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Theorem issibf 30842
Description: The predicate "𝐹 is a simple function" relative to the Bochner integral. (Contributed by Thierry Arnoux, 19-Feb-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
issibf (𝜑 → (𝐹 ∈ dom (𝑊sitg𝑀) ↔ (𝐹 ∈ (dom 𝑀MblFnM𝑆) ∧ ran 𝐹 ∈ Fin ∧ ∀𝑥 ∈ (ran 𝐹 ∖ { 0 })(𝑀‘(𝐹 “ {𝑥})) ∈ (0[,)+∞))))
Distinct variable groups:   𝑥,𝐹   𝑥,𝑀   𝑥,𝑊   𝑥, 0
Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)   𝑆(𝑥)   · (𝑥)   𝐻(𝑥)   𝐽(𝑥)   𝑉(𝑥)

Proof of Theorem issibf
Dummy variables 𝑓 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sitgval.b . . . . . . . . 9 𝐵 = (Base‘𝑊)
2 sitgval.j . . . . . . . . 9 𝐽 = (TopOpen‘𝑊)
3 sitgval.s . . . . . . . . 9 𝑆 = (sigaGen‘𝐽)
4 sitgval.0 . . . . . . . . 9 0 = (0g𝑊)
5 sitgval.x . . . . . . . . 9 · = ( ·𝑠𝑊)
6 sitgval.h . . . . . . . . 9 𝐻 = (ℝHom‘(Scalar‘𝑊))
7 sitgval.1 . . . . . . . . 9 (𝜑𝑊𝑉)
8 sitgval.2 . . . . . . . . 9 (𝜑𝑀 ran measures)
91, 2, 3, 4, 5, 6, 7, 8sitgval 30841 . . . . . . . 8 (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)))))
109dmeqd 5494 . . . . . . 7 (𝜑 → dom (𝑊sitg𝑀) = dom (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)))))
11 eqid 2765 . . . . . . . 8 (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))))
1211dmmpt 5816 . . . . . . 7 dom (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))) ∈ V}
1310, 12syl6eq 2815 . . . . . 6 (𝜑 → dom (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))) ∈ V})
1413eleq2d 2830 . . . . 5 (𝜑 → (𝐹 ∈ dom (𝑊sitg𝑀) ↔ 𝐹 ∈ {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}))
15 rneq 5519 . . . . . . . . . 10 (𝑓 = 𝐹 → ran 𝑓 = ran 𝐹)
1615difeq1d 3889 . . . . . . . . 9 (𝑓 = 𝐹 → (ran 𝑓 ∖ { 0 }) = (ran 𝐹 ∖ { 0 }))
17 cnveq 5464 . . . . . . . . . . . . 13 (𝑓 = 𝐹𝑓 = 𝐹)
1817imaeq1d 5647 . . . . . . . . . . . 12 (𝑓 = 𝐹 → (𝑓 “ {𝑥}) = (𝐹 “ {𝑥}))
1918fveq2d 6379 . . . . . . . . . . 11 (𝑓 = 𝐹 → (𝑀‘(𝑓 “ {𝑥})) = (𝑀‘(𝐹 “ {𝑥})))
2019fveq2d 6379 . . . . . . . . . 10 (𝑓 = 𝐹 → (𝐻‘(𝑀‘(𝑓 “ {𝑥}))) = (𝐻‘(𝑀‘(𝐹 “ {𝑥}))))
2120oveq1d 6857 . . . . . . . . 9 (𝑓 = 𝐹 → ((𝐻‘(𝑀‘(𝑓 “ {𝑥}))) · 𝑥) = ((𝐻‘(𝑀‘(𝐹 “ {𝑥}))) · 𝑥))
2216, 21mpteq12dv 4892 . . . . . . . 8 (𝑓 = 𝐹 → (𝑥 ∈ (ran 𝑓 ∖ { 0 }) ↦ ((𝐻‘(𝑀‘(𝑓 “ {𝑥}))) · 𝑥)) = (𝑥 ∈ (ran 𝐹 ∖ { 0 }) ↦ ((𝐻‘(𝑀‘(𝐹 “ {𝑥}))) · 𝑥)))
2322oveq2d 6858 . . . . . . 7 (𝑓 = 𝐹 → (𝑊 Σg (𝑥 ∈ (ran 𝑓 ∖ { 0 }) ↦ ((𝐻‘(𝑀‘(𝑓 “ {𝑥}))) · 𝑥))) = (𝑊 Σg (𝑥 ∈ (ran 𝐹 ∖ { 0 }) ↦ ((𝐻‘(𝑀‘(𝐹 “ {𝑥}))) · 𝑥))))
2423eleq1d 2829 . . . . . 6 (𝑓 = 𝐹 → ((𝑊 Σg (𝑥 ∈ (ran 𝑓 ∖ { 0 }) ↦ ((𝐻‘(𝑀‘(𝑓 “ {𝑥}))) · 𝑥))) ∈ V ↔ (𝑊 Σg (𝑥 ∈ (ran 𝐹 ∖ { 0 }) ↦ ((𝐻‘(𝑀‘(𝐹 “ {𝑥}))) · 𝑥))) ∈ V))
2524elrab 3519 . . . . 5 (𝐹 ∈ {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} ↔ (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))
2614, 25syl6bb 278 . . . 4 (𝜑 → (𝐹 ∈ dom (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))) ∈ V)))
27 ovex 6874 . . . . 5 (𝑊 Σg (𝑥 ∈ (ran 𝐹 ∖ { 0 }) ↦ ((𝐻‘(𝑀‘(𝐹 “ {𝑥}))) · 𝑥))) ∈ V
2827biantru 525 . . . 4 (𝐹 ∈ {𝑔 ∈ (dom 𝑀MblFnM𝑆) ∣ (ran 𝑔 ∈ Fin ∧ ∀𝑥 ∈ (ran 𝑔 ∖ { 0 })(𝑀‘(𝑔 “ {𝑥})) ∈ (0[,)+∞))} ↔ (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))
2926, 28syl6bbr 280 . . 3 (𝜑 → (𝐹 ∈ dom (𝑊sitg𝑀) ↔ 𝐹 ∈ {𝑔 ∈ (dom 𝑀MblFnM𝑆) ∣ (ran 𝑔 ∈ Fin ∧ ∀𝑥 ∈ (ran 𝑔 ∖ { 0 })(𝑀‘(𝑔 “ {𝑥})) ∈ (0[,)+∞))}))
30 rneq 5519 . . . . . 6 (𝑔 = 𝐹 → ran 𝑔 = ran 𝐹)
3130eleq1d 2829 . . . . 5 (𝑔 = 𝐹 → (ran 𝑔 ∈ Fin ↔ ran 𝐹 ∈ Fin))
3230difeq1d 3889 . . . . . 6 (𝑔 = 𝐹 → (ran 𝑔 ∖ { 0 }) = (ran 𝐹 ∖ { 0 }))
33 cnveq 5464 . . . . . . . . 9 (𝑔 = 𝐹𝑔 = 𝐹)
3433imaeq1d 5647 . . . . . . . 8 (𝑔 = 𝐹 → (𝑔 “ {𝑥}) = (𝐹 “ {𝑥}))
3534fveq2d 6379 . . . . . . 7 (𝑔 = 𝐹 → (𝑀‘(𝑔 “ {𝑥})) = (𝑀‘(𝐹 “ {𝑥})))
3635eleq1d 2829 . . . . . 6 (𝑔 = 𝐹 → ((𝑀‘(𝑔 “ {𝑥})) ∈ (0[,)+∞) ↔ (𝑀‘(𝐹 “ {𝑥})) ∈ (0[,)+∞)))
3732, 36raleqbidv 3300 . . . . 5 (𝑔 = 𝐹 → (∀𝑥 ∈ (ran 𝑔 ∖ { 0 })(𝑀‘(𝑔 “ {𝑥})) ∈ (0[,)+∞) ↔ ∀𝑥 ∈ (ran 𝐹 ∖ { 0 })(𝑀‘(𝐹 “ {𝑥})) ∈ (0[,)+∞)))
3831, 37anbi12d 624 . . . 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[,)+∞))))
3938elrab 3519 . . 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[,)+∞))))
4029, 39syl6bb 278 . 2 (𝜑 → (𝐹 ∈ dom (𝑊sitg𝑀) ↔ (𝐹 ∈ (dom 𝑀MblFnM𝑆) ∧ (ran 𝐹 ∈ Fin ∧ ∀𝑥 ∈ (ran 𝐹 ∖ { 0 })(𝑀‘(𝐹 “ {𝑥})) ∈ (0[,)+∞)))))
41 3anass 1116 . 2 ((𝐹 ∈ (dom 𝑀MblFnM𝑆) ∧ ran 𝐹 ∈ Fin ∧ ∀𝑥 ∈ (ran 𝐹 ∖ { 0 })(𝑀‘(𝐹 “ {𝑥})) ∈ (0[,)+∞)) ↔ (𝐹 ∈ (dom 𝑀MblFnM𝑆) ∧ (ran 𝐹 ∈ Fin ∧ ∀𝑥 ∈ (ran 𝐹 ∖ { 0 })(𝑀‘(𝐹 “ {𝑥})) ∈ (0[,)+∞))))
4240, 41syl6bbr 280 1 (𝜑 → (𝐹 ∈ dom (𝑊sitg𝑀) ↔ (𝐹 ∈ (dom 𝑀MblFnM𝑆) ∧ ran 𝐹 ∈ Fin ∧ ∀𝑥 ∈ (ran 𝐹 ∖ { 0 })(𝑀‘(𝐹 “ {𝑥})) ∈ (0[,)+∞))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  w3a 1107   = wceq 1652  wcel 2155  wral 3055  {crab 3059  Vcvv 3350  cdif 3729  {csn 4334   cuni 4594  cmpt 4888  ccnv 5276  dom cdm 5277  ran crn 5278  cima 5280  cfv 6068  (class class class)co 6842  Fincfn 8160  0cc0 10189  +∞cpnf 10325  [,)cico 12379  Basecbs 16130  Scalarcsca 16217   ·𝑠 cvsca 16218  TopOpenctopn 16348  0gc0g 16366   Σg cgsu 16367  ℝHomcrrh 30484  sigaGencsigagen 30648  measurescmeas 30705  MblFnMcmbfm 30759  sitgcsitg 30838
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pr 5062
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-id 5185  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-sitg 30839
This theorem is referenced by:  sibf0  30843  sibfmbl  30844  sibfrn  30846  sibfima  30847  sibfof  30849
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