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Theorem aean 31679
 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 4229 . . . . . 6 ({𝑥𝑂 ∣ ¬ 𝜑} ∪ {𝑥𝑂 ∣ ¬ 𝜓}) = {𝑥𝑂 ∣ (¬ 𝜑 ∨ ¬ 𝜓)}
2 ianor 979 . . . . . . 7 (¬ (𝜑𝜓) ↔ (¬ 𝜑 ∨ ¬ 𝜓))
32rabbii 3421 . . . . . 6 {𝑥𝑂 ∣ ¬ (𝜑𝜓)} = {𝑥𝑂 ∣ (¬ 𝜑 ∨ ¬ 𝜓)}
41, 3eqtr4i 2824 . . . . 5 ({𝑥𝑂 ∣ ¬ 𝜑} ∪ {𝑥𝑂 ∣ ¬ 𝜓}) = {𝑥𝑂 ∣ ¬ (𝜑𝜓)}
54fveq2i 6658 . . . 4 (𝑀‘({𝑥𝑂 ∣ ¬ 𝜑} ∪ {𝑥𝑂 ∣ ¬ 𝜓})) = (𝑀‘{𝑥𝑂 ∣ ¬ (𝜑𝜓)})
65eqeq1i 2803 . . 3 ((𝑀‘({𝑥𝑂 ∣ ¬ 𝜑} ∪ {𝑥𝑂 ∣ ¬ 𝜓})) = 0 ↔ (𝑀‘{𝑥𝑂 ∣ ¬ (𝜑𝜓)}) = 0)
7 measbasedom 31637 . . . . . . . . 9 (𝑀 ran measures ↔ 𝑀 ∈ (measures‘dom 𝑀))
87biimpi 219 . . . . . . . 8 (𝑀 ran measures → 𝑀 ∈ (measures‘dom 𝑀))
983ad2ant1 1130 . . . . . . 7 ((𝑀 ran measures ∧ {𝑥𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) → 𝑀 ∈ (measures‘dom 𝑀))
109adantr 484 . . . . . 6 (((𝑀 ran measures ∧ {𝑥𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) ∧ (𝑀‘({𝑥𝑂 ∣ ¬ 𝜑} ∪ {𝑥𝑂 ∣ ¬ 𝜓})) = 0) → 𝑀 ∈ (measures‘dom 𝑀))
11 simp2 1134 . . . . . . 7 ((𝑀 ran measures ∧ {𝑥𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) → {𝑥𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀)
1211adantr 484 . . . . . 6 (((𝑀 ran measures ∧ {𝑥𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) ∧ (𝑀‘({𝑥𝑂 ∣ ¬ 𝜑} ∪ {𝑥𝑂 ∣ ¬ 𝜓})) = 0) → {𝑥𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀)
13 dmmeas 31636 . . . . . . . . . 10 (𝑀 ran measures → dom 𝑀 ran sigAlgebra)
14 unelsiga 31569 . . . . . . . . . 10 ((dom 𝑀 ran sigAlgebra ∧ {𝑥𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) → ({𝑥𝑂 ∣ ¬ 𝜑} ∪ {𝑥𝑂 ∣ ¬ 𝜓}) ∈ dom 𝑀)
1513, 14syl3an1 1160 . . . . . . . . 9 ((𝑀 ran measures ∧ {𝑥𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) → ({𝑥𝑂 ∣ ¬ 𝜑} ∪ {𝑥𝑂 ∣ ¬ 𝜓}) ∈ dom 𝑀)
16 ssun1 4102 . . . . . . . . . 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 31650 . . . . . . . 8 ((𝑀 ran measures ∧ {𝑥𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) → (𝑀‘{𝑥𝑂 ∣ ¬ 𝜑}) ≤ (𝑀‘({𝑥𝑂 ∣ ¬ 𝜑} ∪ {𝑥𝑂 ∣ ¬ 𝜓})))
1918adantr 484 . . . . . . 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 488 . . . . . . 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 5060 . . . . . 6 (((𝑀 ran measures ∧ {𝑥𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) ∧ (𝑀‘({𝑥𝑂 ∣ ¬ 𝜑} ∪ {𝑥𝑂 ∣ ¬ 𝜓})) = 0) → (𝑀‘{𝑥𝑂 ∣ ¬ 𝜑}) ≤ 0)
22 measle0 31643 . . . . . 6 ((𝑀 ∈ (measures‘dom 𝑀) ∧ {𝑥𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ (𝑀‘{𝑥𝑂 ∣ ¬ 𝜑}) ≤ 0) → (𝑀‘{𝑥𝑂 ∣ ¬ 𝜑}) = 0)
2310, 12, 21, 22syl3anc 1368 . . . . 5 (((𝑀 ran measures ∧ {𝑥𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) ∧ (𝑀‘({𝑥𝑂 ∣ ¬ 𝜑} ∪ {𝑥𝑂 ∣ ¬ 𝜓})) = 0) → (𝑀‘{𝑥𝑂 ∣ ¬ 𝜑}) = 0)
24 simp3 1135 . . . . . . 7 ((𝑀 ran measures ∧ {𝑥𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) → {𝑥𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀)
2524adantr 484 . . . . . 6 (((𝑀 ran measures ∧ {𝑥𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) ∧ (𝑀‘({𝑥𝑂 ∣ ¬ 𝜑} ∪ {𝑥𝑂 ∣ ¬ 𝜓})) = 0) → {𝑥𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀)
26 ssun2 4103 . . . . . . . . . 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 31650 . . . . . . . 8 ((𝑀 ran measures ∧ {𝑥𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) → (𝑀‘{𝑥𝑂 ∣ ¬ 𝜓}) ≤ (𝑀‘({𝑥𝑂 ∣ ¬ 𝜑} ∪ {𝑥𝑂 ∣ ¬ 𝜓})))
2928adantr 484 . . . . . . 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 5060 . . . . . 6 (((𝑀 ran measures ∧ {𝑥𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) ∧ (𝑀‘({𝑥𝑂 ∣ ¬ 𝜑} ∪ {𝑥𝑂 ∣ ¬ 𝜓})) = 0) → (𝑀‘{𝑥𝑂 ∣ ¬ 𝜓}) ≤ 0)
31 measle0 31643 . . . . . 6 ((𝑀 ∈ (measures‘dom 𝑀) ∧ {𝑥𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀 ∧ (𝑀‘{𝑥𝑂 ∣ ¬ 𝜓}) ≤ 0) → (𝑀‘{𝑥𝑂 ∣ ¬ 𝜓}) = 0)
3210, 25, 30, 31syl3anc 1368 . . . . 5 (((𝑀 ran measures ∧ {𝑥𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) ∧ (𝑀‘({𝑥𝑂 ∣ ¬ 𝜑} ∪ {𝑥𝑂 ∣ ¬ 𝜓})) = 0) → (𝑀‘{𝑥𝑂 ∣ ¬ 𝜓}) = 0)
3323, 32jca 515 . . . 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 484 . . . . 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 31632 . . . . . . 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 484 . . . . . 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 484 . . . . . 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 1368 . . . . 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 31651 . . . . . 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 770 . . . . . . . 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 772 . . . . . . . 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 7163 . . . . . . 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 10695 . . . . . . . 8 0 ∈ ℝ*
45 xaddid1 12642 . . . . . . . 8 (0 ∈ ℝ* → (0 +𝑒 0) = 0)
4644, 45ax-mp 5 . . . . . . 7 (0 +𝑒 0) = 0
4743, 46eqtrdi 2849 . . . . . 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 5060 . . . . 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 31643 . . . . 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 1368 . . . 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 800 . . 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 288 . 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 31677 . . 3 (𝑀 ran measures → ({𝑥𝑂 ∣ (𝜑𝜓)}a.e.𝑀 ↔ (𝑀‘{𝑥𝑂 ∣ ¬ (𝜑𝜓)}) = 0))
55543ad2ant1 1130 . 2 ((𝑀 ran measures ∧ {𝑥𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) → ({𝑥𝑂 ∣ (𝜑𝜓)}a.e.𝑀 ↔ (𝑀‘{𝑥𝑂 ∣ ¬ (𝜑𝜓)}) = 0))
5653braew 31677 . . . 4 (𝑀 ran measures → ({𝑥𝑂𝜑}a.e.𝑀 ↔ (𝑀‘{𝑥𝑂 ∣ ¬ 𝜑}) = 0))
5753braew 31677 . . . 4 (𝑀 ran measures → ({𝑥𝑂𝜓}a.e.𝑀 ↔ (𝑀‘{𝑥𝑂 ∣ ¬ 𝜓}) = 0))
5856, 57anbi12d 633 . . 3 (𝑀 ran measures → (({𝑥𝑂𝜑}a.e.𝑀 ∧ {𝑥𝑂𝜓}a.e.𝑀) ↔ ((𝑀‘{𝑥𝑂 ∣ ¬ 𝜑}) = 0 ∧ (𝑀‘{𝑥𝑂 ∣ ¬ 𝜓}) = 0)))
59583ad2ant1 1130 . 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 314 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 209   ∧ wa 399   ∨ wo 844   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  {crab 3110   ∪ cun 3881   ⊆ wss 3883  ∪ cuni 4804   class class class wbr 5034  dom cdm 5523  ran crn 5524  ‘cfv 6332  (class class class)co 7145  0cc0 10544  ℝ*cxr 10681   ≤ cle 10683   +𝑒 cxad 12513  sigAlgebracsiga 31543  measurescmeas 31630  a.e.cae 31672 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5158  ax-sep 5171  ax-nul 5178  ax-pow 5235  ax-pr 5299  ax-un 7454  ax-inf2 9106  ax-ac2 9892  ax-cnex 10600  ax-resscn 10601  ax-1cn 10602  ax-icn 10603  ax-addcl 10604  ax-addrcl 10605  ax-mulcl 10606  ax-mulrcl 10607  ax-mulcom 10608  ax-addass 10609  ax-mulass 10610  ax-distr 10611  ax-i2m1 10612  ax-1ne0 10613  ax-1rid 10614  ax-rnegex 10615  ax-rrecex 10616  ax-cnre 10617  ax-pre-lttri 10618  ax-pre-lttrn 10619  ax-pre-ltadd 10620  ax-pre-mulgt0 10621  ax-pre-sup 10622  ax-addf 10623  ax-mulf 10624 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3444  df-sbc 3723  df-csb 3831  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4805  df-int 4843  df-iun 4887  df-iin 4888  df-disj 5000  df-br 5035  df-opab 5097  df-mpt 5115  df-tr 5141  df-id 5429  df-eprel 5434  df-po 5442  df-so 5443  df-fr 5482  df-se 5483  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6123  df-ord 6169  df-on 6170  df-lim 6171  df-suc 6172  df-iota 6291  df-fun 6334  df-fn 6335  df-f 6336  df-f1 6337  df-fo 6338  df-f1o 6339  df-fv 6340  df-isom 6341  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-of 7400  df-om 7574  df-1st 7684  df-2nd 7685  df-supp 7827  df-wrecs 7948  df-recs 8009  df-rdg 8047  df-1o 8103  df-2o 8104  df-oadd 8107  df-er 8290  df-map 8409  df-pm 8410  df-ixp 8463  df-en 8511  df-dom 8512  df-sdom 8513  df-fin 8514  df-fsupp 8836  df-fi 8877  df-sup 8908  df-inf 8909  df-oi 8976  df-dju 9332  df-card 9370  df-acn 9373  df-ac 9545  df-pnf 10684  df-mnf 10685  df-xr 10686  df-ltxr 10687  df-le 10688  df-sub 10879  df-neg 10880  df-div 11305  df-nn 11644  df-2 11706  df-3 11707  df-4 11708  df-5 11709  df-6 11710  df-7 11711  df-8 11712  df-9 11713  df-n0 11904  df-z 11990  df-dec 12107  df-uz 12252  df-q 12357  df-rp 12398  df-xneg 12515  df-xadd 12516  df-xmul 12517  df-ioo 12750  df-ioc 12751  df-ico 12752  df-icc 12753  df-fz 12906  df-fzo 13049  df-fl 13177  df-mod 13253  df-seq 13385  df-exp 13446  df-fac 13650  df-bc 13679  df-hash 13707  df-shft 14438  df-cj 14470  df-re 14471  df-im 14472  df-sqrt 14606  df-abs 14607  df-limsup 14840  df-clim 14857  df-rlim 14858  df-sum 15055  df-ef 15433  df-sin 15435  df-cos 15436  df-pi 15438  df-struct 16497  df-ndx 16498  df-slot 16499  df-base 16501  df-sets 16502  df-ress 16503  df-plusg 16590  df-mulr 16591  df-starv 16592  df-sca 16593  df-vsca 16594  df-ip 16595  df-tset 16596  df-ple 16597  df-ds 16599  df-unif 16600  df-hom 16601  df-cco 16602  df-rest 16708  df-topn 16709  df-0g 16727  df-gsum 16728  df-topgen 16729  df-pt 16730  df-prds 16733  df-ordt 16786  df-xrs 16787  df-qtop 16792  df-imas 16793  df-xps 16795  df-mre 16869  df-mrc 16870  df-acs 16872  df-ps 17822  df-tsr 17823  df-plusf 17863  df-mgm 17864  df-sgrp 17913  df-mnd 17924  df-mhm 17968  df-submnd 17969  df-grp 18118  df-minusg 18119  df-sbg 18120  df-mulg 18238  df-subg 18289  df-cntz 18460  df-cmn 18921  df-abl 18922  df-mgp 19254  df-ur 19266  df-ring 19313  df-cring 19314  df-subrg 19547  df-abv 19602  df-lmod 19650  df-scaf 19651  df-sra 19958  df-rgmod 19959  df-psmet 20104  df-xmet 20105  df-met 20106  df-bl 20107  df-mopn 20108  df-fbas 20109  df-fg 20110  df-cnfld 20113  df-top 21540  df-topon 21557  df-topsp 21579  df-bases 21592  df-cld 21665  df-ntr 21666  df-cls 21667  df-nei 21744  df-lp 21782  df-perf 21783  df-cn 21873  df-cnp 21874  df-haus 21961  df-tx 22208  df-hmeo 22401  df-fil 22492  df-fm 22584  df-flim 22585  df-flf 22586  df-tmd 22718  df-tgp 22719  df-tsms 22773  df-trg 22806  df-xms 22968  df-ms 22969  df-tms 22970  df-nm 23230  df-ngp 23231  df-nrg 23233  df-nlm 23234  df-ii 23523  df-cncf 23524  df-limc 24510  df-dv 24511  df-log 25192  df-esum 31463  df-siga 31544  df-meas 31631  df-ae 31674 This theorem is referenced by: (None)
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