Theorem sitgf 34541

Description: The integral for simple functions is itself a function. (Contributed by Thierry Arnoux, 13-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)
sitgf.1	⊢ ((𝜑 ∧ 𝑓 ∈ dom (𝑊sitg𝑀)) → ((𝑊sitg𝑀)‘𝑓) ∈ 𝐵)

Assertion

Ref	Expression
sitgf	⊢ (𝜑 → (𝑊sitg𝑀):dom (𝑊sitg𝑀)⟶𝐵)

Distinct variable groups:   𝐵,𝑓   𝑓,𝐻   𝑓,𝑀   𝑆,𝑓   𝑓,𝑊   0 ,𝑓   · ,𝑓   𝜑,𝑓

Allowed substitution hints:   𝐽(𝑓)   𝑉(𝑓)

Proof of Theorem sitgf

Dummy variables 𝑔 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		funmpt 6526	. . . 4 ⊢ Fun (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))))
2		sitgval.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑊)
3		sitgval.j	. . . . . 6 ⊢ 𝐽 = (TopOpen‘𝑊)
4		sitgval.s	. . . . . 6 ⊢ 𝑆 = (sigaGen‘𝐽)
5		sitgval.0	. . . . . 6 ⊢ 0 = (0g‘𝑊)
6		sitgval.x	. . . . . 6 ⊢ · = ( ·𝑠 ‘𝑊)
7		sitgval.h	. . . . . 6 ⊢ 𝐻 = (ℝHom‘(Scalar‘𝑊))
8		sitgval.1	. . . . . 6 ⊢ (𝜑 → 𝑊 ∈ 𝑉)
9		sitgval.2	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ∪ ran measures)
10	2, 3, 4, 5, 6, 7, 8, 9	sitgval 34526	. . . . 5 ⊢ (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)))))
11	10	funeqd 6510	. . . 4 ⊢ (𝜑 → (Fun (𝑊sitg𝑀) ↔ Fun (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	1, 11	mpbiri 260	. . 3 ⊢ (𝜑 → Fun (𝑊sitg𝑀))
13	12	funfnd 6519	. 2 ⊢ (𝜑 → (𝑊sitg𝑀) Fn dom (𝑊sitg𝑀))
14		sitgf.1	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ dom (𝑊sitg𝑀)) → ((𝑊sitg𝑀)‘𝑓) ∈ 𝐵)
15	14	ralrimiva 3133	. . 3 ⊢ (𝜑 → ∀𝑓 ∈ dom (𝑊sitg𝑀)((𝑊sitg𝑀)‘𝑓) ∈ 𝐵)
16		fnfvrnss 7065	. . 3 ⊢ (((𝑊sitg𝑀) Fn dom (𝑊sitg𝑀) ∧ ∀𝑓 ∈ dom (𝑊sitg𝑀)((𝑊sitg𝑀)‘𝑓) ∈ 𝐵) → ran (𝑊sitg𝑀) ⊆ 𝐵)
17	13, 15, 16	syl2anc 591	. 2 ⊢ (𝜑 → ran (𝑊sitg𝑀) ⊆ 𝐵)
18		df-f 6492	. 2 ⊢ ((𝑊sitg𝑀):dom (𝑊sitg𝑀)⟶𝐵 ↔ ((𝑊sitg𝑀) Fn dom (𝑊sitg𝑀) ∧ ran (𝑊sitg𝑀) ⊆ 𝐵))
19	13, 17, 18	sylanbrc 590	1 ⊢ (𝜑 → (𝑊sitg𝑀):dom (𝑊sitg𝑀)⟶𝐵)

Syntax hints:   → wi 4   ∧ wa 397   = wceq 1548   ∈ wcel 2121  ∀wral 3055  {crab 3393   ∖ cdif 3881   ⊆ wss 3884  {csn 4557  ∪ cuni 4840   ↦ cmpt 5155  ^◡ccnv 5619  dom cdm 5620  ran crn 5621   “ cima 5623  Fun wfun 6482   Fn wfn 6483  ⟶wf 6484  ‘cfv 6488  (class class class)co 7359  Fincfn 8887  0cc0 11034  +∞cpnf 11172  [,)cico 13295  Basecbs 17174  Scalarcsca 17218   ·_𝑠 cvsca 17219  TopOpenctopn 17379  0_gc0g 17397   Σ_g cgsu 17398  ℝHomcrrh 34187  sigaGencsigagen 34332  measurescmeas 34389  MblFnMcmbfm 34443  sitgcsitg 34523

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 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-rep 5201  ax-sep 5220  ax-nul 5230  ax-pr 5364

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-ral 3056  df-rex 3066  df-reu 3347  df-rab 3394  df-v 3435  df-sbc 3725  df-csb 3833  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4264  df-if 4457  df-pw 4533  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4841  df-iun 4925  df-br 5075  df-opab 5137  df-mpt 5156  df-id 5515  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-iota 6444  df-fun 6490  df-fn 6491  df-f 6492  df-f1 6493  df-fo 6494  df-f1o 6495  df-fv 6496  df-ov 7362  df-oprab 7363  df-mpo 7364  df-sitg 34524

This theorem is referenced by: (None)