Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
||
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 147 | . . . . . . 7 ⊢ (𝜑 → (¬ 𝜓 → 𝑥 ∈ ∅)) |
3 | 2 | ralrimivw 2967 | . . . . . 6 ⊢ (𝜑 → ∀𝑥 ∈ 𝑂 (¬ 𝜓 → 𝑥 ∈ ∅)) |
4 | rabss 3679 | . . . . . 6 ⊢ ({𝑥 ∈ 𝑂 ∣ ¬ 𝜓} ⊆ ∅ ↔ ∀𝑥 ∈ 𝑂 (¬ 𝜓 → 𝑥 ∈ ∅)) | |
5 | 3, 4 | sylibr 224 | . . . . 5 ⊢ (𝜑 → {𝑥 ∈ 𝑂 ∣ ¬ 𝜓} ⊆ ∅) |
6 | ss0 3974 | . . . . 5 ⊢ ({𝑥 ∈ 𝑂 ∣ ¬ 𝜓} ⊆ ∅ → {𝑥 ∈ 𝑂 ∣ ¬ 𝜓} = ∅) | |
7 | 5, 6 | syl 17 | . . . 4 ⊢ (𝜑 → {𝑥 ∈ 𝑂 ∣ ¬ 𝜓} = ∅) |
8 | 7 | fveq2d 6195 | . . 3 ⊢ (𝜑 → (𝑀‘{𝑥 ∈ 𝑂 ∣ ¬ 𝜓}) = (𝑀‘∅)) |
9 | truae.2 | . . . 4 ⊢ (𝜑 → 𝑀 ∈ ∪ ran measures) | |
10 | measbasedom 30265 | . . . . 5 ⊢ (𝑀 ∈ ∪ ran measures ↔ 𝑀 ∈ (measures‘dom 𝑀)) | |
11 | measvnul 30269 | . . . . 5 ⊢ (𝑀 ∈ (measures‘dom 𝑀) → (𝑀‘∅) = 0) | |
12 | 10, 11 | sylbi 207 | . . . 4 ⊢ (𝑀 ∈ ∪ ran measures → (𝑀‘∅) = 0) |
13 | 9, 12 | syl 17 | . . 3 ⊢ (𝜑 → (𝑀‘∅) = 0) |
14 | 8, 13 | eqtrd 2656 | . 2 ⊢ (𝜑 → (𝑀‘{𝑥 ∈ 𝑂 ∣ ¬ 𝜓}) = 0) |
15 | truae.1 | . . . 4 ⊢ ∪ dom 𝑀 = 𝑂 | |
16 | 15 | braew 30305 | . . 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 247 | 1 ⊢ (𝜑 → {𝑥 ∈ 𝑂 ∣ 𝜓}a.e.𝑀) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 = wceq 1483 ∈ wcel 1990 ∀wral 2912 {crab 2916 ⊆ wss 3574 ∅c0 3915 ∪ cuni 4436 class class class wbr 4653 dom cdm 5114 ran crn 5115 ‘cfv 5888 0cc0 9936 measurescmeas 30258 a.e.cae 30300 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-fv 5896 df-ov 6653 df-esum 30090 df-meas 30259 df-ae 30302 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |