Theorem sitgfval 29174

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 29165	. 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 463	. . . . . 6 |- ( ( ph /\ f = F ) -> f = F )
11	10	rneqd 5062	. . . . 5 |- ( ( ph /\ f = F ) -> ran f = ran F )
12	11	difeq1d 3550	. . . 4 |- ( ( ph /\ f = F ) -> ( ran f \ { .0. } ) = ( ran F \ { .0. } ) )
13	10	cnveqd 5010	. . . . . . . 8 |- ( ( ph /\ f = F ) -> `' f = `' F )
14	13	imaeq1d 5167	. . . . . . 7 |- ( ( ph /\ f = F ) -> ( `' f " { x } ) = ( `' F " { x } ) )
15	14	fveq2d 5869	. . . . . 6 |- ( ( ph /\ f = F ) -> ( M ` ( `' f " { x } ) ) = ( M ` ( `' F " { x } ) ) )
16	15	fveq2d 5869	. . . . 5 |- ( ( ph /\ f = F ) -> ( H ` ( M ` ( `' f " { x } ) ) ) = ( H ` ( M ` ( `' F " { x } ) ) ) )
17	16	oveq1d 6305	. . . 4 |- ( ( ph /\ f = F ) -> ( ( H ` ( M ` ( `' f " { x } ) ) ) .x. x ) = ( ( H ` ( M ` ( `' F " { x } ) ) ) .x. x ) )
18	12, 17	mpteq12dv 4481	. . 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 6306	. 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 29168	. . . 4 |- ( ph -> F e. ( dom MMblFnM S ) )
22	1, 2, 3, 4, 5, 6, 7, 8, 20	sibfrn 29170	. . . 4 |- ( ph -> ran F e. Fin )
23	1, 2, 3, 4, 5, 6, 7, 8, 20	sibfima 29171	. . . . 5 |- ( ( ph /\ x e. ( ran F \ { .0. } ) ) -> ( M ` ( `' F " { x } ) ) e. ( 0 [,) +oo ) )
24	23	ralrimiva 2802	. . . 4 |- ( ph -> A. x e. ( ran F \ { .0. } ) ( M ` ( `' F " { x } ) ) e. ( 0 [,) +oo ) )
25	21, 22, 24	jca32 538	. . 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 5060	. . . . . 6 |- ( g = F -> ran g = ran F )
27	26	eleq1d 2513	. . . . 5 |- ( g = F -> ( ran g e. Fin <-> ran F e. Fin ) )
28	26	difeq1d 3550	. . . . . 6 |- ( g = F -> ( ran g \ { .0. } ) = ( ran F \ { .0. } ) )
29		cnveq 5008	. . . . . . . . 9 |- ( g = F -> `' g = `' F )
30	29	imaeq1d 5167	. . . . . . . 8 |- ( g = F -> ( `' g " { x } ) = ( `' F " { x } ) )
31	30	fveq2d 5869	. . . . . . 7 |- ( g = F -> ( M ` ( `' g " { x } ) ) = ( M ` ( `' F " { x } ) ) )
32	31	eleq1d 2513	. . . . . 6 |- ( g = F -> ( ( M ` ( `' g " { x } ) ) e. ( 0 [,) +oo ) <-> ( M ` ( `' F " { x } ) ) e. ( 0 [,) +oo ) ) )
33	28, 32	raleqbidv 3001	. . . . 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 717	. . . 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 3196	. . 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 216	. 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 6318	. . 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 5954	1 |- ( ph -> ( ( Wsitg M ) ` F ) = ( W gsumg ( x e. ( ran F \ { .0. } ) |-> ( ( H ` ( M ` ( `' F " { x } ) ) ) .x. x ) ) ) )

Syntax hints:   -> wi 4   /\ wa 371   = wceq 1444   e. wcel 1887   A.wral 2737   {crab 2741   _Vcvv 3045   \ cdif 3401   {csn 3968   U.cuni 4198   |-> cmpt 4461   `'ccnv 4833   dom cdm 4834   ran crn 4835   "cima 4837   ` cfv 5582  (class class class)co 6290   Fincfn 7569   0cc0 9539   +oocpnf 9672   [,)cico 11637   Basecbs 15121  Scalarcsca 15193   .scvsca 15194   TopOpenctopn 15320   0gc0g 15338   gsum_g cgsu 15339  RRHomcrrh 28797  sigaGencsigagen 28960  measurescmeas 29017  MblFnMcmbfm 29072  sitgcsitg 29162

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pr 4639

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-reu 2744  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-sitg 29163

This theorem is referenced by:  sitgclg  29175  sitg0  29179