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Theorem sitgf 31833
 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 6373 . . . 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 31818 . . . . 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 6357 . . . 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 261 . . 3 (𝜑 → Fun (𝑊sitg𝑀))
1312funfnd 6366 . 2 (𝜑 → (𝑊sitg𝑀) Fn dom (𝑊sitg𝑀))
14 sitgf.1 . . . 4 ((𝜑𝑓 ∈ dom (𝑊sitg𝑀)) → ((𝑊sitg𝑀)‘𝑓) ∈ 𝐵)
1514ralrimiva 3113 . . 3 (𝜑 → ∀𝑓 ∈ dom (𝑊sitg𝑀)((𝑊sitg𝑀)‘𝑓) ∈ 𝐵)
16 fnfvrnss 6875 . . 3 (((𝑊sitg𝑀) Fn dom (𝑊sitg𝑀) ∧ ∀𝑓 ∈ dom (𝑊sitg𝑀)((𝑊sitg𝑀)‘𝑓) ∈ 𝐵) → ran (𝑊sitg𝑀) ⊆ 𝐵)
1713, 15, 16syl2anc 587 . 2 (𝜑 → ran (𝑊sitg𝑀) ⊆ 𝐵)
18 df-f 6339 . 2 ((𝑊sitg𝑀):dom (𝑊sitg𝑀)⟶𝐵 ↔ ((𝑊sitg𝑀) Fn dom (𝑊sitg𝑀) ∧ ran (𝑊sitg𝑀) ⊆ 𝐵))
1913, 17, 18sylanbrc 586 1 (𝜑 → (𝑊sitg𝑀):dom (𝑊sitg𝑀)⟶𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3070  {crab 3074   ∖ cdif 3855   ⊆ wss 3858  {csn 4522  ∪ cuni 4798   ↦ cmpt 5112  ◡ccnv 5523  dom cdm 5524  ran crn 5525   “ cima 5527  Fun wfun 6329   Fn wfn 6330  ⟶wf 6331  ‘cfv 6335  (class class class)co 7150  Fincfn 8527  0cc0 10575  +∞cpnf 10710  [,)cico 12781  Basecbs 16541  Scalarcsca 16626   ·𝑠 cvsca 16627  TopOpenctopn 16753  0gc0g 16771   Σg cgsu 16772  ℝHomcrrh 31462  sigaGencsigagen 31625  measurescmeas 31682  MblFnMcmbfm 31736  sitgcsitg 31815 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5156  ax-sep 5169  ax-nul 5176  ax-pr 5298 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3697  df-csb 3806  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-iun 4885  df-br 5033  df-opab 5095  df-mpt 5113  df-id 5430  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-f1 6340  df-fo 6341  df-f1o 6342  df-fv 6343  df-ov 7153  df-oprab 7154  df-mpo 7155  df-sitg 31816 This theorem is referenced by: (None)
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