Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > meaf | Structured version Visualization version GIF version |
Description: A measure is a function that maps to nonnegative extended reals. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
Ref | Expression |
---|---|
meaf.m | ⊢ (𝜑 → 𝑀 ∈ Meas) |
meaf.s | ⊢ 𝑆 = dom 𝑀 |
Ref | Expression |
---|---|
meaf | ⊢ (𝜑 → 𝑀:𝑆⟶(0[,]+∞)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | meaf.m | . . . . 5 ⊢ (𝜑 → 𝑀 ∈ Meas) | |
2 | ismea 39344 | . . . . 5 ⊢ (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 207 | . . . 4 ⊢ (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 | simpld 474 | . . 3 ⊢ (𝜑 → ((𝑀:dom 𝑀⟶(0[,]+∞) ∧ dom 𝑀 ∈ SAlg) ∧ (𝑀‘∅) = 0)) |
5 | 4 | simplld 787 | . 2 ⊢ (𝜑 → 𝑀:dom 𝑀⟶(0[,]+∞)) |
6 | meaf.s | . . . 4 ⊢ 𝑆 = dom 𝑀 | |
7 | 6 | a1i 11 | . . 3 ⊢ (𝜑 → 𝑆 = dom 𝑀) |
8 | 7 | feq2d 5944 | . 2 ⊢ (𝜑 → (𝑀:𝑆⟶(0[,]+∞) ↔ 𝑀:dom 𝑀⟶(0[,]+∞))) |
9 | 5, 8 | mpbird 246 | 1 ⊢ (𝜑 → 𝑀:𝑆⟶(0[,]+∞)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ∅c0 3874 𝒫 cpw 4108 ∪ cuni 4372 Disj wdisj 4553 class class class wbr 4583 dom cdm 5038 ↾ cres 5040 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ωcom 6957 ≼ cdom 7839 0cc0 9815 +∞cpnf 9950 [,]cicc 12049 SAlgcsalg 39204 Σ^csumge0 39255 Meascmea 39342 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-mea 39343 |
This theorem is referenced by: meacl 39351 meadjun 39355 meadjiunlem 39358 meadjiun 39359 |
Copyright terms: Public domain | W3C validator |