Theorem aean 32212

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 4239	. . . . . 6 ⊢ ({𝑥 ∈ 𝑂 ∣ ¬ 𝜑} ∪ {𝑥 ∈ 𝑂 ∣ ¬ 𝜓}) = {𝑥 ∈ 𝑂 ∣ (¬ 𝜑 ∨ ¬ 𝜓)}
2		ianor 979	. . . . . . 7 ⊢ (¬ (𝜑 ∧ 𝜓) ↔ (¬ 𝜑 ∨ ¬ 𝜓))
3	2	rabbii 3408	. . . . . 6 ⊢ {𝑥 ∈ 𝑂 ∣ ¬ (𝜑 ∧ 𝜓)} = {𝑥 ∈ 𝑂 ∣ (¬ 𝜑 ∨ ¬ 𝜓)}
4	1, 3	eqtr4i 2769	. . . . 5 ⊢ ({𝑥 ∈ 𝑂 ∣ ¬ 𝜑} ∪ {𝑥 ∈ 𝑂 ∣ ¬ 𝜓}) = {𝑥 ∈ 𝑂 ∣ ¬ (𝜑 ∧ 𝜓)}
5	4	fveq2i 6777	. . . 4 ⊢ (𝑀‘({𝑥 ∈ 𝑂 ∣ ¬ 𝜑} ∪ {𝑥 ∈ 𝑂 ∣ ¬ 𝜓})) = (𝑀‘{𝑥 ∈ 𝑂 ∣ ¬ (𝜑 ∧ 𝜓)})
6	5	eqeq1i 2743	. . 3 ⊢ ((𝑀‘({𝑥 ∈ 𝑂 ∣ ¬ 𝜑} ∪ {𝑥 ∈ 𝑂 ∣ ¬ 𝜓})) = 0 ↔ (𝑀‘{𝑥 ∈ 𝑂 ∣ ¬ (𝜑 ∧ 𝜓)}) = 0)
7		measbasedom 32170	. . . . . . . . 9 ⊢ (𝑀 ∈ ∪ ran measures ↔ 𝑀 ∈ (measures‘dom 𝑀))
8	7	biimpi 215	. . . . . . . 8 ⊢ (𝑀 ∈ ∪ ran measures → 𝑀 ∈ (measures‘dom 𝑀))
9	8	3ad2ant1 1132	. . . . . . 7 ⊢ ((𝑀 ∈ ∪ ran measures ∧ {𝑥 ∈ 𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥 ∈ 𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) → 𝑀 ∈ (measures‘dom 𝑀))
10	9	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))
11		simp2 1136	. . . . . . 7 ⊢ ((𝑀 ∈ ∪ ran measures ∧ {𝑥 ∈ 𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥 ∈ 𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) → {𝑥 ∈ 𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀)
12	11	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)
13		dmmeas 32169	. . . . . . . . . 10 ⊢ (𝑀 ∈ ∪ ran measures → dom 𝑀 ∈ ∪ ran sigAlgebra)
14		unelsiga 32102	. . . . . . . . . 10 ⊢ ((dom 𝑀 ∈ ∪ ran sigAlgebra ∧ {𝑥 ∈ 𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥 ∈ 𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) → ({𝑥 ∈ 𝑂 ∣ ¬ 𝜑} ∪ {𝑥 ∈ 𝑂 ∣ ¬ 𝜓}) ∈ dom 𝑀)
15	13, 14	syl3an1 1162	. . . . . . . . 9 ⊢ ((𝑀 ∈ ∪ ran measures ∧ {𝑥 ∈ 𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥 ∈ 𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) → ({𝑥 ∈ 𝑂 ∣ ¬ 𝜑} ∪ {𝑥 ∈ 𝑂 ∣ ¬ 𝜓}) ∈ dom 𝑀)
16		ssun1 4106	. . . . . . . . . 10 ⊢ {𝑥 ∈ 𝑂 ∣ ¬ 𝜑} ⊆ ({𝑥 ∈ 𝑂 ∣ ¬ 𝜑} ∪ {𝑥 ∈ 𝑂 ∣ ¬ 𝜓})
17	16	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}))
18	9, 11, 15, 17	measssd 32183	. . . . . . . 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})))
19	18	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})))
20		simpr 485	. . . . . . 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)
21	19, 20	breqtrd 5100	. . . . . 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)
22		measle0 32176	. . . . . 6 ⊢ ((𝑀 ∈ (measures‘dom 𝑀) ∧ {𝑥 ∈ 𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ (𝑀‘{𝑥 ∈ 𝑂 ∣ ¬ 𝜑}) ≤ 0) → (𝑀‘{𝑥 ∈ 𝑂 ∣ ¬ 𝜑}) = 0)
23	10, 12, 21, 22	syl3anc 1370	. . . . 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)
24		simp3 1137	. . . . . . 7 ⊢ ((𝑀 ∈ ∪ ran measures ∧ {𝑥 ∈ 𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥 ∈ 𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) → {𝑥 ∈ 𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀)
25	24	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)
26		ssun2 4107	. . . . . . . . . 10 ⊢ {𝑥 ∈ 𝑂 ∣ ¬ 𝜓} ⊆ ({𝑥 ∈ 𝑂 ∣ ¬ 𝜑} ∪ {𝑥 ∈ 𝑂 ∣ ¬ 𝜓})
27	26	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}))
28	9, 24, 15, 27	measssd 32183	. . . . . . . 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})))
29	28	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})))
30	29, 20	breqtrd 5100	. . . . . 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)
31		measle0 32176	. . . . . 6 ⊢ ((𝑀 ∈ (measures‘dom 𝑀) ∧ {𝑥 ∈ 𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀 ∧ (𝑀‘{𝑥 ∈ 𝑂 ∣ ¬ 𝜓}) ≤ 0) → (𝑀‘{𝑥 ∈ 𝑂 ∣ ¬ 𝜓}) = 0)
32	10, 25, 30, 31	syl3anc 1370	. . . . 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)
33	23, 32	jca 512	. . . 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))
34	9	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))
35		measbase 32165	. . . . . . 7 ⊢ (𝑀 ∈ (measures‘dom 𝑀) → dom 𝑀 ∈ ∪ ran sigAlgebra)
36	34, 35	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)
37	11	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)
38	24	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)
39	36, 37, 38, 14	syl3anc 1370	. . . . 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)
40	34, 37, 38	measunl 32184	. . . . . 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})))
41		simprl 768	. . . . . . . 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)
42		simprr 770	. . . . . . . 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)
43	41, 42	oveq12d 7293	. . . . . . 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))
44		0xr 11022	. . . . . . . 8 ⊢ 0 ∈ ℝ*
45		xaddid1 12975	. . . . . . . 8 ⊢ (0 ∈ ℝ* → (0 +𝑒 0) = 0)
46	44, 45	ax-mp 5	. . . . . . 7 ⊢ (0 +𝑒 0) = 0
47	43, 46	eqtrdi 2794	. . . . . 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)
48	40, 47	breqtrd 5100	. . . . 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)
49		measle0 32176	. . . . 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)
50	34, 39, 48, 49	syl3anc 1370	. . . 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)
51	33, 50	impbida 798	. . 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)))
52	6, 51	bitr3id 285	. 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)))
53		aean.1	. . . 4 ⊢ ∪ dom 𝑀 = 𝑂
54	53	braew 32210	. . 3 ⊢ (𝑀 ∈ ∪ ran measures → ({𝑥 ∈ 𝑂 ∣ (𝜑 ∧ 𝜓)}a.e.𝑀 ↔ (𝑀‘{𝑥 ∈ 𝑂 ∣ ¬ (𝜑 ∧ 𝜓)}) = 0))
55	54	3ad2ant1 1132	. 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))
56	53	braew 32210	. . . 4 ⊢ (𝑀 ∈ ∪ ran measures → ({𝑥 ∈ 𝑂 ∣ 𝜑}a.e.𝑀 ↔ (𝑀‘{𝑥 ∈ 𝑂 ∣ ¬ 𝜑}) = 0))
57	53	braew 32210	. . . 4 ⊢ (𝑀 ∈ ∪ ran measures → ({𝑥 ∈ 𝑂 ∣ 𝜓}a.e.𝑀 ↔ (𝑀‘{𝑥 ∈ 𝑂 ∣ ¬ 𝜓}) = 0))
58	56, 57	anbi12d 631	. . 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)))
59	58	3ad2ant1 1132	. 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)))
60	52, 55, 59	3bitr4d 311	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 205   ∧ wa 396   ∨ wo 844   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106  {crab 3068   ∪ cun 3885   ⊆ wss 3887  ∪ cuni 4839   class class class wbr 5074  dom cdm 5589  ran crn 5590  ‘cfv 6433  (class class class)co 7275  0cc0 10871  ℝ^*cxr 11008   ≤ cle 11010   +_𝑒 cxad 12846  sigAlgebracsiga 32076  measurescmeas 32163  a.e.cae 32205

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-inf2 9399  ax-ac2 10219  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948  ax-pre-sup 10949  ax-addf 10950  ax-mulf 10951

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-iin 4927  df-disj 5040  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-se 5545  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-isom 6442  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-of 7533  df-om 7713  df-1st 7831  df-2nd 7832  df-supp 7978  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-2o 8298  df-er 8498  df-map 8617  df-pm 8618  df-ixp 8686  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-fsupp 9129  df-fi 9170  df-sup 9201  df-inf 9202  df-oi 9269  df-dju 9659  df-card 9697  df-acn 9700  df-ac 9872  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-div 11633  df-nn 11974  df-2 12036  df-3 12037  df-4 12038  df-5 12039  df-6 12040  df-7 12041  df-8 12042  df-9 12043  df-n0 12234  df-z 12320  df-dec 12438  df-uz 12583  df-q 12689  df-rp 12731  df-xneg 12848  df-xadd 12849  df-xmul 12850  df-ioo 13083  df-ioc 13084  df-ico 13085  df-icc 13086  df-fz 13240  df-fzo 13383  df-fl 13512  df-mod 13590  df-seq 13722  df-exp 13783  df-fac 13988  df-bc 14017  df-hash 14045  df-shft 14778  df-cj 14810  df-re 14811  df-im 14812  df-sqrt 14946  df-abs 14947  df-limsup 15180  df-clim 15197  df-rlim 15198  df-sum 15398  df-ef 15777  df-sin 15779  df-cos 15780  df-pi 15782  df-struct 16848  df-sets 16865  df-slot 16883  df-ndx 16895  df-base 16913  df-ress 16942  df-plusg 16975  df-mulr 16976  df-starv 16977  df-sca 16978  df-vsca 16979  df-ip 16980  df-tset 16981  df-ple 16982  df-ds 16984  df-unif 16985  df-hom 16986  df-cco 16987  df-rest 17133  df-topn 17134  df-0g 17152  df-gsum 17153  df-topgen 17154  df-pt 17155  df-prds 17158  df-ordt 17212  df-xrs 17213  df-qtop 17218  df-imas 17219  df-xps 17221  df-mre 17295  df-mrc 17296  df-acs 17298  df-ps 18284  df-tsr 18285  df-plusf 18325  df-mgm 18326  df-sgrp 18375  df-mnd 18386  df-mhm 18430  df-submnd 18431  df-grp 18580  df-minusg 18581  df-sbg 18582  df-mulg 18701  df-subg 18752  df-cntz 18923  df-cmn 19388  df-abl 19389  df-mgp 19721  df-ur 19738  df-ring 19785  df-cring 19786  df-subrg 20022  df-abv 20077  df-lmod 20125  df-scaf 20126  df-sra 20434  df-rgmod 20435  df-psmet 20589  df-xmet 20590  df-met 20591  df-bl 20592  df-mopn 20593  df-fbas 20594  df-fg 20595  df-cnfld 20598  df-top 22043  df-topon 22060  df-topsp 22082  df-bases 22096  df-cld 22170  df-ntr 22171  df-cls 22172  df-nei 22249  df-lp 22287  df-perf 22288  df-cn 22378  df-cnp 22379  df-haus 22466  df-tx 22713  df-hmeo 22906  df-fil 22997  df-fm 23089  df-flim 23090  df-flf 23091  df-tmd 23223  df-tgp 23224  df-tsms 23278  df-trg 23311  df-xms 23473  df-ms 23474  df-tms 23475  df-nm 23738  df-ngp 23739  df-nrg 23741  df-nlm 23742  df-ii 24040  df-cncf 24041  df-limc 25030  df-dv 25031  df-log 25712  df-esum 31996  df-siga 32077  df-meas 32164  df-ae 32207

This theorem is referenced by: (None)