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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 154 | . . . . . . 7 ⊢ (𝜑 → (¬ 𝜓 → 𝑥 ∈ ∅)) |
3 | 2 | ralrimivw 3183 | . . . . . 6 ⊢ (𝜑 → ∀𝑥 ∈ 𝑂 (¬ 𝜓 → 𝑥 ∈ ∅)) |
4 | rabss 4047 | . . . . . 6 ⊢ ({𝑥 ∈ 𝑂 ∣ ¬ 𝜓} ⊆ ∅ ↔ ∀𝑥 ∈ 𝑂 (¬ 𝜓 → 𝑥 ∈ ∅)) | |
5 | 3, 4 | sylibr 236 | . . . . 5 ⊢ (𝜑 → {𝑥 ∈ 𝑂 ∣ ¬ 𝜓} ⊆ ∅) |
6 | ss0 4351 | . . . . 5 ⊢ ({𝑥 ∈ 𝑂 ∣ ¬ 𝜓} ⊆ ∅ → {𝑥 ∈ 𝑂 ∣ ¬ 𝜓} = ∅) | |
7 | 5, 6 | syl 17 | . . . 4 ⊢ (𝜑 → {𝑥 ∈ 𝑂 ∣ ¬ 𝜓} = ∅) |
8 | 7 | fveq2d 6673 | . . 3 ⊢ (𝜑 → (𝑀‘{𝑥 ∈ 𝑂 ∣ ¬ 𝜓}) = (𝑀‘∅)) |
9 | truae.2 | . . . 4 ⊢ (𝜑 → 𝑀 ∈ ∪ ran measures) | |
10 | measbasedom 31461 | . . . . 5 ⊢ (𝑀 ∈ ∪ ran measures ↔ 𝑀 ∈ (measures‘dom 𝑀)) | |
11 | measvnul 31465 | . . . . 5 ⊢ (𝑀 ∈ (measures‘dom 𝑀) → (𝑀‘∅) = 0) | |
12 | 10, 11 | sylbi 219 | . . . 4 ⊢ (𝑀 ∈ ∪ ran measures → (𝑀‘∅) = 0) |
13 | 9, 12 | syl 17 | . . 3 ⊢ (𝜑 → (𝑀‘∅) = 0) |
14 | 8, 13 | eqtrd 2856 | . 2 ⊢ (𝜑 → (𝑀‘{𝑥 ∈ 𝑂 ∣ ¬ 𝜓}) = 0) |
15 | truae.1 | . . . 4 ⊢ ∪ dom 𝑀 = 𝑂 | |
16 | 15 | braew 31501 | . . 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 259 | 1 ⊢ (𝜑 → {𝑥 ∈ 𝑂 ∣ 𝜓}a.e.𝑀) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 = wceq 1533 ∈ wcel 2110 ∀wral 3138 {crab 3142 ⊆ wss 3935 ∅c0 4290 ∪ cuni 4837 class class class wbr 5065 dom cdm 5554 ran crn 5555 ‘cfv 6354 0cc0 10536 measurescmeas 31454 a.e.cae 31496 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-br 5066 df-opab 5128 df-mpt 5146 df-id 5459 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-fv 6362 df-ov 7158 df-esum 31287 df-meas 31455 df-ae 31498 |
This theorem is referenced by: (None) |
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