Theorem sitmval 30411

Description: Value of the simple function integral metric for a given space W and measure M. (Contributed by Thierry Arnoux, 30-Jan-2018.)

Hypotheses

Ref	Expression
sitmval.d	|- D = ( dist ` W )
sitmval.1	|- ( ph -> W e. V )
sitmval.2	|- ( ph -> M e. U. ran measures )

Assertion

Ref	Expression
sitmval	|- ( ph -> ( Wsitm M ) = ( f e. dom ( Wsitg M ) , g e. dom ( Wsitg M ) |-> ( ( ( RR*s ↾s ( 0 [,] +oo ) )sitg M ) ` ( f oF D g ) ) ) )

Distinct variable groups:   f, g, M   f, W, g

Allowed substitution hints:   ph( f, g)   D( f, g)   V( f, g)

Proof of Theorem sitmval

Dummy variables w m are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		sitmval.1	. . 3 |- ( ph -> W e. V )
2		elex 3212	. . 3 |- ( W e. V -> W e. _V )
3	1, 2	syl 17	. 2 |- ( ph -> W e. _V )
4		sitmval.2	. 2 |- ( ph -> M e. U. ran measures )
5		oveq1 6657	. . . . 5 |- ( w = W -> ( wsitg m ) = ( Wsitg m ) )
6	5	dmeqd 5326	. . . 4 |- ( w = W -> dom ( wsitg m ) = dom ( Wsitg m ) )
7		fveq2 6191	. . . . . . 7 |- ( w = W -> ( dist ` w ) = ( dist ` W ) )
8		ofeq 6899	. . . . . . 7 |- ( ( dist ` w ) = ( dist ` W ) -> oF ( dist ` w ) = oF ( dist ` W ) )
9	7, 8	syl 17	. . . . . 6 |- ( w = W -> oF ( dist ` w ) = oF ( dist ` W ) )
10	9	oveqd 6667	. . . . 5 |- ( w = W -> ( f oF ( dist ` w ) g ) = ( f oF ( dist ` W ) g ) )
11	10	fveq2d 6195	. . . 4 |- ( w = W -> ( ( ( RR*s ↾s ( 0 [,] +oo ) )sitg m ) ` ( f oF ( dist ` w ) g ) ) = ( ( ( RR*s ↾s ( 0 [,] +oo ) )sitg m ) ` ( f oF ( dist ` W ) g ) ) )
12	6, 6, 11	mpt2eq123dv 6717	. . 3 |- ( w = W -> ( f e. dom ( wsitg m ) , g e. dom ( wsitg m ) |-> ( ( ( RR*s ↾s ( 0 [,] +oo ) )sitg m ) ` ( f oF ( dist ` w ) g ) ) ) = ( f e. dom ( Wsitg m ) , g e. dom ( Wsitg m ) |-> ( ( ( RR*s ↾s ( 0 [,] +oo ) )sitg m ) ` ( f oF ( dist ` W ) g ) ) ) )
13		oveq2 6658	. . . . 5 |- ( m = M -> ( Wsitg m ) = ( Wsitg M ) )
14	13	dmeqd 5326	. . . 4 |- ( m = M -> dom ( Wsitg m ) = dom ( Wsitg M ) )
15		oveq2 6658	. . . . 5 |- ( m = M -> ( ( RR*s ↾s ( 0 [,] +oo ) )sitg m ) = ( ( RR*s ↾s ( 0 [,] +oo ) )sitg M ) )
16		sitmval.d	. . . . . . . 8 |- D = ( dist ` W )
17	16	eqcomi 2631	. . . . . . 7 |- ( dist ` W ) = D
18		ofeq 6899	. . . . . . 7 |- ( ( dist ` W ) = D -> oF ( dist ` W ) = oF D )
19	17, 18	mp1i 13	. . . . . 6 |- ( m = M -> oF ( dist ` W ) = oF D )
20	19	oveqd 6667	. . . . 5 |- ( m = M -> ( f oF ( dist ` W ) g ) = ( f oF D g ) )
21	15, 20	fveq12d 6197	. . . 4 |- ( m = M -> ( ( ( RR*s ↾s ( 0 [,] +oo ) )sitg m ) ` ( f oF ( dist ` W ) g ) ) = ( ( ( RR*s ↾s ( 0 [,] +oo ) )sitg M ) ` ( f oF D g ) ) )
22	14, 14, 21	mpt2eq123dv 6717	. . 3 |- ( m = M -> ( f e. dom ( Wsitg m ) , g e. dom ( Wsitg m ) |-> ( ( ( RR*s ↾s ( 0 [,] +oo ) )sitg m ) ` ( f oF ( dist ` W ) g ) ) ) = ( f e. dom ( Wsitg M ) , g e. dom ( Wsitg M ) |-> ( ( ( RR*s ↾s ( 0 [,] +oo ) )sitg M ) ` ( f oF D g ) ) ) )
23		df-sitm 30393	. . 3 |- sitm = ( w e. _V , m e. U. ran measures |-> ( f e. dom ( wsitg m ) , g e. dom ( wsitg m ) |-> ( ( ( RR*s ↾s ( 0 [,] +oo ) )sitg m ) ` ( f oF ( dist ` w ) g ) ) ) )
24		ovex 6678	. . . . 5 |- ( Wsitg M ) e. _V
25	24	dmex 7099	. . . 4 |- dom ( Wsitg M ) e. _V
26	25, 25	mpt2ex 7247	. . 3 |- ( f e. dom ( Wsitg M ) , g e. dom ( Wsitg M ) |-> ( ( ( RR*s ↾s ( 0 [,] +oo ) )sitg M ) ` ( f oF D g ) ) ) e. _V
27	12, 22, 23, 26	ovmpt2 6796	. 2 |- ( ( W e. _V /\ M e. U. ran measures ) -> ( Wsitm M ) = ( f e. dom ( Wsitg M ) , g e. dom ( Wsitg M ) |-> ( ( ( RR*s ↾s ( 0 [,] +oo ) )sitg M ) ` ( f oF D g ) ) ) )
28	3, 4, 27	syl2anc 693	1 |- ( ph -> ( Wsitm M ) = ( f e. dom ( Wsitg M ) , g e. dom ( Wsitg M ) |-> ( ( ( RR*s ↾s ( 0 [,] +oo ) )sitg M ) ` ( f oF D g ) ) ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   _Vcvv 3200   U.cuni 4436   dom cdm 5114   ran crn 5115   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   oFcof 6895   0cc0 9936   +oocpnf 10071   [,]cicc 12178   ↾_s cress 15858   distcds 15950   RR*scxrs 16160  measurescmeas 30258  sitmcsitm 30390  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-8 1992  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-pow 4843  ax-pr 4906  ax-un 6949

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-pw 4160  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-of 6897  df-1st 7168  df-2nd 7169  df-sitm 30393

This theorem is referenced by:  sitmfval  30412  sitmf  30414