Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > meage0 | Structured version Visualization version GIF version |
Description: If the measure of a measurable set is greater than or equal to 0. (Contributed by Glauco Siliprandi, 8-Apr-2021.) |
Ref | Expression |
---|---|
meage0.m | ⊢ (𝜑 → 𝑀 ∈ Meas) |
meage0.a | ⊢ (𝜑 → 𝐴 ∈ dom 𝑀) |
Ref | Expression |
---|---|
meage0 | ⊢ (𝜑 → 0 ≤ (𝑀‘𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0xr 10676 | . . 3 ⊢ 0 ∈ ℝ* | |
2 | 1 | a1i 11 | . 2 ⊢ (𝜑 → 0 ∈ ℝ*) |
3 | pnfxr 10683 | . . 3 ⊢ +∞ ∈ ℝ* | |
4 | 3 | a1i 11 | . 2 ⊢ (𝜑 → +∞ ∈ ℝ*) |
5 | meage0.m | . . 3 ⊢ (𝜑 → 𝑀 ∈ Meas) | |
6 | eqid 2818 | . . 3 ⊢ dom 𝑀 = dom 𝑀 | |
7 | meage0.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ dom 𝑀) | |
8 | 5, 6, 7 | meacl 42617 | . 2 ⊢ (𝜑 → (𝑀‘𝐴) ∈ (0[,]+∞)) |
9 | iccgelb 12781 | . 2 ⊢ ((0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ (𝑀‘𝐴) ∈ (0[,]+∞)) → 0 ≤ (𝑀‘𝐴)) | |
10 | 2, 4, 8, 9 | syl3anc 1363 | 1 ⊢ (𝜑 → 0 ≤ (𝑀‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2105 class class class wbr 5057 dom cdm 5548 ‘cfv 6348 (class class class)co 7145 0cc0 10525 +∞cpnf 10660 ℝ*cxr 10662 ≤ cle 10664 [,]cicc 12729 Meascmea 42608 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-addrcl 10586 ax-rnegex 10596 ax-cnre 10598 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 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-ov 7148 df-oprab 7149 df-mpo 7150 df-pnf 10665 df-xr 10667 df-icc 12733 df-mea 42609 |
This theorem is referenced by: meassre 42636 meale0eq0 42637 meaiuninclem 42639 |
Copyright terms: Public domain | W3C validator |