Theorem sibf0 35033

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 34901	. . . 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 6919	. . . . . 6 ⊢ 𝐽 ∈ V
7	6	a1i 11	. . . . 5 ⊢ (𝜑 → 𝐽 ∈ V)
8	7	sgsiga 34842	. . . 4 ⊢ (𝜑 → (sigaGen‘𝐽) ∈ ∪ ran sigAlgebra)
9	4, 8	eqeltrid 2868	. . 3 ⊢ (𝜑 → 𝑆 ∈ ∪ ran sigAlgebra)
10		fconstmpt 5717	. . . 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 18965	. . . . 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 23281	. . . . . 6 ⊢ (𝑊 ∈ TopSp → 𝐵 = ∪ 𝐽)
19	17, 18	syl 18	. . . . 5 ⊢ (𝜑 → 𝐵 = ∪ 𝐽)
20	4	unieqi 4882	. . . . . 6 ⊢ ∪ 𝑆 = ∪ (sigaGen‘𝐽)
21		unisg 34843	. . . . . . 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 2866	. . 3 ⊢ (𝜑 → 0 ∈ ∪ 𝑆)
26	3, 9, 11, 25	mbfmcst 34958	. 2 ⊢ (𝜑 → (∪ dom 𝑀 × { 0 }) ∈ (dom 𝑀MblFnM𝑆))
27		xpeq1 5669	. . . . . . . 8 ⊢ (∪ dom 𝑀 = ∅ → (∪ dom 𝑀 × { 0 }) = (∅ × { 0 }))
28		0xp 5755	. . . . . . . 8 ⊢ (∅ × { 0 }) = ∅
29	27, 28	eqtrdi 2814	. . . . . . 7 ⊢ (∪ dom 𝑀 = ∅ → (∪ dom 𝑀 × { 0 }) = ∅)
30	29	rneqd 5926	. . . . . 6 ⊢ (∪ dom 𝑀 = ∅ → ran (∪ dom 𝑀 × { 0 }) = ran ∅)
31		rn0 5914	. . . . . 6 ⊢ ran ∅ = ∅
32	30, 31	eqtrdi 2814	. . . . 5 ⊢ (∪ dom 𝑀 = ∅ → ran (∪ dom 𝑀 × { 0 }) = ∅)
33		0fi 9095	. . . . 5 ⊢ ∅ ∈ Fin
34	32, 33	eqeltrdi 2872	. . . 4 ⊢ (∪ dom 𝑀 = ∅ → ran (∪ dom 𝑀 × { 0 }) ∈ Fin)
35		rnxp 6173	. . . . 5 ⊢ (∪ dom 𝑀 ≠ ∅ → ran (∪ dom 𝑀 × { 0 }) = { 0 })
36		snfi 9096	. . . . 5 ⊢ { 0 } ∈ Fin
37	35, 36	eqeltrdi 2872	. . . 4 ⊢ (∪ dom 𝑀 ≠ ∅ → ran (∪ dom 𝑀 × { 0 }) ∈ Fin)
38	34, 37	pm2.61ine 3042	. . 3 ⊢ ran (∪ dom 𝑀 × { 0 }) ∈ Fin
39	38	a1i 11	. 2 ⊢ (𝜑 → ran (∪ dom 𝑀 × { 0 }) ∈ Fin)
40		noel 4286	. . . . . 6 ⊢ ¬ 𝑥 ∈ ∅
41	32	difeq1d 4065	. . . . . . . . 9 ⊢ (∪ dom 𝑀 = ∅ → (ran (∪ dom 𝑀 × { 0 }) ∖ { 0 }) = (∅ ∖ { 0 }))
42		0dif 4359	. . . . . . . . 9 ⊢ (∅ ∖ { 0 }) = ∅
43	41, 42	eqtrdi 2814	. . . . . . . 8 ⊢ (∪ dom 𝑀 = ∅ → (ran (∪ dom 𝑀 × { 0 }) ∖ { 0 }) = ∅)
44	35	difeq1d 4065	. . . . . . . . 9 ⊢ (∪ dom 𝑀 ≠ ∅ → (ran (∪ dom 𝑀 × { 0 }) ∖ { 0 }) = ({ 0 } ∖ { 0 }))
45		difid 4328	. . . . . . . . 9 ⊢ ({ 0 } ∖ { 0 }) = ∅
46	44, 45	eqtrdi 2814	. . . . . . . 8 ⊢ (∪ dom 𝑀 ≠ ∅ → (ran (∪ dom 𝑀 × { 0 }) ∖ { 0 }) = ∅)
47	43, 46	pm2.61ine 3042	. . . . . . 7 ⊢ (ran (∪ dom 𝑀 × { 0 }) ∖ { 0 }) = ∅
48	47	eleq2i 2856	. . . . . 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 487	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (ran (∪ dom 𝑀 × { 0 }) ∖ { 0 })) → (𝑀‘(◡(∪ dom 𝑀 × { 0 }) “ {𝑥})) ∈ (0[,)+∞))
52	51	ralrimiva 3158	. 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 35032	. 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 2147   ≠ wne 2959  ∀wral 3080  Vcvv 3454   ∖ cdif 4048  ∅c0 4281  {csn 4588  ∪ cuni 4870   ↦ cmpt 5189   × cxp 5653  ^◡ccnv 5654  dom cdm 5655  ran crn 5656   “ cima 5658  ‘cfv 6555  (class class class)co 7440  Fincfn 8998  0cc0 11224  +∞cpnf 11364  [,)cico 13502  Basecbs 17413  Scalarcsca 17457   ·_𝑠 cvsca 17458  TopOpenctopn 17618  0_gc0g 17636  Mndcmnd 18949  TopSpctps 23277  ℝHomcrrh 34692  sigAlgebracsiga 34807  sigaGencsigagen 34838  measurescmeas 34895  MblFnMcmbfm 34949  sitgcsitg 35028

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7772

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-ord 6380  df-on 6381  df-lim 6382  df-suc 6383  df-iota 6509  df-fun 6557  df-fn 6558  df-f 6559  df-f1 6560  df-fo 6561  df-f1o 6562  df-fv 6563  df-riota 7397  df-ov 7443  df-oprab 7444  df-mpo 7445  df-om 7899  df-1o 8501  df-map 8874  df-en 8999  df-fin 9002  df-0g 17638  df-mgm 18842  df-sgrp 18934  df-mnd 18950  df-top 23239  df-topon 23256  df-topsp 23278  df-esum 34727  df-siga 34808  df-sigagen 34839  df-meas 34896  df-mbfm 34950  df-sitg 35029

This theorem is used by:  sitg0  35045