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Mirrors > Home > MPE Home > Th. List > Mathboxes > measvnul | Structured version Visualization version GIF version |
Description: The measure of the empty set is always zero. (Contributed by Thierry Arnoux, 26-Dec-2016.) |
Ref | Expression |
---|---|
measvnul | ⊢ (𝑀 ∈ (measures‘𝑆) → (𝑀‘∅) = 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | measbase 31456 | . . . 4 ⊢ (𝑀 ∈ (measures‘𝑆) → 𝑆 ∈ ∪ ran sigAlgebra) | |
2 | ismeas 31458 | . . . 4 ⊢ (𝑆 ∈ ∪ ran sigAlgebra → (𝑀 ∈ (measures‘𝑆) ↔ (𝑀:𝑆⟶(0[,]+∞) ∧ (𝑀‘∅) = 0 ∧ ∀𝑦 ∈ 𝒫 𝑆((𝑦 ≼ ω ∧ Disj 𝑥 ∈ 𝑦 𝑥) → (𝑀‘∪ 𝑦) = Σ*𝑥 ∈ 𝑦(𝑀‘𝑥))))) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ (𝑀 ∈ (measures‘𝑆) → (𝑀 ∈ (measures‘𝑆) ↔ (𝑀:𝑆⟶(0[,]+∞) ∧ (𝑀‘∅) = 0 ∧ ∀𝑦 ∈ 𝒫 𝑆((𝑦 ≼ ω ∧ Disj 𝑥 ∈ 𝑦 𝑥) → (𝑀‘∪ 𝑦) = Σ*𝑥 ∈ 𝑦(𝑀‘𝑥))))) |
4 | 3 | ibi 269 | . 2 ⊢ (𝑀 ∈ (measures‘𝑆) → (𝑀:𝑆⟶(0[,]+∞) ∧ (𝑀‘∅) = 0 ∧ ∀𝑦 ∈ 𝒫 𝑆((𝑦 ≼ ω ∧ Disj 𝑥 ∈ 𝑦 𝑥) → (𝑀‘∪ 𝑦) = Σ*𝑥 ∈ 𝑦(𝑀‘𝑥)))) |
5 | 4 | simp2d 1139 | 1 ⊢ (𝑀 ∈ (measures‘𝑆) → (𝑀‘∅) = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ∀wral 3138 ∅c0 4290 𝒫 cpw 4538 ∪ cuni 4837 Disj wdisj 5030 class class class wbr 5065 ran crn 5555 ⟶wf 6350 ‘cfv 6354 (class class class)co 7155 ωcom 7579 ≼ cdom 8506 0cc0 10536 +∞cpnf 10671 [,]cicc 12740 Σ*cesum 31286 sigAlgebracsiga 31367 measurescmeas 31454 |
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-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-fv 6362 df-ov 7158 df-esum 31287 df-meas 31455 |
This theorem is referenced by: measxun2 31469 measvunilem0 31472 measssd 31474 measinb 31480 measres 31481 measdivcst 31483 measdivcstALTV 31484 truae 31502 probnul 31672 |
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