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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 4476 | . . . . . 6 ⊢ (𝐴 ∪ (𝐵 ∖ 𝐴)) = (𝐴 ∪ 𝐵) | |
| 2 | 1 | eqcomi 2745 | . . . . 5 ⊢ (𝐴 ∪ 𝐵) = (𝐴 ∪ (𝐵 ∖ 𝐴)) | 
| 3 | 2 | fveq2i 6908 | . . . 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 46472 | . . . . 5 ⊢ (𝜑 → 𝑆 ∈ SAlg) | 
| 9 | meaunle.4 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑆) | |
| 10 | saldifcl2 46348 | . . . . 5 ⊢ ((𝑆 ∈ SAlg ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ∈ 𝑆) → (𝐵 ∖ 𝐴) ∈ 𝑆) | |
| 11 | 8, 9, 7, 10 | syl3anc 1372 | . . . 4 ⊢ (𝜑 → (𝐵 ∖ 𝐴) ∈ 𝑆) | 
| 12 | disjdif 4471 | . . . . 5 ⊢ (𝐴 ∩ (𝐵 ∖ 𝐴)) = ∅ | |
| 13 | 12 | a1i 11 | . . . 4 ⊢ (𝜑 → (𝐴 ∩ (𝐵 ∖ 𝐴)) = ∅) | 
| 14 | 5, 6, 7, 11, 13 | meadjun 46482 | . . 3 ⊢ (𝜑 → (𝑀‘(𝐴 ∪ (𝐵 ∖ 𝐴))) = ((𝑀‘𝐴) +𝑒 (𝑀‘(𝐵 ∖ 𝐴)))) | 
| 15 | 4, 14 | eqtrd 2776 | . 2 ⊢ (𝜑 → (𝑀‘(𝐴 ∪ 𝐵)) = ((𝑀‘𝐴) +𝑒 (𝑀‘(𝐵 ∖ 𝐴)))) | 
| 16 | 5, 6, 11 | meaxrcl 46481 | . . 3 ⊢ (𝜑 → (𝑀‘(𝐵 ∖ 𝐴)) ∈ ℝ*) | 
| 17 | 5, 6, 9 | meaxrcl 46481 | . . 3 ⊢ (𝜑 → (𝑀‘𝐵) ∈ ℝ*) | 
| 18 | 5, 6, 7 | meaxrcl 46481 | . . 3 ⊢ (𝜑 → (𝑀‘𝐴) ∈ ℝ*) | 
| 19 | difssd 4136 | . . . 4 ⊢ (𝜑 → (𝐵 ∖ 𝐴) ⊆ 𝐵) | |
| 20 | 5, 6, 11, 9, 19 | meassle 46483 | . . 3 ⊢ (𝜑 → (𝑀‘(𝐵 ∖ 𝐴)) ≤ (𝑀‘𝐵)) | 
| 21 | 16, 17, 18, 20 | xleadd2d 45343 | . 2 ⊢ (𝜑 → ((𝑀‘𝐴) +𝑒 (𝑀‘(𝐵 ∖ 𝐴))) ≤ ((𝑀‘𝐴) +𝑒 (𝑀‘𝐵))) | 
| 22 | 15, 21 | eqbrtrd 5164 | 1 ⊢ (𝜑 → (𝑀‘(𝐴 ∪ 𝐵)) ≤ ((𝑀‘𝐴) +𝑒 (𝑀‘𝐵))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2107 ∖ cdif 3947 ∪ cun 3948 ∩ cin 3949 ∅c0 4332 class class class wbr 5142 dom cdm 5684 ‘cfv 6560 (class class class)co 7432 ≤ cle 11297 +𝑒 cxad 13153 SAlgcsalg 46328 Meascmea 46469 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-inf2 9682 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 ax-pre-sup 11234 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-int 4946 df-iun 4992 df-disj 5110 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-se 5637 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-isom 6569 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-om 7889 df-1st 8015 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-1o 8507 df-2o 8508 df-er 8746 df-en 8987 df-dom 8988 df-sdom 8989 df-fin 8990 df-sup 9483 df-oi 9551 df-card 9980 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-div 11922 df-nn 12268 df-2 12330 df-3 12331 df-n0 12529 df-z 12616 df-uz 12880 df-rp 13036 df-xadd 13156 df-ico 13394 df-icc 13395 df-fz 13549 df-fzo 13696 df-seq 14044 df-exp 14104 df-hash 14371 df-cj 15139 df-re 15140 df-im 15141 df-sqrt 15275 df-abs 15276 df-clim 15525 df-sum 15724 df-salg 46329 df-sumge0 46383 df-mea 46470 | 
| This theorem is referenced by: (None) | 
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