Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > meale0eq0 | Structured version Visualization version GIF version |
Description: A measure that is less than or equal to 0 is 0. (Contributed by Glauco Siliprandi, 8-Apr-2021.) |
Ref | Expression |
---|---|
meale0eq0.m | ⊢ (𝜑 → 𝑀 ∈ Meas) |
meale0eq0.a | ⊢ (𝜑 → 𝐴 ∈ dom 𝑀) |
meale0eq0.l | ⊢ (𝜑 → (𝑀‘𝐴) ≤ 0) |
Ref | Expression |
---|---|
meale0eq0 | ⊢ (𝜑 → (𝑀‘𝐴) = 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | meale0eq0.m | . . 3 ⊢ (𝜑 → 𝑀 ∈ Meas) | |
2 | eqid 2739 | . . 3 ⊢ dom 𝑀 = dom 𝑀 | |
3 | meale0eq0.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ dom 𝑀) | |
4 | 1, 2, 3 | meaxrcl 43581 | . 2 ⊢ (𝜑 → (𝑀‘𝐴) ∈ ℝ*) |
5 | 0xr 10778 | . . 3 ⊢ 0 ∈ ℝ* | |
6 | 5 | a1i 11 | . 2 ⊢ (𝜑 → 0 ∈ ℝ*) |
7 | meale0eq0.l | . 2 ⊢ (𝜑 → (𝑀‘𝐴) ≤ 0) | |
8 | 1, 3 | meage0 43595 | . 2 ⊢ (𝜑 → 0 ≤ (𝑀‘𝐴)) |
9 | 4, 6, 7, 8 | xrletrid 12643 | 1 ⊢ (𝜑 → (𝑀‘𝐴) = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2114 class class class wbr 5040 dom cdm 5535 ‘cfv 6349 0cc0 10627 ℝ*cxr 10764 ≤ cle 10766 Meascmea 43569 |
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 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5242 ax-pr 5306 ax-un 7491 ax-cnex 10683 ax-resscn 10684 ax-1cn 10685 ax-addrcl 10688 ax-rnegex 10698 ax-cnre 10700 ax-pre-lttri 10701 ax-pre-lttrn 10702 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rab 3063 df-v 3402 df-sbc 3686 df-csb 3801 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4222 df-if 4425 df-pw 4500 df-sn 4527 df-pr 4529 df-op 4533 df-uni 4807 df-iun 4893 df-br 5041 df-opab 5103 df-mpt 5121 df-id 5439 df-po 5452 df-so 5453 df-xp 5541 df-rel 5542 df-cnv 5543 df-co 5544 df-dm 5545 df-rn 5546 df-res 5547 df-ima 5548 df-iota 6307 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-ov 7185 df-oprab 7186 df-mpo 7187 df-1st 7726 df-2nd 7727 df-er 8332 df-en 8568 df-dom 8569 df-sdom 8570 df-pnf 10767 df-mnf 10768 df-xr 10769 df-ltxr 10770 df-le 10771 df-icc 12840 df-mea 43570 |
This theorem is referenced by: (None) |
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