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Theorem sitgf 29736
 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 5840 . . . 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 29721 . . . . 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 5825 . . . 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 247 . . 3 (𝜑 → Fun (𝑊sitg𝑀))
13 funfn 5833 . . 3 (Fun (𝑊sitg𝑀) ↔ (𝑊sitg𝑀) Fn dom (𝑊sitg𝑀))
1412, 13sylib 207 . 2 (𝜑 → (𝑊sitg𝑀) Fn dom (𝑊sitg𝑀))
15 sitgf.1 . . . 4 ((𝜑𝑓 ∈ dom (𝑊sitg𝑀)) → ((𝑊sitg𝑀)‘𝑓) ∈ 𝐵)
1615ralrimiva 2949 . . 3 (𝜑 → ∀𝑓 ∈ dom (𝑊sitg𝑀)((𝑊sitg𝑀)‘𝑓) ∈ 𝐵)
17 fnfvrnss 6297 . . 3 (((𝑊sitg𝑀) Fn dom (𝑊sitg𝑀) ∧ ∀𝑓 ∈ dom (𝑊sitg𝑀)((𝑊sitg𝑀)‘𝑓) ∈ 𝐵) → ran (𝑊sitg𝑀) ⊆ 𝐵)
1814, 16, 17syl2anc 691 . 2 (𝜑 → ran (𝑊sitg𝑀) ⊆ 𝐵)
19 df-f 5808 . 2 ((𝑊sitg𝑀):dom (𝑊sitg𝑀)⟶𝐵 ↔ ((𝑊sitg𝑀) Fn dom (𝑊sitg𝑀) ∧ ran (𝑊sitg𝑀) ⊆ 𝐵))
2014, 18, 19sylanbrc 695 1 (𝜑 → (𝑊sitg𝑀):dom (𝑊sitg𝑀)⟶𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  {crab 2900   ∖ cdif 3537   ⊆ wss 3540  {csn 4125  ∪ cuni 4372   ↦ cmpt 4643  ◡ccnv 5037  dom cdm 5038  ran crn 5039   “ cima 5041  Fun wfun 5798   Fn wfn 5799  ⟶wf 5800  ‘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   Σg cgsu 15924  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: (None)
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