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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 43332 | . . . . 5 β’ (π β π β V) |
6 | 5, 4 | ssexd 5286 | . . . 4 β’ (π β πΈ β V) |
7 | elpwg 4568 | . . . 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 257 | . 2 β’ (π β πΈ β π« π) |
10 | 1 | adantr 482 | . . . . . 6 β’ ((π β§ π β π« π) β π β OutMeas) |
11 | 4 | adantr 482 | . . . . . 6 β’ ((π β§ π β π« π) β πΈ β π) |
12 | caragencmpl.z | . . . . . . 7 β’ (π β (πβπΈ) = 0) | |
13 | 12 | adantr 482 | . . . . . 6 β’ ((π β§ π β π« π) β (πβπΈ) = 0) |
14 | inss2 4194 | . . . . . . 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 44849 | . . . . 5 β’ ((π β§ π β π« π) β (πβ(π β© πΈ)) = 0) |
17 | 16 | oveq1d 7377 | . . . 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 4097 | . . . . . . . 8 β’ (π β π« π β (π β πΈ) β π) | |
19 | elpwi 4572 | . . . . . . . 8 β’ (π β π« π β π β π) | |
20 | 18, 19 | sstrd 3959 | . . . . . . 7 β’ (π β π« π β (π β πΈ) β π) |
21 | 20 | adantl 483 | . . . . . 6 β’ ((π β§ π β π« π) β (π β πΈ) β π) |
22 | 10, 2, 21 | omexrcl 44822 | . . . . 5 β’ ((π β§ π β π« π) β (πβ(π β πΈ)) β β*) |
23 | xaddid2 13168 | . . . . 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 2777 | . . 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 483 | . . . 4 β’ ((π β§ π β π« π) β π β π) |
27 | 18 | adantl 483 | . . . 4 β’ ((π β§ π β π« π) β (π β πΈ) β π) |
28 | 10, 2, 26, 27 | omessle 44813 | . . 3 β’ ((π β§ π β π« π) β (πβ(π β πΈ)) β€ (πβπ)) |
29 | 25, 28 | eqbrtrd 5132 | . 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 44847 | 1 β’ (π β πΈ β π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 397 = wceq 1542 β wcel 2107 Vcvv 3448 β cdif 3912 β© cin 3914 β wss 3915 π« cpw 4565 βͺ cuni 4870 dom cdm 5638 βcfv 6501 (class class class)co 7362 0cc0 11058 β*cxr 11195 β€ cle 11197 +π cxad 13038 OutMeascome 44804 CaraGenccaragen 44806 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5247 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-inf2 9584 ax-cnex 11114 ax-resscn 11115 ax-1cn 11116 ax-icn 11117 ax-addcl 11118 ax-addrcl 11119 ax-mulcl 11120 ax-mulrcl 11121 ax-mulcom 11122 ax-addass 11123 ax-mulass 11124 ax-distr 11125 ax-i2m1 11126 ax-1ne0 11127 ax-1rid 11128 ax-rnegex 11129 ax-rrecex 11130 ax-cnre 11131 ax-pre-lttri 11132 ax-pre-lttrn 11133 ax-pre-ltadd 11134 ax-pre-mulgt0 11135 ax-pre-sup 11136 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rmo 3356 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-int 4913 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-se 5594 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-isom 6510 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7808 df-1st 7926 df-2nd 7927 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-1o 8417 df-er 8655 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-sup 9385 df-oi 9453 df-card 9882 df-pnf 11198 df-mnf 11199 df-xr 11200 df-ltxr 11201 df-le 11202 df-sub 11394 df-neg 11395 df-div 11820 df-nn 12161 df-2 12223 df-3 12224 df-n0 12421 df-z 12507 df-uz 12771 df-rp 12923 df-xadd 13041 df-ico 13277 df-icc 13278 df-fz 13432 df-fzo 13575 df-seq 13914 df-exp 13975 df-hash 14238 df-cj 14991 df-re 14992 df-im 14993 df-sqrt 15127 df-abs 15128 df-clim 15377 df-sum 15578 df-sumge0 44678 df-ome 44805 df-caragen 44807 |
This theorem is referenced by: voncmpl 44936 |
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