Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > meacl | Structured version Visualization version GIF version |
Description: The measure of a set is a nonnegative extended real. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
Ref | Expression |
---|---|
meacl.1 | ⊢ (𝜑 → 𝑀 ∈ Meas) |
meacl.2 | ⊢ 𝑆 = dom 𝑀 |
meacl.3 | ⊢ (𝜑 → 𝐴 ∈ 𝑆) |
Ref | Expression |
---|---|
meacl | ⊢ (𝜑 → (𝑀‘𝐴) ∈ (0[,]+∞)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | meacl.1 | . . 3 ⊢ (𝜑 → 𝑀 ∈ Meas) | |
2 | meacl.2 | . . 3 ⊢ 𝑆 = dom 𝑀 | |
3 | 1, 2 | meaf 42742 | . 2 ⊢ (𝜑 → 𝑀:𝑆⟶(0[,]+∞)) |
4 | meacl.3 | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝑆) | |
5 | 3, 4 | ffvelrnd 6854 | 1 ⊢ (𝜑 → (𝑀‘𝐴) ∈ (0[,]+∞)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 dom cdm 5557 ‘cfv 6357 (class class class)co 7158 0cc0 10539 +∞cpnf 10674 [,]cicc 12744 Meascmea 42738 |
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 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-mea 42739 |
This theorem is referenced by: meaxrcl 42750 meassle 42752 meaiunlelem 42757 meage0 42764 voncl 42955 |
Copyright terms: Public domain | W3C validator |