Theorem sitmfval 30412

Description: Value of the integral distance between two simple functions. (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 )
sitmfval.1	|- ( ph -> F e. dom ( Wsitg M ) )
sitmfval.2	|- ( ph -> G e. dom ( Wsitg M ) )

Assertion

Ref	Expression
sitmfval	|- ( ph -> ( F ( Wsitm M ) G ) = ( ( ( RR*s ↾s ( 0 [,] +oo ) )sitg M ) ` ( F oF D G ) ) )

Proof of Theorem sitmfval

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

Step	Hyp	Ref	Expression
1		sitmval.d	. . 3 |- D = ( dist ` W )
2		sitmval.1	. . 3 |- ( ph -> W e. V )
3		sitmval.2	. . 3 |- ( ph -> M e. U. ran measures )
4	1, 2, 3	sitmval 30411	. 2 |- ( ph -> ( Wsitm M ) = ( f e. dom ( Wsitg M ) , g e. dom ( Wsitg M ) |-> ( ( ( RR*s ↾s ( 0 [,] +oo ) )sitg M ) ` ( f oF D g ) ) ) )
5		simprl 794	. . . 4 |- ( ( ph /\ ( f = F /\ g = G ) ) -> f = F )
6		simprr 796	. . . 4 |- ( ( ph /\ ( f = F /\ g = G ) ) -> g = G )
7	5, 6	oveq12d 6668	. . 3 |- ( ( ph /\ ( f = F /\ g = G ) ) -> ( f oF D g ) = ( F oF D G ) )
8	7	fveq2d 6195	. 2 |- ( ( ph /\ ( f = F /\ g = G ) ) -> ( ( ( RR*s ↾s ( 0 [,] +oo ) )sitg M ) ` ( f oF D g ) ) = ( ( ( RR*s ↾s ( 0 [,] +oo ) )sitg M ) ` ( F oF D G ) ) )
9		sitmfval.1	. 2 |- ( ph -> F e. dom ( Wsitg M ) )
10		sitmfval.2	. 2 |- ( ph -> G e. dom ( Wsitg M ) )
11		fvexd 6203	. 2 |- ( ph -> ( ( ( RR*s ↾s ( 0 [,] +oo ) )sitg M ) ` ( F oF D G ) ) e. _V )
12	4, 8, 9, 10, 11	ovmpt2d 6788	1 |- ( ph -> ( F ( Wsitm M ) G ) = ( ( ( RR*s ↾s ( 0 [,] +oo ) )sitg M ) ` ( F oF D G ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   U.cuni 4436   dom cdm 5114   ran crn 5115   ` cfv 5888  (class class class)co 6650   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:  sitmcl  30413  sitmf  30414