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Theorem sitgval 30522
Description: Value of the simple function integral builder for a given space 𝑊 and measure 𝑀. (Contributed by Thierry Arnoux, 30-Jan-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)
Assertion
Ref Expression
sitgval (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)))))
Distinct variable groups:   𝐵,𝑓   𝑓,𝑔,𝑥   𝑓,𝐻   𝑓,𝑀,𝑔,𝑥   𝑆,𝑓,𝑔   𝑓,𝑊,𝑔,𝑥   0 ,𝑓,𝑔,𝑥   · ,𝑓
Allowed substitution hints:   𝜑(𝑥,𝑓,𝑔)   𝐵(𝑥,𝑔)   𝑆(𝑥)   · (𝑥,𝑔)   𝐻(𝑥,𝑔)   𝐽(𝑥,𝑓,𝑔)   𝑉(𝑥,𝑓,𝑔)

Proof of Theorem sitgval
Dummy variables 𝑚 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sitgval.1 . . 3 (𝜑𝑊𝑉)
21elexd 3245 . 2 (𝜑𝑊 ∈ V)
3 sitgval.2 . 2 (𝜑𝑀 ran measures)
4 fveq2 6229 . . . . . . . 8 (𝑤 = 𝑊 → (TopOpen‘𝑤) = (TopOpen‘𝑊))
54fveq2d 6233 . . . . . . 7 (𝑤 = 𝑊 → (sigaGen‘(TopOpen‘𝑤)) = (sigaGen‘(TopOpen‘𝑊)))
6 sitgval.s . . . . . . . 8 𝑆 = (sigaGen‘𝐽)
7 sitgval.j . . . . . . . . 9 𝐽 = (TopOpen‘𝑊)
87fveq2i 6232 . . . . . . . 8 (sigaGen‘𝐽) = (sigaGen‘(TopOpen‘𝑊))
96, 8eqtri 2673 . . . . . . 7 𝑆 = (sigaGen‘(TopOpen‘𝑊))
105, 9syl6eqr 2703 . . . . . 6 (𝑤 = 𝑊 → (sigaGen‘(TopOpen‘𝑤)) = 𝑆)
1110oveq2d 6706 . . . . 5 (𝑤 = 𝑊 → (dom 𝑚MblFnM(sigaGen‘(TopOpen‘𝑤))) = (dom 𝑚MblFnM𝑆))
12 fveq2 6229 . . . . . . . . . 10 (𝑤 = 𝑊 → (0g𝑤) = (0g𝑊))
13 sitgval.0 . . . . . . . . . 10 0 = (0g𝑊)
1412, 13syl6eqr 2703 . . . . . . . . 9 (𝑤 = 𝑊 → (0g𝑤) = 0 )
1514sneqd 4222 . . . . . . . 8 (𝑤 = 𝑊 → {(0g𝑤)} = { 0 })
1615difeq2d 3761 . . . . . . 7 (𝑤 = 𝑊 → (ran 𝑔 ∖ {(0g𝑤)}) = (ran 𝑔 ∖ { 0 }))
1716raleqdv 3174 . . . . . 6 (𝑤 = 𝑊 → (∀𝑥 ∈ (ran 𝑔 ∖ {(0g𝑤)})(𝑚‘(𝑔 “ {𝑥})) ∈ (0[,)+∞) ↔ ∀𝑥 ∈ (ran 𝑔 ∖ { 0 })(𝑚‘(𝑔 “ {𝑥})) ∈ (0[,)+∞)))
1817anbi2d 740 . . . . 5 (𝑤 = 𝑊 → ((ran 𝑔 ∈ Fin ∧ ∀𝑥 ∈ (ran 𝑔 ∖ {(0g𝑤)})(𝑚‘(𝑔 “ {𝑥})) ∈ (0[,)+∞)) ↔ (ran 𝑔 ∈ Fin ∧ ∀𝑥 ∈ (ran 𝑔 ∖ { 0 })(𝑚‘(𝑔 “ {𝑥})) ∈ (0[,)+∞))))
1911, 18rabeqbidv 3226 . . . 4 (𝑤 = 𝑊 → {𝑔 ∈ (dom 𝑚MblFnM(sigaGen‘(TopOpen‘𝑤))) ∣ (ran 𝑔 ∈ Fin ∧ ∀𝑥 ∈ (ran 𝑔 ∖ {(0g𝑤)})(𝑚‘(𝑔 “ {𝑥})) ∈ (0[,)+∞))} = {𝑔 ∈ (dom 𝑚MblFnM𝑆) ∣ (ran 𝑔 ∈ Fin ∧ ∀𝑥 ∈ (ran 𝑔 ∖ { 0 })(𝑚‘(𝑔 “ {𝑥})) ∈ (0[,)+∞))})
20 id 22 . . . . 5 (𝑤 = 𝑊𝑤 = 𝑊)
2115difeq2d 3761 . . . . . 6 (𝑤 = 𝑊 → (ran 𝑓 ∖ {(0g𝑤)}) = (ran 𝑓 ∖ { 0 }))
22 fveq2 6229 . . . . . . . 8 (𝑤 = 𝑊 → ( ·𝑠𝑤) = ( ·𝑠𝑊))
23 sitgval.x . . . . . . . 8 · = ( ·𝑠𝑊)
2422, 23syl6eqr 2703 . . . . . . 7 (𝑤 = 𝑊 → ( ·𝑠𝑤) = · )
25 fveq2 6229 . . . . . . . . . 10 (𝑤 = 𝑊 → (Scalar‘𝑤) = (Scalar‘𝑊))
2625fveq2d 6233 . . . . . . . . 9 (𝑤 = 𝑊 → (ℝHom‘(Scalar‘𝑤)) = (ℝHom‘(Scalar‘𝑊)))
27 sitgval.h . . . . . . . . 9 𝐻 = (ℝHom‘(Scalar‘𝑊))
2826, 27syl6eqr 2703 . . . . . . . 8 (𝑤 = 𝑊 → (ℝHom‘(Scalar‘𝑤)) = 𝐻)
2928fveq1d 6231 . . . . . . 7 (𝑤 = 𝑊 → ((ℝHom‘(Scalar‘𝑤))‘(𝑚‘(𝑓 “ {𝑥}))) = (𝐻‘(𝑚‘(𝑓 “ {𝑥}))))
30 eqidd 2652 . . . . . . 7 (𝑤 = 𝑊𝑥 = 𝑥)
3124, 29, 30oveq123d 6711 . . . . . 6 (𝑤 = 𝑊 → (((ℝHom‘(Scalar‘𝑤))‘(𝑚‘(𝑓 “ {𝑥})))( ·𝑠𝑤)𝑥) = ((𝐻‘(𝑚‘(𝑓 “ {𝑥}))) · 𝑥))
3221, 31mpteq12dv 4766 . . . . 5 (𝑤 = 𝑊 → (𝑥 ∈ (ran 𝑓 ∖ {(0g𝑤)}) ↦ (((ℝHom‘(Scalar‘𝑤))‘(𝑚‘(𝑓 “ {𝑥})))( ·𝑠𝑤)𝑥)) = (𝑥 ∈ (ran 𝑓 ∖ { 0 }) ↦ ((𝐻‘(𝑚‘(𝑓 “ {𝑥}))) · 𝑥)))
3320, 32oveq12d 6708 . . . 4 (𝑤 = 𝑊 → (𝑤 Σg (𝑥 ∈ (ran 𝑓 ∖ {(0g𝑤)}) ↦ (((ℝHom‘(Scalar‘𝑤))‘(𝑚‘(𝑓 “ {𝑥})))( ·𝑠𝑤)𝑥))) = (𝑊 Σg (𝑥 ∈ (ran 𝑓 ∖ { 0 }) ↦ ((𝐻‘(𝑚‘(𝑓 “ {𝑥}))) · 𝑥))))
3419, 33mpteq12dv 4766 . . 3 (𝑤 = 𝑊 → (𝑓 ∈ {𝑔 ∈ (dom 𝑚MblFnM(sigaGen‘(TopOpen‘𝑤))) ∣ (ran 𝑔 ∈ Fin ∧ ∀𝑥 ∈ (ran 𝑔 ∖ {(0g𝑤)})(𝑚‘(𝑔 “ {𝑥})) ∈ (0[,)+∞))} ↦ (𝑤 Σg (𝑥 ∈ (ran 𝑓 ∖ {(0g𝑤)}) ↦ (((ℝHom‘(Scalar‘𝑤))‘(𝑚‘(𝑓 “ {𝑥})))( ·𝑠𝑤)𝑥)))) = (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)))))
35 dmeq 5356 . . . . . 6 (𝑚 = 𝑀 → dom 𝑚 = dom 𝑀)
3635oveq1d 6705 . . . . 5 (𝑚 = 𝑀 → (dom 𝑚MblFnM𝑆) = (dom 𝑀MblFnM𝑆))
37 fveq1 6228 . . . . . . . 8 (𝑚 = 𝑀 → (𝑚‘(𝑔 “ {𝑥})) = (𝑀‘(𝑔 “ {𝑥})))
3837eleq1d 2715 . . . . . . 7 (𝑚 = 𝑀 → ((𝑚‘(𝑔 “ {𝑥})) ∈ (0[,)+∞) ↔ (𝑀‘(𝑔 “ {𝑥})) ∈ (0[,)+∞)))
3938ralbidv 3015 . . . . . 6 (𝑚 = 𝑀 → (∀𝑥 ∈ (ran 𝑔 ∖ { 0 })(𝑚‘(𝑔 “ {𝑥})) ∈ (0[,)+∞) ↔ ∀𝑥 ∈ (ran 𝑔 ∖ { 0 })(𝑀‘(𝑔 “ {𝑥})) ∈ (0[,)+∞)))
4039anbi2d 740 . . . . 5 (𝑚 = 𝑀 → ((ran 𝑔 ∈ Fin ∧ ∀𝑥 ∈ (ran 𝑔 ∖ { 0 })(𝑚‘(𝑔 “ {𝑥})) ∈ (0[,)+∞)) ↔ (ran 𝑔 ∈ Fin ∧ ∀𝑥 ∈ (ran 𝑔 ∖ { 0 })(𝑀‘(𝑔 “ {𝑥})) ∈ (0[,)+∞))))
4136, 40rabeqbidv 3226 . . . 4 (𝑚 = 𝑀 → {𝑔 ∈ (dom 𝑚MblFnM𝑆) ∣ (ran 𝑔 ∈ Fin ∧ ∀𝑥 ∈ (ran 𝑔 ∖ { 0 })(𝑚‘(𝑔 “ {𝑥})) ∈ (0[,)+∞))} = {𝑔 ∈ (dom 𝑀MblFnM𝑆) ∣ (ran 𝑔 ∈ Fin ∧ ∀𝑥 ∈ (ran 𝑔 ∖ { 0 })(𝑀‘(𝑔 “ {𝑥})) ∈ (0[,)+∞))})
42 simpl 472 . . . . . . . . 9 ((𝑚 = 𝑀𝑥 ∈ (ran 𝑓 ∖ { 0 })) → 𝑚 = 𝑀)
4342fveq1d 6231 . . . . . . . 8 ((𝑚 = 𝑀𝑥 ∈ (ran 𝑓 ∖ { 0 })) → (𝑚‘(𝑓 “ {𝑥})) = (𝑀‘(𝑓 “ {𝑥})))
4443fveq2d 6233 . . . . . . 7 ((𝑚 = 𝑀𝑥 ∈ (ran 𝑓 ∖ { 0 })) → (𝐻‘(𝑚‘(𝑓 “ {𝑥}))) = (𝐻‘(𝑀‘(𝑓 “ {𝑥}))))
4544oveq1d 6705 . . . . . 6 ((𝑚 = 𝑀𝑥 ∈ (ran 𝑓 ∖ { 0 })) → ((𝐻‘(𝑚‘(𝑓 “ {𝑥}))) · 𝑥) = ((𝐻‘(𝑀‘(𝑓 “ {𝑥}))) · 𝑥))
4645mpteq2dva 4777 . . . . 5 (𝑚 = 𝑀 → (𝑥 ∈ (ran 𝑓 ∖ { 0 }) ↦ ((𝐻‘(𝑚‘(𝑓 “ {𝑥}))) · 𝑥)) = (𝑥 ∈ (ran 𝑓 ∖ { 0 }) ↦ ((𝐻‘(𝑀‘(𝑓 “ {𝑥}))) · 𝑥)))
4746oveq2d 6706 . . . 4 (𝑚 = 𝑀 → (𝑊 Σg (𝑥 ∈ (ran 𝑓 ∖ { 0 }) ↦ ((𝐻‘(𝑚‘(𝑓 “ {𝑥}))) · 𝑥))) = (𝑊 Σg (𝑥 ∈ (ran 𝑓 ∖ { 0 }) ↦ ((𝐻‘(𝑀‘(𝑓 “ {𝑥}))) · 𝑥))))
4841, 47mpteq12dv 4766 . . 3 (𝑚 = 𝑀 → (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)))) = (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)))))
49 df-sitg 30520 . . 3 sitg = (𝑤 ∈ V, 𝑚 ran measures ↦ (𝑓 ∈ {𝑔 ∈ (dom 𝑚MblFnM(sigaGen‘(TopOpen‘𝑤))) ∣ (ran 𝑔 ∈ Fin ∧ ∀𝑥 ∈ (ran 𝑔 ∖ {(0g𝑤)})(𝑚‘(𝑔 “ {𝑥})) ∈ (0[,)+∞))} ↦ (𝑤 Σg (𝑥 ∈ (ran 𝑓 ∖ {(0g𝑤)}) ↦ (((ℝHom‘(Scalar‘𝑤))‘(𝑚‘(𝑓 “ {𝑥})))( ·𝑠𝑤)𝑥)))))
50 ovex 6718 . . . 4 (dom 𝑀MblFnM𝑆) ∈ V
5150mptrabex 6529 . . 3 (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)))) ∈ V
5234, 48, 49, 51ovmpt2 6838 . 2 ((𝑊 ∈ V ∧ 𝑀 ran measures) → (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)))))
532, 3, 52syl2anc 694 1 (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)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wcel 2030  wral 2941  {crab 2945  Vcvv 3231  cdif 3604  {csn 4210   cuni 4468  cmpt 4762  ccnv 5142  dom cdm 5143  ran crn 5144  cima 5146  cfv 5926  (class class class)co 6690  Fincfn 7997  0cc0 9974  +∞cpnf 10109  [,)cico 12215  Basecbs 15904  Scalarcsca 15991   ·𝑠 cvsca 15992  TopOpenctopn 16129  0gc0g 16147   Σg cgsu 16148  ℝHomcrrh 30165  sigaGencsigagen 30329  measurescmeas 30386  MblFnMcmbfm 30440  sitgcsitg 30519
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-sitg 30520
This theorem is referenced by:  issibf  30523  sitgfval  30531  sitgf  30537
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