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Theorem indsum 27829
Description: Finite sum of a product with the indicator function / Cartesian product with the indicator function. (Contributed by Thierry Arnoux, 14-Aug-2017.)
Hypotheses
Ref Expression
indsum.1  |-  ( ph  ->  O  e.  Fin )
indsum.2  |-  ( ph  ->  A  C_  O )
indsum.3  |-  ( (
ph  /\  x  e.  O )  ->  B  e.  CC )
Assertion
Ref Expression
indsum  |-  ( ph  -> 
sum_ x  e.  O  ( ( ( (𝟭 `  O ) `  A
) `  x )  x.  B )  =  sum_ x  e.  A  B )
Distinct variable groups:    x, A    x, O    ph, x
Allowed substitution hint:    B( x)

Proof of Theorem indsum
StepHypRef Expression
1 indsum.2 . . 3  |-  ( ph  ->  A  C_  O )
21sselda 3509 . . . 4  |-  ( (
ph  /\  x  e.  A )  ->  x  e.  O )
3 pr01ssre 27824 . . . . . . 7  |-  { 0 ,  1 }  C_  RR
4 indsum.1 . . . . . . . . 9  |-  ( ph  ->  O  e.  Fin )
5 indf 27822 . . . . . . . . 9  |-  ( ( O  e.  Fin  /\  A  C_  O )  -> 
( (𝟭 `  O ) `  A ) : O --> { 0 ,  1 } )
64, 1, 5syl2anc 661 . . . . . . . 8  |-  ( ph  ->  ( (𝟭 `  O
) `  A ) : O --> { 0 ,  1 } )
76ffvelrnda 6031 . . . . . . 7  |-  ( (
ph  /\  x  e.  O )  ->  (
( (𝟭 `  O ) `  A ) `  x
)  e.  { 0 ,  1 } )
83, 7sseldi 3507 . . . . . 6  |-  ( (
ph  /\  x  e.  O )  ->  (
( (𝟭 `  O ) `  A ) `  x
)  e.  RR )
98recnd 9632 . . . . 5  |-  ( (
ph  /\  x  e.  O )  ->  (
( (𝟭 `  O ) `  A ) `  x
)  e.  CC )
10 indsum.3 . . . . 5  |-  ( (
ph  /\  x  e.  O )  ->  B  e.  CC )
119, 10mulcld 9626 . . . 4  |-  ( (
ph  /\  x  e.  O )  ->  (
( ( (𝟭 `  O
) `  A ) `  x )  x.  B
)  e.  CC )
122, 11syldan 470 . . 3  |-  ( (
ph  /\  x  e.  A )  ->  (
( ( (𝟭 `  O
) `  A ) `  x )  x.  B
)  e.  CC )
134adantr 465 . . . . . 6  |-  ( (
ph  /\  x  e.  ( O  \  A ) )  ->  O  e.  Fin )
141adantr 465 . . . . . 6  |-  ( (
ph  /\  x  e.  ( O  \  A ) )  ->  A  C_  O
)
15 simpr 461 . . . . . 6  |-  ( (
ph  /\  x  e.  ( O  \  A ) )  ->  x  e.  ( O  \  A ) )
16 ind0 27826 . . . . . 6  |-  ( ( O  e.  Fin  /\  A  C_  O  /\  x  e.  ( O  \  A
) )  ->  (
( (𝟭 `  O ) `  A ) `  x
)  =  0 )
1713, 14, 15, 16syl3anc 1228 . . . . 5  |-  ( (
ph  /\  x  e.  ( O  \  A ) )  ->  ( (
(𝟭 `  O ) `  A ) `  x
)  =  0 )
1817oveq1d 6309 . . . 4  |-  ( (
ph  /\  x  e.  ( O  \  A ) )  ->  ( (
( (𝟭 `  O ) `  A ) `  x
)  x.  B )  =  ( 0  x.  B ) )
19 difssd 3637 . . . . . 6  |-  ( ph  ->  ( O  \  A
)  C_  O )
2019sselda 3509 . . . . 5  |-  ( (
ph  /\  x  e.  ( O  \  A ) )  ->  x  e.  O )
2110mul02d 9787 . . . . 5  |-  ( (
ph  /\  x  e.  O )  ->  (
0  x.  B )  =  0 )
2220, 21syldan 470 . . . 4  |-  ( (
ph  /\  x  e.  ( O  \  A ) )  ->  ( 0  x.  B )  =  0 )
2318, 22eqtrd 2508 . . 3  |-  ( (
ph  /\  x  e.  ( O  \  A ) )  ->  ( (
( (𝟭 `  O ) `  A ) `  x
)  x.  B )  =  0 )
241, 12, 23, 4fsumss 13522 . 2  |-  ( ph  -> 
sum_ x  e.  A  ( ( ( (𝟭 `  O ) `  A
) `  x )  x.  B )  =  sum_ x  e.  O  ( ( ( (𝟭 `  O
) `  A ) `  x )  x.  B
) )
254adantr 465 . . . . . 6  |-  ( (
ph  /\  x  e.  A )  ->  O  e.  Fin )
261adantr 465 . . . . . 6  |-  ( (
ph  /\  x  e.  A )  ->  A  C_  O )
27 simpr 461 . . . . . 6  |-  ( (
ph  /\  x  e.  A )  ->  x  e.  A )
28 ind1 27825 . . . . . 6  |-  ( ( O  e.  Fin  /\  A  C_  O  /\  x  e.  A )  ->  (
( (𝟭 `  O ) `  A ) `  x
)  =  1 )
2925, 26, 27, 28syl3anc 1228 . . . . 5  |-  ( (
ph  /\  x  e.  A )  ->  (
( (𝟭 `  O ) `  A ) `  x
)  =  1 )
3029oveq1d 6309 . . . 4  |-  ( (
ph  /\  x  e.  A )  ->  (
( ( (𝟭 `  O
) `  A ) `  x )  x.  B
)  =  ( 1  x.  B ) )
3110mulid2d 9624 . . . . 5  |-  ( (
ph  /\  x  e.  O )  ->  (
1  x.  B )  =  B )
322, 31syldan 470 . . . 4  |-  ( (
ph  /\  x  e.  A )  ->  (
1  x.  B )  =  B )
3330, 32eqtrd 2508 . . 3  |-  ( (
ph  /\  x  e.  A )  ->  (
( ( (𝟭 `  O
) `  A ) `  x )  x.  B
)  =  B )
3433sumeq2dv 13500 . 2  |-  ( ph  -> 
sum_ x  e.  A  ( ( ( (𝟭 `  O ) `  A
) `  x )  x.  B )  =  sum_ x  e.  A  B )
3524, 34eqtr3d 2510 1  |-  ( ph  -> 
sum_ x  e.  O  ( ( ( (𝟭 `  O ) `  A
) `  x )  x.  B )  =  sum_ x  e.  A  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767    \ cdif 3478    C_ wss 3481   {cpr 4034   -->wf 5589   ` cfv 5593  (class class class)co 6294   Fincfn 7526   CCcc 9500   RRcr 9501   0cc0 9502   1c1 9503    x. cmul 9507   sum_csu 13483  𝟭cind 27817
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 4563  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691  ax-un 6586  ax-inf2 8068  ax-cnex 9558  ax-resscn 9559  ax-1cn 9560  ax-icn 9561  ax-addcl 9562  ax-addrcl 9563  ax-mulcl 9564  ax-mulrcl 9565  ax-mulcom 9566  ax-addass 9567  ax-mulass 9568  ax-distr 9569  ax-i2m1 9570  ax-1ne0 9571  ax-1rid 9572  ax-rnegex 9573  ax-rrecex 9574  ax-cnre 9575  ax-pre-lttri 9576  ax-pre-lttrn 9577  ax-pre-ltadd 9578  ax-pre-mulgt0 9579  ax-pre-sup 9580
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 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4251  df-int 4288  df-iun 4332  df-br 4453  df-opab 4511  df-mpt 4512  df-tr 4546  df-eprel 4796  df-id 4800  df-po 4805  df-so 4806  df-fr 4843  df-se 4844  df-we 4845  df-ord 4886  df-on 4887  df-lim 4888  df-suc 4889  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-isom 5602  df-riota 6255  df-ov 6297  df-oprab 6298  df-mpt2 6299  df-om 6695  df-1st 6794  df-2nd 6795  df-recs 7052  df-rdg 7086  df-1o 7140  df-oadd 7144  df-er 7321  df-en 7527  df-dom 7528  df-sdom 7529  df-fin 7530  df-sup 7911  df-oi 7945  df-card 8330  df-pnf 9640  df-mnf 9641  df-xr 9642  df-ltxr 9643  df-le 9644  df-sub 9817  df-neg 9818  df-div 10217  df-nn 10547  df-2 10604  df-3 10605  df-n0 10806  df-z 10875  df-uz 11093  df-rp 11231  df-fz 11683  df-fzo 11803  df-seq 12086  df-exp 12145  df-hash 12384  df-cj 12907  df-re 12908  df-im 12909  df-sqrt 13043  df-abs 13044  df-clim 13286  df-sum 13484  df-ind 27818
This theorem is referenced by: (None)
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