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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 4483 | . . . . . 6 ⊢ (𝐴 ∪ (𝐵 ∖ 𝐴)) = (𝐴 ∪ 𝐵) | |
2 | 1 | eqcomi 2744 | . . . . 5 ⊢ (𝐴 ∪ 𝐵) = (𝐴 ∪ (𝐵 ∖ 𝐴)) |
3 | 2 | fveq2i 6910 | . . . 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 46408 | . . . . 5 ⊢ (𝜑 → 𝑆 ∈ SAlg) |
9 | meaunle.4 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑆) | |
10 | saldifcl2 46284 | . . . . 5 ⊢ ((𝑆 ∈ SAlg ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ∈ 𝑆) → (𝐵 ∖ 𝐴) ∈ 𝑆) | |
11 | 8, 9, 7, 10 | syl3anc 1370 | . . . 4 ⊢ (𝜑 → (𝐵 ∖ 𝐴) ∈ 𝑆) |
12 | disjdif 4478 | . . . . 5 ⊢ (𝐴 ∩ (𝐵 ∖ 𝐴)) = ∅ | |
13 | 12 | a1i 11 | . . . 4 ⊢ (𝜑 → (𝐴 ∩ (𝐵 ∖ 𝐴)) = ∅) |
14 | 5, 6, 7, 11, 13 | meadjun 46418 | . . 3 ⊢ (𝜑 → (𝑀‘(𝐴 ∪ (𝐵 ∖ 𝐴))) = ((𝑀‘𝐴) +𝑒 (𝑀‘(𝐵 ∖ 𝐴)))) |
15 | 4, 14 | eqtrd 2775 | . 2 ⊢ (𝜑 → (𝑀‘(𝐴 ∪ 𝐵)) = ((𝑀‘𝐴) +𝑒 (𝑀‘(𝐵 ∖ 𝐴)))) |
16 | 5, 6, 11 | meaxrcl 46417 | . . 3 ⊢ (𝜑 → (𝑀‘(𝐵 ∖ 𝐴)) ∈ ℝ*) |
17 | 5, 6, 9 | meaxrcl 46417 | . . 3 ⊢ (𝜑 → (𝑀‘𝐵) ∈ ℝ*) |
18 | 5, 6, 7 | meaxrcl 46417 | . . 3 ⊢ (𝜑 → (𝑀‘𝐴) ∈ ℝ*) |
19 | difssd 4147 | . . . 4 ⊢ (𝜑 → (𝐵 ∖ 𝐴) ⊆ 𝐵) | |
20 | 5, 6, 11, 9, 19 | meassle 46419 | . . 3 ⊢ (𝜑 → (𝑀‘(𝐵 ∖ 𝐴)) ≤ (𝑀‘𝐵)) |
21 | 16, 17, 18, 20 | xleadd2d 45277 | . 2 ⊢ (𝜑 → ((𝑀‘𝐴) +𝑒 (𝑀‘(𝐵 ∖ 𝐴))) ≤ ((𝑀‘𝐴) +𝑒 (𝑀‘𝐵))) |
22 | 15, 21 | eqbrtrd 5170 | 1 ⊢ (𝜑 → (𝑀‘(𝐴 ∪ 𝐵)) ≤ ((𝑀‘𝐴) +𝑒 (𝑀‘𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 ∖ cdif 3960 ∪ cun 3961 ∩ cin 3962 ∅c0 4339 class class class wbr 5148 dom cdm 5689 ‘cfv 6563 (class class class)co 7431 ≤ cle 11294 +𝑒 cxad 13150 SAlgcsalg 46264 Meascmea 46405 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-inf2 9679 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-disj 5116 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-sup 9480 df-oi 9548 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-n0 12525 df-z 12612 df-uz 12877 df-rp 13033 df-xadd 13153 df-ico 13390 df-icc 13391 df-fz 13545 df-fzo 13692 df-seq 14040 df-exp 14100 df-hash 14367 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-clim 15521 df-sum 15720 df-salg 46265 df-sumge0 46319 df-mea 46406 |
This theorem is referenced by: (None) |
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