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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 146 | . . . . . . 7 ⊢ (𝜑 → (¬ 𝜓 → 𝑥 ∈ ∅)) |
3 | 2 | ralrimivw 2950 | . . . . . 6 ⊢ (𝜑 → ∀𝑥 ∈ 𝑂 (¬ 𝜓 → 𝑥 ∈ ∅)) |
4 | rabss 3642 | . . . . . 6 ⊢ ({𝑥 ∈ 𝑂 ∣ ¬ 𝜓} ⊆ ∅ ↔ ∀𝑥 ∈ 𝑂 (¬ 𝜓 → 𝑥 ∈ ∅)) | |
5 | 3, 4 | sylibr 223 | . . . . 5 ⊢ (𝜑 → {𝑥 ∈ 𝑂 ∣ ¬ 𝜓} ⊆ ∅) |
6 | ss0 3926 | . . . . 5 ⊢ ({𝑥 ∈ 𝑂 ∣ ¬ 𝜓} ⊆ ∅ → {𝑥 ∈ 𝑂 ∣ ¬ 𝜓} = ∅) | |
7 | 5, 6 | syl 17 | . . . 4 ⊢ (𝜑 → {𝑥 ∈ 𝑂 ∣ ¬ 𝜓} = ∅) |
8 | 7 | fveq2d 6107 | . . 3 ⊢ (𝜑 → (𝑀‘{𝑥 ∈ 𝑂 ∣ ¬ 𝜓}) = (𝑀‘∅)) |
9 | truae.2 | . . . 4 ⊢ (𝜑 → 𝑀 ∈ ∪ ran measures) | |
10 | measbasedom 29592 | . . . . 5 ⊢ (𝑀 ∈ ∪ ran measures ↔ 𝑀 ∈ (measures‘dom 𝑀)) | |
11 | measvnul 29596 | . . . . 5 ⊢ (𝑀 ∈ (measures‘dom 𝑀) → (𝑀‘∅) = 0) | |
12 | 10, 11 | sylbi 206 | . . . 4 ⊢ (𝑀 ∈ ∪ ran measures → (𝑀‘∅) = 0) |
13 | 9, 12 | syl 17 | . . 3 ⊢ (𝜑 → (𝑀‘∅) = 0) |
14 | 8, 13 | eqtrd 2644 | . 2 ⊢ (𝜑 → (𝑀‘{𝑥 ∈ 𝑂 ∣ ¬ 𝜓}) = 0) |
15 | truae.1 | . . . 4 ⊢ ∪ dom 𝑀 = 𝑂 | |
16 | 15 | braew 29632 | . . 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 246 | 1 ⊢ (𝜑 → {𝑥 ∈ 𝑂 ∣ 𝜓}a.e.𝑀) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 = wceq 1475 ∈ wcel 1977 ∀wral 2896 {crab 2900 ⊆ wss 3540 ∅c0 3874 ∪ cuni 4372 class class class wbr 4583 dom cdm 5038 ran crn 5039 ‘cfv 5804 0cc0 9815 measurescmeas 29585 a.e.cae 29627 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-fv 5812 df-ov 6552 df-esum 29417 df-meas 29586 df-ae 29629 |
This theorem is referenced by: (None) |
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