Theorem aean 30307

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 ∧ {𝑥 ∈ 𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥 ∈ 𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) → ({𝑥 ∈ 𝑂 ∣ (𝜑 ∧ 𝜓)}a.e.𝑀 ↔ ({𝑥 ∈ 𝑂 ∣ 𝜑}a.e.𝑀 ∧ {𝑥 ∈ 𝑂 ∣ 𝜓}a.e.𝑀)))

Distinct variable group:   𝑥,𝑂

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)   𝑀(𝑥)

Proof of Theorem aean

Step	Hyp	Ref	Expression
1		unrab 3898	. . . . . 6 ⊢ ({𝑥 ∈ 𝑂 ∣ ¬ 𝜑} ∪ {𝑥 ∈ 𝑂 ∣ ¬ 𝜓}) = {𝑥 ∈ 𝑂 ∣ (¬ 𝜑 ∨ ¬ 𝜓)}
2		ianor 509	. . . . . . . 8 ⊢ (¬ (𝜑 ∧ 𝜓) ↔ (¬ 𝜑 ∨ ¬ 𝜓))
3	2	a1i 11	. . . . . . 7 ⊢ (𝑥 ∈ 𝑂 → (¬ (𝜑 ∧ 𝜓) ↔ (¬ 𝜑 ∨ ¬ 𝜓)))
4	3	rabbiia 3185	. . . . . 6 ⊢ {𝑥 ∈ 𝑂 ∣ ¬ (𝜑 ∧ 𝜓)} = {𝑥 ∈ 𝑂 ∣ (¬ 𝜑 ∨ ¬ 𝜓)}
5	1, 4	eqtr4i 2647	. . . . 5 ⊢ ({𝑥 ∈ 𝑂 ∣ ¬ 𝜑} ∪ {𝑥 ∈ 𝑂 ∣ ¬ 𝜓}) = {𝑥 ∈ 𝑂 ∣ ¬ (𝜑 ∧ 𝜓)}
6	5	fveq2i 6194	. . . 4 ⊢ (𝑀‘({𝑥 ∈ 𝑂 ∣ ¬ 𝜑} ∪ {𝑥 ∈ 𝑂 ∣ ¬ 𝜓})) = (𝑀‘{𝑥 ∈ 𝑂 ∣ ¬ (𝜑 ∧ 𝜓)})
7	6	eqeq1i 2627	. . 3 ⊢ ((𝑀‘({𝑥 ∈ 𝑂 ∣ ¬ 𝜑} ∪ {𝑥 ∈ 𝑂 ∣ ¬ 𝜓})) = 0 ↔ (𝑀‘{𝑥 ∈ 𝑂 ∣ ¬ (𝜑 ∧ 𝜓)}) = 0)
8		measbasedom 30265	. . . . . . . . 9 ⊢ (𝑀 ∈ ∪ ran measures ↔ 𝑀 ∈ (measures‘dom 𝑀))
9	8	biimpi 206	. . . . . . . 8 ⊢ (𝑀 ∈ ∪ ran measures → 𝑀 ∈ (measures‘dom 𝑀))
10	9	3ad2ant1 1082	. . . . . . 7 ⊢ ((𝑀 ∈ ∪ ran measures ∧ {𝑥 ∈ 𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥 ∈ 𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) → 𝑀 ∈ (measures‘dom 𝑀))
11	10	adantr 481	. . . . . 6 ⊢ (((𝑀 ∈ ∪ ran measures ∧ {𝑥 ∈ 𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥 ∈ 𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) ∧ (𝑀‘({𝑥 ∈ 𝑂 ∣ ¬ 𝜑} ∪ {𝑥 ∈ 𝑂 ∣ ¬ 𝜓})) = 0) → 𝑀 ∈ (measures‘dom 𝑀))
12		simp2 1062	. . . . . . 7 ⊢ ((𝑀 ∈ ∪ ran measures ∧ {𝑥 ∈ 𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥 ∈ 𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) → {𝑥 ∈ 𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀)
13	12	adantr 481	. . . . . 6 ⊢ (((𝑀 ∈ ∪ ran measures ∧ {𝑥 ∈ 𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥 ∈ 𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) ∧ (𝑀‘({𝑥 ∈ 𝑂 ∣ ¬ 𝜑} ∪ {𝑥 ∈ 𝑂 ∣ ¬ 𝜓})) = 0) → {𝑥 ∈ 𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀)
14		dmmeas 30264	. . . . . . . . . 10 ⊢ (𝑀 ∈ ∪ ran measures → dom 𝑀 ∈ ∪ ran sigAlgebra)
15		unelsiga 30197	. . . . . . . . . 10 ⊢ ((dom 𝑀 ∈ ∪ ran sigAlgebra ∧ {𝑥 ∈ 𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥 ∈ 𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) → ({𝑥 ∈ 𝑂 ∣ ¬ 𝜑} ∪ {𝑥 ∈ 𝑂 ∣ ¬ 𝜓}) ∈ dom 𝑀)
16	14, 15	syl3an1 1359	. . . . . . . . 9 ⊢ ((𝑀 ∈ ∪ ran measures ∧ {𝑥 ∈ 𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥 ∈ 𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) → ({𝑥 ∈ 𝑂 ∣ ¬ 𝜑} ∪ {𝑥 ∈ 𝑂 ∣ ¬ 𝜓}) ∈ dom 𝑀)
17		ssun1 3776	. . . . . . . . . 10 ⊢ {𝑥 ∈ 𝑂 ∣ ¬ 𝜑} ⊆ ({𝑥 ∈ 𝑂 ∣ ¬ 𝜑} ∪ {𝑥 ∈ 𝑂 ∣ ¬ 𝜓})
18	17	a1i 11	. . . . . . . . 9 ⊢ ((𝑀 ∈ ∪ ran measures ∧ {𝑥 ∈ 𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥 ∈ 𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) → {𝑥 ∈ 𝑂 ∣ ¬ 𝜑} ⊆ ({𝑥 ∈ 𝑂 ∣ ¬ 𝜑} ∪ {𝑥 ∈ 𝑂 ∣ ¬ 𝜓}))
19	10, 12, 16, 18	measssd 30278	. . . . . . . 8 ⊢ ((𝑀 ∈ ∪ ran measures ∧ {𝑥 ∈ 𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥 ∈ 𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) → (𝑀‘{𝑥 ∈ 𝑂 ∣ ¬ 𝜑}) ≤ (𝑀‘({𝑥 ∈ 𝑂 ∣ ¬ 𝜑} ∪ {𝑥 ∈ 𝑂 ∣ ¬ 𝜓})))
20	19	adantr 481	. . . . . . 7 ⊢ (((𝑀 ∈ ∪ ran measures ∧ {𝑥 ∈ 𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥 ∈ 𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) ∧ (𝑀‘({𝑥 ∈ 𝑂 ∣ ¬ 𝜑} ∪ {𝑥 ∈ 𝑂 ∣ ¬ 𝜓})) = 0) → (𝑀‘{𝑥 ∈ 𝑂 ∣ ¬ 𝜑}) ≤ (𝑀‘({𝑥 ∈ 𝑂 ∣ ¬ 𝜑} ∪ {𝑥 ∈ 𝑂 ∣ ¬ 𝜓})))
21		simpr 477	. . . . . . 7 ⊢ (((𝑀 ∈ ∪ ran measures ∧ {𝑥 ∈ 𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥 ∈ 𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) ∧ (𝑀‘({𝑥 ∈ 𝑂 ∣ ¬ 𝜑} ∪ {𝑥 ∈ 𝑂 ∣ ¬ 𝜓})) = 0) → (𝑀‘({𝑥 ∈ 𝑂 ∣ ¬ 𝜑} ∪ {𝑥 ∈ 𝑂 ∣ ¬ 𝜓})) = 0)
22	20, 21	breqtrd 4679	. . . . . 6 ⊢ (((𝑀 ∈ ∪ ran measures ∧ {𝑥 ∈ 𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥 ∈ 𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) ∧ (𝑀‘({𝑥 ∈ 𝑂 ∣ ¬ 𝜑} ∪ {𝑥 ∈ 𝑂 ∣ ¬ 𝜓})) = 0) → (𝑀‘{𝑥 ∈ 𝑂 ∣ ¬ 𝜑}) ≤ 0)
23		measle0 30271	. . . . . 6 ⊢ ((𝑀 ∈ (measures‘dom 𝑀) ∧ {𝑥 ∈ 𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ (𝑀‘{𝑥 ∈ 𝑂 ∣ ¬ 𝜑}) ≤ 0) → (𝑀‘{𝑥 ∈ 𝑂 ∣ ¬ 𝜑}) = 0)
24	11, 13, 22, 23	syl3anc 1326	. . . . 5 ⊢ (((𝑀 ∈ ∪ ran measures ∧ {𝑥 ∈ 𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥 ∈ 𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) ∧ (𝑀‘({𝑥 ∈ 𝑂 ∣ ¬ 𝜑} ∪ {𝑥 ∈ 𝑂 ∣ ¬ 𝜓})) = 0) → (𝑀‘{𝑥 ∈ 𝑂 ∣ ¬ 𝜑}) = 0)
25		simp3 1063	. . . . . . 7 ⊢ ((𝑀 ∈ ∪ ran measures ∧ {𝑥 ∈ 𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥 ∈ 𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) → {𝑥 ∈ 𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀)
26	25	adantr 481	. . . . . 6 ⊢ (((𝑀 ∈ ∪ ran measures ∧ {𝑥 ∈ 𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥 ∈ 𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) ∧ (𝑀‘({𝑥 ∈ 𝑂 ∣ ¬ 𝜑} ∪ {𝑥 ∈ 𝑂 ∣ ¬ 𝜓})) = 0) → {𝑥 ∈ 𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀)
27		ssun2 3777	. . . . . . . . . 10 ⊢ {𝑥 ∈ 𝑂 ∣ ¬ 𝜓} ⊆ ({𝑥 ∈ 𝑂 ∣ ¬ 𝜑} ∪ {𝑥 ∈ 𝑂 ∣ ¬ 𝜓})
28	27	a1i 11	. . . . . . . . 9 ⊢ ((𝑀 ∈ ∪ ran measures ∧ {𝑥 ∈ 𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥 ∈ 𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) → {𝑥 ∈ 𝑂 ∣ ¬ 𝜓} ⊆ ({𝑥 ∈ 𝑂 ∣ ¬ 𝜑} ∪ {𝑥 ∈ 𝑂 ∣ ¬ 𝜓}))
29	10, 25, 16, 28	measssd 30278	. . . . . . . 8 ⊢ ((𝑀 ∈ ∪ ran measures ∧ {𝑥 ∈ 𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥 ∈ 𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) → (𝑀‘{𝑥 ∈ 𝑂 ∣ ¬ 𝜓}) ≤ (𝑀‘({𝑥 ∈ 𝑂 ∣ ¬ 𝜑} ∪ {𝑥 ∈ 𝑂 ∣ ¬ 𝜓})))
30	29	adantr 481	. . . . . . 7 ⊢ (((𝑀 ∈ ∪ ran measures ∧ {𝑥 ∈ 𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥 ∈ 𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) ∧ (𝑀‘({𝑥 ∈ 𝑂 ∣ ¬ 𝜑} ∪ {𝑥 ∈ 𝑂 ∣ ¬ 𝜓})) = 0) → (𝑀‘{𝑥 ∈ 𝑂 ∣ ¬ 𝜓}) ≤ (𝑀‘({𝑥 ∈ 𝑂 ∣ ¬ 𝜑} ∪ {𝑥 ∈ 𝑂 ∣ ¬ 𝜓})))
31	30, 21	breqtrd 4679	. . . . . 6 ⊢ (((𝑀 ∈ ∪ ran measures ∧ {𝑥 ∈ 𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥 ∈ 𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) ∧ (𝑀‘({𝑥 ∈ 𝑂 ∣ ¬ 𝜑} ∪ {𝑥 ∈ 𝑂 ∣ ¬ 𝜓})) = 0) → (𝑀‘{𝑥 ∈ 𝑂 ∣ ¬ 𝜓}) ≤ 0)
32		measle0 30271	. . . . . 6 ⊢ ((𝑀 ∈ (measures‘dom 𝑀) ∧ {𝑥 ∈ 𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀 ∧ (𝑀‘{𝑥 ∈ 𝑂 ∣ ¬ 𝜓}) ≤ 0) → (𝑀‘{𝑥 ∈ 𝑂 ∣ ¬ 𝜓}) = 0)
33	11, 26, 31, 32	syl3anc 1326	. . . . 5 ⊢ (((𝑀 ∈ ∪ ran measures ∧ {𝑥 ∈ 𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥 ∈ 𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) ∧ (𝑀‘({𝑥 ∈ 𝑂 ∣ ¬ 𝜑} ∪ {𝑥 ∈ 𝑂 ∣ ¬ 𝜓})) = 0) → (𝑀‘{𝑥 ∈ 𝑂 ∣ ¬ 𝜓}) = 0)
34	24, 33	jca 554	. . . 4 ⊢ (((𝑀 ∈ ∪ ran measures ∧ {𝑥 ∈ 𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥 ∈ 𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) ∧ (𝑀‘({𝑥 ∈ 𝑂 ∣ ¬ 𝜑} ∪ {𝑥 ∈ 𝑂 ∣ ¬ 𝜓})) = 0) → ((𝑀‘{𝑥 ∈ 𝑂 ∣ ¬ 𝜑}) = 0 ∧ (𝑀‘{𝑥 ∈ 𝑂 ∣ ¬ 𝜓}) = 0))
35	10	adantr 481	. . . . 5 ⊢ (((𝑀 ∈ ∪ ran measures ∧ {𝑥 ∈ 𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥 ∈ 𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) ∧ ((𝑀‘{𝑥 ∈ 𝑂 ∣ ¬ 𝜑}) = 0 ∧ (𝑀‘{𝑥 ∈ 𝑂 ∣ ¬ 𝜓}) = 0)) → 𝑀 ∈ (measures‘dom 𝑀))
36		measbase 30260	. . . . . . 7 ⊢ (𝑀 ∈ (measures‘dom 𝑀) → dom 𝑀 ∈ ∪ ran sigAlgebra)
37	35, 36	syl 17	. . . . . 6 ⊢ (((𝑀 ∈ ∪ ran measures ∧ {𝑥 ∈ 𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥 ∈ 𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) ∧ ((𝑀‘{𝑥 ∈ 𝑂 ∣ ¬ 𝜑}) = 0 ∧ (𝑀‘{𝑥 ∈ 𝑂 ∣ ¬ 𝜓}) = 0)) → dom 𝑀 ∈ ∪ ran sigAlgebra)
38	12	adantr 481	. . . . . 6 ⊢ (((𝑀 ∈ ∪ ran measures ∧ {𝑥 ∈ 𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥 ∈ 𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) ∧ ((𝑀‘{𝑥 ∈ 𝑂 ∣ ¬ 𝜑}) = 0 ∧ (𝑀‘{𝑥 ∈ 𝑂 ∣ ¬ 𝜓}) = 0)) → {𝑥 ∈ 𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀)
39	25	adantr 481	. . . . . 6 ⊢ (((𝑀 ∈ ∪ ran measures ∧ {𝑥 ∈ 𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥 ∈ 𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) ∧ ((𝑀‘{𝑥 ∈ 𝑂 ∣ ¬ 𝜑}) = 0 ∧ (𝑀‘{𝑥 ∈ 𝑂 ∣ ¬ 𝜓}) = 0)) → {𝑥 ∈ 𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀)
40	37, 38, 39, 15	syl3anc 1326	. . . . 5 ⊢ (((𝑀 ∈ ∪ ran measures ∧ {𝑥 ∈ 𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥 ∈ 𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) ∧ ((𝑀‘{𝑥 ∈ 𝑂 ∣ ¬ 𝜑}) = 0 ∧ (𝑀‘{𝑥 ∈ 𝑂 ∣ ¬ 𝜓}) = 0)) → ({𝑥 ∈ 𝑂 ∣ ¬ 𝜑} ∪ {𝑥 ∈ 𝑂 ∣ ¬ 𝜓}) ∈ dom 𝑀)
41	35, 38, 39	measunl 30279	. . . . . 6 ⊢ (((𝑀 ∈ ∪ ran measures ∧ {𝑥 ∈ 𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥 ∈ 𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) ∧ ((𝑀‘{𝑥 ∈ 𝑂 ∣ ¬ 𝜑}) = 0 ∧ (𝑀‘{𝑥 ∈ 𝑂 ∣ ¬ 𝜓}) = 0)) → (𝑀‘({𝑥 ∈ 𝑂 ∣ ¬ 𝜑} ∪ {𝑥 ∈ 𝑂 ∣ ¬ 𝜓})) ≤ ((𝑀‘{𝑥 ∈ 𝑂 ∣ ¬ 𝜑}) +𝑒 (𝑀‘{𝑥 ∈ 𝑂 ∣ ¬ 𝜓})))
42		simprl 794	. . . . . . . 8 ⊢ (((𝑀 ∈ ∪ ran measures ∧ {𝑥 ∈ 𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥 ∈ 𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) ∧ ((𝑀‘{𝑥 ∈ 𝑂 ∣ ¬ 𝜑}) = 0 ∧ (𝑀‘{𝑥 ∈ 𝑂 ∣ ¬ 𝜓}) = 0)) → (𝑀‘{𝑥 ∈ 𝑂 ∣ ¬ 𝜑}) = 0)
43		simprr 796	. . . . . . . 8 ⊢ (((𝑀 ∈ ∪ ran measures ∧ {𝑥 ∈ 𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥 ∈ 𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) ∧ ((𝑀‘{𝑥 ∈ 𝑂 ∣ ¬ 𝜑}) = 0 ∧ (𝑀‘{𝑥 ∈ 𝑂 ∣ ¬ 𝜓}) = 0)) → (𝑀‘{𝑥 ∈ 𝑂 ∣ ¬ 𝜓}) = 0)
44	42, 43	oveq12d 6668	. . . . . . 7 ⊢ (((𝑀 ∈ ∪ ran measures ∧ {𝑥 ∈ 𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥 ∈ 𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) ∧ ((𝑀‘{𝑥 ∈ 𝑂 ∣ ¬ 𝜑}) = 0 ∧ (𝑀‘{𝑥 ∈ 𝑂 ∣ ¬ 𝜓}) = 0)) → ((𝑀‘{𝑥 ∈ 𝑂 ∣ ¬ 𝜑}) +𝑒 (𝑀‘{𝑥 ∈ 𝑂 ∣ ¬ 𝜓})) = (0 +𝑒 0))
45		0xr 10086	. . . . . . . 8 ⊢ 0 ∈ ℝ*
46		xaddid1 12072	. . . . . . . 8 ⊢ (0 ∈ ℝ* → (0 +𝑒 0) = 0)
47	45, 46	ax-mp 5	. . . . . . 7 ⊢ (0 +𝑒 0) = 0
48	44, 47	syl6eq 2672	. . . . . 6 ⊢ (((𝑀 ∈ ∪ ran measures ∧ {𝑥 ∈ 𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥 ∈ 𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) ∧ ((𝑀‘{𝑥 ∈ 𝑂 ∣ ¬ 𝜑}) = 0 ∧ (𝑀‘{𝑥 ∈ 𝑂 ∣ ¬ 𝜓}) = 0)) → ((𝑀‘{𝑥 ∈ 𝑂 ∣ ¬ 𝜑}) +𝑒 (𝑀‘{𝑥 ∈ 𝑂 ∣ ¬ 𝜓})) = 0)
49	41, 48	breqtrd 4679	. . . . 5 ⊢ (((𝑀 ∈ ∪ ran measures ∧ {𝑥 ∈ 𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥 ∈ 𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) ∧ ((𝑀‘{𝑥 ∈ 𝑂 ∣ ¬ 𝜑}) = 0 ∧ (𝑀‘{𝑥 ∈ 𝑂 ∣ ¬ 𝜓}) = 0)) → (𝑀‘({𝑥 ∈ 𝑂 ∣ ¬ 𝜑} ∪ {𝑥 ∈ 𝑂 ∣ ¬ 𝜓})) ≤ 0)
50		measle0 30271	. . . . 5 ⊢ ((𝑀 ∈ (measures‘dom 𝑀) ∧ ({𝑥 ∈ 𝑂 ∣ ¬ 𝜑} ∪ {𝑥 ∈ 𝑂 ∣ ¬ 𝜓}) ∈ dom 𝑀 ∧ (𝑀‘({𝑥 ∈ 𝑂 ∣ ¬ 𝜑} ∪ {𝑥 ∈ 𝑂 ∣ ¬ 𝜓})) ≤ 0) → (𝑀‘({𝑥 ∈ 𝑂 ∣ ¬ 𝜑} ∪ {𝑥 ∈ 𝑂 ∣ ¬ 𝜓})) = 0)
51	35, 40, 49, 50	syl3anc 1326	. . . 4 ⊢ (((𝑀 ∈ ∪ ran measures ∧ {𝑥 ∈ 𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥 ∈ 𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) ∧ ((𝑀‘{𝑥 ∈ 𝑂 ∣ ¬ 𝜑}) = 0 ∧ (𝑀‘{𝑥 ∈ 𝑂 ∣ ¬ 𝜓}) = 0)) → (𝑀‘({𝑥 ∈ 𝑂 ∣ ¬ 𝜑} ∪ {𝑥 ∈ 𝑂 ∣ ¬ 𝜓})) = 0)
52	34, 51	impbida 877	. . 3 ⊢ ((𝑀 ∈ ∪ ran measures ∧ {𝑥 ∈ 𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥 ∈ 𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) → ((𝑀‘({𝑥 ∈ 𝑂 ∣ ¬ 𝜑} ∪ {𝑥 ∈ 𝑂 ∣ ¬ 𝜓})) = 0 ↔ ((𝑀‘{𝑥 ∈ 𝑂 ∣ ¬ 𝜑}) = 0 ∧ (𝑀‘{𝑥 ∈ 𝑂 ∣ ¬ 𝜓}) = 0)))
53	7, 52	syl5bbr 274	. 2 ⊢ ((𝑀 ∈ ∪ ran measures ∧ {𝑥 ∈ 𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥 ∈ 𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) → ((𝑀‘{𝑥 ∈ 𝑂 ∣ ¬ (𝜑 ∧ 𝜓)}) = 0 ↔ ((𝑀‘{𝑥 ∈ 𝑂 ∣ ¬ 𝜑}) = 0 ∧ (𝑀‘{𝑥 ∈ 𝑂 ∣ ¬ 𝜓}) = 0)))
54		aean.1	. . . 4 ⊢ ∪ dom 𝑀 = 𝑂
55	54	braew 30305	. . 3 ⊢ (𝑀 ∈ ∪ ran measures → ({𝑥 ∈ 𝑂 ∣ (𝜑 ∧ 𝜓)}a.e.𝑀 ↔ (𝑀‘{𝑥 ∈ 𝑂 ∣ ¬ (𝜑 ∧ 𝜓)}) = 0))
56	55	3ad2ant1 1082	. 2 ⊢ ((𝑀 ∈ ∪ ran measures ∧ {𝑥 ∈ 𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥 ∈ 𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) → ({𝑥 ∈ 𝑂 ∣ (𝜑 ∧ 𝜓)}a.e.𝑀 ↔ (𝑀‘{𝑥 ∈ 𝑂 ∣ ¬ (𝜑 ∧ 𝜓)}) = 0))
57	54	braew 30305	. . . 4 ⊢ (𝑀 ∈ ∪ ran measures → ({𝑥 ∈ 𝑂 ∣ 𝜑}a.e.𝑀 ↔ (𝑀‘{𝑥 ∈ 𝑂 ∣ ¬ 𝜑}) = 0))
58	54	braew 30305	. . . 4 ⊢ (𝑀 ∈ ∪ ran measures → ({𝑥 ∈ 𝑂 ∣ 𝜓}a.e.𝑀 ↔ (𝑀‘{𝑥 ∈ 𝑂 ∣ ¬ 𝜓}) = 0))
59	57, 58	anbi12d 747	. . 3 ⊢ (𝑀 ∈ ∪ ran measures → (({𝑥 ∈ 𝑂 ∣ 𝜑}a.e.𝑀 ∧ {𝑥 ∈ 𝑂 ∣ 𝜓}a.e.𝑀) ↔ ((𝑀‘{𝑥 ∈ 𝑂 ∣ ¬ 𝜑}) = 0 ∧ (𝑀‘{𝑥 ∈ 𝑂 ∣ ¬ 𝜓}) = 0)))
60	59	3ad2ant1 1082	. 2 ⊢ ((𝑀 ∈ ∪ ran measures ∧ {𝑥 ∈ 𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥 ∈ 𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) → (({𝑥 ∈ 𝑂 ∣ 𝜑}a.e.𝑀 ∧ {𝑥 ∈ 𝑂 ∣ 𝜓}a.e.𝑀) ↔ ((𝑀‘{𝑥 ∈ 𝑂 ∣ ¬ 𝜑}) = 0 ∧ (𝑀‘{𝑥 ∈ 𝑂 ∣ ¬ 𝜓}) = 0)))
61	53, 56, 60	3bitr4d 300	1 ⊢ ((𝑀 ∈ ∪ ran measures ∧ {𝑥 ∈ 𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥 ∈ 𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) → ({𝑥 ∈ 𝑂 ∣ (𝜑 ∧ 𝜓)}a.e.𝑀 ↔ ({𝑥 ∈ 𝑂 ∣ 𝜑}a.e.𝑀 ∧ {𝑥 ∈ 𝑂 ∣ 𝜓}a.e.𝑀)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∨ wo 383   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  {crab 2916   ∪ cun 3572   ⊆ wss 3574  ∪ cuni 4436   class class class wbr 4653  dom cdm 5114  ran crn 5115  ‘cfv 5888  (class class class)co 6650  0cc0 9936  ℝ^*cxr 10073   ≤ cle 10075   +_𝑒 cxad 11944  sigAlgebracsiga 30170  measurescmeas 30258  a.e.cae 30300

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-ac2 9285  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-addf 10015  ax-mulf 10016

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-disj 4621  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-fi 8317  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-acn 8768  df-ac 8939  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ioo 12179  df-ioc 12180  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-fac 13061  df-bc 13090  df-hash 13118  df-shft 13807  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-limsup 14202  df-clim 14219  df-rlim 14220  df-sum 14417  df-ef 14798  df-sin 14800  df-cos 14801  df-pi 14803  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-hom 15966  df-cco 15967  df-rest 16083  df-topn 16084  df-0g 16102  df-gsum 16103  df-topgen 16104  df-pt 16105  df-prds 16108  df-ordt 16161  df-xrs 16162  df-qtop 16167  df-imas 16168  df-xps 16170  df-mre 16246  df-mrc 16247  df-acs 16249  df-ps 17200  df-tsr 17201  df-plusf 17241  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-mhm 17335  df-submnd 17336  df-grp 17425  df-minusg 17426  df-sbg 17427  df-mulg 17541  df-subg 17591  df-cntz 17750  df-cmn 18195  df-abl 18196  df-mgp 18490  df-ur 18502  df-ring 18549  df-cring 18550  df-subrg 18778  df-abv 18817  df-lmod 18865  df-scaf 18866  df-sra 19172  df-rgmod 19173  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-fbas 19743  df-fg 19744  df-cnfld 19747  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cld 20823  df-ntr 20824  df-cls 20825  df-nei 20902  df-lp 20940  df-perf 20941  df-cn 21031  df-cnp 21032  df-haus 21119  df-tx 21365  df-hmeo 21558  df-fil 21650  df-fm 21742  df-flim 21743  df-flf 21744  df-tmd 21876  df-tgp 21877  df-tsms 21930  df-trg 21963  df-xms 22125  df-ms 22126  df-tms 22127  df-nm 22387  df-ngp 22388  df-nrg 22390  df-nlm 22391  df-ii 22680  df-cncf 22681  df-limc 23630  df-dv 23631  df-log 24303  df-esum 30090  df-siga 30171  df-meas 30259  df-ae 30302

This theorem is referenced by: (None)