Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > meaxrcl | Structured version Visualization version GIF version |
Description: The measure of a set is an extended real. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
Ref | Expression |
---|---|
meaxrcl.1 | ⊢ (𝜑 → 𝑀 ∈ Meas) |
meaxrcl.2 | ⊢ 𝑆 = dom 𝑀 |
meaxrcl.3 | ⊢ (𝜑 → 𝐴 ∈ 𝑆) |
Ref | Expression |
---|---|
meaxrcl | ⊢ (𝜑 → (𝑀‘𝐴) ∈ ℝ*) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iccssxr 12905 | . 2 ⊢ (0[,]+∞) ⊆ ℝ* | |
2 | meaxrcl.1 | . . 3 ⊢ (𝜑 → 𝑀 ∈ Meas) | |
3 | meaxrcl.2 | . . 3 ⊢ 𝑆 = dom 𝑀 | |
4 | meaxrcl.3 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑆) | |
5 | 2, 3, 4 | meacl 43530 | . 2 ⊢ (𝜑 → (𝑀‘𝐴) ∈ (0[,]+∞)) |
6 | 1, 5 | sseldi 3876 | 1 ⊢ (𝜑 → (𝑀‘𝐴) ∈ ℝ*) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2113 dom cdm 5526 ‘cfv 6340 (class class class)co 7171 0cc0 10616 +∞cpnf 10751 ℝ*cxr 10753 [,]cicc 12825 Meascmea 43521 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-rep 5155 ax-sep 5168 ax-nul 5175 ax-pr 5297 ax-un 7480 ax-cnex 10672 ax-resscn 10673 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-ral 3058 df-rex 3059 df-reu 3060 df-rab 3062 df-v 3400 df-sbc 3683 df-csb 3792 df-dif 3847 df-un 3849 df-in 3851 df-ss 3861 df-nul 4213 df-if 4416 df-pw 4491 df-sn 4518 df-pr 4520 df-op 4524 df-uni 4798 df-iun 4884 df-br 5032 df-opab 5094 df-mpt 5112 df-id 5430 df-xp 5532 df-rel 5533 df-cnv 5534 df-co 5535 df-dm 5536 df-rn 5537 df-res 5538 df-ima 5539 df-iota 6298 df-fun 6342 df-fn 6343 df-f 6344 df-f1 6345 df-fo 6346 df-f1o 6347 df-fv 6348 df-ov 7174 df-oprab 7175 df-mpo 7176 df-1st 7715 df-2nd 7716 df-xr 10758 df-icc 12829 df-mea 43522 |
This theorem is referenced by: meassle 43535 meaunle 43536 meassre 43549 meale0eq0 43550 meaiuninclem 43552 meaiuninc3v 43556 |
Copyright terms: Public domain | W3C validator |