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Theorem sibfmbl 29517
Description: A simple function is measurable. (Contributed by Thierry Arnoux, 19-Feb-2018.)
Hypotheses
Ref Expression
sitgval.b 𝐵 = (Base‘𝑊)
sitgval.j 𝐽 = (TopOpen‘𝑊)
sitgval.s 𝑆 = (sigaGen‘𝐽)
sitgval.0 0 = (0g𝑊)
sitgval.x · = ( ·𝑠𝑊)
sitgval.h 𝐻 = (ℝHom‘(Scalar‘𝑊))
sitgval.1 (𝜑𝑊𝑉)
sitgval.2 (𝜑𝑀 ran measures)
sibfmbl.1 (𝜑𝐹 ∈ dom (𝑊sitg𝑀))
Assertion
Ref Expression
sibfmbl (𝜑𝐹 ∈ (dom 𝑀MblFnM𝑆))

Proof of Theorem sibfmbl
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 sibfmbl.1 . . 3 (𝜑𝐹 ∈ dom (𝑊sitg𝑀))
2 sitgval.b . . . 4 𝐵 = (Base‘𝑊)
3 sitgval.j . . . 4 𝐽 = (TopOpen‘𝑊)
4 sitgval.s . . . 4 𝑆 = (sigaGen‘𝐽)
5 sitgval.0 . . . 4 0 = (0g𝑊)
6 sitgval.x . . . 4 · = ( ·𝑠𝑊)
7 sitgval.h . . . 4 𝐻 = (ℝHom‘(Scalar‘𝑊))
8 sitgval.1 . . . 4 (𝜑𝑊𝑉)
9 sitgval.2 . . . 4 (𝜑𝑀 ran measures)
102, 3, 4, 5, 6, 7, 8, 9issibf 29515 . . 3 (𝜑 → (𝐹 ∈ dom (𝑊sitg𝑀) ↔ (𝐹 ∈ (dom 𝑀MblFnM𝑆) ∧ ran 𝐹 ∈ Fin ∧ ∀𝑥 ∈ (ran 𝐹 ∖ { 0 })(𝑀‘(𝐹 “ {𝑥})) ∈ (0[,)+∞))))
111, 10mpbid 220 . 2 (𝜑 → (𝐹 ∈ (dom 𝑀MblFnM𝑆) ∧ ran 𝐹 ∈ Fin ∧ ∀𝑥 ∈ (ran 𝐹 ∖ { 0 })(𝑀‘(𝐹 “ {𝑥})) ∈ (0[,)+∞)))
1211simp1d 1065 1 (𝜑𝐹 ∈ (dom 𝑀MblFnM𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1030   = wceq 1474  wcel 1976  wral 2895  cdif 3536  {csn 4124   cuni 4366  ccnv 5026  dom cdm 5027  ran crn 5028  cima 5030  cfv 5789  (class class class)co 6526  Fincfn 7818  0cc0 9792  +∞cpnf 9927  [,)cico 12006  Basecbs 15643  Scalarcsca 15719   ·𝑠 cvsca 15720  TopOpenctopn 15853  0gc0g 15871  ℝHomcrrh 29158  sigaGencsigagen 29321  measurescmeas 29378  MblFnMcmbfm 29432  sitgcsitg 29511
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4711  ax-pr 4827
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4942  df-xp 5033  df-rel 5034  df-cnv 5035  df-co 5036  df-dm 5037  df-rn 5038  df-res 5039  df-ima 5040  df-iota 5753  df-fun 5791  df-fn 5792  df-f 5793  df-f1 5794  df-fo 5795  df-f1o 5796  df-fv 5797  df-ov 6529  df-oprab 6530  df-mpt2 6531  df-sitg 29512
This theorem is referenced by:  sibff  29518  sibfinima  29521  sibfof  29522  sitgfval  29523  sitgclg  29524
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