Metamath Proof Explorer


Theorem carsgval

Description: Value of the Caratheodory sigma-Algebra construction function. (Contributed by Thierry Arnoux, 17-May-2020)

Ref Expression
Hypotheses carsgval.1 φ O V
carsgval.2 φ M : 𝒫 O 0 +∞
Assertion carsgval φ toCaraSiga M = a 𝒫 O | e 𝒫 O M e a + 𝑒 M e a = M e

Proof

Step Hyp Ref Expression
1 carsgval.1 φ O V
2 carsgval.2 φ M : 𝒫 O 0 +∞
3 df-carsg toCaraSiga = m V a 𝒫 dom m | e 𝒫 dom m m e a + 𝑒 m e a = m e
4 simpr φ m = M m = M
5 4 dmeqd φ m = M dom m = dom M
6 2 fdmd φ dom M = 𝒫 O
7 6 adantr φ m = M dom M = 𝒫 O
8 5 7 eqtrd φ m = M dom m = 𝒫 O
9 8 unieqd φ m = M dom m = 𝒫 O
10 unipw 𝒫 O = O
11 9 10 eqtrdi φ m = M dom m = O
12 11 pweqd φ m = M 𝒫 dom m = 𝒫 O
13 fveq1 m = M m e a = M e a
14 fveq1 m = M m e a = M e a
15 13 14 oveq12d m = M m e a + 𝑒 m e a = M e a + 𝑒 M e a
16 fveq1 m = M m e = M e
17 15 16 eqeq12d m = M m e a + 𝑒 m e a = m e M e a + 𝑒 M e a = M e
18 17 adantl φ m = M m e a + 𝑒 m e a = m e M e a + 𝑒 M e a = M e
19 12 18 raleqbidv φ m = M e 𝒫 dom m m e a + 𝑒 m e a = m e e 𝒫 O M e a + 𝑒 M e a = M e
20 12 19 rabeqbidv φ m = M a 𝒫 dom m | e 𝒫 dom m m e a + 𝑒 m e a = m e = a 𝒫 O | e 𝒫 O M e a + 𝑒 M e a = M e
21 1 pwexd φ 𝒫 O V
22 2 21 fexd φ M V
23 pwexg O V 𝒫 O V
24 rabexg 𝒫 O V a 𝒫 O | e 𝒫 O M e a + 𝑒 M e a = M e V
25 1 23 24 3syl φ a 𝒫 O | e 𝒫 O M e a + 𝑒 M e a = M e V
26 3 20 22 25 fvmptd2 φ toCaraSiga M = a 𝒫 O | e 𝒫 O M e a + 𝑒 M e a = M e