Theorem sitgfval 34094

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 34085	. 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 483	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 = 𝐹) → 𝑓 = 𝐹)
11	10	rneqd 5940	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 = 𝐹) → ran 𝑓 = ran 𝐹)
12	11	difeq1d 4117	. . . 4 ⊢ ((𝜑 ∧ 𝑓 = 𝐹) → (ran 𝑓 ∖ { 0 }) = (ran 𝐹 ∖ { 0 }))
13	10	cnveqd 5878	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 = 𝐹) → ◡𝑓 = ◡𝐹)
14	13	imaeq1d 6063	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 = 𝐹) → (◡𝑓 “ {𝑥}) = (◡𝐹 “ {𝑥}))
15	14	fveq2d 6900	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 = 𝐹) → (𝑀‘(◡𝑓 “ {𝑥})) = (𝑀‘(◡𝐹 “ {𝑥})))
16	15	fveq2d 6900	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 = 𝐹) → (𝐻‘(𝑀‘(◡𝑓 “ {𝑥}))) = (𝐻‘(𝑀‘(◡𝐹 “ {𝑥}))))
17	16	oveq1d 7434	. . . 4 ⊢ ((𝜑 ∧ 𝑓 = 𝐹) → ((𝐻‘(𝑀‘(◡𝑓 “ {𝑥}))) · 𝑥) = ((𝐻‘(𝑀‘(◡𝐹 “ {𝑥}))) · 𝑥))
18	12, 17	mpteq12dv 5240	. . 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 7435	. 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 34088	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (dom 𝑀MblFnM𝑆))
22	1, 2, 3, 4, 5, 6, 7, 8, 20	sibfrn 34090	. . . 4 ⊢ (𝜑 → ran 𝐹 ∈ Fin)
23	1, 2, 3, 4, 5, 6, 7, 8, 20	sibfima 34091	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (ran 𝐹 ∖ { 0 })) → (𝑀‘(◡𝐹 “ {𝑥})) ∈ (0[,)+∞))
24	23	ralrimiva 3135	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ (ran 𝐹 ∖ { 0 })(𝑀‘(◡𝐹 “ {𝑥})) ∈ (0[,)+∞))
25	21, 22, 24	jca32 514	. . 3 ⊢ (𝜑 → (𝐹 ∈ (dom 𝑀MblFnM𝑆) ∧ (ran 𝐹 ∈ Fin ∧ ∀𝑥 ∈ (ran 𝐹 ∖ { 0 })(𝑀‘(◡𝐹 “ {𝑥})) ∈ (0[,)+∞))))
26		rneq 5938	. . . . . 6 ⊢ (𝑔 = 𝐹 → ran 𝑔 = ran 𝐹)
27	26	eleq1d 2810	. . . . 5 ⊢ (𝑔 = 𝐹 → (ran 𝑔 ∈ Fin ↔ ran 𝐹 ∈ Fin))
28	26	difeq1d 4117	. . . . . 6 ⊢ (𝑔 = 𝐹 → (ran 𝑔 ∖ { 0 }) = (ran 𝐹 ∖ { 0 }))
29		cnveq 5876	. . . . . . . . 9 ⊢ (𝑔 = 𝐹 → ◡𝑔 = ◡𝐹)
30	29	imaeq1d 6063	. . . . . . . 8 ⊢ (𝑔 = 𝐹 → (◡𝑔 “ {𝑥}) = (◡𝐹 “ {𝑥}))
31	30	fveq2d 6900	. . . . . . 7 ⊢ (𝑔 = 𝐹 → (𝑀‘(◡𝑔 “ {𝑥})) = (𝑀‘(◡𝐹 “ {𝑥})))
32	31	eleq1d 2810	. . . . . 6 ⊢ (𝑔 = 𝐹 → ((𝑀‘(◡𝑔 “ {𝑥})) ∈ (0[,)+∞) ↔ (𝑀‘(◡𝐹 “ {𝑥})) ∈ (0[,)+∞)))
33	28, 32	raleqbidv 3329	. . . . 5 ⊢ (𝑔 = 𝐹 → (∀𝑥 ∈ (ran 𝑔 ∖ { 0 })(𝑀‘(◡𝑔 “ {𝑥})) ∈ (0[,)+∞) ↔ ∀𝑥 ∈ (ran 𝐹 ∖ { 0 })(𝑀‘(◡𝐹 “ {𝑥})) ∈ (0[,)+∞)))
34	27, 33	anbi12d 630	. . . 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 3679	. . 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 233	. 2 ⊢ (𝜑 → 𝐹 ∈ {𝑔 ∈ (dom 𝑀MblFnM𝑆) ∣ (ran 𝑔 ∈ Fin ∧ ∀𝑥 ∈ (ran 𝑔 ∖ { 0 })(𝑀‘(◡𝑔 “ {𝑥})) ∈ (0[,)+∞))})
37		ovexd 7454	. 2 ⊢ (𝜑 → (𝑊 Σg (𝑥 ∈ (ran 𝐹 ∖ { 0 }) ↦ ((𝐻‘(𝑀‘(◡𝐹 “ {𝑥}))) · 𝑥))) ∈ V)
38	9, 19, 36, 37	fvmptd 7011	1 ⊢ (𝜑 → ((𝑊sitg𝑀)‘𝐹) = (𝑊 Σg (𝑥 ∈ (ran 𝐹 ∖ { 0 }) ↦ ((𝐻‘(𝑀‘(◡𝐹 “ {𝑥}))) · 𝑥))))

Syntax hints:   → wi 4   ∧ wa 394   = wceq 1533   ∈ wcel 2098  ∀wral 3050  {crab 3418  Vcvv 3461   ∖ cdif 3941  {csn 4630  ∪ cuni 4909   ↦ cmpt 5232  ^◡ccnv 5677  dom cdm 5678  ran crn 5679   “ cima 5681  ‘cfv 6549  (class class class)co 7419  Fincfn 8964  0cc0 11145  +∞cpnf 11282  [,)cico 13366  Basecbs 17188  Scalarcsca 17244   ·_𝑠 cvsca 17245  TopOpenctopn 17411  0_gc0g 17429   Σ_g cgsu 17430  ℝHomcrrh 33727  sigaGencsigagen 33890  measurescmeas 33947  MblFnMcmbfm 34001  sitgcsitg 34082

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pr 5429

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-ov 7422  df-oprab 7423  df-mpo 7424  df-sitg 34083

This theorem is referenced by:  sitgclg  34095  sitg0  34099