Theorem sitgfval 26679

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 26670	. 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 461	. . . . . 6 |- ( ( ph /\ f = F ) -> f = F )
11	10	rneqd 5062	. . . . 5 |- ( ( ph /\ f = F ) -> ran f = ran F )
12	11	difeq1d 3468	. . . 4 |- ( ( ph /\ f = F ) -> ( ran f \ { .0. } ) = ( ran F \ { .0. } ) )
13	10	cnveqd 5010	. . . . . . . 8 |- ( ( ph /\ f = F ) -> `' f = `' F )
14	13	imaeq1d 5163	. . . . . . 7 |- ( ( ph /\ f = F ) -> ( `' f " { x } ) = ( `' F " { x } ) )
15	14	fveq2d 5690	. . . . . 6 |- ( ( ph /\ f = F ) -> ( M ` ( `' f " { x } ) ) = ( M ` ( `' F " { x } ) ) )
16	15	fveq2d 5690	. . . . 5 |- ( ( ph /\ f = F ) -> ( H ` ( M ` ( `' f " { x } ) ) ) = ( H ` ( M ` ( `' F " { x } ) ) ) )
17	16	oveq1d 6101	. . . 4 |- ( ( ph /\ f = F ) -> ( ( H ` ( M ` ( `' f " { x } ) ) ) .x. x ) = ( ( H ` ( M ` ( `' F " { x } ) ) ) .x. x ) )
18	12, 17	mpteq12dv 4365	. . 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 6102	. 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 26673	. . . 4 |- ( ph -> F e. ( dom MMblFnM S ) )
22	1, 2, 3, 4, 5, 6, 7, 8, 20	sibfrn 26675	. . . 4 |- ( ph -> ran F e. Fin )
23	1, 2, 3, 4, 5, 6, 7, 8, 20	sibfima 26676	. . . . 5 |- ( ( ph /\ x e. ( ran F \ { .0. } ) ) -> ( M ` ( `' F " { x } ) ) e. ( 0 [,) +oo ) )
24	23	ralrimiva 2794	. . . 4 |- ( ph -> A. x e. ( ran F \ { .0. } ) ( M ` ( `' F " { x } ) ) e. ( 0 [,) +oo ) )
25	21, 22, 24	jca32 535	. . 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 2504	. . . . 5 |- ( g = F -> ( ran g e. Fin <-> ran F e. Fin ) )
28	26	difeq1d 3468	. . . . . 6 |- ( g = F -> ( ran g \ { .0. } ) = ( ran F \ { .0. } ) )
29		cnveq 5008	. . . . . . . . 9 |- ( g = F -> `' g = `' F )
30	29	imaeq1d 5163	. . . . . . . 8 |- ( g = F -> ( `' g " { x } ) = ( `' F " { x } ) )
31	30	fveq2d 5690	. . . . . . 7 |- ( g = F -> ( M ` ( `' g " { x } ) ) = ( M ` ( `' F " { x } ) ) )
32	31	eleq1d 2504	. . . . . 6 |- ( g = F -> ( ( M ` ( `' g " { x } ) ) e. ( 0 [,) +oo ) <-> ( M ` ( `' F " { x } ) ) e. ( 0 [,) +oo ) ) )
33	28, 32	raleqbidv 2926	. . . . 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 710	. . . 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 3112	. . 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 212	. 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 6111	. . 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 5774	1 |- ( ph -> ( ( Wsitg M ) ` F ) = ( W gsumg ( x e. ( ran F \ { .0. } ) |-> ( ( H ` ( M ` ( `' F " { x } ) ) ) .x. x ) ) ) )

Syntax hints:   -> wi 4   /\ wa 369   = wceq 1369   e. wcel 1756   A.wral 2710   {crab 2714   _Vcvv 2967   \ cdif 3320   {csn 3872   U.cuni 4086   e. cmpt 4345   `'ccnv 4834   dom cdm 4835   ran crn 4836   "cima 4838   ` cfv 5413  (class class class)co 6086   Fincfn 7302   0cc0 9274   +oocpnf 9407   [,)cico 11294   Basecbs 14166  Scalarcsca 14233   .scvsca 14234   TopOpenctopn 14352   0gc0g 14370   gsum_g cgsu 14371  RRHomcrrh 26374  sigaGencsigagen 26533  measurescmeas 26561  MblFnMcmbfm 26617  sitgcsitg 26667

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pr 4526

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-reu 2717  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-sitg 26668

This theorem is referenced by:  sitgclg  26680  sitg0  26684