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Theorem sibfmbl 29724
 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 29722 . . 3 (𝜑 → (𝐹 ∈ dom (𝑊sitg𝑀) ↔ (𝐹 ∈ (dom 𝑀MblFnM𝑆) ∧ ran 𝐹 ∈ Fin ∧ ∀𝑥 ∈ (ran 𝐹 ∖ { 0 })(𝑀‘(𝐹 “ {𝑥})) ∈ (0[,)+∞))))
111, 10mpbid 221 . 2 (𝜑 → (𝐹 ∈ (dom 𝑀MblFnM𝑆) ∧ ran 𝐹 ∈ Fin ∧ ∀𝑥 ∈ (ran 𝐹 ∖ { 0 })(𝑀‘(𝐹 “ {𝑥})) ∈ (0[,)+∞)))
1211simp1d 1066 1 (𝜑𝐹 ∈ (dom 𝑀MblFnM𝑆))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896   ∖ cdif 3537  {csn 4125  ∪ cuni 4372  ◡ccnv 5037  dom cdm 5038  ran crn 5039   “ cima 5041  ‘cfv 5804  (class class class)co 6549  Fincfn 7841  0cc0 9815  +∞cpnf 9950  [,)cico 12048  Basecbs 15695  Scalarcsca 15771   ·𝑠 cvsca 15772  TopOpenctopn 15905  0gc0g 15923  ℝHomcrrh 29365  sigaGencsigagen 29528  measurescmeas 29585  MblFnMcmbfm 29639  sitgcsitg 29718 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-sitg 29719 This theorem is referenced by:  sibff  29725  sibfinima  29728  sibfof  29729  sitgfval  29730  sitgclg  29731
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