Theorem sibf0 34733

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 34600	. . . 4 ⊢ (𝑀 ∈ ∪ ran measures → dom 𝑀 ∈ ∪ ran sigAlgebra)
3	1, 2	syl 18	. . 3 ⊢ (𝜑 → dom 𝑀 ∈ ∪ ran sigAlgebra)
4		sitgval.s	. . . 4 ⊢ 𝑆 = (sigaGen‘𝐽)
5		sitgval.j	. . . . . . 7 ⊢ 𝐽 = (TopOpen‘𝑊)
6	5	fvexi 6895	. . . . . 6 ⊢ 𝐽 ∈ V
7	6	a1i 11	. . . . 5 ⊢ (𝜑 → 𝐽 ∈ V)
8	7	sgsiga 34541	. . . 4 ⊢ (𝜑 → (sigaGen‘𝐽) ∈ ∪ ran sigAlgebra)
9	4, 8	eqeltrid 2867	. . 3 ⊢ (𝜑 → 𝑆 ∈ ∪ ran sigAlgebra)
10		fconstmpt 5723	. . . 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 18811	. . . . 5 ⊢ (𝑊 ∈ Mnd → 0 ∈ 𝐵)
16	12, 15	syl 18	. . . 4 ⊢ (𝜑 → 0 ∈ 𝐵)
17		sibf0.1	. . . . . 6 ⊢ (𝜑 → 𝑊 ∈ TopSp)
18	13, 5	tpsuni 23102	. . . . . 6 ⊢ (𝑊 ∈ TopSp → 𝐵 = ∪ 𝐽)
19	17, 18	syl 18	. . . . 5 ⊢ (𝜑 → 𝐵 = ∪ 𝐽)
20	4	unieqi 4884	. . . . . 6 ⊢ ∪ 𝑆 = ∪ (sigaGen‘𝐽)
21		unisg 34542	. . . . . . 7 ⊢ (𝐽 ∈ V → ∪ (sigaGen‘𝐽) = ∪ 𝐽)
22	6, 21	mp1i 14	. . . . . 6 ⊢ (𝜑 → ∪ (sigaGen‘𝐽) = ∪ 𝐽)
23	20, 22	eqtrid 2810	. . . . 5 ⊢ (𝜑 → ∪ 𝑆 = ∪ 𝐽)
24	19, 23	eqtr4d 2801	. . . 4 ⊢ (𝜑 → 𝐵 = ∪ 𝑆)
25	16, 24	eleqtrd 2865	. . 3 ⊢ (𝜑 → 0 ∈ ∪ 𝑆)
26	3, 9, 11, 25	mbfmcst 34658	. 2 ⊢ (𝜑 → (∪ dom 𝑀 × { 0 }) ∈ (dom 𝑀MblFnM𝑆))
27		xpeq1 5675	. . . . . . . 8 ⊢ (∪ dom 𝑀 = ∅ → (∪ dom 𝑀 × { 0 }) = (∅ × { 0 }))
28		0xp 5760	. . . . . . . 8 ⊢ (∅ × { 0 }) = ∅
29	27, 28	eqtrdi 2814	. . . . . . 7 ⊢ (∪ dom 𝑀 = ∅ → (∪ dom 𝑀 × { 0 }) = ∅)
30	29	rneqd 5928	. . . . . 6 ⊢ (∪ dom 𝑀 = ∅ → ran (∪ dom 𝑀 × { 0 }) = ran ∅)
31		rn0 5916	. . . . . 6 ⊢ ran ∅ = ∅
32	30, 31	eqtrdi 2814	. . . . 5 ⊢ (∪ dom 𝑀 = ∅ → ran (∪ dom 𝑀 × { 0 }) = ∅)
33		0fi 9035	. . . . 5 ⊢ ∅ ∈ Fin
34	32, 33	eqeltrdi 2871	. . . 4 ⊢ (∪ dom 𝑀 = ∅ → ran (∪ dom 𝑀 × { 0 }) ∈ Fin)
35		rnxp 6168	. . . . 5 ⊢ (∪ dom 𝑀 ≠ ∅ → ran (∪ dom 𝑀 × { 0 }) = { 0 })
36		snfi 9036	. . . . 5 ⊢ { 0 } ∈ Fin
37	35, 36	eqeltrdi 2871	. . . 4 ⊢ (∪ dom 𝑀 ≠ ∅ → ran (∪ dom 𝑀 × { 0 }) ∈ Fin)
38	34, 37	pm2.61ine 3041	. . 3 ⊢ ran (∪ dom 𝑀 × { 0 }) ∈ Fin
39	38	a1i 11	. 2 ⊢ (𝜑 → ran (∪ dom 𝑀 × { 0 }) ∈ Fin)
40		noel 4291	. . . . . 6 ⊢ ¬ 𝑥 ∈ ∅
41	32	difeq1d 4080	. . . . . . . . 9 ⊢ (∪ dom 𝑀 = ∅ → (ran (∪ dom 𝑀 × { 0 }) ∖ { 0 }) = (∅ ∖ { 0 }))
42		0dif 4363	. . . . . . . . 9 ⊢ (∅ ∖ { 0 }) = ∅
43	41, 42	eqtrdi 2814	. . . . . . . 8 ⊢ (∪ dom 𝑀 = ∅ → (ran (∪ dom 𝑀 × { 0 }) ∖ { 0 }) = ∅)
44	35	difeq1d 4080	. . . . . . . . 9 ⊢ (∪ dom 𝑀 ≠ ∅ → (ran (∪ dom 𝑀 × { 0 }) ∖ { 0 }) = ({ 0 } ∖ { 0 }))
45		difid 4332	. . . . . . . . 9 ⊢ ({ 0 } ∖ { 0 }) = ∅
46	44, 45	eqtrdi 2814	. . . . . . . 8 ⊢ (∪ dom 𝑀 ≠ ∅ → (ran (∪ dom 𝑀 × { 0 }) ∖ { 0 }) = ∅)
47	43, 46	pm2.61ine 3041	. . . . . . 7 ⊢ (ran (∪ dom 𝑀 × { 0 }) ∖ { 0 }) = ∅
48	47	eleq2i 2855	. . . . . 6 ⊢ (𝑥 ∈ (ran (∪ dom 𝑀 × { 0 }) ∖ { 0 }) ↔ 𝑥 ∈ ∅)
49	40, 48	mtbir 326	. . . . 5 ⊢ ¬ 𝑥 ∈ (ran (∪ dom 𝑀 × { 0 }) ∖ { 0 })
50	49	pm2.21i 120	. . . 4 ⊢ (𝑥 ∈ (ran (∪ dom 𝑀 × { 0 }) ∖ { 0 }) → (𝑀‘(◡(∪ dom 𝑀 × { 0 }) “ {𝑥})) ∈ (0[,)+∞))
51	50	adantl 486	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (ran (∪ dom 𝑀 × { 0 }) ∖ { 0 })) → (𝑀‘(◡(∪ dom 𝑀 × { 0 }) “ {𝑥})) ∈ (0[,)+∞))
52	51	ralrimiva 3157	. 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 34732	. 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 1361	1 ⊢ (𝜑 → (∪ dom 𝑀 × { 0 }) ∈ dom (𝑊sitg𝑀))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2143   ≠ wne 2958  ∀wral 3079  Vcvv 3455   ∖ cdif 3902  ∅c0 4286  {csn 4589  ∪ cuni 4872   ↦ cmpt 5192   × cxp 5659  ^◡ccnv 5660  dom cdm 5661  ran crn 5662   “ cima 5664  ‘cfv 6536  (class class class)co 7410  Fincfn 8939  0cc0 11104  +∞cpnf 11244  [,)cico 13378  Basecbs 17273  Scalarcsca 17317   ·_𝑠 cvsca 17318  TopOpenctopn 17478  0_gc0g 17496  Mndcmnd 18796  TopSpctps 23098  ℝHomcrrh 34392  sigAlgebracsiga 34507  sigaGencsigagen 34537  measurescmeas 34594  MblFnMcmbfm 34648  sitgcsitg 34728

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-int 4913  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7859  df-1o 8449  df-map 8822  df-en 8940  df-fin 8943  df-0g 17498  df-mgm 18702  df-sgrp 18781  df-mnd 18797  df-top 23060  df-topon 23077  df-topsp 23099  df-esum 34427  df-siga 34508  df-sigagen 34538  df-meas 34595  df-mbfm 34649  df-sitg 34729

This theorem is used by:  sitg0  34745