Theorem sitgfval 34339

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 34330	. 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 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 = 𝐹) → 𝑓 = 𝐹)
11	10	rneqd 5905	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 = 𝐹) → ran 𝑓 = ran 𝐹)
12	11	difeq1d 4091	. . . 4 ⊢ ((𝜑 ∧ 𝑓 = 𝐹) → (ran 𝑓 ∖ { 0 }) = (ran 𝐹 ∖ { 0 }))
13	10	cnveqd 5842	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 = 𝐹) → ◡𝑓 = ◡𝐹)
14	13	imaeq1d 6033	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 = 𝐹) → (◡𝑓 “ {𝑥}) = (◡𝐹 “ {𝑥}))
15	14	fveq2d 6865	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 = 𝐹) → (𝑀‘(◡𝑓 “ {𝑥})) = (𝑀‘(◡𝐹 “ {𝑥})))
16	15	fveq2d 6865	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 = 𝐹) → (𝐻‘(𝑀‘(◡𝑓 “ {𝑥}))) = (𝐻‘(𝑀‘(◡𝐹 “ {𝑥}))))
17	16	oveq1d 7405	. . . 4 ⊢ ((𝜑 ∧ 𝑓 = 𝐹) → ((𝐻‘(𝑀‘(◡𝑓 “ {𝑥}))) · 𝑥) = ((𝐻‘(𝑀‘(◡𝐹 “ {𝑥}))) · 𝑥))
18	12, 17	mpteq12dv 5197	. . 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 7406	. 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 34333	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (dom 𝑀MblFnM𝑆))
22	1, 2, 3, 4, 5, 6, 7, 8, 20	sibfrn 34335	. . . 4 ⊢ (𝜑 → ran 𝐹 ∈ Fin)
23	1, 2, 3, 4, 5, 6, 7, 8, 20	sibfima 34336	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (ran 𝐹 ∖ { 0 })) → (𝑀‘(◡𝐹 “ {𝑥})) ∈ (0[,)+∞))
24	23	ralrimiva 3126	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ (ran 𝐹 ∖ { 0 })(𝑀‘(◡𝐹 “ {𝑥})) ∈ (0[,)+∞))
25	21, 22, 24	jca32 515	. . 3 ⊢ (𝜑 → (𝐹 ∈ (dom 𝑀MblFnM𝑆) ∧ (ran 𝐹 ∈ Fin ∧ ∀𝑥 ∈ (ran 𝐹 ∖ { 0 })(𝑀‘(◡𝐹 “ {𝑥})) ∈ (0[,)+∞))))
26		rneq 5903	. . . . . 6 ⊢ (𝑔 = 𝐹 → ran 𝑔 = ran 𝐹)
27	26	eleq1d 2814	. . . . 5 ⊢ (𝑔 = 𝐹 → (ran 𝑔 ∈ Fin ↔ ran 𝐹 ∈ Fin))
28	26	difeq1d 4091	. . . . . 6 ⊢ (𝑔 = 𝐹 → (ran 𝑔 ∖ { 0 }) = (ran 𝐹 ∖ { 0 }))
29		cnveq 5840	. . . . . . . . 9 ⊢ (𝑔 = 𝐹 → ◡𝑔 = ◡𝐹)
30	29	imaeq1d 6033	. . . . . . . 8 ⊢ (𝑔 = 𝐹 → (◡𝑔 “ {𝑥}) = (◡𝐹 “ {𝑥}))
31	30	fveq2d 6865	. . . . . . 7 ⊢ (𝑔 = 𝐹 → (𝑀‘(◡𝑔 “ {𝑥})) = (𝑀‘(◡𝐹 “ {𝑥})))
32	31	eleq1d 2814	. . . . . 6 ⊢ (𝑔 = 𝐹 → ((𝑀‘(◡𝑔 “ {𝑥})) ∈ (0[,)+∞) ↔ (𝑀‘(◡𝐹 “ {𝑥})) ∈ (0[,)+∞)))
33	28, 32	raleqbidv 3321	. . . . 5 ⊢ (𝑔 = 𝐹 → (∀𝑥 ∈ (ran 𝑔 ∖ { 0 })(𝑀‘(◡𝑔 “ {𝑥})) ∈ (0[,)+∞) ↔ ∀𝑥 ∈ (ran 𝐹 ∖ { 0 })(𝑀‘(◡𝐹 “ {𝑥})) ∈ (0[,)+∞)))
34	27, 33	anbi12d 632	. . . 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 3662	. . 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 234	. 2 ⊢ (𝜑 → 𝐹 ∈ {𝑔 ∈ (dom 𝑀MblFnM𝑆) ∣ (ran 𝑔 ∈ Fin ∧ ∀𝑥 ∈ (ran 𝑔 ∖ { 0 })(𝑀‘(◡𝑔 “ {𝑥})) ∈ (0[,)+∞))})
37		ovexd 7425	. 2 ⊢ (𝜑 → (𝑊 Σg (𝑥 ∈ (ran 𝐹 ∖ { 0 }) ↦ ((𝐻‘(𝑀‘(◡𝐹 “ {𝑥}))) · 𝑥))) ∈ V)
38	9, 19, 36, 37	fvmptd 6978	1 ⊢ (𝜑 → ((𝑊sitg𝑀)‘𝐹) = (𝑊 Σg (𝑥 ∈ (ran 𝐹 ∖ { 0 }) ↦ ((𝐻‘(𝑀‘(◡𝐹 “ {𝑥}))) · 𝑥))))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3045  {crab 3408  Vcvv 3450   ∖ cdif 3914  {csn 4592  ∪ cuni 4874   ↦ cmpt 5191  ^◡ccnv 5640  dom cdm 5641  ran crn 5642   “ cima 5644  ‘cfv 6514  (class class class)co 7390  Fincfn 8921  0cc0 11075  +∞cpnf 11212  [,)cico 13315  Basecbs 17186  Scalarcsca 17230   ·_𝑠 cvsca 17231  TopOpenctopn 17391  0_gc0g 17409   Σ_g cgsu 17410  ℝHomcrrh 33990  sigaGencsigagen 34135  measurescmeas 34192  MblFnMcmbfm 34246  sitgcsitg 34327

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pr 5390

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-ov 7393  df-oprab 7394  df-mpo 7395  df-sitg 34328

This theorem is referenced by:  sitgclg  34340  sitg0  34344