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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > meassle | Structured version Visualization version GIF version |
Description: The measure of a set is greater than or equal to the measure of a subset, Property 112C (b) of [Fremlin1] p. 15. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
Ref | Expression |
---|---|
meassle.m | ⊢ (𝜑 → 𝑀 ∈ Meas) |
meassle.x | ⊢ 𝑆 = dom 𝑀 |
meassle.a | ⊢ (𝜑 → 𝐴 ∈ 𝑆) |
meassle.b | ⊢ (𝜑 → 𝐵 ∈ 𝑆) |
meassle.ss | ⊢ (𝜑 → 𝐴 ⊆ 𝐵) |
Ref | Expression |
---|---|
meassle | ⊢ (𝜑 → (𝑀‘𝐴) ≤ (𝑀‘𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | meassle.m | . . . 4 ⊢ (𝜑 → 𝑀 ∈ Meas) | |
2 | meassle.x | . . . 4 ⊢ 𝑆 = dom 𝑀 | |
3 | meassle.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑆) | |
4 | 1, 2, 3 | meaxrcl 45746 | . . 3 ⊢ (𝜑 → (𝑀‘𝐴) ∈ ℝ*) |
5 | 1, 2 | dmmeasal 45737 | . . . . 5 ⊢ (𝜑 → 𝑆 ∈ SAlg) |
6 | meassle.b | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑆) | |
7 | saldifcl2 45613 | . . . . 5 ⊢ ((𝑆 ∈ SAlg ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ∈ 𝑆) → (𝐵 ∖ 𝐴) ∈ 𝑆) | |
8 | 5, 6, 3, 7 | syl3anc 1368 | . . . 4 ⊢ (𝜑 → (𝐵 ∖ 𝐴) ∈ 𝑆) |
9 | 1, 2, 8 | meacl 45743 | . . 3 ⊢ (𝜑 → (𝑀‘(𝐵 ∖ 𝐴)) ∈ (0[,]+∞)) |
10 | 4, 9 | xadd0ge 44599 | . 2 ⊢ (𝜑 → (𝑀‘𝐴) ≤ ((𝑀‘𝐴) +𝑒 (𝑀‘(𝐵 ∖ 𝐴)))) |
11 | meassle.ss | . . . . . 6 ⊢ (𝜑 → 𝐴 ⊆ 𝐵) | |
12 | undif 4476 | . . . . . . 7 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∪ (𝐵 ∖ 𝐴)) = 𝐵) | |
13 | 12 | biimpi 215 | . . . . . 6 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∪ (𝐵 ∖ 𝐴)) = 𝐵) |
14 | 11, 13 | syl 17 | . . . . 5 ⊢ (𝜑 → (𝐴 ∪ (𝐵 ∖ 𝐴)) = 𝐵) |
15 | 14 | fveq2d 6889 | . . . 4 ⊢ (𝜑 → (𝑀‘(𝐴 ∪ (𝐵 ∖ 𝐴))) = (𝑀‘𝐵)) |
16 | 15 | eqcomd 2732 | . . 3 ⊢ (𝜑 → (𝑀‘𝐵) = (𝑀‘(𝐴 ∪ (𝐵 ∖ 𝐴)))) |
17 | disjdif 4466 | . . . . 5 ⊢ (𝐴 ∩ (𝐵 ∖ 𝐴)) = ∅ | |
18 | 17 | a1i 11 | . . . 4 ⊢ (𝜑 → (𝐴 ∩ (𝐵 ∖ 𝐴)) = ∅) |
19 | 1, 2, 3, 8, 18 | meadjun 45747 | . . 3 ⊢ (𝜑 → (𝑀‘(𝐴 ∪ (𝐵 ∖ 𝐴))) = ((𝑀‘𝐴) +𝑒 (𝑀‘(𝐵 ∖ 𝐴)))) |
20 | 16, 19 | eqtr2d 2767 | . 2 ⊢ (𝜑 → ((𝑀‘𝐴) +𝑒 (𝑀‘(𝐵 ∖ 𝐴))) = (𝑀‘𝐵)) |
21 | 10, 20 | breqtrd 5167 | 1 ⊢ (𝜑 → (𝑀‘𝐴) ≤ (𝑀‘𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ∖ cdif 3940 ∪ cun 3941 ∩ cin 3942 ⊆ wss 3943 ∅c0 4317 class class class wbr 5141 dom cdm 5669 ‘cfv 6537 (class class class)co 7405 ≤ cle 11253 +𝑒 cxad 13096 SAlgcsalg 45593 Meascmea 45734 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-inf2 9638 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-disj 5107 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-sup 9439 df-oi 9507 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-n0 12477 df-z 12563 df-uz 12827 df-rp 12981 df-xadd 13099 df-ico 13336 df-icc 13337 df-fz 13491 df-fzo 13634 df-seq 13973 df-exp 14033 df-hash 14296 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-clim 15438 df-sum 15639 df-salg 45594 df-sumge0 45648 df-mea 45735 |
This theorem is referenced by: meaunle 45749 meaiunlelem 45753 meassre 45762 meaiuninclem 45765 meaiuninc3v 45769 meaiininclem 45771 vonioolem2 45966 |
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