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Theorem aean 30637
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 4041 . . . . . 6 ({𝑥𝑂 ∣ ¬ 𝜑} ∪ {𝑥𝑂 ∣ ¬ 𝜓}) = {𝑥𝑂 ∣ (¬ 𝜑 ∨ ¬ 𝜓)}
2 ianor 510 . . . . . . 7 (¬ (𝜑𝜓) ↔ (¬ 𝜑 ∨ ¬ 𝜓))
32rabbii 3325 . . . . . 6 {𝑥𝑂 ∣ ¬ (𝜑𝜓)} = {𝑥𝑂 ∣ (¬ 𝜑 ∨ ¬ 𝜓)}
41, 3eqtr4i 2785 . . . . 5 ({𝑥𝑂 ∣ ¬ 𝜑} ∪ {𝑥𝑂 ∣ ¬ 𝜓}) = {𝑥𝑂 ∣ ¬ (𝜑𝜓)}
54fveq2i 6356 . . . 4 (𝑀‘({𝑥𝑂 ∣ ¬ 𝜑} ∪ {𝑥𝑂 ∣ ¬ 𝜓})) = (𝑀‘{𝑥𝑂 ∣ ¬ (𝜑𝜓)})
65eqeq1i 2765 . . 3 ((𝑀‘({𝑥𝑂 ∣ ¬ 𝜑} ∪ {𝑥𝑂 ∣ ¬ 𝜓})) = 0 ↔ (𝑀‘{𝑥𝑂 ∣ ¬ (𝜑𝜓)}) = 0)
7 measbasedom 30595 . . . . . . . . 9 (𝑀 ran measures ↔ 𝑀 ∈ (measures‘dom 𝑀))
87biimpi 206 . . . . . . . 8 (𝑀 ran measures → 𝑀 ∈ (measures‘dom 𝑀))
983ad2ant1 1128 . . . . . . 7 ((𝑀 ran measures ∧ {𝑥𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) → 𝑀 ∈ (measures‘dom 𝑀))
109adantr 472 . . . . . 6 (((𝑀 ran measures ∧ {𝑥𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) ∧ (𝑀‘({𝑥𝑂 ∣ ¬ 𝜑} ∪ {𝑥𝑂 ∣ ¬ 𝜓})) = 0) → 𝑀 ∈ (measures‘dom 𝑀))
11 simp2 1132 . . . . . . 7 ((𝑀 ran measures ∧ {𝑥𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) → {𝑥𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀)
1211adantr 472 . . . . . 6 (((𝑀 ran measures ∧ {𝑥𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) ∧ (𝑀‘({𝑥𝑂 ∣ ¬ 𝜑} ∪ {𝑥𝑂 ∣ ¬ 𝜓})) = 0) → {𝑥𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀)
13 dmmeas 30594 . . . . . . . . . 10 (𝑀 ran measures → dom 𝑀 ran sigAlgebra)
14 unelsiga 30527 . . . . . . . . . 10 ((dom 𝑀 ran sigAlgebra ∧ {𝑥𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) → ({𝑥𝑂 ∣ ¬ 𝜑} ∪ {𝑥𝑂 ∣ ¬ 𝜓}) ∈ dom 𝑀)
1513, 14syl3an1 1167 . . . . . . . . 9 ((𝑀 ran measures ∧ {𝑥𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) → ({𝑥𝑂 ∣ ¬ 𝜑} ∪ {𝑥𝑂 ∣ ¬ 𝜓}) ∈ dom 𝑀)
16 ssun1 3919 . . . . . . . . . 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 30608 . . . . . . . 8 ((𝑀 ran measures ∧ {𝑥𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) → (𝑀‘{𝑥𝑂 ∣ ¬ 𝜑}) ≤ (𝑀‘({𝑥𝑂 ∣ ¬ 𝜑} ∪ {𝑥𝑂 ∣ ¬ 𝜓})))
1918adantr 472 . . . . . . 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 479 . . . . . . 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 4830 . . . . . 6 (((𝑀 ran measures ∧ {𝑥𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) ∧ (𝑀‘({𝑥𝑂 ∣ ¬ 𝜑} ∪ {𝑥𝑂 ∣ ¬ 𝜓})) = 0) → (𝑀‘{𝑥𝑂 ∣ ¬ 𝜑}) ≤ 0)
22 measle0 30601 . . . . . 6 ((𝑀 ∈ (measures‘dom 𝑀) ∧ {𝑥𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ (𝑀‘{𝑥𝑂 ∣ ¬ 𝜑}) ≤ 0) → (𝑀‘{𝑥𝑂 ∣ ¬ 𝜑}) = 0)
2310, 12, 21, 22syl3anc 1477 . . . . 5 (((𝑀 ran measures ∧ {𝑥𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) ∧ (𝑀‘({𝑥𝑂 ∣ ¬ 𝜑} ∪ {𝑥𝑂 ∣ ¬ 𝜓})) = 0) → (𝑀‘{𝑥𝑂 ∣ ¬ 𝜑}) = 0)
24 simp3 1133 . . . . . . 7 ((𝑀 ran measures ∧ {𝑥𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) → {𝑥𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀)
2524adantr 472 . . . . . 6 (((𝑀 ran measures ∧ {𝑥𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) ∧ (𝑀‘({𝑥𝑂 ∣ ¬ 𝜑} ∪ {𝑥𝑂 ∣ ¬ 𝜓})) = 0) → {𝑥𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀)
26 ssun2 3920 . . . . . . . . . 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 30608 . . . . . . . 8 ((𝑀 ran measures ∧ {𝑥𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) → (𝑀‘{𝑥𝑂 ∣ ¬ 𝜓}) ≤ (𝑀‘({𝑥𝑂 ∣ ¬ 𝜑} ∪ {𝑥𝑂 ∣ ¬ 𝜓})))
2928adantr 472 . . . . . . 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 4830 . . . . . 6 (((𝑀 ran measures ∧ {𝑥𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) ∧ (𝑀‘({𝑥𝑂 ∣ ¬ 𝜑} ∪ {𝑥𝑂 ∣ ¬ 𝜓})) = 0) → (𝑀‘{𝑥𝑂 ∣ ¬ 𝜓}) ≤ 0)
31 measle0 30601 . . . . . 6 ((𝑀 ∈ (measures‘dom 𝑀) ∧ {𝑥𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀 ∧ (𝑀‘{𝑥𝑂 ∣ ¬ 𝜓}) ≤ 0) → (𝑀‘{𝑥𝑂 ∣ ¬ 𝜓}) = 0)
3210, 25, 30, 31syl3anc 1477 . . . . 5 (((𝑀 ran measures ∧ {𝑥𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) ∧ (𝑀‘({𝑥𝑂 ∣ ¬ 𝜑} ∪ {𝑥𝑂 ∣ ¬ 𝜓})) = 0) → (𝑀‘{𝑥𝑂 ∣ ¬ 𝜓}) = 0)
3323, 32jca 555 . . . 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 472 . . . . 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 30590 . . . . . . 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 472 . . . . . 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 472 . . . . . 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 1477 . . . . 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 30609 . . . . . 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 811 . . . . . . . 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 813 . . . . . . . 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 6832 . . . . . . 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 10298 . . . . . . . 8 0 ∈ ℝ*
45 xaddid1 12285 . . . . . . . 8 (0 ∈ ℝ* → (0 +𝑒 0) = 0)
4644, 45ax-mp 5 . . . . . . 7 (0 +𝑒 0) = 0
4743, 46syl6eq 2810 . . . . . 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 4830 . . . . 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 30601 . . . . 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 1477 . . . 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 913 . . 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, 51syl5bbr 274 . 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 30635 . . 3 (𝑀 ran measures → ({𝑥𝑂 ∣ (𝜑𝜓)}a.e.𝑀 ↔ (𝑀‘{𝑥𝑂 ∣ ¬ (𝜑𝜓)}) = 0))
55543ad2ant1 1128 . 2 ((𝑀 ran measures ∧ {𝑥𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) → ({𝑥𝑂 ∣ (𝜑𝜓)}a.e.𝑀 ↔ (𝑀‘{𝑥𝑂 ∣ ¬ (𝜑𝜓)}) = 0))
5653braew 30635 . . . 4 (𝑀 ran measures → ({𝑥𝑂𝜑}a.e.𝑀 ↔ (𝑀‘{𝑥𝑂 ∣ ¬ 𝜑}) = 0))
5753braew 30635 . . . 4 (𝑀 ran measures → ({𝑥𝑂𝜓}a.e.𝑀 ↔ (𝑀‘{𝑥𝑂 ∣ ¬ 𝜓}) = 0))
5856, 57anbi12d 749 . . 3 (𝑀 ran measures → (({𝑥𝑂𝜑}a.e.𝑀 ∧ {𝑥𝑂𝜓}a.e.𝑀) ↔ ((𝑀‘{𝑥𝑂 ∣ ¬ 𝜑}) = 0 ∧ (𝑀‘{𝑥𝑂 ∣ ¬ 𝜓}) = 0)))
59583ad2ant1 1128 . 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 300 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 196  wo 382  wa 383  w3a 1072   = wceq 1632  wcel 2139  {crab 3054  cun 3713  wss 3715   cuni 4588   class class class wbr 4804  dom cdm 5266  ran crn 5267  cfv 6049  (class class class)co 6814  0cc0 10148  *cxr 10285  cle 10287   +𝑒 cxad 12157  sigAlgebracsiga 30500  measurescmeas 30588  a.e.cae 30630
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115  ax-inf2 8713  ax-ac2 9497  ax-cnex 10204  ax-resscn 10205  ax-1cn 10206  ax-icn 10207  ax-addcl 10208  ax-addrcl 10209  ax-mulcl 10210  ax-mulrcl 10211  ax-mulcom 10212  ax-addass 10213  ax-mulass 10214  ax-distr 10215  ax-i2m1 10216  ax-1ne0 10217  ax-1rid 10218  ax-rnegex 10219  ax-rrecex 10220  ax-cnre 10221  ax-pre-lttri 10222  ax-pre-lttrn 10223  ax-pre-ltadd 10224  ax-pre-mulgt0 10225  ax-pre-sup 10226  ax-addf 10227  ax-mulf 10228
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-fal 1638  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-iin 4675  df-disj 4773  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-se 5226  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-isom 6058  df-riota 6775  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-of 7063  df-om 7232  df-1st 7334  df-2nd 7335  df-supp 7465  df-wrecs 7577  df-recs 7638  df-rdg 7676  df-1o 7730  df-2o 7731  df-oadd 7734  df-er 7913  df-map 8027  df-pm 8028  df-ixp 8077  df-en 8124  df-dom 8125  df-sdom 8126  df-fin 8127  df-fsupp 8443  df-fi 8484  df-sup 8515  df-inf 8516  df-oi 8582  df-card 8975  df-acn 8978  df-ac 9149  df-cda 9202  df-pnf 10288  df-mnf 10289  df-xr 10290  df-ltxr 10291  df-le 10292  df-sub 10480  df-neg 10481  df-div 10897  df-nn 11233  df-2 11291  df-3 11292  df-4 11293  df-5 11294  df-6 11295  df-7 11296  df-8 11297  df-9 11298  df-n0 11505  df-z 11590  df-dec 11706  df-uz 11900  df-q 12002  df-rp 12046  df-xneg 12159  df-xadd 12160  df-xmul 12161  df-ioo 12392  df-ioc 12393  df-ico 12394  df-icc 12395  df-fz 12540  df-fzo 12680  df-fl 12807  df-mod 12883  df-seq 13016  df-exp 13075  df-fac 13275  df-bc 13304  df-hash 13332  df-shft 14026  df-cj 14058  df-re 14059  df-im 14060  df-sqrt 14194  df-abs 14195  df-limsup 14421  df-clim 14438  df-rlim 14439  df-sum 14636  df-ef 15017  df-sin 15019  df-cos 15020  df-pi 15022  df-struct 16081  df-ndx 16082  df-slot 16083  df-base 16085  df-sets 16086  df-ress 16087  df-plusg 16176  df-mulr 16177  df-starv 16178  df-sca 16179  df-vsca 16180  df-ip 16181  df-tset 16182  df-ple 16183  df-ds 16186  df-unif 16187  df-hom 16188  df-cco 16189  df-rest 16305  df-topn 16306  df-0g 16324  df-gsum 16325  df-topgen 16326  df-pt 16327  df-prds 16330  df-ordt 16383  df-xrs 16384  df-qtop 16389  df-imas 16390  df-xps 16392  df-mre 16468  df-mrc 16469  df-acs 16471  df-ps 17421  df-tsr 17422  df-plusf 17462  df-mgm 17463  df-sgrp 17505  df-mnd 17516  df-mhm 17556  df-submnd 17557  df-grp 17646  df-minusg 17647  df-sbg 17648  df-mulg 17762  df-subg 17812  df-cntz 17970  df-cmn 18415  df-abl 18416  df-mgp 18710  df-ur 18722  df-ring 18769  df-cring 18770  df-subrg 19000  df-abv 19039  df-lmod 19087  df-scaf 19088  df-sra 19394  df-rgmod 19395  df-psmet 19960  df-xmet 19961  df-met 19962  df-bl 19963  df-mopn 19964  df-fbas 19965  df-fg 19966  df-cnfld 19969  df-top 20921  df-topon 20938  df-topsp 20959  df-bases 20972  df-cld 21045  df-ntr 21046  df-cls 21047  df-nei 21124  df-lp 21162  df-perf 21163  df-cn 21253  df-cnp 21254  df-haus 21341  df-tx 21587  df-hmeo 21780  df-fil 21871  df-fm 21963  df-flim 21964  df-flf 21965  df-tmd 22097  df-tgp 22098  df-tsms 22151  df-trg 22184  df-xms 22346  df-ms 22347  df-tms 22348  df-nm 22608  df-ngp 22609  df-nrg 22611  df-nlm 22612  df-ii 22901  df-cncf 22902  df-limc 23849  df-dv 23850  df-log 24523  df-esum 30420  df-siga 30501  df-meas 30589  df-ae 30632
This theorem is referenced by: (None)
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