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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 43332 | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ V) |
5 | ssexg 5285 | . . . . . 6 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ V) → 𝐴 ∈ V) | |
6 | 1, 4, 5 | syl2anc 585 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ V) |
7 | omeunle.b | . . . . . 6 ⊢ (𝜑 → 𝐵 ⊆ 𝑋) | |
8 | ssexg 5285 | . . . . . 6 ⊢ ((𝐵 ⊆ 𝑋 ∧ 𝑋 ∈ V) → 𝐵 ∈ V) | |
9 | 7, 4, 8 | syl2anc 585 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ V) |
10 | uniprg 4887 | . . . . 5 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ∪ {𝐴, 𝐵} = (𝐴 ∪ 𝐵)) | |
11 | 6, 9, 10 | syl2anc 585 | . . . 4 ⊢ (𝜑 → ∪ {𝐴, 𝐵} = (𝐴 ∪ 𝐵)) |
12 | 11 | eqcomd 2743 | . . 3 ⊢ (𝜑 → (𝐴 ∪ 𝐵) = ∪ {𝐴, 𝐵}) |
13 | 12 | fveq2d 6851 | . 2 ⊢ (𝜑 → (𝑂‘(𝐴 ∪ 𝐵)) = (𝑂‘∪ {𝐴, 𝐵})) |
14 | iccssxr 13354 | . . . 4 ⊢ (0[,]+∞) ⊆ ℝ* | |
15 | 1, 7 | unssd 4151 | . . . . . 6 ⊢ (𝜑 → (𝐴 ∪ 𝐵) ⊆ 𝑋) |
16 | 11, 15 | eqsstrd 3987 | . . . . 5 ⊢ (𝜑 → ∪ {𝐴, 𝐵} ⊆ 𝑋) |
17 | 2, 3, 16 | omecl 44818 | . . . 4 ⊢ (𝜑 → (𝑂‘∪ {𝐴, 𝐵}) ∈ (0[,]+∞)) |
18 | 14, 17 | sselid 3947 | . . 3 ⊢ (𝜑 → (𝑂‘∪ {𝐴, 𝐵}) ∈ ℝ*) |
19 | prfi 9273 | . . . . . 6 ⊢ {𝐴, 𝐵} ∈ Fin | |
20 | 19 | elexi 3467 | . . . . 5 ⊢ {𝐴, 𝐵} ∈ V |
21 | 20 | a1i 11 | . . . 4 ⊢ (𝜑 → {𝐴, 𝐵} ∈ V) |
22 | 2, 3 | omef 44811 | . . . . 5 ⊢ (𝜑 → 𝑂:𝒫 𝑋⟶(0[,]+∞)) |
23 | elpwg 4568 | . . . . . . . . 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 4568 | . . . . . . . . 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 513 | . . . . . 6 ⊢ (𝜑 → (𝐴 ∈ 𝒫 𝑋 ∧ 𝐵 ∈ 𝒫 𝑋)) |
30 | prssg 4784 | . . . . . . 7 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ((𝐴 ∈ 𝒫 𝑋 ∧ 𝐵 ∈ 𝒫 𝑋) ↔ {𝐴, 𝐵} ⊆ 𝒫 𝑋)) | |
31 | 6, 9, 30 | syl2anc 585 | . . . . . 6 ⊢ (𝜑 → ((𝐴 ∈ 𝒫 𝑋 ∧ 𝐵 ∈ 𝒫 𝑋) ↔ {𝐴, 𝐵} ⊆ 𝒫 𝑋)) |
32 | 29, 31 | mpbid 231 | . . . . 5 ⊢ (𝜑 → {𝐴, 𝐵} ⊆ 𝒫 𝑋) |
33 | 22, 32 | fssresd 6714 | . . . 4 ⊢ (𝜑 → (𝑂 ↾ {𝐴, 𝐵}):{𝐴, 𝐵}⟶(0[,]+∞)) |
34 | 21, 33 | sge0xrcl 44700 | . . 3 ⊢ (𝜑 → (Σ^‘(𝑂 ↾ {𝐴, 𝐵})) ∈ ℝ*) |
35 | 2, 3, 1 | omecl 44818 | . . . . 5 ⊢ (𝜑 → (𝑂‘𝐴) ∈ (0[,]+∞)) |
36 | 14, 35 | sselid 3947 | . . . 4 ⊢ (𝜑 → (𝑂‘𝐴) ∈ ℝ*) |
37 | 2, 3, 7 | omecl 44818 | . . . . 5 ⊢ (𝜑 → (𝑂‘𝐵) ∈ (0[,]+∞)) |
38 | 14, 37 | sselid 3947 | . . . 4 ⊢ (𝜑 → (𝑂‘𝐵) ∈ ℝ*) |
39 | 36, 38 | xaddcld 13227 | . . 3 ⊢ (𝜑 → ((𝑂‘𝐴) +𝑒 (𝑂‘𝐵)) ∈ ℝ*) |
40 | isfinite 9595 | . . . . . . . 8 ⊢ ({𝐴, 𝐵} ∈ Fin ↔ {𝐴, 𝐵} ≺ ω) | |
41 | 40 | biimpi 215 | . . . . . . 7 ⊢ ({𝐴, 𝐵} ∈ Fin → {𝐴, 𝐵} ≺ ω) |
42 | sdomdom 8927 | . . . . . . 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 44820 | . . 3 ⊢ (𝜑 → (𝑂‘∪ {𝐴, 𝐵}) ≤ (Σ^‘(𝑂 ↾ {𝐴, 𝐵}))) |
47 | 22, 32 | feqresmpt 6916 | . . . . 5 ⊢ (𝜑 → (𝑂 ↾ {𝐴, 𝐵}) = (𝑘 ∈ {𝐴, 𝐵} ↦ (𝑂‘𝑘))) |
48 | 47 | fveq2d 6851 | . . . 4 ⊢ (𝜑 → (Σ^‘(𝑂 ↾ {𝐴, 𝐵})) = (Σ^‘(𝑘 ∈ {𝐴, 𝐵} ↦ (𝑂‘𝑘)))) |
49 | fveq2 6847 | . . . . 5 ⊢ (𝑘 = 𝐴 → (𝑂‘𝑘) = (𝑂‘𝐴)) | |
50 | fveq2 6847 | . . . . 5 ⊢ (𝑘 = 𝐵 → (𝑂‘𝑘) = (𝑂‘𝐵)) | |
51 | 6, 9, 35, 37, 49, 50 | sge0prle 44716 | . . . 4 ⊢ (𝜑 → (Σ^‘(𝑘 ∈ {𝐴, 𝐵} ↦ (𝑂‘𝑘))) ≤ ((𝑂‘𝐴) +𝑒 (𝑂‘𝐵))) |
52 | 48, 51 | eqbrtrd 5132 | . . 3 ⊢ (𝜑 → (Σ^‘(𝑂 ↾ {𝐴, 𝐵})) ≤ ((𝑂‘𝐴) +𝑒 (𝑂‘𝐵))) |
53 | 18, 34, 39, 46, 52 | xrletrd 13088 | . 2 ⊢ (𝜑 → (𝑂‘∪ {𝐴, 𝐵}) ≤ ((𝑂‘𝐴) +𝑒 (𝑂‘𝐵))) |
54 | 13, 53 | eqbrtrd 5132 | 1 ⊢ (𝜑 → (𝑂‘(𝐴 ∪ 𝐵)) ≤ ((𝑂‘𝐴) +𝑒 (𝑂‘𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 Vcvv 3448 ∪ cun 3913 ⊆ wss 3915 𝒫 cpw 4565 {cpr 4593 ∪ cuni 4870 class class class wbr 5110 ↦ cmpt 5193 dom cdm 5638 ↾ cres 5640 ‘cfv 6501 (class class class)co 7362 ωcom 7807 ≼ cdom 8888 ≺ csdm 8889 Fincfn 8890 0cc0 11058 +∞cpnf 11193 ℝ*cxr 11195 ≤ cle 11197 +𝑒 cxad 13038 [,]cicc 13274 Σ^csumge0 44677 OutMeascome 44804 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5247 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-inf2 9584 ax-cnex 11114 ax-resscn 11115 ax-1cn 11116 ax-icn 11117 ax-addcl 11118 ax-addrcl 11119 ax-mulcl 11120 ax-mulrcl 11121 ax-mulcom 11122 ax-addass 11123 ax-mulass 11124 ax-distr 11125 ax-i2m1 11126 ax-1ne0 11127 ax-1rid 11128 ax-rnegex 11129 ax-rrecex 11130 ax-cnre 11131 ax-pre-lttri 11132 ax-pre-lttrn 11133 ax-pre-ltadd 11134 ax-pre-mulgt0 11135 ax-pre-sup 11136 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rmo 3356 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-int 4913 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-se 5594 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-isom 6510 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7808 df-1st 7926 df-2nd 7927 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-1o 8417 df-er 8655 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-sup 9385 df-oi 9453 df-card 9882 df-pnf 11198 df-mnf 11199 df-xr 11200 df-ltxr 11201 df-le 11202 df-sub 11394 df-neg 11395 df-div 11820 df-nn 12161 df-2 12223 df-3 12224 df-n0 12421 df-z 12507 df-uz 12771 df-rp 12923 df-xadd 13041 df-ico 13277 df-icc 13278 df-fz 13432 df-fzo 13575 df-seq 13914 df-exp 13975 df-hash 14238 df-cj 14991 df-re 14992 df-im 14993 df-sqrt 15127 df-abs 15128 df-clim 15377 df-sum 15578 df-sumge0 44678 df-ome 44805 |
This theorem is referenced by: omelesplit 44833 |
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