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Theorem itg1val2 19578
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 19577 . . 3  |-  ( F  e.  dom  S.1  ->  ( S.1 `  F )  =  sum_ x  e.  ( ran  F  \  {
0 } ) ( x  x.  ( vol `  ( `' F " { x } ) ) ) )
21adantr 453 . 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 965 . . 3  |-  ( ( F  e.  dom  S.1  /\  ( A  e.  Fin  /\  ( ran  F  \  { 0 } ) 
C_  A  /\  A  C_  ( RR  \  {
0 } ) ) )  ->  ( ran  F 
\  { 0 } )  C_  A )
43sselda 3350 . . . 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 966 . . . . . . . 8  |-  ( ( F  e.  dom  S.1  /\  ( A  e.  Fin  /\  ( ran  F  \  { 0 } ) 
C_  A  /\  A  C_  ( RR  \  {
0 } ) ) )  ->  A  C_  ( RR  \  { 0 } ) )
65sselda 3350 . . . . . . 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 3471 . . . . . . 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 19574 . . . . . . . 8  |-  ( ( F  e.  dom  S.1  /\  x  e.  ( RR 
\  { 0 } ) )  ->  ( vol `  ( `' F " { x } ) )  e.  RR )
109adantlr 697 . . . . . . 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 458 . . . . . 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 9118 . . . . 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 9116 . . . 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 458 . . 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 19570 . . . . . . . . . 10  |-  ( F  e.  dom  S.1  ->  F : RR --> RR )
1615ad2antrr 708 . . . . . . . . 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 5593 . . . . . . . . . 10  |-  ( F : RR --> RR  ->  F  Fn  RR )
18 dffn3 5600 . . . . . . . . . 10  |-  ( F  Fn  RR  <->  F : RR
--> ran  F )
1917, 18sylib 190 . . . . . . . . 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 3472 . . . . . . . . . . 11  |-  ( x  e.  ( A  \ 
( ran  F  \  {
0 } ) )  ->  -.  x  e.  ( ran  F  \  {
0 } ) )
2221adantl 454 . . . . . . . . . 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 1002 . . . . . . . . . . . . . . 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 3487 . . . . . . . . . . . . . 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 449 . . . . . . . . . . . . . 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 3351 . . . . . . . . . . . . 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 3472 . . . . . . . . . . . . 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 495 . . . . . . . . . . 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 3332 . . . . . . . . . . 11  |-  ( x  e.  ( ran  F  \  { 0 } )  <-> 
( x  e.  ran  F  /\  -.  x  e. 
{ 0 } ) )
3129, 30syl6rbbr 257 . . . . . . . . . 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 293 . . . . . . . . 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 3870 . . . . . . . . 9  |-  ( ( ran  F  i^i  {
x } )  =  (/) 
<->  -.  x  e.  ran  F )
3432, 33sylibr 205 . . . . . . . 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 5623 . . . . . . . 8  |-  ( ( F : RR --> ran  F  /\  ( ran  F  i^i  { x } )  =  (/) )  ->  ( `' F " { x } )  =  (/) )
3620, 34, 35syl2anc 644 . . . . . . 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 5734 . . . . . 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 19436 . . . . . . . 8  |-  (/)  e.  dom  vol
39 mblvol 19428 . . . . . . . 8  |-  ( (/)  e.  dom  vol  ->  ( vol `  (/) )  =  ( vol * `  (/) ) )
4038, 39ax-mp 8 . . . . . . 7  |-  ( vol `  (/) )  =  ( vol * `  (/) )
41 ovol0 19391 . . . . . . 7  |-  ( vol
* `  (/) )  =  0
4240, 41eqtri 2458 . . . . . 6  |-  ( vol `  (/) )  =  0
4337, 42syl6eq 2486 . . . . 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 6099 . . . 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 3471 . . . . . . 7  |-  ( x  e.  ( A  \ 
( ran  F  \  {
0 } ) )  ->  x  e.  A
)
4645, 8sylan2 462 . . . . . 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 9116 . . . . 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 9267 . . . 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 2470 . . 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 964 . . 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 12521 . 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 2470 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 set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    \ cdif 3319    i^i cin 3321    C_ wss 3322   (/)c0 3630   {csn 3816   `'ccnv 4879   dom cdm 4880   ran crn 4881   "cima 4883    Fn wfn 5451   -->wf 5452   ` cfv 5456  (class class class)co 6083   Fincfn 7111   CCcc 8990   RRcr 8991   0cc0 8992    x. cmul 8997   sum_csu 12481   vol *covol 19361   volcvol 19362   S.1citg1 19509
This theorem is referenced by:  itg1addlem4  19593  itg1climres  19608
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-inf2 7598  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069  ax-pre-sup 9070
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-se 4544  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-isom 5465  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-of 6307  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-1o 6726  df-2o 6727  df-oadd 6730  df-er 6907  df-map 7022  df-pm 7023  df-en 7112  df-dom 7113  df-sdom 7114  df-fin 7115  df-sup 7448  df-oi 7481  df-card 7828  df-cda 8050  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-div 9680  df-nn 10003  df-2 10060  df-3 10061  df-n0 10224  df-z 10285  df-uz 10491  df-q 10577  df-rp 10615  df-xadd 10713  df-ioo 10922  df-ico 10924  df-icc 10925  df-fz 11046  df-fzo 11138  df-fl 11204  df-seq 11326  df-exp 11385  df-hash 11621  df-cj 11906  df-re 11907  df-im 11908  df-sqr 12042  df-abs 12043  df-clim 12284  df-sum 12482  df-xmet 16697  df-met 16698  df-ovol 19363  df-vol 19364  df-mbf 19514  df-itg1 19515
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