Theorem sitgf 34845

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 6575	. . . 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 34830	. . . . 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 6559	. . . 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 261	. . 3 ⊢ (𝜑 → Fun (𝑊sitg𝑀))
13	12	funfnd 6568	. 2 ⊢ (𝜑 → (𝑊sitg𝑀) Fn dom (𝑊sitg𝑀))
14		sitgf.1	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ dom (𝑊sitg𝑀)) → ((𝑊sitg𝑀)‘𝑓) ∈ 𝐵)
15	14	ralrimiva 3156	. . 3 ⊢ (𝜑 → ∀𝑓 ∈ dom (𝑊sitg𝑀)((𝑊sitg𝑀)‘𝑓) ∈ 𝐵)
16		fnfvrnss 7117	. . 3 ⊢ (((𝑊sitg𝑀) Fn dom (𝑊sitg𝑀) ∧ ∀𝑓 ∈ dom (𝑊sitg𝑀)((𝑊sitg𝑀)‘𝑓) ∈ 𝐵) → ran (𝑊sitg𝑀) ⊆ 𝐵)
17	13, 15, 16	syl2anc 596	. 2 ⊢ (𝜑 → ran (𝑊sitg𝑀) ⊆ 𝐵)
18		df-f 6541	. 2 ⊢ ((𝑊sitg𝑀):dom (𝑊sitg𝑀)⟶𝐵 ↔ ((𝑊sitg𝑀) Fn dom (𝑊sitg𝑀) ∧ ran (𝑊sitg𝑀) ⊆ 𝐵))
19	13, 17, 18	sylanbrc 595	1 ⊢ (𝜑 → (𝑊sitg𝑀):dom (𝑊sitg𝑀)⟶𝐵)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3078  {crab 3414   ∖ cdif 3899   ⊆ wss 3902  {csn 4587  ∪ cuni 4870   ↦ cmpt 5190  ^◡ccnv 5658  dom cdm 5659  ran crn 5660   “ cima 5662  Fun wfun 6531   Fn wfn 6532  ⟶wf 6533  ‘cfv 6537  (class class class)co 7416  Fincfn 8955  0cc0 11127  +∞cpnf 11267  [,)cico 13402  Basecbs 17305  Scalarcsca 17349   ·_𝑠 cvsca 17350  TopOpenctopn 17510  0_gc0g 17528   Σ_g cgsu 17529  ℝHomcrrh 34490  sigaGencsigagen 34636  measurescmeas 34693  MblFnMcmbfm 34747  sitgcsitg 34827

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pr 5402

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7419  df-oprab 7420  df-mpo 7421  df-sitg 34828

This theorem is used by: (None)