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Theorem sitgf 29542
Description: The integral for simple functions is itself a function. (Contributed by Thierry Arnoux, 13-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)
sitgf.1 ((𝜑𝑓 ∈ dom (𝑊sitg𝑀)) → ((𝑊sitg𝑀)‘𝑓) ∈ 𝐵)
Assertion
Ref Expression
sitgf (𝜑 → (𝑊sitg𝑀):dom (𝑊sitg𝑀)⟶𝐵)
Distinct variable groups:   𝐵,𝑓   𝑓,𝐻   𝑓,𝑀   𝑆,𝑓   𝑓,𝑊   0 ,𝑓   · ,𝑓   𝜑,𝑓
Allowed substitution hints:   𝐽(𝑓)   𝑉(𝑓)

Proof of Theorem sitgf
Dummy variables 𝑔 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 funmpt 5826 . . . 4 Fun (𝑓 ∈ {𝑔 ∈ (dom 𝑀MblFnM𝑆) ∣ (ran 𝑔 ∈ Fin ∧ ∀𝑥 ∈ (ran 𝑔 ∖ { 0 })(𝑀‘(𝑔 “ {𝑥})) ∈ (0[,)+∞))} ↦ (𝑊 Σg (𝑥 ∈ (ran 𝑓 ∖ { 0 }) ↦ ((𝐻‘(𝑀‘(𝑓 “ {𝑥}))) · 𝑥))))
2 sitgval.b . . . . . 6 𝐵 = (Base‘𝑊)
3 sitgval.j . . . . . 6 𝐽 = (TopOpen‘𝑊)
4 sitgval.s . . . . . 6 𝑆 = (sigaGen‘𝐽)
5 sitgval.0 . . . . . 6 0 = (0g𝑊)
6 sitgval.x . . . . . 6 · = ( ·𝑠𝑊)
7 sitgval.h . . . . . 6 𝐻 = (ℝHom‘(Scalar‘𝑊))
8 sitgval.1 . . . . . 6 (𝜑𝑊𝑉)
9 sitgval.2 . . . . . 6 (𝜑𝑀 ran measures)
102, 3, 4, 5, 6, 7, 8, 9sitgval 29527 . . . . 5 (𝜑 → (𝑊sitg𝑀) = (𝑓 ∈ {𝑔 ∈ (dom 𝑀MblFnM𝑆) ∣ (ran 𝑔 ∈ Fin ∧ ∀𝑥 ∈ (ran 𝑔 ∖ { 0 })(𝑀‘(𝑔 “ {𝑥})) ∈ (0[,)+∞))} ↦ (𝑊 Σg (𝑥 ∈ (ran 𝑓 ∖ { 0 }) ↦ ((𝐻‘(𝑀‘(𝑓 “ {𝑥}))) · 𝑥)))))
1110funeqd 5811 . . . 4 (𝜑 → (Fun (𝑊sitg𝑀) ↔ Fun (𝑓 ∈ {𝑔 ∈ (dom 𝑀MblFnM𝑆) ∣ (ran 𝑔 ∈ Fin ∧ ∀𝑥 ∈ (ran 𝑔 ∖ { 0 })(𝑀‘(𝑔 “ {𝑥})) ∈ (0[,)+∞))} ↦ (𝑊 Σg (𝑥 ∈ (ran 𝑓 ∖ { 0 }) ↦ ((𝐻‘(𝑀‘(𝑓 “ {𝑥}))) · 𝑥))))))
121, 11mpbiri 246 . . 3 (𝜑 → Fun (𝑊sitg𝑀))
13 funfn 5819 . . 3 (Fun (𝑊sitg𝑀) ↔ (𝑊sitg𝑀) Fn dom (𝑊sitg𝑀))
1412, 13sylib 206 . 2 (𝜑 → (𝑊sitg𝑀) Fn dom (𝑊sitg𝑀))
15 sitgf.1 . . . 4 ((𝜑𝑓 ∈ dom (𝑊sitg𝑀)) → ((𝑊sitg𝑀)‘𝑓) ∈ 𝐵)
1615ralrimiva 2948 . . 3 (𝜑 → ∀𝑓 ∈ dom (𝑊sitg𝑀)((𝑊sitg𝑀)‘𝑓) ∈ 𝐵)
17 fnfvrnss 6282 . . 3 (((𝑊sitg𝑀) Fn dom (𝑊sitg𝑀) ∧ ∀𝑓 ∈ dom (𝑊sitg𝑀)((𝑊sitg𝑀)‘𝑓) ∈ 𝐵) → ran (𝑊sitg𝑀) ⊆ 𝐵)
1814, 16, 17syl2anc 690 . 2 (𝜑 → ran (𝑊sitg𝑀) ⊆ 𝐵)
19 df-f 5794 . 2 ((𝑊sitg𝑀):dom (𝑊sitg𝑀)⟶𝐵 ↔ ((𝑊sitg𝑀) Fn dom (𝑊sitg𝑀) ∧ ran (𝑊sitg𝑀) ⊆ 𝐵))
2014, 18, 19sylanbrc 694 1 (𝜑 → (𝑊sitg𝑀):dom (𝑊sitg𝑀)⟶𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  wcel 1976  wral 2895  {crab 2899  cdif 3536  wss 3539  {csn 4124   cuni 4366  cmpt 4637  ccnv 5027  dom cdm 5028  ran crn 5029  cima 5031  Fun wfun 5784   Fn wfn 5785  wf 5786  cfv 5790  (class class class)co 6527  Fincfn 7818  0cc0 9792  +∞cpnf 9927  [,)cico 12004  Basecbs 15641  Scalarcsca 15717   ·𝑠 cvsca 15718  TopOpenctopn 15851  0gc0g 15869   Σg cgsu 15870  ℝHomcrrh 29171  sigaGencsigagen 29334  measurescmeas 29391  MblFnMcmbfm 29445  sitgcsitg 29524
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 2032  ax-13 2232  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pr 4828
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 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-sitg 29525
This theorem is referenced by: (None)
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