Theorem sitg0 30408

Description: The integral of the constant zero function is zero. (Contributed by Thierry Arnoux, 13-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 )
sitg0.1	|- ( ph -> W e. TopSp )
sitg0.2	|- ( ph -> W e. Mnd )

Assertion

Ref	Expression
sitg0	|- ( ph -> ( ( Wsitg M ) ` ( U. dom M X. { .0. } ) ) = .0. )

Proof of Theorem sitg0

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sitgval.b	. . 3 |- B = ( Base ` W )
2		sitgval.j	. . 3 |- J = ( TopOpen ` W )
3		sitgval.s	. . 3 |- S = (sigaGen ` J )
4		sitgval.0	. . 3 |- .0. = ( 0g ` W )
5		sitgval.x	. . 3 |- .x. = ( .s ` W )
6		sitgval.h	. . 3 |- H = (RRHom ` (Scalar ` W ) )
7		sitgval.1	. . 3 |- ( ph -> W e. V )
8		sitgval.2	. . 3 |- ( ph -> M e. U. ran measures )
9		sitg0.1	. . . 4 |- ( ph -> W e. TopSp )
10		sitg0.2	. . . 4 |- ( ph -> W e. Mnd )
11	1, 2, 3, 4, 5, 6, 7, 8, 9, 10	sibf0 30396	. . 3 |- ( ph -> ( U. dom M X. { .0. } ) e. dom ( Wsitg M ) )
12	1, 2, 3, 4, 5, 6, 7, 8, 11	sitgfval 30403	. 2 |- ( ph -> ( ( Wsitg M ) ` ( U. dom M X. { .0. } ) ) = ( W gsumg ( x e. ( ran ( U. dom M X. { .0. } ) \ { .0. } ) |-> ( ( H ` ( M ` ( `' ( U. dom M X. { .0. } ) " { x } ) ) ) .x. x ) ) ) )
13		rnxpss 5566	. . . . . . 7 |- ran ( U. dom M X. { .0. } ) C_ { .0. }
14		ssdif0 3942	. . . . . . 7 |- ( ran ( U. dom M X. { .0. } ) C_ { .0. } <-> ( ran ( U. dom M X. { .0. } ) \ { .0. } ) = (/) )
15	13, 14	mpbi 220	. . . . . 6 |- ( ran ( U. dom M X. { .0. } ) \ { .0. } ) = (/)
16		mpteq1 4737	. . . . . 6 |- ( ( ran ( U. dom M X. { .0. } ) \ { .0. } ) = (/) -> ( x e. ( ran ( U. dom M X. { .0. } ) \ { .0. } ) |-> ( ( H ` ( M ` ( `' ( U. dom M X. { .0. } ) " { x } ) ) ) .x. x ) ) = ( x e. (/) |-> ( ( H ` ( M ` ( `' ( U. dom M X. { .0. } ) " { x } ) ) ) .x. x ) ) )
17	15, 16	ax-mp 5	. . . . 5 |- ( x e. ( ran ( U. dom M X. { .0. } ) \ { .0. } ) |-> ( ( H ` ( M ` ( `' ( U. dom M X. { .0. } ) " { x } ) ) ) .x. x ) ) = ( x e. (/) |-> ( ( H ` ( M ` ( `' ( U. dom M X. { .0. } ) " { x } ) ) ) .x. x ) )
18		mpt0 6021	. . . . 5 |- ( x e. (/) |-> ( ( H ` ( M ` ( `' ( U. dom M X. { .0. } ) " { x } ) ) ) .x. x ) ) = (/)
19	17, 18	eqtri 2644	. . . 4 |- ( x e. ( ran ( U. dom M X. { .0. } ) \ { .0. } ) |-> ( ( H ` ( M ` ( `' ( U. dom M X. { .0. } ) " { x } ) ) ) .x. x ) ) = (/)
20	19	oveq2i 6661	. . 3 |- ( W gsumg ( x e. ( ran ( U. dom M X. { .0. } ) \ { .0. } ) |-> ( ( H ` ( M ` ( `' ( U. dom M X. { .0. } ) " { x } ) ) ) .x. x ) ) ) = ( W gsumg (/) )
21	4	gsum0 17278	. . 3 |- ( W gsumg (/) ) = .0.
22	20, 21	eqtri 2644	. 2 |- ( W gsumg ( x e. ( ran ( U. dom M X. { .0. } ) \ { .0. } ) |-> ( ( H ` ( M ` ( `' ( U. dom M X. { .0. } ) " { x } ) ) ) .x. x ) ) ) = .0.
23	12, 22	syl6eq 2672	1 |- ( ph -> ( ( Wsitg M ) ` ( U. dom M X. { .0. } ) ) = .0. )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   \ cdif 3571   C_ wss 3574   (/)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   Basecbs 15857  Scalarcsca 15944   .scvsca 15945   TopOpenctopn 16082   0gc0g 16100   gsum_g cgsu 16101   Mndcmnd 17294   TopSpctps 20736  RRHomcrrh 30037  sigaGencsigagen 30201  measurescmeas 30258  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-pred 5680  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-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-map 7859  df-en 7956  df-fin 7959  df-seq 12802  df-0g 16102  df-gsum 16103  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: (None)