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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > vonvol2 | Structured version Visualization version GIF version |
Description: The 1-dimensional Lebesgue measure agrees with the Lebesgue measure on subsets of Real numbers. (Contributed by Glauco Siliprandi, 3-Mar-2021.) |
Ref | Expression |
---|---|
vonvol2.f | β’ β²ππ |
vonvol2.a | β’ (π β π΄ β π) |
vonvol2.x | β’ (π β π β dom (volnβ{π΄})) |
vonvol2.y | β’ π = βͺ π β π ran π |
Ref | Expression |
---|---|
vonvol2 | β’ (π β ((volnβ{π΄})βπ) = (volβπ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vonvol2.a | . . 3 β’ (π β π΄ β π) | |
2 | vonvol2.f | . . . . . . 7 β’ β²ππ | |
3 | snfi 8991 | . . . . . . . . 9 β’ {π΄} β Fin | |
4 | 3 | a1i 11 | . . . . . . . 8 β’ (π β {π΄} β Fin) |
5 | vonvol2.x | . . . . . . . 8 β’ (π β π β dom (volnβ{π΄})) | |
6 | 4, 5 | vonmblss2 44969 | . . . . . . 7 β’ (π β π β (β βm {π΄})) |
7 | vonvol2.y | . . . . . . 7 β’ π = βͺ π β π ran π | |
8 | 2, 1, 6, 7 | ssmapsn 43524 | . . . . . 6 β’ (π β π = (π βm {π΄})) |
9 | 8 | eqcomd 2739 | . . . . 5 β’ (π β (π βm {π΄}) = π) |
10 | 9, 5 | eqeltrd 2834 | . . . 4 β’ (π β (π βm {π΄}) β dom (volnβ{π΄})) |
11 | 6 | adantr 482 | . . . . . . . . . 10 β’ ((π β§ π β π) β π β (β βm {π΄})) |
12 | simpr 486 | . . . . . . . . . 10 β’ ((π β§ π β π) β π β π) | |
13 | 11, 12 | sseldd 3946 | . . . . . . . . 9 β’ ((π β§ π β π) β π β (β βm {π΄})) |
14 | elmapi 8790 | . . . . . . . . 9 β’ (π β (β βm {π΄}) β π:{π΄}βΆβ) | |
15 | frn 6676 | . . . . . . . . 9 β’ (π:{π΄}βΆβ β ran π β β) | |
16 | 13, 14, 15 | 3syl 18 | . . . . . . . 8 β’ ((π β§ π β π) β ran π β β) |
17 | 16 | ralrimiva 3140 | . . . . . . 7 β’ (π β βπ β π ran π β β) |
18 | iunss 5006 | . . . . . . 7 β’ (βͺ π β π ran π β β β βπ β π ran π β β) | |
19 | 17, 18 | sylibr 233 | . . . . . 6 β’ (π β βͺ π β π ran π β β) |
20 | 7, 19 | eqsstrid 3993 | . . . . 5 β’ (π β π β β) |
21 | 1, 20 | vonvolmbl 44988 | . . . 4 β’ (π β ((π βm {π΄}) β dom (volnβ{π΄}) β π β dom vol)) |
22 | 10, 21 | mpbid 231 | . . 3 β’ (π β π β dom vol) |
23 | 1, 22 | vonvol 44989 | . 2 β’ (π β ((volnβ{π΄})β(π βm {π΄})) = (volβπ)) |
24 | 9 | eqcomd 2739 | . . 3 β’ (π β π = (π βm {π΄})) |
25 | 24 | fveq2d 6847 | . 2 β’ (π β ((volnβ{π΄})βπ) = ((volnβ{π΄})β(π βm {π΄}))) |
26 | eqidd 2734 | . 2 β’ (π β (volβπ) = (volβπ)) | |
27 | 23, 25, 26 | 3eqtr4d 2783 | 1 β’ (π β ((volnβ{π΄})βπ) = (volβπ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 β²wnfc 2884 βwral 3061 β wss 3911 {csn 4587 βͺ ciun 4955 dom cdm 5634 ran crn 5635 βΆwf 6493 βcfv 6497 (class class class)co 7358 βm cmap 8768 Fincfn 8886 βcr 11055 volcvol 24843 volncvoln 44865 |
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 2704 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-inf2 9582 ax-cc 10376 ax-ac2 10404 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 ax-pre-sup 11134 ax-addf 11135 ax-mulf 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 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-uni 4867 df-int 4909 df-iun 4957 df-disj 5072 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-se 5590 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-isom 6506 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-of 7618 df-om 7804 df-1st 7922 df-2nd 7923 df-tpos 8158 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-2o 8414 df-er 8651 df-map 8770 df-pm 8771 df-ixp 8839 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-fi 9352 df-sup 9383 df-inf 9384 df-oi 9451 df-dju 9842 df-card 9880 df-acn 9883 df-ac 10057 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-div 11818 df-nn 12159 df-2 12221 df-3 12222 df-4 12223 df-5 12224 df-6 12225 df-7 12226 df-8 12227 df-9 12228 df-n0 12419 df-z 12505 df-dec 12624 df-uz 12769 df-q 12879 df-rp 12921 df-xneg 13038 df-xadd 13039 df-xmul 13040 df-ioo 13274 df-ico 13276 df-icc 13277 df-fz 13431 df-fzo 13574 df-fl 13703 df-seq 13913 df-exp 13974 df-hash 14237 df-cj 14990 df-re 14991 df-im 14992 df-sqrt 15126 df-abs 15127 df-clim 15376 df-rlim 15377 df-sum 15577 df-prod 15794 df-struct 17024 df-sets 17041 df-slot 17059 df-ndx 17071 df-base 17089 df-ress 17118 df-plusg 17151 df-mulr 17152 df-starv 17153 df-tset 17157 df-ple 17158 df-ds 17160 df-unif 17161 df-rest 17309 df-0g 17328 df-topgen 17330 df-mgm 18502 df-sgrp 18551 df-mnd 18562 df-grp 18756 df-minusg 18757 df-subg 18930 df-cmn 19569 df-abl 19570 df-mgp 19902 df-ur 19919 df-ring 19971 df-cring 19972 df-oppr 20054 df-dvdsr 20075 df-unit 20076 df-invr 20106 df-dvr 20117 df-drng 20199 df-psmet 20804 df-xmet 20805 df-met 20806 df-bl 20807 df-mopn 20808 df-cnfld 20813 df-top 22259 df-topon 22276 df-bases 22312 df-cmp 22754 df-ovol 24844 df-vol 24845 df-sumge0 44690 df-ome 44817 df-caragen 44819 df-ovoln 44864 df-voln 44866 |
This theorem is referenced by: (None) |
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