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 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 38242 | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ V) |
5 | ssexg 4732 | . . . . . 6 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ V) → 𝐴 ∈ V) | |
6 | 1, 4, 5 | syl2anc 691 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ V) |
7 | omeunle.b | . . . . . 6 ⊢ (𝜑 → 𝐵 ⊆ 𝑋) | |
8 | ssexg 4732 | . . . . . 6 ⊢ ((𝐵 ⊆ 𝑋 ∧ 𝑋 ∈ V) → 𝐵 ∈ V) | |
9 | 7, 4, 8 | syl2anc 691 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ V) |
10 | uniprg 4386 | . . . . 5 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ∪ {𝐴, 𝐵} = (𝐴 ∪ 𝐵)) | |
11 | 6, 9, 10 | syl2anc 691 | . . . 4 ⊢ (𝜑 → ∪ {𝐴, 𝐵} = (𝐴 ∪ 𝐵)) |
12 | 11 | eqcomd 2616 | . . 3 ⊢ (𝜑 → (𝐴 ∪ 𝐵) = ∪ {𝐴, 𝐵}) |
13 | 12 | fveq2d 6107 | . 2 ⊢ (𝜑 → (𝑂‘(𝐴 ∪ 𝐵)) = (𝑂‘∪ {𝐴, 𝐵})) |
14 | iccssxr 12127 | . . . 4 ⊢ (0[,]+∞) ⊆ ℝ* | |
15 | 1, 7 | unssd 3751 | . . . . . 6 ⊢ (𝜑 → (𝐴 ∪ 𝐵) ⊆ 𝑋) |
16 | 11, 15 | eqsstrd 3602 | . . . . 5 ⊢ (𝜑 → ∪ {𝐴, 𝐵} ⊆ 𝑋) |
17 | 2, 3, 16 | omecl 39393 | . . . 4 ⊢ (𝜑 → (𝑂‘∪ {𝐴, 𝐵}) ∈ (0[,]+∞)) |
18 | 14, 17 | sseldi 3566 | . . 3 ⊢ (𝜑 → (𝑂‘∪ {𝐴, 𝐵}) ∈ ℝ*) |
19 | prfi 8120 | . . . . . 6 ⊢ {𝐴, 𝐵} ∈ Fin | |
20 | 19 | elexi 3186 | . . . . 5 ⊢ {𝐴, 𝐵} ∈ V |
21 | 20 | a1i 11 | . . . 4 ⊢ (𝜑 → {𝐴, 𝐵} ∈ V) |
22 | 2, 3 | omef 39386 | . . . . 5 ⊢ (𝜑 → 𝑂:𝒫 𝑋⟶(0[,]+∞)) |
23 | elpwg 4116 | . . . . . . . . 9 ⊢ (𝐴 ∈ V → (𝐴 ∈ 𝒫 𝑋 ↔ 𝐴 ⊆ 𝑋)) | |
24 | 6, 23 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → (𝐴 ∈ 𝒫 𝑋 ↔ 𝐴 ⊆ 𝑋)) |
25 | 1, 24 | mpbird 246 | . . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ 𝒫 𝑋) |
26 | elpwg 4116 | . . . . . . . . 9 ⊢ (𝐵 ∈ V → (𝐵 ∈ 𝒫 𝑋 ↔ 𝐵 ⊆ 𝑋)) | |
27 | 9, 26 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → (𝐵 ∈ 𝒫 𝑋 ↔ 𝐵 ⊆ 𝑋)) |
28 | 7, 27 | mpbird 246 | . . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ 𝒫 𝑋) |
29 | 25, 28 | jca 553 | . . . . . 6 ⊢ (𝜑 → (𝐴 ∈ 𝒫 𝑋 ∧ 𝐵 ∈ 𝒫 𝑋)) |
30 | prssg 4290 | . . . . . . 7 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ((𝐴 ∈ 𝒫 𝑋 ∧ 𝐵 ∈ 𝒫 𝑋) ↔ {𝐴, 𝐵} ⊆ 𝒫 𝑋)) | |
31 | 6, 9, 30 | syl2anc 691 | . . . . . 6 ⊢ (𝜑 → ((𝐴 ∈ 𝒫 𝑋 ∧ 𝐵 ∈ 𝒫 𝑋) ↔ {𝐴, 𝐵} ⊆ 𝒫 𝑋)) |
32 | 29, 31 | mpbid 221 | . . . . 5 ⊢ (𝜑 → {𝐴, 𝐵} ⊆ 𝒫 𝑋) |
33 | 22, 32 | fssresd 5984 | . . . 4 ⊢ (𝜑 → (𝑂 ↾ {𝐴, 𝐵}):{𝐴, 𝐵}⟶(0[,]+∞)) |
34 | 21, 33 | sge0xrcl 39278 | . . 3 ⊢ (𝜑 → (Σ^‘(𝑂 ↾ {𝐴, 𝐵})) ∈ ℝ*) |
35 | 2, 3, 1 | omecl 39393 | . . . . 5 ⊢ (𝜑 → (𝑂‘𝐴) ∈ (0[,]+∞)) |
36 | 14, 35 | sseldi 3566 | . . . 4 ⊢ (𝜑 → (𝑂‘𝐴) ∈ ℝ*) |
37 | 2, 3, 7 | omecl 39393 | . . . . 5 ⊢ (𝜑 → (𝑂‘𝐵) ∈ (0[,]+∞)) |
38 | 14, 37 | sseldi 3566 | . . . 4 ⊢ (𝜑 → (𝑂‘𝐵) ∈ ℝ*) |
39 | 36, 38 | xaddcld 12003 | . . 3 ⊢ (𝜑 → ((𝑂‘𝐴) +𝑒 (𝑂‘𝐵)) ∈ ℝ*) |
40 | isfinite 8432 | . . . . . . . 8 ⊢ ({𝐴, 𝐵} ∈ Fin ↔ {𝐴, 𝐵} ≺ ω) | |
41 | 40 | biimpi 205 | . . . . . . 7 ⊢ ({𝐴, 𝐵} ∈ Fin → {𝐴, 𝐵} ≺ ω) |
42 | sdomdom 7869 | . . . . . . 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 39395 | . . 3 ⊢ (𝜑 → (𝑂‘∪ {𝐴, 𝐵}) ≤ (Σ^‘(𝑂 ↾ {𝐴, 𝐵}))) |
47 | 22, 32 | feqresmpt 6160 | . . . . 5 ⊢ (𝜑 → (𝑂 ↾ {𝐴, 𝐵}) = (𝑘 ∈ {𝐴, 𝐵} ↦ (𝑂‘𝑘))) |
48 | 47 | fveq2d 6107 | . . . 4 ⊢ (𝜑 → (Σ^‘(𝑂 ↾ {𝐴, 𝐵})) = (Σ^‘(𝑘 ∈ {𝐴, 𝐵} ↦ (𝑂‘𝑘)))) |
49 | fveq2 6103 | . . . . 5 ⊢ (𝑘 = 𝐴 → (𝑂‘𝑘) = (𝑂‘𝐴)) | |
50 | fveq2 6103 | . . . . 5 ⊢ (𝑘 = 𝐵 → (𝑂‘𝑘) = (𝑂‘𝐵)) | |
51 | 6, 9, 35, 37, 49, 50 | sge0prle 39294 | . . . 4 ⊢ (𝜑 → (Σ^‘(𝑘 ∈ {𝐴, 𝐵} ↦ (𝑂‘𝑘))) ≤ ((𝑂‘𝐴) +𝑒 (𝑂‘𝐵))) |
52 | 48, 51 | eqbrtrd 4605 | . . 3 ⊢ (𝜑 → (Σ^‘(𝑂 ↾ {𝐴, 𝐵})) ≤ ((𝑂‘𝐴) +𝑒 (𝑂‘𝐵))) |
53 | 18, 34, 39, 46, 52 | xrletrd 11869 | . 2 ⊢ (𝜑 → (𝑂‘∪ {𝐴, 𝐵}) ≤ ((𝑂‘𝐴) +𝑒 (𝑂‘𝐵))) |
54 | 13, 53 | eqbrtrd 4605 | 1 ⊢ (𝜑 → (𝑂‘(𝐴 ∪ 𝐵)) ≤ ((𝑂‘𝐴) +𝑒 (𝑂‘𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ∪ cun 3538 ⊆ wss 3540 𝒫 cpw 4108 {cpr 4127 ∪ cuni 4372 class class class wbr 4583 ↦ cmpt 4643 dom cdm 5038 ↾ cres 5040 ‘cfv 5804 (class class class)co 6549 ωcom 6957 ≼ cdom 7839 ≺ csdm 7840 Fincfn 7841 0cc0 9815 +∞cpnf 9950 ℝ*cxr 9952 ≤ cle 9954 +𝑒 cxad 11820 [,]cicc 12049 Σ^csumge0 39255 OutMeascome 39379 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-sup 8231 df-oi 8298 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-n0 11170 df-z 11255 df-uz 11564 df-rp 11709 df-xadd 11823 df-ico 12052 df-icc 12053 df-fz 12198 df-fzo 12335 df-seq 12664 df-exp 12723 df-hash 12980 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-clim 14067 df-sum 14265 df-sumge0 39256 df-ome 39380 |
This theorem is referenced by: omelesplit 39408 |
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