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Theorem sitgval 32932
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 3465 . 2 (𝜑𝑊 ∈ V)
3 sitgval.2 . 2 (𝜑𝑀 ran measures)
4 2fveq3 6847 . . . . . . 7 (𝑤 = 𝑊 → (sigaGen‘(TopOpen‘𝑤)) = (sigaGen‘(TopOpen‘𝑊)))
5 sitgval.s . . . . . . . 8 𝑆 = (sigaGen‘𝐽)
6 sitgval.j . . . . . . . . 9 𝐽 = (TopOpen‘𝑊)
76fveq2i 6845 . . . . . . . 8 (sigaGen‘𝐽) = (sigaGen‘(TopOpen‘𝑊))
85, 7eqtri 2764 . . . . . . 7 𝑆 = (sigaGen‘(TopOpen‘𝑊))
94, 8eqtr4di 2794 . . . . . 6 (𝑤 = 𝑊 → (sigaGen‘(TopOpen‘𝑤)) = 𝑆)
109oveq2d 7373 . . . . 5 (𝑤 = 𝑊 → (dom 𝑚MblFnM(sigaGen‘(TopOpen‘𝑤))) = (dom 𝑚MblFnM𝑆))
11 fveq2 6842 . . . . . . . . . 10 (𝑤 = 𝑊 → (0g𝑤) = (0g𝑊))
12 sitgval.0 . . . . . . . . . 10 0 = (0g𝑊)
1311, 12eqtr4di 2794 . . . . . . . . 9 (𝑤 = 𝑊 → (0g𝑤) = 0 )
1413sneqd 4598 . . . . . . . 8 (𝑤 = 𝑊 → {(0g𝑤)} = { 0 })
1514difeq2d 4082 . . . . . . 7 (𝑤 = 𝑊 → (ran 𝑔 ∖ {(0g𝑤)}) = (ran 𝑔 ∖ { 0 }))
1615raleqdv 3313 . . . . . 6 (𝑤 = 𝑊 → (∀𝑥 ∈ (ran 𝑔 ∖ {(0g𝑤)})(𝑚‘(𝑔 “ {𝑥})) ∈ (0[,)+∞) ↔ ∀𝑥 ∈ (ran 𝑔 ∖ { 0 })(𝑚‘(𝑔 “ {𝑥})) ∈ (0[,)+∞)))
1716anbi2d 629 . . . . 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 3424 . . . 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 4082 . . . . . 6 (𝑤 = 𝑊 → (ran 𝑓 ∖ {(0g𝑤)}) = (ran 𝑓 ∖ { 0 }))
21 fveq2 6842 . . . . . . . 8 (𝑤 = 𝑊 → ( ·𝑠𝑤) = ( ·𝑠𝑊))
22 sitgval.x . . . . . . . 8 · = ( ·𝑠𝑊)
2321, 22eqtr4di 2794 . . . . . . 7 (𝑤 = 𝑊 → ( ·𝑠𝑤) = · )
24 2fveq3 6847 . . . . . . . . 9 (𝑤 = 𝑊 → (ℝHom‘(Scalar‘𝑤)) = (ℝHom‘(Scalar‘𝑊)))
25 sitgval.h . . . . . . . . 9 𝐻 = (ℝHom‘(Scalar‘𝑊))
2624, 25eqtr4di 2794 . . . . . . . 8 (𝑤 = 𝑊 → (ℝHom‘(Scalar‘𝑤)) = 𝐻)
2726fveq1d 6844 . . . . . . 7 (𝑤 = 𝑊 → ((ℝHom‘(Scalar‘𝑤))‘(𝑚‘(𝑓 “ {𝑥}))) = (𝐻‘(𝑚‘(𝑓 “ {𝑥}))))
28 eqidd 2737 . . . . . . 7 (𝑤 = 𝑊𝑥 = 𝑥)
2923, 27, 28oveq123d 7378 . . . . . 6 (𝑤 = 𝑊 → (((ℝHom‘(Scalar‘𝑤))‘(𝑚‘(𝑓 “ {𝑥})))( ·𝑠𝑤)𝑥) = ((𝐻‘(𝑚‘(𝑓 “ {𝑥}))) · 𝑥))
3020, 29mpteq12dv 5196 . . . . 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 7375 . . . 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 5196 . . 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 5859 . . . . . 6 (𝑚 = 𝑀 → dom 𝑚 = dom 𝑀)
3433oveq1d 7372 . . . . 5 (𝑚 = 𝑀 → (dom 𝑚MblFnM𝑆) = (dom 𝑀MblFnM𝑆))
35 fveq1 6841 . . . . . . . 8 (𝑚 = 𝑀 → (𝑚‘(𝑔 “ {𝑥})) = (𝑀‘(𝑔 “ {𝑥})))
3635eleq1d 2822 . . . . . . 7 (𝑚 = 𝑀 → ((𝑚‘(𝑔 “ {𝑥})) ∈ (0[,)+∞) ↔ (𝑀‘(𝑔 “ {𝑥})) ∈ (0[,)+∞)))
3736ralbidv 3174 . . . . . 6 (𝑚 = 𝑀 → (∀𝑥 ∈ (ran 𝑔 ∖ { 0 })(𝑚‘(𝑔 “ {𝑥})) ∈ (0[,)+∞) ↔ ∀𝑥 ∈ (ran 𝑔 ∖ { 0 })(𝑀‘(𝑔 “ {𝑥})) ∈ (0[,)+∞)))
3837anbi2d 629 . . . . 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 3424 . . . 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 6844 . . . . . . . 8 ((𝑚 = 𝑀𝑥 ∈ (ran 𝑓 ∖ { 0 })) → (𝑚‘(𝑓 “ {𝑥})) = (𝑀‘(𝑓 “ {𝑥})))
4241fveq2d 6846 . . . . . . 7 ((𝑚 = 𝑀𝑥 ∈ (ran 𝑓 ∖ { 0 })) → (𝐻‘(𝑚‘(𝑓 “ {𝑥}))) = (𝐻‘(𝑀‘(𝑓 “ {𝑥}))))
4342oveq1d 7372 . . . . . 6 ((𝑚 = 𝑀𝑥 ∈ (ran 𝑓 ∖ { 0 })) → ((𝐻‘(𝑚‘(𝑓 “ {𝑥}))) · 𝑥) = ((𝐻‘(𝑀‘(𝑓 “ {𝑥}))) · 𝑥))
4443mpteq2dva 5205 . . . . 5 (𝑚 = 𝑀 → (𝑥 ∈ (ran 𝑓 ∖ { 0 }) ↦ ((𝐻‘(𝑚‘(𝑓 “ {𝑥}))) · 𝑥)) = (𝑥 ∈ (ran 𝑓 ∖ { 0 }) ↦ ((𝐻‘(𝑀‘(𝑓 “ {𝑥}))) · 𝑥)))
4544oveq2d 7373 . . . 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 5196 . . 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 32930 . . 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 7390 . . . 4 (dom 𝑀MblFnM𝑆) ∈ V
4948mptrabex 7175 . . 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 7515 . 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 1541  wcel 2106  wral 3064  {crab 3407  Vcvv 3445  cdif 3907  {csn 4586   cuni 4865  cmpt 5188  ccnv 5632  dom cdm 5633  ran crn 5634  cima 5636  cfv 6496  (class class class)co 7357  Fincfn 8883  0cc0 11051  +∞cpnf 11186  [,)cico 13266  Basecbs 17083  Scalarcsca 17136   ·𝑠 cvsca 17137  TopOpenctopn 17303  0gc0g 17321   Σg cgsu 17322  ℝHomcrrh 32574  sigaGencsigagen 32737  measurescmeas 32794  MblFnMcmbfm 32848  sitgcsitg 32929
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pr 5384
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-sitg 32930
This theorem is referenced by:  issibf  32933  sitgfval  32941  sitgf  32947
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