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Theorem esummulc1 27912
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 2467 . . 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 2467 . . . 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 27748 . . 3  |-  ( ph  ->  ( z  e.  ( 0 [,] +oo )  |->  ( z xe C ) )  e.  ( ( (ordTop `  <_  )t  ( 0 [,] +oo ) )  Cn  (
(ordTop `  <_  )t  ( 0 [,] +oo ) ) ) )
7 eqidd 2468 . . . 4  |-  ( ph  ->  ( z  e.  ( 0 [,] +oo )  |->  ( z xe C ) )  =  ( z  e.  ( 0 [,] +oo )  |->  ( z xe C ) ) )
8 oveq1 6302 . . . . 5  |-  ( z  =  0  ->  (
z xe C )  =  ( 0 xe C ) )
9 icossxr 11621 . . . . . . 7  |-  ( 0 [,) +oo )  C_  RR*
109, 5sseldi 3507 . . . . . 6  |-  ( ph  ->  C  e.  RR* )
11 xmul02 11472 . . . . . 6  |-  ( C  e.  RR*  ->  ( 0 xe C )  =  0 )
1210, 11syl 16 . . . . 5  |-  ( ph  ->  ( 0 xe C )  =  0 )
138, 12sylan9eqr 2530 . . . 4  |-  ( (
ph  /\  z  = 
0 )  ->  (
z xe C )  =  0 )
14 0e0iccpnf 11643 . . . . 5  |-  0  e.  ( 0 [,] +oo )
1514a1i 11 . . . 4  |-  ( ph  ->  0  e.  ( 0 [,] +oo ) )
167, 13, 15, 15fvmptd 5962 . . 3  |-  ( ph  ->  ( ( z  e.  ( 0 [,] +oo )  |->  ( z xe C ) ) `
 0 )  =  0 )
17 simp2 997 . . . . 5  |-  ( (
ph  /\  x  e.  ( 0 [,] +oo )  /\  y  e.  ( 0 [,] +oo )
)  ->  x  e.  ( 0 [,] +oo ) )
18 simp3 998 . . . . 5  |-  ( (
ph  /\  x  e.  ( 0 [,] +oo )  /\  y  e.  ( 0 [,] +oo )
)  ->  y  e.  ( 0 [,] +oo ) )
19 icossicc 11623 . . . . . 6  |-  ( 0 [,) +oo )  C_  ( 0 [,] +oo )
2053ad2ant1 1017 . . . . . 6  |-  ( (
ph  /\  x  e.  ( 0 [,] +oo )  /\  y  e.  ( 0 [,] +oo )
)  ->  C  e.  ( 0 [,) +oo ) )
2119, 20sseldi 3507 . . . . 5  |-  ( (
ph  /\  x  e.  ( 0 [,] +oo )  /\  y  e.  ( 0 [,] +oo )
)  ->  C  e.  ( 0 [,] +oo ) )
22 xrge0adddir 27506 . . . . 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 1228 . . . 4  |-  ( (
ph  /\  x  e.  ( 0 [,] +oo )  /\  y  e.  ( 0 [,] +oo )
)  ->  ( (
x +e y ) xe C )  =  ( ( x xe C ) +e ( y xe C ) ) )
24 eqidd 2468 . . . . 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 6310 . . . . 5  |-  ( ( ( ph  /\  x  e.  ( 0 [,] +oo )  /\  y  e.  ( 0 [,] +oo )
)  /\  z  =  ( x +e
y ) )  -> 
( z xe C )  =  ( ( x +e
y ) xe C ) )
27 ge0xaddcl 11646 . . . . . 6  |-  ( ( x  e.  ( 0 [,] +oo )  /\  y  e.  ( 0 [,] +oo ) )  ->  ( x +e y )  e.  ( 0 [,] +oo ) )
28273adant1 1014 . . . . 5  |-  ( (
ph  /\  x  e.  ( 0 [,] +oo )  /\  y  e.  ( 0 [,] +oo )
)  ->  ( x +e y )  e.  ( 0 [,] +oo ) )
29 ovex 6320 . . . . . 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 5962 . . . 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 6310 . . . . . 6  |-  ( ( ( ph  /\  x  e.  ( 0 [,] +oo )  /\  y  e.  ( 0 [,] +oo )
)  /\  z  =  x )  ->  (
z xe C )  =  ( x xe C ) )
34 ovex 6320 . . . . . . 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 5962 . . . . 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 6310 . . . . . 6  |-  ( ( ( ph  /\  x  e.  ( 0 [,] +oo )  /\  y  e.  ( 0 [,] +oo )
)  /\  z  =  y )  ->  (
z xe C )  =  ( y xe C ) )
39 ovex 6320 . . . . . . 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 5962 . . . . 5  |-  ( (
ph  /\  x  e.  ( 0 [,] +oo )  /\  y  e.  ( 0 [,] +oo )
)  ->  ( (
z  e.  ( 0 [,] +oo )  |->  ( z xe C ) ) `  y
)  =  ( y xe C ) )
4236, 41oveq12d 6313 . . . 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 2518 . . 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 27911 . 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 6310 . . 3  |-  ( (
ph  /\  z  = Σ* k  e.  A B )  -> 
( z xe C )  =  (Σ* k  e.  A B xe C ) )
473ralrimiva 2881 . . . 4  |-  ( ph  ->  A. k  e.  A  B  e.  ( 0 [,] +oo ) )
48 nfcv 2629 . . . . 5  |-  F/_ k A
4948esumcl 27868 . . . 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 6320 . . . 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 5962 . 2  |-  ( ph  ->  ( ( z  e.  ( 0 [,] +oo )  |->  ( z xe C ) ) `
Σ* k  e.  A B )  =  (Σ* k  e.  A B xe C ) )
54 eqidd 2468 . . . 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 6310 . . . 4  |-  ( ( ( ph  /\  k  e.  A )  /\  z  =  B )  ->  (
z xe C )  =  ( B xe C ) )
57 ovex 6320 . . . . 5  |-  ( B xe C )  e.  _V
5857a1i 11 . . . 4  |-  ( (
ph  /\  k  e.  A )  ->  ( B xe C )  e.  _V )
5954, 56, 3, 58fvmptd 5962 . . 3  |-  ( (
ph  /\  k  e.  A )  ->  (
( z  e.  ( 0 [,] +oo )  |->  ( z xe C ) ) `  B )  =  ( B xe C ) )
6059esumeq2dv 27876 . 2  |-  ( ph  -> Σ* k  e.  A ( ( z  e.  ( 0 [,] +oo )  |->  ( z xe C ) ) `  B
)  = Σ* k  e.  A
( B xe C ) )
6144, 53, 603eqtr3d 2516 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 973    = wceq 1379    e. wcel 1767   A.wral 2817   _Vcvv 3118    |-> cmpt 4511   ` cfv 5594  (class class class)co 6295   0cc0 9504   +oocpnf 9637   RR*cxr 9639    <_ cle 9641   +ecxad 11328   xecxmu 11329   [,)cico 11543   [,]cicc 11544   ↾t crest 14693  ordTopcordt 14771  Σ*cesum 27865
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 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-pre-sup 9582
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 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-iin 4334  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-se 4845  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-of 6535  df-om 6696  df-1st 6795  df-2nd 6796  df-supp 6914  df-recs 7054  df-rdg 7088  df-1o 7142  df-oadd 7146  df-er 7323  df-map 7434  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-fsupp 7842  df-fi 7883  df-sup 7913  df-oi 7947  df-card 8332  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-div 10219  df-nn 10549  df-2 10606  df-3 10607  df-4 10608  df-5 10609  df-6 10610  df-7 10611  df-8 10612  df-9 10613  df-10 10614  df-n0 10808  df-z 10877  df-dec 10989  df-uz 11095  df-q 11195  df-rp 11233  df-xneg 11330  df-xadd 11331  df-xmul 11332  df-ioo 11545  df-ioc 11546  df-ico 11547  df-icc 11548  df-fz 11685  df-fzo 11805  df-seq 12088  df-hash 12386  df-struct 14509  df-ndx 14510  df-slot 14511  df-base 14512  df-sets 14513  df-ress 14514  df-plusg 14585  df-mulr 14586  df-tset 14591  df-ple 14592  df-ds 14594  df-rest 14695  df-topn 14696  df-0g 14714  df-gsum 14715  df-topgen 14716  df-ordt 14773  df-xrs 14774  df-mre 14858  df-mrc 14859  df-acs 14861  df-ps 15704  df-tsr 15705  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-mhm 15839  df-submnd 15840  df-cntz 16227  df-cmn 16673  df-fbas 18286  df-fg 18287  df-top 19268  df-bases 19270  df-topon 19271  df-topsp 19272  df-ntr 19389  df-nei 19467  df-cn 19596  df-cnp 19597  df-haus 19684  df-fil 20215  df-fm 20307  df-flim 20308  df-flf 20309  df-tsms 20493  df-esum 27866
This theorem is referenced by:  esummulc2  27913  esumdivc  27914
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