| Mathbox for Glauco Siliprandi |
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| Mirrors > Home > MPE Home > Th. List > Mathboxes > omeunle | Structured version Visualization version GIF version | ||
| Description: The outer measure of the union of two sets is less than or equal to the sum of the measures, Remark 113B (c) of [Fremlin1] p. 19. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
| Ref | Expression |
|---|---|
| omeunle.o | ⊢ (𝜑 → 𝑂 ∈ OutMeas) |
| omeunle.x | ⊢ 𝑋 = ∪ dom 𝑂 |
| omeunle.a | ⊢ (𝜑 → 𝐴 ⊆ 𝑋) |
| omeunle.b | ⊢ (𝜑 → 𝐵 ⊆ 𝑋) |
| Ref | Expression |
|---|---|
| omeunle | ⊢ (𝜑 → (𝑂‘(𝐴 ∪ 𝐵)) ≤ ((𝑂‘𝐴) +𝑒 (𝑂‘𝐵))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | omeunle.a | . . . . . 6 ⊢ (𝜑 → 𝐴 ⊆ 𝑋) | |
| 2 | omeunle.o | . . . . . . 7 ⊢ (𝜑 → 𝑂 ∈ OutMeas) | |
| 3 | omeunle.x | . . . . . . 7 ⊢ 𝑋 = ∪ dom 𝑂 | |
| 4 | 2, 3 | unidmex 45055 | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ V) |
| 5 | ssexg 5323 | . . . . . 6 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ V) → 𝐴 ∈ V) | |
| 6 | 1, 4, 5 | syl2anc 584 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ V) |
| 7 | omeunle.b | . . . . . 6 ⊢ (𝜑 → 𝐵 ⊆ 𝑋) | |
| 8 | ssexg 5323 | . . . . . 6 ⊢ ((𝐵 ⊆ 𝑋 ∧ 𝑋 ∈ V) → 𝐵 ∈ V) | |
| 9 | 7, 4, 8 | syl2anc 584 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ V) |
| 10 | uniprg 4923 | . . . . 5 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ∪ {𝐴, 𝐵} = (𝐴 ∪ 𝐵)) | |
| 11 | 6, 9, 10 | syl2anc 584 | . . . 4 ⊢ (𝜑 → ∪ {𝐴, 𝐵} = (𝐴 ∪ 𝐵)) |
| 12 | 11 | eqcomd 2743 | . . 3 ⊢ (𝜑 → (𝐴 ∪ 𝐵) = ∪ {𝐴, 𝐵}) |
| 13 | 12 | fveq2d 6910 | . 2 ⊢ (𝜑 → (𝑂‘(𝐴 ∪ 𝐵)) = (𝑂‘∪ {𝐴, 𝐵})) |
| 14 | iccssxr 13470 | . . . 4 ⊢ (0[,]+∞) ⊆ ℝ* | |
| 15 | 1, 7 | unssd 4192 | . . . . . 6 ⊢ (𝜑 → (𝐴 ∪ 𝐵) ⊆ 𝑋) |
| 16 | 11, 15 | eqsstrd 4018 | . . . . 5 ⊢ (𝜑 → ∪ {𝐴, 𝐵} ⊆ 𝑋) |
| 17 | 2, 3, 16 | omecl 46518 | . . . 4 ⊢ (𝜑 → (𝑂‘∪ {𝐴, 𝐵}) ∈ (0[,]+∞)) |
| 18 | 14, 17 | sselid 3981 | . . 3 ⊢ (𝜑 → (𝑂‘∪ {𝐴, 𝐵}) ∈ ℝ*) |
| 19 | prfi 9363 | . . . . . 6 ⊢ {𝐴, 𝐵} ∈ Fin | |
| 20 | 19 | elexi 3503 | . . . . 5 ⊢ {𝐴, 𝐵} ∈ V |
| 21 | 20 | a1i 11 | . . . 4 ⊢ (𝜑 → {𝐴, 𝐵} ∈ V) |
| 22 | 2, 3 | omef 46511 | . . . . 5 ⊢ (𝜑 → 𝑂:𝒫 𝑋⟶(0[,]+∞)) |
| 23 | elpwg 4603 | . . . . . . . . 9 ⊢ (𝐴 ∈ V → (𝐴 ∈ 𝒫 𝑋 ↔ 𝐴 ⊆ 𝑋)) | |
| 24 | 6, 23 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → (𝐴 ∈ 𝒫 𝑋 ↔ 𝐴 ⊆ 𝑋)) |
| 25 | 1, 24 | mpbird 257 | . . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ 𝒫 𝑋) |
| 26 | elpwg 4603 | . . . . . . . . 9 ⊢ (𝐵 ∈ V → (𝐵 ∈ 𝒫 𝑋 ↔ 𝐵 ⊆ 𝑋)) | |
| 27 | 9, 26 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → (𝐵 ∈ 𝒫 𝑋 ↔ 𝐵 ⊆ 𝑋)) |
| 28 | 7, 27 | mpbird 257 | . . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ 𝒫 𝑋) |
| 29 | 25, 28 | jca 511 | . . . . . 6 ⊢ (𝜑 → (𝐴 ∈ 𝒫 𝑋 ∧ 𝐵 ∈ 𝒫 𝑋)) |
| 30 | prssg 4819 | . . . . . . 7 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ((𝐴 ∈ 𝒫 𝑋 ∧ 𝐵 ∈ 𝒫 𝑋) ↔ {𝐴, 𝐵} ⊆ 𝒫 𝑋)) | |
| 31 | 6, 9, 30 | syl2anc 584 | . . . . . 6 ⊢ (𝜑 → ((𝐴 ∈ 𝒫 𝑋 ∧ 𝐵 ∈ 𝒫 𝑋) ↔ {𝐴, 𝐵} ⊆ 𝒫 𝑋)) |
| 32 | 29, 31 | mpbid 232 | . . . . 5 ⊢ (𝜑 → {𝐴, 𝐵} ⊆ 𝒫 𝑋) |
| 33 | 22, 32 | fssresd 6775 | . . . 4 ⊢ (𝜑 → (𝑂 ↾ {𝐴, 𝐵}):{𝐴, 𝐵}⟶(0[,]+∞)) |
| 34 | 21, 33 | sge0xrcl 46400 | . . 3 ⊢ (𝜑 → (Σ^‘(𝑂 ↾ {𝐴, 𝐵})) ∈ ℝ*) |
| 35 | 2, 3, 1 | omecl 46518 | . . . . 5 ⊢ (𝜑 → (𝑂‘𝐴) ∈ (0[,]+∞)) |
| 36 | 14, 35 | sselid 3981 | . . . 4 ⊢ (𝜑 → (𝑂‘𝐴) ∈ ℝ*) |
| 37 | 2, 3, 7 | omecl 46518 | . . . . 5 ⊢ (𝜑 → (𝑂‘𝐵) ∈ (0[,]+∞)) |
| 38 | 14, 37 | sselid 3981 | . . . 4 ⊢ (𝜑 → (𝑂‘𝐵) ∈ ℝ*) |
| 39 | 36, 38 | xaddcld 13343 | . . 3 ⊢ (𝜑 → ((𝑂‘𝐴) +𝑒 (𝑂‘𝐵)) ∈ ℝ*) |
| 40 | isfinite 9692 | . . . . . . . 8 ⊢ ({𝐴, 𝐵} ∈ Fin ↔ {𝐴, 𝐵} ≺ ω) | |
| 41 | 40 | biimpi 216 | . . . . . . 7 ⊢ ({𝐴, 𝐵} ∈ Fin → {𝐴, 𝐵} ≺ ω) |
| 42 | sdomdom 9020 | . . . . . . 7 ⊢ ({𝐴, 𝐵} ≺ ω → {𝐴, 𝐵} ≼ ω) | |
| 43 | 41, 42 | syl 17 | . . . . . 6 ⊢ ({𝐴, 𝐵} ∈ Fin → {𝐴, 𝐵} ≼ ω) |
| 44 | 19, 43 | ax-mp 5 | . . . . 5 ⊢ {𝐴, 𝐵} ≼ ω |
| 45 | 44 | a1i 11 | . . . 4 ⊢ (𝜑 → {𝐴, 𝐵} ≼ ω) |
| 46 | 2, 3, 32, 45 | omeunile 46520 | . . 3 ⊢ (𝜑 → (𝑂‘∪ {𝐴, 𝐵}) ≤ (Σ^‘(𝑂 ↾ {𝐴, 𝐵}))) |
| 47 | 22, 32 | feqresmpt 6978 | . . . . 5 ⊢ (𝜑 → (𝑂 ↾ {𝐴, 𝐵}) = (𝑘 ∈ {𝐴, 𝐵} ↦ (𝑂‘𝑘))) |
| 48 | 47 | fveq2d 6910 | . . . 4 ⊢ (𝜑 → (Σ^‘(𝑂 ↾ {𝐴, 𝐵})) = (Σ^‘(𝑘 ∈ {𝐴, 𝐵} ↦ (𝑂‘𝑘)))) |
| 49 | fveq2 6906 | . . . . 5 ⊢ (𝑘 = 𝐴 → (𝑂‘𝑘) = (𝑂‘𝐴)) | |
| 50 | fveq2 6906 | . . . . 5 ⊢ (𝑘 = 𝐵 → (𝑂‘𝑘) = (𝑂‘𝐵)) | |
| 51 | 6, 9, 35, 37, 49, 50 | sge0prle 46416 | . . . 4 ⊢ (𝜑 → (Σ^‘(𝑘 ∈ {𝐴, 𝐵} ↦ (𝑂‘𝑘))) ≤ ((𝑂‘𝐴) +𝑒 (𝑂‘𝐵))) |
| 52 | 48, 51 | eqbrtrd 5165 | . . 3 ⊢ (𝜑 → (Σ^‘(𝑂 ↾ {𝐴, 𝐵})) ≤ ((𝑂‘𝐴) +𝑒 (𝑂‘𝐵))) |
| 53 | 18, 34, 39, 46, 52 | xrletrd 13204 | . 2 ⊢ (𝜑 → (𝑂‘∪ {𝐴, 𝐵}) ≤ ((𝑂‘𝐴) +𝑒 (𝑂‘𝐵))) |
| 54 | 13, 53 | eqbrtrd 5165 | 1 ⊢ (𝜑 → (𝑂‘(𝐴 ∪ 𝐵)) ≤ ((𝑂‘𝐴) +𝑒 (𝑂‘𝐵))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ∪ cun 3949 ⊆ wss 3951 𝒫 cpw 4600 {cpr 4628 ∪ cuni 4907 class class class wbr 5143 ↦ cmpt 5225 dom cdm 5685 ↾ cres 5687 ‘cfv 6561 (class class class)co 7431 ωcom 7887 ≼ cdom 8983 ≺ csdm 8984 Fincfn 8985 0cc0 11155 +∞cpnf 11292 ℝ*cxr 11294 ≤ cle 11296 +𝑒 cxad 13152 [,]cicc 13390 Σ^csumge0 46377 OutMeascome 46504 |
| 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-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-sumge0 46378 df-ome 46505 |
| This theorem is referenced by: omelesplit 46533 |
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