Theorem aean 34811

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 4260	. . . . . 6 ⊢ ({𝑥 ∈ 𝑂 ∣ ¬ 𝜑} ∪ {𝑥 ∈ 𝑂 ∣ ¬ 𝜓}) = {𝑥 ∈ 𝑂 ∣ (¬ 𝜑 ∨ ¬ 𝜓)}
2		ianor 997	. . . . . . 7 ⊢ (¬ (𝜑 ∧ 𝜓) ↔ (¬ 𝜑 ∨ ¬ 𝜓))
3	2	rabbii 3417	. . . . . 6 ⊢ {𝑥 ∈ 𝑂 ∣ ¬ (𝜑 ∧ 𝜓)} = {𝑥 ∈ 𝑂 ∣ (¬ 𝜑 ∨ ¬ 𝜓)}
4	1, 3	eqtr4i 2786	. . . . 5 ⊢ ({𝑥 ∈ 𝑂 ∣ ¬ 𝜑} ∪ {𝑥 ∈ 𝑂 ∣ ¬ 𝜓}) = {𝑥 ∈ 𝑂 ∣ ¬ (𝜑 ∧ 𝜓)}
5	4	fveq2i 6876	. . . 4 ⊢ (𝑀‘({𝑥 ∈ 𝑂 ∣ ¬ 𝜑} ∪ {𝑥 ∈ 𝑂 ∣ ¬ 𝜓})) = (𝑀‘{𝑥 ∈ 𝑂 ∣ ¬ (𝜑 ∧ 𝜓)})
6	5	eqeq1i 2765	. . 3 ⊢ ((𝑀‘({𝑥 ∈ 𝑂 ∣ ¬ 𝜑} ∪ {𝑥 ∈ 𝑂 ∣ ¬ 𝜓})) = 0 ↔ (𝑀‘{𝑥 ∈ 𝑂 ∣ ¬ (𝜑 ∧ 𝜓)}) = 0)
7		measbasedom 34769	. . . . . . . . 9 ⊢ (𝑀 ∈ ∪ ran measures ↔ 𝑀 ∈ (measures‘dom 𝑀))
8	7	biimpi 219	. . . . . . . 8 ⊢ (𝑀 ∈ ∪ ran measures → 𝑀 ∈ (measures‘dom 𝑀))
9	8	3ad2ant1 1151	. . . . . . 7 ⊢ ((𝑀 ∈ ∪ ran measures ∧ {𝑥 ∈ 𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥 ∈ 𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) → 𝑀 ∈ (measures‘dom 𝑀))
10	9	adantr 486	. . . . . 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 1155	. . . . . . 7 ⊢ ((𝑀 ∈ ∪ ran measures ∧ {𝑥 ∈ 𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥 ∈ 𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) → {𝑥 ∈ 𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀)
12	11	adantr 486	. . . . . 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 34768	. . . . . . . . . 10 ⊢ (𝑀 ∈ ∪ ran measures → dom 𝑀 ∈ ∪ ran sigAlgebra)
14		unelsiga 34700	. . . . . . . . . 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 1181	. . . . . . . . 9 ⊢ ((𝑀 ∈ ∪ ran measures ∧ {𝑥 ∈ 𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥 ∈ 𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) → ({𝑥 ∈ 𝑂 ∣ ¬ 𝜑} ∪ {𝑥 ∈ 𝑂 ∣ ¬ 𝜓}) ∈ dom 𝑀)
16		ssun1 4123	. . . . . . . . . 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 34782	. . . . . . . 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 486	. . . . . . 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 490	. . . . . . 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 5130	. . . . . 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 34775	. . . . . 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 1398	. . . . 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 1156	. . . . . . 7 ⊢ ((𝑀 ∈ ∪ ran measures ∧ {𝑥 ∈ 𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥 ∈ 𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) → {𝑥 ∈ 𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀)
25	24	adantr 486	. . . . . 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 4124	. . . . . . . . . 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 34782	. . . . . . . 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 486	. . . . . . 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 5130	. . . . . 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 34775	. . . . . 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 1398	. . . . 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 521	. . . 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 486	. . . . 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 34764	. . . . . . 7 ⊢ (𝑀 ∈ (measures‘dom 𝑀) → dom 𝑀 ∈ ∪ ran sigAlgebra)
36	34, 35	syl 18	. . . . . 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 486	. . . . . 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 486	. . . . . 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 1398	. . . . 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 34783	. . . . . 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 783	. . . . . . . 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 785	. . . . . . . 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 7426	. . . . . . 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 11328	. . . . . . . 8 ⊢ 0 ∈ ℝ*
45		xaddrid 13341	. . . . . . . 8 ⊢ (0 ∈ ℝ* → (0 +𝑒 0) = 0)
46	44, 45	ax-mp 5	. . . . . . 7 ⊢ (0 +𝑒 0) = 0
47	43, 46	eqtrdi 2811	. . . . . 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 5130	. . . . 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 34775	. . . . 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 1398	. . . 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 813	. . 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 288	. 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 34809	. . 3 ⊢ (𝑀 ∈ ∪ ran measures → ({𝑥 ∈ 𝑂 ∣ (𝜑 ∧ 𝜓)}a.e.𝑀 ↔ (𝑀‘{𝑥 ∈ 𝑂 ∣ ¬ (𝜑 ∧ 𝜓)}) = 0))
55	54	3ad2ant1 1151	. 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 34809	. . . 4 ⊢ (𝑀 ∈ ∪ ran measures → ({𝑥 ∈ 𝑂 ∣ 𝜑}a.e.𝑀 ↔ (𝑀‘{𝑥 ∈ 𝑂 ∣ ¬ 𝜑}) = 0))
57	53	braew 34809	. . . 4 ⊢ (𝑀 ∈ ∪ ran measures → ({𝑥 ∈ 𝑂 ∣ 𝜓}a.e.𝑀 ↔ (𝑀‘{𝑥 ∈ 𝑂 ∣ ¬ 𝜓}) = 0))
58	56, 57	anbi12d 644	. . 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 1151	. 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 314	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)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  {crab 3412   ∪ cun 3896   ⊆ wss 3898  ∪ cuni 4866   class class class wbr 5102  dom cdm 5647  ran crn 5648  ‘cfv 6527  (class class class)co 7408  0cc0 11172  ℝ^*cxr 11314   ≤ cle 11316   +_𝑒 cxad 13209  sigAlgebracsiga 34674  measurescmeas 34762  a.e.cae 34804

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7734  ax-inf2 9620  ax-cnex 11228  ax-resscn 11229  ax-1cn 11230  ax-icn 11231  ax-addcl 11232  ax-addrcl 11233  ax-mulcl 11234  ax-mulrcl 11235  ax-mulcom 11236  ax-addass 11237  ax-mulass 11238  ax-distr 11239  ax-i2m1 11240  ax-1ne0 11241  ax-1rid 11242  ax-rnegex 11243  ax-rrecex 11244  ax-cnre 11245  ax-pre-lttri 11246  ax-pre-lttrn 11247  ax-pre-ltadd 11248  ax-pre-mulgt0 11249  ax-pre-sup 11250  ax-addf 11251  ax-mulf 11252

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-tp 4588  df-op 4590  df-uni 4867  df-int 4907  df-iun 4952  df-iin 4953  df-disj 5070  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-se 5601  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-pred 6293  df-ord 6354  df-on 6355  df-lim 6356  df-suc 6357  df-iota 6483  df-fun 6529  df-fn 6530  df-f 6531  df-f1 6532  df-fo 6533  df-f1o 6534  df-fv 6535  df-isom 6536  df-riota 7365  df-ov 7411  df-oprab 7412  df-mpo 7413  df-of 7676  df-om 7861  df-1st 7984  df-2nd 7985  df-supp 8156  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-1o 8454  df-2o 8455  df-er 8695  df-map 8827  df-pm 8828  df-ixp 8904  df-en 8952  df-dom 8953  df-sdom 8954  df-fin 8955  df-fsupp 9332  df-fi 9381  df-sup 9412  df-inf 9413  df-oi 9482  df-dju 9954  df-card 9992  df-pnf 11317  df-mnf 11318  df-xr 11319  df-ltxr 11320  df-le 11321  df-sub 11515  df-neg 11516  df-div 11944  df-nn 12306  df-2 12375  df-3 12376  df-4 12377  df-5 12378  df-6 12379  df-7 12380  df-8 12381  df-9 12382  df-n0 12577  df-z 12664  df-dec 12785  df-uz 12936  df-q 13046  df-rp 13091  df-xneg 13211  df-xadd 13212  df-xmul 13213  df-ioo 13450  df-ioc 13451  df-ico 13452  df-icc 13453  df-fz 13610  df-fzo 13758  df-fl 13901  df-mod 13979  df-seq 14114  df-exp 14174  df-fac 14386  df-bc 14415  df-hash 14443  df-shft 15188  df-cj 15234  df-re 15235  df-im 15236  df-sqrt 15370  df-abs 15371  df-limsup 15606  df-clim 15623  df-rlim 15624  df-sum 15822  df-ef 16201  df-sin 16203  df-cos 16204  df-pi 16206  df-struct 17287  df-sets 17304  df-slot 17322  df-ndx 17334  df-base 17350  df-ress 17371  df-plusg 17403  df-mulr 17404  df-starv 17405  df-sca 17406  df-vsca 17407  df-ip 17408  df-tset 17409  df-ple 17410  df-ds 17412  df-unif 17413  df-hom 17414  df-cco 17415  df-rest 17555  df-topn 17556  df-0g 17574  df-gsum 17575  df-topgen 17576  df-pt 17577  df-prds 17580  df-ordt 17635  df-xrs 17636  df-qtop 17641  df-imas 17642  df-xps 17644  df-mre 17718  df-mrc 17719  df-acs 17721  df-ps 18702  df-tsr 18703  df-plusf 18777  df-mgm 18778  df-sgrp 18870  df-mnd 18886  df-mhm 18940  df-submnd 18941  df-grp 19109  df-minusg 19110  df-sbg 19111  df-mulg 19240  df-subg 19295  df-cntz 19493  df-cmn 19958  df-abl 19959  df-mgp 20323  df-rng 20337  df-ur 20370  df-ring 20423  df-cring 20424  df-subrng 20760  df-subrg 20784  df-abv 21028  df-lmod 21099  df-scaf 21100  df-sra 21410  df-rgmod 21411  df-psmet 21632  df-xmet 21633  df-met 21634  df-bl 21635  df-mopn 21636  df-fbas 21637  df-fg 21638  df-cnfld 21641  df-top 23174  df-topon 23191  df-topsp 23213  df-bases 23226  df-cld 23299  df-ntr 23300  df-cls 23301  df-nei 23378  df-lp 23416  df-perf 23417  df-cn 23507  df-cnp 23508  df-haus 23595  df-tx 23843  df-hmeo 24036  df-fil 24127  df-fm 24219  df-flim 24220  df-flf 24221  df-tmd 24353  df-tgp 24354  df-tsms 24408  df-trg 24441  df-xms 24601  df-ms 24602  df-tms 24603  df-nm 24863  df-ngp 24864  df-nrg 24866  df-nlm 24867  df-ii 25160  df-cncf 25161  df-limc 26148  df-dv 26149  df-log 26848  df-esum 34594  df-siga 34675  df-meas 34763  df-ae 34806

This theorem is used by: (None)