Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > mea0 | Structured version Visualization version GIF version |
Description: The measure of the empty set is always 0 . (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
Ref | Expression |
---|---|
mea0.1 | ⊢ (𝜑 → 𝑀 ∈ Meas) |
Ref | Expression |
---|---|
mea0 | ⊢ (𝜑 → (𝑀‘∅) = 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mea0.1 | . . 3 ⊢ (𝜑 → 𝑀 ∈ Meas) | |
2 | ismea 42753 | . . 3 ⊢ (𝑀 ∈ Meas ↔ (((𝑀:dom 𝑀⟶(0[,]+∞) ∧ dom 𝑀 ∈ SAlg) ∧ (𝑀‘∅) = 0) ∧ ∀𝑥 ∈ 𝒫 dom 𝑀((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → (𝑀‘∪ 𝑥) = (Σ^‘(𝑀 ↾ 𝑥))))) | |
3 | 1, 2 | sylib 220 | . 2 ⊢ (𝜑 → (((𝑀:dom 𝑀⟶(0[,]+∞) ∧ dom 𝑀 ∈ SAlg) ∧ (𝑀‘∅) = 0) ∧ ∀𝑥 ∈ 𝒫 dom 𝑀((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → (𝑀‘∪ 𝑥) = (Σ^‘(𝑀 ↾ 𝑥))))) |
4 | 3 | simplrd 768 | 1 ⊢ (𝜑 → (𝑀‘∅) = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∀wral 3138 ∅c0 4291 𝒫 cpw 4539 ∪ cuni 4838 Disj wdisj 5031 class class class wbr 5066 dom cdm 5555 ↾ cres 5557 ⟶wf 6351 ‘cfv 6355 (class class class)co 7156 ωcom 7580 ≼ cdom 8507 0cc0 10537 +∞cpnf 10672 [,]cicc 12742 SAlgcsalg 42613 Σ^csumge0 42664 Meascmea 42751 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-mea 42752 |
This theorem is referenced by: meadjun 42764 meadjiunlem 42767 vonioo 42984 vonicc 42987 |
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