| 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 46476 | . . 3 ⊢ (𝜑 → (𝑀‘𝐴) ∈ ℝ*) |
| 5 | 1, 2 | dmmeasal 46467 | . . . . 5 ⊢ (𝜑 → 𝑆 ∈ SAlg) |
| 6 | meassle.b | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑆) | |
| 7 | saldifcl2 46343 | . . . . 5 ⊢ ((𝑆 ∈ SAlg ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ∈ 𝑆) → (𝐵 ∖ 𝐴) ∈ 𝑆) | |
| 8 | 5, 6, 3, 7 | syl3anc 1373 | . . . 4 ⊢ (𝜑 → (𝐵 ∖ 𝐴) ∈ 𝑆) |
| 9 | 1, 2, 8 | meacl 46473 | . . 3 ⊢ (𝜑 → (𝑀‘(𝐵 ∖ 𝐴)) ∈ (0[,]+∞)) |
| 10 | 4, 9 | xadd0ge 45332 | . 2 ⊢ (𝜑 → (𝑀‘𝐴) ≤ ((𝑀‘𝐴) +𝑒 (𝑀‘(𝐵 ∖ 𝐴)))) |
| 11 | meassle.ss | . . . . . 6 ⊢ (𝜑 → 𝐴 ⊆ 𝐵) | |
| 12 | undif 4482 | . . . . . . 7 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∪ (𝐵 ∖ 𝐴)) = 𝐵) | |
| 13 | 12 | biimpi 216 | . . . . . 6 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∪ (𝐵 ∖ 𝐴)) = 𝐵) |
| 14 | 11, 13 | syl 17 | . . . . 5 ⊢ (𝜑 → (𝐴 ∪ (𝐵 ∖ 𝐴)) = 𝐵) |
| 15 | 14 | fveq2d 6910 | . . . 4 ⊢ (𝜑 → (𝑀‘(𝐴 ∪ (𝐵 ∖ 𝐴))) = (𝑀‘𝐵)) |
| 16 | 15 | eqcomd 2743 | . . 3 ⊢ (𝜑 → (𝑀‘𝐵) = (𝑀‘(𝐴 ∪ (𝐵 ∖ 𝐴)))) |
| 17 | disjdif 4472 | . . . . 5 ⊢ (𝐴 ∩ (𝐵 ∖ 𝐴)) = ∅ | |
| 18 | 17 | a1i 11 | . . . 4 ⊢ (𝜑 → (𝐴 ∩ (𝐵 ∖ 𝐴)) = ∅) |
| 19 | 1, 2, 3, 8, 18 | meadjun 46477 | . . 3 ⊢ (𝜑 → (𝑀‘(𝐴 ∪ (𝐵 ∖ 𝐴))) = ((𝑀‘𝐴) +𝑒 (𝑀‘(𝐵 ∖ 𝐴)))) |
| 20 | 16, 19 | eqtr2d 2778 | . 2 ⊢ (𝜑 → ((𝑀‘𝐴) +𝑒 (𝑀‘(𝐵 ∖ 𝐴))) = (𝑀‘𝐵)) |
| 21 | 10, 20 | breqtrd 5169 | 1 ⊢ (𝜑 → (𝑀‘𝐴) ≤ (𝑀‘𝐵)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ∖ cdif 3948 ∪ cun 3949 ∩ cin 3950 ⊆ wss 3951 ∅c0 4333 class class class wbr 5143 dom cdm 5685 ‘cfv 6561 (class class class)co 7431 ≤ cle 11296 +𝑒 cxad 13152 SAlgcsalg 46323 Meascmea 46464 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-disj 5111 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-sup 9482 df-oi 9550 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-n0 12527 df-z 12614 df-uz 12879 df-rp 13035 df-xadd 13155 df-ico 13393 df-icc 13394 df-fz 13548 df-fzo 13695 df-seq 14043 df-exp 14103 df-hash 14370 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-clim 15524 df-sum 15723 df-salg 46324 df-sumge0 46378 df-mea 46465 |
| This theorem is referenced by: meaunle 46479 meaiunlelem 46483 meassre 46492 meaiuninclem 46495 meaiuninc3v 46499 meaiininclem 46501 vonioolem2 46696 |
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