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Mathbox for Glauco Siliprandi |
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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 4474 | . . . . 5 ⊢ ((𝐴 ∩ 𝐸) ∪ (𝐴 ∖ 𝐸)) = 𝐴 | |
2 | 1 | eqcomi 2734 | . . . 4 ⊢ 𝐴 = ((𝐴 ∩ 𝐸) ∪ (𝐴 ∖ 𝐸)) |
3 | 2 | a1i 11 | . . 3 ⊢ (𝜑 → 𝐴 = ((𝐴 ∩ 𝐸) ∪ (𝐴 ∖ 𝐸))) |
4 | 3 | fveq2d 6895 | . 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 4232 | . . . 4 ⊢ (𝐴 ⊆ 𝑋 → (𝐴 ∩ 𝐸) ⊆ 𝑋) | |
9 | 7, 8 | syl 17 | . . 3 ⊢ (𝜑 → (𝐴 ∩ 𝐸) ⊆ 𝑋) |
10 | 7 | ssdifssd 4135 | . . 3 ⊢ (𝜑 → (𝐴 ∖ 𝐸) ⊆ 𝑋) |
11 | 5, 6, 9, 10 | omeunle 45966 | . 2 ⊢ (𝜑 → (𝑂‘((𝐴 ∩ 𝐸) ∪ (𝐴 ∖ 𝐸))) ≤ ((𝑂‘(𝐴 ∩ 𝐸)) +𝑒 (𝑂‘(𝐴 ∖ 𝐸)))) |
12 | 4, 11 | eqbrtrd 5165 | 1 ⊢ (𝜑 → (𝑂‘𝐴) ≤ ((𝑂‘(𝐴 ∩ 𝐸)) +𝑒 (𝑂‘(𝐴 ∖ 𝐸)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ∖ cdif 3937 ∪ cun 3938 ∩ cin 3939 ⊆ wss 3940 ∪ cuni 4903 class class class wbr 5143 dom cdm 5672 ‘cfv 6542 (class class class)co 7415 ≤ cle 11277 +𝑒 cxad 13120 OutMeascome 45939 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7737 ax-inf2 9662 ax-cnex 11192 ax-resscn 11193 ax-1cn 11194 ax-icn 11195 ax-addcl 11196 ax-addrcl 11197 ax-mulcl 11198 ax-mulrcl 11199 ax-mulcom 11200 ax-addass 11201 ax-mulass 11202 ax-distr 11203 ax-i2m1 11204 ax-1ne0 11205 ax-1rid 11206 ax-rnegex 11207 ax-rrecex 11208 ax-cnre 11209 ax-pre-lttri 11210 ax-pre-lttrn 11211 ax-pre-ltadd 11212 ax-pre-mulgt0 11213 ax-pre-sup 11214 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3960 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-om 7868 df-1st 7989 df-2nd 7990 df-frecs 8283 df-wrecs 8314 df-recs 8388 df-rdg 8427 df-1o 8483 df-er 8721 df-en 8961 df-dom 8962 df-sdom 8963 df-fin 8964 df-sup 9463 df-oi 9531 df-card 9960 df-pnf 11278 df-mnf 11279 df-xr 11280 df-ltxr 11281 df-le 11282 df-sub 11474 df-neg 11475 df-div 11900 df-nn 12241 df-2 12303 df-3 12304 df-n0 12501 df-z 12587 df-uz 12851 df-rp 13005 df-xadd 13123 df-ico 13360 df-icc 13361 df-fz 13515 df-fzo 13658 df-seq 13997 df-exp 14057 df-hash 14320 df-cj 15076 df-re 15077 df-im 15078 df-sqrt 15212 df-abs 15213 df-clim 15462 df-sum 15663 df-sumge0 45813 df-ome 45940 |
This theorem is referenced by: carageniuncl 45973 caragenel2d 45982 |
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