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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > caragencmpl | Structured version Visualization version GIF version |
Description: A measure built with the Caratheodory's construction is complete. See Definition 112Df of [Fremlin1] p. 19. This is Exercise 113Xa of [Fremlin1] p. 21. (Contributed by Glauco Siliprandi, 24-Dec-2020.) |
Ref | Expression |
---|---|
caragencmpl.o | β’ (π β π β OutMeas) |
caragencmpl.x | β’ π = βͺ dom π |
caragencmpl.e | β’ (π β πΈ β π) |
caragencmpl.z | β’ (π β (πβπΈ) = 0) |
caragencmpl.s | β’ π = (CaraGenβπ) |
Ref | Expression |
---|---|
caragencmpl | β’ (π β πΈ β π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | caragencmpl.o | . 2 β’ (π β π β OutMeas) | |
2 | caragencmpl.x | . 2 β’ π = βͺ dom π | |
3 | caragencmpl.s | . 2 β’ π = (CaraGenβπ) | |
4 | caragencmpl.e | . . 3 β’ (π β πΈ β π) | |
5 | 1, 2 | unidmex 43722 | . . . . 5 β’ (π β π β V) |
6 | 5, 4 | ssexd 5323 | . . . 4 β’ (π β πΈ β V) |
7 | elpwg 4604 | . . . 4 β’ (πΈ β V β (πΈ β π« π β πΈ β π)) | |
8 | 6, 7 | syl 17 | . . 3 β’ (π β (πΈ β π« π β πΈ β π)) |
9 | 4, 8 | mpbird 256 | . 2 β’ (π β πΈ β π« π) |
10 | 1 | adantr 481 | . . . . . 6 β’ ((π β§ π β π« π) β π β OutMeas) |
11 | 4 | adantr 481 | . . . . . 6 β’ ((π β§ π β π« π) β πΈ β π) |
12 | caragencmpl.z | . . . . . . 7 β’ (π β (πβπΈ) = 0) | |
13 | 12 | adantr 481 | . . . . . 6 β’ ((π β§ π β π« π) β (πβπΈ) = 0) |
14 | inss2 4228 | . . . . . . 7 β’ (π β© πΈ) β πΈ | |
15 | 14 | a1i 11 | . . . . . 6 β’ ((π β§ π β π« π) β (π β© πΈ) β πΈ) |
16 | 10, 2, 11, 13, 15 | omess0 45236 | . . . . 5 β’ ((π β§ π β π« π) β (πβ(π β© πΈ)) = 0) |
17 | 16 | oveq1d 7420 | . . . 4 β’ ((π β§ π β π« π) β ((πβ(π β© πΈ)) +π (πβ(π β πΈ))) = (0 +π (πβ(π β πΈ)))) |
18 | difssd 4131 | . . . . . . . 8 β’ (π β π« π β (π β πΈ) β π) | |
19 | elpwi 4608 | . . . . . . . 8 β’ (π β π« π β π β π) | |
20 | 18, 19 | sstrd 3991 | . . . . . . 7 β’ (π β π« π β (π β πΈ) β π) |
21 | 20 | adantl 482 | . . . . . 6 β’ ((π β§ π β π« π) β (π β πΈ) β π) |
22 | 10, 2, 21 | omexrcl 45209 | . . . . 5 β’ ((π β§ π β π« π) β (πβ(π β πΈ)) β β*) |
23 | xaddlid 13217 | . . . . 5 β’ ((πβ(π β πΈ)) β β* β (0 +π (πβ(π β πΈ))) = (πβ(π β πΈ))) | |
24 | 22, 23 | syl 17 | . . . 4 β’ ((π β§ π β π« π) β (0 +π (πβ(π β πΈ))) = (πβ(π β πΈ))) |
25 | 17, 24 | eqtrd 2772 | . . 3 β’ ((π β§ π β π« π) β ((πβ(π β© πΈ)) +π (πβ(π β πΈ))) = (πβ(π β πΈ))) |
26 | 19 | adantl 482 | . . . 4 β’ ((π β§ π β π« π) β π β π) |
27 | 18 | adantl 482 | . . . 4 β’ ((π β§ π β π« π) β (π β πΈ) β π) |
28 | 10, 2, 26, 27 | omessle 45200 | . . 3 β’ ((π β§ π β π« π) β (πβ(π β πΈ)) β€ (πβπ)) |
29 | 25, 28 | eqbrtrd 5169 | . 2 β’ ((π β§ π β π« π) β ((πβ(π β© πΈ)) +π (πβ(π β πΈ))) β€ (πβπ)) |
30 | 1, 2, 3, 9, 29 | caragenel2d 45234 | 1 β’ (π β πΈ β π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 396 = wceq 1541 β wcel 2106 Vcvv 3474 β cdif 3944 β© cin 3946 β wss 3947 π« cpw 4601 βͺ cuni 4907 dom cdm 5675 βcfv 6540 (class class class)co 7405 0cc0 11106 β*cxr 11243 β€ cle 11245 +π cxad 13086 OutMeascome 45191 CaraGenccaragen 45193 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-inf2 9632 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-sup 9433 df-oi 9501 df-card 9930 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-n0 12469 df-z 12555 df-uz 12819 df-rp 12971 df-xadd 13089 df-ico 13326 df-icc 13327 df-fz 13481 df-fzo 13624 df-seq 13963 df-exp 14024 df-hash 14287 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-clim 15428 df-sum 15629 df-sumge0 45065 df-ome 45192 df-caragen 45194 |
This theorem is referenced by: voncmpl 45323 |
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