Step |
Hyp |
Ref |
Expression |
1 |
|
meadjun.m |
⊢ ( 𝜑 → 𝑀 ∈ Meas ) |
2 |
|
meadjun.x |
⊢ 𝑆 = dom 𝑀 |
3 |
|
meadjun.a |
⊢ ( 𝜑 → 𝐴 ∈ 𝑆 ) |
4 |
|
meadjun.b |
⊢ ( 𝜑 → 𝐵 ∈ 𝑆 ) |
5 |
|
meadjun.dj |
⊢ ( 𝜑 → ( 𝐴 ∩ 𝐵 ) = ∅ ) |
6 |
|
iccssxr |
⊢ ( 0 [,] +∞ ) ⊆ ℝ* |
7 |
1 2
|
meaf |
⊢ ( 𝜑 → 𝑀 : 𝑆 ⟶ ( 0 [,] +∞ ) ) |
8 |
7 4
|
ffvelrnd |
⊢ ( 𝜑 → ( 𝑀 ‘ 𝐵 ) ∈ ( 0 [,] +∞ ) ) |
9 |
6 8
|
sseldi |
⊢ ( 𝜑 → ( 𝑀 ‘ 𝐵 ) ∈ ℝ* ) |
10 |
|
xaddid2 |
⊢ ( ( 𝑀 ‘ 𝐵 ) ∈ ℝ* → ( 0 +𝑒 ( 𝑀 ‘ 𝐵 ) ) = ( 𝑀 ‘ 𝐵 ) ) |
11 |
9 10
|
syl |
⊢ ( 𝜑 → ( 0 +𝑒 ( 𝑀 ‘ 𝐵 ) ) = ( 𝑀 ‘ 𝐵 ) ) |
12 |
11
|
eqcomd |
⊢ ( 𝜑 → ( 𝑀 ‘ 𝐵 ) = ( 0 +𝑒 ( 𝑀 ‘ 𝐵 ) ) ) |
13 |
12
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐴 = ∅ ) → ( 𝑀 ‘ 𝐵 ) = ( 0 +𝑒 ( 𝑀 ‘ 𝐵 ) ) ) |
14 |
|
uneq1 |
⊢ ( 𝐴 = ∅ → ( 𝐴 ∪ 𝐵 ) = ( ∅ ∪ 𝐵 ) ) |
15 |
|
0un |
⊢ ( ∅ ∪ 𝐵 ) = 𝐵 |
16 |
15
|
a1i |
⊢ ( 𝐴 = ∅ → ( ∅ ∪ 𝐵 ) = 𝐵 ) |
17 |
14 16
|
eqtrd |
⊢ ( 𝐴 = ∅ → ( 𝐴 ∪ 𝐵 ) = 𝐵 ) |
18 |
17
|
fveq2d |
⊢ ( 𝐴 = ∅ → ( 𝑀 ‘ ( 𝐴 ∪ 𝐵 ) ) = ( 𝑀 ‘ 𝐵 ) ) |
19 |
18
|
adantl |
⊢ ( ( 𝜑 ∧ 𝐴 = ∅ ) → ( 𝑀 ‘ ( 𝐴 ∪ 𝐵 ) ) = ( 𝑀 ‘ 𝐵 ) ) |
20 |
|
fveq2 |
⊢ ( 𝐴 = ∅ → ( 𝑀 ‘ 𝐴 ) = ( 𝑀 ‘ ∅ ) ) |
21 |
20
|
adantl |
⊢ ( ( 𝜑 ∧ 𝐴 = ∅ ) → ( 𝑀 ‘ 𝐴 ) = ( 𝑀 ‘ ∅ ) ) |
22 |
1
|
mea0 |
⊢ ( 𝜑 → ( 𝑀 ‘ ∅ ) = 0 ) |
23 |
22
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐴 = ∅ ) → ( 𝑀 ‘ ∅ ) = 0 ) |
24 |
21 23
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝐴 = ∅ ) → ( 𝑀 ‘ 𝐴 ) = 0 ) |
25 |
24
|
oveq1d |
⊢ ( ( 𝜑 ∧ 𝐴 = ∅ ) → ( ( 𝑀 ‘ 𝐴 ) +𝑒 ( 𝑀 ‘ 𝐵 ) ) = ( 0 +𝑒 ( 𝑀 ‘ 𝐵 ) ) ) |
26 |
13 19 25
|
3eqtr4d |
⊢ ( ( 𝜑 ∧ 𝐴 = ∅ ) → ( 𝑀 ‘ ( 𝐴 ∪ 𝐵 ) ) = ( ( 𝑀 ‘ 𝐴 ) +𝑒 ( 𝑀 ‘ 𝐵 ) ) ) |
27 |
|
simpl |
⊢ ( ( 𝜑 ∧ ¬ 𝐴 = ∅ ) → 𝜑 ) |
28 |
5
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ¬ 𝐴 = ∅ ) ∧ 𝐴 = 𝐵 ) → ( 𝐴 ∩ 𝐵 ) = ∅ ) |
29 |
|
inidm |
⊢ ( 𝐴 ∩ 𝐴 ) = 𝐴 |
30 |
29
|
eqcomi |
⊢ 𝐴 = ( 𝐴 ∩ 𝐴 ) |
31 |
|
ineq2 |
⊢ ( 𝐴 = 𝐵 → ( 𝐴 ∩ 𝐴 ) = ( 𝐴 ∩ 𝐵 ) ) |
32 |
30 31
|
syl5req |
⊢ ( 𝐴 = 𝐵 → ( 𝐴 ∩ 𝐵 ) = 𝐴 ) |
33 |
32
|
adantl |
⊢ ( ( ¬ 𝐴 = ∅ ∧ 𝐴 = 𝐵 ) → ( 𝐴 ∩ 𝐵 ) = 𝐴 ) |
34 |
|
neqne |
⊢ ( ¬ 𝐴 = ∅ → 𝐴 ≠ ∅ ) |
35 |
34
|
adantr |
⊢ ( ( ¬ 𝐴 = ∅ ∧ 𝐴 = 𝐵 ) → 𝐴 ≠ ∅ ) |
36 |
33 35
|
eqnetrd |
⊢ ( ( ¬ 𝐴 = ∅ ∧ 𝐴 = 𝐵 ) → ( 𝐴 ∩ 𝐵 ) ≠ ∅ ) |
37 |
36
|
neneqd |
⊢ ( ( ¬ 𝐴 = ∅ ∧ 𝐴 = 𝐵 ) → ¬ ( 𝐴 ∩ 𝐵 ) = ∅ ) |
38 |
37
|
adantll |
⊢ ( ( ( 𝜑 ∧ ¬ 𝐴 = ∅ ) ∧ 𝐴 = 𝐵 ) → ¬ ( 𝐴 ∩ 𝐵 ) = ∅ ) |
39 |
28 38
|
pm2.65da |
⊢ ( ( 𝜑 ∧ ¬ 𝐴 = ∅ ) → ¬ 𝐴 = 𝐵 ) |
40 |
39
|
neqned |
⊢ ( ( 𝜑 ∧ ¬ 𝐴 = ∅ ) → 𝐴 ≠ 𝐵 ) |
41 |
|
uniprg |
⊢ ( ( 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ) → ∪ { 𝐴 , 𝐵 } = ( 𝐴 ∪ 𝐵 ) ) |
42 |
3 4 41
|
syl2anc |
⊢ ( 𝜑 → ∪ { 𝐴 , 𝐵 } = ( 𝐴 ∪ 𝐵 ) ) |
43 |
42
|
eqcomd |
⊢ ( 𝜑 → ( 𝐴 ∪ 𝐵 ) = ∪ { 𝐴 , 𝐵 } ) |
44 |
43
|
fveq2d |
⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝐴 ∪ 𝐵 ) ) = ( 𝑀 ‘ ∪ { 𝐴 , 𝐵 } ) ) |
45 |
44
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐴 ≠ 𝐵 ) → ( 𝑀 ‘ ( 𝐴 ∪ 𝐵 ) ) = ( 𝑀 ‘ ∪ { 𝐴 , 𝐵 } ) ) |
46 |
3 4
|
prssd |
⊢ ( 𝜑 → { 𝐴 , 𝐵 } ⊆ 𝑆 ) |
47 |
|
prfi |
⊢ { 𝐴 , 𝐵 } ∈ Fin |
48 |
|
isfinite |
⊢ ( { 𝐴 , 𝐵 } ∈ Fin ↔ { 𝐴 , 𝐵 } ≺ ω ) |
49 |
48
|
biimpi |
⊢ ( { 𝐴 , 𝐵 } ∈ Fin → { 𝐴 , 𝐵 } ≺ ω ) |
50 |
|
sdomdom |
⊢ ( { 𝐴 , 𝐵 } ≺ ω → { 𝐴 , 𝐵 } ≼ ω ) |
51 |
49 50
|
syl |
⊢ ( { 𝐴 , 𝐵 } ∈ Fin → { 𝐴 , 𝐵 } ≼ ω ) |
52 |
47 51
|
ax-mp |
⊢ { 𝐴 , 𝐵 } ≼ ω |
53 |
52
|
a1i |
⊢ ( 𝜑 → { 𝐴 , 𝐵 } ≼ ω ) |
54 |
|
disjxsn |
⊢ Disj 𝑥 ∈ { 𝐵 } 𝑥 |
55 |
54
|
a1i |
⊢ ( 𝐴 = 𝐵 → Disj 𝑥 ∈ { 𝐵 } 𝑥 ) |
56 |
|
preq1 |
⊢ ( 𝐴 = 𝐵 → { 𝐴 , 𝐵 } = { 𝐵 , 𝐵 } ) |
57 |
|
dfsn2 |
⊢ { 𝐵 } = { 𝐵 , 𝐵 } |
58 |
57
|
eqcomi |
⊢ { 𝐵 , 𝐵 } = { 𝐵 } |
59 |
58
|
a1i |
⊢ ( 𝐴 = 𝐵 → { 𝐵 , 𝐵 } = { 𝐵 } ) |
60 |
56 59
|
eqtrd |
⊢ ( 𝐴 = 𝐵 → { 𝐴 , 𝐵 } = { 𝐵 } ) |
61 |
60
|
disjeq1d |
⊢ ( 𝐴 = 𝐵 → ( Disj 𝑥 ∈ { 𝐴 , 𝐵 } 𝑥 ↔ Disj 𝑥 ∈ { 𝐵 } 𝑥 ) ) |
62 |
55 61
|
mpbird |
⊢ ( 𝐴 = 𝐵 → Disj 𝑥 ∈ { 𝐴 , 𝐵 } 𝑥 ) |
63 |
62
|
adantl |
⊢ ( ( 𝜑 ∧ 𝐴 = 𝐵 ) → Disj 𝑥 ∈ { 𝐴 , 𝐵 } 𝑥 ) |
64 |
|
simpl |
⊢ ( ( 𝜑 ∧ ¬ 𝐴 = 𝐵 ) → 𝜑 ) |
65 |
|
neqne |
⊢ ( ¬ 𝐴 = 𝐵 → 𝐴 ≠ 𝐵 ) |
66 |
65
|
adantl |
⊢ ( ( 𝜑 ∧ ¬ 𝐴 = 𝐵 ) → 𝐴 ≠ 𝐵 ) |
67 |
5
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐴 ≠ 𝐵 ) → ( 𝐴 ∩ 𝐵 ) = ∅ ) |
68 |
3
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐴 ≠ 𝐵 ) → 𝐴 ∈ 𝑆 ) |
69 |
4
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐴 ≠ 𝐵 ) → 𝐵 ∈ 𝑆 ) |
70 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝐴 ≠ 𝐵 ) → 𝐴 ≠ 𝐵 ) |
71 |
|
id |
⊢ ( 𝑥 = 𝐴 → 𝑥 = 𝐴 ) |
72 |
|
id |
⊢ ( 𝑥 = 𝐵 → 𝑥 = 𝐵 ) |
73 |
71 72
|
disjprg |
⊢ ( ( 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵 ) → ( Disj 𝑥 ∈ { 𝐴 , 𝐵 } 𝑥 ↔ ( 𝐴 ∩ 𝐵 ) = ∅ ) ) |
74 |
68 69 70 73
|
syl3anc |
⊢ ( ( 𝜑 ∧ 𝐴 ≠ 𝐵 ) → ( Disj 𝑥 ∈ { 𝐴 , 𝐵 } 𝑥 ↔ ( 𝐴 ∩ 𝐵 ) = ∅ ) ) |
75 |
67 74
|
mpbird |
⊢ ( ( 𝜑 ∧ 𝐴 ≠ 𝐵 ) → Disj 𝑥 ∈ { 𝐴 , 𝐵 } 𝑥 ) |
76 |
64 66 75
|
syl2anc |
⊢ ( ( 𝜑 ∧ ¬ 𝐴 = 𝐵 ) → Disj 𝑥 ∈ { 𝐴 , 𝐵 } 𝑥 ) |
77 |
63 76
|
pm2.61dan |
⊢ ( 𝜑 → Disj 𝑥 ∈ { 𝐴 , 𝐵 } 𝑥 ) |
78 |
1 2 46 53 77
|
meadjuni |
⊢ ( 𝜑 → ( 𝑀 ‘ ∪ { 𝐴 , 𝐵 } ) = ( Σ^ ‘ ( 𝑀 ↾ { 𝐴 , 𝐵 } ) ) ) |
79 |
78
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐴 ≠ 𝐵 ) → ( 𝑀 ‘ ∪ { 𝐴 , 𝐵 } ) = ( Σ^ ‘ ( 𝑀 ↾ { 𝐴 , 𝐵 } ) ) ) |
80 |
7 3
|
ffvelrnd |
⊢ ( 𝜑 → ( 𝑀 ‘ 𝐴 ) ∈ ( 0 [,] +∞ ) ) |
81 |
80
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐴 ≠ 𝐵 ) → ( 𝑀 ‘ 𝐴 ) ∈ ( 0 [,] +∞ ) ) |
82 |
8
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐴 ≠ 𝐵 ) → ( 𝑀 ‘ 𝐵 ) ∈ ( 0 [,] +∞ ) ) |
83 |
|
fveq2 |
⊢ ( 𝑥 = 𝐴 → ( 𝑀 ‘ 𝑥 ) = ( 𝑀 ‘ 𝐴 ) ) |
84 |
|
fveq2 |
⊢ ( 𝑥 = 𝐵 → ( 𝑀 ‘ 𝑥 ) = ( 𝑀 ‘ 𝐵 ) ) |
85 |
68 69 81 82 83 84 70
|
sge0pr |
⊢ ( ( 𝜑 ∧ 𝐴 ≠ 𝐵 ) → ( Σ^ ‘ ( 𝑥 ∈ { 𝐴 , 𝐵 } ↦ ( 𝑀 ‘ 𝑥 ) ) ) = ( ( 𝑀 ‘ 𝐴 ) +𝑒 ( 𝑀 ‘ 𝐵 ) ) ) |
86 |
7 46
|
fssresd |
⊢ ( 𝜑 → ( 𝑀 ↾ { 𝐴 , 𝐵 } ) : { 𝐴 , 𝐵 } ⟶ ( 0 [,] +∞ ) ) |
87 |
86
|
feqmptd |
⊢ ( 𝜑 → ( 𝑀 ↾ { 𝐴 , 𝐵 } ) = ( 𝑥 ∈ { 𝐴 , 𝐵 } ↦ ( ( 𝑀 ↾ { 𝐴 , 𝐵 } ) ‘ 𝑥 ) ) ) |
88 |
|
fvres |
⊢ ( 𝑥 ∈ { 𝐴 , 𝐵 } → ( ( 𝑀 ↾ { 𝐴 , 𝐵 } ) ‘ 𝑥 ) = ( 𝑀 ‘ 𝑥 ) ) |
89 |
88
|
mpteq2ia |
⊢ ( 𝑥 ∈ { 𝐴 , 𝐵 } ↦ ( ( 𝑀 ↾ { 𝐴 , 𝐵 } ) ‘ 𝑥 ) ) = ( 𝑥 ∈ { 𝐴 , 𝐵 } ↦ ( 𝑀 ‘ 𝑥 ) ) |
90 |
89
|
a1i |
⊢ ( 𝜑 → ( 𝑥 ∈ { 𝐴 , 𝐵 } ↦ ( ( 𝑀 ↾ { 𝐴 , 𝐵 } ) ‘ 𝑥 ) ) = ( 𝑥 ∈ { 𝐴 , 𝐵 } ↦ ( 𝑀 ‘ 𝑥 ) ) ) |
91 |
87 90
|
eqtrd |
⊢ ( 𝜑 → ( 𝑀 ↾ { 𝐴 , 𝐵 } ) = ( 𝑥 ∈ { 𝐴 , 𝐵 } ↦ ( 𝑀 ‘ 𝑥 ) ) ) |
92 |
91
|
fveq2d |
⊢ ( 𝜑 → ( Σ^ ‘ ( 𝑀 ↾ { 𝐴 , 𝐵 } ) ) = ( Σ^ ‘ ( 𝑥 ∈ { 𝐴 , 𝐵 } ↦ ( 𝑀 ‘ 𝑥 ) ) ) ) |
93 |
92
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐴 ≠ 𝐵 ) → ( Σ^ ‘ ( 𝑀 ↾ { 𝐴 , 𝐵 } ) ) = ( Σ^ ‘ ( 𝑥 ∈ { 𝐴 , 𝐵 } ↦ ( 𝑀 ‘ 𝑥 ) ) ) ) |
94 |
|
eqidd |
⊢ ( ( 𝜑 ∧ 𝐴 ≠ 𝐵 ) → ( ( 𝑀 ‘ 𝐴 ) +𝑒 ( 𝑀 ‘ 𝐵 ) ) = ( ( 𝑀 ‘ 𝐴 ) +𝑒 ( 𝑀 ‘ 𝐵 ) ) ) |
95 |
85 93 94
|
3eqtr4d |
⊢ ( ( 𝜑 ∧ 𝐴 ≠ 𝐵 ) → ( Σ^ ‘ ( 𝑀 ↾ { 𝐴 , 𝐵 } ) ) = ( ( 𝑀 ‘ 𝐴 ) +𝑒 ( 𝑀 ‘ 𝐵 ) ) ) |
96 |
45 79 95
|
3eqtrd |
⊢ ( ( 𝜑 ∧ 𝐴 ≠ 𝐵 ) → ( 𝑀 ‘ ( 𝐴 ∪ 𝐵 ) ) = ( ( 𝑀 ‘ 𝐴 ) +𝑒 ( 𝑀 ‘ 𝐵 ) ) ) |
97 |
27 40 96
|
syl2anc |
⊢ ( ( 𝜑 ∧ ¬ 𝐴 = ∅ ) → ( 𝑀 ‘ ( 𝐴 ∪ 𝐵 ) ) = ( ( 𝑀 ‘ 𝐴 ) +𝑒 ( 𝑀 ‘ 𝐵 ) ) ) |
98 |
26 97
|
pm2.61dan |
⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝐴 ∪ 𝐵 ) ) = ( ( 𝑀 ‘ 𝐴 ) +𝑒 ( 𝑀 ‘ 𝐵 ) ) ) |