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 than 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 4425 | . . . . . 6 ⊢ (𝐴 ∪ (𝐵 ∖ 𝐴)) = (𝐴 ∪ 𝐵) | |
2 | 1 | eqcomi 2830 | . . . . 5 ⊢ (𝐴 ∪ 𝐵) = (𝐴 ∪ (𝐵 ∖ 𝐴)) |
3 | 2 | fveq2i 6668 | . . . 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 42727 | . . . . 5 ⊢ (𝜑 → 𝑆 ∈ SAlg) |
9 | meaunle.4 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑆) | |
10 | saldifcl2 42604 | . . . . 5 ⊢ ((𝑆 ∈ SAlg ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ∈ 𝑆) → (𝐵 ∖ 𝐴) ∈ 𝑆) | |
11 | 8, 9, 7, 10 | syl3anc 1367 | . . . 4 ⊢ (𝜑 → (𝐵 ∖ 𝐴) ∈ 𝑆) |
12 | disjdif 4421 | . . . . 5 ⊢ (𝐴 ∩ (𝐵 ∖ 𝐴)) = ∅ | |
13 | 12 | a1i 11 | . . . 4 ⊢ (𝜑 → (𝐴 ∩ (𝐵 ∖ 𝐴)) = ∅) |
14 | 5, 6, 7, 11, 13 | meadjun 42737 | . . 3 ⊢ (𝜑 → (𝑀‘(𝐴 ∪ (𝐵 ∖ 𝐴))) = ((𝑀‘𝐴) +𝑒 (𝑀‘(𝐵 ∖ 𝐴)))) |
15 | 4, 14 | eqtrd 2856 | . 2 ⊢ (𝜑 → (𝑀‘(𝐴 ∪ 𝐵)) = ((𝑀‘𝐴) +𝑒 (𝑀‘(𝐵 ∖ 𝐴)))) |
16 | 5, 6, 11 | meaxrcl 42736 | . . 3 ⊢ (𝜑 → (𝑀‘(𝐵 ∖ 𝐴)) ∈ ℝ*) |
17 | 5, 6, 9 | meaxrcl 42736 | . . 3 ⊢ (𝜑 → (𝑀‘𝐵) ∈ ℝ*) |
18 | 5, 6, 7 | meaxrcl 42736 | . . 3 ⊢ (𝜑 → (𝑀‘𝐴) ∈ ℝ*) |
19 | difssd 4109 | . . . 4 ⊢ (𝜑 → (𝐵 ∖ 𝐴) ⊆ 𝐵) | |
20 | 5, 6, 11, 9, 19 | meassle 42738 | . . 3 ⊢ (𝜑 → (𝑀‘(𝐵 ∖ 𝐴)) ≤ (𝑀‘𝐵)) |
21 | 16, 17, 18, 20 | xleadd2d 41587 | . 2 ⊢ (𝜑 → ((𝑀‘𝐴) +𝑒 (𝑀‘(𝐵 ∖ 𝐴))) ≤ ((𝑀‘𝐴) +𝑒 (𝑀‘𝐵))) |
22 | 15, 21 | eqbrtrd 5081 | 1 ⊢ (𝜑 → (𝑀‘(𝐴 ∪ 𝐵)) ≤ ((𝑀‘𝐴) +𝑒 (𝑀‘𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 ∖ cdif 3933 ∪ cun 3934 ∩ cin 3935 ∅c0 4291 class class class wbr 5059 dom cdm 5550 ‘cfv 6350 (class class class)co 7150 ≤ cle 10670 +𝑒 cxad 12499 SAlgcsalg 42586 Meascmea 42724 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-inf2 9098 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-int 4870 df-iun 4914 df-disj 5025 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-se 5510 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-isom 6359 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-sup 8900 df-oi 8968 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-2 11694 df-3 11695 df-n0 11892 df-z 11976 df-uz 12238 df-rp 12384 df-xadd 12502 df-ico 12738 df-icc 12739 df-fz 12887 df-fzo 13028 df-seq 13364 df-exp 13424 df-hash 13685 df-cj 14452 df-re 14453 df-im 14454 df-sqrt 14588 df-abs 14589 df-clim 14839 df-sum 15037 df-salg 42587 df-sumge0 42638 df-mea 42725 |
This theorem is referenced by: (None) |
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