Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  esummulc1 Unicode version

Theorem esummulc1 24424
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 x e C )  = Σ* k  e.  A ( B x e 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 2404 . . 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 2404 . . . 4  |-  ( z  e.  ( 0 [,] 
+oo )  |->  ( z x e C ) )  =  ( z  e.  ( 0 [,] 
+oo )  |->  ( z x e C ) )
5 esummulc2.c . . . 4  |-  ( ph  ->  C  e.  ( 0 [,)  +oo ) )
61, 4, 5xrge0mulc1cn 24280 . . 3  |-  ( ph  ->  ( z  e.  ( 0 [,]  +oo )  |->  ( z x e C ) )  e.  ( ( (ordTop `  <_  )t  ( 0 [,]  +oo ) )  Cn  (
(ordTop `  <_  )t  ( 0 [,]  +oo ) ) ) )
7 eqidd 2405 . . . 4  |-  ( ph  ->  ( z  e.  ( 0 [,]  +oo )  |->  ( z x e C ) )  =  ( z  e.  ( 0 [,]  +oo )  |->  ( z x e C ) ) )
8 oveq1 6047 . . . . 5  |-  ( z  =  0  ->  (
z x e C )  =  ( 0 x e C ) )
9 icossxr 10951 . . . . . . 7  |-  ( 0 [,)  +oo )  C_  RR*
109, 5sseldi 3306 . . . . . 6  |-  ( ph  ->  C  e.  RR* )
11 xmul02 10803 . . . . . 6  |-  ( C  e.  RR*  ->  ( 0 x e C )  =  0 )
1210, 11syl 16 . . . . 5  |-  ( ph  ->  ( 0 x e C )  =  0 )
138, 12sylan9eqr 2458 . . . 4  |-  ( (
ph  /\  z  = 
0 )  ->  (
z x e C )  =  0 )
14 0xr 9087 . . . . . 6  |-  0  e.  RR*
15 pnfxr 10669 . . . . . 6  |-  +oo  e.  RR*
16 pnfge 10683 . . . . . . 7  |-  ( 0  e.  RR*  ->  0  <_  +oo )
1714, 16ax-mp 8 . . . . . 6  |-  0  <_  +oo
18 lbicc2 10969 . . . . . 6  |-  ( ( 0  e.  RR*  /\  +oo  e.  RR*  /\  0  <_  +oo )  ->  0  e.  ( 0 [,]  +oo ) )
1914, 15, 17, 18mp3an 1279 . . . . 5  |-  0  e.  ( 0 [,]  +oo )
2019a1i 11 . . . 4  |-  ( ph  ->  0  e.  ( 0 [,]  +oo ) )
217, 13, 20, 20fvmptd 5769 . . 3  |-  ( ph  ->  ( ( z  e.  ( 0 [,]  +oo )  |->  ( z x e C ) ) `
 0 )  =  0 )
22 simp2 958 . . . . 5  |-  ( (
ph  /\  x  e.  ( 0 [,]  +oo )  /\  y  e.  ( 0 [,]  +oo )
)  ->  x  e.  ( 0 [,]  +oo ) )
23 simp3 959 . . . . 5  |-  ( (
ph  /\  x  e.  ( 0 [,]  +oo )  /\  y  e.  ( 0 [,]  +oo )
)  ->  y  e.  ( 0 [,]  +oo ) )
24 icossicc 24082 . . . . . 6  |-  ( 0 [,)  +oo )  C_  (
0 [,]  +oo )
2553ad2ant1 978 . . . . . 6  |-  ( (
ph  /\  x  e.  ( 0 [,]  +oo )  /\  y  e.  ( 0 [,]  +oo )
)  ->  C  e.  ( 0 [,)  +oo ) )
2624, 25sseldi 3306 . . . . 5  |-  ( (
ph  /\  x  e.  ( 0 [,]  +oo )  /\  y  e.  ( 0 [,]  +oo )
)  ->  C  e.  ( 0 [,]  +oo ) )
27 xrge0adddir 24168 . . . . 5  |-  ( ( x  e.  ( 0 [,]  +oo )  /\  y  e.  ( 0 [,]  +oo )  /\  C  e.  ( 0 [,]  +oo )
)  ->  ( (
x + e y ) x e C )  =  ( ( x x e C ) + e ( y x e C ) ) )
2822, 23, 26, 27syl3anc 1184 . . . 4  |-  ( (
ph  /\  x  e.  ( 0 [,]  +oo )  /\  y  e.  ( 0 [,]  +oo )
)  ->  ( (
x + e y ) x e C )  =  ( ( x x e C ) + e ( y x e C ) ) )
29 eqidd 2405 . . . . 5  |-  ( (
ph  /\  x  e.  ( 0 [,]  +oo )  /\  y  e.  ( 0 [,]  +oo )
)  ->  ( z  e.  ( 0 [,]  +oo )  |->  ( z x e C ) )  =  ( z  e.  ( 0 [,]  +oo )  |->  ( z x e C ) ) )
30 simpr 448 . . . . . 6  |-  ( ( ( ph  /\  x  e.  ( 0 [,]  +oo )  /\  y  e.  ( 0 [,]  +oo )
)  /\  z  =  ( x + e
y ) )  -> 
z  =  ( x + e y ) )
3130oveq1d 6055 . . . . 5  |-  ( ( ( ph  /\  x  e.  ( 0 [,]  +oo )  /\  y  e.  ( 0 [,]  +oo )
)  /\  z  =  ( x + e
y ) )  -> 
( z x e C )  =  ( ( x + e
y ) x e C ) )
32 ge0xaddcl 10967 . . . . . 6  |-  ( ( x  e.  ( 0 [,]  +oo )  /\  y  e.  ( 0 [,]  +oo ) )  ->  (
x + e y )  e.  ( 0 [,]  +oo ) )
33323adant1 975 . . . . 5  |-  ( (
ph  /\  x  e.  ( 0 [,]  +oo )  /\  y  e.  ( 0 [,]  +oo )
)  ->  ( x + e y )  e.  ( 0 [,]  +oo ) )
34 ovex 6065 . . . . . 6  |-  ( ( x + e y ) x e C )  e.  _V
3534a1i 11 . . . . 5  |-  ( (
ph  /\  x  e.  ( 0 [,]  +oo )  /\  y  e.  ( 0 [,]  +oo )
)  ->  ( (
x + e y ) x e C )  e.  _V )
3629, 31, 33, 35fvmptd 5769 . . . 4  |-  ( (
ph  /\  x  e.  ( 0 [,]  +oo )  /\  y  e.  ( 0 [,]  +oo )
)  ->  ( (
z  e.  ( 0 [,]  +oo )  |->  ( z x e C ) ) `  ( x + e y ) )  =  ( ( x + e y ) x e C ) )
37 simpr 448 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  ( 0 [,]  +oo )  /\  y  e.  ( 0 [,]  +oo )
)  /\  z  =  x )  ->  z  =  x )
3837oveq1d 6055 . . . . . 6  |-  ( ( ( ph  /\  x  e.  ( 0 [,]  +oo )  /\  y  e.  ( 0 [,]  +oo )
)  /\  z  =  x )  ->  (
z x e C )  =  ( x x e C ) )
39 ovex 6065 . . . . . . 7  |-  ( x x e C )  e.  _V
4039a1i 11 . . . . . 6  |-  ( (
ph  /\  x  e.  ( 0 [,]  +oo )  /\  y  e.  ( 0 [,]  +oo )
)  ->  ( x x e C )  e. 
_V )
4129, 38, 22, 40fvmptd 5769 . . . . 5  |-  ( (
ph  /\  x  e.  ( 0 [,]  +oo )  /\  y  e.  ( 0 [,]  +oo )
)  ->  ( (
z  e.  ( 0 [,]  +oo )  |->  ( z x e C ) ) `  x )  =  ( x x e C ) )
42 simpr 448 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  ( 0 [,]  +oo )  /\  y  e.  ( 0 [,]  +oo )
)  /\  z  =  y )  ->  z  =  y )
4342oveq1d 6055 . . . . . 6  |-  ( ( ( ph  /\  x  e.  ( 0 [,]  +oo )  /\  y  e.  ( 0 [,]  +oo )
)  /\  z  =  y )  ->  (
z x e C )  =  ( y x e C ) )
44 ovex 6065 . . . . . . 7  |-  ( y x e C )  e.  _V
4544a1i 11 . . . . . 6  |-  ( (
ph  /\  x  e.  ( 0 [,]  +oo )  /\  y  e.  ( 0 [,]  +oo )
)  ->  ( y x e C )  e. 
_V )
4629, 43, 23, 45fvmptd 5769 . . . . 5  |-  ( (
ph  /\  x  e.  ( 0 [,]  +oo )  /\  y  e.  ( 0 [,]  +oo )
)  ->  ( (
z  e.  ( 0 [,]  +oo )  |->  ( z x e C ) ) `  y )  =  ( y x e C ) )
4741, 46oveq12d 6058 . . . 4  |-  ( (
ph  /\  x  e.  ( 0 [,]  +oo )  /\  y  e.  ( 0 [,]  +oo )
)  ->  ( (
( z  e.  ( 0 [,]  +oo )  |->  ( z x e C ) ) `  x ) + e
( ( z  e.  ( 0 [,]  +oo )  |->  ( z x e C ) ) `
 y ) )  =  ( ( x x e C ) + e ( y x e C ) ) )
4828, 36, 473eqtr4d 2446 . . 3  |-  ( (
ph  /\  x  e.  ( 0 [,]  +oo )  /\  y  e.  ( 0 [,]  +oo )
)  ->  ( (
z  e.  ( 0 [,]  +oo )  |->  ( z x e C ) ) `  ( x + e y ) )  =  ( ( ( z  e.  ( 0 [,]  +oo )  |->  ( z x e C ) ) `  x ) + e
( ( z  e.  ( 0 [,]  +oo )  |->  ( z x e C ) ) `
 y ) ) )
491, 2, 3, 6, 21, 48esumcocn 24423 . 2  |-  ( ph  ->  ( ( z  e.  ( 0 [,]  +oo )  |->  ( z x e C ) ) `
Σ* k  e.  A B )  = Σ* k  e.  A
( ( z  e.  ( 0 [,]  +oo )  |->  ( z x e C ) ) `
 B ) )
50 simpr 448 . . . 4  |-  ( (
ph  /\  z  = Σ* k  e.  A B )  -> 
z  = Σ* k  e.  A B )
5150oveq1d 6055 . . 3  |-  ( (
ph  /\  z  = Σ* k  e.  A B )  -> 
( z x e C )  =  (Σ* k  e.  A B x e C ) )
523ralrimiva 2749 . . . 4  |-  ( ph  ->  A. k  e.  A  B  e.  ( 0 [,]  +oo ) )
53 nfcv 2540 . . . . 5  |-  F/_ k A
5453esumcl 24380 . . . 4  |-  ( ( A  e.  V  /\  A. k  e.  A  B  e.  ( 0 [,]  +oo ) )  -> Σ* k  e.  A B  e.  ( 0 [,]  +oo ) )
552, 52, 54syl2anc 643 . . 3  |-  ( ph  -> Σ* k  e.  A B  e.  ( 0 [,]  +oo ) )
56 ovex 6065 . . . 4  |-  (Σ* k  e.  A B x e C )  e.  _V
5756a1i 11 . . 3  |-  ( ph  ->  (Σ* k  e.  A B x e C )  e.  _V )
587, 51, 55, 57fvmptd 5769 . 2  |-  ( ph  ->  ( ( z  e.  ( 0 [,]  +oo )  |->  ( z x e C ) ) `
Σ* k  e.  A B )  =  (Σ* k  e.  A B x e C ) )
59 eqidd 2405 . . . 4  |-  ( (
ph  /\  k  e.  A )  ->  (
z  e.  ( 0 [,]  +oo )  |->  ( z x e C ) )  =  ( z  e.  ( 0 [,] 
+oo )  |->  ( z x e C ) ) )
60 simpr 448 . . . . 5  |-  ( ( ( ph  /\  k  e.  A )  /\  z  =  B )  ->  z  =  B )
6160oveq1d 6055 . . . 4  |-  ( ( ( ph  /\  k  e.  A )  /\  z  =  B )  ->  (
z x e C )  =  ( B x e C ) )
62 ovex 6065 . . . . 5  |-  ( B x e C )  e.  _V
6362a1i 11 . . . 4  |-  ( (
ph  /\  k  e.  A )  ->  ( B x e C )  e.  _V )
6459, 61, 3, 63fvmptd 5769 . . 3  |-  ( (
ph  /\  k  e.  A )  ->  (
( z  e.  ( 0 [,]  +oo )  |->  ( z x e C ) ) `  B )  =  ( B x e C ) )
6564esumeq2dv 24388 . 2  |-  ( ph  -> Σ* k  e.  A ( ( z  e.  ( 0 [,]  +oo )  |->  ( z x e C ) ) `  B )  = Σ* k  e.  A ( B x e C ) )
6649, 58, 653eqtr3d 2444 1  |-  ( ph  ->  (Σ* k  e.  A B x e C )  = Σ* k  e.  A ( B x e C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   A.wral 2666   _Vcvv 2916   class class class wbr 4172    e. cmpt 4226   ` cfv 5413  (class class class)co 6040   0cc0 8946    +oocpnf 9073   RR*cxr 9075    <_ cle 9077   + ecxad 10664   x ecxmu 10665   [,)cico 10874   [,]cicc 10875   ↾t crest 13603  ordTopcordt 13676  Σ*cesum 24377
This theorem is referenced by:  esummulc2  24425  esumdivc  24426
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-map 6979  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-fi 7374  df-sup 7404  df-oi 7435  df-card 7782  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-q 10531  df-rp 10569  df-xneg 10666  df-xadd 10667  df-xmul 10668  df-ioo 10876  df-ioc 10877  df-ico 10878  df-icc 10879  df-fz 11000  df-fzo 11091  df-seq 11279  df-hash 11574  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-mulr 13498  df-tset 13503  df-ple 13504  df-ds 13506  df-rest 13605  df-topn 13606  df-topgen 13622  df-ordt 13680  df-xrs 13681  df-0g 13682  df-gsum 13683  df-mre 13766  df-mrc 13767  df-acs 13769  df-ps 14584  df-tsr 14585  df-mnd 14645  df-mhm 14693  df-submnd 14694  df-cntz 15071  df-cmn 15369  df-fbas 16654  df-fg 16655  df-top 16918  df-bases 16920  df-topon 16921  df-topsp 16922  df-ntr 17039  df-nei 17117  df-cn 17245  df-cnp 17246  df-haus 17333  df-fil 17831  df-fm 17923  df-flim 17924  df-flf 17925  df-tsms 18109  df-esum 24378
  Copyright terms: Public domain W3C validator