Mathbox for Glauco Siliprandi |
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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 42724 | . . . . 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 220 | . . . 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 497 | . . 3 ⊢ (𝜑 → ((𝑀:dom 𝑀⟶(0[,]+∞) ∧ dom 𝑀 ∈ SAlg) ∧ (𝑀‘∅) = 0)) |
5 | 4 | simplld 766 | . 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 6493 | . 2 ⊢ (𝜑 → (𝑀:𝑆⟶(0[,]+∞) ↔ 𝑀:dom 𝑀⟶(0[,]+∞))) |
9 | 5, 8 | mpbird 259 | 1 ⊢ (𝜑 → 𝑀:𝑆⟶(0[,]+∞)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1531 ∈ wcel 2108 ∀wral 3136 ∅c0 4289 𝒫 cpw 4537 ∪ cuni 4830 Disj wdisj 5022 class class class wbr 5057 dom cdm 5548 ↾ cres 5550 ⟶wf 6344 ‘cfv 6348 (class class class)co 7148 ωcom 7572 ≼ cdom 8499 0cc0 10529 +∞cpnf 10664 [,]cicc 12733 SAlgcsalg 42584 Σ^csumge0 42635 Meascmea 42722 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-ral 3141 df-rex 3142 df-reu 3143 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-op 4566 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-mea 42723 |
This theorem is referenced by: meacl 42731 meadjun 42735 meadjiunlem 42738 meadjiun 42739 |
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