Theorem sitgfval 29002

Description: Value of the Bochner integral for a simple function F. (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 )
sibfmbl.1	|- ( ph -> F e. dom ( Wsitg M ) )

Assertion

Ref	Expression
sitgfval	|- ( ph -> ( ( Wsitg M ) ` F ) = ( W gsumg ( x e. ( ran F \ { .0. } ) |-> ( ( H ` ( M ` ( `' F " { x } ) ) ) .x. x ) ) ) )

Distinct variable groups:   x, F   x, M   x, W   x, .0.   ph, x

Allowed substitution hints:   B( x)   S( x)   .x. ( x)   H( x)   J( x)   V( x)

Proof of Theorem sitgfval

Dummy variables f g are mutually distinct and 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	1, 2, 3, 4, 5, 6, 7, 8	sitgval 28993	. 2 |- ( 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 ) ) ) ) )
10		simpr 462	. . . . . 6 |- ( ( ph /\ f = F ) -> f = F )
11	10	rneqd 5082	. . . . 5 |- ( ( ph /\ f = F ) -> ran f = ran F )
12	11	difeq1d 3588	. . . 4 |- ( ( ph /\ f = F ) -> ( ran f \ { .0. } ) = ( ran F \ { .0. } ) )
13	10	cnveqd 5030	. . . . . . . 8 |- ( ( ph /\ f = F ) -> `' f = `' F )
14	13	imaeq1d 5187	. . . . . . 7 |- ( ( ph /\ f = F ) -> ( `' f " { x } ) = ( `' F " { x } ) )
15	14	fveq2d 5885	. . . . . 6 |- ( ( ph /\ f = F ) -> ( M ` ( `' f " { x } ) ) = ( M ` ( `' F " { x } ) ) )
16	15	fveq2d 5885	. . . . 5 |- ( ( ph /\ f = F ) -> ( H ` ( M ` ( `' f " { x } ) ) ) = ( H ` ( M ` ( `' F " { x } ) ) ) )
17	16	oveq1d 6320	. . . 4 |- ( ( ph /\ f = F ) -> ( ( H ` ( M ` ( `' f " { x } ) ) ) .x. x ) = ( ( H ` ( M ` ( `' F " { x } ) ) ) .x. x ) )
18	12, 17	mpteq12dv 4504	. . 3 |- ( ( ph /\ f = F ) -> ( x e. ( ran f \ { .0. } ) |-> ( ( H ` ( M ` ( `' f " { x } ) ) ) .x. x ) ) = ( x e. ( ran F \ { .0. } ) |-> ( ( H ` ( M ` ( `' F " { x } ) ) ) .x. x ) ) )
19	18	oveq2d 6321	. 2 |- ( ( ph /\ f = F ) -> ( 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 ) ) ) )
20		sibfmbl.1	. . . . 5 |- ( ph -> F e. dom ( Wsitg M ) )
21	1, 2, 3, 4, 5, 6, 7, 8, 20	sibfmbl 28996	. . . 4 |- ( ph -> F e. ( dom MMblFnM S ) )
22	1, 2, 3, 4, 5, 6, 7, 8, 20	sibfrn 28998	. . . 4 |- ( ph -> ran F e. Fin )
23	1, 2, 3, 4, 5, 6, 7, 8, 20	sibfima 28999	. . . . 5 |- ( ( ph /\ x e. ( ran F \ { .0. } ) ) -> ( M ` ( `' F " { x } ) ) e. ( 0 [,) +oo ) )
24	23	ralrimiva 2846	. . . 4 |- ( ph -> A. x e. ( ran F \ { .0. } ) ( M ` ( `' F " { x } ) ) e. ( 0 [,) +oo ) )
25	21, 22, 24	jca32 537	. . 3 |- ( ph -> ( F e. ( dom MMblFnM S ) /\ ( ran F e. Fin /\ A. x e. ( ran F \ { .0. } ) ( M ` ( `' F " { x } ) ) e. ( 0 [,) +oo ) ) ) )
26		rneq 5080	. . . . . 6 |- ( g = F -> ran g = ran F )
27	26	eleq1d 2498	. . . . 5 |- ( g = F -> ( ran g e. Fin <-> ran F e. Fin ) )
28	26	difeq1d 3588	. . . . . 6 |- ( g = F -> ( ran g \ { .0. } ) = ( ran F \ { .0. } ) )
29		cnveq 5028	. . . . . . . . 9 |- ( g = F -> `' g = `' F )
30	29	imaeq1d 5187	. . . . . . . 8 |- ( g = F -> ( `' g " { x } ) = ( `' F " { x } ) )
31	30	fveq2d 5885	. . . . . . 7 |- ( g = F -> ( M ` ( `' g " { x } ) ) = ( M ` ( `' F " { x } ) ) )
32	31	eleq1d 2498	. . . . . 6 |- ( g = F -> ( ( M ` ( `' g " { x } ) ) e. ( 0 [,) +oo ) <-> ( M ` ( `' F " { x } ) ) e. ( 0 [,) +oo ) ) )
33	28, 32	raleqbidv 3046	. . . . 5 |- ( g = F -> ( A. x e. ( ran g \ { .0. } ) ( M ` ( `' g " { x } ) ) e. ( 0 [,) +oo ) <-> A. x e. ( ran F \ { .0. } ) ( M ` ( `' F " { x } ) ) e. ( 0 [,) +oo ) ) )
34	27, 33	anbi12d 715	. . . 4 |- ( g = F -> ( ( ran g e. Fin /\ A. x e. ( ran g \ { .0. } ) ( M ` ( `' g " { x } ) ) e. ( 0 [,) +oo ) ) <-> ( ran F e. Fin /\ A. x e. ( ran F \ { .0. } ) ( M ` ( `' F " { x } ) ) e. ( 0 [,) +oo ) ) ) )
35	34	elrab 3235	. . 3 |- ( F e. { g e. ( dom MMblFnM S ) | ( ran g e. Fin /\ A. x e. ( ran g \ { .0. } ) ( M ` ( `' g " { x } ) ) e. ( 0 [,) +oo ) ) } <-> ( F e. ( dom MMblFnM S ) /\ ( ran F e. Fin /\ A. x e. ( ran F \ { .0. } ) ( M ` ( `' F " { x } ) ) e. ( 0 [,) +oo ) ) ) )
36	25, 35	sylibr 215	. 2 |- ( ph -> F e. { g e. ( dom MMblFnM S ) | ( ran g e. Fin /\ A. x e. ( ran g \ { .0. } ) ( M ` ( `' g " { x } ) ) e. ( 0 [,) +oo ) ) } )
37		ovex 6333	. . 3 |- ( W gsumg ( x e. ( ran F \ { .0. } ) |-> ( ( H ` ( M ` ( `' F " { x } ) ) ) .x. x ) ) ) e. _V
38	37	a1i 11	. 2 |- ( ph -> ( W gsumg ( x e. ( ran F \ { .0. } ) |-> ( ( H ` ( M ` ( `' F " { x } ) ) ) .x. x ) ) ) e. _V )
39	9, 19, 36, 38	fvmptd 5970	1 |- ( ph -> ( ( Wsitg M ) ` F ) = ( W gsumg ( x e. ( ran F \ { .0. } ) |-> ( ( H ` ( M ` ( `' F " { x } ) ) ) .x. x ) ) ) )

Syntax hints:   -> wi 4   /\ wa 370   = wceq 1437   e. wcel 1870   A.wral 2782   {crab 2786   _Vcvv 3087   \ cdif 3439   {csn 4002   U.cuni 4222   |-> cmpt 4484   `'ccnv 4853   dom cdm 4854   ran crn 4855   "cima 4857   ` cfv 5601  (class class class)co 6305   Fincfn 7577   0cc0 9538   +oocpnf 9671   [,)cico 11637   Basecbs 15084  Scalarcsca 15155   .scvsca 15156   TopOpenctopn 15279   0gc0g 15297   gsum_g cgsu 15298  RRHomcrrh 28636  sigaGencsigagen 28799  measurescmeas 28856  MblFnMcmbfm 28911  sitgcsitg 28990

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pr 4661

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-sitg 28991

This theorem is referenced by:  sitgclg  29003  sitg0  29007