Theorem sibf0 30524

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 30392	. . . 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		fvex 6239	. . . . . . 7 ⊢ (TopOpen‘𝑊) ∈ V
7	5, 6	eqeltri 2726	. . . . . 6 ⊢ 𝐽 ∈ V
8	7	a1i 11	. . . . 5 ⊢ (𝜑 → 𝐽 ∈ V)
9	8	sgsiga 30333	. . . 4 ⊢ (𝜑 → (sigaGen‘𝐽) ∈ ∪ ran sigAlgebra)
10	4, 9	syl5eqel 2734	. . 3 ⊢ (𝜑 → 𝑆 ∈ ∪ ran sigAlgebra)
11		fconstmpt 5197	. . . 4 ⊢ (∪ dom 𝑀 × { 0 }) = (𝑥 ∈ ∪ dom 𝑀 ↦ 0 )
12	11	a1i 11	. . 3 ⊢ (𝜑 → (∪ dom 𝑀 × { 0 }) = (𝑥 ∈ ∪ dom 𝑀 ↦ 0 ))
13		sibf0.2	. . . . 5 ⊢ (𝜑 → 𝑊 ∈ Mnd)
14		sitgval.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑊)
15		sitgval.0	. . . . . 6 ⊢ 0 = (0g‘𝑊)
16	14, 15	mndidcl 17355	. . . . 5 ⊢ (𝑊 ∈ Mnd → 0 ∈ 𝐵)
17	13, 16	syl 17	. . . 4 ⊢ (𝜑 → 0 ∈ 𝐵)
18		sibf0.1	. . . . . 6 ⊢ (𝜑 → 𝑊 ∈ TopSp)
19	14, 5	tpsuni 20788	. . . . . 6 ⊢ (𝑊 ∈ TopSp → 𝐵 = ∪ 𝐽)
20	18, 19	syl 17	. . . . 5 ⊢ (𝜑 → 𝐵 = ∪ 𝐽)
21	4	unieqi 4477	. . . . . 6 ⊢ ∪ 𝑆 = ∪ (sigaGen‘𝐽)
22		unisg 30334	. . . . . . 7 ⊢ (𝐽 ∈ V → ∪ (sigaGen‘𝐽) = ∪ 𝐽)
23	7, 22	mp1i 13	. . . . . 6 ⊢ (𝜑 → ∪ (sigaGen‘𝐽) = ∪ 𝐽)
24	21, 23	syl5eq 2697	. . . . 5 ⊢ (𝜑 → ∪ 𝑆 = ∪ 𝐽)
25	20, 24	eqtr4d 2688	. . . 4 ⊢ (𝜑 → 𝐵 = ∪ 𝑆)
26	17, 25	eleqtrd 2732	. . 3 ⊢ (𝜑 → 0 ∈ ∪ 𝑆)
27	3, 10, 12, 26	mbfmcst 30449	. 2 ⊢ (𝜑 → (∪ dom 𝑀 × { 0 }) ∈ (dom 𝑀MblFnM𝑆))
28		xpeq1 5157	. . . . . . . 8 ⊢ (∪ dom 𝑀 = ∅ → (∪ dom 𝑀 × { 0 }) = (∅ × { 0 }))
29		0xp 5233	. . . . . . . 8 ⊢ (∅ × { 0 }) = ∅
30	28, 29	syl6eq 2701	. . . . . . 7 ⊢ (∪ dom 𝑀 = ∅ → (∪ dom 𝑀 × { 0 }) = ∅)
31	30	rneqd 5385	. . . . . 6 ⊢ (∪ dom 𝑀 = ∅ → ran (∪ dom 𝑀 × { 0 }) = ran ∅)
32		rn0 5409	. . . . . 6 ⊢ ran ∅ = ∅
33	31, 32	syl6eq 2701	. . . . 5 ⊢ (∪ dom 𝑀 = ∅ → ran (∪ dom 𝑀 × { 0 }) = ∅)
34		0fin 8229	. . . . 5 ⊢ ∅ ∈ Fin
35	33, 34	syl6eqel 2738	. . . 4 ⊢ (∪ dom 𝑀 = ∅ → ran (∪ dom 𝑀 × { 0 }) ∈ Fin)
36		rnxp 5599	. . . . 5 ⊢ (∪ dom 𝑀 ≠ ∅ → ran (∪ dom 𝑀 × { 0 }) = { 0 })
37		snfi 8079	. . . . 5 ⊢ { 0 } ∈ Fin
38	36, 37	syl6eqel 2738	. . . 4 ⊢ (∪ dom 𝑀 ≠ ∅ → ran (∪ dom 𝑀 × { 0 }) ∈ Fin)
39	35, 38	pm2.61ine 2906	. . 3 ⊢ ran (∪ dom 𝑀 × { 0 }) ∈ Fin
40	39	a1i 11	. 2 ⊢ (𝜑 → ran (∪ dom 𝑀 × { 0 }) ∈ Fin)
41		noel 3952	. . . . . 6 ⊢ ¬ 𝑥 ∈ ∅
42	33	difeq1d 3760	. . . . . . . . 9 ⊢ (∪ dom 𝑀 = ∅ → (ran (∪ dom 𝑀 × { 0 }) ∖ { 0 }) = (∅ ∖ { 0 }))
43		0dif 4010	. . . . . . . . 9 ⊢ (∅ ∖ { 0 }) = ∅
44	42, 43	syl6eq 2701	. . . . . . . 8 ⊢ (∪ dom 𝑀 = ∅ → (ran (∪ dom 𝑀 × { 0 }) ∖ { 0 }) = ∅)
45	36	difeq1d 3760	. . . . . . . . 9 ⊢ (∪ dom 𝑀 ≠ ∅ → (ran (∪ dom 𝑀 × { 0 }) ∖ { 0 }) = ({ 0 } ∖ { 0 }))
46		difid 3981	. . . . . . . . 9 ⊢ ({ 0 } ∖ { 0 }) = ∅
47	45, 46	syl6eq 2701	. . . . . . . 8 ⊢ (∪ dom 𝑀 ≠ ∅ → (ran (∪ dom 𝑀 × { 0 }) ∖ { 0 }) = ∅)
48	44, 47	pm2.61ine 2906	. . . . . . 7 ⊢ (ran (∪ dom 𝑀 × { 0 }) ∖ { 0 }) = ∅
49	48	eleq2i 2722	. . . . . 6 ⊢ (𝑥 ∈ (ran (∪ dom 𝑀 × { 0 }) ∖ { 0 }) ↔ 𝑥 ∈ ∅)
50	41, 49	mtbir 312	. . . . 5 ⊢ ¬ 𝑥 ∈ (ran (∪ dom 𝑀 × { 0 }) ∖ { 0 })
51	50	pm2.21i 116	. . . 4 ⊢ (𝑥 ∈ (ran (∪ dom 𝑀 × { 0 }) ∖ { 0 }) → (𝑀‘(◡(∪ dom 𝑀 × { 0 }) “ {𝑥})) ∈ (0[,)+∞))
52	51	adantl 481	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (ran (∪ dom 𝑀 × { 0 }) ∖ { 0 })) → (𝑀‘(◡(∪ dom 𝑀 × { 0 }) “ {𝑥})) ∈ (0[,)+∞))
53	52	ralrimiva 2995	. 2 ⊢ (𝜑 → ∀𝑥 ∈ (ran (∪ dom 𝑀 × { 0 }) ∖ { 0 })(𝑀‘(◡(∪ dom 𝑀 × { 0 }) “ {𝑥})) ∈ (0[,)+∞))
54		sitgval.x	. . 3 ⊢ · = ( ·𝑠 ‘𝑊)
55		sitgval.h	. . 3 ⊢ 𝐻 = (ℝHom‘(Scalar‘𝑊))
56		sitgval.1	. . 3 ⊢ (𝜑 → 𝑊 ∈ 𝑉)
57	14, 5, 4, 15, 54, 55, 56, 1	issibf 30523	. 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[,)+∞))))
58	27, 40, 53, 57	mpbir3and 1264	1 ⊢ (𝜑 → (∪ dom 𝑀 × { 0 }) ∈ dom (𝑊sitg𝑀))

Syntax hints:   → wi 4   = wceq 1523   ∈ wcel 2030   ≠ wne 2823  ∀wral 2941  Vcvv 3231   ∖ cdif 3604  ∅c0 3948  {csn 4210  ∪ cuni 4468   ↦ cmpt 4762   × cxp 5141  ^◡ccnv 5142  dom cdm 5143  ran crn 5144   “ cima 5146  ‘cfv 5926  (class class class)co 6690  Fincfn 7997  0cc0 9974  +∞cpnf 10109  [,)cico 12215  Basecbs 15904  Scalarcsca 15991   ·_𝑠 cvsca 15992  TopOpenctopn 16129  0_gc0g 16147  Mndcmnd 17341  TopSpctps 20784  ℝHomcrrh 30165  sigAlgebracsiga 30298  sigaGencsigagen 30329  measurescmeas 30386  MblFnMcmbfm 30440  sitgcsitg 30519

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1o 7605  df-map 7901  df-en 7998  df-fin 8001  df-0g 16149  df-mgm 17289  df-sgrp 17331  df-mnd 17342  df-top 20747  df-topon 20764  df-topsp 20785  df-esum 30218  df-siga 30299  df-sigagen 30330  df-meas 30387  df-mbfm 30441  df-sitg 30520

This theorem is referenced by:  sitg0  30536