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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > omeiunlempt | Structured version Visualization version GIF version |
Description: The outer measure of the indexed union of a countable set is the less than or equal to the extended sum of the outer measures. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
Ref | Expression |
---|---|
omeiunlempt.nph | β’ β²ππ |
omeiunlempt.o | β’ (π β π β OutMeas) |
omeiunlempt.x | β’ π = βͺ dom π |
omeiunlempt.z | β’ π = (β€β₯βπ) |
omeiunlempt.e | β’ ((π β§ π β π) β πΈ β π) |
Ref | Expression |
---|---|
omeiunlempt | β’ (π β (πββͺ π β π πΈ) β€ (Ξ£^β(π β π β¦ (πβπΈ)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | omeiunlempt.nph | . . 3 β’ β²ππ | |
2 | nfmpt1 5255 | . . 3 β’ β²π(π β π β¦ πΈ) | |
3 | omeiunlempt.o | . . 3 β’ (π β π β OutMeas) | |
4 | omeiunlempt.x | . . 3 β’ π = βͺ dom π | |
5 | omeiunlempt.z | . . 3 β’ π = (β€β₯βπ) | |
6 | omeiunlempt.e | . . . . 5 β’ ((π β§ π β π) β πΈ β π) | |
7 | 3, 4 | unidmex 43722 | . . . . . . . 8 β’ (π β π β V) |
8 | 7 | adantr 481 | . . . . . . 7 β’ ((π β§ π β π) β π β V) |
9 | ssexg 5322 | . . . . . . 7 β’ ((πΈ β π β§ π β V) β πΈ β V) | |
10 | 6, 8, 9 | syl2anc 584 | . . . . . 6 β’ ((π β§ π β π) β πΈ β V) |
11 | elpwg 4604 | . . . . . 6 β’ (πΈ β V β (πΈ β π« π β πΈ β π)) | |
12 | 10, 11 | syl 17 | . . . . 5 β’ ((π β§ π β π) β (πΈ β π« π β πΈ β π)) |
13 | 6, 12 | mpbird 256 | . . . 4 β’ ((π β§ π β π) β πΈ β π« π) |
14 | eqid 2732 | . . . 4 β’ (π β π β¦ πΈ) = (π β π β¦ πΈ) | |
15 | 1, 13, 14 | fmptdf 7113 | . . 3 β’ (π β (π β π β¦ πΈ):πβΆπ« π) |
16 | 1, 2, 3, 4, 5, 15 | omeiunle 45219 | . 2 β’ (π β (πββͺ π β π ((π β π β¦ πΈ)βπ)) β€ (Ξ£^β(π β π β¦ (πβ((π β π β¦ πΈ)βπ))))) |
17 | simpr 485 | . . . . . . 7 β’ ((π β§ π β π) β π β π) | |
18 | 14 | fvmpt2 7006 | . . . . . . 7 β’ ((π β π β§ πΈ β V) β ((π β π β¦ πΈ)βπ) = πΈ) |
19 | 17, 10, 18 | syl2anc 584 | . . . . . 6 β’ ((π β§ π β π) β ((π β π β¦ πΈ)βπ) = πΈ) |
20 | 19 | eqcomd 2738 | . . . . 5 β’ ((π β§ π β π) β πΈ = ((π β π β¦ πΈ)βπ)) |
21 | 1, 20 | iuneq2df 43718 | . . . 4 β’ (π β βͺ π β π πΈ = βͺ π β π ((π β π β¦ πΈ)βπ)) |
22 | 21 | fveq2d 6892 | . . 3 β’ (π β (πββͺ π β π πΈ) = (πββͺ π β π ((π β π β¦ πΈ)βπ))) |
23 | 20 | fveq2d 6892 | . . . . 5 β’ ((π β§ π β π) β (πβπΈ) = (πβ((π β π β¦ πΈ)βπ))) |
24 | 1, 23 | mpteq2da 5245 | . . . 4 β’ (π β (π β π β¦ (πβπΈ)) = (π β π β¦ (πβ((π β π β¦ πΈ)βπ)))) |
25 | 24 | fveq2d 6892 | . . 3 β’ (π β (Ξ£^β(π β π β¦ (πβπΈ))) = (Ξ£^β(π β π β¦ (πβ((π β π β¦ πΈ)βπ))))) |
26 | 22, 25 | breq12d 5160 | . 2 β’ (π β ((πββͺ π β π πΈ) β€ (Ξ£^β(π β π β¦ (πβπΈ))) β (πββͺ π β π ((π β π β¦ πΈ)βπ)) β€ (Ξ£^β(π β π β¦ (πβ((π β π β¦ πΈ)βπ)))))) |
27 | 16, 26 | mpbird 256 | 1 β’ (π β (πββͺ π β π πΈ) β€ (Ξ£^β(π β π β¦ (πβπΈ)))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 396 = wceq 1541 β²wnf 1785 β wcel 2106 Vcvv 3474 β wss 3947 π« cpw 4601 βͺ cuni 4907 βͺ ciun 4996 class class class wbr 5147 β¦ cmpt 5230 dom cdm 5675 βcfv 6540 β€ cle 11245 β€β₯cuz 12818 Ξ£^csumge0 45064 OutMeascome 45191 |
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-ac2 10454 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-oadd 8466 df-omul 8467 df-er 8699 df-map 8818 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-sup 9433 df-oi 9501 df-card 9930 df-acn 9933 df-ac 10107 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-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 |
This theorem is referenced by: (None) |
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