Theorem sibf0 34493

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 34360	. . . 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 6849	. . . . . 6 ⊢ 𝐽 ∈ V
7	6	a1i 11	. . . . 5 ⊢ (𝜑 → 𝐽 ∈ V)
8	7	sgsiga 34301	. . . 4 ⊢ (𝜑 → (sigaGen‘𝐽) ∈ ∪ ran sigAlgebra)
9	4, 8	eqeltrid 2841	. . 3 ⊢ (𝜑 → 𝑆 ∈ ∪ ran sigAlgebra)
10		fconstmpt 5687	. . . 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 18678	. . . . 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 22884	. . . . . 6 ⊢ (𝑊 ∈ TopSp → 𝐵 = ∪ 𝐽)
19	17, 18	syl 17	. . . . 5 ⊢ (𝜑 → 𝐵 = ∪ 𝐽)
20	4	unieqi 4876	. . . . . 6 ⊢ ∪ 𝑆 = ∪ (sigaGen‘𝐽)
21		unisg 34302	. . . . . . 7 ⊢ (𝐽 ∈ V → ∪ (sigaGen‘𝐽) = ∪ 𝐽)
22	6, 21	mp1i 13	. . . . . 6 ⊢ (𝜑 → ∪ (sigaGen‘𝐽) = ∪ 𝐽)
23	20, 22	eqtrid 2784	. . . . 5 ⊢ (𝜑 → ∪ 𝑆 = ∪ 𝐽)
24	19, 23	eqtr4d 2775	. . . 4 ⊢ (𝜑 → 𝐵 = ∪ 𝑆)
25	16, 24	eleqtrd 2839	. . 3 ⊢ (𝜑 → 0 ∈ ∪ 𝑆)
26	3, 9, 11, 25	mbfmcst 34418	. 2 ⊢ (𝜑 → (∪ dom 𝑀 × { 0 }) ∈ (dom 𝑀MblFnM𝑆))
27		xpeq1 5639	. . . . . . . 8 ⊢ (∪ dom 𝑀 = ∅ → (∪ dom 𝑀 × { 0 }) = (∅ × { 0 }))
28		0xp 5724	. . . . . . . 8 ⊢ (∅ × { 0 }) = ∅
29	27, 28	eqtrdi 2788	. . . . . . 7 ⊢ (∪ dom 𝑀 = ∅ → (∪ dom 𝑀 × { 0 }) = ∅)
30	29	rneqd 5888	. . . . . 6 ⊢ (∪ dom 𝑀 = ∅ → ran (∪ dom 𝑀 × { 0 }) = ran ∅)
31		rn0 5876	. . . . . 6 ⊢ ran ∅ = ∅
32	30, 31	eqtrdi 2788	. . . . 5 ⊢ (∪ dom 𝑀 = ∅ → ran (∪ dom 𝑀 × { 0 }) = ∅)
33		0fi 8983	. . . . 5 ⊢ ∅ ∈ Fin
34	32, 33	eqeltrdi 2845	. . . 4 ⊢ (∪ dom 𝑀 = ∅ → ran (∪ dom 𝑀 × { 0 }) ∈ Fin)
35		rnxp 6129	. . . . 5 ⊢ (∪ dom 𝑀 ≠ ∅ → ran (∪ dom 𝑀 × { 0 }) = { 0 })
36		snfi 8984	. . . . 5 ⊢ { 0 } ∈ Fin
37	35, 36	eqeltrdi 2845	. . . 4 ⊢ (∪ dom 𝑀 ≠ ∅ → ran (∪ dom 𝑀 × { 0 }) ∈ Fin)
38	34, 37	pm2.61ine 3016	. . 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 4078	. . . . . . . . 9 ⊢ (∪ dom 𝑀 = ∅ → (ran (∪ dom 𝑀 × { 0 }) ∖ { 0 }) = (∅ ∖ { 0 }))
42		0dif 4358	. . . . . . . . 9 ⊢ (∅ ∖ { 0 }) = ∅
43	41, 42	eqtrdi 2788	. . . . . . . 8 ⊢ (∪ dom 𝑀 = ∅ → (ran (∪ dom 𝑀 × { 0 }) ∖ { 0 }) = ∅)
44	35	difeq1d 4078	. . . . . . . . 9 ⊢ (∪ dom 𝑀 ≠ ∅ → (ran (∪ dom 𝑀 × { 0 }) ∖ { 0 }) = ({ 0 } ∖ { 0 }))
45		difid 4329	. . . . . . . . 9 ⊢ ({ 0 } ∖ { 0 }) = ∅
46	44, 45	eqtrdi 2788	. . . . . . . 8 ⊢ (∪ dom 𝑀 ≠ ∅ → (ran (∪ dom 𝑀 × { 0 }) ∖ { 0 }) = ∅)
47	43, 46	pm2.61ine 3016	. . . . . . 7 ⊢ (ran (∪ dom 𝑀 × { 0 }) ∖ { 0 }) = ∅
48	47	eleq2i 2829	. . . . . 6 ⊢ (𝑥 ∈ (ran (∪ dom 𝑀 × { 0 }) ∖ { 0 }) ↔ 𝑥 ∈ ∅)
49	40, 48	mtbir 323	. . . . 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 481	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (ran (∪ dom 𝑀 × { 0 }) ∖ { 0 })) → (𝑀‘(◡(∪ dom 𝑀 × { 0 }) “ {𝑥})) ∈ (0[,)+∞))
52	51	ralrimiva 3129	. 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 34492	. 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 1344	1 ⊢ (𝜑 → (∪ dom 𝑀 × { 0 }) ∈ dom (𝑊sitg𝑀))

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  ∀wral 3052  Vcvv 3441   ∖ cdif 3899  ∅c0 4286  {csn 4581  ∪ cuni 4864   ↦ cmpt 5180   × cxp 5623  ^◡ccnv 5624  dom cdm 5625  ran crn 5626   “ cima 5628  ‘cfv 6493  (class class class)co 7360  Fincfn 8887  0cc0 11030  +∞cpnf 11167  [,)cico 13267  Basecbs 17140  Scalarcsca 17184   ·_𝑠 cvsca 17185  TopOpenctopn 17345  0_gc0g 17363  Mndcmnd 18663  TopSpctps 22880  ℝHomcrrh 34152  sigAlgebracsiga 34267  sigaGencsigagen 34297  measurescmeas 34354  MblFnMcmbfm 34408  sitgcsitg 34488

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5225  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rmo 3351  df-reu 3352  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-int 4904  df-iun 4949  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  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 17365  df-mgm 18569  df-sgrp 18648  df-mnd 18664  df-top 22842  df-topon 22859  df-topsp 22881  df-esum 34187  df-siga 34268  df-sigagen 34298  df-meas 34355  df-mbfm 34409  df-sitg 34489

This theorem is referenced by:  sitg0  34505