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Theorem itg1val2 21842
Description: The value of the integral on simple functions. (Contributed by Mario Carneiro, 26-Jun-2014.)
Assertion
Ref Expression
itg1val2  |-  ( ( F  e.  dom  S.1  /\  ( A  e.  Fin  /\  ( ran  F  \  { 0 } ) 
C_  A  /\  A  C_  ( RR  \  {
0 } ) ) )  ->  ( S.1 `  F )  =  sum_ x  e.  A  ( x  x.  ( vol `  ( `' F " { x } ) ) ) )
Distinct variable groups:    x, F    x, A

Proof of Theorem itg1val2
StepHypRef Expression
1 itg1val 21841 . . 3  |-  ( F  e.  dom  S.1  ->  ( S.1 `  F )  =  sum_ x  e.  ( ran  F  \  {
0 } ) ( x  x.  ( vol `  ( `' F " { x } ) ) ) )
21adantr 465 . 2  |-  ( ( F  e.  dom  S.1  /\  ( A  e.  Fin  /\  ( ran  F  \  { 0 } ) 
C_  A  /\  A  C_  ( RR  \  {
0 } ) ) )  ->  ( S.1 `  F )  =  sum_ x  e.  ( ran  F  \  { 0 } ) ( x  x.  ( vol `  ( `' F " { x } ) ) ) )
3 simpr2 1003 . . 3  |-  ( ( F  e.  dom  S.1  /\  ( A  e.  Fin  /\  ( ran  F  \  { 0 } ) 
C_  A  /\  A  C_  ( RR  \  {
0 } ) ) )  ->  ( ran  F 
\  { 0 } )  C_  A )
43sselda 3504 . . . 4  |-  ( ( ( F  e.  dom  S.1 
/\  ( A  e. 
Fin  /\  ( ran  F 
\  { 0 } )  C_  A  /\  A  C_  ( RR  \  { 0 } ) ) )  /\  x  e.  ( ran  F  \  { 0 } ) )  ->  x  e.  A )
5 simpr3 1004 . . . . . . . 8  |-  ( ( F  e.  dom  S.1  /\  ( A  e.  Fin  /\  ( ran  F  \  { 0 } ) 
C_  A  /\  A  C_  ( RR  \  {
0 } ) ) )  ->  A  C_  ( RR  \  { 0 } ) )
65sselda 3504 . . . . . . 7  |-  ( ( ( F  e.  dom  S.1 
/\  ( A  e. 
Fin  /\  ( ran  F 
\  { 0 } )  C_  A  /\  A  C_  ( RR  \  { 0 } ) ) )  /\  x  e.  A )  ->  x  e.  ( RR  \  {
0 } ) )
7 eldifi 3626 . . . . . . 7  |-  ( x  e.  ( RR  \  { 0 } )  ->  x  e.  RR )
86, 7syl 16 . . . . . 6  |-  ( ( ( F  e.  dom  S.1 
/\  ( A  e. 
Fin  /\  ( ran  F 
\  { 0 } )  C_  A  /\  A  C_  ( RR  \  { 0 } ) ) )  /\  x  e.  A )  ->  x  e.  RR )
9 i1fima2sn 21838 . . . . . . . 8  |-  ( ( F  e.  dom  S.1  /\  x  e.  ( RR 
\  { 0 } ) )  ->  ( vol `  ( `' F " { x } ) )  e.  RR )
109adantlr 714 . . . . . . 7  |-  ( ( ( F  e.  dom  S.1 
/\  ( A  e. 
Fin  /\  ( ran  F 
\  { 0 } )  C_  A  /\  A  C_  ( RR  \  { 0 } ) ) )  /\  x  e.  ( RR  \  {
0 } ) )  ->  ( vol `  ( `' F " { x } ) )  e.  RR )
116, 10syldan 470 . . . . . 6  |-  ( ( ( F  e.  dom  S.1 
/\  ( A  e. 
Fin  /\  ( ran  F 
\  { 0 } )  C_  A  /\  A  C_  ( RR  \  { 0 } ) ) )  /\  x  e.  A )  ->  ( vol `  ( `' F " { x } ) )  e.  RR )
128, 11remulcld 9623 . . . . 5  |-  ( ( ( F  e.  dom  S.1 
/\  ( A  e. 
Fin  /\  ( ran  F 
\  { 0 } )  C_  A  /\  A  C_  ( RR  \  { 0 } ) ) )  /\  x  e.  A )  ->  (
x  x.  ( vol `  ( `' F " { x } ) ) )  e.  RR )
1312recnd 9621 . . . 4  |-  ( ( ( F  e.  dom  S.1 
/\  ( A  e. 
Fin  /\  ( ran  F 
\  { 0 } )  C_  A  /\  A  C_  ( RR  \  { 0 } ) ) )  /\  x  e.  A )  ->  (
x  x.  ( vol `  ( `' F " { x } ) ) )  e.  CC )
144, 13syldan 470 . . 3  |-  ( ( ( F  e.  dom  S.1 
/\  ( A  e. 
Fin  /\  ( ran  F 
\  { 0 } )  C_  A  /\  A  C_  ( RR  \  { 0 } ) ) )  /\  x  e.  ( ran  F  \  { 0 } ) )  ->  ( x  x.  ( vol `  ( `' F " { x } ) ) )  e.  CC )
15 i1ff 21834 . . . . . . . . . 10  |-  ( F  e.  dom  S.1  ->  F : RR --> RR )
1615ad2antrr 725 . . . . . . . . 9  |-  ( ( ( F  e.  dom  S.1 
/\  ( A  e. 
Fin  /\  ( ran  F 
\  { 0 } )  C_  A  /\  A  C_  ( RR  \  { 0 } ) ) )  /\  x  e.  ( A  \  ( ran  F  \  { 0 } ) ) )  ->  F : RR --> RR )
17 ffn 5730 . . . . . . . . . 10  |-  ( F : RR --> RR  ->  F  Fn  RR )
18 dffn3 5737 . . . . . . . . . 10  |-  ( F  Fn  RR  <->  F : RR
--> ran  F )
1917, 18sylib 196 . . . . . . . . 9  |-  ( F : RR --> RR  ->  F : RR --> ran  F
)
2016, 19syl 16 . . . . . . . 8  |-  ( ( ( F  e.  dom  S.1 
/\  ( A  e. 
Fin  /\  ( ran  F 
\  { 0 } )  C_  A  /\  A  C_  ( RR  \  { 0 } ) ) )  /\  x  e.  ( A  \  ( ran  F  \  { 0 } ) ) )  ->  F : RR --> ran  F )
21 eldifn 3627 . . . . . . . . . . 11  |-  ( x  e.  ( A  \ 
( ran  F  \  {
0 } ) )  ->  -.  x  e.  ( ran  F  \  {
0 } ) )
2221adantl 466 . . . . . . . . . 10  |-  ( ( ( F  e.  dom  S.1 
/\  ( A  e. 
Fin  /\  ( ran  F 
\  { 0 } )  C_  A  /\  A  C_  ( RR  \  { 0 } ) ) )  /\  x  e.  ( A  \  ( ran  F  \  { 0 } ) ) )  ->  -.  x  e.  ( ran  F  \  {
0 } ) )
23 simplr3 1040 . . . . . . . . . . . . . . 15  |-  ( ( ( F  e.  dom  S.1 
/\  ( A  e. 
Fin  /\  ( ran  F 
\  { 0 } )  C_  A  /\  A  C_  ( RR  \  { 0 } ) ) )  /\  x  e.  ( A  \  ( ran  F  \  { 0 } ) ) )  ->  A  C_  ( RR  \  { 0 } ) )
2423ssdifssd 3642 . . . . . . . . . . . . . 14  |-  ( ( ( F  e.  dom  S.1 
/\  ( A  e. 
Fin  /\  ( ran  F 
\  { 0 } )  C_  A  /\  A  C_  ( RR  \  { 0 } ) ) )  /\  x  e.  ( A  \  ( ran  F  \  { 0 } ) ) )  ->  ( A  \ 
( ran  F  \  {
0 } ) ) 
C_  ( RR  \  { 0 } ) )
25 simpr 461 . . . . . . . . . . . . . 14  |-  ( ( ( F  e.  dom  S.1 
/\  ( A  e. 
Fin  /\  ( ran  F 
\  { 0 } )  C_  A  /\  A  C_  ( RR  \  { 0 } ) ) )  /\  x  e.  ( A  \  ( ran  F  \  { 0 } ) ) )  ->  x  e.  ( A  \  ( ran 
F  \  { 0 } ) ) )
2624, 25sseldd 3505 . . . . . . . . . . . . 13  |-  ( ( ( F  e.  dom  S.1 
/\  ( A  e. 
Fin  /\  ( ran  F 
\  { 0 } )  C_  A  /\  A  C_  ( RR  \  { 0 } ) ) )  /\  x  e.  ( A  \  ( ran  F  \  { 0 } ) ) )  ->  x  e.  ( RR  \  { 0 } ) )
27 eldifn 3627 . . . . . . . . . . . . 13  |-  ( x  e.  ( RR  \  { 0 } )  ->  -.  x  e.  { 0 } )
2826, 27syl 16 . . . . . . . . . . . 12  |-  ( ( ( F  e.  dom  S.1 
/\  ( A  e. 
Fin  /\  ( ran  F 
\  { 0 } )  C_  A  /\  A  C_  ( RR  \  { 0 } ) ) )  /\  x  e.  ( A  \  ( ran  F  \  { 0 } ) ) )  ->  -.  x  e.  { 0 } )
2928biantrud 507 . . . . . . . . . . 11  |-  ( ( ( F  e.  dom  S.1 
/\  ( A  e. 
Fin  /\  ( ran  F 
\  { 0 } )  C_  A  /\  A  C_  ( RR  \  { 0 } ) ) )  /\  x  e.  ( A  \  ( ran  F  \  { 0 } ) ) )  ->  ( x  e. 
ran  F  <->  ( x  e. 
ran  F  /\  -.  x  e.  { 0 } ) ) )
30 eldif 3486 . . . . . . . . . . 11  |-  ( x  e.  ( ran  F  \  { 0 } )  <-> 
( x  e.  ran  F  /\  -.  x  e. 
{ 0 } ) )
3129, 30syl6rbbr 264 . . . . . . . . . 10  |-  ( ( ( F  e.  dom  S.1 
/\  ( A  e. 
Fin  /\  ( ran  F 
\  { 0 } )  C_  A  /\  A  C_  ( RR  \  { 0 } ) ) )  /\  x  e.  ( A  \  ( ran  F  \  { 0 } ) ) )  ->  ( x  e.  ( ran  F  \  { 0 } )  <-> 
x  e.  ran  F
) )
3222, 31mtbid 300 . . . . . . . . 9  |-  ( ( ( F  e.  dom  S.1 
/\  ( A  e. 
Fin  /\  ( ran  F 
\  { 0 } )  C_  A  /\  A  C_  ( RR  \  { 0 } ) ) )  /\  x  e.  ( A  \  ( ran  F  \  { 0 } ) ) )  ->  -.  x  e.  ran  F )
33 disjsn 4088 . . . . . . . . 9  |-  ( ( ran  F  i^i  {
x } )  =  (/) 
<->  -.  x  e.  ran  F )
3432, 33sylibr 212 . . . . . . . 8  |-  ( ( ( F  e.  dom  S.1 
/\  ( A  e. 
Fin  /\  ( ran  F 
\  { 0 } )  C_  A  /\  A  C_  ( RR  \  { 0 } ) ) )  /\  x  e.  ( A  \  ( ran  F  \  { 0 } ) ) )  ->  ( ran  F  i^i  { x } )  =  (/) )
35 fimacnvdisj 5762 . . . . . . . 8  |-  ( ( F : RR --> ran  F  /\  ( ran  F  i^i  { x } )  =  (/) )  ->  ( `' F " { x } )  =  (/) )
3620, 34, 35syl2anc 661 . . . . . . 7  |-  ( ( ( F  e.  dom  S.1 
/\  ( A  e. 
Fin  /\  ( ran  F 
\  { 0 } )  C_  A  /\  A  C_  ( RR  \  { 0 } ) ) )  /\  x  e.  ( A  \  ( ran  F  \  { 0 } ) ) )  ->  ( `' F " { x } )  =  (/) )
3736fveq2d 5869 . . . . . 6  |-  ( ( ( F  e.  dom  S.1 
/\  ( A  e. 
Fin  /\  ( ran  F 
\  { 0 } )  C_  A  /\  A  C_  ( RR  \  { 0 } ) ) )  /\  x  e.  ( A  \  ( ran  F  \  { 0 } ) ) )  ->  ( vol `  ( `' F " { x } ) )  =  ( vol `  (/) ) )
38 0mbl 21701 . . . . . . . 8  |-  (/)  e.  dom  vol
39 mblvol 21692 . . . . . . . 8  |-  ( (/)  e.  dom  vol  ->  ( vol `  (/) )  =  ( vol* `  (/) ) )
4038, 39ax-mp 5 . . . . . . 7  |-  ( vol `  (/) )  =  ( vol* `  (/) )
41 ovol0 21655 . . . . . . 7  |-  ( vol* `  (/) )  =  0
4240, 41eqtri 2496 . . . . . 6  |-  ( vol `  (/) )  =  0
4337, 42syl6eq 2524 . . . . 5  |-  ( ( ( F  e.  dom  S.1 
/\  ( A  e. 
Fin  /\  ( ran  F 
\  { 0 } )  C_  A  /\  A  C_  ( RR  \  { 0 } ) ) )  /\  x  e.  ( A  \  ( ran  F  \  { 0 } ) ) )  ->  ( vol `  ( `' F " { x } ) )  =  0 )
4443oveq2d 6299 . . . 4  |-  ( ( ( F  e.  dom  S.1 
/\  ( A  e. 
Fin  /\  ( ran  F 
\  { 0 } )  C_  A  /\  A  C_  ( RR  \  { 0 } ) ) )  /\  x  e.  ( A  \  ( ran  F  \  { 0 } ) ) )  ->  ( x  x.  ( vol `  ( `' F " { x } ) ) )  =  ( x  x.  0 ) )
45 eldifi 3626 . . . . . . 7  |-  ( x  e.  ( A  \ 
( ran  F  \  {
0 } ) )  ->  x  e.  A
)
4645, 8sylan2 474 . . . . . 6  |-  ( ( ( F  e.  dom  S.1 
/\  ( A  e. 
Fin  /\  ( ran  F 
\  { 0 } )  C_  A  /\  A  C_  ( RR  \  { 0 } ) ) )  /\  x  e.  ( A  \  ( ran  F  \  { 0 } ) ) )  ->  x  e.  RR )
4746recnd 9621 . . . . 5  |-  ( ( ( F  e.  dom  S.1 
/\  ( A  e. 
Fin  /\  ( ran  F 
\  { 0 } )  C_  A  /\  A  C_  ( RR  \  { 0 } ) ) )  /\  x  e.  ( A  \  ( ran  F  \  { 0 } ) ) )  ->  x  e.  CC )
4847mul01d 9777 . . . 4  |-  ( ( ( F  e.  dom  S.1 
/\  ( A  e. 
Fin  /\  ( ran  F 
\  { 0 } )  C_  A  /\  A  C_  ( RR  \  { 0 } ) ) )  /\  x  e.  ( A  \  ( ran  F  \  { 0 } ) ) )  ->  ( x  x.  0 )  =  0 )
4944, 48eqtrd 2508 . . 3  |-  ( ( ( F  e.  dom  S.1 
/\  ( A  e. 
Fin  /\  ( ran  F 
\  { 0 } )  C_  A  /\  A  C_  ( RR  \  { 0 } ) ) )  /\  x  e.  ( A  \  ( ran  F  \  { 0 } ) ) )  ->  ( x  x.  ( vol `  ( `' F " { x } ) ) )  =  0 )
50 simpr1 1002 . . 3  |-  ( ( F  e.  dom  S.1  /\  ( A  e.  Fin  /\  ( ran  F  \  { 0 } ) 
C_  A  /\  A  C_  ( RR  \  {
0 } ) ) )  ->  A  e.  Fin )
513, 14, 49, 50fsumss 13509 . 2  |-  ( ( F  e.  dom  S.1  /\  ( A  e.  Fin  /\  ( ran  F  \  { 0 } ) 
C_  A  /\  A  C_  ( RR  \  {
0 } ) ) )  ->  sum_ x  e.  ( ran  F  \  { 0 } ) ( x  x.  ( vol `  ( `' F " { x } ) ) )  =  sum_ x  e.  A  ( x  x.  ( vol `  ( `' F " { x } ) ) ) )
522, 51eqtrd 2508 1  |-  ( ( F  e.  dom  S.1  /\  ( A  e.  Fin  /\  ( ran  F  \  { 0 } ) 
C_  A  /\  A  C_  ( RR  \  {
0 } ) ) )  ->  ( S.1 `  F )  =  sum_ x  e.  A  ( x  x.  ( vol `  ( `' F " { x } ) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    \ cdif 3473    i^i cin 3475    C_ wss 3476   (/)c0 3785   {csn 4027   `'ccnv 4998   dom cdm 4999   ran crn 5000   "cima 5002    Fn wfn 5582   -->wf 5583   ` cfv 5587  (class class class)co 6283   Fincfn 7516   CCcc 9489   RRcr 9490   0cc0 9491    x. cmul 9496   sum_csu 13470   vol*covol 21625   volcvol 21626   S.1citg1 21775
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-8 1769  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-pow 4625  ax-pr 4686  ax-un 6575  ax-inf2 8057  ax-cnex 9547  ax-resscn 9548  ax-1cn 9549  ax-icn 9550  ax-addcl 9551  ax-addrcl 9552  ax-mulcl 9553  ax-mulrcl 9554  ax-mulcom 9555  ax-addass 9556  ax-mulass 9557  ax-distr 9558  ax-i2m1 9559  ax-1ne0 9560  ax-1rid 9561  ax-rnegex 9562  ax-rrecex 9563  ax-cnre 9564  ax-pre-lttri 9565  ax-pre-lttrn 9566  ax-pre-ltadd 9567  ax-pre-mulgt0 9568  ax-pre-sup 9569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  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-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  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-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  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 5550  df-fun 5589  df-fn 5590  df-f 5591  df-f1 5592  df-fo 5593  df-f1o 5594  df-fv 5595  df-isom 5596  df-riota 6244  df-ov 6286  df-oprab 6287  df-mpt2 6288  df-of 6523  df-om 6680  df-1st 6784  df-2nd 6785  df-recs 7042  df-rdg 7076  df-1o 7130  df-2o 7131  df-oadd 7134  df-er 7311  df-map 7422  df-pm 7423  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-sup 7900  df-oi 7934  df-card 8319  df-cda 8547  df-pnf 9629  df-mnf 9630  df-xr 9631  df-ltxr 9632  df-le 9633  df-sub 9806  df-neg 9807  df-div 10206  df-nn 10536  df-2 10593  df-3 10594  df-n0 10795  df-z 10864  df-uz 11082  df-q 11182  df-rp 11220  df-xadd 11318  df-ioo 11532  df-ico 11534  df-icc 11535  df-fz 11672  df-fzo 11792  df-fl 11896  df-seq 12075  df-exp 12134  df-hash 12373  df-cj 12894  df-re 12895  df-im 12896  df-sqrt 13030  df-abs 13031  df-clim 13273  df-sum 13471  df-xmet 18199  df-met 18200  df-ovol 21627  df-vol 21628  df-mbf 21779  df-itg1 21780
This theorem is referenced by:  itg1addlem4  21857  itg1climres  21872
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