Theorem sitgf 34367

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 6525	. . . 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 34352	. . . . 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 6509	. . . 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 258	. . 3 ⊢ (𝜑 → Fun (𝑊sitg𝑀))
13	12	funfnd 6518	. 2 ⊢ (𝜑 → (𝑊sitg𝑀) Fn dom (𝑊sitg𝑀))
14		sitgf.1	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ dom (𝑊sitg𝑀)) → ((𝑊sitg𝑀)‘𝑓) ∈ 𝐵)
15	14	ralrimiva 3124	. . 3 ⊢ (𝜑 → ∀𝑓 ∈ dom (𝑊sitg𝑀)((𝑊sitg𝑀)‘𝑓) ∈ 𝐵)
16		fnfvrnss 7060	. . 3 ⊢ (((𝑊sitg𝑀) Fn dom (𝑊sitg𝑀) ∧ ∀𝑓 ∈ dom (𝑊sitg𝑀)((𝑊sitg𝑀)‘𝑓) ∈ 𝐵) → ran (𝑊sitg𝑀) ⊆ 𝐵)
17	13, 15, 16	syl2anc 584	. 2 ⊢ (𝜑 → ran (𝑊sitg𝑀) ⊆ 𝐵)
18		df-f 6491	. 2 ⊢ ((𝑊sitg𝑀):dom (𝑊sitg𝑀)⟶𝐵 ↔ ((𝑊sitg𝑀) Fn dom (𝑊sitg𝑀) ∧ ran (𝑊sitg𝑀) ⊆ 𝐵))
19	13, 17, 18	sylanbrc 583	1 ⊢ (𝜑 → (𝑊sitg𝑀):dom (𝑊sitg𝑀)⟶𝐵)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111  ∀wral 3047  {crab 3395   ∖ cdif 3894   ⊆ wss 3897  {csn 4575  ∪ cuni 4858   ↦ cmpt 5174  ^◡ccnv 5618  dom cdm 5619  ran crn 5620   “ cima 5622  Fun wfun 6481   Fn wfn 6482  ⟶wf 6483  ‘cfv 6487  (class class class)co 7352  Fincfn 8875  0cc0 11012  +∞cpnf 11149  [,)cico 13253  Basecbs 17126  Scalarcsca 17170   ·_𝑠 cvsca 17171  TopOpenctopn 17331  0_gc0g 17349   Σ_g cgsu 17350  ℝHomcrrh 34013  sigaGencsigagen 34158  measurescmeas 34215  MblFnMcmbfm 34269  sitgcsitg 34349

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 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5219  ax-sep 5236  ax-nul 5246  ax-pr 5372

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-iun 4943  df-br 5094  df-opab 5156  df-mpt 5175  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6443  df-fun 6489  df-fn 6490  df-f 6491  df-f1 6492  df-fo 6493  df-f1o 6494  df-fv 6495  df-ov 7355  df-oprab 7356  df-mpo 7357  df-sitg 34350

This theorem is referenced by: (None)