Theorem aean 34247

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 4263	. . . . . 6 ⊢ ({𝑥 ∈ 𝑂 ∣ ¬ 𝜑} ∪ {𝑥 ∈ 𝑂 ∣ ¬ 𝜓}) = {𝑥 ∈ 𝑂 ∣ (¬ 𝜑 ∨ ¬ 𝜓)}
2		ianor 983	. . . . . . 7 ⊢ (¬ (𝜑 ∧ 𝜓) ↔ (¬ 𝜑 ∨ ¬ 𝜓))
3	2	rabbii 3398	. . . . . 6 ⊢ {𝑥 ∈ 𝑂 ∣ ¬ (𝜑 ∧ 𝜓)} = {𝑥 ∈ 𝑂 ∣ (¬ 𝜑 ∨ ¬ 𝜓)}
4	1, 3	eqtr4i 2756	. . . . 5 ⊢ ({𝑥 ∈ 𝑂 ∣ ¬ 𝜑} ∪ {𝑥 ∈ 𝑂 ∣ ¬ 𝜓}) = {𝑥 ∈ 𝑂 ∣ ¬ (𝜑 ∧ 𝜓)}
5	4	fveq2i 6820	. . . 4 ⊢ (𝑀‘({𝑥 ∈ 𝑂 ∣ ¬ 𝜑} ∪ {𝑥 ∈ 𝑂 ∣ ¬ 𝜓})) = (𝑀‘{𝑥 ∈ 𝑂 ∣ ¬ (𝜑 ∧ 𝜓)})
6	5	eqeq1i 2735	. . 3 ⊢ ((𝑀‘({𝑥 ∈ 𝑂 ∣ ¬ 𝜑} ∪ {𝑥 ∈ 𝑂 ∣ ¬ 𝜓})) = 0 ↔ (𝑀‘{𝑥 ∈ 𝑂 ∣ ¬ (𝜑 ∧ 𝜓)}) = 0)
7		measbasedom 34205	. . . . . . . . 9 ⊢ (𝑀 ∈ ∪ ran measures ↔ 𝑀 ∈ (measures‘dom 𝑀))
8	7	biimpi 216	. . . . . . . 8 ⊢ (𝑀 ∈ ∪ ran measures → 𝑀 ∈ (measures‘dom 𝑀))
9	8	3ad2ant1 1133	. . . . . . 7 ⊢ ((𝑀 ∈ ∪ ran measures ∧ {𝑥 ∈ 𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥 ∈ 𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) → 𝑀 ∈ (measures‘dom 𝑀))
10	9	adantr 480	. . . . . 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 1137	. . . . . . 7 ⊢ ((𝑀 ∈ ∪ ran measures ∧ {𝑥 ∈ 𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥 ∈ 𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) → {𝑥 ∈ 𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀)
12	11	adantr 480	. . . . . 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 34204	. . . . . . . . . 10 ⊢ (𝑀 ∈ ∪ ran measures → dom 𝑀 ∈ ∪ ran sigAlgebra)
14		unelsiga 34137	. . . . . . . . . 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 1163	. . . . . . . . 9 ⊢ ((𝑀 ∈ ∪ ran measures ∧ {𝑥 ∈ 𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥 ∈ 𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) → ({𝑥 ∈ 𝑂 ∣ ¬ 𝜑} ∪ {𝑥 ∈ 𝑂 ∣ ¬ 𝜓}) ∈ dom 𝑀)
16		ssun1 4126	. . . . . . . . . 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 34218	. . . . . . . 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 480	. . . . . . 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 484	. . . . . . 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 5115	. . . . . 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 34211	. . . . . 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 1373	. . . . 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 1138	. . . . . . 7 ⊢ ((𝑀 ∈ ∪ ran measures ∧ {𝑥 ∈ 𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥 ∈ 𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) → {𝑥 ∈ 𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀)
25	24	adantr 480	. . . . . 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 4127	. . . . . . . . . 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 34218	. . . . . . . 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 480	. . . . . . 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 5115	. . . . . 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 34211	. . . . . 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 1373	. . . . 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 511	. . . 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 480	. . . . 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 34200	. . . . . . 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 480	. . . . . 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 480	. . . . . 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 1373	. . . . 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 34219	. . . . . 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 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 ∣ ¬ 𝜑}) = 0)
42		simprr 772	. . . . . . . 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 7359	. . . . . . 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 11151	. . . . . . . 8 ⊢ 0 ∈ ℝ*
45		xaddrid 13132	. . . . . . . 8 ⊢ (0 ∈ ℝ* → (0 +𝑒 0) = 0)
46	44, 45	ax-mp 5	. . . . . . 7 ⊢ (0 +𝑒 0) = 0
47	43, 46	eqtrdi 2781	. . . . . 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 5115	. . . . 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 34211	. . . . 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 1373	. . . 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 800	. . 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 34245	. . 3 ⊢ (𝑀 ∈ ∪ ran measures → ({𝑥 ∈ 𝑂 ∣ (𝜑 ∧ 𝜓)}a.e.𝑀 ↔ (𝑀‘{𝑥 ∈ 𝑂 ∣ ¬ (𝜑 ∧ 𝜓)}) = 0))
55	54	3ad2ant1 1133	. 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 34245	. . . 4 ⊢ (𝑀 ∈ ∪ ran measures → ({𝑥 ∈ 𝑂 ∣ 𝜑}a.e.𝑀 ↔ (𝑀‘{𝑥 ∈ 𝑂 ∣ ¬ 𝜑}) = 0))
57	53	braew 34245	. . . 4 ⊢ (𝑀 ∈ ∪ ran measures → ({𝑥 ∈ 𝑂 ∣ 𝜓}a.e.𝑀 ↔ (𝑀‘{𝑥 ∈ 𝑂 ∣ ¬ 𝜓}) = 0))
58	56, 57	anbi12d 632	. . 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 1133	. 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 206   ∧ wa 395   ∨ wo 847   ∧ w3a 1086   = wceq 1541   ∈ wcel 2110  {crab 3393   ∪ cun 3898   ⊆ wss 3900  ∪ cuni 4857   class class class wbr 5089  dom cdm 5614  ran crn 5615  ‘cfv 6477  (class class class)co 7341  0cc0 10998  ℝ^*cxr 11137   ≤ cle 11139   +_𝑒 cxad 13001  sigAlgebracsiga 34111  measurescmeas 34198  a.e.cae 34240

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2112  ax-9 2120  ax-10 2143  ax-11 2159  ax-12 2179  ax-ext 2702  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7663  ax-inf2 9526  ax-ac2 10346  ax-cnex 11054  ax-resscn 11055  ax-1cn 11056  ax-icn 11057  ax-addcl 11058  ax-addrcl 11059  ax-mulcl 11060  ax-mulrcl 11061  ax-mulcom 11062  ax-addass 11063  ax-mulass 11064  ax-distr 11065  ax-i2m1 11066  ax-1ne0 11067  ax-1rid 11068  ax-rnegex 11069  ax-rrecex 11070  ax-cnre 11071  ax-pre-lttri 11072  ax-pre-lttrn 11073  ax-pre-ltadd 11074  ax-pre-mulgt0 11075  ax-pre-sup 11076  ax-addf 11077  ax-mulf 11078

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-rmo 3344  df-reu 3345  df-rab 3394  df-v 3436  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-pss 3920  df-nul 4282  df-if 4474  df-pw 4550  df-sn 4575  df-pr 4577  df-tp 4579  df-op 4581  df-uni 4858  df-int 4896  df-iun 4941  df-iin 4942  df-disj 5057  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-se 5568  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6244  df-ord 6305  df-on 6306  df-lim 6307  df-suc 6308  df-iota 6433  df-fun 6479  df-fn 6480  df-f 6481  df-f1 6482  df-fo 6483  df-f1o 6484  df-fv 6485  df-isom 6486  df-riota 7298  df-ov 7344  df-oprab 7345  df-mpo 7346  df-of 7605  df-om 7792  df-1st 7916  df-2nd 7917  df-supp 8086  df-frecs 8206  df-wrecs 8237  df-recs 8286  df-rdg 8324  df-1o 8380  df-2o 8381  df-er 8617  df-map 8747  df-pm 8748  df-ixp 8817  df-en 8865  df-dom 8866  df-sdom 8867  df-fin 8868  df-fsupp 9241  df-fi 9290  df-sup 9321  df-inf 9322  df-oi 9391  df-dju 9786  df-card 9824  df-acn 9827  df-ac 9999  df-pnf 11140  df-mnf 11141  df-xr 11142  df-ltxr 11143  df-le 11144  df-sub 11338  df-neg 11339  df-div 11767  df-nn 12118  df-2 12180  df-3 12181  df-4 12182  df-5 12183  df-6 12184  df-7 12185  df-8 12186  df-9 12187  df-n0 12374  df-z 12461  df-dec 12581  df-uz 12725  df-q 12839  df-rp 12883  df-xneg 13003  df-xadd 13004  df-xmul 13005  df-ioo 13241  df-ioc 13242  df-ico 13243  df-icc 13244  df-fz 13400  df-fzo 13547  df-fl 13688  df-mod 13766  df-seq 13901  df-exp 13961  df-fac 14173  df-bc 14202  df-hash 14230  df-shft 14966  df-cj 14998  df-re 14999  df-im 15000  df-sqrt 15134  df-abs 15135  df-limsup 15370  df-clim 15387  df-rlim 15388  df-sum 15586  df-ef 15966  df-sin 15968  df-cos 15969  df-pi 15971  df-struct 17050  df-sets 17067  df-slot 17085  df-ndx 17097  df-base 17113  df-ress 17134  df-plusg 17166  df-mulr 17167  df-starv 17168  df-sca 17169  df-vsca 17170  df-ip 17171  df-tset 17172  df-ple 17173  df-ds 17175  df-unif 17176  df-hom 17177  df-cco 17178  df-rest 17318  df-topn 17319  df-0g 17337  df-gsum 17338  df-topgen 17339  df-pt 17340  df-prds 17343  df-ordt 17397  df-xrs 17398  df-qtop 17403  df-imas 17404  df-xps 17406  df-mre 17480  df-mrc 17481  df-acs 17483  df-ps 18464  df-tsr 18465  df-plusf 18539  df-mgm 18540  df-sgrp 18619  df-mnd 18635  df-mhm 18683  df-submnd 18684  df-grp 18841  df-minusg 18842  df-sbg 18843  df-mulg 18973  df-subg 19028  df-cntz 19222  df-cmn 19687  df-abl 19688  df-mgp 20052  df-rng 20064  df-ur 20093  df-ring 20146  df-cring 20147  df-subrng 20454  df-subrg 20478  df-abv 20717  df-lmod 20788  df-scaf 20789  df-sra 21100  df-rgmod 21101  df-psmet 21276  df-xmet 21277  df-met 21278  df-bl 21279  df-mopn 21280  df-fbas 21281  df-fg 21282  df-cnfld 21285  df-top 22802  df-topon 22819  df-topsp 22841  df-bases 22854  df-cld 22927  df-ntr 22928  df-cls 22929  df-nei 23006  df-lp 23044  df-perf 23045  df-cn 23135  df-cnp 23136  df-haus 23223  df-tx 23470  df-hmeo 23663  df-fil 23754  df-fm 23846  df-flim 23847  df-flf 23848  df-tmd 23980  df-tgp 23981  df-tsms 24035  df-trg 24068  df-xms 24228  df-ms 24229  df-tms 24230  df-nm 24490  df-ngp 24491  df-nrg 24493  df-nlm 24494  df-ii 24790  df-cncf 24791  df-limc 25787  df-dv 25788  df-log 26485  df-esum 34031  df-siga 34112  df-meas 34199  df-ae 34242

This theorem is referenced by: (None)