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Theorem esummulc1 26466
Description: An extended sum multiplied by a constant. (Contributed by Thierry Arnoux, 6-Jul-2017.)
Hypotheses
Ref Expression
esummulc2.a  |-  ( ph  ->  A  e.  V )
esummulc2.b  |-  ( (
ph  /\  k  e.  A )  ->  B  e.  ( 0 [,] +oo ) )
esummulc2.c  |-  ( ph  ->  C  e.  ( 0 [,) +oo ) )
Assertion
Ref Expression
esummulc1  |-  ( ph  ->  (Σ* k  e.  A B xe C )  = Σ* k  e.  A ( B xe C ) )
Distinct variable groups:    A, k    C, k    k, V    ph, k
Allowed substitution hint:    B( k)

Proof of Theorem esummulc1
Dummy variables  z  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2441 . . 3  |-  ( (ordTop `  <_  )t  ( 0 [,] +oo ) )  =  ( (ordTop `  <_  )t  ( 0 [,] +oo ) )
2 esummulc2.a . . 3  |-  ( ph  ->  A  e.  V )
3 esummulc2.b . . 3  |-  ( (
ph  /\  k  e.  A )  ->  B  e.  ( 0 [,] +oo ) )
4 eqid 2441 . . . 4  |-  ( z  e.  ( 0 [,] +oo )  |->  ( z xe C ) )  =  ( z  e.  ( 0 [,] +oo )  |->  ( z xe C ) )
5 esummulc2.c . . . 4  |-  ( ph  ->  C  e.  ( 0 [,) +oo ) )
61, 4, 5xrge0mulc1cn 26307 . . 3  |-  ( ph  ->  ( z  e.  ( 0 [,] +oo )  |->  ( z xe C ) )  e.  ( ( (ordTop `  <_  )t  ( 0 [,] +oo ) )  Cn  (
(ordTop `  <_  )t  ( 0 [,] +oo ) ) ) )
7 eqidd 2442 . . . 4  |-  ( ph  ->  ( z  e.  ( 0 [,] +oo )  |->  ( z xe C ) )  =  ( z  e.  ( 0 [,] +oo )  |->  ( z xe C ) ) )
8 oveq1 6097 . . . . 5  |-  ( z  =  0  ->  (
z xe C )  =  ( 0 xe C ) )
9 icossxr 11376 . . . . . . 7  |-  ( 0 [,) +oo )  C_  RR*
109, 5sseldi 3351 . . . . . 6  |-  ( ph  ->  C  e.  RR* )
11 xmul02 11227 . . . . . 6  |-  ( C  e.  RR*  ->  ( 0 xe C )  =  0 )
1210, 11syl 16 . . . . 5  |-  ( ph  ->  ( 0 xe C )  =  0 )
138, 12sylan9eqr 2495 . . . 4  |-  ( (
ph  /\  z  = 
0 )  ->  (
z xe C )  =  0 )
14 0e0iccpnf 11392 . . . . 5  |-  0  e.  ( 0 [,] +oo )
1514a1i 11 . . . 4  |-  ( ph  ->  0  e.  ( 0 [,] +oo ) )
167, 13, 15, 15fvmptd 5776 . . 3  |-  ( ph  ->  ( ( z  e.  ( 0 [,] +oo )  |->  ( z xe C ) ) `
 0 )  =  0 )
17 simp2 984 . . . . 5  |-  ( (
ph  /\  x  e.  ( 0 [,] +oo )  /\  y  e.  ( 0 [,] +oo )
)  ->  x  e.  ( 0 [,] +oo ) )
18 simp3 985 . . . . 5  |-  ( (
ph  /\  x  e.  ( 0 [,] +oo )  /\  y  e.  ( 0 [,] +oo )
)  ->  y  e.  ( 0 [,] +oo ) )
19 icossicc 25991 . . . . . 6  |-  ( 0 [,) +oo )  C_  ( 0 [,] +oo )
2053ad2ant1 1004 . . . . . 6  |-  ( (
ph  /\  x  e.  ( 0 [,] +oo )  /\  y  e.  ( 0 [,] +oo )
)  ->  C  e.  ( 0 [,) +oo ) )
2119, 20sseldi 3351 . . . . 5  |-  ( (
ph  /\  x  e.  ( 0 [,] +oo )  /\  y  e.  ( 0 [,] +oo )
)  ->  C  e.  ( 0 [,] +oo ) )
22 xrge0adddir 26088 . . . . 5  |-  ( ( x  e.  ( 0 [,] +oo )  /\  y  e.  ( 0 [,] +oo )  /\  C  e.  ( 0 [,] +oo ) )  ->  ( ( x +e y ) xe C )  =  ( ( x xe C ) +e ( y xe C ) ) )
2317, 18, 21, 22syl3anc 1213 . . . 4  |-  ( (
ph  /\  x  e.  ( 0 [,] +oo )  /\  y  e.  ( 0 [,] +oo )
)  ->  ( (
x +e y ) xe C )  =  ( ( x xe C ) +e ( y xe C ) ) )
24 eqidd 2442 . . . . 5  |-  ( (
ph  /\  x  e.  ( 0 [,] +oo )  /\  y  e.  ( 0 [,] +oo )
)  ->  ( z  e.  ( 0 [,] +oo )  |->  ( z xe C ) )  =  ( z  e.  ( 0 [,] +oo )  |->  ( z xe C ) ) )
25 simpr 458 . . . . . 6  |-  ( ( ( ph  /\  x  e.  ( 0 [,] +oo )  /\  y  e.  ( 0 [,] +oo )
)  /\  z  =  ( x +e
y ) )  -> 
z  =  ( x +e y ) )
2625oveq1d 6105 . . . . 5  |-  ( ( ( ph  /\  x  e.  ( 0 [,] +oo )  /\  y  e.  ( 0 [,] +oo )
)  /\  z  =  ( x +e
y ) )  -> 
( z xe C )  =  ( ( x +e
y ) xe C ) )
27 ge0xaddcl 11395 . . . . . 6  |-  ( ( x  e.  ( 0 [,] +oo )  /\  y  e.  ( 0 [,] +oo ) )  ->  ( x +e y )  e.  ( 0 [,] +oo ) )
28273adant1 1001 . . . . 5  |-  ( (
ph  /\  x  e.  ( 0 [,] +oo )  /\  y  e.  ( 0 [,] +oo )
)  ->  ( x +e y )  e.  ( 0 [,] +oo ) )
29 ovex 6115 . . . . . 6  |-  ( ( x +e y ) xe C )  e.  _V
3029a1i 11 . . . . 5  |-  ( (
ph  /\  x  e.  ( 0 [,] +oo )  /\  y  e.  ( 0 [,] +oo )
)  ->  ( (
x +e y ) xe C )  e.  _V )
3124, 26, 28, 30fvmptd 5776 . . . 4  |-  ( (
ph  /\  x  e.  ( 0 [,] +oo )  /\  y  e.  ( 0 [,] +oo )
)  ->  ( (
z  e.  ( 0 [,] +oo )  |->  ( z xe C ) ) `  (
x +e y ) )  =  ( ( x +e
y ) xe C ) )
32 simpr 458 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  ( 0 [,] +oo )  /\  y  e.  ( 0 [,] +oo )
)  /\  z  =  x )  ->  z  =  x )
3332oveq1d 6105 . . . . . 6  |-  ( ( ( ph  /\  x  e.  ( 0 [,] +oo )  /\  y  e.  ( 0 [,] +oo )
)  /\  z  =  x )  ->  (
z xe C )  =  ( x xe C ) )
34 ovex 6115 . . . . . . 7  |-  ( x xe C )  e.  _V
3534a1i 11 . . . . . 6  |-  ( (
ph  /\  x  e.  ( 0 [,] +oo )  /\  y  e.  ( 0 [,] +oo )
)  ->  ( x xe C )  e.  _V )
3624, 33, 17, 35fvmptd 5776 . . . . 5  |-  ( (
ph  /\  x  e.  ( 0 [,] +oo )  /\  y  e.  ( 0 [,] +oo )
)  ->  ( (
z  e.  ( 0 [,] +oo )  |->  ( z xe C ) ) `  x
)  =  ( x xe C ) )
37 simpr 458 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  ( 0 [,] +oo )  /\  y  e.  ( 0 [,] +oo )
)  /\  z  =  y )  ->  z  =  y )
3837oveq1d 6105 . . . . . 6  |-  ( ( ( ph  /\  x  e.  ( 0 [,] +oo )  /\  y  e.  ( 0 [,] +oo )
)  /\  z  =  y )  ->  (
z xe C )  =  ( y xe C ) )
39 ovex 6115 . . . . . . 7  |-  ( y xe C )  e.  _V
4039a1i 11 . . . . . 6  |-  ( (
ph  /\  x  e.  ( 0 [,] +oo )  /\  y  e.  ( 0 [,] +oo )
)  ->  ( y xe C )  e.  _V )
4124, 38, 18, 40fvmptd 5776 . . . . 5  |-  ( (
ph  /\  x  e.  ( 0 [,] +oo )  /\  y  e.  ( 0 [,] +oo )
)  ->  ( (
z  e.  ( 0 [,] +oo )  |->  ( z xe C ) ) `  y
)  =  ( y xe C ) )
4236, 41oveq12d 6108 . . . 4  |-  ( (
ph  /\  x  e.  ( 0 [,] +oo )  /\  y  e.  ( 0 [,] +oo )
)  ->  ( (
( z  e.  ( 0 [,] +oo )  |->  ( z xe C ) ) `  x ) +e
( ( z  e.  ( 0 [,] +oo )  |->  ( z xe C ) ) `
 y ) )  =  ( ( x xe C ) +e ( y xe C ) ) )
4323, 31, 423eqtr4d 2483 . . 3  |-  ( (
ph  /\  x  e.  ( 0 [,] +oo )  /\  y  e.  ( 0 [,] +oo )
)  ->  ( (
z  e.  ( 0 [,] +oo )  |->  ( z xe C ) ) `  (
x +e y ) )  =  ( ( ( z  e.  ( 0 [,] +oo )  |->  ( z xe C ) ) `
 x ) +e ( ( z  e.  ( 0 [,] +oo )  |->  ( z xe C ) ) `  y ) ) )
441, 2, 3, 6, 16, 43esumcocn 26465 . 2  |-  ( ph  ->  ( ( z  e.  ( 0 [,] +oo )  |->  ( z xe C ) ) `
Σ* k  e.  A B )  = Σ* k  e.  A
( ( z  e.  ( 0 [,] +oo )  |->  ( z xe C ) ) `
 B ) )
45 simpr 458 . . . 4  |-  ( (
ph  /\  z  = Σ* k  e.  A B )  -> 
z  = Σ* k  e.  A B )
4645oveq1d 6105 . . 3  |-  ( (
ph  /\  z  = Σ* k  e.  A B )  -> 
( z xe C )  =  (Σ* k  e.  A B xe C ) )
473ralrimiva 2797 . . . 4  |-  ( ph  ->  A. k  e.  A  B  e.  ( 0 [,] +oo ) )
48 nfcv 2577 . . . . 5  |-  F/_ k A
4948esumcl 26422 . . . 4  |-  ( ( A  e.  V  /\  A. k  e.  A  B  e.  ( 0 [,] +oo ) )  -> Σ* k  e.  A B  e.  ( 0 [,] +oo ) )
502, 47, 49syl2anc 656 . . 3  |-  ( ph  -> Σ* k  e.  A B  e.  ( 0 [,] +oo ) )
51 ovex 6115 . . . 4  |-  (Σ* k  e.  A B xe C )  e.  _V
5251a1i 11 . . 3  |-  ( ph  ->  (Σ* k  e.  A B xe C )  e.  _V )
537, 46, 50, 52fvmptd 5776 . 2  |-  ( ph  ->  ( ( z  e.  ( 0 [,] +oo )  |->  ( z xe C ) ) `
Σ* k  e.  A B )  =  (Σ* k  e.  A B xe C ) )
54 eqidd 2442 . . . 4  |-  ( (
ph  /\  k  e.  A )  ->  (
z  e.  ( 0 [,] +oo )  |->  ( z xe C ) )  =  ( z  e.  ( 0 [,] +oo )  |->  ( z xe C ) ) )
55 simpr 458 . . . . 5  |-  ( ( ( ph  /\  k  e.  A )  /\  z  =  B )  ->  z  =  B )
5655oveq1d 6105 . . . 4  |-  ( ( ( ph  /\  k  e.  A )  /\  z  =  B )  ->  (
z xe C )  =  ( B xe C ) )
57 ovex 6115 . . . . 5  |-  ( B xe C )  e.  _V
5857a1i 11 . . . 4  |-  ( (
ph  /\  k  e.  A )  ->  ( B xe C )  e.  _V )
5954, 56, 3, 58fvmptd 5776 . . 3  |-  ( (
ph  /\  k  e.  A )  ->  (
( z  e.  ( 0 [,] +oo )  |->  ( z xe C ) ) `  B )  =  ( B xe C ) )
6059esumeq2dv 26430 . 2  |-  ( ph  -> Σ* k  e.  A ( ( z  e.  ( 0 [,] +oo )  |->  ( z xe C ) ) `  B
)  = Σ* k  e.  A
( B xe C ) )
6144, 53, 603eqtr3d 2481 1  |-  ( ph  ->  (Σ* k  e.  A B xe C )  = Σ* k  e.  A ( B xe C ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 960    = wceq 1364    e. wcel 1761   A.wral 2713   _Vcvv 2970    e. cmpt 4347   ` cfv 5415  (class class class)co 6090   0cc0 9278   +oocpnf 9411   RR*cxr 9413    <_ cle 9415   +ecxad 11083   xecxmu 11084   [,)cico 11298   [,]cicc 11299   ↾t crest 14355  ordTopcordt 14433  Σ*cesum 26419
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-pre-sup 9356
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-fal 1370  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-iin 4171  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-se 4676  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-isom 5424  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-of 6319  df-om 6476  df-1st 6576  df-2nd 6577  df-supp 6690  df-recs 6828  df-rdg 6862  df-1o 6916  df-oadd 6920  df-er 7097  df-map 7212  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-fsupp 7617  df-fi 7657  df-sup 7687  df-oi 7720  df-card 8105  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-nn 10319  df-2 10376  df-3 10377  df-4 10378  df-5 10379  df-6 10380  df-7 10381  df-8 10382  df-9 10383  df-10 10384  df-n0 10576  df-z 10643  df-dec 10752  df-uz 10858  df-q 10950  df-rp 10988  df-xneg 11085  df-xadd 11086  df-xmul 11087  df-ioo 11300  df-ioc 11301  df-ico 11302  df-icc 11303  df-fz 11434  df-fzo 11545  df-seq 11803  df-hash 12100  df-struct 14172  df-ndx 14173  df-slot 14174  df-base 14175  df-sets 14176  df-ress 14177  df-plusg 14247  df-mulr 14248  df-tset 14253  df-ple 14254  df-ds 14256  df-rest 14357  df-topn 14358  df-0g 14376  df-gsum 14377  df-topgen 14378  df-ordt 14435  df-xrs 14436  df-mre 14520  df-mrc 14521  df-acs 14523  df-ps 15366  df-tsr 15367  df-mnd 15411  df-mhm 15460  df-submnd 15461  df-cntz 15828  df-cmn 16272  df-fbas 17773  df-fg 17774  df-top 18462  df-bases 18464  df-topon 18465  df-topsp 18466  df-ntr 18583  df-nei 18661  df-cn 18790  df-cnp 18791  df-haus 18878  df-fil 19378  df-fm 19470  df-flim 19471  df-flf 19472  df-tsms 19656  df-esum 26420
This theorem is referenced by:  esummulc2  26467  esumdivc  26468
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