Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > meaunle | Structured version Visualization version GIF version |
Description: The measure of the union of two sets is less or equal to the sum of the measures, Property 112C (c) of [Fremlin1] p. 15. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
Ref | Expression |
---|---|
meaunle.1 | ⊢ (𝜑 → 𝑀 ∈ Meas) |
meaunle.2 | ⊢ 𝑆 = dom 𝑀 |
meaunle.3 | ⊢ (𝜑 → 𝐴 ∈ 𝑆) |
meaunle.4 | ⊢ (𝜑 → 𝐵 ∈ 𝑆) |
Ref | Expression |
---|---|
meaunle | ⊢ (𝜑 → (𝑀‘(𝐴 ∪ 𝐵)) ≤ ((𝑀‘𝐴) +𝑒 (𝑀‘𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | undif2 3996 | . . . . . 6 ⊢ (𝐴 ∪ (𝐵 ∖ 𝐴)) = (𝐴 ∪ 𝐵) | |
2 | 1 | eqcomi 2619 | . . . . 5 ⊢ (𝐴 ∪ 𝐵) = (𝐴 ∪ (𝐵 ∖ 𝐴)) |
3 | 2 | fveq2i 6106 | . . . 4 ⊢ (𝑀‘(𝐴 ∪ 𝐵)) = (𝑀‘(𝐴 ∪ (𝐵 ∖ 𝐴))) |
4 | 3 | a1i 11 | . . 3 ⊢ (𝜑 → (𝑀‘(𝐴 ∪ 𝐵)) = (𝑀‘(𝐴 ∪ (𝐵 ∖ 𝐴)))) |
5 | meaunle.1 | . . . 4 ⊢ (𝜑 → 𝑀 ∈ Meas) | |
6 | meaunle.2 | . . . 4 ⊢ 𝑆 = dom 𝑀 | |
7 | meaunle.3 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑆) | |
8 | 5, 6 | dmmeasal 39345 | . . . . 5 ⊢ (𝜑 → 𝑆 ∈ SAlg) |
9 | meaunle.4 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑆) | |
10 | saldifcl2 39222 | . . . . 5 ⊢ ((𝑆 ∈ SAlg ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ∈ 𝑆) → (𝐵 ∖ 𝐴) ∈ 𝑆) | |
11 | 8, 9, 7, 10 | syl3anc 1318 | . . . 4 ⊢ (𝜑 → (𝐵 ∖ 𝐴) ∈ 𝑆) |
12 | disjdif 3992 | . . . . 5 ⊢ (𝐴 ∩ (𝐵 ∖ 𝐴)) = ∅ | |
13 | 12 | a1i 11 | . . . 4 ⊢ (𝜑 → (𝐴 ∩ (𝐵 ∖ 𝐴)) = ∅) |
14 | 5, 6, 7, 11, 13 | meadjun 39355 | . . 3 ⊢ (𝜑 → (𝑀‘(𝐴 ∪ (𝐵 ∖ 𝐴))) = ((𝑀‘𝐴) +𝑒 (𝑀‘(𝐵 ∖ 𝐴)))) |
15 | 4, 14 | eqtrd 2644 | . 2 ⊢ (𝜑 → (𝑀‘(𝐴 ∪ 𝐵)) = ((𝑀‘𝐴) +𝑒 (𝑀‘(𝐵 ∖ 𝐴)))) |
16 | 5, 6, 11 | meaxrcl 39354 | . . 3 ⊢ (𝜑 → (𝑀‘(𝐵 ∖ 𝐴)) ∈ ℝ*) |
17 | 5, 6, 9 | meaxrcl 39354 | . . 3 ⊢ (𝜑 → (𝑀‘𝐵) ∈ ℝ*) |
18 | 5, 6, 7 | meaxrcl 39354 | . . 3 ⊢ (𝜑 → (𝑀‘𝐴) ∈ ℝ*) |
19 | difssd 3700 | . . . 4 ⊢ (𝜑 → (𝐵 ∖ 𝐴) ⊆ 𝐵) | |
20 | 5, 6, 11, 9, 19 | meassle 39356 | . . 3 ⊢ (𝜑 → (𝑀‘(𝐵 ∖ 𝐴)) ≤ (𝑀‘𝐵)) |
21 | 16, 17, 18, 20 | xleadd2d 38484 | . 2 ⊢ (𝜑 → ((𝑀‘𝐴) +𝑒 (𝑀‘(𝐵 ∖ 𝐴))) ≤ ((𝑀‘𝐴) +𝑒 (𝑀‘𝐵))) |
22 | 15, 21 | eqbrtrd 4605 | 1 ⊢ (𝜑 → (𝑀‘(𝐴 ∪ 𝐵)) ≤ ((𝑀‘𝐴) +𝑒 (𝑀‘𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 ∖ cdif 3537 ∪ cun 3538 ∩ cin 3539 ∅c0 3874 class class class wbr 4583 dom cdm 5038 ‘cfv 5804 (class class class)co 6549 ≤ cle 9954 +𝑒 cxad 11820 SAlgcsalg 39204 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-inf2 8421 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-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 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-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 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-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-disj 4554 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 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-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 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-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-sup 8231 df-oi 8298 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-n0 11170 df-z 11255 df-uz 11564 df-rp 11709 df-xadd 11823 df-ico 12052 df-icc 12053 df-fz 12198 df-fzo 12335 df-seq 12664 df-exp 12723 df-hash 12980 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-clim 14067 df-sum 14265 df-salg 39205 df-sumge0 39256 df-mea 39343 |
This theorem is referenced by: (None) |
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