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Theorem aean 32318
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 4249 . . . . . 6 ({𝑥𝑂 ∣ ¬ 𝜑} ∪ {𝑥𝑂 ∣ ¬ 𝜓}) = {𝑥𝑂 ∣ (¬ 𝜑 ∨ ¬ 𝜓)}
2 ianor 979 . . . . . . 7 (¬ (𝜑𝜓) ↔ (¬ 𝜑 ∨ ¬ 𝜓))
32rabbii 3410 . . . . . 6 {𝑥𝑂 ∣ ¬ (𝜑𝜓)} = {𝑥𝑂 ∣ (¬ 𝜑 ∨ ¬ 𝜓)}
41, 3eqtr4i 2768 . . . . 5 ({𝑥𝑂 ∣ ¬ 𝜑} ∪ {𝑥𝑂 ∣ ¬ 𝜓}) = {𝑥𝑂 ∣ ¬ (𝜑𝜓)}
54fveq2i 6812 . . . 4 (𝑀‘({𝑥𝑂 ∣ ¬ 𝜑} ∪ {𝑥𝑂 ∣ ¬ 𝜓})) = (𝑀‘{𝑥𝑂 ∣ ¬ (𝜑𝜓)})
65eqeq1i 2742 . . 3 ((𝑀‘({𝑥𝑂 ∣ ¬ 𝜑} ∪ {𝑥𝑂 ∣ ¬ 𝜓})) = 0 ↔ (𝑀‘{𝑥𝑂 ∣ ¬ (𝜑𝜓)}) = 0)
7 measbasedom 32276 . . . . . . . . 9 (𝑀 ran measures ↔ 𝑀 ∈ (measures‘dom 𝑀))
87biimpi 215 . . . . . . . 8 (𝑀 ran measures → 𝑀 ∈ (measures‘dom 𝑀))
983ad2ant1 1132 . . . . . . 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 1136 . . . . . . 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 32275 . . . . . . . . . 10 (𝑀 ran measures → dom 𝑀 ran sigAlgebra)
14 unelsiga 32208 . . . . . . . . . 10 ((dom 𝑀 ran sigAlgebra ∧ {𝑥𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) → ({𝑥𝑂 ∣ ¬ 𝜑} ∪ {𝑥𝑂 ∣ ¬ 𝜓}) ∈ dom 𝑀)
1513, 14syl3an1 1162 . . . . . . . . 9 ((𝑀 ran measures ∧ {𝑥𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) → ({𝑥𝑂 ∣ ¬ 𝜑} ∪ {𝑥𝑂 ∣ ¬ 𝜓}) ∈ dom 𝑀)
16 ssun1 4116 . . . . . . . . . 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 32289 . . . . . . . 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 5111 . . . . . 6 (((𝑀 ran measures ∧ {𝑥𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) ∧ (𝑀‘({𝑥𝑂 ∣ ¬ 𝜑} ∪ {𝑥𝑂 ∣ ¬ 𝜓})) = 0) → (𝑀‘{𝑥𝑂 ∣ ¬ 𝜑}) ≤ 0)
22 measle0 32282 . . . . . 6 ((𝑀 ∈ (measures‘dom 𝑀) ∧ {𝑥𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ (𝑀‘{𝑥𝑂 ∣ ¬ 𝜑}) ≤ 0) → (𝑀‘{𝑥𝑂 ∣ ¬ 𝜑}) = 0)
2310, 12, 21, 22syl3anc 1370 . . . . 5 (((𝑀 ran measures ∧ {𝑥𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) ∧ (𝑀‘({𝑥𝑂 ∣ ¬ 𝜑} ∪ {𝑥𝑂 ∣ ¬ 𝜓})) = 0) → (𝑀‘{𝑥𝑂 ∣ ¬ 𝜑}) = 0)
24 simp3 1137 . . . . . . 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 4117 . . . . . . . . . 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 32289 . . . . . . . 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 5111 . . . . . 6 (((𝑀 ran measures ∧ {𝑥𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) ∧ (𝑀‘({𝑥𝑂 ∣ ¬ 𝜑} ∪ {𝑥𝑂 ∣ ¬ 𝜓})) = 0) → (𝑀‘{𝑥𝑂 ∣ ¬ 𝜓}) ≤ 0)
31 measle0 32282 . . . . . 6 ((𝑀 ∈ (measures‘dom 𝑀) ∧ {𝑥𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀 ∧ (𝑀‘{𝑥𝑂 ∣ ¬ 𝜓}) ≤ 0) → (𝑀‘{𝑥𝑂 ∣ ¬ 𝜓}) = 0)
3210, 25, 30, 31syl3anc 1370 . . . . 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 32271 . . . . . . 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 1370 . . . . 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 32290 . . . . . 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 768 . . . . . . . 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 770 . . . . . . . 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 7331 . . . . . . 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 11092 . . . . . . . 8 0 ∈ ℝ*
45 xaddid1 13045 . . . . . . . 8 (0 ∈ ℝ* → (0 +𝑒 0) = 0)
4644, 45ax-mp 5 . . . . . . 7 (0 +𝑒 0) = 0
4743, 46eqtrdi 2793 . . . . . 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 5111 . . . . 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 32282 . . . . 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 1370 . . . 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 798 . . 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 32316 . . 3 (𝑀 ran measures → ({𝑥𝑂 ∣ (𝜑𝜓)}a.e.𝑀 ↔ (𝑀‘{𝑥𝑂 ∣ ¬ (𝜑𝜓)}) = 0))
55543ad2ant1 1132 . 2 ((𝑀 ran measures ∧ {𝑥𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) → ({𝑥𝑂 ∣ (𝜑𝜓)}a.e.𝑀 ↔ (𝑀‘{𝑥𝑂 ∣ ¬ (𝜑𝜓)}) = 0))
5653braew 32316 . . . 4 (𝑀 ran measures → ({𝑥𝑂𝜑}a.e.𝑀 ↔ (𝑀‘{𝑥𝑂 ∣ ¬ 𝜑}) = 0))
5753braew 32316 . . . 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 1132 . 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 844  w3a 1086   = wceq 1540  wcel 2105  {crab 3404  cun 3894  wss 3896   cuni 4848   class class class wbr 5085  dom cdm 5605  ran crn 5606  cfv 6463  (class class class)co 7313  0cc0 10941  *cxr 11078  cle 11080   +𝑒 cxad 12916  sigAlgebracsiga 32182  measurescmeas 32269  a.e.cae 32311
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2708  ax-rep 5222  ax-sep 5236  ax-nul 5243  ax-pow 5301  ax-pr 5365  ax-un 7626  ax-inf2 9467  ax-ac2 10289  ax-cnex 10997  ax-resscn 10998  ax-1cn 10999  ax-icn 11000  ax-addcl 11001  ax-addrcl 11002  ax-mulcl 11003  ax-mulrcl 11004  ax-mulcom 11005  ax-addass 11006  ax-mulass 11007  ax-distr 11008  ax-i2m1 11009  ax-1ne0 11010  ax-1rid 11011  ax-rnegex 11012  ax-rrecex 11013  ax-cnre 11014  ax-pre-lttri 11015  ax-pre-lttrn 11016  ax-pre-ltadd 11017  ax-pre-mulgt0 11018  ax-pre-sup 11019  ax-addf 11020  ax-mulf 11021
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3350  df-reu 3351  df-rab 3405  df-v 3443  df-sbc 3726  df-csb 3842  df-dif 3899  df-un 3901  df-in 3903  df-ss 3913  df-pss 3915  df-nul 4267  df-if 4470  df-pw 4545  df-sn 4570  df-pr 4572  df-tp 4574  df-op 4576  df-uni 4849  df-int 4891  df-iun 4937  df-iin 4938  df-disj 5051  df-br 5086  df-opab 5148  df-mpt 5169  df-tr 5203  df-id 5505  df-eprel 5511  df-po 5519  df-so 5520  df-fr 5560  df-se 5561  df-we 5562  df-xp 5611  df-rel 5612  df-cnv 5613  df-co 5614  df-dm 5615  df-rn 5616  df-res 5617  df-ima 5618  df-pred 6222  df-ord 6289  df-on 6290  df-lim 6291  df-suc 6292  df-iota 6415  df-fun 6465  df-fn 6466  df-f 6467  df-f1 6468  df-fo 6469  df-f1o 6470  df-fv 6471  df-isom 6472  df-riota 7270  df-ov 7316  df-oprab 7317  df-mpo 7318  df-of 7571  df-om 7756  df-1st 7874  df-2nd 7875  df-supp 8023  df-frecs 8142  df-wrecs 8173  df-recs 8247  df-rdg 8286  df-1o 8342  df-2o 8343  df-er 8544  df-map 8663  df-pm 8664  df-ixp 8732  df-en 8780  df-dom 8781  df-sdom 8782  df-fin 8783  df-fsupp 9197  df-fi 9238  df-sup 9269  df-inf 9270  df-oi 9337  df-dju 9727  df-card 9765  df-acn 9768  df-ac 9942  df-pnf 11081  df-mnf 11082  df-xr 11083  df-ltxr 11084  df-le 11085  df-sub 11277  df-neg 11278  df-div 11703  df-nn 12044  df-2 12106  df-3 12107  df-4 12108  df-5 12109  df-6 12110  df-7 12111  df-8 12112  df-9 12113  df-n0 12304  df-z 12390  df-dec 12508  df-uz 12653  df-q 12759  df-rp 12801  df-xneg 12918  df-xadd 12919  df-xmul 12920  df-ioo 13153  df-ioc 13154  df-ico 13155  df-icc 13156  df-fz 13310  df-fzo 13453  df-fl 13582  df-mod 13660  df-seq 13792  df-exp 13853  df-fac 14058  df-bc 14087  df-hash 14115  df-shft 14847  df-cj 14879  df-re 14880  df-im 14881  df-sqrt 15015  df-abs 15016  df-limsup 15249  df-clim 15266  df-rlim 15267  df-sum 15467  df-ef 15846  df-sin 15848  df-cos 15849  df-pi 15851  df-struct 16915  df-sets 16932  df-slot 16950  df-ndx 16962  df-base 16980  df-ress 17009  df-plusg 17042  df-mulr 17043  df-starv 17044  df-sca 17045  df-vsca 17046  df-ip 17047  df-tset 17048  df-ple 17049  df-ds 17051  df-unif 17052  df-hom 17053  df-cco 17054  df-rest 17200  df-topn 17201  df-0g 17219  df-gsum 17220  df-topgen 17221  df-pt 17222  df-prds 17225  df-ordt 17279  df-xrs 17280  df-qtop 17285  df-imas 17286  df-xps 17288  df-mre 17362  df-mrc 17363  df-acs 17365  df-ps 18351  df-tsr 18352  df-plusf 18392  df-mgm 18393  df-sgrp 18442  df-mnd 18453  df-mhm 18497  df-submnd 18498  df-grp 18647  df-minusg 18648  df-sbg 18649  df-mulg 18768  df-subg 18819  df-cntz 18990  df-cmn 19455  df-abl 19456  df-mgp 19788  df-ur 19805  df-ring 19852  df-cring 19853  df-subrg 20093  df-abv 20148  df-lmod 20196  df-scaf 20197  df-sra 20505  df-rgmod 20506  df-psmet 20660  df-xmet 20661  df-met 20662  df-bl 20663  df-mopn 20664  df-fbas 20665  df-fg 20666  df-cnfld 20669  df-top 22114  df-topon 22131  df-topsp 22153  df-bases 22167  df-cld 22241  df-ntr 22242  df-cls 22243  df-nei 22320  df-lp 22358  df-perf 22359  df-cn 22449  df-cnp 22450  df-haus 22537  df-tx 22784  df-hmeo 22977  df-fil 23068  df-fm 23160  df-flim 23161  df-flf 23162  df-tmd 23294  df-tgp 23295  df-tsms 23349  df-trg 23382  df-xms 23544  df-ms 23545  df-tms 23546  df-nm 23809  df-ngp 23810  df-nrg 23812  df-nlm 23813  df-ii 24111  df-cncf 24112  df-limc 25101  df-dv 25102  df-log 25783  df-esum 32102  df-siga 32183  df-meas 32270  df-ae 32313
This theorem is referenced by: (None)
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