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 41332 | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ V) |
5 | ssexg 5227 | . . . . . 6 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ V) → 𝐴 ∈ V) | |
6 | 1, 4, 5 | syl2anc 586 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ V) |
7 | omeunle.b | . . . . . 6 ⊢ (𝜑 → 𝐵 ⊆ 𝑋) | |
8 | ssexg 5227 | . . . . . 6 ⊢ ((𝐵 ⊆ 𝑋 ∧ 𝑋 ∈ V) → 𝐵 ∈ V) | |
9 | 7, 4, 8 | syl2anc 586 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ V) |
10 | uniprg 4856 | . . . . 5 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ∪ {𝐴, 𝐵} = (𝐴 ∪ 𝐵)) | |
11 | 6, 9, 10 | syl2anc 586 | . . . 4 ⊢ (𝜑 → ∪ {𝐴, 𝐵} = (𝐴 ∪ 𝐵)) |
12 | 11 | eqcomd 2827 | . . 3 ⊢ (𝜑 → (𝐴 ∪ 𝐵) = ∪ {𝐴, 𝐵}) |
13 | 12 | fveq2d 6674 | . 2 ⊢ (𝜑 → (𝑂‘(𝐴 ∪ 𝐵)) = (𝑂‘∪ {𝐴, 𝐵})) |
14 | iccssxr 12820 | . . . 4 ⊢ (0[,]+∞) ⊆ ℝ* | |
15 | 1, 7 | unssd 4162 | . . . . . 6 ⊢ (𝜑 → (𝐴 ∪ 𝐵) ⊆ 𝑋) |
16 | 11, 15 | eqsstrd 4005 | . . . . 5 ⊢ (𝜑 → ∪ {𝐴, 𝐵} ⊆ 𝑋) |
17 | 2, 3, 16 | omecl 42805 | . . . 4 ⊢ (𝜑 → (𝑂‘∪ {𝐴, 𝐵}) ∈ (0[,]+∞)) |
18 | 14, 17 | sseldi 3965 | . . 3 ⊢ (𝜑 → (𝑂‘∪ {𝐴, 𝐵}) ∈ ℝ*) |
19 | prfi 8793 | . . . . . 6 ⊢ {𝐴, 𝐵} ∈ Fin | |
20 | 19 | elexi 3513 | . . . . 5 ⊢ {𝐴, 𝐵} ∈ V |
21 | 20 | a1i 11 | . . . 4 ⊢ (𝜑 → {𝐴, 𝐵} ∈ V) |
22 | 2, 3 | omef 42798 | . . . . 5 ⊢ (𝜑 → 𝑂:𝒫 𝑋⟶(0[,]+∞)) |
23 | elpwg 4542 | . . . . . . . . 9 ⊢ (𝐴 ∈ V → (𝐴 ∈ 𝒫 𝑋 ↔ 𝐴 ⊆ 𝑋)) | |
24 | 6, 23 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → (𝐴 ∈ 𝒫 𝑋 ↔ 𝐴 ⊆ 𝑋)) |
25 | 1, 24 | mpbird 259 | . . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ 𝒫 𝑋) |
26 | elpwg 4542 | . . . . . . . . 9 ⊢ (𝐵 ∈ V → (𝐵 ∈ 𝒫 𝑋 ↔ 𝐵 ⊆ 𝑋)) | |
27 | 9, 26 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → (𝐵 ∈ 𝒫 𝑋 ↔ 𝐵 ⊆ 𝑋)) |
28 | 7, 27 | mpbird 259 | . . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ 𝒫 𝑋) |
29 | 25, 28 | jca 514 | . . . . . 6 ⊢ (𝜑 → (𝐴 ∈ 𝒫 𝑋 ∧ 𝐵 ∈ 𝒫 𝑋)) |
30 | prssg 4752 | . . . . . . 7 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ((𝐴 ∈ 𝒫 𝑋 ∧ 𝐵 ∈ 𝒫 𝑋) ↔ {𝐴, 𝐵} ⊆ 𝒫 𝑋)) | |
31 | 6, 9, 30 | syl2anc 586 | . . . . . 6 ⊢ (𝜑 → ((𝐴 ∈ 𝒫 𝑋 ∧ 𝐵 ∈ 𝒫 𝑋) ↔ {𝐴, 𝐵} ⊆ 𝒫 𝑋)) |
32 | 29, 31 | mpbid 234 | . . . . 5 ⊢ (𝜑 → {𝐴, 𝐵} ⊆ 𝒫 𝑋) |
33 | 22, 32 | fssresd 6545 | . . . 4 ⊢ (𝜑 → (𝑂 ↾ {𝐴, 𝐵}):{𝐴, 𝐵}⟶(0[,]+∞)) |
34 | 21, 33 | sge0xrcl 42687 | . . 3 ⊢ (𝜑 → (Σ^‘(𝑂 ↾ {𝐴, 𝐵})) ∈ ℝ*) |
35 | 2, 3, 1 | omecl 42805 | . . . . 5 ⊢ (𝜑 → (𝑂‘𝐴) ∈ (0[,]+∞)) |
36 | 14, 35 | sseldi 3965 | . . . 4 ⊢ (𝜑 → (𝑂‘𝐴) ∈ ℝ*) |
37 | 2, 3, 7 | omecl 42805 | . . . . 5 ⊢ (𝜑 → (𝑂‘𝐵) ∈ (0[,]+∞)) |
38 | 14, 37 | sseldi 3965 | . . . 4 ⊢ (𝜑 → (𝑂‘𝐵) ∈ ℝ*) |
39 | 36, 38 | xaddcld 12695 | . . 3 ⊢ (𝜑 → ((𝑂‘𝐴) +𝑒 (𝑂‘𝐵)) ∈ ℝ*) |
40 | isfinite 9115 | . . . . . . . 8 ⊢ ({𝐴, 𝐵} ∈ Fin ↔ {𝐴, 𝐵} ≺ ω) | |
41 | 40 | biimpi 218 | . . . . . . 7 ⊢ ({𝐴, 𝐵} ∈ Fin → {𝐴, 𝐵} ≺ ω) |
42 | sdomdom 8537 | . . . . . . 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 42807 | . . 3 ⊢ (𝜑 → (𝑂‘∪ {𝐴, 𝐵}) ≤ (Σ^‘(𝑂 ↾ {𝐴, 𝐵}))) |
47 | 22, 32 | feqresmpt 6734 | . . . . 5 ⊢ (𝜑 → (𝑂 ↾ {𝐴, 𝐵}) = (𝑘 ∈ {𝐴, 𝐵} ↦ (𝑂‘𝑘))) |
48 | 47 | fveq2d 6674 | . . . 4 ⊢ (𝜑 → (Σ^‘(𝑂 ↾ {𝐴, 𝐵})) = (Σ^‘(𝑘 ∈ {𝐴, 𝐵} ↦ (𝑂‘𝑘)))) |
49 | fveq2 6670 | . . . . 5 ⊢ (𝑘 = 𝐴 → (𝑂‘𝑘) = (𝑂‘𝐴)) | |
50 | fveq2 6670 | . . . . 5 ⊢ (𝑘 = 𝐵 → (𝑂‘𝑘) = (𝑂‘𝐵)) | |
51 | 6, 9, 35, 37, 49, 50 | sge0prle 42703 | . . . 4 ⊢ (𝜑 → (Σ^‘(𝑘 ∈ {𝐴, 𝐵} ↦ (𝑂‘𝑘))) ≤ ((𝑂‘𝐴) +𝑒 (𝑂‘𝐵))) |
52 | 48, 51 | eqbrtrd 5088 | . . 3 ⊢ (𝜑 → (Σ^‘(𝑂 ↾ {𝐴, 𝐵})) ≤ ((𝑂‘𝐴) +𝑒 (𝑂‘𝐵))) |
53 | 18, 34, 39, 46, 52 | xrletrd 12556 | . 2 ⊢ (𝜑 → (𝑂‘∪ {𝐴, 𝐵}) ≤ ((𝑂‘𝐴) +𝑒 (𝑂‘𝐵))) |
54 | 13, 53 | eqbrtrd 5088 | 1 ⊢ (𝜑 → (𝑂‘(𝐴 ∪ 𝐵)) ≤ ((𝑂‘𝐴) +𝑒 (𝑂‘𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 Vcvv 3494 ∪ cun 3934 ⊆ wss 3936 𝒫 cpw 4539 {cpr 4569 ∪ cuni 4838 class class class wbr 5066 ↦ cmpt 5146 dom cdm 5555 ↾ cres 5557 ‘cfv 6355 (class class class)co 7156 ωcom 7580 ≼ cdom 8507 ≺ csdm 8508 Fincfn 8509 0cc0 10537 +∞cpnf 10672 ℝ*cxr 10674 ≤ cle 10676 +𝑒 cxad 12506 [,]cicc 12742 Σ^csumge0 42664 OutMeascome 42791 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-inf2 9104 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-sup 8906 df-oi 8974 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-n0 11899 df-z 11983 df-uz 12245 df-rp 12391 df-xadd 12509 df-ico 12745 df-icc 12746 df-fz 12894 df-fzo 13035 df-seq 13371 df-exp 13431 df-hash 13692 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-clim 14845 df-sum 15043 df-sumge0 42665 df-ome 42792 |
This theorem is referenced by: omelesplit 42820 |
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