Theorem sibf0 30396

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	|- .x. = ( .s ` W )
sitgval.h	|- H = (RRHom ` (Scalar ` W ) )
sitgval.1	|- ( ph -> W e. V )
sitgval.2	|- ( ph -> M e. U. ran measures )
sibf0.1	|- ( ph -> W e. TopSp )
sibf0.2	|- ( ph -> W e. Mnd )

Assertion

Ref	Expression
sibf0	|- ( ph -> ( U. dom M X. { .0. } ) e. dom ( Wsitg M ) )

Proof of Theorem sibf0

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sitgval.2	. . . 4 |- ( ph -> M e. U. ran measures )
2		dmmeas 30264	. . . 4 |- ( M e. U. ran measures -> dom M e. U. ran sigAlgebra )
3	1, 2	syl 17	. . 3 |- ( ph -> dom M e. U. ran sigAlgebra )
4		sitgval.s	. . . 4 |- S = (sigaGen ` J )
5		sitgval.j	. . . . . . 7 |- J = ( TopOpen ` W )
6		fvex 6201	. . . . . . 7 |- ( TopOpen ` W ) e. _V
7	5, 6	eqeltri 2697	. . . . . 6 |- J e. _V
8	7	a1i 11	. . . . 5 |- ( ph -> J e. _V )
9	8	sgsiga 30205	. . . 4 |- ( ph -> (sigaGen ` J ) e. U. ran sigAlgebra )
10	4, 9	syl5eqel 2705	. . 3 |- ( ph -> S e. U. ran sigAlgebra )
11		fconstmpt 5163	. . . 4 |- ( U. dom M X. { .0. } ) = ( x e. U. dom M |-> .0. )
12	11	a1i 11	. . 3 |- ( ph -> ( U. dom M X. { .0. } ) = ( x e. U. dom M |-> .0. ) )
13		sibf0.2	. . . . 5 |- ( ph -> W e. Mnd )
14		sitgval.b	. . . . . 6 |- B = ( Base ` W )
15		sitgval.0	. . . . . 6 |- .0. = ( 0g ` W )
16	14, 15	mndidcl 17308	. . . . 5 |- ( W e. Mnd -> .0. e. B )
17	13, 16	syl 17	. . . 4 |- ( ph -> .0. e. B )
18		sibf0.1	. . . . . 6 |- ( ph -> W e. TopSp )
19	14, 5	tpsuni 20740	. . . . . 6 |- ( W e. TopSp -> B = U. J )
20	18, 19	syl 17	. . . . 5 |- ( ph -> B = U. J )
21	4	unieqi 4445	. . . . . 6 |- U. S = U. (sigaGen ` J )
22		unisg 30206	. . . . . . 7 |- ( J e. _V -> U. (sigaGen ` J ) = U. J )
23	7, 22	mp1i 13	. . . . . 6 |- ( ph -> U. (sigaGen ` J ) = U. J )
24	21, 23	syl5eq 2668	. . . . 5 |- ( ph -> U. S = U. J )
25	20, 24	eqtr4d 2659	. . . 4 |- ( ph -> B = U. S )
26	17, 25	eleqtrd 2703	. . 3 |- ( ph -> .0. e. U. S )
27	3, 10, 12, 26	mbfmcst 30321	. 2 |- ( ph -> ( U. dom M X. { .0. } ) e. ( dom MMblFnM S ) )
28		xpeq1 5128	. . . . . . . 8 |- ( U. dom M = (/) -> ( U. dom M X. { .0. } ) = ( (/) X. { .0. } ) )
29		0xp 5199	. . . . . . . 8 |- ( (/) X. { .0. } ) = (/)
30	28, 29	syl6eq 2672	. . . . . . 7 |- ( U. dom M = (/) -> ( U. dom M X. { .0. } ) = (/) )
31	30	rneqd 5353	. . . . . 6 |- ( U. dom M = (/) -> ran ( U. dom M X. { .0. } ) = ran (/) )
32		rn0 5377	. . . . . 6 |- ran (/) = (/)
33	31, 32	syl6eq 2672	. . . . 5 |- ( U. dom M = (/) -> ran ( U. dom M X. { .0. } ) = (/) )
34		0fin 8188	. . . . 5 |- (/) e. Fin
35	33, 34	syl6eqel 2709	. . . 4 |- ( U. dom M = (/) -> ran ( U. dom M X. { .0. } ) e. Fin )
36		rnxp 5564	. . . . 5 |- ( U. dom M =/= (/) -> ran ( U. dom M X. { .0. } ) = { .0. } )
37		snfi 8038	. . . . 5 |- { .0. } e. Fin
38	36, 37	syl6eqel 2709	. . . 4 |- ( U. dom M =/= (/) -> ran ( U. dom M X. { .0. } ) e. Fin )
39	35, 38	pm2.61ine 2877	. . 3 |- ran ( U. dom M X. { .0. } ) e. Fin
40	39	a1i 11	. 2 |- ( ph -> ran ( U. dom M X. { .0. } ) e. Fin )
41		noel 3919	. . . . . 6 |- -. x e. (/)
42	33	difeq1d 3727	. . . . . . . . 9 |- ( U. dom M = (/) -> ( ran ( U. dom M X. { .0. } ) \ { .0. } ) = ( (/) \ { .0. } ) )
43		0dif 3977	. . . . . . . . 9 |- ( (/) \ { .0. } ) = (/)
44	42, 43	syl6eq 2672	. . . . . . . 8 |- ( U. dom M = (/) -> ( ran ( U. dom M X. { .0. } ) \ { .0. } ) = (/) )
45	36	difeq1d 3727	. . . . . . . . 9 |- ( U. dom M =/= (/) -> ( ran ( U. dom M X. { .0. } ) \ { .0. } ) = ( { .0. } \ { .0. } ) )
46		difid 3948	. . . . . . . . 9 |- ( { .0. } \ { .0. } ) = (/)
47	45, 46	syl6eq 2672	. . . . . . . 8 |- ( U. dom M =/= (/) -> ( ran ( U. dom M X. { .0. } ) \ { .0. } ) = (/) )
48	44, 47	pm2.61ine 2877	. . . . . . 7 |- ( ran ( U. dom M X. { .0. } ) \ { .0. } ) = (/)
49	48	eleq2i 2693	. . . . . 6 |- ( x e. ( ran ( U. dom M X. { .0. } ) \ { .0. } ) <-> x e. (/) )
50	41, 49	mtbir 313	. . . . 5 |- -. x e. ( ran ( U. dom M X. { .0. } ) \ { .0. } )
51	50	pm2.21i 116	. . . 4 |- ( x e. ( ran ( U. dom M X. { .0. } ) \ { .0. } ) -> ( M ` ( `' ( U. dom M X. { .0. } ) " { x } ) ) e. ( 0 [,) +oo ) )
52	51	adantl 482	. . 3 |- ( ( ph /\ x e. ( ran ( U. dom M X. { .0. } ) \ { .0. } ) ) -> ( M ` ( `' ( U. dom M X. { .0. } ) " { x } ) ) e. ( 0 [,) +oo ) )
53	52	ralrimiva 2966	. 2 |- ( ph -> A. x e. ( ran ( U. dom M X. { .0. } ) \ { .0. } ) ( M ` ( `' ( U. dom M X. { .0. } ) " { x } ) ) e. ( 0 [,) +oo ) )
54		sitgval.x	. . 3 |- .x. = ( .s ` W )
55		sitgval.h	. . 3 |- H = (RRHom ` (Scalar ` W ) )
56		sitgval.1	. . 3 |- ( ph -> W e. V )
57	14, 5, 4, 15, 54, 55, 56, 1	issibf 30395	. 2 |- ( ph -> ( ( U. dom M X. { .0. } ) e. dom ( Wsitg M ) <-> ( ( U. dom M X. { .0. } ) e. ( dom MMblFnM S ) /\ ran ( U. dom M X. { .0. } ) e. Fin /\ A. x e. ( ran ( U. dom M X. { .0. } ) \ { .0. } ) ( M ` ( `' ( U. dom M X. { .0. } ) " { x } ) ) e. ( 0 [,) +oo ) ) ) )
58	27, 40, 53, 57	mpbir3and 1245	1 |- ( ph -> ( U. dom M X. { .0. } ) e. dom ( Wsitg M ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   _Vcvv 3200   \ cdif 3571   (/)c0 3915   {csn 4177   U.cuni 4436   |-> cmpt 4729   X. cxp 5112   `'ccnv 5113   dom cdm 5114   ran crn 5115   "cima 5117   ` cfv 5888  (class class class)co 6650   Fincfn 7955   0cc0 9936   +oocpnf 10071   [,)cico 12177   Basecbs 15857  Scalarcsca 15944   .scvsca 15945   TopOpenctopn 16082   0gc0g 16100   Mndcmnd 17294   TopSpctps 20736  RRHomcrrh 30037  sigAlgebracsiga 30170  sigaGencsigagen 30201  measurescmeas 30258  MblFnMcmbfm 30312  sitgcsitg 30391

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1o 7560  df-map 7859  df-en 7956  df-fin 7959  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-top 20699  df-topon 20716  df-topsp 20737  df-esum 30090  df-siga 30171  df-sigagen 30202  df-meas 30259  df-mbfm 30313  df-sitg 30392

This theorem is referenced by:  sitg0  30408