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Theorem ismeas 28036
 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 3102 . . 3 measures
21a1i 11 . 2 sigAlgebra measures
3 simp1 995 . . 3 Disj Σ*
4 ovex 6305 . . . 4
5 fex2 6736 . . . . . 6 sigAlgebra
653expb 1196 . . . . 5 sigAlgebra
76expcom 435 . . . 4 sigAlgebra
84, 7mpan2 671 . . 3 sigAlgebra
93, 8syl5 32 . 2 sigAlgebra Disj Σ*
10 df-meas 28033 . . . 4 measures sigAlgebra Disj Σ*
11 vex 3096 . . . . . 6
12 mapex 7424 . . . . . 6
1311, 4, 12mp2an 672 . . . . 5
14 simp1 995 . . . . . 6 Disj Σ*
1514ss2abi 3554 . . . . 5 Disj Σ*
1613, 15ssexi 4578 . . . 4 Disj Σ*
17 simpr 461 . . . . . 6
18 simpl 457 . . . . . 6
1917, 18feq12d 5706 . . . . 5
20 fveq1 5851 . . . . . . 7
2120eqeq1d 2443 . . . . . 6
2221adantl 466 . . . . 5
2318pweqd 3998 . . . . . 6
24 fveq1 5851 . . . . . . . . 9
25 fveq1 5851 . . . . . . . . . 10
2625esumeq2sdv 27918 . . . . . . . . 9 Σ* Σ*
2724, 26eqeq12d 2463 . . . . . . . 8 Σ* Σ*
2827imbi2d 316 . . . . . . 7 Disj Σ* Disj Σ*
2928adantl 466 . . . . . 6 Disj Σ* Disj Σ*
3023, 29raleqbidv 3052 . . . . 5 Disj Σ* Disj Σ*
3119, 22, 303anbi123d 1298 . . . 4 Disj Σ* Disj Σ*
3210, 16, 31abfmpel 27358 . . 3 sigAlgebra measures Disj Σ*
3332ex 434 . 2 sigAlgebra measures Disj Σ*
342, 9, 33pm5.21ndd 354 1 sigAlgebra measures Disj Σ*
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 184   wa 369   w3a 972   wceq 1381   wcel 1802  cab 2426  wral 2791  cvv 3093  c0 3767  cpw 3993  cuni 4230  Disj wdisj 4403   class class class wbr 4433   crn 4986  wf 5570  cfv 5574  (class class class)co 6277  com 6681   cdom 7512  cc0 9490   cpnf 9623  cicc 11536  Σ*cesum 27906  sigAlgebracsiga 27973  measurescmeas 28032 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 974  df-tru 1384  df-fal 1387  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-op 4017  df-uni 4231  df-br 4434  df-opab 4492  df-mpt 4493  df-id 4781  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-fv 5582  df-ov 6280  df-esum 27907  df-meas 28033 This theorem is referenced by:  measbasedom  28039  measfrge0  28040  measvnul  28043  measvun  28046  measinb  28058  measres  28059  measdivcstOLD  28061  measdivcst  28062  cntmeas  28063  volmeas  28069  ddemeas  28074  dstrvprob  28276
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