Theorem sitgfval 34740

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

Step	Hyp	Ref	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 = (0g‘𝑊)
5		sitgval.x	. . 3 ⊢ · = ( ·𝑠 ‘𝑊)
6		sitgval.h	. . 3 ⊢ 𝐻 = (ℝHom‘(Scalar‘𝑊))
7		sitgval.1	. . 3 ⊢ (𝜑 → 𝑊 ∈ 𝑉)
8		sitgval.2	. . 3 ⊢ (𝜑 → 𝑀 ∈ ∪ ran measures)
9	1, 2, 3, 4, 5, 6, 7, 8	sitgval 34731	. 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 489	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 = 𝐹) → 𝑓 = 𝐹)
11	10	rneqd 5928	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 = 𝐹) → ran 𝑓 = ran 𝐹)
12	11	difeq1d 4080	. . . 4 ⊢ ((𝜑 ∧ 𝑓 = 𝐹) → (ran 𝑓 ∖ { 0 }) = (ran 𝐹 ∖ { 0 }))
13	10	cnveqd 5861	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 = 𝐹) → ◡𝑓 = ◡𝐹)
14	13	imaeq1d 6061	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 = 𝐹) → (◡𝑓 “ {𝑥}) = (◡𝐹 “ {𝑥}))
15	14	fveq2d 6885	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 = 𝐹) → (𝑀‘(◡𝑓 “ {𝑥})) = (𝑀‘(◡𝐹 “ {𝑥})))
16	15	fveq2d 6885	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 = 𝐹) → (𝐻‘(𝑀‘(◡𝑓 “ {𝑥}))) = (𝐻‘(𝑀‘(◡𝐹 “ {𝑥}))))
17	16	oveq1d 7425	. . . 4 ⊢ ((𝜑 ∧ 𝑓 = 𝐹) → ((𝐻‘(𝑀‘(◡𝑓 “ {𝑥}))) · 𝑥) = ((𝐻‘(𝑀‘(◡𝐹 “ {𝑥}))) · 𝑥))
18	12, 17	mpteq12dv 5198	. . 3 ⊢ ((𝜑 ∧ 𝑓 = 𝐹) → (𝑥 ∈ (ran 𝑓 ∖ { 0 }) ↦ ((𝐻‘(𝑀‘(◡𝑓 “ {𝑥}))) · 𝑥)) = (𝑥 ∈ (ran 𝐹 ∖ { 0 }) ↦ ((𝐻‘(𝑀‘(◡𝐹 “ {𝑥}))) · 𝑥)))
19	18	oveq2d 7426	. 2 ⊢ ((𝜑 ∧ 𝑓 = 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 ⊢ (𝜑 → 𝐹 ∈ dom (𝑊sitg𝑀))
21	1, 2, 3, 4, 5, 6, 7, 8, 20	sibfmbl 34734	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (dom 𝑀MblFnM𝑆))
22	1, 2, 3, 4, 5, 6, 7, 8, 20	sibfrn 34736	. . . 4 ⊢ (𝜑 → ran 𝐹 ∈ Fin)
23	1, 2, 3, 4, 5, 6, 7, 8, 20	sibfima 34737	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (ran 𝐹 ∖ { 0 })) → (𝑀‘(◡𝐹 “ {𝑥})) ∈ (0[,)+∞))
24	23	ralrimiva 3157	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ (ran 𝐹 ∖ { 0 })(𝑀‘(◡𝐹 “ {𝑥})) ∈ (0[,)+∞))
25	21, 22, 24	jca32 524	. . 3 ⊢ (𝜑 → (𝐹 ∈ (dom 𝑀MblFnM𝑆) ∧ (ran 𝐹 ∈ Fin ∧ ∀𝑥 ∈ (ran 𝐹 ∖ { 0 })(𝑀‘(◡𝐹 “ {𝑥})) ∈ (0[,)+∞))))
26		rneq 5926	. . . . . 6 ⊢ (𝑔 = 𝐹 → ran 𝑔 = ran 𝐹)
27	26	eleq1d 2848	. . . . 5 ⊢ (𝑔 = 𝐹 → (ran 𝑔 ∈ Fin ↔ ran 𝐹 ∈ Fin))
28	26	difeq1d 4080	. . . . . 6 ⊢ (𝑔 = 𝐹 → (ran 𝑔 ∖ { 0 }) = (ran 𝐹 ∖ { 0 }))
29		cnveq 5859	. . . . . . . . 9 ⊢ (𝑔 = 𝐹 → ◡𝑔 = ◡𝐹)
30	29	imaeq1d 6061	. . . . . . . 8 ⊢ (𝑔 = 𝐹 → (◡𝑔 “ {𝑥}) = (◡𝐹 “ {𝑥}))
31	30	fveq2d 6885	. . . . . . 7 ⊢ (𝑔 = 𝐹 → (𝑀‘(◡𝑔 “ {𝑥})) = (𝑀‘(◡𝐹 “ {𝑥})))
32	31	eleq1d 2848	. . . . . 6 ⊢ (𝑔 = 𝐹 → ((𝑀‘(◡𝑔 “ {𝑥})) ∈ (0[,)+∞) ↔ (𝑀‘(◡𝐹 “ {𝑥})) ∈ (0[,)+∞)))
33	28, 32	raleqbidv 3338	. . . . 5 ⊢ (𝑔 = 𝐹 → (∀𝑥 ∈ (ran 𝑔 ∖ { 0 })(𝑀‘(◡𝑔 “ {𝑥})) ∈ (0[,)+∞) ↔ ∀𝑥 ∈ (ran 𝐹 ∖ { 0 })(𝑀‘(◡𝐹 “ {𝑥})) ∈ (0[,)+∞)))
34	27, 33	anbi12d 643	. . . 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[,)+∞))))
35	34	elrab 3650	. . 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[,)+∞))))
36	25, 35	sylibr 237	. 2 ⊢ (𝜑 → 𝐹 ∈ {𝑔 ∈ (dom 𝑀MblFnM𝑆) ∣ (ran 𝑔 ∈ Fin ∧ ∀𝑥 ∈ (ran 𝑔 ∖ { 0 })(𝑀‘(◡𝑔 “ {𝑥})) ∈ (0[,)+∞))})
37		ovexd 7445	. 2 ⊢ (𝜑 → (𝑊 Σg (𝑥 ∈ (ran 𝐹 ∖ { 0 }) ↦ ((𝐻‘(𝑀‘(◡𝐹 “ {𝑥}))) · 𝑥))) ∈ V)
38	9, 19, 36, 37	fvmptd 6997	1 ⊢ (𝜑 → ((𝑊sitg𝑀)‘𝐹) = (𝑊 Σg (𝑥 ∈ (ran 𝐹 ∖ { 0 }) ↦ ((𝐻‘(𝑀‘(◡𝐹 “ {𝑥}))) · 𝑥))))

This proof depends on syntax axioms:   → wi 4   ∧ wa 400   = wceq 1570   ∈ wcel 2143  ∀wral 3079  {crab 3416  Vcvv 3455   ∖ cdif 3902  {csn 4589  ∪ cuni 4872   ↦ cmpt 5192  ^◡ccnv 5660  dom cdm 5661  ran crn 5662   “ cima 5664  ‘cfv 6536  (class class class)co 7410  Fincfn 8939  0cc0 11104  +∞cpnf 11244  [,)cico 13378  Basecbs 17273  Scalarcsca 17317   ·_𝑠 cvsca 17318  TopOpenctopn 17478  0_gc0g 17496   Σ_g cgsu 17497  ℝHomcrrh 34392  sigaGencsigagen 34537  measurescmeas 34594  MblFnMcmbfm 34648  sitgcsitg 34728

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pr 5404

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-sitg 34729

This theorem is used by:  sitgclg  34741  sitg0  34745