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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 5218 | . . 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 43332 | . . . . . . . 8 β’ (π β π β V) |
8 | 7 | adantr 482 | . . . . . . 7 β’ ((π β§ π β π) β π β V) |
9 | ssexg 5285 | . . . . . . 7 β’ ((πΈ β π β§ π β V) β πΈ β V) | |
10 | 6, 8, 9 | syl2anc 585 | . . . . . 6 β’ ((π β§ π β π) β πΈ β V) |
11 | elpwg 4568 | . . . . . 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 257 | . . . 4 β’ ((π β§ π β π) β πΈ β π« π) |
14 | eqid 2737 | . . . 4 β’ (π β π β¦ πΈ) = (π β π β¦ πΈ) | |
15 | 1, 13, 14 | fmptdf 7070 | . . 3 β’ (π β (π β π β¦ πΈ):πβΆπ« π) |
16 | 1, 2, 3, 4, 5, 15 | omeiunle 44832 | . 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 486 | . . . . . . 7 β’ ((π β§ π β π) β π β π) | |
18 | 14 | fvmpt2 6964 | . . . . . . 7 β’ ((π β π β§ πΈ β V) β ((π β π β¦ πΈ)βπ) = πΈ) |
19 | 17, 10, 18 | syl2anc 585 | . . . . . 6 β’ ((π β§ π β π) β ((π β π β¦ πΈ)βπ) = πΈ) |
20 | 19 | eqcomd 2743 | . . . . 5 β’ ((π β§ π β π) β πΈ = ((π β π β¦ πΈ)βπ)) |
21 | 1, 20 | iuneq2df 43328 | . . . 4 β’ (π β βͺ π β π πΈ = βͺ π β π ((π β π β¦ πΈ)βπ)) |
22 | 21 | fveq2d 6851 | . . 3 β’ (π β (πββͺ π β π πΈ) = (πββͺ π β π ((π β π β¦ πΈ)βπ))) |
23 | 20 | fveq2d 6851 | . . . . 5 β’ ((π β§ π β π) β (πβπΈ) = (πβ((π β π β¦ πΈ)βπ))) |
24 | 1, 23 | mpteq2da 5208 | . . . 4 β’ (π β (π β π β¦ (πβπΈ)) = (π β π β¦ (πβ((π β π β¦ πΈ)βπ)))) |
25 | 24 | fveq2d 6851 | . . 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 5123 | . 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 257 | 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 397 = wceq 1542 β²wnf 1786 β wcel 2107 Vcvv 3448 β wss 3915 π« cpw 4565 βͺ cuni 4870 βͺ ciun 4959 class class class wbr 5110 β¦ cmpt 5193 dom cdm 5638 βcfv 6501 β€ cle 11197 β€β₯cuz 12770 Ξ£^csumge0 44677 OutMeascome 44804 |
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-ac2 10406 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-oadd 8421 df-omul 8422 df-er 8655 df-map 8774 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-sup 9385 df-oi 9453 df-card 9882 df-acn 9885 df-ac 10059 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-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 |
This theorem is referenced by: (None) |
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