Theorem sitgfval 32414

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 32405	. 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 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 = 𝐹) → 𝑓 = 𝐹)
11	10	rneqd 5864	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 = 𝐹) → ran 𝑓 = ran 𝐹)
12	11	difeq1d 4066	. . . 4 ⊢ ((𝜑 ∧ 𝑓 = 𝐹) → (ran 𝑓 ∖ { 0 }) = (ran 𝐹 ∖ { 0 }))
13	10	cnveqd 5802	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 = 𝐹) → ◡𝑓 = ◡𝐹)
14	13	imaeq1d 5983	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 = 𝐹) → (◡𝑓 “ {𝑥}) = (◡𝐹 “ {𝑥}))
15	14	fveq2d 6813	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 = 𝐹) → (𝑀‘(◡𝑓 “ {𝑥})) = (𝑀‘(◡𝐹 “ {𝑥})))
16	15	fveq2d 6813	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 = 𝐹) → (𝐻‘(𝑀‘(◡𝑓 “ {𝑥}))) = (𝐻‘(𝑀‘(◡𝐹 “ {𝑥}))))
17	16	oveq1d 7328	. . . 4 ⊢ ((𝜑 ∧ 𝑓 = 𝐹) → ((𝐻‘(𝑀‘(◡𝑓 “ {𝑥}))) · 𝑥) = ((𝐻‘(𝑀‘(◡𝐹 “ {𝑥}))) · 𝑥))
18	12, 17	mpteq12dv 5176	. . 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 7329	. 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 32408	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (dom 𝑀MblFnM𝑆))
22	1, 2, 3, 4, 5, 6, 7, 8, 20	sibfrn 32410	. . . 4 ⊢ (𝜑 → ran 𝐹 ∈ Fin)
23	1, 2, 3, 4, 5, 6, 7, 8, 20	sibfima 32411	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (ran 𝐹 ∖ { 0 })) → (𝑀‘(◡𝐹 “ {𝑥})) ∈ (0[,)+∞))
24	23	ralrimiva 3140	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ (ran 𝐹 ∖ { 0 })(𝑀‘(◡𝐹 “ {𝑥})) ∈ (0[,)+∞))
25	21, 22, 24	jca32 516	. . 3 ⊢ (𝜑 → (𝐹 ∈ (dom 𝑀MblFnM𝑆) ∧ (ran 𝐹 ∈ Fin ∧ ∀𝑥 ∈ (ran 𝐹 ∖ { 0 })(𝑀‘(◡𝐹 “ {𝑥})) ∈ (0[,)+∞))))
26		rneq 5862	. . . . . 6 ⊢ (𝑔 = 𝐹 → ran 𝑔 = ran 𝐹)
27	26	eleq1d 2822	. . . . 5 ⊢ (𝑔 = 𝐹 → (ran 𝑔 ∈ Fin ↔ ran 𝐹 ∈ Fin))
28	26	difeq1d 4066	. . . . . 6 ⊢ (𝑔 = 𝐹 → (ran 𝑔 ∖ { 0 }) = (ran 𝐹 ∖ { 0 }))
29		cnveq 5800	. . . . . . . . 9 ⊢ (𝑔 = 𝐹 → ◡𝑔 = ◡𝐹)
30	29	imaeq1d 5983	. . . . . . . 8 ⊢ (𝑔 = 𝐹 → (◡𝑔 “ {𝑥}) = (◡𝐹 “ {𝑥}))
31	30	fveq2d 6813	. . . . . . 7 ⊢ (𝑔 = 𝐹 → (𝑀‘(◡𝑔 “ {𝑥})) = (𝑀‘(◡𝐹 “ {𝑥})))
32	31	eleq1d 2822	. . . . . 6 ⊢ (𝑔 = 𝐹 → ((𝑀‘(◡𝑔 “ {𝑥})) ∈ (0[,)+∞) ↔ (𝑀‘(◡𝐹 “ {𝑥})) ∈ (0[,)+∞)))
33	28, 32	raleqbidv 3316	. . . . 5 ⊢ (𝑔 = 𝐹 → (∀𝑥 ∈ (ran 𝑔 ∖ { 0 })(𝑀‘(◡𝑔 “ {𝑥})) ∈ (0[,)+∞) ↔ ∀𝑥 ∈ (ran 𝐹 ∖ { 0 })(𝑀‘(◡𝐹 “ {𝑥})) ∈ (0[,)+∞)))
34	27, 33	anbi12d 631	. . . 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 3633	. . 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 7348	. 2 ⊢ (𝜑 → (𝑊 Σg (𝑥 ∈ (ran 𝐹 ∖ { 0 }) ↦ ((𝐻‘(𝑀‘(◡𝐹 “ {𝑥}))) · 𝑥))) ∈ V)
38	9, 19, 36, 37	fvmptd 6919	1 ⊢ (𝜑 → ((𝑊sitg𝑀)‘𝐹) = (𝑊 Σg (𝑥 ∈ (ran 𝐹 ∖ { 0 }) ↦ ((𝐻‘(𝑀‘(◡𝐹 “ {𝑥}))) · 𝑥))))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1540   ∈ wcel 2105  ∀wral 3062  {crab 3404  Vcvv 3441   ∖ cdif 3893  {csn 4569  ∪ cuni 4848   ↦ cmpt 5168  ^◡ccnv 5604  dom cdm 5605  ran crn 5606   “ cima 5608  ‘cfv 6463  (class class class)co 7313  Fincfn 8779  0cc0 10941  +∞cpnf 11076  [,)cico 13151  Basecbs 16979  Scalarcsca 17032   ·_𝑠 cvsca 17033  TopOpenctopn 17199  0_gc0g 17217   Σ_g cgsu 17218  ℝHomcrrh 32049  sigaGencsigagen 32212  measurescmeas 32269  MblFnMcmbfm 32323  sitgcsitg 32402

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2708  ax-rep 5222  ax-sep 5236  ax-nul 5243  ax-pr 5365

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3351  df-rab 3405  df-v 3443  df-sbc 3726  df-csb 3842  df-dif 3899  df-un 3901  df-in 3903  df-ss 3913  df-nul 4267  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4849  df-iun 4937  df-br 5086  df-opab 5148  df-mpt 5169  df-id 5505  df-xp 5611  df-rel 5612  df-cnv 5613  df-co 5614  df-dm 5615  df-rn 5616  df-res 5617  df-ima 5618  df-iota 6415  df-fun 6465  df-fn 6466  df-f 6467  df-f1 6468  df-fo 6469  df-f1o 6470  df-fv 6471  df-ov 7316  df-oprab 7317  df-mpo 7318  df-sitg 32403

This theorem is referenced by:  sitgclg  32415  sitg0  32419