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Theorem sitgval 34166
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 3485 . 2 (𝜑𝑊 ∈ V)
3 sitgval.2 . 2 (𝜑𝑀 ran measures)
4 2fveq3 6906 . . . . . . 7 (𝑤 = 𝑊 → (sigaGen‘(TopOpen‘𝑤)) = (sigaGen‘(TopOpen‘𝑊)))
5 sitgval.s . . . . . . . 8 𝑆 = (sigaGen‘𝐽)
6 sitgval.j . . . . . . . . 9 𝐽 = (TopOpen‘𝑊)
76fveq2i 6904 . . . . . . . 8 (sigaGen‘𝐽) = (sigaGen‘(TopOpen‘𝑊))
85, 7eqtri 2754 . . . . . . 7 𝑆 = (sigaGen‘(TopOpen‘𝑊))
94, 8eqtr4di 2784 . . . . . 6 (𝑤 = 𝑊 → (sigaGen‘(TopOpen‘𝑤)) = 𝑆)
109oveq2d 7440 . . . . 5 (𝑤 = 𝑊 → (dom 𝑚MblFnM(sigaGen‘(TopOpen‘𝑤))) = (dom 𝑚MblFnM𝑆))
11 fveq2 6901 . . . . . . . . . 10 (𝑤 = 𝑊 → (0g𝑤) = (0g𝑊))
12 sitgval.0 . . . . . . . . . 10 0 = (0g𝑊)
1311, 12eqtr4di 2784 . . . . . . . . 9 (𝑤 = 𝑊 → (0g𝑤) = 0 )
1413sneqd 4645 . . . . . . . 8 (𝑤 = 𝑊 → {(0g𝑤)} = { 0 })
1514difeq2d 4121 . . . . . . 7 (𝑤 = 𝑊 → (ran 𝑔 ∖ {(0g𝑤)}) = (ran 𝑔 ∖ { 0 }))
1615raleqdv 3315 . . . . . 6 (𝑤 = 𝑊 → (∀𝑥 ∈ (ran 𝑔 ∖ {(0g𝑤)})(𝑚‘(𝑔 “ {𝑥})) ∈ (0[,)+∞) ↔ ∀𝑥 ∈ (ran 𝑔 ∖ { 0 })(𝑚‘(𝑔 “ {𝑥})) ∈ (0[,)+∞)))
1716anbi2d 628 . . . . 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[,)+∞))))
1810, 17rabeqbidv 3437 . . . 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[,)+∞))})
19 id 22 . . . . 5 (𝑤 = 𝑊𝑤 = 𝑊)
2014difeq2d 4121 . . . . . 6 (𝑤 = 𝑊 → (ran 𝑓 ∖ {(0g𝑤)}) = (ran 𝑓 ∖ { 0 }))
21 fveq2 6901 . . . . . . . 8 (𝑤 = 𝑊 → ( ·𝑠𝑤) = ( ·𝑠𝑊))
22 sitgval.x . . . . . . . 8 · = ( ·𝑠𝑊)
2321, 22eqtr4di 2784 . . . . . . 7 (𝑤 = 𝑊 → ( ·𝑠𝑤) = · )
24 2fveq3 6906 . . . . . . . . 9 (𝑤 = 𝑊 → (ℝHom‘(Scalar‘𝑤)) = (ℝHom‘(Scalar‘𝑊)))
25 sitgval.h . . . . . . . . 9 𝐻 = (ℝHom‘(Scalar‘𝑊))
2624, 25eqtr4di 2784 . . . . . . . 8 (𝑤 = 𝑊 → (ℝHom‘(Scalar‘𝑤)) = 𝐻)
2726fveq1d 6903 . . . . . . 7 (𝑤 = 𝑊 → ((ℝHom‘(Scalar‘𝑤))‘(𝑚‘(𝑓 “ {𝑥}))) = (𝐻‘(𝑚‘(𝑓 “ {𝑥}))))
28 eqidd 2727 . . . . . . 7 (𝑤 = 𝑊𝑥 = 𝑥)
2923, 27, 28oveq123d 7445 . . . . . 6 (𝑤 = 𝑊 → (((ℝHom‘(Scalar‘𝑤))‘(𝑚‘(𝑓 “ {𝑥})))( ·𝑠𝑤)𝑥) = ((𝐻‘(𝑚‘(𝑓 “ {𝑥}))) · 𝑥))
3020, 29mpteq12dv 5244 . . . . 5 (𝑤 = 𝑊 → (𝑥 ∈ (ran 𝑓 ∖ {(0g𝑤)}) ↦ (((ℝHom‘(Scalar‘𝑤))‘(𝑚‘(𝑓 “ {𝑥})))( ·𝑠𝑤)𝑥)) = (𝑥 ∈ (ran 𝑓 ∖ { 0 }) ↦ ((𝐻‘(𝑚‘(𝑓 “ {𝑥}))) · 𝑥)))
3119, 30oveq12d 7442 . . . 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))))
3218, 31mpteq12dv 5244 . . 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)))))
33 dmeq 5910 . . . . . 6 (𝑚 = 𝑀 → dom 𝑚 = dom 𝑀)
3433oveq1d 7439 . . . . 5 (𝑚 = 𝑀 → (dom 𝑚MblFnM𝑆) = (dom 𝑀MblFnM𝑆))
35 fveq1 6900 . . . . . . . 8 (𝑚 = 𝑀 → (𝑚‘(𝑔 “ {𝑥})) = (𝑀‘(𝑔 “ {𝑥})))
3635eleq1d 2811 . . . . . . 7 (𝑚 = 𝑀 → ((𝑚‘(𝑔 “ {𝑥})) ∈ (0[,)+∞) ↔ (𝑀‘(𝑔 “ {𝑥})) ∈ (0[,)+∞)))
3736ralbidv 3168 . . . . . 6 (𝑚 = 𝑀 → (∀𝑥 ∈ (ran 𝑔 ∖ { 0 })(𝑚‘(𝑔 “ {𝑥})) ∈ (0[,)+∞) ↔ ∀𝑥 ∈ (ran 𝑔 ∖ { 0 })(𝑀‘(𝑔 “ {𝑥})) ∈ (0[,)+∞)))
3837anbi2d 628 . . . . 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[,)+∞))))
3934, 38rabeqbidv 3437 . . . 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[,)+∞))})
40 simpl 481 . . . . . . . . 9 ((𝑚 = 𝑀𝑥 ∈ (ran 𝑓 ∖ { 0 })) → 𝑚 = 𝑀)
4140fveq1d 6903 . . . . . . . 8 ((𝑚 = 𝑀𝑥 ∈ (ran 𝑓 ∖ { 0 })) → (𝑚‘(𝑓 “ {𝑥})) = (𝑀‘(𝑓 “ {𝑥})))
4241fveq2d 6905 . . . . . . 7 ((𝑚 = 𝑀𝑥 ∈ (ran 𝑓 ∖ { 0 })) → (𝐻‘(𝑚‘(𝑓 “ {𝑥}))) = (𝐻‘(𝑀‘(𝑓 “ {𝑥}))))
4342oveq1d 7439 . . . . . 6 ((𝑚 = 𝑀𝑥 ∈ (ran 𝑓 ∖ { 0 })) → ((𝐻‘(𝑚‘(𝑓 “ {𝑥}))) · 𝑥) = ((𝐻‘(𝑀‘(𝑓 “ {𝑥}))) · 𝑥))
4443mpteq2dva 5253 . . . . 5 (𝑚 = 𝑀 → (𝑥 ∈ (ran 𝑓 ∖ { 0 }) ↦ ((𝐻‘(𝑚‘(𝑓 “ {𝑥}))) · 𝑥)) = (𝑥 ∈ (ran 𝑓 ∖ { 0 }) ↦ ((𝐻‘(𝑀‘(𝑓 “ {𝑥}))) · 𝑥)))
4544oveq2d 7440 . . . 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))))
4639, 45mpteq12dv 5244 . . 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)))))
47 df-sitg 34164 . . 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)))))
48 ovex 7457 . . . 4 (dom 𝑀MblFnM𝑆) ∈ V
4948mptrabex 7242 . . 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
5032, 46, 47, 49ovmpo 7586 . 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)))))
512, 3, 50syl2anc 582 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 394   = wceq 1534  wcel 2099  wral 3051  {crab 3419  Vcvv 3462  cdif 3944  {csn 4633   cuni 4913  cmpt 5236  ccnv 5681  dom cdm 5682  ran crn 5683  cima 5685  cfv 6554  (class class class)co 7424  Fincfn 8974  0cc0 11158  +∞cpnf 11295  [,)cico 13380  Basecbs 17213  Scalarcsca 17269   ·𝑠 cvsca 17270  TopOpenctopn 17436  0gc0g 17454   Σg cgsu 17455  ℝHomcrrh 33808  sigaGencsigagen 33971  measurescmeas 34028  MblFnMcmbfm 34082  sitgcsitg 34163
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5290  ax-sep 5304  ax-nul 5311  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-iun 5003  df-br 5154  df-opab 5216  df-mpt 5237  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-ov 7427  df-oprab 7428  df-mpo 7429  df-sitg 34164
This theorem is referenced by:  issibf  34167  sitgfval  34175  sitgf  34181
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