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Theorem esummulc1 26545
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 2443 . . 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 2443 . . . 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 26386 . . 3  |-  ( ph  ->  ( z  e.  ( 0 [,] +oo )  |->  ( z xe C ) )  e.  ( ( (ordTop `  <_  )t  ( 0 [,] +oo ) )  Cn  (
(ordTop `  <_  )t  ( 0 [,] +oo ) ) ) )
7 eqidd 2444 . . . 4  |-  ( ph  ->  ( z  e.  ( 0 [,] +oo )  |->  ( z xe C ) )  =  ( z  e.  ( 0 [,] +oo )  |->  ( z xe C ) ) )
8 oveq1 6113 . . . . 5  |-  ( z  =  0  ->  (
z xe C )  =  ( 0 xe C ) )
9 icossxr 11395 . . . . . . 7  |-  ( 0 [,) +oo )  C_  RR*
109, 5sseldi 3369 . . . . . 6  |-  ( ph  ->  C  e.  RR* )
11 xmul02 11246 . . . . . 6  |-  ( C  e.  RR*  ->  ( 0 xe C )  =  0 )
1210, 11syl 16 . . . . 5  |-  ( ph  ->  ( 0 xe C )  =  0 )
138, 12sylan9eqr 2497 . . . 4  |-  ( (
ph  /\  z  = 
0 )  ->  (
z xe C )  =  0 )
14 0e0iccpnf 11411 . . . . 5  |-  0  e.  ( 0 [,] +oo )
1514a1i 11 . . . 4  |-  ( ph  ->  0  e.  ( 0 [,] +oo ) )
167, 13, 15, 15fvmptd 5794 . . 3  |-  ( ph  ->  ( ( z  e.  ( 0 [,] +oo )  |->  ( z xe C ) ) `
 0 )  =  0 )
17 simp2 989 . . . . 5  |-  ( (
ph  /\  x  e.  ( 0 [,] +oo )  /\  y  e.  ( 0 [,] +oo )
)  ->  x  e.  ( 0 [,] +oo ) )
18 simp3 990 . . . . 5  |-  ( (
ph  /\  x  e.  ( 0 [,] +oo )  /\  y  e.  ( 0 [,] +oo )
)  ->  y  e.  ( 0 [,] +oo ) )
19 icossicc 26073 . . . . . 6  |-  ( 0 [,) +oo )  C_  ( 0 [,] +oo )
2053ad2ant1 1009 . . . . . 6  |-  ( (
ph  /\  x  e.  ( 0 [,] +oo )  /\  y  e.  ( 0 [,] +oo )
)  ->  C  e.  ( 0 [,) +oo ) )
2119, 20sseldi 3369 . . . . 5  |-  ( (
ph  /\  x  e.  ( 0 [,] +oo )  /\  y  e.  ( 0 [,] +oo )
)  ->  C  e.  ( 0 [,] +oo ) )
22 xrge0adddir 26170 . . . . 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 1218 . . . 4  |-  ( (
ph  /\  x  e.  ( 0 [,] +oo )  /\  y  e.  ( 0 [,] +oo )
)  ->  ( (
x +e y ) xe C )  =  ( ( x xe C ) +e ( y xe C ) ) )
24 eqidd 2444 . . . . 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 461 . . . . . 6  |-  ( ( ( ph  /\  x  e.  ( 0 [,] +oo )  /\  y  e.  ( 0 [,] +oo )
)  /\  z  =  ( x +e
y ) )  -> 
z  =  ( x +e y ) )
2625oveq1d 6121 . . . . 5  |-  ( ( ( ph  /\  x  e.  ( 0 [,] +oo )  /\  y  e.  ( 0 [,] +oo )
)  /\  z  =  ( x +e
y ) )  -> 
( z xe C )  =  ( ( x +e
y ) xe C ) )
27 ge0xaddcl 11414 . . . . . 6  |-  ( ( x  e.  ( 0 [,] +oo )  /\  y  e.  ( 0 [,] +oo ) )  ->  ( x +e y )  e.  ( 0 [,] +oo ) )
28273adant1 1006 . . . . 5  |-  ( (
ph  /\  x  e.  ( 0 [,] +oo )  /\  y  e.  ( 0 [,] +oo )
)  ->  ( x +e y )  e.  ( 0 [,] +oo ) )
29 ovex 6131 . . . . . 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 5794 . . . 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 461 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  ( 0 [,] +oo )  /\  y  e.  ( 0 [,] +oo )
)  /\  z  =  x )  ->  z  =  x )
3332oveq1d 6121 . . . . . 6  |-  ( ( ( ph  /\  x  e.  ( 0 [,] +oo )  /\  y  e.  ( 0 [,] +oo )
)  /\  z  =  x )  ->  (
z xe C )  =  ( x xe C ) )
34 ovex 6131 . . . . . . 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 5794 . . . . 5  |-  ( (
ph  /\  x  e.  ( 0 [,] +oo )  /\  y  e.  ( 0 [,] +oo )
)  ->  ( (
z  e.  ( 0 [,] +oo )  |->  ( z xe C ) ) `  x
)  =  ( x xe C ) )
37 simpr 461 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  ( 0 [,] +oo )  /\  y  e.  ( 0 [,] +oo )
)  /\  z  =  y )  ->  z  =  y )
3837oveq1d 6121 . . . . . 6  |-  ( ( ( ph  /\  x  e.  ( 0 [,] +oo )  /\  y  e.  ( 0 [,] +oo )
)  /\  z  =  y )  ->  (
z xe C )  =  ( y xe C ) )
39 ovex 6131 . . . . . . 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 5794 . . . . 5  |-  ( (
ph  /\  x  e.  ( 0 [,] +oo )  /\  y  e.  ( 0 [,] +oo )
)  ->  ( (
z  e.  ( 0 [,] +oo )  |->  ( z xe C ) ) `  y
)  =  ( y xe C ) )
4236, 41oveq12d 6124 . . . 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 2485 . . 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 26544 . 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 461 . . . 4  |-  ( (
ph  /\  z  = Σ* k  e.  A B )  -> 
z  = Σ* k  e.  A B )
4645oveq1d 6121 . . 3  |-  ( (
ph  /\  z  = Σ* k  e.  A B )  -> 
( z xe C )  =  (Σ* k  e.  A B xe C ) )
473ralrimiva 2814 . . . 4  |-  ( ph  ->  A. k  e.  A  B  e.  ( 0 [,] +oo ) )
48 nfcv 2589 . . . . 5  |-  F/_ k A
4948esumcl 26501 . . . 4  |-  ( ( A  e.  V  /\  A. k  e.  A  B  e.  ( 0 [,] +oo ) )  -> Σ* k  e.  A B  e.  ( 0 [,] +oo ) )
502, 47, 49syl2anc 661 . . 3  |-  ( ph  -> Σ* k  e.  A B  e.  ( 0 [,] +oo ) )
51 ovex 6131 . . . 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 5794 . 2  |-  ( ph  ->  ( ( z  e.  ( 0 [,] +oo )  |->  ( z xe C ) ) `
Σ* k  e.  A B )  =  (Σ* k  e.  A B xe C ) )
54 eqidd 2444 . . . 4  |-  ( (
ph  /\  k  e.  A )  ->  (
z  e.  ( 0 [,] +oo )  |->  ( z xe C ) )  =  ( z  e.  ( 0 [,] +oo )  |->  ( z xe C ) ) )
55 simpr 461 . . . . 5  |-  ( ( ( ph  /\  k  e.  A )  /\  z  =  B )  ->  z  =  B )
5655oveq1d 6121 . . . 4  |-  ( ( ( ph  /\  k  e.  A )  /\  z  =  B )  ->  (
z xe C )  =  ( B xe C ) )
57 ovex 6131 . . . . 5  |-  ( B xe C )  e.  _V
5857a1i 11 . . . 4  |-  ( (
ph  /\  k  e.  A )  ->  ( B xe C )  e.  _V )
5954, 56, 3, 58fvmptd 5794 . . 3  |-  ( (
ph  /\  k  e.  A )  ->  (
( z  e.  ( 0 [,] +oo )  |->  ( z xe C ) ) `  B )  =  ( B xe C ) )
6059esumeq2dv 26509 . 2  |-  ( ph  -> Σ* k  e.  A ( ( z  e.  ( 0 [,] +oo )  |->  ( z xe C ) ) `  B
)  = Σ* k  e.  A
( B xe C ) )
6144, 53, 603eqtr3d 2483 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 965    = wceq 1369    e. wcel 1756   A.wral 2730   _Vcvv 2987    e. cmpt 4365   ` cfv 5433  (class class class)co 6106   0cc0 9297   +oocpnf 9430   RR*cxr 9432    <_ cle 9434   +ecxad 11102   xecxmu 11103   [,)cico 11317   [,]cicc 11318   ↾t crest 14374  ordTopcordt 14452  Σ*cesum 26498
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4418  ax-sep 4428  ax-nul 4436  ax-pow 4485  ax-pr 4546  ax-un 6387  ax-cnex 9353  ax-resscn 9354  ax-1cn 9355  ax-icn 9356  ax-addcl 9357  ax-addrcl 9358  ax-mulcl 9359  ax-mulrcl 9360  ax-mulcom 9361  ax-addass 9362  ax-mulass 9363  ax-distr 9364  ax-i2m1 9365  ax-1ne0 9366  ax-1rid 9367  ax-rnegex 9368  ax-rrecex 9369  ax-cnre 9370  ax-pre-lttri 9371  ax-pre-lttrn 9372  ax-pre-ltadd 9373  ax-pre-mulgt0 9374  ax-pre-sup 9375
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2735  df-rex 2736  df-reu 2737  df-rmo 2738  df-rab 2739  df-v 2989  df-sbc 3202  df-csb 3304  df-dif 3346  df-un 3348  df-in 3350  df-ss 3357  df-pss 3359  df-nul 3653  df-if 3807  df-pw 3877  df-sn 3893  df-pr 3895  df-tp 3897  df-op 3899  df-uni 4107  df-int 4144  df-iun 4188  df-iin 4189  df-br 4308  df-opab 4366  df-mpt 4367  df-tr 4401  df-eprel 4647  df-id 4651  df-po 4656  df-so 4657  df-fr 4694  df-se 4695  df-we 4696  df-ord 4737  df-on 4738  df-lim 4739  df-suc 4740  df-xp 4861  df-rel 4862  df-cnv 4863  df-co 4864  df-dm 4865  df-rn 4866  df-res 4867  df-ima 4868  df-iota 5396  df-fun 5435  df-fn 5436  df-f 5437  df-f1 5438  df-fo 5439  df-f1o 5440  df-fv 5441  df-isom 5442  df-riota 6067  df-ov 6109  df-oprab 6110  df-mpt2 6111  df-of 6335  df-om 6492  df-1st 6592  df-2nd 6593  df-supp 6706  df-recs 6847  df-rdg 6881  df-1o 6935  df-oadd 6939  df-er 7116  df-map 7231  df-en 7326  df-dom 7327  df-sdom 7328  df-fin 7329  df-fsupp 7636  df-fi 7676  df-sup 7706  df-oi 7739  df-card 8124  df-pnf 9435  df-mnf 9436  df-xr 9437  df-ltxr 9438  df-le 9439  df-sub 9612  df-neg 9613  df-div 10009  df-nn 10338  df-2 10395  df-3 10396  df-4 10397  df-5 10398  df-6 10399  df-7 10400  df-8 10401  df-9 10402  df-10 10403  df-n0 10595  df-z 10662  df-dec 10771  df-uz 10877  df-q 10969  df-rp 11007  df-xneg 11104  df-xadd 11105  df-xmul 11106  df-ioo 11319  df-ioc 11320  df-ico 11321  df-icc 11322  df-fz 11453  df-fzo 11564  df-seq 11822  df-hash 12119  df-struct 14191  df-ndx 14192  df-slot 14193  df-base 14194  df-sets 14195  df-ress 14196  df-plusg 14266  df-mulr 14267  df-tset 14272  df-ple 14273  df-ds 14275  df-rest 14376  df-topn 14377  df-0g 14395  df-gsum 14396  df-topgen 14397  df-ordt 14454  df-xrs 14455  df-mre 14539  df-mrc 14540  df-acs 14542  df-ps 15385  df-tsr 15386  df-mnd 15430  df-mhm 15479  df-submnd 15480  df-cntz 15850  df-cmn 16294  df-fbas 17829  df-fg 17830  df-top 18518  df-bases 18520  df-topon 18521  df-topsp 18522  df-ntr 18639  df-nei 18717  df-cn 18846  df-cnp 18847  df-haus 18934  df-fil 19434  df-fm 19526  df-flim 19527  df-flf 19528  df-tsms 19712  df-esum 26499
This theorem is referenced by:  esummulc2  26546  esumdivc  26547
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