meage0 - Mathbox for Glauco Siliprandi

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Theorem meage0 39368

Description: If the measure of a measurable set is greater than or equal to 0. (Contributed by Glauco Siliprandi, 8-Apr-2021.)

**Hypotheses**
Ref	Expression
meage0.m	⊢ (𝜑 → 𝑀 ∈ Meas)
meage0.a	⊢ (𝜑 → 𝐴 ∈ dom 𝑀)

**Assertion**
Ref	Expression
meage0	⊢ (𝜑 → 0 ≤ (𝑀‘𝐴))

**Proof of Theorem meage0**
Step	Hyp	Ref	Expression
1		0xr 9965	. . 3 ⊢ 0 ∈ ℝ^*
2	1	a1i 11	. 2 ⊢ (𝜑 → 0 ∈ ℝ^*)
3		pnfxr 9971	. . 3 ⊢ +∞ ∈ ℝ^*
4	3	a1i 11	. 2 ⊢ (𝜑 → +∞ ∈ ℝ^*)
5		meage0.m	. . 3 ⊢ (𝜑 → 𝑀 ∈ Meas)
6		eqid 2610	. . 3 ⊢ dom 𝑀 = dom 𝑀
7		meage0.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ dom 𝑀)
8	5, 6, 7	meacl 39351	. 2 ⊢ (𝜑 → (𝑀‘𝐴) ∈ (0[,]+∞))
9		iccgelb 12101	. 2 ⊢ ((0 ∈ ℝ^* ∧ +∞ ∈ ℝ^* ∧ (𝑀‘𝐴) ∈ (0[,]+∞)) → 0 ≤ (𝑀‘𝐴))
10	2, 4, 8, 9	syl3anc 1318	1 ⊢ (𝜑 → 0 ≤ (𝑀‘𝐴))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∈ wcel 1977 class class class wbr 4583 dom cdm 5038 ‘cfv 5804 (class class class)co 6549 0cc0 9815 +∞cpnf 9950 ℝ^*cxr 9952 ≤ cle 9954 [,]cicc 12049 Meascmea 39342

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-i2m1 9883 ax-1ne0 9884 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-pnf 9955 df-xr 9957 df-icc 12053 df-mea 39343

This theorem is referenced by: meassre 39370 meale0eq0 39371 meaiuninclem 39373

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