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Theorem sitgfval 30712
Description: Value of the Bochner integral for a simple function 𝐹. (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)
sibfmbl.1 (𝜑𝐹 ∈ dom (𝑊sitg𝑀))
Assertion
Ref Expression
sitgfval (𝜑 → ((𝑊sitg𝑀)‘𝐹) = (𝑊 Σg (𝑥 ∈ (ran 𝐹 ∖ { 0 }) ↦ ((𝐻‘(𝑀‘(𝐹 “ {𝑥}))) · 𝑥))))
Distinct variable groups:   𝑥,𝐹   𝑥,𝑀   𝑥,𝑊   𝑥, 0   𝜑,𝑥
Allowed substitution hints:   𝐵(𝑥)   𝑆(𝑥)   · (𝑥)   𝐻(𝑥)   𝐽(𝑥)   𝑉(𝑥)

Proof of Theorem sitgfval
Dummy variables 𝑓 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sitgval.b . . 3 𝐵 = (Base‘𝑊)
2 sitgval.j . . 3 𝐽 = (TopOpen‘𝑊)
3 sitgval.s . . 3 𝑆 = (sigaGen‘𝐽)
4 sitgval.0 . . 3 0 = (0g𝑊)
5 sitgval.x . . 3 · = ( ·𝑠𝑊)
6 sitgval.h . . 3 𝐻 = (ℝHom‘(Scalar‘𝑊))
7 sitgval.1 . . 3 (𝜑𝑊𝑉)
8 sitgval.2 . . 3 (𝜑𝑀 ran measures)
91, 2, 3, 4, 5, 6, 7, 8sitgval 30703 . 2 (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)))))
10 simpr 479 . . . . . 6 ((𝜑𝑓 = 𝐹) → 𝑓 = 𝐹)
1110rneqd 5508 . . . . 5 ((𝜑𝑓 = 𝐹) → ran 𝑓 = ran 𝐹)
1211difeq1d 3870 . . . 4 ((𝜑𝑓 = 𝐹) → (ran 𝑓 ∖ { 0 }) = (ran 𝐹 ∖ { 0 }))
1310cnveqd 5453 . . . . . . . 8 ((𝜑𝑓 = 𝐹) → 𝑓 = 𝐹)
1413imaeq1d 5623 . . . . . . 7 ((𝜑𝑓 = 𝐹) → (𝑓 “ {𝑥}) = (𝐹 “ {𝑥}))
1514fveq2d 6356 . . . . . 6 ((𝜑𝑓 = 𝐹) → (𝑀‘(𝑓 “ {𝑥})) = (𝑀‘(𝐹 “ {𝑥})))
1615fveq2d 6356 . . . . 5 ((𝜑𝑓 = 𝐹) → (𝐻‘(𝑀‘(𝑓 “ {𝑥}))) = (𝐻‘(𝑀‘(𝐹 “ {𝑥}))))
1716oveq1d 6828 . . . 4 ((𝜑𝑓 = 𝐹) → ((𝐻‘(𝑀‘(𝑓 “ {𝑥}))) · 𝑥) = ((𝐻‘(𝑀‘(𝐹 “ {𝑥}))) · 𝑥))
1812, 17mpteq12dv 4885 . . 3 ((𝜑𝑓 = 𝐹) → (𝑥 ∈ (ran 𝑓 ∖ { 0 }) ↦ ((𝐻‘(𝑀‘(𝑓 “ {𝑥}))) · 𝑥)) = (𝑥 ∈ (ran 𝐹 ∖ { 0 }) ↦ ((𝐻‘(𝑀‘(𝐹 “ {𝑥}))) · 𝑥)))
1918oveq2d 6829 . 2 ((𝜑𝑓 = 𝐹) → (𝑊 Σg (𝑥 ∈ (ran 𝑓 ∖ { 0 }) ↦ ((𝐻‘(𝑀‘(𝑓 “ {𝑥}))) · 𝑥))) = (𝑊 Σg (𝑥 ∈ (ran 𝐹 ∖ { 0 }) ↦ ((𝐻‘(𝑀‘(𝐹 “ {𝑥}))) · 𝑥))))
20 sibfmbl.1 . . . . 5 (𝜑𝐹 ∈ dom (𝑊sitg𝑀))
211, 2, 3, 4, 5, 6, 7, 8, 20sibfmbl 30706 . . . 4 (𝜑𝐹 ∈ (dom 𝑀MblFnM𝑆))
221, 2, 3, 4, 5, 6, 7, 8, 20sibfrn 30708 . . . 4 (𝜑 → ran 𝐹 ∈ Fin)
231, 2, 3, 4, 5, 6, 7, 8, 20sibfima 30709 . . . . 5 ((𝜑𝑥 ∈ (ran 𝐹 ∖ { 0 })) → (𝑀‘(𝐹 “ {𝑥})) ∈ (0[,)+∞))
2423ralrimiva 3104 . . . 4 (𝜑 → ∀𝑥 ∈ (ran 𝐹 ∖ { 0 })(𝑀‘(𝐹 “ {𝑥})) ∈ (0[,)+∞))
2521, 22, 24jca32 559 . . 3 (𝜑 → (𝐹 ∈ (dom 𝑀MblFnM𝑆) ∧ (ran 𝐹 ∈ Fin ∧ ∀𝑥 ∈ (ran 𝐹 ∖ { 0 })(𝑀‘(𝐹 “ {𝑥})) ∈ (0[,)+∞))))
26 rneq 5506 . . . . . 6 (𝑔 = 𝐹 → ran 𝑔 = ran 𝐹)
2726eleq1d 2824 . . . . 5 (𝑔 = 𝐹 → (ran 𝑔 ∈ Fin ↔ ran 𝐹 ∈ Fin))
2826difeq1d 3870 . . . . . 6 (𝑔 = 𝐹 → (ran 𝑔 ∖ { 0 }) = (ran 𝐹 ∖ { 0 }))
29 cnveq 5451 . . . . . . . . 9 (𝑔 = 𝐹𝑔 = 𝐹)
3029imaeq1d 5623 . . . . . . . 8 (𝑔 = 𝐹 → (𝑔 “ {𝑥}) = (𝐹 “ {𝑥}))
3130fveq2d 6356 . . . . . . 7 (𝑔 = 𝐹 → (𝑀‘(𝑔 “ {𝑥})) = (𝑀‘(𝐹 “ {𝑥})))
3231eleq1d 2824 . . . . . 6 (𝑔 = 𝐹 → ((𝑀‘(𝑔 “ {𝑥})) ∈ (0[,)+∞) ↔ (𝑀‘(𝐹 “ {𝑥})) ∈ (0[,)+∞)))
3328, 32raleqbidv 3291 . . . . 5 (𝑔 = 𝐹 → (∀𝑥 ∈ (ran 𝑔 ∖ { 0 })(𝑀‘(𝑔 “ {𝑥})) ∈ (0[,)+∞) ↔ ∀𝑥 ∈ (ran 𝐹 ∖ { 0 })(𝑀‘(𝐹 “ {𝑥})) ∈ (0[,)+∞)))
3427, 33anbi12d 749 . . . 4 (𝑔 = 𝐹 → ((ran 𝑔 ∈ Fin ∧ ∀𝑥 ∈ (ran 𝑔 ∖ { 0 })(𝑀‘(𝑔 “ {𝑥})) ∈ (0[,)+∞)) ↔ (ran 𝐹 ∈ Fin ∧ ∀𝑥 ∈ (ran 𝐹 ∖ { 0 })(𝑀‘(𝐹 “ {𝑥})) ∈ (0[,)+∞))))
3534elrab 3504 . . 3 (𝐹 ∈ {𝑔 ∈ (dom 𝑀MblFnM𝑆) ∣ (ran 𝑔 ∈ Fin ∧ ∀𝑥 ∈ (ran 𝑔 ∖ { 0 })(𝑀‘(𝑔 “ {𝑥})) ∈ (0[,)+∞))} ↔ (𝐹 ∈ (dom 𝑀MblFnM𝑆) ∧ (ran 𝐹 ∈ Fin ∧ ∀𝑥 ∈ (ran 𝐹 ∖ { 0 })(𝑀‘(𝐹 “ {𝑥})) ∈ (0[,)+∞))))
3625, 35sylibr 224 . 2 (𝜑𝐹 ∈ {𝑔 ∈ (dom 𝑀MblFnM𝑆) ∣ (ran 𝑔 ∈ Fin ∧ ∀𝑥 ∈ (ran 𝑔 ∖ { 0 })(𝑀‘(𝑔 “ {𝑥})) ∈ (0[,)+∞))})
37 ovexd 6843 . 2 (𝜑 → (𝑊 Σg (𝑥 ∈ (ran 𝐹 ∖ { 0 }) ↦ ((𝐻‘(𝑀‘(𝐹 “ {𝑥}))) · 𝑥))) ∈ V)
389, 19, 36, 37fvmptd 6450 1 (𝜑 → ((𝑊sitg𝑀)‘𝐹) = (𝑊 Σg (𝑥 ∈ (ran 𝐹 ∖ { 0 }) ↦ ((𝐻‘(𝑀‘(𝐹 “ {𝑥}))) · 𝑥))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1632  wcel 2139  wral 3050  {crab 3054  Vcvv 3340  cdif 3712  {csn 4321   cuni 4588  cmpt 4881  ccnv 5265  dom cdm 5266  ran crn 5267  cima 5269  cfv 6049  (class class class)co 6813  Fincfn 8121  0cc0 10128  +∞cpnf 10263  [,)cico 12370  Basecbs 16059  Scalarcsca 16146   ·𝑠 cvsca 16147  TopOpenctopn 16284  0gc0g 16302   Σg cgsu 16303  ℝHomcrrh 30346  sigaGencsigagen 30510  measurescmeas 30567  MblFnMcmbfm 30621  sitgcsitg 30700
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-sitg 30701
This theorem is referenced by:  sitgclg  30713  sitg0  30717
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