Theorem aean 34643

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 4267	. . . . . 6 ⊢ ({𝑥 ∈ 𝑂 ∣ ¬ 𝜑} ∪ {𝑥 ∈ 𝑂 ∣ ¬ 𝜓}) = {𝑥 ∈ 𝑂 ∣ (¬ 𝜑 ∨ ¬ 𝜓)}
2		ianor 996	. . . . . . 7 ⊢ (¬ (𝜑 ∧ 𝜓) ↔ (¬ 𝜑 ∨ ¬ 𝜓))
3	2	rabbii 3420	. . . . . 6 ⊢ {𝑥 ∈ 𝑂 ∣ ¬ (𝜑 ∧ 𝜓)} = {𝑥 ∈ 𝑂 ∣ (¬ 𝜑 ∨ ¬ 𝜓)}
4	1, 3	eqtr4i 2788	. . . . 5 ⊢ ({𝑥 ∈ 𝑂 ∣ ¬ 𝜑} ∪ {𝑥 ∈ 𝑂 ∣ ¬ 𝜓}) = {𝑥 ∈ 𝑂 ∣ ¬ (𝜑 ∧ 𝜓)}
5	4	fveq2i 6884	. . . 4 ⊢ (𝑀‘({𝑥 ∈ 𝑂 ∣ ¬ 𝜑} ∪ {𝑥 ∈ 𝑂 ∣ ¬ 𝜓})) = (𝑀‘{𝑥 ∈ 𝑂 ∣ ¬ (𝜑 ∧ 𝜓)})
6	5	eqeq1i 2767	. . 3 ⊢ ((𝑀‘({𝑥 ∈ 𝑂 ∣ ¬ 𝜑} ∪ {𝑥 ∈ 𝑂 ∣ ¬ 𝜓})) = 0 ↔ (𝑀‘{𝑥 ∈ 𝑂 ∣ ¬ (𝜑 ∧ 𝜓)}) = 0)
7		measbasedom 34601	. . . . . . . . 9 ⊢ (𝑀 ∈ ∪ ran measures ↔ 𝑀 ∈ (measures‘dom 𝑀))
8	7	biimpi 219	. . . . . . . 8 ⊢ (𝑀 ∈ ∪ ran measures → 𝑀 ∈ (measures‘dom 𝑀))
9	8	3ad2ant1 1150	. . . . . . 7 ⊢ ((𝑀 ∈ ∪ ran measures ∧ {𝑥 ∈ 𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥 ∈ 𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) → 𝑀 ∈ (measures‘dom 𝑀))
10	9	adantr 485	. . . . . 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 1154	. . . . . . 7 ⊢ ((𝑀 ∈ ∪ ran measures ∧ {𝑥 ∈ 𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥 ∈ 𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) → {𝑥 ∈ 𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀)
12	11	adantr 485	. . . . . 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 34600	. . . . . . . . . 10 ⊢ (𝑀 ∈ ∪ ran measures → dom 𝑀 ∈ ∪ ran sigAlgebra)
14		unelsiga 34533	. . . . . . . . . 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 1180	. . . . . . . . 9 ⊢ ((𝑀 ∈ ∪ ran measures ∧ {𝑥 ∈ 𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥 ∈ 𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) → ({𝑥 ∈ 𝑂 ∣ ¬ 𝜑} ∪ {𝑥 ∈ 𝑂 ∣ ¬ 𝜓}) ∈ dom 𝑀)
16		ssun1 4130	. . . . . . . . . 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 34614	. . . . . . . 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 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}) ≤ (𝑀‘({𝑥 ∈ 𝑂 ∣ ¬ 𝜑} ∪ {𝑥 ∈ 𝑂 ∣ ¬ 𝜓})))
20		simpr 489	. . . . . . 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 5136	. . . . . 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 34607	. . . . . 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 1397	. . . . 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 1155	. . . . . . 7 ⊢ ((𝑀 ∈ ∪ ran measures ∧ {𝑥 ∈ 𝑂 ∣ ¬ 𝜑} ∈ dom 𝑀 ∧ {𝑥 ∈ 𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀) → {𝑥 ∈ 𝑂 ∣ ¬ 𝜓} ∈ dom 𝑀)
25	24	adantr 485	. . . . . 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 4131	. . . . . . . . . 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 34614	. . . . . . . 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 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 ∣ ¬ 𝜓}) ≤ (𝑀‘({𝑥 ∈ 𝑂 ∣ ¬ 𝜑} ∪ {𝑥 ∈ 𝑂 ∣ ¬ 𝜓})))
30	29, 20	breqtrd 5136	. . . . . 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 34607	. . . . . 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 1397	. . . . 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 520	. . . 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 485	. . . . 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 34596	. . . . . . 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 485	. . . . . 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 485	. . . . . 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 1397	. . . . 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 34615	. . . . . 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 782	. . . . . . . 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 784	. . . . . . . 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 7430	. . . . . . 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 11262	. . . . . . . 8 ⊢ 0 ∈ ℝ*
45		xaddrid 13273	. . . . . . . 8 ⊢ (0 ∈ ℝ* → (0 +𝑒 0) = 0)
46	44, 45	ax-mp 5	. . . . . . 7 ⊢ (0 +𝑒 0) = 0
47	43, 46	eqtrdi 2813	. . . . . 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 5136	. . . . 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 34607	. . . . 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 1397	. . . 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 812	. . 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 34641	. . 3 ⊢ (𝑀 ∈ ∪ ran measures → ({𝑥 ∈ 𝑂 ∣ (𝜑 ∧ 𝜓)}a.e.𝑀 ↔ (𝑀‘{𝑥 ∈ 𝑂 ∣ ¬ (𝜑 ∧ 𝜓)}) = 0))
55	54	3ad2ant1 1150	. 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 34641	. . . 4 ⊢ (𝑀 ∈ ∪ ran measures → ({𝑥 ∈ 𝑂 ∣ 𝜑}a.e.𝑀 ↔ (𝑀‘{𝑥 ∈ 𝑂 ∣ ¬ 𝜑}) = 0))
57	53	braew 34641	. . . 4 ⊢ (𝑀 ∈ ∪ ran measures → ({𝑥 ∈ 𝑂 ∣ 𝜓}a.e.𝑀 ↔ (𝑀‘{𝑥 ∈ 𝑂 ∣ ¬ 𝜓}) = 0))
58	56, 57	anbi12d 643	. . 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 1150	. 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 400   ∨ wo 860   ∧ w3a 1102   = wceq 1569   ∈ wcel 2142  {crab 3415   ∪ cun 3902   ⊆ wss 3904  ∪ cuni 4871   class class class wbr 5108  dom cdm 5660  ran crn 5661  ‘cfv 6536  (class class class)co 7412  0cc0 11106  ℝ^*cxr 11248   ≤ cle 11250   +_𝑒 cxad 13141  sigAlgebracsiga 34507  measurescmeas 34594  a.e.cae 34636

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pow 5335  ax-pr 5403  ax-un 7734  ax-inf2 9608  ax-ac2 10453  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183  ax-pre-sup 11184  ax-addf 11185  ax-mulf 11186

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1103  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3064  df-ral 3079  df-rex 3089  df-rmo 3368  df-reu 3369  df-rab 3416  df-v 3456  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4595  df-uni 4872  df-int 4912  df-iun 4957  df-iin 4958  df-disj 5076  df-br 5109  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5555  df-eprel 5560  df-po 5568  df-so 5569  df-fr 5613  df-se 5614  df-we 5615  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-isom 6545  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-of 7676  df-om 7861  df-1st 7984  df-2nd 7985  df-supp 8155  df-frecs 8276  df-wrecs 8307  df-recs 8356  df-rdg 8395  df-1o 8451  df-2o 8452  df-er 8692  df-map 8824  df-pm 8825  df-ixp 8894  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-fsupp 9320  df-fi 9369  df-sup 9400  df-inf 9401  df-oi 9470  df-dju 9894  df-card 9932  df-acn 9935  df-ac 10107  df-pnf 11251  df-mnf 11252  df-xr 11253  df-ltxr 11254  df-le 11255  df-sub 11449  df-neg 11450  df-div 11878  df-nn 12240  df-2 12309  df-3 12310  df-4 12311  df-5 12312  df-6 12313  df-7 12314  df-8 12315  df-9 12316  df-n0 12511  df-z 12598  df-dec 12718  df-uz 12869  df-q 12979  df-rp 13023  df-xneg 13143  df-xadd 13144  df-xmul 13145  df-ioo 13382  df-ioc 13383  df-ico 13384  df-icc 13385  df-fz 13542  df-fzo 13690  df-fl 13832  df-mod 13910  df-seq 14045  df-exp 14105  df-fac 14317  df-bc 14346  df-hash 14374  df-shft 15111  df-cj 15157  df-re 15158  df-im 15159  df-sqrt 15293  df-abs 15294  df-limsup 15529  df-clim 15546  df-rlim 15547  df-sum 15745  df-ef 16127  df-sin 16129  df-cos 16130  df-pi 16132  df-struct 17213  df-sets 17230  df-slot 17248  df-ndx 17260  df-base 17276  df-ress 17297  df-plusg 17329  df-mulr 17330  df-starv 17331  df-sca 17332  df-vsca 17333  df-ip 17334  df-tset 17335  df-ple 17336  df-ds 17338  df-unif 17339  df-hom 17340  df-cco 17341  df-rest 17481  df-topn 17482  df-0g 17500  df-gsum 17501  df-topgen 17502  df-pt 17503  df-prds 17506  df-ordt 17561  df-xrs 17562  df-qtop 17567  df-imas 17568  df-xps 17570  df-mre 17644  df-mrc 17645  df-acs 17647  df-ps 18628  df-tsr 18629  df-plusf 18703  df-mgm 18704  df-sgrp 18783  df-mnd 18799  df-mhm 18847  df-submnd 18848  df-grp 19009  df-minusg 19010  df-sbg 19011  df-mulg 19140  df-subg 19195  df-cntz 19393  df-cmn 19858  df-abl 19859  df-mgp 20223  df-rng 20237  df-ur 20270  df-ring 20323  df-cring 20324  df-subrng 20656  df-subrg 20680  df-abv 20923  df-lmod 20994  df-scaf 20995  df-sra 21305  df-rgmod 21306  df-psmet 21525  df-xmet 21526  df-met 21527  df-bl 21528  df-mopn 21529  df-fbas 21530  df-fg 21531  df-cnfld 21534  df-top 23062  df-topon 23079  df-topsp 23101  df-bases 23114  df-cld 23187  df-ntr 23188  df-cls 23189  df-nei 23266  df-lp 23304  df-perf 23305  df-cn 23395  df-cnp 23396  df-haus 23483  df-tx 23730  df-hmeo 23923  df-fil 24014  df-fm 24106  df-flim 24107  df-flf 24108  df-tmd 24240  df-tgp 24241  df-tsms 24295  df-trg 24328  df-xms 24488  df-ms 24489  df-tms 24490  df-nm 24750  df-ngp 24751  df-nrg 24753  df-nlm 24754  df-ii 25047  df-cncf 25048  df-limc 26036  df-dv 26037  df-log 26732  df-esum 34427  df-siga 34508  df-meas 34595  df-ae 34638

This theorem is used by: (None)