Theorem sibf0 34530

Description: The constant zero function is a simple function. (Contributed by Thierry Arnoux, 4-Mar-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)
sibf0.1	⊢ (𝜑 → 𝑊 ∈ TopSp)
sibf0.2	⊢ (𝜑 → 𝑊 ∈ Mnd)

Assertion

Ref	Expression
sibf0	⊢ (𝜑 → (∪ dom 𝑀 × { 0 }) ∈ dom (𝑊sitg𝑀))

Proof of Theorem sibf0

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sitgval.2	. . . 4 ⊢ (𝜑 → 𝑀 ∈ ∪ ran measures)
2		dmmeas 34397	. . . 4 ⊢ (𝑀 ∈ ∪ ran measures → dom 𝑀 ∈ ∪ ran sigAlgebra)
3	1, 2	syl 17	. . 3 ⊢ (𝜑 → dom 𝑀 ∈ ∪ ran sigAlgebra)
4		sitgval.s	. . . 4 ⊢ 𝑆 = (sigaGen‘𝐽)
5		sitgval.j	. . . . . . 7 ⊢ 𝐽 = (TopOpen‘𝑊)
6	5	fvexi 6845	. . . . . 6 ⊢ 𝐽 ∈ V
7	6	a1i 11	. . . . 5 ⊢ (𝜑 → 𝐽 ∈ V)
8	7	sgsiga 34338	. . . 4 ⊢ (𝜑 → (sigaGen‘𝐽) ∈ ∪ ran sigAlgebra)
9	4, 8	eqeltrid 2845	. . 3 ⊢ (𝜑 → 𝑆 ∈ ∪ ran sigAlgebra)
10		fconstmpt 5683	. . . 4 ⊢ (∪ dom 𝑀 × { 0 }) = (𝑥 ∈ ∪ dom 𝑀 ↦ 0 )
11	10	a1i 11	. . 3 ⊢ (𝜑 → (∪ dom 𝑀 × { 0 }) = (𝑥 ∈ ∪ dom 𝑀 ↦ 0 ))
12		sibf0.2	. . . . 5 ⊢ (𝜑 → 𝑊 ∈ Mnd)
13		sitgval.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑊)
14		sitgval.0	. . . . . 6 ⊢ 0 = (0g‘𝑊)
15	13, 14	mndidcl 18712	. . . . 5 ⊢ (𝑊 ∈ Mnd → 0 ∈ 𝐵)
16	12, 15	syl 17	. . . 4 ⊢ (𝜑 → 0 ∈ 𝐵)
17		sibf0.1	. . . . . 6 ⊢ (𝜑 → 𝑊 ∈ TopSp)
18	13, 5	tpsuni 22923	. . . . . 6 ⊢ (𝑊 ∈ TopSp → 𝐵 = ∪ 𝐽)
19	17, 18	syl 17	. . . . 5 ⊢ (𝜑 → 𝐵 = ∪ 𝐽)
20	4	unieqi 4853	. . . . . 6 ⊢ ∪ 𝑆 = ∪ (sigaGen‘𝐽)
21		unisg 34339	. . . . . . 7 ⊢ (𝐽 ∈ V → ∪ (sigaGen‘𝐽) = ∪ 𝐽)
22	6, 21	mp1i 13	. . . . . 6 ⊢ (𝜑 → ∪ (sigaGen‘𝐽) = ∪ 𝐽)
23	20, 22	eqtrid 2788	. . . . 5 ⊢ (𝜑 → ∪ 𝑆 = ∪ 𝐽)
24	19, 23	eqtr4d 2779	. . . 4 ⊢ (𝜑 → 𝐵 = ∪ 𝑆)
25	16, 24	eleqtrd 2843	. . 3 ⊢ (𝜑 → 0 ∈ ∪ 𝑆)
26	3, 9, 11, 25	mbfmcst 34455	. 2 ⊢ (𝜑 → (∪ dom 𝑀 × { 0 }) ∈ (dom 𝑀MblFnM𝑆))
27		xpeq1 5635	. . . . . . . 8 ⊢ (∪ dom 𝑀 = ∅ → (∪ dom 𝑀 × { 0 }) = (∅ × { 0 }))
28		0xp 5720	. . . . . . . 8 ⊢ (∅ × { 0 }) = ∅
29	27, 28	eqtrdi 2792	. . . . . . 7 ⊢ (∪ dom 𝑀 = ∅ → (∪ dom 𝑀 × { 0 }) = ∅)
30	29	rneqd 5887	. . . . . 6 ⊢ (∪ dom 𝑀 = ∅ → ran (∪ dom 𝑀 × { 0 }) = ran ∅)
31		rn0 5875	. . . . . 6 ⊢ ran ∅ = ∅
32	30, 31	eqtrdi 2792	. . . . 5 ⊢ (∪ dom 𝑀 = ∅ → ran (∪ dom 𝑀 × { 0 }) = ∅)
33		0fi 8983	. . . . 5 ⊢ ∅ ∈ Fin
34	32, 33	eqeltrdi 2849	. . . 4 ⊢ (∪ dom 𝑀 = ∅ → ran (∪ dom 𝑀 × { 0 }) ∈ Fin)
35		rnxp 6125	. . . . 5 ⊢ (∪ dom 𝑀 ≠ ∅ → ran (∪ dom 𝑀 × { 0 }) = { 0 })
36		snfi 8984	. . . . 5 ⊢ { 0 } ∈ Fin
37	35, 36	eqeltrdi 2849	. . . 4 ⊢ (∪ dom 𝑀 ≠ ∅ → ran (∪ dom 𝑀 × { 0 }) ∈ Fin)
38	34, 37	pm2.61ine 3019	. . 3 ⊢ ran (∪ dom 𝑀 × { 0 }) ∈ Fin
39	38	a1i 11	. 2 ⊢ (𝜑 → ran (∪ dom 𝑀 × { 0 }) ∈ Fin)
40		noel 4269	. . . . . 6 ⊢ ¬ 𝑥 ∈ ∅
41	32	difeq1d 4059	. . . . . . . . 9 ⊢ (∪ dom 𝑀 = ∅ → (ran (∪ dom 𝑀 × { 0 }) ∖ { 0 }) = (∅ ∖ { 0 }))
42		0dif 4336	. . . . . . . . 9 ⊢ (∅ ∖ { 0 }) = ∅
43	41, 42	eqtrdi 2792	. . . . . . . 8 ⊢ (∪ dom 𝑀 = ∅ → (ran (∪ dom 𝑀 × { 0 }) ∖ { 0 }) = ∅)
44	35	difeq1d 4059	. . . . . . . . 9 ⊢ (∪ dom 𝑀 ≠ ∅ → (ran (∪ dom 𝑀 × { 0 }) ∖ { 0 }) = ({ 0 } ∖ { 0 }))
45		difid 4307	. . . . . . . . 9 ⊢ ({ 0 } ∖ { 0 }) = ∅
46	44, 45	eqtrdi 2792	. . . . . . . 8 ⊢ (∪ dom 𝑀 ≠ ∅ → (ran (∪ dom 𝑀 × { 0 }) ∖ { 0 }) = ∅)
47	43, 46	pm2.61ine 3019	. . . . . . 7 ⊢ (ran (∪ dom 𝑀 × { 0 }) ∖ { 0 }) = ∅
48	47	eleq2i 2833	. . . . . 6 ⊢ (𝑥 ∈ (ran (∪ dom 𝑀 × { 0 }) ∖ { 0 }) ↔ 𝑥 ∈ ∅)
49	40, 48	mtbir 325	. . . . 5 ⊢ ¬ 𝑥 ∈ (ran (∪ dom 𝑀 × { 0 }) ∖ { 0 })
50	49	pm2.21i 119	. . . 4 ⊢ (𝑥 ∈ (ran (∪ dom 𝑀 × { 0 }) ∖ { 0 }) → (𝑀‘(◡(∪ dom 𝑀 × { 0 }) “ {𝑥})) ∈ (0[,)+∞))
51	50	adantl 483	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (ran (∪ dom 𝑀 × { 0 }) ∖ { 0 })) → (𝑀‘(◡(∪ dom 𝑀 × { 0 }) “ {𝑥})) ∈ (0[,)+∞))
52	51	ralrimiva 3133	. 2 ⊢ (𝜑 → ∀𝑥 ∈ (ran (∪ dom 𝑀 × { 0 }) ∖ { 0 })(𝑀‘(◡(∪ dom 𝑀 × { 0 }) “ {𝑥})) ∈ (0[,)+∞))
53		sitgval.x	. . 3 ⊢ · = ( ·𝑠 ‘𝑊)
54		sitgval.h	. . 3 ⊢ 𝐻 = (ℝHom‘(Scalar‘𝑊))
55		sitgval.1	. . 3 ⊢ (𝜑 → 𝑊 ∈ 𝑉)
56	13, 5, 4, 14, 53, 54, 55, 1	issibf 34529	. 2 ⊢ (𝜑 → ((∪ dom 𝑀 × { 0 }) ∈ dom (𝑊sitg𝑀) ↔ ((∪ dom 𝑀 × { 0 }) ∈ (dom 𝑀MblFnM𝑆) ∧ ran (∪ dom 𝑀 × { 0 }) ∈ Fin ∧ ∀𝑥 ∈ (ran (∪ dom 𝑀 × { 0 }) ∖ { 0 })(𝑀‘(◡(∪ dom 𝑀 × { 0 }) “ {𝑥})) ∈ (0[,)+∞))))
57	26, 39, 52, 56	mpbir3and 1350	1 ⊢ (𝜑 → (∪ dom 𝑀 × { 0 }) ∈ dom (𝑊sitg𝑀))

Syntax hints:   → wi 4   = wceq 1548   ∈ wcel 2121   ≠ wne 2936  ∀wral 3055  Vcvv 3433   ∖ cdif 3882  ∅c0 4264  {csn 4558  ∪ cuni 4841   ↦ cmpt 5156   × cxp 5619  ^◡ccnv 5620  dom cdm 5621  ran crn 5622   “ cima 5624  ‘cfv 6489  (class class class)co 7360  Fincfn 8887  0cc0 11033  +∞cpnf 11171  [,)cico 13295  Basecbs 17174  Scalarcsca 17218   ·_𝑠 cvsca 17219  TopOpenctopn 17379  0_gc0g 17397  Mndcmnd 18697  TopSpctps 22919  ℝHomcrrh 34189  sigAlgebracsiga 34304  sigaGencsigagen 34334  measurescmeas 34391  MblFnMcmbfm 34445  sitgcsitg 34525

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 5202  ax-sep 5221  ax-nul 5231  ax-pow 5297  ax-pr 5365  ax-un 7682

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3or 1094  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-rmo 3346  df-reu 3347  df-rab 3394  df-v 3435  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-pss 3905  df-nul 4265  df-if 4458  df-pw 4534  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-int 4881  df-iun 4926  df-br 5076  df-opab 5138  df-mpt 5157  df-tr 5183  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-ord 6317  df-on 6318  df-lim 6319  df-suc 6320  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-1o 8399  df-map 8769  df-en 8888  df-fin 8891  df-0g 17399  df-mgm 18603  df-sgrp 18682  df-mnd 18698  df-top 22881  df-topon 22898  df-topsp 22920  df-esum 34224  df-siga 34305  df-sigagen 34335  df-meas 34392  df-mbfm 34446  df-sitg 34526

This theorem is referenced by:  sitg0  34542