Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  aean Structured version   Visualization version   GIF version

Theorem aean 32843
Description: A conjunction holds almost everywhere if and only if both its terms do. (Contributed by Thierry Arnoux, 20-Oct-2017.)
Hypothesis
Ref Expression
aean.1 dom 𝑀 = 𝑂
Assertion
Ref Expression
aean ((𝑀 ran measures ∧ {𝑥𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) → ({𝑥𝑂 ∣ (𝜑𝜓)}a.e.𝑀 ↔ ({𝑥𝑂𝜑}a.e.𝑀 ∧ {𝑥𝑂𝜓}a.e.𝑀)))
Distinct variable group:   𝑥,𝑂
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)   𝑀(𝑥)

Proof of Theorem aean
StepHypRef Expression
1 unrab 4265 . . . . . 6 ({𝑥𝑂 ∣ ¬ 𝜑} ∪ {𝑥𝑂 ∣ ¬ 𝜓}) = {𝑥𝑂 ∣ (¬ 𝜑 ∨ ¬ 𝜓)}
2 ianor 980 . . . . . . 7 (¬ (𝜑𝜓) ↔ (¬ 𝜑 ∨ ¬ 𝜓))
32rabbii 3413 . . . . . 6 {𝑥𝑂 ∣ ¬ (𝜑𝜓)} = {𝑥𝑂 ∣ (¬ 𝜑 ∨ ¬ 𝜓)}
41, 3eqtr4i 2767 . . . . 5 ({𝑥𝑂 ∣ ¬ 𝜑} ∪ {𝑥𝑂 ∣ ¬ 𝜓}) = {𝑥𝑂 ∣ ¬ (𝜑𝜓)}
54fveq2i 6845 . . . 4 (𝑀‘({𝑥𝑂 ∣ ¬ 𝜑} ∪ {𝑥𝑂 ∣ ¬ 𝜓})) = (𝑀‘{𝑥𝑂 ∣ ¬ (𝜑𝜓)})
65eqeq1i 2741 . . 3 ((𝑀‘({𝑥𝑂 ∣ ¬ 𝜑} ∪ {𝑥𝑂 ∣ ¬ 𝜓})) = 0 ↔ (𝑀‘{𝑥𝑂 ∣ ¬ (𝜑𝜓)}) = 0)
7 measbasedom 32801 . . . . . . . . 9 (𝑀 ran measures ↔ 𝑀 ∈ (measures‘dom 𝑀))
87biimpi 215 . . . . . . . 8 (𝑀 ran measures → 𝑀 ∈ (measures‘dom 𝑀))
983ad2ant1 1133 . . . . . . 7 ((𝑀 ran measures ∧ {𝑥𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) → 𝑀 ∈ (measures‘dom 𝑀))
109adantr 481 . . . . . 6 (((𝑀 ran measures ∧ {𝑥𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) ∧ (𝑀‘({𝑥𝑂 ∣ ¬ 𝜑} ∪ {𝑥𝑂 ∣ ¬ 𝜓})) = 0) → 𝑀 ∈ (measures‘dom 𝑀))
11 simp2 1137 . . . . . . 7 ((𝑀 ran measures ∧ {𝑥𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) → {𝑥𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀)
1211adantr 481 . . . . . 6 (((𝑀 ran measures ∧ {𝑥𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) ∧ (𝑀‘({𝑥𝑂 ∣ ¬ 𝜑} ∪ {𝑥𝑂 ∣ ¬ 𝜓})) = 0) → {𝑥𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀)
13 dmmeas 32800 . . . . . . . . . 10 (𝑀 ran measures → dom 𝑀 ran sigAlgebra)
14 unelsiga 32733 . . . . . . . . . 10 ((dom 𝑀 ran sigAlgebra ∧ {𝑥𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) → ({𝑥𝑂 ∣ ¬ 𝜑} ∪ {𝑥𝑂 ∣ ¬ 𝜓}) ∈ dom 𝑀)
1513, 14syl3an1 1163 . . . . . . . . 9 ((𝑀 ran measures ∧ {𝑥𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) → ({𝑥𝑂 ∣ ¬ 𝜑} ∪ {𝑥𝑂 ∣ ¬ 𝜓}) ∈ dom 𝑀)
16 ssun1 4132 . . . . . . . . . 10 {𝑥𝑂 ∣ ¬ 𝜑} ⊆ ({𝑥𝑂 ∣ ¬ 𝜑} ∪ {𝑥𝑂 ∣ ¬ 𝜓})
1716a1i 11 . . . . . . . . 9 ((𝑀 ran measures ∧ {𝑥𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) → {𝑥𝑂 ∣ ¬ 𝜑} ⊆ ({𝑥𝑂 ∣ ¬ 𝜑} ∪ {𝑥𝑂 ∣ ¬ 𝜓}))
189, 11, 15, 17measssd 32814 . . . . . . . 8 ((𝑀 ran measures ∧ {𝑥𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) → (𝑀‘{𝑥𝑂 ∣ ¬ 𝜑}) ≤ (𝑀‘({𝑥𝑂 ∣ ¬ 𝜑} ∪ {𝑥𝑂 ∣ ¬ 𝜓})))
1918adantr 481 . . . . . . 7 (((𝑀 ran measures ∧ {𝑥𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) ∧ (𝑀‘({𝑥𝑂 ∣ ¬ 𝜑} ∪ {𝑥𝑂 ∣ ¬ 𝜓})) = 0) → (𝑀‘{𝑥𝑂 ∣ ¬ 𝜑}) ≤ (𝑀‘({𝑥𝑂 ∣ ¬ 𝜑} ∪ {𝑥𝑂 ∣ ¬ 𝜓})))
20 simpr 485 . . . . . . 7 (((𝑀 ran measures ∧ {𝑥𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) ∧ (𝑀‘({𝑥𝑂 ∣ ¬ 𝜑} ∪ {𝑥𝑂 ∣ ¬ 𝜓})) = 0) → (𝑀‘({𝑥𝑂 ∣ ¬ 𝜑} ∪ {𝑥𝑂 ∣ ¬ 𝜓})) = 0)
2119, 20breqtrd 5131 . . . . . 6 (((𝑀 ran measures ∧ {𝑥𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) ∧ (𝑀‘({𝑥𝑂 ∣ ¬ 𝜑} ∪ {𝑥𝑂 ∣ ¬ 𝜓})) = 0) → (𝑀‘{𝑥𝑂 ∣ ¬ 𝜑}) ≤ 0)
22 measle0 32807 . . . . . 6 ((𝑀 ∈ (measures‘dom 𝑀) ∧ {𝑥𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ (𝑀‘{𝑥𝑂 ∣ ¬ 𝜑}) ≤ 0) → (𝑀‘{𝑥𝑂 ∣ ¬ 𝜑}) = 0)
2310, 12, 21, 22syl3anc 1371 . . . . 5 (((𝑀 ran measures ∧ {𝑥𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) ∧ (𝑀‘({𝑥𝑂 ∣ ¬ 𝜑} ∪ {𝑥𝑂 ∣ ¬ 𝜓})) = 0) → (𝑀‘{𝑥𝑂 ∣ ¬ 𝜑}) = 0)
24 simp3 1138 . . . . . . 7 ((𝑀 ran measures ∧ {𝑥𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) → {𝑥𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀)
2524adantr 481 . . . . . 6 (((𝑀 ran measures ∧ {𝑥𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) ∧ (𝑀‘({𝑥𝑂 ∣ ¬ 𝜑} ∪ {𝑥𝑂 ∣ ¬ 𝜓})) = 0) → {𝑥𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀)
26 ssun2 4133 . . . . . . . . . 10 {𝑥𝑂 ∣ ¬ 𝜓} ⊆ ({𝑥𝑂 ∣ ¬ 𝜑} ∪ {𝑥𝑂 ∣ ¬ 𝜓})
2726a1i 11 . . . . . . . . 9 ((𝑀 ran measures ∧ {𝑥𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) → {𝑥𝑂 ∣ ¬ 𝜓} ⊆ ({𝑥𝑂 ∣ ¬ 𝜑} ∪ {𝑥𝑂 ∣ ¬ 𝜓}))
289, 24, 15, 27measssd 32814 . . . . . . . 8 ((𝑀 ran measures ∧ {𝑥𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) → (𝑀‘{𝑥𝑂 ∣ ¬ 𝜓}) ≤ (𝑀‘({𝑥𝑂 ∣ ¬ 𝜑} ∪ {𝑥𝑂 ∣ ¬ 𝜓})))
2928adantr 481 . . . . . . 7 (((𝑀 ran measures ∧ {𝑥𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) ∧ (𝑀‘({𝑥𝑂 ∣ ¬ 𝜑} ∪ {𝑥𝑂 ∣ ¬ 𝜓})) = 0) → (𝑀‘{𝑥𝑂 ∣ ¬ 𝜓}) ≤ (𝑀‘({𝑥𝑂 ∣ ¬ 𝜑} ∪ {𝑥𝑂 ∣ ¬ 𝜓})))
3029, 20breqtrd 5131 . . . . . 6 (((𝑀 ran measures ∧ {𝑥𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) ∧ (𝑀‘({𝑥𝑂 ∣ ¬ 𝜑} ∪ {𝑥𝑂 ∣ ¬ 𝜓})) = 0) → (𝑀‘{𝑥𝑂 ∣ ¬ 𝜓}) ≤ 0)
31 measle0 32807 . . . . . 6 ((𝑀 ∈ (measures‘dom 𝑀) ∧ {𝑥𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀 ∧ (𝑀‘{𝑥𝑂 ∣ ¬ 𝜓}) ≤ 0) → (𝑀‘{𝑥𝑂 ∣ ¬ 𝜓}) = 0)
3210, 25, 30, 31syl3anc 1371 . . . . 5 (((𝑀 ran measures ∧ {𝑥𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) ∧ (𝑀‘({𝑥𝑂 ∣ ¬ 𝜑} ∪ {𝑥𝑂 ∣ ¬ 𝜓})) = 0) → (𝑀‘{𝑥𝑂 ∣ ¬ 𝜓}) = 0)
3323, 32jca 512 . . . 4 (((𝑀 ran measures ∧ {𝑥𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) ∧ (𝑀‘({𝑥𝑂 ∣ ¬ 𝜑} ∪ {𝑥𝑂 ∣ ¬ 𝜓})) = 0) → ((𝑀‘{𝑥𝑂 ∣ ¬ 𝜑}) = 0 ∧ (𝑀‘{𝑥𝑂 ∣ ¬ 𝜓}) = 0))
349adantr 481 . . . . 5 (((𝑀 ran measures ∧ {𝑥𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) ∧ ((𝑀‘{𝑥𝑂 ∣ ¬ 𝜑}) = 0 ∧ (𝑀‘{𝑥𝑂 ∣ ¬ 𝜓}) = 0)) → 𝑀 ∈ (measures‘dom 𝑀))
35 measbase 32796 . . . . . . 7 (𝑀 ∈ (measures‘dom 𝑀) → dom 𝑀 ran sigAlgebra)
3634, 35syl 17 . . . . . 6 (((𝑀 ran measures ∧ {𝑥𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) ∧ ((𝑀‘{𝑥𝑂 ∣ ¬ 𝜑}) = 0 ∧ (𝑀‘{𝑥𝑂 ∣ ¬ 𝜓}) = 0)) → dom 𝑀 ran sigAlgebra)
3711adantr 481 . . . . . 6 (((𝑀 ran measures ∧ {𝑥𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) ∧ ((𝑀‘{𝑥𝑂 ∣ ¬ 𝜑}) = 0 ∧ (𝑀‘{𝑥𝑂 ∣ ¬ 𝜓}) = 0)) → {𝑥𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀)
3824adantr 481 . . . . . 6 (((𝑀 ran measures ∧ {𝑥𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) ∧ ((𝑀‘{𝑥𝑂 ∣ ¬ 𝜑}) = 0 ∧ (𝑀‘{𝑥𝑂 ∣ ¬ 𝜓}) = 0)) → {𝑥𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀)
3936, 37, 38, 14syl3anc 1371 . . . . 5 (((𝑀 ran measures ∧ {𝑥𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) ∧ ((𝑀‘{𝑥𝑂 ∣ ¬ 𝜑}) = 0 ∧ (𝑀‘{𝑥𝑂 ∣ ¬ 𝜓}) = 0)) → ({𝑥𝑂 ∣ ¬ 𝜑} ∪ {𝑥𝑂 ∣ ¬ 𝜓}) ∈ dom 𝑀)
4034, 37, 38measunl 32815 . . . . . 6 (((𝑀 ran measures ∧ {𝑥𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) ∧ ((𝑀‘{𝑥𝑂 ∣ ¬ 𝜑}) = 0 ∧ (𝑀‘{𝑥𝑂 ∣ ¬ 𝜓}) = 0)) → (𝑀‘({𝑥𝑂 ∣ ¬ 𝜑} ∪ {𝑥𝑂 ∣ ¬ 𝜓})) ≤ ((𝑀‘{𝑥𝑂 ∣ ¬ 𝜑}) +𝑒 (𝑀‘{𝑥𝑂 ∣ ¬ 𝜓})))
41 simprl 769 . . . . . . . 8 (((𝑀 ran measures ∧ {𝑥𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) ∧ ((𝑀‘{𝑥𝑂 ∣ ¬ 𝜑}) = 0 ∧ (𝑀‘{𝑥𝑂 ∣ ¬ 𝜓}) = 0)) → (𝑀‘{𝑥𝑂 ∣ ¬ 𝜑}) = 0)
42 simprr 771 . . . . . . . 8 (((𝑀 ran measures ∧ {𝑥𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) ∧ ((𝑀‘{𝑥𝑂 ∣ ¬ 𝜑}) = 0 ∧ (𝑀‘{𝑥𝑂 ∣ ¬ 𝜓}) = 0)) → (𝑀‘{𝑥𝑂 ∣ ¬ 𝜓}) = 0)
4341, 42oveq12d 7375 . . . . . . 7 (((𝑀 ran measures ∧ {𝑥𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) ∧ ((𝑀‘{𝑥𝑂 ∣ ¬ 𝜑}) = 0 ∧ (𝑀‘{𝑥𝑂 ∣ ¬ 𝜓}) = 0)) → ((𝑀‘{𝑥𝑂 ∣ ¬ 𝜑}) +𝑒 (𝑀‘{𝑥𝑂 ∣ ¬ 𝜓})) = (0 +𝑒 0))
44 0xr 11202 . . . . . . . 8 0 ∈ ℝ*
45 xaddid1 13160 . . . . . . . 8 (0 ∈ ℝ* → (0 +𝑒 0) = 0)
4644, 45ax-mp 5 . . . . . . 7 (0 +𝑒 0) = 0
4743, 46eqtrdi 2792 . . . . . 6 (((𝑀 ran measures ∧ {𝑥𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) ∧ ((𝑀‘{𝑥𝑂 ∣ ¬ 𝜑}) = 0 ∧ (𝑀‘{𝑥𝑂 ∣ ¬ 𝜓}) = 0)) → ((𝑀‘{𝑥𝑂 ∣ ¬ 𝜑}) +𝑒 (𝑀‘{𝑥𝑂 ∣ ¬ 𝜓})) = 0)
4840, 47breqtrd 5131 . . . . 5 (((𝑀 ran measures ∧ {𝑥𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) ∧ ((𝑀‘{𝑥𝑂 ∣ ¬ 𝜑}) = 0 ∧ (𝑀‘{𝑥𝑂 ∣ ¬ 𝜓}) = 0)) → (𝑀‘({𝑥𝑂 ∣ ¬ 𝜑} ∪ {𝑥𝑂 ∣ ¬ 𝜓})) ≤ 0)
49 measle0 32807 . . . . 5 ((𝑀 ∈ (measures‘dom 𝑀) ∧ ({𝑥𝑂 ∣ ¬ 𝜑} ∪ {𝑥𝑂 ∣ ¬ 𝜓}) ∈ dom 𝑀 ∧ (𝑀‘({𝑥𝑂 ∣ ¬ 𝜑} ∪ {𝑥𝑂 ∣ ¬ 𝜓})) ≤ 0) → (𝑀‘({𝑥𝑂 ∣ ¬ 𝜑} ∪ {𝑥𝑂 ∣ ¬ 𝜓})) = 0)
5034, 39, 48, 49syl3anc 1371 . . . 4 (((𝑀 ran measures ∧ {𝑥𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) ∧ ((𝑀‘{𝑥𝑂 ∣ ¬ 𝜑}) = 0 ∧ (𝑀‘{𝑥𝑂 ∣ ¬ 𝜓}) = 0)) → (𝑀‘({𝑥𝑂 ∣ ¬ 𝜑} ∪ {𝑥𝑂 ∣ ¬ 𝜓})) = 0)
5133, 50impbida 799 . . 3 ((𝑀 ran measures ∧ {𝑥𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) → ((𝑀‘({𝑥𝑂 ∣ ¬ 𝜑} ∪ {𝑥𝑂 ∣ ¬ 𝜓})) = 0 ↔ ((𝑀‘{𝑥𝑂 ∣ ¬ 𝜑}) = 0 ∧ (𝑀‘{𝑥𝑂 ∣ ¬ 𝜓}) = 0)))
526, 51bitr3id 284 . 2 ((𝑀 ran measures ∧ {𝑥𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) → ((𝑀‘{𝑥𝑂 ∣ ¬ (𝜑𝜓)}) = 0 ↔ ((𝑀‘{𝑥𝑂 ∣ ¬ 𝜑}) = 0 ∧ (𝑀‘{𝑥𝑂 ∣ ¬ 𝜓}) = 0)))
53 aean.1 . . . 4 dom 𝑀 = 𝑂
5453braew 32841 . . 3 (𝑀 ran measures → ({𝑥𝑂 ∣ (𝜑𝜓)}a.e.𝑀 ↔ (𝑀‘{𝑥𝑂 ∣ ¬ (𝜑𝜓)}) = 0))
55543ad2ant1 1133 . 2 ((𝑀 ran measures ∧ {𝑥𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) → ({𝑥𝑂 ∣ (𝜑𝜓)}a.e.𝑀 ↔ (𝑀‘{𝑥𝑂 ∣ ¬ (𝜑𝜓)}) = 0))
5653braew 32841 . . . 4 (𝑀 ran measures → ({𝑥𝑂𝜑}a.e.𝑀 ↔ (𝑀‘{𝑥𝑂 ∣ ¬ 𝜑}) = 0))
5753braew 32841 . . . 4 (𝑀 ran measures → ({𝑥𝑂𝜓}a.e.𝑀 ↔ (𝑀‘{𝑥𝑂 ∣ ¬ 𝜓}) = 0))
5856, 57anbi12d 631 . . 3 (𝑀 ran measures → (({𝑥𝑂𝜑}a.e.𝑀 ∧ {𝑥𝑂𝜓}a.e.𝑀) ↔ ((𝑀‘{𝑥𝑂 ∣ ¬ 𝜑}) = 0 ∧ (𝑀‘{𝑥𝑂 ∣ ¬ 𝜓}) = 0)))
59583ad2ant1 1133 . 2 ((𝑀 ran measures ∧ {𝑥𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) → (({𝑥𝑂𝜑}a.e.𝑀 ∧ {𝑥𝑂𝜓}a.e.𝑀) ↔ ((𝑀‘{𝑥𝑂 ∣ ¬ 𝜑}) = 0 ∧ (𝑀‘{𝑥𝑂 ∣ ¬ 𝜓}) = 0)))
6052, 55, 593bitr4d 310 1 ((𝑀 ran measures ∧ {𝑥𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) → ({𝑥𝑂 ∣ (𝜑𝜓)}a.e.𝑀 ↔ ({𝑥𝑂𝜑}a.e.𝑀 ∧ {𝑥𝑂𝜓}a.e.𝑀)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  wo 845  w3a 1087   = wceq 1541  wcel 2106  {crab 3407  cun 3908  wss 3910   cuni 4865   class class class wbr 5105  dom cdm 5633  ran crn 5634  cfv 6496  (class class class)co 7357  0cc0 11051  *cxr 11188  cle 11190   +𝑒 cxad 13031  sigAlgebracsiga 32707  measurescmeas 32794  a.e.cae 32836
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-inf2 9577  ax-ac2 10399  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129  ax-addf 11130  ax-mulf 11131
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-iin 4957  df-disj 5071  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-of 7617  df-om 7803  df-1st 7921  df-2nd 7922  df-supp 8093  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-2o 8413  df-er 8648  df-map 8767  df-pm 8768  df-ixp 8836  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fsupp 9306  df-fi 9347  df-sup 9378  df-inf 9379  df-oi 9446  df-dju 9837  df-card 9875  df-acn 9878  df-ac 10052  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-3 12217  df-4 12218  df-5 12219  df-6 12220  df-7 12221  df-8 12222  df-9 12223  df-n0 12414  df-z 12500  df-dec 12619  df-uz 12764  df-q 12874  df-rp 12916  df-xneg 13033  df-xadd 13034  df-xmul 13035  df-ioo 13268  df-ioc 13269  df-ico 13270  df-icc 13271  df-fz 13425  df-fzo 13568  df-fl 13697  df-mod 13775  df-seq 13907  df-exp 13968  df-fac 14174  df-bc 14203  df-hash 14231  df-shft 14952  df-cj 14984  df-re 14985  df-im 14986  df-sqrt 15120  df-abs 15121  df-limsup 15353  df-clim 15370  df-rlim 15371  df-sum 15571  df-ef 15950  df-sin 15952  df-cos 15953  df-pi 15955  df-struct 17019  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-plusg 17146  df-mulr 17147  df-starv 17148  df-sca 17149  df-vsca 17150  df-ip 17151  df-tset 17152  df-ple 17153  df-ds 17155  df-unif 17156  df-hom 17157  df-cco 17158  df-rest 17304  df-topn 17305  df-0g 17323  df-gsum 17324  df-topgen 17325  df-pt 17326  df-prds 17329  df-ordt 17383  df-xrs 17384  df-qtop 17389  df-imas 17390  df-xps 17392  df-mre 17466  df-mrc 17467  df-acs 17469  df-ps 18455  df-tsr 18456  df-plusf 18496  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-mhm 18601  df-submnd 18602  df-grp 18751  df-minusg 18752  df-sbg 18753  df-mulg 18873  df-subg 18925  df-cntz 19097  df-cmn 19564  df-abl 19565  df-mgp 19897  df-ur 19914  df-ring 19966  df-cring 19967  df-subrg 20220  df-abv 20276  df-lmod 20324  df-scaf 20325  df-sra 20633  df-rgmod 20634  df-psmet 20788  df-xmet 20789  df-met 20790  df-bl 20791  df-mopn 20792  df-fbas 20793  df-fg 20794  df-cnfld 20797  df-top 22243  df-topon 22260  df-topsp 22282  df-bases 22296  df-cld 22370  df-ntr 22371  df-cls 22372  df-nei 22449  df-lp 22487  df-perf 22488  df-cn 22578  df-cnp 22579  df-haus 22666  df-tx 22913  df-hmeo 23106  df-fil 23197  df-fm 23289  df-flim 23290  df-flf 23291  df-tmd 23423  df-tgp 23424  df-tsms 23478  df-trg 23511  df-xms 23673  df-ms 23674  df-tms 23675  df-nm 23938  df-ngp 23939  df-nrg 23941  df-nlm 23942  df-ii 24240  df-cncf 24241  df-limc 25230  df-dv 25231  df-log 25912  df-esum 32627  df-siga 32708  df-meas 32795  df-ae 32838
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator