Theorem sitgval 30394

Description: Value of the simple function integral builder for a given space W and measure M. (Contributed by Thierry Arnoux, 30-Jan-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 )

Assertion

Ref	Expression
sitgval	|- ( ph -> ( Wsitg M ) = ( f e. { g e. ( dom MMblFnM S ) | ( ran g e. Fin /\ A. x e. ( ran g \ { .0. } ) ( M ` ( `' g " { x } ) ) e. ( 0 [,) +oo ) ) } |-> ( W gsumg ( x e. ( ran f \ { .0. } ) |-> ( ( H ` ( M ` ( `' f " { x } ) ) ) .x. x ) ) ) ) )

Distinct variable groups:   B, f   f, g, x   f, H   f, M, g, x   S, f, g   f, W, g, x   .0. , f, g, x   .x. , f

Allowed substitution hints:   ph( x, f, g)   B( x, g)   S( x)   .x. ( x, g)   H( x, g)   J( x, f, g)   V( x, f, g)

Proof of Theorem sitgval

Dummy variables m w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		sitgval.1	. . 3 |- ( ph -> W e. V )
2	1	elexd 3214	. 2 |- ( ph -> W e. _V )
3		sitgval.2	. 2 |- ( ph -> M e. U. ran measures )
4		fveq2 6191	. . . . . . . 8 |- ( w = W -> ( TopOpen ` w ) = ( TopOpen ` W ) )
5	4	fveq2d 6195	. . . . . . 7 |- ( w = W -> (sigaGen ` ( TopOpen ` w ) ) = (sigaGen ` ( TopOpen ` W ) ) )
6		sitgval.s	. . . . . . . 8 |- S = (sigaGen ` J )
7		sitgval.j	. . . . . . . . 9 |- J = ( TopOpen ` W )
8	7	fveq2i 6194	. . . . . . . 8 |- (sigaGen ` J ) = (sigaGen ` ( TopOpen ` W ) )
9	6, 8	eqtri 2644	. . . . . . 7 |- S = (sigaGen ` ( TopOpen ` W ) )
10	5, 9	syl6eqr 2674	. . . . . 6 |- ( w = W -> (sigaGen ` ( TopOpen ` w ) ) = S )
11	10	oveq2d 6666	. . . . 5 |- ( w = W -> ( dom mMblFnM (sigaGen ` ( TopOpen ` w ) ) ) = ( dom mMblFnM S ) )
12		fveq2 6191	. . . . . . . . . 10 |- ( w = W -> ( 0g ` w ) = ( 0g ` W ) )
13		sitgval.0	. . . . . . . . . 10 |- .0. = ( 0g ` W )
14	12, 13	syl6eqr 2674	. . . . . . . . 9 |- ( w = W -> ( 0g ` w ) = .0. )
15	14	sneqd 4189	. . . . . . . 8 |- ( w = W -> { ( 0g ` w ) } = { .0. } )
16	15	difeq2d 3728	. . . . . . 7 |- ( w = W -> ( ran g \ { ( 0g ` w ) } ) = ( ran g \ { .0. } ) )
17	16	raleqdv 3144	. . . . . 6 |- ( w = W -> ( A. x e. ( ran g \ { ( 0g ` w ) } ) ( m ` ( `' g " { x } ) ) e. ( 0 [,) +oo ) <-> A. x e. ( ran g \ { .0. } ) ( m ` ( `' g " { x } ) ) e. ( 0 [,) +oo ) ) )
18	17	anbi2d 740	. . . . 5 |- ( w = W -> ( ( ran g e. Fin /\ A. x e. ( ran g \ { ( 0g ` w ) } ) ( m ` ( `' g " { x } ) ) e. ( 0 [,) +oo ) ) <-> ( ran g e. Fin /\ A. x e. ( ran g \ { .0. } ) ( m ` ( `' g " { x } ) ) e. ( 0 [,) +oo ) ) ) )
19	11, 18	rabeqbidv 3195	. . . 4 |- ( w = W -> { g e. ( dom mMblFnM (sigaGen ` ( TopOpen ` w ) ) ) | ( ran g e. Fin /\ A. x e. ( ran g \ { ( 0g ` w ) } ) ( m ` ( `' g " { x } ) ) e. ( 0 [,) +oo ) ) } = { g e. ( dom mMblFnM S ) | ( ran g e. Fin /\ A. x e. ( ran g \ { .0. } ) ( m ` ( `' g " { x } ) ) e. ( 0 [,) +oo ) ) } )
20		id 22	. . . . 5 |- ( w = W -> w = W )
21	15	difeq2d 3728	. . . . . 6 |- ( w = W -> ( ran f \ { ( 0g ` w ) } ) = ( ran f \ { .0. } ) )
22		fveq2 6191	. . . . . . . 8 |- ( w = W -> ( .s ` w ) = ( .s ` W ) )
23		sitgval.x	. . . . . . . 8 |- .x. = ( .s ` W )
24	22, 23	syl6eqr 2674	. . . . . . 7 |- ( w = W -> ( .s ` w ) = .x. )
25		fveq2 6191	. . . . . . . . . 10 |- ( w = W -> (Scalar ` w ) = (Scalar ` W ) )
26	25	fveq2d 6195	. . . . . . . . 9 |- ( w = W -> (RRHom ` (Scalar ` w ) ) = (RRHom ` (Scalar ` W ) ) )
27		sitgval.h	. . . . . . . . 9 |- H = (RRHom ` (Scalar ` W ) )
28	26, 27	syl6eqr 2674	. . . . . . . 8 |- ( w = W -> (RRHom ` (Scalar ` w ) ) = H )
29	28	fveq1d 6193	. . . . . . 7 |- ( w = W -> ( (RRHom ` (Scalar ` w ) ) ` ( m ` ( `' f " { x } ) ) ) = ( H ` ( m ` ( `' f " { x } ) ) ) )
30		eqidd 2623	. . . . . . 7 |- ( w = W -> x = x )
31	24, 29, 30	oveq123d 6671	. . . . . 6 |- ( w = W -> ( ( (RRHom ` (Scalar ` w ) ) ` ( m ` ( `' f " { x } ) ) ) ( .s ` w ) x ) = ( ( H ` ( m ` ( `' f " { x } ) ) ) .x. x ) )
32	21, 31	mpteq12dv 4733	. . . . 5 |- ( w = W -> ( x e. ( ran f \ { ( 0g ` w ) } ) |-> ( ( (RRHom ` (Scalar ` w ) ) ` ( m ` ( `' f " { x } ) ) ) ( .s ` w ) x ) ) = ( x e. ( ran f \ { .0. } ) |-> ( ( H ` ( m ` ( `' f " { x } ) ) ) .x. x ) ) )
33	20, 32	oveq12d 6668	. . . 4 |- ( w = W -> ( w gsumg ( x e. ( ran f \ { ( 0g ` w ) } ) |-> ( ( (RRHom ` (Scalar ` w ) ) ` ( m ` ( `' f " { x } ) ) ) ( .s ` w ) x ) ) ) = ( W gsumg ( x e. ( ran f \ { .0. } ) |-> ( ( H ` ( m ` ( `' f " { x } ) ) ) .x. x ) ) ) )
34	19, 33	mpteq12dv 4733	. . 3 |- ( w = W -> ( f e. { g e. ( dom mMblFnM (sigaGen ` ( TopOpen ` w ) ) ) | ( ran g e. Fin /\ A. x e. ( ran g \ { ( 0g ` w ) } ) ( m ` ( `' g " { x } ) ) e. ( 0 [,) +oo ) ) } |-> ( w gsumg ( x e. ( ran f \ { ( 0g ` w ) } ) |-> ( ( (RRHom ` (Scalar ` w ) ) ` ( m ` ( `' f " { x } ) ) ) ( .s ` w ) x ) ) ) ) = ( f e. { g e. ( dom mMblFnM S ) | ( ran g e. Fin /\ A. x e. ( ran g \ { .0. } ) ( m ` ( `' g " { x } ) ) e. ( 0 [,) +oo ) ) } |-> ( W gsumg ( x e. ( ran f \ { .0. } ) |-> ( ( H ` ( m ` ( `' f " { x } ) ) ) .x. x ) ) ) ) )
35		dmeq 5324	. . . . . 6 |- ( m = M -> dom m = dom M )
36	35	oveq1d 6665	. . . . 5 |- ( m = M -> ( dom mMblFnM S ) = ( dom MMblFnM S ) )
37		fveq1 6190	. . . . . . . 8 |- ( m = M -> ( m ` ( `' g " { x } ) ) = ( M ` ( `' g " { x } ) ) )
38	37	eleq1d 2686	. . . . . . 7 |- ( m = M -> ( ( m ` ( `' g " { x } ) ) e. ( 0 [,) +oo ) <-> ( M ` ( `' g " { x } ) ) e. ( 0 [,) +oo ) ) )
39	38	ralbidv 2986	. . . . . 6 |- ( m = M -> ( A. x e. ( ran g \ { .0. } ) ( m ` ( `' g " { x } ) ) e. ( 0 [,) +oo ) <-> A. x e. ( ran g \ { .0. } ) ( M ` ( `' g " { x } ) ) e. ( 0 [,) +oo ) ) )
40	39	anbi2d 740	. . . . 5 |- ( m = M -> ( ( ran g e. Fin /\ A. x e. ( ran g \ { .0. } ) ( m ` ( `' g " { x } ) ) e. ( 0 [,) +oo ) ) <-> ( ran g e. Fin /\ A. x e. ( ran g \ { .0. } ) ( M ` ( `' g " { x } ) ) e. ( 0 [,) +oo ) ) ) )
41	36, 40	rabeqbidv 3195	. . . 4 |- ( m = M -> { g e. ( dom mMblFnM S ) | ( ran g e. Fin /\ A. x e. ( ran g \ { .0. } ) ( m ` ( `' g " { x } ) ) e. ( 0 [,) +oo ) ) } = { g e. ( dom MMblFnM S ) | ( ran g e. Fin /\ A. x e. ( ran g \ { .0. } ) ( M ` ( `' g " { x } ) ) e. ( 0 [,) +oo ) ) } )
42		simpl 473	. . . . . . . . 9 |- ( ( m = M /\ x e. ( ran f \ { .0. } ) ) -> m = M )
43	42	fveq1d 6193	. . . . . . . 8 |- ( ( m = M /\ x e. ( ran f \ { .0. } ) ) -> ( m ` ( `' f " { x } ) ) = ( M ` ( `' f " { x } ) ) )
44	43	fveq2d 6195	. . . . . . 7 |- ( ( m = M /\ x e. ( ran f \ { .0. } ) ) -> ( H ` ( m ` ( `' f " { x } ) ) ) = ( H ` ( M ` ( `' f " { x } ) ) ) )
45	44	oveq1d 6665	. . . . . 6 |- ( ( m = M /\ x e. ( ran f \ { .0. } ) ) -> ( ( H ` ( m ` ( `' f " { x } ) ) ) .x. x ) = ( ( H ` ( M ` ( `' f " { x } ) ) ) .x. x ) )
46	45	mpteq2dva 4744	. . . . 5 |- ( m = M -> ( x e. ( ran f \ { .0. } ) |-> ( ( H ` ( m ` ( `' f " { x } ) ) ) .x. x ) ) = ( x e. ( ran f \ { .0. } ) |-> ( ( H ` ( M ` ( `' f " { x } ) ) ) .x. x ) ) )
47	46	oveq2d 6666	. . . 4 |- ( m = M -> ( W gsumg ( x e. ( ran f \ { .0. } ) |-> ( ( H ` ( m ` ( `' f " { x } ) ) ) .x. x ) ) ) = ( W gsumg ( x e. ( ran f \ { .0. } ) |-> ( ( H ` ( M ` ( `' f " { x } ) ) ) .x. x ) ) ) )
48	41, 47	mpteq12dv 4733	. . 3 |- ( m = M -> ( f e. { g e. ( dom mMblFnM S ) | ( ran g e. Fin /\ A. x e. ( ran g \ { .0. } ) ( m ` ( `' g " { x } ) ) e. ( 0 [,) +oo ) ) } |-> ( W gsumg ( x e. ( ran f \ { .0. } ) |-> ( ( H ` ( m ` ( `' f " { x } ) ) ) .x. x ) ) ) ) = ( f e. { g e. ( dom MMblFnM S ) | ( ran g e. Fin /\ A. x e. ( ran g \ { .0. } ) ( M ` ( `' g " { x } ) ) e. ( 0 [,) +oo ) ) } |-> ( W gsumg ( x e. ( ran f \ { .0. } ) |-> ( ( H ` ( M ` ( `' f " { x } ) ) ) .x. x ) ) ) ) )
49		df-sitg 30392	. . 3 |- sitg = ( w e. _V , m e. U. ran measures |-> ( f e. { g e. ( dom mMblFnM (sigaGen ` ( TopOpen ` w ) ) ) | ( ran g e. Fin /\ A. x e. ( ran g \ { ( 0g ` w ) } ) ( m ` ( `' g " { x } ) ) e. ( 0 [,) +oo ) ) } |-> ( w gsumg ( x e. ( ran f \ { ( 0g ` w ) } ) |-> ( ( (RRHom ` (Scalar ` w ) ) ` ( m ` ( `' f " { x } ) ) ) ( .s ` w ) x ) ) ) ) )
50		ovex 6678	. . . 4 |- ( dom MMblFnM S ) e. _V
51	50	mptrabex 6488	. . 3 |- ( f e. { g e. ( dom MMblFnM S ) | ( ran g e. Fin /\ A. x e. ( ran g \ { .0. } ) ( M ` ( `' g " { x } ) ) e. ( 0 [,) +oo ) ) } |-> ( W gsumg ( x e. ( ran f \ { .0. } ) |-> ( ( H ` ( M ` ( `' f " { x } ) ) ) .x. x ) ) ) ) e. _V
52	34, 48, 49, 51	ovmpt2 6796	. 2 |- ( ( W e. _V /\ M e. U. ran measures ) -> ( Wsitg M ) = ( f e. { g e. ( dom MMblFnM S ) | ( ran g e. Fin /\ A. x e. ( ran g \ { .0. } ) ( M ` ( `' g " { x } ) ) e. ( 0 [,) +oo ) ) } |-> ( W gsumg ( x e. ( ran f \ { .0. } ) |-> ( ( H ` ( M ` ( `' f " { x } ) ) ) .x. x ) ) ) ) )
53	2, 3, 52	syl2anc 693	1 |- ( ph -> ( Wsitg M ) = ( f e. { g e. ( dom MMblFnM S ) | ( ran g e. Fin /\ A. x e. ( ran g \ { .0. } ) ( M ` ( `' g " { x } ) ) e. ( 0 [,) +oo ) ) } |-> ( W gsumg ( x e. ( ran f \ { .0. } ) |-> ( ( H ` ( M ` ( `' f " { x } ) ) ) .x. x ) ) ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   {crab 2916   _Vcvv 3200   \ cdif 3571   {csn 4177   U.cuni 4436   |-> cmpt 4729   `'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   gsum_g cgsu 16101  RRHomcrrh 30037  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-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-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  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-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  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-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-sitg 30392

This theorem is referenced by:  issibf  30395  sitgfval  30403  sitgf  30409