| Step |
Hyp |
Ref |
Expression |
| 1 |
|
funmpt |
⊢ Fun ( 𝑥 ∈ 𝑆 ↦ ( ( 𝑀 ‘ 𝑥 ) /𝑒 𝐴 ) ) |
| 2 |
|
ovex |
⊢ ( ( 𝑀 ‘ 𝑥 ) /𝑒 𝐴 ) ∈ V |
| 3 |
2
|
rgenw |
⊢ ∀ 𝑥 ∈ 𝑆 ( ( 𝑀 ‘ 𝑥 ) /𝑒 𝐴 ) ∈ V |
| 4 |
|
dmmptg |
⊢ ( ∀ 𝑥 ∈ 𝑆 ( ( 𝑀 ‘ 𝑥 ) /𝑒 𝐴 ) ∈ V → dom ( 𝑥 ∈ 𝑆 ↦ ( ( 𝑀 ‘ 𝑥 ) /𝑒 𝐴 ) ) = 𝑆 ) |
| 5 |
3 4
|
ax-mp |
⊢ dom ( 𝑥 ∈ 𝑆 ↦ ( ( 𝑀 ‘ 𝑥 ) /𝑒 𝐴 ) ) = 𝑆 |
| 6 |
|
df-fn |
⊢ ( ( 𝑥 ∈ 𝑆 ↦ ( ( 𝑀 ‘ 𝑥 ) /𝑒 𝐴 ) ) Fn 𝑆 ↔ ( Fun ( 𝑥 ∈ 𝑆 ↦ ( ( 𝑀 ‘ 𝑥 ) /𝑒 𝐴 ) ) ∧ dom ( 𝑥 ∈ 𝑆 ↦ ( ( 𝑀 ‘ 𝑥 ) /𝑒 𝐴 ) ) = 𝑆 ) ) |
| 7 |
1 5 6
|
mpbir2an |
⊢ ( 𝑥 ∈ 𝑆 ↦ ( ( 𝑀 ‘ 𝑥 ) /𝑒 𝐴 ) ) Fn 𝑆 |
| 8 |
7
|
a1i |
⊢ ( ( 𝑀 ∈ ( measures ‘ 𝑆 ) ∧ 𝐴 ∈ ℝ+ ) → ( 𝑥 ∈ 𝑆 ↦ ( ( 𝑀 ‘ 𝑥 ) /𝑒 𝐴 ) ) Fn 𝑆 ) |
| 9 |
|
vex |
⊢ 𝑦 ∈ V |
| 10 |
|
eqid |
⊢ ( 𝑥 ∈ 𝑆 ↦ ( ( 𝑀 ‘ 𝑥 ) /𝑒 𝐴 ) ) = ( 𝑥 ∈ 𝑆 ↦ ( ( 𝑀 ‘ 𝑥 ) /𝑒 𝐴 ) ) |
| 11 |
10
|
elrnmpt |
⊢ ( 𝑦 ∈ V → ( 𝑦 ∈ ran ( 𝑥 ∈ 𝑆 ↦ ( ( 𝑀 ‘ 𝑥 ) /𝑒 𝐴 ) ) ↔ ∃ 𝑥 ∈ 𝑆 𝑦 = ( ( 𝑀 ‘ 𝑥 ) /𝑒 𝐴 ) ) ) |
| 12 |
9 11
|
ax-mp |
⊢ ( 𝑦 ∈ ran ( 𝑥 ∈ 𝑆 ↦ ( ( 𝑀 ‘ 𝑥 ) /𝑒 𝐴 ) ) ↔ ∃ 𝑥 ∈ 𝑆 𝑦 = ( ( 𝑀 ‘ 𝑥 ) /𝑒 𝐴 ) ) |
| 13 |
|
measfrge0 |
⊢ ( 𝑀 ∈ ( measures ‘ 𝑆 ) → 𝑀 : 𝑆 ⟶ ( 0 [,] +∞ ) ) |
| 14 |
|
ffvelrn |
⊢ ( ( 𝑀 : 𝑆 ⟶ ( 0 [,] +∞ ) ∧ 𝑥 ∈ 𝑆 ) → ( 𝑀 ‘ 𝑥 ) ∈ ( 0 [,] +∞ ) ) |
| 15 |
13 14
|
sylan |
⊢ ( ( 𝑀 ∈ ( measures ‘ 𝑆 ) ∧ 𝑥 ∈ 𝑆 ) → ( 𝑀 ‘ 𝑥 ) ∈ ( 0 [,] +∞ ) ) |
| 16 |
15
|
adantlr |
⊢ ( ( ( 𝑀 ∈ ( measures ‘ 𝑆 ) ∧ 𝐴 ∈ ℝ+ ) ∧ 𝑥 ∈ 𝑆 ) → ( 𝑀 ‘ 𝑥 ) ∈ ( 0 [,] +∞ ) ) |
| 17 |
|
simplr |
⊢ ( ( ( 𝑀 ∈ ( measures ‘ 𝑆 ) ∧ 𝐴 ∈ ℝ+ ) ∧ 𝑥 ∈ 𝑆 ) → 𝐴 ∈ ℝ+ ) |
| 18 |
16 17
|
xrpxdivcld |
⊢ ( ( ( 𝑀 ∈ ( measures ‘ 𝑆 ) ∧ 𝐴 ∈ ℝ+ ) ∧ 𝑥 ∈ 𝑆 ) → ( ( 𝑀 ‘ 𝑥 ) /𝑒 𝐴 ) ∈ ( 0 [,] +∞ ) ) |
| 19 |
|
eleq1a |
⊢ ( ( ( 𝑀 ‘ 𝑥 ) /𝑒 𝐴 ) ∈ ( 0 [,] +∞ ) → ( 𝑦 = ( ( 𝑀 ‘ 𝑥 ) /𝑒 𝐴 ) → 𝑦 ∈ ( 0 [,] +∞ ) ) ) |
| 20 |
18 19
|
syl |
⊢ ( ( ( 𝑀 ∈ ( measures ‘ 𝑆 ) ∧ 𝐴 ∈ ℝ+ ) ∧ 𝑥 ∈ 𝑆 ) → ( 𝑦 = ( ( 𝑀 ‘ 𝑥 ) /𝑒 𝐴 ) → 𝑦 ∈ ( 0 [,] +∞ ) ) ) |
| 21 |
20
|
rexlimdva |
⊢ ( ( 𝑀 ∈ ( measures ‘ 𝑆 ) ∧ 𝐴 ∈ ℝ+ ) → ( ∃ 𝑥 ∈ 𝑆 𝑦 = ( ( 𝑀 ‘ 𝑥 ) /𝑒 𝐴 ) → 𝑦 ∈ ( 0 [,] +∞ ) ) ) |
| 22 |
12 21
|
syl5bi |
⊢ ( ( 𝑀 ∈ ( measures ‘ 𝑆 ) ∧ 𝐴 ∈ ℝ+ ) → ( 𝑦 ∈ ran ( 𝑥 ∈ 𝑆 ↦ ( ( 𝑀 ‘ 𝑥 ) /𝑒 𝐴 ) ) → 𝑦 ∈ ( 0 [,] +∞ ) ) ) |
| 23 |
22
|
ssrdv |
⊢ ( ( 𝑀 ∈ ( measures ‘ 𝑆 ) ∧ 𝐴 ∈ ℝ+ ) → ran ( 𝑥 ∈ 𝑆 ↦ ( ( 𝑀 ‘ 𝑥 ) /𝑒 𝐴 ) ) ⊆ ( 0 [,] +∞ ) ) |
| 24 |
|
df-f |
⊢ ( ( 𝑥 ∈ 𝑆 ↦ ( ( 𝑀 ‘ 𝑥 ) /𝑒 𝐴 ) ) : 𝑆 ⟶ ( 0 [,] +∞ ) ↔ ( ( 𝑥 ∈ 𝑆 ↦ ( ( 𝑀 ‘ 𝑥 ) /𝑒 𝐴 ) ) Fn 𝑆 ∧ ran ( 𝑥 ∈ 𝑆 ↦ ( ( 𝑀 ‘ 𝑥 ) /𝑒 𝐴 ) ) ⊆ ( 0 [,] +∞ ) ) ) |
| 25 |
8 23 24
|
sylanbrc |
⊢ ( ( 𝑀 ∈ ( measures ‘ 𝑆 ) ∧ 𝐴 ∈ ℝ+ ) → ( 𝑥 ∈ 𝑆 ↦ ( ( 𝑀 ‘ 𝑥 ) /𝑒 𝐴 ) ) : 𝑆 ⟶ ( 0 [,] +∞ ) ) |
| 26 |
|
measbase |
⊢ ( 𝑀 ∈ ( measures ‘ 𝑆 ) → 𝑆 ∈ ∪ ran sigAlgebra ) |
| 27 |
|
0elsiga |
⊢ ( 𝑆 ∈ ∪ ran sigAlgebra → ∅ ∈ 𝑆 ) |
| 28 |
26 27
|
syl |
⊢ ( 𝑀 ∈ ( measures ‘ 𝑆 ) → ∅ ∈ 𝑆 ) |
| 29 |
28
|
adantr |
⊢ ( ( 𝑀 ∈ ( measures ‘ 𝑆 ) ∧ 𝐴 ∈ ℝ+ ) → ∅ ∈ 𝑆 ) |
| 30 |
|
ovex |
⊢ ( ( 𝑀 ‘ ∅ ) /𝑒 𝐴 ) ∈ V |
| 31 |
29 30
|
jctir |
⊢ ( ( 𝑀 ∈ ( measures ‘ 𝑆 ) ∧ 𝐴 ∈ ℝ+ ) → ( ∅ ∈ 𝑆 ∧ ( ( 𝑀 ‘ ∅ ) /𝑒 𝐴 ) ∈ V ) ) |
| 32 |
|
fveq2 |
⊢ ( 𝑥 = ∅ → ( 𝑀 ‘ 𝑥 ) = ( 𝑀 ‘ ∅ ) ) |
| 33 |
32
|
oveq1d |
⊢ ( 𝑥 = ∅ → ( ( 𝑀 ‘ 𝑥 ) /𝑒 𝐴 ) = ( ( 𝑀 ‘ ∅ ) /𝑒 𝐴 ) ) |
| 34 |
33 10
|
fvmptg |
⊢ ( ( ∅ ∈ 𝑆 ∧ ( ( 𝑀 ‘ ∅ ) /𝑒 𝐴 ) ∈ V ) → ( ( 𝑥 ∈ 𝑆 ↦ ( ( 𝑀 ‘ 𝑥 ) /𝑒 𝐴 ) ) ‘ ∅ ) = ( ( 𝑀 ‘ ∅ ) /𝑒 𝐴 ) ) |
| 35 |
31 34
|
syl |
⊢ ( ( 𝑀 ∈ ( measures ‘ 𝑆 ) ∧ 𝐴 ∈ ℝ+ ) → ( ( 𝑥 ∈ 𝑆 ↦ ( ( 𝑀 ‘ 𝑥 ) /𝑒 𝐴 ) ) ‘ ∅ ) = ( ( 𝑀 ‘ ∅ ) /𝑒 𝐴 ) ) |
| 36 |
|
measvnul |
⊢ ( 𝑀 ∈ ( measures ‘ 𝑆 ) → ( 𝑀 ‘ ∅ ) = 0 ) |
| 37 |
36
|
oveq1d |
⊢ ( 𝑀 ∈ ( measures ‘ 𝑆 ) → ( ( 𝑀 ‘ ∅ ) /𝑒 𝐴 ) = ( 0 /𝑒 𝐴 ) ) |
| 38 |
|
xdiv0rp |
⊢ ( 𝐴 ∈ ℝ+ → ( 0 /𝑒 𝐴 ) = 0 ) |
| 39 |
37 38
|
sylan9eq |
⊢ ( ( 𝑀 ∈ ( measures ‘ 𝑆 ) ∧ 𝐴 ∈ ℝ+ ) → ( ( 𝑀 ‘ ∅ ) /𝑒 𝐴 ) = 0 ) |
| 40 |
35 39
|
eqtrd |
⊢ ( ( 𝑀 ∈ ( measures ‘ 𝑆 ) ∧ 𝐴 ∈ ℝ+ ) → ( ( 𝑥 ∈ 𝑆 ↦ ( ( 𝑀 ‘ 𝑥 ) /𝑒 𝐴 ) ) ‘ ∅ ) = 0 ) |
| 41 |
|
simpll |
⊢ ( ( ( ( 𝑀 ∈ ( measures ‘ 𝑆 ) ∧ 𝐴 ∈ ℝ+ ) ∧ 𝑦 ∈ 𝒫 𝑆 ) ∧ ( 𝑦 ≼ ω ∧ Disj 𝑧 ∈ 𝑦 𝑧 ) ) → ( 𝑀 ∈ ( measures ‘ 𝑆 ) ∧ 𝐴 ∈ ℝ+ ) ) |
| 42 |
|
simplr |
⊢ ( ( ( ( 𝑀 ∈ ( measures ‘ 𝑆 ) ∧ 𝐴 ∈ ℝ+ ) ∧ 𝑦 ∈ 𝒫 𝑆 ) ∧ ( 𝑦 ≼ ω ∧ Disj 𝑧 ∈ 𝑦 𝑧 ) ) → 𝑦 ∈ 𝒫 𝑆 ) |
| 43 |
|
simprl |
⊢ ( ( ( ( 𝑀 ∈ ( measures ‘ 𝑆 ) ∧ 𝐴 ∈ ℝ+ ) ∧ 𝑦 ∈ 𝒫 𝑆 ) ∧ ( 𝑦 ≼ ω ∧ Disj 𝑧 ∈ 𝑦 𝑧 ) ) → 𝑦 ≼ ω ) |
| 44 |
|
simprr |
⊢ ( ( ( ( 𝑀 ∈ ( measures ‘ 𝑆 ) ∧ 𝐴 ∈ ℝ+ ) ∧ 𝑦 ∈ 𝒫 𝑆 ) ∧ ( 𝑦 ≼ ω ∧ Disj 𝑧 ∈ 𝑦 𝑧 ) ) → Disj 𝑧 ∈ 𝑦 𝑧 ) |
| 45 |
42 43 44
|
3jca |
⊢ ( ( ( ( 𝑀 ∈ ( measures ‘ 𝑆 ) ∧ 𝐴 ∈ ℝ+ ) ∧ 𝑦 ∈ 𝒫 𝑆 ) ∧ ( 𝑦 ≼ ω ∧ Disj 𝑧 ∈ 𝑦 𝑧 ) ) → ( 𝑦 ∈ 𝒫 𝑆 ∧ 𝑦 ≼ ω ∧ Disj 𝑧 ∈ 𝑦 𝑧 ) ) |
| 46 |
9
|
a1i |
⊢ ( ( ( 𝑀 ∈ ( measures ‘ 𝑆 ) ∧ 𝐴 ∈ ℝ+ ) ∧ 𝑦 ∈ 𝒫 𝑆 ) → 𝑦 ∈ V ) |
| 47 |
|
simplll |
⊢ ( ( ( ( 𝑀 ∈ ( measures ‘ 𝑆 ) ∧ 𝐴 ∈ ℝ+ ) ∧ 𝑦 ∈ 𝒫 𝑆 ) ∧ 𝑧 ∈ 𝑦 ) → 𝑀 ∈ ( measures ‘ 𝑆 ) ) |
| 48 |
|
simplr |
⊢ ( ( ( ( 𝑀 ∈ ( measures ‘ 𝑆 ) ∧ 𝐴 ∈ ℝ+ ) ∧ 𝑦 ∈ 𝒫 𝑆 ) ∧ 𝑧 ∈ 𝑦 ) → 𝑦 ∈ 𝒫 𝑆 ) |
| 49 |
|
simpr |
⊢ ( ( ( ( 𝑀 ∈ ( measures ‘ 𝑆 ) ∧ 𝐴 ∈ ℝ+ ) ∧ 𝑦 ∈ 𝒫 𝑆 ) ∧ 𝑧 ∈ 𝑦 ) → 𝑧 ∈ 𝑦 ) |
| 50 |
|
elpwg |
⊢ ( 𝑦 ∈ V → ( 𝑦 ∈ 𝒫 𝑆 ↔ 𝑦 ⊆ 𝑆 ) ) |
| 51 |
9 50
|
ax-mp |
⊢ ( 𝑦 ∈ 𝒫 𝑆 ↔ 𝑦 ⊆ 𝑆 ) |
| 52 |
|
ssel2 |
⊢ ( ( 𝑦 ⊆ 𝑆 ∧ 𝑧 ∈ 𝑦 ) → 𝑧 ∈ 𝑆 ) |
| 53 |
51 52
|
sylanb |
⊢ ( ( 𝑦 ∈ 𝒫 𝑆 ∧ 𝑧 ∈ 𝑦 ) → 𝑧 ∈ 𝑆 ) |
| 54 |
48 49 53
|
syl2anc |
⊢ ( ( ( ( 𝑀 ∈ ( measures ‘ 𝑆 ) ∧ 𝐴 ∈ ℝ+ ) ∧ 𝑦 ∈ 𝒫 𝑆 ) ∧ 𝑧 ∈ 𝑦 ) → 𝑧 ∈ 𝑆 ) |
| 55 |
|
measvxrge0 |
⊢ ( ( 𝑀 ∈ ( measures ‘ 𝑆 ) ∧ 𝑧 ∈ 𝑆 ) → ( 𝑀 ‘ 𝑧 ) ∈ ( 0 [,] +∞ ) ) |
| 56 |
47 54 55
|
syl2anc |
⊢ ( ( ( ( 𝑀 ∈ ( measures ‘ 𝑆 ) ∧ 𝐴 ∈ ℝ+ ) ∧ 𝑦 ∈ 𝒫 𝑆 ) ∧ 𝑧 ∈ 𝑦 ) → ( 𝑀 ‘ 𝑧 ) ∈ ( 0 [,] +∞ ) ) |
| 57 |
|
simplr |
⊢ ( ( ( 𝑀 ∈ ( measures ‘ 𝑆 ) ∧ 𝐴 ∈ ℝ+ ) ∧ 𝑦 ∈ 𝒫 𝑆 ) → 𝐴 ∈ ℝ+ ) |
| 58 |
46 56 57
|
esumdivc |
⊢ ( ( ( 𝑀 ∈ ( measures ‘ 𝑆 ) ∧ 𝐴 ∈ ℝ+ ) ∧ 𝑦 ∈ 𝒫 𝑆 ) → ( Σ* 𝑧 ∈ 𝑦 ( 𝑀 ‘ 𝑧 ) /𝑒 𝐴 ) = Σ* 𝑧 ∈ 𝑦 ( ( 𝑀 ‘ 𝑧 ) /𝑒 𝐴 ) ) |
| 59 |
58
|
3ad2antr1 |
⊢ ( ( ( 𝑀 ∈ ( measures ‘ 𝑆 ) ∧ 𝐴 ∈ ℝ+ ) ∧ ( 𝑦 ∈ 𝒫 𝑆 ∧ 𝑦 ≼ ω ∧ Disj 𝑧 ∈ 𝑦 𝑧 ) ) → ( Σ* 𝑧 ∈ 𝑦 ( 𝑀 ‘ 𝑧 ) /𝑒 𝐴 ) = Σ* 𝑧 ∈ 𝑦 ( ( 𝑀 ‘ 𝑧 ) /𝑒 𝐴 ) ) |
| 60 |
26
|
ad2antrr |
⊢ ( ( ( 𝑀 ∈ ( measures ‘ 𝑆 ) ∧ 𝐴 ∈ ℝ+ ) ∧ ( 𝑦 ∈ 𝒫 𝑆 ∧ 𝑦 ≼ ω ∧ Disj 𝑧 ∈ 𝑦 𝑧 ) ) → 𝑆 ∈ ∪ ran sigAlgebra ) |
| 61 |
|
simpr1 |
⊢ ( ( ( 𝑀 ∈ ( measures ‘ 𝑆 ) ∧ 𝐴 ∈ ℝ+ ) ∧ ( 𝑦 ∈ 𝒫 𝑆 ∧ 𝑦 ≼ ω ∧ Disj 𝑧 ∈ 𝑦 𝑧 ) ) → 𝑦 ∈ 𝒫 𝑆 ) |
| 62 |
|
simpr2 |
⊢ ( ( ( 𝑀 ∈ ( measures ‘ 𝑆 ) ∧ 𝐴 ∈ ℝ+ ) ∧ ( 𝑦 ∈ 𝒫 𝑆 ∧ 𝑦 ≼ ω ∧ Disj 𝑧 ∈ 𝑦 𝑧 ) ) → 𝑦 ≼ ω ) |
| 63 |
|
sigaclcu |
⊢ ( ( 𝑆 ∈ ∪ ran sigAlgebra ∧ 𝑦 ∈ 𝒫 𝑆 ∧ 𝑦 ≼ ω ) → ∪ 𝑦 ∈ 𝑆 ) |
| 64 |
60 61 62 63
|
syl3anc |
⊢ ( ( ( 𝑀 ∈ ( measures ‘ 𝑆 ) ∧ 𝐴 ∈ ℝ+ ) ∧ ( 𝑦 ∈ 𝒫 𝑆 ∧ 𝑦 ≼ ω ∧ Disj 𝑧 ∈ 𝑦 𝑧 ) ) → ∪ 𝑦 ∈ 𝑆 ) |
| 65 |
|
fveq2 |
⊢ ( 𝑥 = ∪ 𝑦 → ( 𝑀 ‘ 𝑥 ) = ( 𝑀 ‘ ∪ 𝑦 ) ) |
| 66 |
65
|
oveq1d |
⊢ ( 𝑥 = ∪ 𝑦 → ( ( 𝑀 ‘ 𝑥 ) /𝑒 𝐴 ) = ( ( 𝑀 ‘ ∪ 𝑦 ) /𝑒 𝐴 ) ) |
| 67 |
66 10 2
|
fvmpt3i |
⊢ ( ∪ 𝑦 ∈ 𝑆 → ( ( 𝑥 ∈ 𝑆 ↦ ( ( 𝑀 ‘ 𝑥 ) /𝑒 𝐴 ) ) ‘ ∪ 𝑦 ) = ( ( 𝑀 ‘ ∪ 𝑦 ) /𝑒 𝐴 ) ) |
| 68 |
64 67
|
syl |
⊢ ( ( ( 𝑀 ∈ ( measures ‘ 𝑆 ) ∧ 𝐴 ∈ ℝ+ ) ∧ ( 𝑦 ∈ 𝒫 𝑆 ∧ 𝑦 ≼ ω ∧ Disj 𝑧 ∈ 𝑦 𝑧 ) ) → ( ( 𝑥 ∈ 𝑆 ↦ ( ( 𝑀 ‘ 𝑥 ) /𝑒 𝐴 ) ) ‘ ∪ 𝑦 ) = ( ( 𝑀 ‘ ∪ 𝑦 ) /𝑒 𝐴 ) ) |
| 69 |
|
simpll |
⊢ ( ( ( 𝑀 ∈ ( measures ‘ 𝑆 ) ∧ 𝐴 ∈ ℝ+ ) ∧ ( 𝑦 ∈ 𝒫 𝑆 ∧ 𝑦 ≼ ω ∧ Disj 𝑧 ∈ 𝑦 𝑧 ) ) → 𝑀 ∈ ( measures ‘ 𝑆 ) ) |
| 70 |
69 61
|
jca |
⊢ ( ( ( 𝑀 ∈ ( measures ‘ 𝑆 ) ∧ 𝐴 ∈ ℝ+ ) ∧ ( 𝑦 ∈ 𝒫 𝑆 ∧ 𝑦 ≼ ω ∧ Disj 𝑧 ∈ 𝑦 𝑧 ) ) → ( 𝑀 ∈ ( measures ‘ 𝑆 ) ∧ 𝑦 ∈ 𝒫 𝑆 ) ) |
| 71 |
|
simpr3 |
⊢ ( ( ( 𝑀 ∈ ( measures ‘ 𝑆 ) ∧ 𝐴 ∈ ℝ+ ) ∧ ( 𝑦 ∈ 𝒫 𝑆 ∧ 𝑦 ≼ ω ∧ Disj 𝑧 ∈ 𝑦 𝑧 ) ) → Disj 𝑧 ∈ 𝑦 𝑧 ) |
| 72 |
62 71
|
jca |
⊢ ( ( ( 𝑀 ∈ ( measures ‘ 𝑆 ) ∧ 𝐴 ∈ ℝ+ ) ∧ ( 𝑦 ∈ 𝒫 𝑆 ∧ 𝑦 ≼ ω ∧ Disj 𝑧 ∈ 𝑦 𝑧 ) ) → ( 𝑦 ≼ ω ∧ Disj 𝑧 ∈ 𝑦 𝑧 ) ) |
| 73 |
|
measvun |
⊢ ( ( 𝑀 ∈ ( measures ‘ 𝑆 ) ∧ 𝑦 ∈ 𝒫 𝑆 ∧ ( 𝑦 ≼ ω ∧ Disj 𝑧 ∈ 𝑦 𝑧 ) ) → ( 𝑀 ‘ ∪ 𝑦 ) = Σ* 𝑧 ∈ 𝑦 ( 𝑀 ‘ 𝑧 ) ) |
| 74 |
73
|
3expia |
⊢ ( ( 𝑀 ∈ ( measures ‘ 𝑆 ) ∧ 𝑦 ∈ 𝒫 𝑆 ) → ( ( 𝑦 ≼ ω ∧ Disj 𝑧 ∈ 𝑦 𝑧 ) → ( 𝑀 ‘ ∪ 𝑦 ) = Σ* 𝑧 ∈ 𝑦 ( 𝑀 ‘ 𝑧 ) ) ) |
| 75 |
74
|
ralrimiva |
⊢ ( 𝑀 ∈ ( measures ‘ 𝑆 ) → ∀ 𝑦 ∈ 𝒫 𝑆 ( ( 𝑦 ≼ ω ∧ Disj 𝑧 ∈ 𝑦 𝑧 ) → ( 𝑀 ‘ ∪ 𝑦 ) = Σ* 𝑧 ∈ 𝑦 ( 𝑀 ‘ 𝑧 ) ) ) |
| 76 |
75
|
r19.21bi |
⊢ ( ( 𝑀 ∈ ( measures ‘ 𝑆 ) ∧ 𝑦 ∈ 𝒫 𝑆 ) → ( ( 𝑦 ≼ ω ∧ Disj 𝑧 ∈ 𝑦 𝑧 ) → ( 𝑀 ‘ ∪ 𝑦 ) = Σ* 𝑧 ∈ 𝑦 ( 𝑀 ‘ 𝑧 ) ) ) |
| 77 |
70 72 76
|
sylc |
⊢ ( ( ( 𝑀 ∈ ( measures ‘ 𝑆 ) ∧ 𝐴 ∈ ℝ+ ) ∧ ( 𝑦 ∈ 𝒫 𝑆 ∧ 𝑦 ≼ ω ∧ Disj 𝑧 ∈ 𝑦 𝑧 ) ) → ( 𝑀 ‘ ∪ 𝑦 ) = Σ* 𝑧 ∈ 𝑦 ( 𝑀 ‘ 𝑧 ) ) |
| 78 |
77
|
oveq1d |
⊢ ( ( ( 𝑀 ∈ ( measures ‘ 𝑆 ) ∧ 𝐴 ∈ ℝ+ ) ∧ ( 𝑦 ∈ 𝒫 𝑆 ∧ 𝑦 ≼ ω ∧ Disj 𝑧 ∈ 𝑦 𝑧 ) ) → ( ( 𝑀 ‘ ∪ 𝑦 ) /𝑒 𝐴 ) = ( Σ* 𝑧 ∈ 𝑦 ( 𝑀 ‘ 𝑧 ) /𝑒 𝐴 ) ) |
| 79 |
68 78
|
eqtrd |
⊢ ( ( ( 𝑀 ∈ ( measures ‘ 𝑆 ) ∧ 𝐴 ∈ ℝ+ ) ∧ ( 𝑦 ∈ 𝒫 𝑆 ∧ 𝑦 ≼ ω ∧ Disj 𝑧 ∈ 𝑦 𝑧 ) ) → ( ( 𝑥 ∈ 𝑆 ↦ ( ( 𝑀 ‘ 𝑥 ) /𝑒 𝐴 ) ) ‘ ∪ 𝑦 ) = ( Σ* 𝑧 ∈ 𝑦 ( 𝑀 ‘ 𝑧 ) /𝑒 𝐴 ) ) |
| 80 |
|
fveq2 |
⊢ ( 𝑥 = 𝑧 → ( 𝑀 ‘ 𝑥 ) = ( 𝑀 ‘ 𝑧 ) ) |
| 81 |
80
|
oveq1d |
⊢ ( 𝑥 = 𝑧 → ( ( 𝑀 ‘ 𝑥 ) /𝑒 𝐴 ) = ( ( 𝑀 ‘ 𝑧 ) /𝑒 𝐴 ) ) |
| 82 |
81 10 2
|
fvmpt3i |
⊢ ( 𝑧 ∈ 𝑆 → ( ( 𝑥 ∈ 𝑆 ↦ ( ( 𝑀 ‘ 𝑥 ) /𝑒 𝐴 ) ) ‘ 𝑧 ) = ( ( 𝑀 ‘ 𝑧 ) /𝑒 𝐴 ) ) |
| 83 |
53 82
|
syl |
⊢ ( ( 𝑦 ∈ 𝒫 𝑆 ∧ 𝑧 ∈ 𝑦 ) → ( ( 𝑥 ∈ 𝑆 ↦ ( ( 𝑀 ‘ 𝑥 ) /𝑒 𝐴 ) ) ‘ 𝑧 ) = ( ( 𝑀 ‘ 𝑧 ) /𝑒 𝐴 ) ) |
| 84 |
83
|
esumeq2dv |
⊢ ( 𝑦 ∈ 𝒫 𝑆 → Σ* 𝑧 ∈ 𝑦 ( ( 𝑥 ∈ 𝑆 ↦ ( ( 𝑀 ‘ 𝑥 ) /𝑒 𝐴 ) ) ‘ 𝑧 ) = Σ* 𝑧 ∈ 𝑦 ( ( 𝑀 ‘ 𝑧 ) /𝑒 𝐴 ) ) |
| 85 |
61 84
|
syl |
⊢ ( ( ( 𝑀 ∈ ( measures ‘ 𝑆 ) ∧ 𝐴 ∈ ℝ+ ) ∧ ( 𝑦 ∈ 𝒫 𝑆 ∧ 𝑦 ≼ ω ∧ Disj 𝑧 ∈ 𝑦 𝑧 ) ) → Σ* 𝑧 ∈ 𝑦 ( ( 𝑥 ∈ 𝑆 ↦ ( ( 𝑀 ‘ 𝑥 ) /𝑒 𝐴 ) ) ‘ 𝑧 ) = Σ* 𝑧 ∈ 𝑦 ( ( 𝑀 ‘ 𝑧 ) /𝑒 𝐴 ) ) |
| 86 |
59 79 85
|
3eqtr4d |
⊢ ( ( ( 𝑀 ∈ ( measures ‘ 𝑆 ) ∧ 𝐴 ∈ ℝ+ ) ∧ ( 𝑦 ∈ 𝒫 𝑆 ∧ 𝑦 ≼ ω ∧ Disj 𝑧 ∈ 𝑦 𝑧 ) ) → ( ( 𝑥 ∈ 𝑆 ↦ ( ( 𝑀 ‘ 𝑥 ) /𝑒 𝐴 ) ) ‘ ∪ 𝑦 ) = Σ* 𝑧 ∈ 𝑦 ( ( 𝑥 ∈ 𝑆 ↦ ( ( 𝑀 ‘ 𝑥 ) /𝑒 𝐴 ) ) ‘ 𝑧 ) ) |
| 87 |
41 45 86
|
syl2anc |
⊢ ( ( ( ( 𝑀 ∈ ( measures ‘ 𝑆 ) ∧ 𝐴 ∈ ℝ+ ) ∧ 𝑦 ∈ 𝒫 𝑆 ) ∧ ( 𝑦 ≼ ω ∧ Disj 𝑧 ∈ 𝑦 𝑧 ) ) → ( ( 𝑥 ∈ 𝑆 ↦ ( ( 𝑀 ‘ 𝑥 ) /𝑒 𝐴 ) ) ‘ ∪ 𝑦 ) = Σ* 𝑧 ∈ 𝑦 ( ( 𝑥 ∈ 𝑆 ↦ ( ( 𝑀 ‘ 𝑥 ) /𝑒 𝐴 ) ) ‘ 𝑧 ) ) |
| 88 |
87
|
ex |
⊢ ( ( ( 𝑀 ∈ ( measures ‘ 𝑆 ) ∧ 𝐴 ∈ ℝ+ ) ∧ 𝑦 ∈ 𝒫 𝑆 ) → ( ( 𝑦 ≼ ω ∧ Disj 𝑧 ∈ 𝑦 𝑧 ) → ( ( 𝑥 ∈ 𝑆 ↦ ( ( 𝑀 ‘ 𝑥 ) /𝑒 𝐴 ) ) ‘ ∪ 𝑦 ) = Σ* 𝑧 ∈ 𝑦 ( ( 𝑥 ∈ 𝑆 ↦ ( ( 𝑀 ‘ 𝑥 ) /𝑒 𝐴 ) ) ‘ 𝑧 ) ) ) |
| 89 |
88
|
ralrimiva |
⊢ ( ( 𝑀 ∈ ( measures ‘ 𝑆 ) ∧ 𝐴 ∈ ℝ+ ) → ∀ 𝑦 ∈ 𝒫 𝑆 ( ( 𝑦 ≼ ω ∧ Disj 𝑧 ∈ 𝑦 𝑧 ) → ( ( 𝑥 ∈ 𝑆 ↦ ( ( 𝑀 ‘ 𝑥 ) /𝑒 𝐴 ) ) ‘ ∪ 𝑦 ) = Σ* 𝑧 ∈ 𝑦 ( ( 𝑥 ∈ 𝑆 ↦ ( ( 𝑀 ‘ 𝑥 ) /𝑒 𝐴 ) ) ‘ 𝑧 ) ) ) |
| 90 |
25 40 89
|
3jca |
⊢ ( ( 𝑀 ∈ ( measures ‘ 𝑆 ) ∧ 𝐴 ∈ ℝ+ ) → ( ( 𝑥 ∈ 𝑆 ↦ ( ( 𝑀 ‘ 𝑥 ) /𝑒 𝐴 ) ) : 𝑆 ⟶ ( 0 [,] +∞ ) ∧ ( ( 𝑥 ∈ 𝑆 ↦ ( ( 𝑀 ‘ 𝑥 ) /𝑒 𝐴 ) ) ‘ ∅ ) = 0 ∧ ∀ 𝑦 ∈ 𝒫 𝑆 ( ( 𝑦 ≼ ω ∧ Disj 𝑧 ∈ 𝑦 𝑧 ) → ( ( 𝑥 ∈ 𝑆 ↦ ( ( 𝑀 ‘ 𝑥 ) /𝑒 𝐴 ) ) ‘ ∪ 𝑦 ) = Σ* 𝑧 ∈ 𝑦 ( ( 𝑥 ∈ 𝑆 ↦ ( ( 𝑀 ‘ 𝑥 ) /𝑒 𝐴 ) ) ‘ 𝑧 ) ) ) ) |
| 91 |
|
ismeas |
⊢ ( 𝑆 ∈ ∪ ran sigAlgebra → ( ( 𝑥 ∈ 𝑆 ↦ ( ( 𝑀 ‘ 𝑥 ) /𝑒 𝐴 ) ) ∈ ( measures ‘ 𝑆 ) ↔ ( ( 𝑥 ∈ 𝑆 ↦ ( ( 𝑀 ‘ 𝑥 ) /𝑒 𝐴 ) ) : 𝑆 ⟶ ( 0 [,] +∞ ) ∧ ( ( 𝑥 ∈ 𝑆 ↦ ( ( 𝑀 ‘ 𝑥 ) /𝑒 𝐴 ) ) ‘ ∅ ) = 0 ∧ ∀ 𝑦 ∈ 𝒫 𝑆 ( ( 𝑦 ≼ ω ∧ Disj 𝑧 ∈ 𝑦 𝑧 ) → ( ( 𝑥 ∈ 𝑆 ↦ ( ( 𝑀 ‘ 𝑥 ) /𝑒 𝐴 ) ) ‘ ∪ 𝑦 ) = Σ* 𝑧 ∈ 𝑦 ( ( 𝑥 ∈ 𝑆 ↦ ( ( 𝑀 ‘ 𝑥 ) /𝑒 𝐴 ) ) ‘ 𝑧 ) ) ) ) ) |
| 92 |
26 91
|
syl |
⊢ ( 𝑀 ∈ ( measures ‘ 𝑆 ) → ( ( 𝑥 ∈ 𝑆 ↦ ( ( 𝑀 ‘ 𝑥 ) /𝑒 𝐴 ) ) ∈ ( measures ‘ 𝑆 ) ↔ ( ( 𝑥 ∈ 𝑆 ↦ ( ( 𝑀 ‘ 𝑥 ) /𝑒 𝐴 ) ) : 𝑆 ⟶ ( 0 [,] +∞ ) ∧ ( ( 𝑥 ∈ 𝑆 ↦ ( ( 𝑀 ‘ 𝑥 ) /𝑒 𝐴 ) ) ‘ ∅ ) = 0 ∧ ∀ 𝑦 ∈ 𝒫 𝑆 ( ( 𝑦 ≼ ω ∧ Disj 𝑧 ∈ 𝑦 𝑧 ) → ( ( 𝑥 ∈ 𝑆 ↦ ( ( 𝑀 ‘ 𝑥 ) /𝑒 𝐴 ) ) ‘ ∪ 𝑦 ) = Σ* 𝑧 ∈ 𝑦 ( ( 𝑥 ∈ 𝑆 ↦ ( ( 𝑀 ‘ 𝑥 ) /𝑒 𝐴 ) ) ‘ 𝑧 ) ) ) ) ) |
| 93 |
92
|
biimprd |
⊢ ( 𝑀 ∈ ( measures ‘ 𝑆 ) → ( ( ( 𝑥 ∈ 𝑆 ↦ ( ( 𝑀 ‘ 𝑥 ) /𝑒 𝐴 ) ) : 𝑆 ⟶ ( 0 [,] +∞ ) ∧ ( ( 𝑥 ∈ 𝑆 ↦ ( ( 𝑀 ‘ 𝑥 ) /𝑒 𝐴 ) ) ‘ ∅ ) = 0 ∧ ∀ 𝑦 ∈ 𝒫 𝑆 ( ( 𝑦 ≼ ω ∧ Disj 𝑧 ∈ 𝑦 𝑧 ) → ( ( 𝑥 ∈ 𝑆 ↦ ( ( 𝑀 ‘ 𝑥 ) /𝑒 𝐴 ) ) ‘ ∪ 𝑦 ) = Σ* 𝑧 ∈ 𝑦 ( ( 𝑥 ∈ 𝑆 ↦ ( ( 𝑀 ‘ 𝑥 ) /𝑒 𝐴 ) ) ‘ 𝑧 ) ) ) → ( 𝑥 ∈ 𝑆 ↦ ( ( 𝑀 ‘ 𝑥 ) /𝑒 𝐴 ) ) ∈ ( measures ‘ 𝑆 ) ) ) |
| 94 |
93
|
adantr |
⊢ ( ( 𝑀 ∈ ( measures ‘ 𝑆 ) ∧ 𝐴 ∈ ℝ+ ) → ( ( ( 𝑥 ∈ 𝑆 ↦ ( ( 𝑀 ‘ 𝑥 ) /𝑒 𝐴 ) ) : 𝑆 ⟶ ( 0 [,] +∞ ) ∧ ( ( 𝑥 ∈ 𝑆 ↦ ( ( 𝑀 ‘ 𝑥 ) /𝑒 𝐴 ) ) ‘ ∅ ) = 0 ∧ ∀ 𝑦 ∈ 𝒫 𝑆 ( ( 𝑦 ≼ ω ∧ Disj 𝑧 ∈ 𝑦 𝑧 ) → ( ( 𝑥 ∈ 𝑆 ↦ ( ( 𝑀 ‘ 𝑥 ) /𝑒 𝐴 ) ) ‘ ∪ 𝑦 ) = Σ* 𝑧 ∈ 𝑦 ( ( 𝑥 ∈ 𝑆 ↦ ( ( 𝑀 ‘ 𝑥 ) /𝑒 𝐴 ) ) ‘ 𝑧 ) ) ) → ( 𝑥 ∈ 𝑆 ↦ ( ( 𝑀 ‘ 𝑥 ) /𝑒 𝐴 ) ) ∈ ( measures ‘ 𝑆 ) ) ) |
| 95 |
90 94
|
mpd |
⊢ ( ( 𝑀 ∈ ( measures ‘ 𝑆 ) ∧ 𝐴 ∈ ℝ+ ) → ( 𝑥 ∈ 𝑆 ↦ ( ( 𝑀 ‘ 𝑥 ) /𝑒 𝐴 ) ) ∈ ( measures ‘ 𝑆 ) ) |