Mathbox for Thierry Arnoux < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ismeas Structured version   Unicode version

Theorem ismeas 24545
 Description: The property of being a measure (Contributed by Thierry Arnoux, 10-Sep-2016.) (Revised by Thierry Arnoux, 19-Oct-2016.)
Assertion
Ref Expression
ismeas sigAlgebra measures Disj Σ*
Distinct variable groups:   ,,   ,
Allowed substitution hint:   ()

Proof of Theorem ismeas
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 2956 . . 3 measures
21a1i 11 . 2 sigAlgebra measures
3 simp1 957 . . 3 Disj Σ*
4 ovex 6098 . . . 4
5 fex2 5595 . . . . . 6 sigAlgebra
653expb 1154 . . . . 5 sigAlgebra
76expcom 425 . . . 4 sigAlgebra
84, 7mpan2 653 . . 3 sigAlgebra
93, 8syl5 30 . 2 sigAlgebra Disj Σ*
10 df-meas 24542 . . . 4 measures sigAlgebra Disj Σ*
11 vex 2951 . . . . . 6
12 mapex 7016 . . . . . 6
1311, 4, 12mp2an 654 . . . . 5
14 simp1 957 . . . . . 6 Disj Σ*
1514ss2abi 3407 . . . . 5 Disj Σ*
1613, 15ssexi 4340 . . . 4 Disj Σ*
17 simpr 448 . . . . . 6
18 simpl 444 . . . . . 6
1917, 18feq12d 5574 . . . . 5
20 fveq1 5719 . . . . . . 7
2120eqeq1d 2443 . . . . . 6
2221adantl 453 . . . . 5
2318pweqd 3796 . . . . . 6
24 fveq1 5719 . . . . . . . . 9
25 fveq1 5719 . . . . . . . . . 10
2625esumeq2sdv 24428 . . . . . . . . 9 Σ* Σ*
2724, 26eqeq12d 2449 . . . . . . . 8 Σ* Σ*
2827imbi2d 308 . . . . . . 7 Disj Σ* Disj Σ*
2928adantl 453 . . . . . 6 Disj Σ* Disj Σ*
3023, 29raleqbidv 2908 . . . . 5 Disj Σ* Disj Σ*
3119, 22, 303anbi123d 1254 . . . 4 Disj Σ* Disj Σ*
3210, 16, 31abfmpel 24059 . . 3 sigAlgebra measures Disj Σ*
3332ex 424 . 2 sigAlgebra measures Disj Σ*
342, 9, 33pm5.21ndd 344 1 sigAlgebra measures Disj Σ*
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177   wa 359   w3a 936   wceq 1652   wcel 1725  cab 2421  wral 2697  cvv 2948  c0 3620  cpw 3791  cuni 4007  Disj wdisj 4174   class class class wbr 4204  com 4837   crn 4871  wf 5442  cfv 5446  (class class class)co 6073   cdom 7099  cc0 8982   cpnf 9109  cicc 10911  Σ*cesum 24416  sigAlgebracsiga 24482  measurescmeas 24541 This theorem is referenced by:  measbasedom  24548  measfrge0  24549  measvnul  24552  measvun  24555  measinb  24567  measres  24568  measdivcstOLD  24570  measdivcst  24571  cntmeas  24572  volmeas  24579  dstrvprob  24721 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-fv 5454  df-ov 6076  df-esum 24417  df-meas 24542
 Copyright terms: Public domain W3C validator