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Theorem sitgval 31489
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 3512 . 2 (𝜑𝑊 ∈ V)
3 sitgval.2 . 2 (𝜑𝑀 ran measures)
4 2fveq3 6668 . . . . . . 7 (𝑤 = 𝑊 → (sigaGen‘(TopOpen‘𝑤)) = (sigaGen‘(TopOpen‘𝑊)))
5 sitgval.s . . . . . . . 8 𝑆 = (sigaGen‘𝐽)
6 sitgval.j . . . . . . . . 9 𝐽 = (TopOpen‘𝑊)
76fveq2i 6666 . . . . . . . 8 (sigaGen‘𝐽) = (sigaGen‘(TopOpen‘𝑊))
85, 7eqtri 2841 . . . . . . 7 𝑆 = (sigaGen‘(TopOpen‘𝑊))
94, 8syl6eqr 2871 . . . . . 6 (𝑤 = 𝑊 → (sigaGen‘(TopOpen‘𝑤)) = 𝑆)
109oveq2d 7161 . . . . 5 (𝑤 = 𝑊 → (dom 𝑚MblFnM(sigaGen‘(TopOpen‘𝑤))) = (dom 𝑚MblFnM𝑆))
11 fveq2 6663 . . . . . . . . . 10 (𝑤 = 𝑊 → (0g𝑤) = (0g𝑊))
12 sitgval.0 . . . . . . . . . 10 0 = (0g𝑊)
1311, 12syl6eqr 2871 . . . . . . . . 9 (𝑤 = 𝑊 → (0g𝑤) = 0 )
1413sneqd 4569 . . . . . . . 8 (𝑤 = 𝑊 → {(0g𝑤)} = { 0 })
1514difeq2d 4096 . . . . . . 7 (𝑤 = 𝑊 → (ran 𝑔 ∖ {(0g𝑤)}) = (ran 𝑔 ∖ { 0 }))
1615raleqdv 3413 . . . . . 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 3483 . . . 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 4096 . . . . . 6 (𝑤 = 𝑊 → (ran 𝑓 ∖ {(0g𝑤)}) = (ran 𝑓 ∖ { 0 }))
21 fveq2 6663 . . . . . . . 8 (𝑤 = 𝑊 → ( ·𝑠𝑤) = ( ·𝑠𝑊))
22 sitgval.x . . . . . . . 8 · = ( ·𝑠𝑊)
2321, 22syl6eqr 2871 . . . . . . 7 (𝑤 = 𝑊 → ( ·𝑠𝑤) = · )
24 2fveq3 6668 . . . . . . . . 9 (𝑤 = 𝑊 → (ℝHom‘(Scalar‘𝑤)) = (ℝHom‘(Scalar‘𝑊)))
25 sitgval.h . . . . . . . . 9 𝐻 = (ℝHom‘(Scalar‘𝑊))
2624, 25syl6eqr 2871 . . . . . . . 8 (𝑤 = 𝑊 → (ℝHom‘(Scalar‘𝑤)) = 𝐻)
2726fveq1d 6665 . . . . . . 7 (𝑤 = 𝑊 → ((ℝHom‘(Scalar‘𝑤))‘(𝑚‘(𝑓 “ {𝑥}))) = (𝐻‘(𝑚‘(𝑓 “ {𝑥}))))
28 eqidd 2819 . . . . . . 7 (𝑤 = 𝑊𝑥 = 𝑥)
2923, 27, 28oveq123d 7166 . . . . . 6 (𝑤 = 𝑊 → (((ℝHom‘(Scalar‘𝑤))‘(𝑚‘(𝑓 “ {𝑥})))( ·𝑠𝑤)𝑥) = ((𝐻‘(𝑚‘(𝑓 “ {𝑥}))) · 𝑥))
3020, 29mpteq12dv 5142 . . . . 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 7163 . . . 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 5142 . . 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 5765 . . . . . 6 (𝑚 = 𝑀 → dom 𝑚 = dom 𝑀)
3433oveq1d 7160 . . . . 5 (𝑚 = 𝑀 → (dom 𝑚MblFnM𝑆) = (dom 𝑀MblFnM𝑆))
35 fveq1 6662 . . . . . . . 8 (𝑚 = 𝑀 → (𝑚‘(𝑔 “ {𝑥})) = (𝑀‘(𝑔 “ {𝑥})))
3635eleq1d 2894 . . . . . . 7 (𝑚 = 𝑀 → ((𝑚‘(𝑔 “ {𝑥})) ∈ (0[,)+∞) ↔ (𝑀‘(𝑔 “ {𝑥})) ∈ (0[,)+∞)))
3736ralbidv 3194 . . . . . 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 3483 . . . 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 483 . . . . . . . . 9 ((𝑚 = 𝑀𝑥 ∈ (ran 𝑓 ∖ { 0 })) → 𝑚 = 𝑀)
4140fveq1d 6665 . . . . . . . 8 ((𝑚 = 𝑀𝑥 ∈ (ran 𝑓 ∖ { 0 })) → (𝑚‘(𝑓 “ {𝑥})) = (𝑀‘(𝑓 “ {𝑥})))
4241fveq2d 6667 . . . . . . 7 ((𝑚 = 𝑀𝑥 ∈ (ran 𝑓 ∖ { 0 })) → (𝐻‘(𝑚‘(𝑓 “ {𝑥}))) = (𝐻‘(𝑀‘(𝑓 “ {𝑥}))))
4342oveq1d 7160 . . . . . 6 ((𝑚 = 𝑀𝑥 ∈ (ran 𝑓 ∖ { 0 })) → ((𝐻‘(𝑚‘(𝑓 “ {𝑥}))) · 𝑥) = ((𝐻‘(𝑀‘(𝑓 “ {𝑥}))) · 𝑥))
4443mpteq2dva 5152 . . . . 5 (𝑚 = 𝑀 → (𝑥 ∈ (ran 𝑓 ∖ { 0 }) ↦ ((𝐻‘(𝑚‘(𝑓 “ {𝑥}))) · 𝑥)) = (𝑥 ∈ (ran 𝑓 ∖ { 0 }) ↦ ((𝐻‘(𝑀‘(𝑓 “ {𝑥}))) · 𝑥)))
4544oveq2d 7161 . . . 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 5142 . . 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 31487 . . 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 7178 . . . 4 (dom 𝑀MblFnM𝑆) ∈ V
4948mptrabex 6979 . . 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 7299 . 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 584 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 396   = wceq 1528  wcel 2105  wral 3135  {crab 3139  Vcvv 3492  cdif 3930  {csn 4557   cuni 4830  cmpt 5137  ccnv 5547  dom cdm 5548  ran crn 5549  cima 5551  cfv 6348  (class class class)co 7145  Fincfn 8497  0cc0 10525  +∞cpnf 10660  [,)cico 12728  Basecbs 16471  Scalarcsca 16556   ·𝑠 cvsca 16557  TopOpenctopn 16683  0gc0g 16701   Σg cgsu 16702  ℝHomcrrh 31133  sigaGencsigagen 31296  measurescmeas 31353  MblFnMcmbfm 31407  sitgcsitg 31486
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-sitg 31487
This theorem is referenced by:  issibf  31490  sitgfval  31498  sitgf  31504
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