MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  seqdistr Structured version   Unicode version

Theorem seqdistr 11841
Description: The distributive property for series. (Contributed by Mario Carneiro, 28-Jul-2013.) (Revised by Mario Carneiro, 27-May-2014.)
Hypotheses
Ref Expression
seqdistr.1  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  .+  y
)  e.  S )
seqdistr.2  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( C T ( x  .+  y ) )  =  ( ( C T x ) 
.+  ( C T y ) ) )
seqdistr.3  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
seqdistr.4  |-  ( (
ph  /\  x  e.  ( M ... N ) )  ->  ( G `  x )  e.  S
)
seqdistr.5  |-  ( (
ph  /\  x  e.  ( M ... N ) )  ->  ( F `  x )  =  ( C T ( G `
 x ) ) )
Assertion
Ref Expression
seqdistr  |-  ( ph  ->  (  seq M ( 
.+  ,  F ) `
 N )  =  ( C T (  seq M (  .+  ,  G ) `  N
) ) )
Distinct variable groups:    x, y, C    x, G, y    x, M, y    x, N, y   
x,  .+ , y    x, F    ph, x, y    x, S, y    x, T, y
Allowed substitution hint:    F( y)

Proof of Theorem seqdistr
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 seqdistr.1 . . 3  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  .+  y
)  e.  S )
2 seqdistr.4 . . 3  |-  ( (
ph  /\  x  e.  ( M ... N ) )  ->  ( G `  x )  e.  S
)
3 seqdistr.3 . . 3  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
4 seqdistr.2 . . . 4  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( C T ( x  .+  y ) )  =  ( ( C T x ) 
.+  ( C T y ) ) )
5 oveq2 6088 . . . . . 6  |-  ( z  =  ( x  .+  y )  ->  ( C T z )  =  ( C T ( x  .+  y ) ) )
6 eqid 2433 . . . . . 6  |-  ( z  e.  S  |->  ( C T z ) )  =  ( z  e.  S  |->  ( C T z ) )
7 ovex 6105 . . . . . 6  |-  ( C T ( x  .+  y ) )  e. 
_V
85, 6, 7fvmpt 5762 . . . . 5  |-  ( ( x  .+  y )  e.  S  ->  (
( z  e.  S  |->  ( C T z ) ) `  (
x  .+  y )
)  =  ( C T ( x  .+  y ) ) )
91, 8syl 16 . . . 4  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( ( z  e.  S  |->  ( C T z ) ) `  ( x  .+  y ) )  =  ( C T ( x  .+  y ) ) )
10 oveq2 6088 . . . . . . 7  |-  ( z  =  x  ->  ( C T z )  =  ( C T x ) )
11 ovex 6105 . . . . . . 7  |-  ( C T x )  e. 
_V
1210, 6, 11fvmpt 5762 . . . . . 6  |-  ( x  e.  S  ->  (
( z  e.  S  |->  ( C T z ) ) `  x
)  =  ( C T x ) )
1312ad2antrl 720 . . . . 5  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( ( z  e.  S  |->  ( C T z ) ) `  x )  =  ( C T x ) )
14 oveq2 6088 . . . . . . 7  |-  ( z  =  y  ->  ( C T z )  =  ( C T y ) )
15 ovex 6105 . . . . . . 7  |-  ( C T y )  e. 
_V
1614, 6, 15fvmpt 5762 . . . . . 6  |-  ( y  e.  S  ->  (
( z  e.  S  |->  ( C T z ) ) `  y
)  =  ( C T y ) )
1716ad2antll 721 . . . . 5  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( ( z  e.  S  |->  ( C T z ) ) `  y )  =  ( C T y ) )
1813, 17oveq12d 6098 . . . 4  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( ( ( z  e.  S  |->  ( C T z ) ) `
 x )  .+  ( ( z  e.  S  |->  ( C T z ) ) `  y ) )  =  ( ( C T x )  .+  ( C T y ) ) )
194, 9, 183eqtr4d 2475 . . 3  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( ( z  e.  S  |->  ( C T z ) ) `  ( x  .+  y ) )  =  ( ( ( z  e.  S  |->  ( C T z ) ) `  x
)  .+  ( (
z  e.  S  |->  ( C T z ) ) `  y ) ) )
20 oveq2 6088 . . . . . 6  |-  ( z  =  ( G `  x )  ->  ( C T z )  =  ( C T ( G `  x ) ) )
21 ovex 6105 . . . . . 6  |-  ( C T ( G `  x ) )  e. 
_V
2220, 6, 21fvmpt 5762 . . . . 5  |-  ( ( G `  x )  e.  S  ->  (
( z  e.  S  |->  ( C T z ) ) `  ( G `  x )
)  =  ( C T ( G `  x ) ) )
232, 22syl 16 . . . 4  |-  ( (
ph  /\  x  e.  ( M ... N ) )  ->  ( (
z  e.  S  |->  ( C T z ) ) `  ( G `
 x ) )  =  ( C T ( G `  x
) ) )
24 seqdistr.5 . . . 4  |-  ( (
ph  /\  x  e.  ( M ... N ) )  ->  ( F `  x )  =  ( C T ( G `
 x ) ) )
2523, 24eqtr4d 2468 . . 3  |-  ( (
ph  /\  x  e.  ( M ... N ) )  ->  ( (
z  e.  S  |->  ( C T z ) ) `  ( G `
 x ) )  =  ( F `  x ) )
261, 2, 3, 19, 25seqhomo 11837 . 2  |-  ( ph  ->  ( ( z  e.  S  |->  ( C T z ) ) `  (  seq M (  .+  ,  G ) `  N
) )  =  (  seq M (  .+  ,  F ) `  N
) )
273, 2, 1seqcl 11810 . . 3  |-  ( ph  ->  (  seq M ( 
.+  ,  G ) `
 N )  e.  S )
28 oveq2 6088 . . . 4  |-  ( z  =  (  seq M
(  .+  ,  G
) `  N )  ->  ( C T z )  =  ( C T (  seq M
(  .+  ,  G
) `  N )
) )
29 ovex 6105 . . . 4  |-  ( C T (  seq M
(  .+  ,  G
) `  N )
)  e.  _V
3028, 6, 29fvmpt 5762 . . 3  |-  ( (  seq M (  .+  ,  G ) `  N
)  e.  S  -> 
( ( z  e.  S  |->  ( C T z ) ) `  (  seq M (  .+  ,  G ) `  N
) )  =  ( C T (  seq M (  .+  ,  G ) `  N
) ) )
3127, 30syl 16 . 2  |-  ( ph  ->  ( ( z  e.  S  |->  ( C T z ) ) `  (  seq M (  .+  ,  G ) `  N
) )  =  ( C T (  seq M (  .+  ,  G ) `  N
) ) )
3226, 31eqtr3d 2467 1  |-  ( ph  ->  (  seq M ( 
.+  ,  F ) `
 N )  =  ( C T (  seq M (  .+  ,  G ) `  N
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1362    e. wcel 1755    e. cmpt 4338   ` cfv 5406  (class class class)co 6080   ZZ>=cuz 10849   ...cfz 11424    seqcseq 11790
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-cnex 9326  ax-resscn 9327  ax-1cn 9328  ax-icn 9329  ax-addcl 9330  ax-addrcl 9331  ax-mulcl 9332  ax-mulrcl 9333  ax-mulcom 9334  ax-addass 9335  ax-mulass 9336  ax-distr 9337  ax-i2m1 9338  ax-1ne0 9339  ax-1rid 9340  ax-rnegex 9341  ax-rrecex 9342  ax-cnre 9343  ax-pre-lttri 9344  ax-pre-lttrn 9345  ax-pre-ltadd 9346  ax-pre-mulgt0 9347
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-uni 4080  df-iun 4161  df-br 4281  df-opab 4339  df-mpt 4340  df-tr 4374  df-eprel 4619  df-id 4623  df-po 4628  df-so 4629  df-fr 4666  df-we 4668  df-ord 4709  df-on 4710  df-lim 4711  df-suc 4712  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-om 6466  df-1st 6566  df-2nd 6567  df-recs 6818  df-rdg 6852  df-er 7089  df-en 7299  df-dom 7300  df-sdom 7301  df-pnf 9408  df-mnf 9409  df-xr 9410  df-ltxr 9411  df-le 9412  df-sub 9585  df-neg 9586  df-nn 10311  df-n0 10568  df-z 10635  df-uz 10850  df-fz 11425  df-seq 11791
This theorem is referenced by:  isermulc2  13119  fsummulc2  13234  stirlinglem7  29721
  Copyright terms: Public domain W3C validator