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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > truae | Structured version Visualization version GIF version |
Description: A truth holds almost everywhere. (Contributed by Thierry Arnoux, 20-Oct-2017.) |
Ref | Expression |
---|---|
truae.1 | ⊢ ∪ dom 𝑀 = 𝑂 |
truae.2 | ⊢ (𝜑 → 𝑀 ∈ ∪ ran measures) |
truae.3 | ⊢ (𝜑 → 𝜓) |
Ref | Expression |
---|---|
truae | ⊢ (𝜑 → {𝑥 ∈ 𝑂 ∣ 𝜓}a.e.𝑀) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | truae.3 | . . . . . . . 8 ⊢ (𝜑 → 𝜓) | |
2 | 1 | pm2.24d 149 | . . . . . . 7 ⊢ (𝜑 → (¬ 𝜓 → 𝑥 ∈ ∅)) |
3 | 2 | ralrimivw 3148 | . . . . . 6 ⊢ (𝜑 → ∀𝑥 ∈ 𝑂 (¬ 𝜓 → 𝑥 ∈ ∅)) |
4 | rabss 3875 | . . . . . 6 ⊢ ({𝑥 ∈ 𝑂 ∣ ¬ 𝜓} ⊆ ∅ ↔ ∀𝑥 ∈ 𝑂 (¬ 𝜓 → 𝑥 ∈ ∅)) | |
5 | 3, 4 | sylibr 226 | . . . . 5 ⊢ (𝜑 → {𝑥 ∈ 𝑂 ∣ ¬ 𝜓} ⊆ ∅) |
6 | ss0 4170 | . . . . 5 ⊢ ({𝑥 ∈ 𝑂 ∣ ¬ 𝜓} ⊆ ∅ → {𝑥 ∈ 𝑂 ∣ ¬ 𝜓} = ∅) | |
7 | 5, 6 | syl 17 | . . . 4 ⊢ (𝜑 → {𝑥 ∈ 𝑂 ∣ ¬ 𝜓} = ∅) |
8 | 7 | fveq2d 6415 | . . 3 ⊢ (𝜑 → (𝑀‘{𝑥 ∈ 𝑂 ∣ ¬ 𝜓}) = (𝑀‘∅)) |
9 | truae.2 | . . . 4 ⊢ (𝜑 → 𝑀 ∈ ∪ ran measures) | |
10 | measbasedom 30781 | . . . . 5 ⊢ (𝑀 ∈ ∪ ran measures ↔ 𝑀 ∈ (measures‘dom 𝑀)) | |
11 | measvnul 30785 | . . . . 5 ⊢ (𝑀 ∈ (measures‘dom 𝑀) → (𝑀‘∅) = 0) | |
12 | 10, 11 | sylbi 209 | . . . 4 ⊢ (𝑀 ∈ ∪ ran measures → (𝑀‘∅) = 0) |
13 | 9, 12 | syl 17 | . . 3 ⊢ (𝜑 → (𝑀‘∅) = 0) |
14 | 8, 13 | eqtrd 2833 | . 2 ⊢ (𝜑 → (𝑀‘{𝑥 ∈ 𝑂 ∣ ¬ 𝜓}) = 0) |
15 | truae.1 | . . . 4 ⊢ ∪ dom 𝑀 = 𝑂 | |
16 | 15 | braew 30821 | . . 3 ⊢ (𝑀 ∈ ∪ ran measures → ({𝑥 ∈ 𝑂 ∣ 𝜓}a.e.𝑀 ↔ (𝑀‘{𝑥 ∈ 𝑂 ∣ ¬ 𝜓}) = 0)) |
17 | 9, 16 | syl 17 | . 2 ⊢ (𝜑 → ({𝑥 ∈ 𝑂 ∣ 𝜓}a.e.𝑀 ↔ (𝑀‘{𝑥 ∈ 𝑂 ∣ ¬ 𝜓}) = 0)) |
18 | 14, 17 | mpbird 249 | 1 ⊢ (𝜑 → {𝑥 ∈ 𝑂 ∣ 𝜓}a.e.𝑀) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 198 = wceq 1653 ∈ wcel 2157 ∀wral 3089 {crab 3093 ⊆ wss 3769 ∅c0 4115 ∪ cuni 4628 class class class wbr 4843 dom cdm 5312 ran crn 5313 ‘cfv 6101 0cc0 10224 measurescmeas 30774 a.e.cae 30816 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-fal 1667 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ral 3094 df-rex 3095 df-rab 3098 df-v 3387 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-op 4375 df-uni 4629 df-br 4844 df-opab 4906 df-mpt 4923 df-id 5220 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-fv 6109 df-ov 6881 df-esum 30606 df-meas 30775 df-ae 30818 |
This theorem is referenced by: (None) |
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