Theorem issibf 32012

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 = (0g‘𝑊)
sitgval.x	⊢ · = ( ·𝑠 ‘𝑊)
sitgval.h	⊢ 𝐻 = (ℝHom‘(Scalar‘𝑊))
sitgval.1	⊢ (𝜑 → 𝑊 ∈ 𝑉)
sitgval.2	⊢ (𝜑 → 𝑀 ∈ ∪ 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.

Step	Hyp	Ref	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 = (0g‘𝑊)
5		sitgval.x	. . . . . . . . 9 ⊢ · = ( ·𝑠 ‘𝑊)
6		sitgval.h	. . . . . . . . 9 ⊢ 𝐻 = (ℝHom‘(Scalar‘𝑊))
7		sitgval.1	. . . . . . . . 9 ⊢ (𝜑 → 𝑊 ∈ 𝑉)
8		sitgval.2	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ ∪ ran measures)
9	1, 2, 3, 4, 5, 6, 7, 8	sitgval 32011	. . . . . . . 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)))))
10	9	dmeqd 5774	. . . . . . 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 2737	. . . . . . . 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))))
12	11	dmmpt 6103	. . . . . . 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}
13	10, 12	eqtrdi 2794	. . . . . 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})
14	13	eleq2d 2823	. . . . 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 5805	. . . . . . . . . 10 ⊢ (𝑓 = 𝐹 → ran 𝑓 = ran 𝐹)
16	15	difeq1d 4036	. . . . . . . . 9 ⊢ (𝑓 = 𝐹 → (ran 𝑓 ∖ { 0 }) = (ran 𝐹 ∖ { 0 }))
17		cnveq 5742	. . . . . . . . . . . . 13 ⊢ (𝑓 = 𝐹 → ◡𝑓 = ◡𝐹)
18	17	imaeq1d 5928	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝐹 → (◡𝑓 “ {𝑥}) = (◡𝐹 “ {𝑥}))
19	18	fveq2d 6721	. . . . . . . . . . 11 ⊢ (𝑓 = 𝐹 → (𝑀‘(◡𝑓 “ {𝑥})) = (𝑀‘(◡𝐹 “ {𝑥})))
20	19	fveq2d 6721	. . . . . . . . . 10 ⊢ (𝑓 = 𝐹 → (𝐻‘(𝑀‘(◡𝑓 “ {𝑥}))) = (𝐻‘(𝑀‘(◡𝐹 “ {𝑥}))))
21	20	oveq1d 7228	. . . . . . . . 9 ⊢ (𝑓 = 𝐹 → ((𝐻‘(𝑀‘(◡𝑓 “ {𝑥}))) · 𝑥) = ((𝐻‘(𝑀‘(◡𝐹 “ {𝑥}))) · 𝑥))
22	16, 21	mpteq12dv 5140	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → (𝑥 ∈ (ran 𝑓 ∖ { 0 }) ↦ ((𝐻‘(𝑀‘(◡𝑓 “ {𝑥}))) · 𝑥)) = (𝑥 ∈ (ran 𝐹 ∖ { 0 }) ↦ ((𝐻‘(𝑀‘(◡𝐹 “ {𝑥}))) · 𝑥)))
23	22	oveq2d 7229	. . . . . . 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))))
24	23	eleq1d 2822	. . . . . 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))
25	24	elrab 3602	. . . . 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))
26	14, 25	bitrdi 290	. . . 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 7246	. . . . 5 ⊢ (𝑊 Σg (𝑥 ∈ (ran 𝐹 ∖ { 0 }) ↦ ((𝐻‘(𝑀‘(◡𝐹 “ {𝑥}))) · 𝑥))) ∈ V
28	27	biantru 533	. . . 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))
29	26, 28	bitr4di 292	. . 3 ⊢ (𝜑 → (𝐹 ∈ dom (𝑊sitg𝑀) ↔ 𝐹 ∈ {𝑔 ∈ (dom 𝑀MblFnM𝑆) ∣ (ran 𝑔 ∈ Fin ∧ ∀𝑥 ∈ (ran 𝑔 ∖ { 0 })(𝑀‘(◡𝑔 “ {𝑥})) ∈ (0[,)+∞))}))
30		rneq 5805	. . . . . 6 ⊢ (𝑔 = 𝐹 → ran 𝑔 = ran 𝐹)
31	30	eleq1d 2822	. . . . 5 ⊢ (𝑔 = 𝐹 → (ran 𝑔 ∈ Fin ↔ ran 𝐹 ∈ Fin))
32	30	difeq1d 4036	. . . . . 6 ⊢ (𝑔 = 𝐹 → (ran 𝑔 ∖ { 0 }) = (ran 𝐹 ∖ { 0 }))
33		cnveq 5742	. . . . . . . . 9 ⊢ (𝑔 = 𝐹 → ◡𝑔 = ◡𝐹)
34	33	imaeq1d 5928	. . . . . . . 8 ⊢ (𝑔 = 𝐹 → (◡𝑔 “ {𝑥}) = (◡𝐹 “ {𝑥}))
35	34	fveq2d 6721	. . . . . . 7 ⊢ (𝑔 = 𝐹 → (𝑀‘(◡𝑔 “ {𝑥})) = (𝑀‘(◡𝐹 “ {𝑥})))
36	35	eleq1d 2822	. . . . . 6 ⊢ (𝑔 = 𝐹 → ((𝑀‘(◡𝑔 “ {𝑥})) ∈ (0[,)+∞) ↔ (𝑀‘(◡𝐹 “ {𝑥})) ∈ (0[,)+∞)))
37	32, 36	raleqbidv 3313	. . . . 5 ⊢ (𝑔 = 𝐹 → (∀𝑥 ∈ (ran 𝑔 ∖ { 0 })(𝑀‘(◡𝑔 “ {𝑥})) ∈ (0[,)+∞) ↔ ∀𝑥 ∈ (ran 𝐹 ∖ { 0 })(𝑀‘(◡𝐹 “ {𝑥})) ∈ (0[,)+∞)))
38	31, 37	anbi12d 634	. . . 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[,)+∞))))
39	38	elrab 3602	. . 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[,)+∞))))
40	29, 39	bitrdi 290	. 2 ⊢ (𝜑 → (𝐹 ∈ dom (𝑊sitg𝑀) ↔ (𝐹 ∈ (dom 𝑀MblFnM𝑆) ∧ (ran 𝐹 ∈ Fin ∧ ∀𝑥 ∈ (ran 𝐹 ∖ { 0 })(𝑀‘(◡𝐹 “ {𝑥})) ∈ (0[,)+∞)))))
41		3anass 1097	. 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[,)+∞))))
42	40, 41	bitr4di 292	1 ⊢ (𝜑 → (𝐹 ∈ dom (𝑊sitg𝑀) ↔ (𝐹 ∈ (dom 𝑀MblFnM𝑆) ∧ ran 𝐹 ∈ Fin ∧ ∀𝑥 ∈ (ran 𝐹 ∖ { 0 })(𝑀‘(◡𝐹 “ {𝑥})) ∈ (0[,)+∞))))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1089   = wceq 1543   ∈ wcel 2110  ∀wral 3061  {crab 3065  Vcvv 3408   ∖ cdif 3863  {csn 4541  ∪ cuni 4819   ↦ cmpt 5135  ^◡ccnv 5550  dom cdm 5551  ran crn 5552   “ cima 5554  ‘cfv 6380  (class class class)co 7213  Fincfn 8626  0cc0 10729  +∞cpnf 10864  [,)cico 12937  Basecbs 16760  Scalarcsca 16805   ·_𝑠 cvsca 16806  TopOpenctopn 16926  0_gc0g 16944   Σ_g cgsu 16945  ℝHomcrrh 31655  sigaGencsigagen 31818  measurescmeas 31875  MblFnMcmbfm 31929  sitgcsitg 32008

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pr 5322

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-ov 7216  df-oprab 7217  df-mpo 7218  df-sitg 32009

This theorem is referenced by:  sibf0  32013  sibfmbl  32014  sibfrn  32016  sibfima  32017  sibfof  32019