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Mirrors > Home > MPE Home > Th. List > mbfneg | Structured version Visualization version GIF version |
Description: The negative of a measurable function is measurable. (Contributed by Mario Carneiro, 31-Jul-2014.) |
Ref | Expression |
---|---|
mbfneg.1 | β’ ((π β§ π₯ β π΄) β π΅ β π) |
mbfneg.2 | β’ (π β (π₯ β π΄ β¦ π΅) β MblFn) |
Ref | Expression |
---|---|
mbfneg | β’ (π β (π₯ β π΄ β¦ -π΅) β MblFn) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2733 | . . . . . 6 β’ (π₯ β π΄ β¦ π΅) = (π₯ β π΄ β¦ π΅) | |
2 | mbfneg.1 | . . . . . 6 β’ ((π β§ π₯ β π΄) β π΅ β π) | |
3 | 1, 2 | dmmptd 6696 | . . . . 5 β’ (π β dom (π₯ β π΄ β¦ π΅) = π΄) |
4 | mbfneg.2 | . . . . . 6 β’ (π β (π₯ β π΄ β¦ π΅) β MblFn) | |
5 | 4 | dmexd 7896 | . . . . 5 β’ (π β dom (π₯ β π΄ β¦ π΅) β V) |
6 | 3, 5 | eqeltrrd 2835 | . . . 4 β’ (π β π΄ β V) |
7 | neg1rr 12327 | . . . . 5 β’ -1 β β | |
8 | 7 | a1i 11 | . . . 4 β’ ((π β§ π₯ β π΄) β -1 β β) |
9 | fconstmpt 5739 | . . . . 5 β’ (π΄ Γ {-1}) = (π₯ β π΄ β¦ -1) | |
10 | 9 | a1i 11 | . . . 4 β’ (π β (π΄ Γ {-1}) = (π₯ β π΄ β¦ -1)) |
11 | eqidd 2734 | . . . 4 β’ (π β (π₯ β π΄ β¦ π΅) = (π₯ β π΄ β¦ π΅)) | |
12 | 6, 8, 2, 10, 11 | offval2 7690 | . . 3 β’ (π β ((π΄ Γ {-1}) βf Β· (π₯ β π΄ β¦ π΅)) = (π₯ β π΄ β¦ (-1 Β· π΅))) |
13 | 4, 2 | mbfmptcl 25153 | . . . . 5 β’ ((π β§ π₯ β π΄) β π΅ β β) |
14 | 13 | mulm1d 11666 | . . . 4 β’ ((π β§ π₯ β π΄) β (-1 Β· π΅) = -π΅) |
15 | 14 | mpteq2dva 5249 | . . 3 β’ (π β (π₯ β π΄ β¦ (-1 Β· π΅)) = (π₯ β π΄ β¦ -π΅)) |
16 | 12, 15 | eqtrd 2773 | . 2 β’ (π β ((π΄ Γ {-1}) βf Β· (π₯ β π΄ β¦ π΅)) = (π₯ β π΄ β¦ -π΅)) |
17 | 7 | a1i 11 | . . 3 β’ (π β -1 β β) |
18 | 13 | fmpttd 7115 | . . 3 β’ (π β (π₯ β π΄ β¦ π΅):π΄βΆβ) |
19 | 4, 17, 18 | mbfmulc2re 25165 | . 2 β’ (π β ((π΄ Γ {-1}) βf Β· (π₯ β π΄ β¦ π΅)) β MblFn) |
20 | 16, 19 | eqeltrrd 2835 | 1 β’ (π β (π₯ β π΄ β¦ -π΅) β MblFn) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 Vcvv 3475 {csn 4629 β¦ cmpt 5232 Γ cxp 5675 dom cdm 5677 (class class class)co 7409 βf cof 7668 βcc 11108 βcr 11109 1c1 11111 Β· cmul 11115 -cneg 11445 MblFncmbf 25131 |
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 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-inf2 9636 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 ax-pre-sup 11188 |
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 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-of 7670 df-om 7856 df-1st 7975 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-2o 8467 df-er 8703 df-map 8822 df-pm 8823 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-sup 9437 df-inf 9438 df-oi 9505 df-dju 9896 df-card 9934 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-div 11872 df-nn 12213 df-2 12275 df-3 12276 df-n0 12473 df-z 12559 df-uz 12823 df-q 12933 df-rp 12975 df-xadd 13093 df-ioo 13328 df-ico 13330 df-icc 13331 df-fz 13485 df-fzo 13628 df-fl 13757 df-seq 13967 df-exp 14028 df-hash 14291 df-cj 15046 df-re 15047 df-im 15048 df-sqrt 15182 df-abs 15183 df-clim 15432 df-sum 15633 df-xmet 20937 df-met 20938 df-ovol 24981 df-vol 24982 df-mbf 25136 |
This theorem is referenced by: mbfposb 25170 mbfsub 25179 mbfinf 25182 mbfi1flimlem 25240 itgreval 25314 ibladd 25338 iblabslem 25345 ibladdnc 36545 itgaddnclem2 36547 itgmulc2nclem2 36555 ftc1anclem6 36566 |
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