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Mirrors > Home > MPE Home > Th. List > Mathboxes > omelesplit | Structured version Visualization version GIF version |
Description: The outer measure of a set 𝐴 is less than or equal to the extended addition of the outer measures of the decomposition induced on 𝐴 by any 𝐸. Step (a) in the proof of Caratheodory's Method, Theorem 113C of [Fremlin1] p. 19. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
Ref | Expression |
---|---|
omelesplit.1 | ⊢ (𝜑 → 𝑂 ∈ OutMeas) |
omelesplit.2 | ⊢ 𝑋 = ∪ dom 𝑂 |
omelesplit.3 | ⊢ (𝜑 → 𝐴 ⊆ 𝑋) |
Ref | Expression |
---|---|
omelesplit | ⊢ (𝜑 → (𝑂‘𝐴) ≤ ((𝑂‘(𝐴 ∩ 𝐸)) +𝑒 (𝑂‘(𝐴 ∖ 𝐸)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | inundif 4385 | . . . . 5 ⊢ ((𝐴 ∩ 𝐸) ∪ (𝐴 ∖ 𝐸)) = 𝐴 | |
2 | 1 | eqcomi 2807 | . . . 4 ⊢ 𝐴 = ((𝐴 ∩ 𝐸) ∪ (𝐴 ∖ 𝐸)) |
3 | 2 | a1i 11 | . . 3 ⊢ (𝜑 → 𝐴 = ((𝐴 ∩ 𝐸) ∪ (𝐴 ∖ 𝐸))) |
4 | 3 | fveq2d 6649 | . 2 ⊢ (𝜑 → (𝑂‘𝐴) = (𝑂‘((𝐴 ∩ 𝐸) ∪ (𝐴 ∖ 𝐸)))) |
5 | omelesplit.1 | . . 3 ⊢ (𝜑 → 𝑂 ∈ OutMeas) | |
6 | omelesplit.2 | . . 3 ⊢ 𝑋 = ∪ dom 𝑂 | |
7 | omelesplit.3 | . . . 4 ⊢ (𝜑 → 𝐴 ⊆ 𝑋) | |
8 | ssinss1 4164 | . . . 4 ⊢ (𝐴 ⊆ 𝑋 → (𝐴 ∩ 𝐸) ⊆ 𝑋) | |
9 | 7, 8 | syl 17 | . . 3 ⊢ (𝜑 → (𝐴 ∩ 𝐸) ⊆ 𝑋) |
10 | 7 | ssdifssd 4070 | . . 3 ⊢ (𝜑 → (𝐴 ∖ 𝐸) ⊆ 𝑋) |
11 | 5, 6, 9, 10 | omeunle 43155 | . 2 ⊢ (𝜑 → (𝑂‘((𝐴 ∩ 𝐸) ∪ (𝐴 ∖ 𝐸))) ≤ ((𝑂‘(𝐴 ∩ 𝐸)) +𝑒 (𝑂‘(𝐴 ∖ 𝐸)))) |
12 | 4, 11 | eqbrtrd 5052 | 1 ⊢ (𝜑 → (𝑂‘𝐴) ≤ ((𝑂‘(𝐴 ∩ 𝐸)) +𝑒 (𝑂‘(𝐴 ∖ 𝐸)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 ∖ cdif 3878 ∪ cun 3879 ∩ cin 3880 ⊆ wss 3881 ∪ cuni 4800 class class class wbr 5030 dom cdm 5519 ‘cfv 6324 (class class class)co 7135 ≤ cle 10665 +𝑒 cxad 12493 OutMeascome 43128 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-inf2 9088 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-sup 8890 df-oi 8958 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-n0 11886 df-z 11970 df-uz 12232 df-rp 12378 df-xadd 12496 df-ico 12732 df-icc 12733 df-fz 12886 df-fzo 13029 df-seq 13365 df-exp 13426 df-hash 13687 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-clim 14837 df-sum 15035 df-sumge0 43002 df-ome 43129 |
This theorem is referenced by: carageniuncl 43162 caragenel2d 43171 |
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