Theorem sibf0 34830

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 34697	. . . 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 6896	. . . . . 6 ⊢ 𝐽 ∈ V
7	6	a1i 11	. . . . 5 ⊢ (𝜑 → 𝐽 ∈ V)
8	7	sgsiga 34638	. . . 4 ⊢ (𝜑 → (sigaGen‘𝐽) ∈ ∪ ran sigAlgebra)
9	4, 8	eqeltrid 2866	. . 3 ⊢ (𝜑 → 𝑆 ∈ ∪ ran sigAlgebra)
10		fconstmpt 5721	. . . 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 18852	. . . . 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 23160	. . . . . 6 ⊢ (𝑊 ∈ TopSp → 𝐵 = ∪ 𝐽)
19	17, 18	syl 18	. . . . 5 ⊢ (𝜑 → 𝐵 = ∪ 𝐽)
20	4	unieqi 4882	. . . . . 6 ⊢ ∪ 𝑆 = ∪ (sigaGen‘𝐽)
21		unisg 34639	. . . . . . 7 ⊢ (𝐽 ∈ V → ∪ (sigaGen‘𝐽) = ∪ 𝐽)
22	6, 21	mp1i 14	. . . . . 6 ⊢ (𝜑 → ∪ (sigaGen‘𝐽) = ∪ 𝐽)
23	20, 22	eqtrid 2809	. . . . 5 ⊢ (𝜑 → ∪ 𝑆 = ∪ 𝐽)
24	19, 23	eqtr4d 2800	. . . 4 ⊢ (𝜑 → 𝐵 = ∪ 𝑆)
25	16, 24	eleqtrd 2864	. . 3 ⊢ (𝜑 → 0 ∈ ∪ 𝑆)
26	3, 9, 11, 25	mbfmcst 34755	. 2 ⊢ (𝜑 → (∪ dom 𝑀 × { 0 }) ∈ (dom 𝑀MblFnM𝑆))
27		xpeq1 5673	. . . . . . . 8 ⊢ (∪ dom 𝑀 = ∅ → (∪ dom 𝑀 × { 0 }) = (∅ × { 0 }))
28		0xp 5758	. . . . . . . 8 ⊢ (∅ × { 0 }) = ∅
29	27, 28	eqtrdi 2813	. . . . . . 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 2813	. . . . 5 ⊢ (∪ dom 𝑀 = ∅ → ran (∪ dom 𝑀 × { 0 }) = ∅)
33		0fi 9052	. . . . 5 ⊢ ∅ ∈ Fin
34	32, 33	eqeltrdi 2870	. . . 4 ⊢ (∪ dom 𝑀 = ∅ → ran (∪ dom 𝑀 × { 0 }) ∈ Fin)
35		rnxp 6167	. . . . 5 ⊢ (∪ dom 𝑀 ≠ ∅ → ran (∪ dom 𝑀 × { 0 }) = { 0 })
36		snfi 9053	. . . . 5 ⊢ { 0 } ∈ Fin
37	35, 36	eqeltrdi 2870	. . . 4 ⊢ (∪ dom 𝑀 ≠ ∅ → ran (∪ dom 𝑀 × { 0 }) ∈ Fin)
38	34, 37	pm2.61ine 3040	. . 3 ⊢ ran (∪ dom 𝑀 × { 0 }) ∈ Fin
39	38	a1i 11	. 2 ⊢ (𝜑 → ran (∪ dom 𝑀 × { 0 }) ∈ Fin)
40		noel 4287	. . . . . 6 ⊢ ¬ 𝑥 ∈ ∅
41	32	difeq1d 4076	. . . . . . . . 9 ⊢ (∪ dom 𝑀 = ∅ → (ran (∪ dom 𝑀 × { 0 }) ∖ { 0 }) = (∅ ∖ { 0 }))
42		0dif 4359	. . . . . . . . 9 ⊢ (∅ ∖ { 0 }) = ∅
43	41, 42	eqtrdi 2813	. . . . . . . 8 ⊢ (∪ dom 𝑀 = ∅ → (ran (∪ dom 𝑀 × { 0 }) ∖ { 0 }) = ∅)
44	35	difeq1d 4076	. . . . . . . . 9 ⊢ (∪ dom 𝑀 ≠ ∅ → (ran (∪ dom 𝑀 × { 0 }) ∖ { 0 }) = ({ 0 } ∖ { 0 }))
45		difid 4328	. . . . . . . . 9 ⊢ ({ 0 } ∖ { 0 }) = ∅
46	44, 45	eqtrdi 2813	. . . . . . . 8 ⊢ (∪ dom 𝑀 ≠ ∅ → (ran (∪ dom 𝑀 × { 0 }) ∖ { 0 }) = ∅)
47	43, 46	pm2.61ine 3040	. . . . . . 7 ⊢ (ran (∪ dom 𝑀 × { 0 }) ∖ { 0 }) = ∅
48	47	eleq2i 2854	. . . . . 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 3156	. 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 34829	. 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 2145   ≠ wne 2957  ∀wral 3078  Vcvv 3453   ∖ cdif 3899  ∅c0 4282  {csn 4587  ∪ cuni 4870   ↦ cmpt 5190   × cxp 5657  ^◡ccnv 5658  dom cdm 5659  ran crn 5660   “ cima 5662  ‘cfv 6537  (class class class)co 7416  Fincfn 8955  0cc0 11125  +∞cpnf 11265  [,)cico 13400  Basecbs 17303  Scalarcsca 17347   ·_𝑠 cvsca 17348  TopOpenctopn 17508  0_gc0g 17526  Mndcmnd 18836  TopSpctps 23156  ℝHomcrrh 34488  sigAlgebracsiga 34603  sigaGencsigagen 34634  measurescmeas 34691  MblFnMcmbfm 34745  sitgcsitg 34825

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7739

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 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-int 4911  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7373  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7866  df-1o 8458  df-map 8831  df-en 8956  df-fin 8959  df-0g 17528  df-mgm 18732  df-sgrp 18821  df-mnd 18837  df-top 23118  df-topon 23135  df-topsp 23157  df-esum 34523  df-siga 34604  df-sigagen 34635  df-meas 34692  df-mbfm 34746  df-sitg 34826

This theorem is used by:  sitg0  34842