ ( ( 𝑀 ∈ ( measures ‘ 𝑆 ) ∧ 𝐴 ∈ 𝑆 ) → ( 𝑀 ‘ 𝐴 ) ∈ ( 0 [,] +∞ ) ) |
15 |
1 2 14
|
syl2anc |
⊢ ( 𝜑 → ( 𝑀 ‘ 𝐴 ) ∈ ( 0 [,] +∞ ) ) |
16 |
|
elxrge0 |
⊢ ( ( 𝑀 ‘ 𝐴 ) ∈ ( 0 [,] +∞ ) ↔ ( ( 𝑀 ‘ 𝐴 ) ∈ ℝ* ∧ 0 ≤ ( 𝑀 ‘ 𝐴 ) ) ) |
17 |
16
|
simplbi |
⊢ ( ( 𝑀 ‘ 𝐴 ) ∈ ( 0 [,] +∞ ) → ( 𝑀 ‘ 𝐴 ) ∈ ℝ* ) |
18 |
15 17
|
syl |
⊢ ( 𝜑 → ( 𝑀 ‘ 𝐴 ) ∈ ℝ* ) |
19 |
11
|
simplbi |
⊢ ( ( 𝑀 ‘ ( 𝐵 ∖ 𝐴 ) ) ∈ ( 0 [,] +∞ ) → ( 𝑀 ‘ ( 𝐵 ∖ 𝐴 ) ) ∈ ℝ* ) |
20 |
10 19
|
syl |
⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝐵 ∖ 𝐴 ) ) ∈ ℝ* ) |
21 |
|
xraddge02 |
⊢ ( ( ( 𝑀 ‘ 𝐴 ) ∈ ℝ* ∧ ( 𝑀 ‘ ( 𝐵 ∖ 𝐴 ) ) ∈ ℝ* ) → ( 0 ≤ ( 𝑀 ‘ ( 𝐵 ∖ 𝐴 ) ) → ( 𝑀 ‘ 𝐴 ) ≤ ( ( 𝑀 ‘ 𝐴 ) +𝑒 ( 𝑀 ‘ ( 𝐵 ∖ 𝐴 ) ) ) ) ) |
22 |
18 20 21
|
syl2anc |
⊢ ( 𝜑 → ( 0 ≤ ( 𝑀 ‘ ( 𝐵 ∖ 𝐴 ) ) → ( 𝑀 ‘ 𝐴 ) ≤ ( ( 𝑀 ‘ 𝐴 ) +𝑒 ( 𝑀 ‘ ( 𝐵 ∖ 𝐴 ) ) ) ) ) |
23 |
13 22
|
mpd |
⊢ ( 𝜑 → ( 𝑀 ‘ 𝐴 ) ≤ ( ( 𝑀 ‘ 𝐴 ) +𝑒 ( 𝑀 ‘ ( 𝐵 ∖ 𝐴 ) ) ) ) |
24 |
|
prssi |
⊢ ( ( 𝐴 ∈ 𝑆 ∧ ( 𝐵 ∖ 𝐴 ) ∈ 𝑆 ) → { 𝐴 , ( 𝐵 ∖ 𝐴 ) } ⊆ 𝑆 ) |
25 |
2 8 24
|
syl2anc |
⊢ ( 𝜑 → { 𝐴 , ( 𝐵 ∖ 𝐴 ) } ⊆ 𝑆 ) |
26 |
|
prex |
⊢ { 𝐴 , ( 𝐵 ∖ 𝐴 ) } ∈ V |
27 |
26
|
elpw |
⊢ ( { 𝐴 , ( 𝐵 ∖ 𝐴 ) } ∈ 𝒫 𝑆 ↔ { 𝐴 , ( 𝐵 ∖ 𝐴 ) } ⊆ 𝑆 ) |
28 |
25 27
|
sylibr |
⊢ ( 𝜑 → { 𝐴 , ( 𝐵 ∖ 𝐴 ) } ∈ 𝒫 𝑆 ) |
29 |
|
prct |
⊢ ( ( 𝐴 ∈ 𝑆 ∧ ( 𝐵 ∖ 𝐴 ) ∈ 𝑆 ) → { 𝐴 , ( 𝐵 ∖ 𝐴 ) } ≼ ω ) |
30 |
2 8 29
|
syl2anc |
⊢ ( 𝜑 → { 𝐴 , ( 𝐵 ∖ 𝐴 ) } ≼ ω ) |
31 |
|
disjdifprg |
⊢ ( ( 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ) → Disj 𝑦 ∈ { ( 𝐵 ∖ 𝐴 ) , 𝐴 } 𝑦 ) |
32 |
2 3 31
|
syl2anc |
⊢ ( 𝜑 → Disj 𝑦 ∈ { ( 𝐵 ∖ 𝐴 ) , 𝐴 } 𝑦 ) |
33 |
|
prcom |
⊢ { ( 𝐵 ∖ 𝐴 ) , 𝐴 } = { 𝐴 , ( 𝐵 ∖ 𝐴 ) } |
34 |
33
|
a1i |
⊢ ( 𝜑 → { ( 𝐵 ∖ 𝐴 ) , 𝐴 } = { 𝐴 , ( 𝐵 ∖ 𝐴 ) } ) |
35 |
34
|
disjeq1d |
⊢ ( 𝜑 → ( Disj 𝑦 ∈ { ( 𝐵 ∖ 𝐴 ) , 𝐴 } 𝑦 ↔ Disj 𝑦 ∈ { 𝐴 , ( 𝐵 ∖ 𝐴 ) } 𝑦 ) ) |
36 |
32 35
|
mpbid |
⊢ ( 𝜑 → Disj 𝑦 ∈ { 𝐴 , ( 𝐵 ∖ 𝐴 ) } 𝑦 ) |
37 |
|
measvun |
⊢ ( ( 𝑀 ∈ ( measures ‘ 𝑆 ) ∧ { 𝐴 , ( 𝐵 ∖ 𝐴 ) } ∈ 𝒫 𝑆 ∧ ( { 𝐴 , ( 𝐵 ∖ 𝐴 ) } ≼ ω ∧ Disj 𝑦 ∈ { 𝐴 , ( 𝐵 ∖ 𝐴 ) } 𝑦 ) ) → ( 𝑀 ‘ ∪ { 𝐴 , ( 𝐵 ∖ 𝐴 ) } ) = Σ* 𝑦 ∈ { 𝐴 , ( 𝐵 ∖ 𝐴 ) } ( 𝑀 ‘ 𝑦 ) ) |
38 |
1 28 30 36 37
|
syl112anc |
⊢ ( 𝜑 → ( 𝑀 ‘ ∪ { 𝐴 , ( 𝐵 ∖ 𝐴 ) } ) = Σ* 𝑦 ∈ { 𝐴 , ( 𝐵 ∖ 𝐴 ) } ( 𝑀 ‘ 𝑦 ) ) |
39 |
|
uniprg |
⊢ ( ( 𝐴 ∈ 𝑆 ∧ ( 𝐵 ∖ 𝐴 ) ∈ 𝑆 ) → ∪ { 𝐴 , ( 𝐵 ∖ 𝐴 ) } = ( 𝐴 ∪ ( 𝐵 ∖ 𝐴 ) ) ) |
40 |
2 8 39
|
syl2anc |
⊢ ( 𝜑 → ∪ { 𝐴 , ( 𝐵 ∖ 𝐴 ) } = ( 𝐴 ∪ ( 𝐵 ∖ 𝐴 ) ) ) |
41 |
|
undif |
⊢ ( 𝐴 ⊆ 𝐵 ↔ ( 𝐴 ∪ ( 𝐵 ∖ 𝐴 ) ) = 𝐵 ) |
42 |
4 41
|
sylib |
⊢ ( 𝜑 → ( 𝐴 ∪ ( 𝐵 ∖ 𝐴 ) ) = 𝐵 ) |
43 |
40 42
|
eqtrd |
⊢ ( 𝜑 → ∪ { 𝐴 , ( 𝐵 ∖ 𝐴 ) } = 𝐵 ) |
44 |
43
|
fveq2d |
⊢ ( 𝜑 → ( 𝑀 ‘ ∪ { 𝐴 , ( 𝐵 ∖ 𝐴 ) } ) = ( 𝑀 ‘ 𝐵 ) ) |
45 |
|
fveq2 |
⊢ ( 𝑦 = 𝐴 → ( 𝑀 ‘ 𝑦 ) = ( 𝑀 ‘ 𝐴 ) ) |
46 |
45
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑦 = 𝐴 ) → ( 𝑀 ‘ 𝑦 ) = ( 𝑀 ‘ 𝐴 ) ) |
47 |
|
fveq2 |
⊢ ( 𝑦 = ( 𝐵 ∖ 𝐴 ) → ( 𝑀 ‘ 𝑦 ) = ( 𝑀 ‘ ( 𝐵 ∖ 𝐴 ) ) ) |
48 |
47
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑦 = ( 𝐵 ∖ 𝐴 ) ) → ( 𝑀 ‘ 𝑦 ) = ( 𝑀 ‘ ( 𝐵 ∖ 𝐴 ) ) ) |
49 |
|
eqimss |
⊢ ( 𝐴 = ( 𝐵 ∖ 𝐴 ) → 𝐴 ⊆ ( 𝐵 ∖ 𝐴 ) ) |
50 |
|
ssdifeq0 |
⊢ ( 𝐴 ⊆ ( 𝐵 ∖ 𝐴 ) ↔ 𝐴 = ∅ ) |
51 |
49 50
|
sylib |
⊢ ( 𝐴 = ( 𝐵 ∖ 𝐴 ) → 𝐴 = ∅ ) |
52 |
51
|
adantl |
⊢ ( ( 𝜑 ∧ 𝐴 = ( 𝐵 ∖ 𝐴 ) ) → 𝐴 = ∅ ) |
53 |
52
|
fveq2d |
⊢ ( ( 𝜑 ∧ 𝐴 = ( 𝐵 ∖ 𝐴 ) ) → ( 𝑀 ‘ 𝐴 ) = ( 𝑀 ‘ ∅ ) ) |
54 |
|
measvnul |
⊢ ( 𝑀 ∈ ( measures ‘ 𝑆 ) → ( 𝑀 ‘ ∅ ) = 0 ) |
55 |
1 54
|
syl |
⊢ ( 𝜑 → ( 𝑀 ‘ ∅ ) = 0 ) |
56 |
55
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐴 = ( 𝐵 ∖ 𝐴 ) ) → ( 𝑀 ‘ ∅ ) = 0 ) |
57 |
53 56
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝐴 = ( 𝐵 ∖ 𝐴 ) ) → ( 𝑀 ‘ 𝐴 ) = 0 ) |
58 |
57
|
orcd |
⊢ ( ( 𝜑 ∧ 𝐴 = ( 𝐵 ∖ 𝐴 ) ) → ( ( 𝑀 ‘ 𝐴 ) = 0 ∨ ( 𝑀 ‘ 𝐴 ) = +∞ ) ) |
59 |
58
|
ex |
⊢ ( 𝜑 → ( 𝐴 = ( 𝐵 ∖ 𝐴 ) → ( ( 𝑀 ‘ 𝐴 ) = 0 ∨ ( 𝑀 ‘ 𝐴 ) = +∞ ) ) ) |
60 |
46 48 2 8 15 10 59
|
esumpr2 |
⊢ ( 𝜑 → Σ* 𝑦 ∈ { 𝐴 , ( 𝐵 ∖ 𝐴 ) } ( 𝑀 ‘ 𝑦 ) = ( ( 𝑀 ‘ 𝐴 ) +𝑒 ( 𝑀 ‘ ( 𝐵 ∖ 𝐴 ) ) ) ) |
61 |
38 44 60
|
3eqtr3d |
⊢ ( 𝜑 → ( 𝑀 ‘ 𝐵 ) = ( ( 𝑀 ‘ 𝐴 ) +𝑒 ( 𝑀 ‘ ( 𝐵 ∖ 𝐴 ) ) ) ) |
62 |
23 61
|
breqtrrd |
⊢ ( 𝜑 → ( 𝑀 ‘ 𝐴 ) ≤ ( 𝑀 ‘ 𝐵 ) ) |