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Theorem aean 30130
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 3880 . . . . . 6 ({𝑥𝑂 ∣ ¬ 𝜑} ∪ {𝑥𝑂 ∣ ¬ 𝜓}) = {𝑥𝑂 ∣ (¬ 𝜑 ∨ ¬ 𝜓)}
2 ianor 509 . . . . . . . 8 (¬ (𝜑𝜓) ↔ (¬ 𝜑 ∨ ¬ 𝜓))
32a1i 11 . . . . . . 7 (𝑥𝑂 → (¬ (𝜑𝜓) ↔ (¬ 𝜑 ∨ ¬ 𝜓)))
43rabbiia 3177 . . . . . 6 {𝑥𝑂 ∣ ¬ (𝜑𝜓)} = {𝑥𝑂 ∣ (¬ 𝜑 ∨ ¬ 𝜓)}
51, 4eqtr4i 2646 . . . . 5 ({𝑥𝑂 ∣ ¬ 𝜑} ∪ {𝑥𝑂 ∣ ¬ 𝜓}) = {𝑥𝑂 ∣ ¬ (𝜑𝜓)}
65fveq2i 6161 . . . 4 (𝑀‘({𝑥𝑂 ∣ ¬ 𝜑} ∪ {𝑥𝑂 ∣ ¬ 𝜓})) = (𝑀‘{𝑥𝑂 ∣ ¬ (𝜑𝜓)})
76eqeq1i 2626 . . 3 ((𝑀‘({𝑥𝑂 ∣ ¬ 𝜑} ∪ {𝑥𝑂 ∣ ¬ 𝜓})) = 0 ↔ (𝑀‘{𝑥𝑂 ∣ ¬ (𝜑𝜓)}) = 0)
8 measbasedom 30088 . . . . . . . . 9 (𝑀 ran measures ↔ 𝑀 ∈ (measures‘dom 𝑀))
98biimpi 206 . . . . . . . 8 (𝑀 ran measures → 𝑀 ∈ (measures‘dom 𝑀))
1093ad2ant1 1080 . . . . . . 7 ((𝑀 ran measures ∧ {𝑥𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) → 𝑀 ∈ (measures‘dom 𝑀))
1110adantr 481 . . . . . 6 (((𝑀 ran measures ∧ {𝑥𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) ∧ (𝑀‘({𝑥𝑂 ∣ ¬ 𝜑} ∪ {𝑥𝑂 ∣ ¬ 𝜓})) = 0) → 𝑀 ∈ (measures‘dom 𝑀))
12 simp2 1060 . . . . . . 7 ((𝑀 ran measures ∧ {𝑥𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) → {𝑥𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀)
1312adantr 481 . . . . . 6 (((𝑀 ran measures ∧ {𝑥𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) ∧ (𝑀‘({𝑥𝑂 ∣ ¬ 𝜑} ∪ {𝑥𝑂 ∣ ¬ 𝜓})) = 0) → {𝑥𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀)
14 dmmeas 30087 . . . . . . . . . 10 (𝑀 ran measures → dom 𝑀 ran sigAlgebra)
15 unelsiga 30020 . . . . . . . . . 10 ((dom 𝑀 ran sigAlgebra ∧ {𝑥𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) → ({𝑥𝑂 ∣ ¬ 𝜑} ∪ {𝑥𝑂 ∣ ¬ 𝜓}) ∈ dom 𝑀)
1614, 15syl3an1 1356 . . . . . . . . 9 ((𝑀 ran measures ∧ {𝑥𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) → ({𝑥𝑂 ∣ ¬ 𝜑} ∪ {𝑥𝑂 ∣ ¬ 𝜓}) ∈ dom 𝑀)
17 ssun1 3760 . . . . . . . . . 10 {𝑥𝑂 ∣ ¬ 𝜑} ⊆ ({𝑥𝑂 ∣ ¬ 𝜑} ∪ {𝑥𝑂 ∣ ¬ 𝜓})
1817a1i 11 . . . . . . . . 9 ((𝑀 ran measures ∧ {𝑥𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) → {𝑥𝑂 ∣ ¬ 𝜑} ⊆ ({𝑥𝑂 ∣ ¬ 𝜑} ∪ {𝑥𝑂 ∣ ¬ 𝜓}))
1910, 12, 16, 18measssd 30101 . . . . . . . 8 ((𝑀 ran measures ∧ {𝑥𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) → (𝑀‘{𝑥𝑂 ∣ ¬ 𝜑}) ≤ (𝑀‘({𝑥𝑂 ∣ ¬ 𝜑} ∪ {𝑥𝑂 ∣ ¬ 𝜓})))
2019adantr 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})))
21 simpr 477 . . . . . . 7 (((𝑀 ran measures ∧ {𝑥𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) ∧ (𝑀‘({𝑥𝑂 ∣ ¬ 𝜑} ∪ {𝑥𝑂 ∣ ¬ 𝜓})) = 0) → (𝑀‘({𝑥𝑂 ∣ ¬ 𝜑} ∪ {𝑥𝑂 ∣ ¬ 𝜓})) = 0)
2220, 21breqtrd 4649 . . . . . 6 (((𝑀 ran measures ∧ {𝑥𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) ∧ (𝑀‘({𝑥𝑂 ∣ ¬ 𝜑} ∪ {𝑥𝑂 ∣ ¬ 𝜓})) = 0) → (𝑀‘{𝑥𝑂 ∣ ¬ 𝜑}) ≤ 0)
23 measle0 30094 . . . . . 6 ((𝑀 ∈ (measures‘dom 𝑀) ∧ {𝑥𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ (𝑀‘{𝑥𝑂 ∣ ¬ 𝜑}) ≤ 0) → (𝑀‘{𝑥𝑂 ∣ ¬ 𝜑}) = 0)
2411, 13, 22, 23syl3anc 1323 . . . . 5 (((𝑀 ran measures ∧ {𝑥𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) ∧ (𝑀‘({𝑥𝑂 ∣ ¬ 𝜑} ∪ {𝑥𝑂 ∣ ¬ 𝜓})) = 0) → (𝑀‘{𝑥𝑂 ∣ ¬ 𝜑}) = 0)
25 simp3 1061 . . . . . . 7 ((𝑀 ran measures ∧ {𝑥𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) → {𝑥𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀)
2625adantr 481 . . . . . 6 (((𝑀 ran measures ∧ {𝑥𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) ∧ (𝑀‘({𝑥𝑂 ∣ ¬ 𝜑} ∪ {𝑥𝑂 ∣ ¬ 𝜓})) = 0) → {𝑥𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀)
27 ssun2 3761 . . . . . . . . . 10 {𝑥𝑂 ∣ ¬ 𝜓} ⊆ ({𝑥𝑂 ∣ ¬ 𝜑} ∪ {𝑥𝑂 ∣ ¬ 𝜓})
2827a1i 11 . . . . . . . . 9 ((𝑀 ran measures ∧ {𝑥𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) → {𝑥𝑂 ∣ ¬ 𝜓} ⊆ ({𝑥𝑂 ∣ ¬ 𝜑} ∪ {𝑥𝑂 ∣ ¬ 𝜓}))
2910, 25, 16, 28measssd 30101 . . . . . . . 8 ((𝑀 ran measures ∧ {𝑥𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) → (𝑀‘{𝑥𝑂 ∣ ¬ 𝜓}) ≤ (𝑀‘({𝑥𝑂 ∣ ¬ 𝜑} ∪ {𝑥𝑂 ∣ ¬ 𝜓})))
3029adantr 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})))
3130, 21breqtrd 4649 . . . . . 6 (((𝑀 ran measures ∧ {𝑥𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) ∧ (𝑀‘({𝑥𝑂 ∣ ¬ 𝜑} ∪ {𝑥𝑂 ∣ ¬ 𝜓})) = 0) → (𝑀‘{𝑥𝑂 ∣ ¬ 𝜓}) ≤ 0)
32 measle0 30094 . . . . . 6 ((𝑀 ∈ (measures‘dom 𝑀) ∧ {𝑥𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀 ∧ (𝑀‘{𝑥𝑂 ∣ ¬ 𝜓}) ≤ 0) → (𝑀‘{𝑥𝑂 ∣ ¬ 𝜓}) = 0)
3311, 26, 31, 32syl3anc 1323 . . . . 5 (((𝑀 ran measures ∧ {𝑥𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) ∧ (𝑀‘({𝑥𝑂 ∣ ¬ 𝜑} ∪ {𝑥𝑂 ∣ ¬ 𝜓})) = 0) → (𝑀‘{𝑥𝑂 ∣ ¬ 𝜓}) = 0)
3424, 33jca 554 . . . 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))
3510adantr 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))
36 measbase 30083 . . . . . . 7 (𝑀 ∈ (measures‘dom 𝑀) → dom 𝑀 ran sigAlgebra)
3735, 36syl 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)
3812adantr 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)
3925adantr 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)
4037, 38, 39, 15syl3anc 1323 . . . . 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)
4135, 38, 39measunl 30102 . . . . . 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})))
42 simprl 793 . . . . . . . 8 (((𝑀 ran measures ∧ {𝑥𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) ∧ ((𝑀‘{𝑥𝑂 ∣ ¬ 𝜑}) = 0 ∧ (𝑀‘{𝑥𝑂 ∣ ¬ 𝜓}) = 0)) → (𝑀‘{𝑥𝑂 ∣ ¬ 𝜑}) = 0)
43 simprr 795 . . . . . . . 8 (((𝑀 ran measures ∧ {𝑥𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) ∧ ((𝑀‘{𝑥𝑂 ∣ ¬ 𝜑}) = 0 ∧ (𝑀‘{𝑥𝑂 ∣ ¬ 𝜓}) = 0)) → (𝑀‘{𝑥𝑂 ∣ ¬ 𝜓}) = 0)
4442, 43oveq12d 6633 . . . . . . 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))
45 0xr 10046 . . . . . . . 8 0 ∈ ℝ*
46 xaddid1 12031 . . . . . . . 8 (0 ∈ ℝ* → (0 +𝑒 0) = 0)
4745, 46ax-mp 5 . . . . . . 7 (0 +𝑒 0) = 0
4844, 47syl6eq 2671 . . . . . 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)
4941, 48breqtrd 4649 . . . . 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)
50 measle0 30094 . . . . 5 ((𝑀 ∈ (measures‘dom 𝑀) ∧ ({𝑥𝑂 ∣ ¬ 𝜑} ∪ {𝑥𝑂 ∣ ¬ 𝜓}) ∈ dom 𝑀 ∧ (𝑀‘({𝑥𝑂 ∣ ¬ 𝜑} ∪ {𝑥𝑂 ∣ ¬ 𝜓})) ≤ 0) → (𝑀‘({𝑥𝑂 ∣ ¬ 𝜑} ∪ {𝑥𝑂 ∣ ¬ 𝜓})) = 0)
5135, 40, 49, 50syl3anc 1323 . . . 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)
5234, 51impbida 876 . . 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)))
537, 52syl5bbr 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)))
54 aean.1 . . . 4 dom 𝑀 = 𝑂
5554braew 30128 . . 3 (𝑀 ran measures → ({𝑥𝑂 ∣ (𝜑𝜓)}a.e.𝑀 ↔ (𝑀‘{𝑥𝑂 ∣ ¬ (𝜑𝜓)}) = 0))
56553ad2ant1 1080 . 2 ((𝑀 ran measures ∧ {𝑥𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) → ({𝑥𝑂 ∣ (𝜑𝜓)}a.e.𝑀 ↔ (𝑀‘{𝑥𝑂 ∣ ¬ (𝜑𝜓)}) = 0))
5754braew 30128 . . . 4 (𝑀 ran measures → ({𝑥𝑂𝜑}a.e.𝑀 ↔ (𝑀‘{𝑥𝑂 ∣ ¬ 𝜑}) = 0))
5854braew 30128 . . . 4 (𝑀 ran measures → ({𝑥𝑂𝜓}a.e.𝑀 ↔ (𝑀‘{𝑥𝑂 ∣ ¬ 𝜓}) = 0))
5957, 58anbi12d 746 . . 3 (𝑀 ran measures → (({𝑥𝑂𝜑}a.e.𝑀 ∧ {𝑥𝑂𝜓}a.e.𝑀) ↔ ((𝑀‘{𝑥𝑂 ∣ ¬ 𝜑}) = 0 ∧ (𝑀‘{𝑥𝑂 ∣ ¬ 𝜓}) = 0)))
60593ad2ant1 1080 . 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)))
6153, 56, 603bitr4d 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 383  wa 384  w3a 1036   = wceq 1480  wcel 1987  {crab 2912  cun 3558  wss 3560   cuni 4409   class class class wbr 4623  dom cdm 5084  ran crn 5085  cfv 5857  (class class class)co 6615  0cc0 9896  *cxr 10033  cle 10035   +𝑒 cxad 11904  sigAlgebracsiga 29993  measurescmeas 30081  a.e.cae 30123
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914  ax-inf2 8498  ax-ac2 9245  ax-cnex 9952  ax-resscn 9953  ax-1cn 9954  ax-icn 9955  ax-addcl 9956  ax-addrcl 9957  ax-mulcl 9958  ax-mulrcl 9959  ax-mulcom 9960  ax-addass 9961  ax-mulass 9962  ax-distr 9963  ax-i2m1 9964  ax-1ne0 9965  ax-1rid 9966  ax-rnegex 9967  ax-rrecex 9968  ax-cnre 9969  ax-pre-lttri 9970  ax-pre-lttrn 9971  ax-pre-ltadd 9972  ax-pre-mulgt0 9973  ax-pre-sup 9974  ax-addf 9975  ax-mulf 9976
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2913  df-rex 2914  df-reu 2915  df-rmo 2916  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-int 4448  df-iun 4494  df-iin 4495  df-disj 4594  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-se 5044  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-isom 5866  df-riota 6576  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-of 6862  df-om 7028  df-1st 7128  df-2nd 7129  df-supp 7256  df-wrecs 7367  df-recs 7428  df-rdg 7466  df-1o 7520  df-2o 7521  df-oadd 7524  df-er 7702  df-map 7819  df-pm 7820  df-ixp 7869  df-en 7916  df-dom 7917  df-sdom 7918  df-fin 7919  df-fsupp 8236  df-fi 8277  df-sup 8308  df-inf 8309  df-oi 8375  df-card 8725  df-acn 8728  df-ac 8899  df-cda 8950  df-pnf 10036  df-mnf 10037  df-xr 10038  df-ltxr 10039  df-le 10040  df-sub 10228  df-neg 10229  df-div 10645  df-nn 10981  df-2 11039  df-3 11040  df-4 11041  df-5 11042  df-6 11043  df-7 11044  df-8 11045  df-9 11046  df-n0 11253  df-z 11338  df-dec 11454  df-uz 11648  df-q 11749  df-rp 11793  df-xneg 11906  df-xadd 11907  df-xmul 11908  df-ioo 12137  df-ioc 12138  df-ico 12139  df-icc 12140  df-fz 12285  df-fzo 12423  df-fl 12549  df-mod 12625  df-seq 12758  df-exp 12817  df-fac 13017  df-bc 13046  df-hash 13074  df-shft 13757  df-cj 13789  df-re 13790  df-im 13791  df-sqrt 13925  df-abs 13926  df-limsup 14152  df-clim 14169  df-rlim 14170  df-sum 14367  df-ef 14742  df-sin 14744  df-cos 14745  df-pi 14747  df-struct 15802  df-ndx 15803  df-slot 15804  df-base 15805  df-sets 15806  df-ress 15807  df-plusg 15894  df-mulr 15895  df-starv 15896  df-sca 15897  df-vsca 15898  df-ip 15899  df-tset 15900  df-ple 15901  df-ds 15904  df-unif 15905  df-hom 15906  df-cco 15907  df-rest 16023  df-topn 16024  df-0g 16042  df-gsum 16043  df-topgen 16044  df-pt 16045  df-prds 16048  df-ordt 16101  df-xrs 16102  df-qtop 16107  df-imas 16108  df-xps 16110  df-mre 16186  df-mrc 16187  df-acs 16189  df-ps 17140  df-tsr 17141  df-plusf 17181  df-mgm 17182  df-sgrp 17224  df-mnd 17235  df-mhm 17275  df-submnd 17276  df-grp 17365  df-minusg 17366  df-sbg 17367  df-mulg 17481  df-subg 17531  df-cntz 17690  df-cmn 18135  df-abl 18136  df-mgp 18430  df-ur 18442  df-ring 18489  df-cring 18490  df-subrg 18718  df-abv 18757  df-lmod 18805  df-scaf 18806  df-sra 19112  df-rgmod 19113  df-psmet 19678  df-xmet 19679  df-met 19680  df-bl 19681  df-mopn 19682  df-fbas 19683  df-fg 19684  df-cnfld 19687  df-top 20639  df-topon 20656  df-topsp 20677  df-bases 20690  df-cld 20763  df-ntr 20764  df-cls 20765  df-nei 20842  df-lp 20880  df-perf 20881  df-cn 20971  df-cnp 20972  df-haus 21059  df-tx 21305  df-hmeo 21498  df-fil 21590  df-fm 21682  df-flim 21683  df-flf 21684  df-tmd 21816  df-tgp 21817  df-tsms 21870  df-trg 21903  df-xms 22065  df-ms 22066  df-tms 22067  df-nm 22327  df-ngp 22328  df-nrg 22330  df-nlm 22331  df-ii 22620  df-cncf 22621  df-limc 23570  df-dv 23571  df-log 24241  df-esum 29913  df-siga 29994  df-meas 30082  df-ae 30125
This theorem is referenced by: (None)
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