Theorem issibf 30395

Description: The predicate " F is a simple function" relative to the Bochner integral. (Contributed by Thierry Arnoux, 19-Feb-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 )

Assertion

Ref	Expression
issibf	|- ( ph -> ( F e. dom ( Wsitg M ) <-> ( F e. ( dom MMblFnM S ) /\ ran F e. Fin /\ A. x e. ( ran F \ { .0. } ) ( M ` ( `' F " { x } ) ) e. ( 0 [,) +oo ) ) ) )

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

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

Proof of Theorem issibf

Dummy variables f g are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		sitgval.b	. . . . . . . . 9 |- B = ( Base ` W )
2		sitgval.j	. . . . . . . . 9 |- J = ( TopOpen ` W )
3		sitgval.s	. . . . . . . . 9 |- S = (sigaGen ` J )
4		sitgval.0	. . . . . . . . 9 |- .0. = ( 0g ` W )
5		sitgval.x	. . . . . . . . 9 |- .x. = ( .s ` W )
6		sitgval.h	. . . . . . . . 9 |- H = (RRHom ` (Scalar ` W ) )
7		sitgval.1	. . . . . . . . 9 |- ( ph -> W e. V )
8		sitgval.2	. . . . . . . . 9 |- ( ph -> M e. U. ran measures )
9	1, 2, 3, 4, 5, 6, 7, 8	sitgval 30394	. . . . . . . 8 |- ( 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	9	dmeqd 5326	. . . . . . 7 |- ( ph -> dom ( Wsitg M ) = dom ( 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 ) ) ) ) )
11		eqid 2622	. . . . . . . 8 |- ( 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 ) ) ) ) = ( 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 ) ) ) )
12	11	dmmpt 5630	. . . . . . 7 |- dom ( 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 ) ) ) ) = { 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 ) ) ) e. _V }
13	10, 12	syl6eq 2672	. . . . . 6 |- ( ph -> dom ( 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 ) ) ) e. _V } )
14	13	eleq2d 2687	. . . . 5 |- ( ph -> ( F e. dom ( Wsitg M ) <-> F e. { 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 ) ) ) e. _V } ) )
15		rneq 5351	. . . . . . . . . 10 |- ( f = F -> ran f = ran F )
16	15	difeq1d 3727	. . . . . . . . 9 |- ( f = F -> ( ran f \ { .0. } ) = ( ran F \ { .0. } ) )
17		cnveq 5296	. . . . . . . . . . . . 13 |- ( f = F -> `' f = `' F )
18	17	imaeq1d 5465	. . . . . . . . . . . 12 |- ( f = F -> ( `' f " { x } ) = ( `' F " { x } ) )
19	18	fveq2d 6195	. . . . . . . . . . 11 |- ( f = F -> ( M ` ( `' f " { x } ) ) = ( M ` ( `' F " { x } ) ) )
20	19	fveq2d 6195	. . . . . . . . . 10 |- ( f = F -> ( H ` ( M ` ( `' f " { x } ) ) ) = ( H ` ( M ` ( `' F " { x } ) ) ) )
21	20	oveq1d 6665	. . . . . . . . 9 |- ( f = F -> ( ( H ` ( M ` ( `' f " { x } ) ) ) .x. x ) = ( ( H ` ( M ` ( `' F " { x } ) ) ) .x. x ) )
22	16, 21	mpteq12dv 4733	. . . . . . . 8 |- ( f = F -> ( x e. ( ran f \ { .0. } ) |-> ( ( H ` ( M ` ( `' f " { x } ) ) ) .x. x ) ) = ( x e. ( ran F \ { .0. } ) |-> ( ( H ` ( M ` ( `' F " { x } ) ) ) .x. x ) ) )
23	22	oveq2d 6666	. . . . . . 7 |- ( 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 ) ) ) )
24	23	eleq1d 2686	. . . . . 6 |- ( f = F -> ( ( W gsumg ( x e. ( ran f \ { .0. } ) |-> ( ( H ` ( M ` ( `' f " { x } ) ) ) .x. x ) ) ) e. _V <-> ( W gsumg ( x e. ( ran F \ { .0. } ) |-> ( ( H ` ( M ` ( `' F " { x } ) ) ) .x. x ) ) ) e. _V ) )
25	24	elrab 3363	. . . . 5 |- ( F e. { 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 ) ) ) e. _V } <-> ( 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 ) ) ) e. _V ) )
26	14, 25	syl6bb 276	. . . 4 |- ( ph -> ( F e. dom ( 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 ) ) ) e. _V ) ) )
27		ovex 6678	. . . . 5 |- ( W gsumg ( x e. ( ran F \ { .0. } ) |-> ( ( H ` ( M ` ( `' F " { x } ) ) ) .x. x ) ) ) e. _V
28	27	biantru 526	. . . 4 |- ( 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. { 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 ) ) ) e. _V ) )
29	26, 28	syl6bbr 278	. . 3 |- ( ph -> ( F e. dom ( 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 ) ) } ) )
30		rneq 5351	. . . . . 6 |- ( g = F -> ran g = ran F )
31	30	eleq1d 2686	. . . . 5 |- ( g = F -> ( ran g e. Fin <-> ran F e. Fin ) )
32	30	difeq1d 3727	. . . . . 6 |- ( g = F -> ( ran g \ { .0. } ) = ( ran F \ { .0. } ) )
33		cnveq 5296	. . . . . . . . 9 |- ( g = F -> `' g = `' F )
34	33	imaeq1d 5465	. . . . . . . 8 |- ( g = F -> ( `' g " { x } ) = ( `' F " { x } ) )
35	34	fveq2d 6195	. . . . . . 7 |- ( g = F -> ( M ` ( `' g " { x } ) ) = ( M ` ( `' F " { x } ) ) )
36	35	eleq1d 2686	. . . . . 6 |- ( g = F -> ( ( M ` ( `' g " { x } ) ) e. ( 0 [,) +oo ) <-> ( M ` ( `' F " { x } ) ) e. ( 0 [,) +oo ) ) )
37	32, 36	raleqbidv 3152	. . . . 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 ) ) )
38	31, 37	anbi12d 747	. . . 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 ) ) ) )
39	38	elrab 3363	. . 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 ) ) ) )
40	29, 39	syl6bb 276	. 2 |- ( ph -> ( F e. dom ( Wsitg M ) <-> ( F e. ( dom MMblFnM S ) /\ ( ran F e. Fin /\ A. x e. ( ran F \ { .0. } ) ( M ` ( `' F " { x } ) ) e. ( 0 [,) +oo ) ) ) ) )
41		3anass 1042	. 2 |- ( ( F e. ( dom MMblFnM S ) /\ ran F e. Fin /\ A. x e. ( ran F \ { .0. } ) ( M ` ( `' F " { 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 ) ) ) )
42	40, 41	syl6bbr 278	1 |- ( ph -> ( F e. dom ( Wsitg M ) <-> ( F e. ( dom MMblFnM S ) /\ ran F e. Fin /\ A. x e. ( ran F \ { .0. } ) ( M ` ( `' F " { x } ) ) e. ( 0 [,) +oo ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   {crab 2916   _Vcvv 3200   \ cdif 3571   {csn 4177   U.cuni 4436   |-> cmpt 4729   `'ccnv 5113   dom cdm 5114   ran crn 5115   "cima 5117   ` cfv 5888  (class class class)co 6650   Fincfn 7955   0cc0 9936   +oocpnf 10071   [,)cico 12177   Basecbs 15857  Scalarcsca 15944   .scvsca 15945   TopOpenctopn 16082   0gc0g 16100   gsum_g cgsu 16101  RRHomcrrh 30037  sigaGencsigagen 30201  measurescmeas 30258  MblFnMcmbfm 30312  sitgcsitg 30391

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-sitg 30392

This theorem is referenced by:  sibf0  30396  sibfmbl  30397  sibfrn  30399  sibfima  30400  sibfof  30402