Theorem sitgfval 26870

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 26861	. 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 5174	. . . . 5 |- ( ( ph /\ f = F ) -> ran f = ran F )
12	11	difeq1d 3580	. . . 4 |- ( ( ph /\ f = F ) -> ( ran f \ { .0. } ) = ( ran F \ { .0. } ) )
13	10	cnveqd 5122	. . . . . . . 8 |- ( ( ph /\ f = F ) -> `' f = `' F )
14	13	imaeq1d 5275	. . . . . . 7 |- ( ( ph /\ f = F ) -> ( `' f " { x } ) = ( `' F " { x } ) )
15	14	fveq2d 5802	. . . . . 6 |- ( ( ph /\ f = F ) -> ( M ` ( `' f " { x } ) ) = ( M ` ( `' F " { x } ) ) )
16	15	fveq2d 5802	. . . . 5 |- ( ( ph /\ f = F ) -> ( H ` ( M ` ( `' f " { x } ) ) ) = ( H ` ( M ` ( `' F " { x } ) ) ) )
17	16	oveq1d 6214	. . . 4 |- ( ( ph /\ f = F ) -> ( ( H ` ( M ` ( `' f " { x } ) ) ) .x. x ) = ( ( H ` ( M ` ( `' F " { x } ) ) ) .x. x ) )
18	12, 17	mpteq12dv 4477	. . 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 6215	. 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 26864	. . . 4 |- ( ph -> F e. ( dom MMblFnM S ) )
22	1, 2, 3, 4, 5, 6, 7, 8, 20	sibfrn 26866	. . . 4 |- ( ph -> ran F e. Fin )
23	1, 2, 3, 4, 5, 6, 7, 8, 20	sibfima 26867	. . . . 5 |- ( ( ph /\ x e. ( ran F \ { .0. } ) ) -> ( M ` ( `' F " { x } ) ) e. ( 0 [,) +oo ) )
24	23	ralrimiva 2829	. . . 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 5172	. . . . . 6 |- ( g = F -> ran g = ran F )
27	26	eleq1d 2523	. . . . 5 |- ( g = F -> ( ran g e. Fin <-> ran F e. Fin ) )
28	26	difeq1d 3580	. . . . . 6 |- ( g = F -> ( ran g \ { .0. } ) = ( ran F \ { .0. } ) )
29		cnveq 5120	. . . . . . . . 9 |- ( g = F -> `' g = `' F )
30	29	imaeq1d 5275	. . . . . . . 8 |- ( g = F -> ( `' g " { x } ) = ( `' F " { x } ) )
31	30	fveq2d 5802	. . . . . . 7 |- ( g = F -> ( M ` ( `' g " { x } ) ) = ( M ` ( `' F " { x } ) ) )
32	31	eleq1d 2523	. . . . . 6 |- ( g = F -> ( ( M ` ( `' g " { x } ) ) e. ( 0 [,) +oo ) <-> ( M ` ( `' F " { x } ) ) e. ( 0 [,) +oo ) ) )
33	28, 32	raleqbidv 3035	. . . . 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 3222	. . 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 6224	. . 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 5887	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 1370   e. wcel 1758   A.wral 2798   {crab 2802   _Vcvv 3076   \ cdif 3432   {csn 3984   U.cuni 4198   |-> cmpt 4457   `'ccnv 4946   dom cdm 4947   ran crn 4948   "cima 4950   ` cfv 5525  (class class class)co 6199   Fincfn 7419   0cc0 9392   +oocpnf 9525   [,)cico 11412   Basecbs 14291  Scalarcsca 14359   .scvsca 14360   TopOpenctopn 14478   0gc0g 14496   gsum_g cgsu 14497  RRHomcrrh 26566  sigaGencsigagen 26725  measurescmeas 26753  MblFnMcmbfm 26808  sitgcsitg 26858

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4510  ax-sep 4520  ax-nul 4528  ax-pr 4638

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-ral 2803  df-rex 2804  df-reu 2805  df-rab 2807  df-v 3078  df-sbc 3293  df-csb 3395  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-nul 3745  df-if 3899  df-sn 3985  df-pr 3987  df-op 3991  df-uni 4199  df-iun 4280  df-br 4400  df-opab 4458  df-mpt 4459  df-id 4743  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-ov 6202  df-oprab 6203  df-mpt2 6204  df-sitg 26859

This theorem is referenced by:  sitgclg  26871  sitg0  26875