Theorem sitgfval 27951

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 27942	. 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 5230	. . . . 5 |- ( ( ph /\ f = F ) -> ran f = ran F )
12	11	difeq1d 3621	. . . 4 |- ( ( ph /\ f = F ) -> ( ran f \ { .0. } ) = ( ran F \ { .0. } ) )
13	10	cnveqd 5178	. . . . . . . 8 |- ( ( ph /\ f = F ) -> `' f = `' F )
14	13	imaeq1d 5336	. . . . . . 7 |- ( ( ph /\ f = F ) -> ( `' f " { x } ) = ( `' F " { x } ) )
15	14	fveq2d 5870	. . . . . 6 |- ( ( ph /\ f = F ) -> ( M ` ( `' f " { x } ) ) = ( M ` ( `' F " { x } ) ) )
16	15	fveq2d 5870	. . . . 5 |- ( ( ph /\ f = F ) -> ( H ` ( M ` ( `' f " { x } ) ) ) = ( H ` ( M ` ( `' F " { x } ) ) ) )
17	16	oveq1d 6299	. . . 4 |- ( ( ph /\ f = F ) -> ( ( H ` ( M ` ( `' f " { x } ) ) ) .x. x ) = ( ( H ` ( M ` ( `' F " { x } ) ) ) .x. x ) )
18	12, 17	mpteq12dv 4525	. . 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 6300	. 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 27945	. . . 4 |- ( ph -> F e. ( dom MMblFnM S ) )
22	1, 2, 3, 4, 5, 6, 7, 8, 20	sibfrn 27947	. . . 4 |- ( ph -> ran F e. Fin )
23	1, 2, 3, 4, 5, 6, 7, 8, 20	sibfima 27948	. . . . 5 |- ( ( ph /\ x e. ( ran F \ { .0. } ) ) -> ( M ` ( `' F " { x } ) ) e. ( 0 [,) +oo ) )
24	23	ralrimiva 2878	. . . 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 5228	. . . . . 6 |- ( g = F -> ran g = ran F )
27	26	eleq1d 2536	. . . . 5 |- ( g = F -> ( ran g e. Fin <-> ran F e. Fin ) )
28	26	difeq1d 3621	. . . . . 6 |- ( g = F -> ( ran g \ { .0. } ) = ( ran F \ { .0. } ) )
29		cnveq 5176	. . . . . . . . 9 |- ( g = F -> `' g = `' F )
30	29	imaeq1d 5336	. . . . . . . 8 |- ( g = F -> ( `' g " { x } ) = ( `' F " { x } ) )
31	30	fveq2d 5870	. . . . . . 7 |- ( g = F -> ( M ` ( `' g " { x } ) ) = ( M ` ( `' F " { x } ) ) )
32	31	eleq1d 2536	. . . . . 6 |- ( g = F -> ( ( M ` ( `' g " { x } ) ) e. ( 0 [,) +oo ) <-> ( M ` ( `' F " { x } ) ) e. ( 0 [,) +oo ) ) )
33	28, 32	raleqbidv 3072	. . . . 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 3261	. . 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 6309	. . 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 5955	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 1379   e. wcel 1767   A.wral 2814   {crab 2818   _Vcvv 3113   \ cdif 3473   {csn 4027   U.cuni 4245   |-> cmpt 4505   `'ccnv 4998   dom cdm 4999   ran crn 5000   "cima 5002   ` cfv 5588  (class class class)co 6284   Fincfn 7516   0cc0 9492   +oocpnf 9625   [,)cico 11531   Basecbs 14490  Scalarcsca 14558   .scvsca 14559   TopOpenctopn 14677   0gc0g 14695   gsum_g cgsu 14696  RRHomcrrh 27638  sigaGencsigagen 27806  measurescmeas 27834  MblFnMcmbfm 27889  sitgcsitg 27939

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pr 4686

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-sitg 27940

This theorem is referenced by:  sitgclg  27952  sitg0  27956