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 42740 | . . 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 1536 ∈ wcel 2113 ∀wral 3141 ∅c0 4294 𝒫 cpw 4542 ∪ cuni 4841 Disj wdisj 5034 class class class wbr 5069 dom cdm 5558 ↾ cres 5560 ⟶wf 6354 ‘cfv 6358 (class class class)co 7159 ωcom 7583 ≼ cdom 8510 0cc0 10540 +∞cpnf 10675 [,]cicc 12744 SAlgcsalg 42600 Σ^csumge0 42651 Meascmea 42738 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pr 5333 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-reu 3148 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-mea 42739 |
This theorem is referenced by: meadjun 42751 meadjiunlem 42754 vonioo 42971 vonicc 42974 |
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