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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 40731 | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ V) |
5 | ssexg 5077 | . . . . . 6 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ V) → 𝐴 ∈ V) | |
6 | 1, 4, 5 | syl2anc 576 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ V) |
7 | omeunle.b | . . . . . 6 ⊢ (𝜑 → 𝐵 ⊆ 𝑋) | |
8 | ssexg 5077 | . . . . . 6 ⊢ ((𝐵 ⊆ 𝑋 ∧ 𝑋 ∈ V) → 𝐵 ∈ V) | |
9 | 7, 4, 8 | syl2anc 576 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ V) |
10 | uniprg 4720 | . . . . 5 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ∪ {𝐴, 𝐵} = (𝐴 ∪ 𝐵)) | |
11 | 6, 9, 10 | syl2anc 576 | . . . 4 ⊢ (𝜑 → ∪ {𝐴, 𝐵} = (𝐴 ∪ 𝐵)) |
12 | 11 | eqcomd 2778 | . . 3 ⊢ (𝜑 → (𝐴 ∪ 𝐵) = ∪ {𝐴, 𝐵}) |
13 | 12 | fveq2d 6497 | . 2 ⊢ (𝜑 → (𝑂‘(𝐴 ∪ 𝐵)) = (𝑂‘∪ {𝐴, 𝐵})) |
14 | iccssxr 12629 | . . . 4 ⊢ (0[,]+∞) ⊆ ℝ* | |
15 | 1, 7 | unssd 4044 | . . . . . 6 ⊢ (𝜑 → (𝐴 ∪ 𝐵) ⊆ 𝑋) |
16 | 11, 15 | eqsstrd 3889 | . . . . 5 ⊢ (𝜑 → ∪ {𝐴, 𝐵} ⊆ 𝑋) |
17 | 2, 3, 16 | omecl 42216 | . . . 4 ⊢ (𝜑 → (𝑂‘∪ {𝐴, 𝐵}) ∈ (0[,]+∞)) |
18 | 14, 17 | sseldi 3850 | . . 3 ⊢ (𝜑 → (𝑂‘∪ {𝐴, 𝐵}) ∈ ℝ*) |
19 | prfi 8582 | . . . . . 6 ⊢ {𝐴, 𝐵} ∈ Fin | |
20 | 19 | elexi 3428 | . . . . 5 ⊢ {𝐴, 𝐵} ∈ V |
21 | 20 | a1i 11 | . . . 4 ⊢ (𝜑 → {𝐴, 𝐵} ∈ V) |
22 | 2, 3 | omef 42209 | . . . . 5 ⊢ (𝜑 → 𝑂:𝒫 𝑋⟶(0[,]+∞)) |
23 | elpwg 4424 | . . . . . . . . 9 ⊢ (𝐴 ∈ V → (𝐴 ∈ 𝒫 𝑋 ↔ 𝐴 ⊆ 𝑋)) | |
24 | 6, 23 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → (𝐴 ∈ 𝒫 𝑋 ↔ 𝐴 ⊆ 𝑋)) |
25 | 1, 24 | mpbird 249 | . . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ 𝒫 𝑋) |
26 | elpwg 4424 | . . . . . . . . 9 ⊢ (𝐵 ∈ V → (𝐵 ∈ 𝒫 𝑋 ↔ 𝐵 ⊆ 𝑋)) | |
27 | 9, 26 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → (𝐵 ∈ 𝒫 𝑋 ↔ 𝐵 ⊆ 𝑋)) |
28 | 7, 27 | mpbird 249 | . . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ 𝒫 𝑋) |
29 | 25, 28 | jca 504 | . . . . . 6 ⊢ (𝜑 → (𝐴 ∈ 𝒫 𝑋 ∧ 𝐵 ∈ 𝒫 𝑋)) |
30 | prssg 4620 | . . . . . . 7 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ((𝐴 ∈ 𝒫 𝑋 ∧ 𝐵 ∈ 𝒫 𝑋) ↔ {𝐴, 𝐵} ⊆ 𝒫 𝑋)) | |
31 | 6, 9, 30 | syl2anc 576 | . . . . . 6 ⊢ (𝜑 → ((𝐴 ∈ 𝒫 𝑋 ∧ 𝐵 ∈ 𝒫 𝑋) ↔ {𝐴, 𝐵} ⊆ 𝒫 𝑋)) |
32 | 29, 31 | mpbid 224 | . . . . 5 ⊢ (𝜑 → {𝐴, 𝐵} ⊆ 𝒫 𝑋) |
33 | 22, 32 | fssresd 6368 | . . . 4 ⊢ (𝜑 → (𝑂 ↾ {𝐴, 𝐵}):{𝐴, 𝐵}⟶(0[,]+∞)) |
34 | 21, 33 | sge0xrcl 42098 | . . 3 ⊢ (𝜑 → (Σ^‘(𝑂 ↾ {𝐴, 𝐵})) ∈ ℝ*) |
35 | 2, 3, 1 | omecl 42216 | . . . . 5 ⊢ (𝜑 → (𝑂‘𝐴) ∈ (0[,]+∞)) |
36 | 14, 35 | sseldi 3850 | . . . 4 ⊢ (𝜑 → (𝑂‘𝐴) ∈ ℝ*) |
37 | 2, 3, 7 | omecl 42216 | . . . . 5 ⊢ (𝜑 → (𝑂‘𝐵) ∈ (0[,]+∞)) |
38 | 14, 37 | sseldi 3850 | . . . 4 ⊢ (𝜑 → (𝑂‘𝐵) ∈ ℝ*) |
39 | 36, 38 | xaddcld 12504 | . . 3 ⊢ (𝜑 → ((𝑂‘𝐴) +𝑒 (𝑂‘𝐵)) ∈ ℝ*) |
40 | isfinite 8903 | . . . . . . . 8 ⊢ ({𝐴, 𝐵} ∈ Fin ↔ {𝐴, 𝐵} ≺ ω) | |
41 | 40 | biimpi 208 | . . . . . . 7 ⊢ ({𝐴, 𝐵} ∈ Fin → {𝐴, 𝐵} ≺ ω) |
42 | sdomdom 8328 | . . . . . . 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 42218 | . . 3 ⊢ (𝜑 → (𝑂‘∪ {𝐴, 𝐵}) ≤ (Σ^‘(𝑂 ↾ {𝐴, 𝐵}))) |
47 | 22, 32 | feqresmpt 6557 | . . . . 5 ⊢ (𝜑 → (𝑂 ↾ {𝐴, 𝐵}) = (𝑘 ∈ {𝐴, 𝐵} ↦ (𝑂‘𝑘))) |
48 | 47 | fveq2d 6497 | . . . 4 ⊢ (𝜑 → (Σ^‘(𝑂 ↾ {𝐴, 𝐵})) = (Σ^‘(𝑘 ∈ {𝐴, 𝐵} ↦ (𝑂‘𝑘)))) |
49 | fveq2 6493 | . . . . 5 ⊢ (𝑘 = 𝐴 → (𝑂‘𝑘) = (𝑂‘𝐴)) | |
50 | fveq2 6493 | . . . . 5 ⊢ (𝑘 = 𝐵 → (𝑂‘𝑘) = (𝑂‘𝐵)) | |
51 | 6, 9, 35, 37, 49, 50 | sge0prle 42114 | . . . 4 ⊢ (𝜑 → (Σ^‘(𝑘 ∈ {𝐴, 𝐵} ↦ (𝑂‘𝑘))) ≤ ((𝑂‘𝐴) +𝑒 (𝑂‘𝐵))) |
52 | 48, 51 | eqbrtrd 4945 | . . 3 ⊢ (𝜑 → (Σ^‘(𝑂 ↾ {𝐴, 𝐵})) ≤ ((𝑂‘𝐴) +𝑒 (𝑂‘𝐵))) |
53 | 18, 34, 39, 46, 52 | xrletrd 12366 | . 2 ⊢ (𝜑 → (𝑂‘∪ {𝐴, 𝐵}) ≤ ((𝑂‘𝐴) +𝑒 (𝑂‘𝐵))) |
54 | 13, 53 | eqbrtrd 4945 | 1 ⊢ (𝜑 → (𝑂‘(𝐴 ∪ 𝐵)) ≤ ((𝑂‘𝐴) +𝑒 (𝑂‘𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 387 = wceq 1507 ∈ wcel 2050 Vcvv 3409 ∪ cun 3821 ⊆ wss 3823 𝒫 cpw 4416 {cpr 4437 ∪ cuni 4706 class class class wbr 4923 ↦ cmpt 5002 dom cdm 5401 ↾ cres 5403 ‘cfv 6182 (class class class)co 6970 ωcom 7390 ≼ cdom 8298 ≺ csdm 8299 Fincfn 8300 0cc0 10329 +∞cpnf 10465 ℝ*cxr 10467 ≤ cle 10469 +𝑒 cxad 12316 [,]cicc 12551 Σ^csumge0 42075 OutMeascome 42202 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2744 ax-rep 5043 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 ax-inf2 8892 ax-cnex 10385 ax-resscn 10386 ax-1cn 10387 ax-icn 10388 ax-addcl 10389 ax-addrcl 10390 ax-mulcl 10391 ax-mulrcl 10392 ax-mulcom 10393 ax-addass 10394 ax-mulass 10395 ax-distr 10396 ax-i2m1 10397 ax-1ne0 10398 ax-1rid 10399 ax-rnegex 10400 ax-rrecex 10401 ax-cnre 10402 ax-pre-lttri 10403 ax-pre-lttrn 10404 ax-pre-ltadd 10405 ax-pre-mulgt0 10406 ax-pre-sup 10407 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-fal 1520 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2753 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-nel 3068 df-ral 3087 df-rex 3088 df-reu 3089 df-rmo 3090 df-rab 3091 df-v 3411 df-sbc 3676 df-csb 3781 df-dif 3826 df-un 3828 df-in 3830 df-ss 3837 df-pss 3839 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4707 df-int 4744 df-iun 4788 df-br 4924 df-opab 4986 df-mpt 5003 df-tr 5025 df-id 5306 df-eprel 5311 df-po 5320 df-so 5321 df-fr 5360 df-se 5361 df-we 5362 df-xp 5407 df-rel 5408 df-cnv 5409 df-co 5410 df-dm 5411 df-rn 5412 df-res 5413 df-ima 5414 df-pred 5980 df-ord 6026 df-on 6027 df-lim 6028 df-suc 6029 df-iota 6146 df-fun 6184 df-fn 6185 df-f 6186 df-f1 6187 df-fo 6188 df-f1o 6189 df-fv 6190 df-isom 6191 df-riota 6931 df-ov 6973 df-oprab 6974 df-mpo 6975 df-om 7391 df-1st 7495 df-2nd 7496 df-wrecs 7744 df-recs 7806 df-rdg 7844 df-1o 7899 df-oadd 7903 df-er 8083 df-en 8301 df-dom 8302 df-sdom 8303 df-fin 8304 df-sup 8695 df-oi 8763 df-card 9156 df-pnf 10470 df-mnf 10471 df-xr 10472 df-ltxr 10473 df-le 10474 df-sub 10666 df-neg 10667 df-div 11093 df-nn 11434 df-2 11497 df-3 11498 df-n0 11702 df-z 11788 df-uz 12053 df-rp 12199 df-xadd 12319 df-ico 12554 df-icc 12555 df-fz 12703 df-fzo 12844 df-seq 13179 df-exp 13239 df-hash 13500 df-cj 14313 df-re 14314 df-im 14315 df-sqrt 14449 df-abs 14450 df-clim 14700 df-sum 14898 df-sumge0 42076 df-ome 42203 |
This theorem is referenced by: omelesplit 42231 |
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