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Theorem seqdistr 12116
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 6285 . . . . . 6  |-  ( z  =  ( x  .+  y )  ->  ( C T z )  =  ( C T ( x  .+  y ) ) )
6 eqid 2462 . . . . . 6  |-  ( z  e.  S  |->  ( C T z ) )  =  ( z  e.  S  |->  ( C T z ) )
7 ovex 6302 . . . . . 6  |-  ( C T ( x  .+  y ) )  e. 
_V
85, 6, 7fvmpt 5943 . . . . 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 6285 . . . . . . 7  |-  ( z  =  x  ->  ( C T z )  =  ( C T x ) )
11 ovex 6302 . . . . . . 7  |-  ( C T x )  e. 
_V
1210, 6, 11fvmpt 5943 . . . . . 6  |-  ( x  e.  S  ->  (
( z  e.  S  |->  ( C T z ) ) `  x
)  =  ( C T x ) )
1312ad2antrl 727 . . . . 5  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( ( z  e.  S  |->  ( C T z ) ) `  x )  =  ( C T x ) )
14 oveq2 6285 . . . . . . 7  |-  ( z  =  y  ->  ( C T z )  =  ( C T y ) )
15 ovex 6302 . . . . . . 7  |-  ( C T y )  e. 
_V
1614, 6, 15fvmpt 5943 . . . . . 6  |-  ( y  e.  S  ->  (
( z  e.  S  |->  ( C T z ) ) `  y
)  =  ( C T y ) )
1716ad2antll 728 . . . . 5  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( ( z  e.  S  |->  ( C T z ) ) `  y )  =  ( C T y ) )
1813, 17oveq12d 6295 . . . 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 2513 . . 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 6285 . . . . . 6  |-  ( z  =  ( G `  x )  ->  ( C T z )  =  ( C T ( G `  x ) ) )
21 ovex 6302 . . . . . 6  |-  ( C T ( G `  x ) )  e. 
_V
2220, 6, 21fvmpt 5943 . . . . 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 2506 . . 3  |-  ( (
ph  /\  x  e.  ( M ... N ) )  ->  ( (
z  e.  S  |->  ( C T z ) ) `  ( G `
 x ) )  =  ( F `  x ) )
261, 2, 3, 19, 25seqhomo 12112 . 2  |-  ( ph  ->  ( ( z  e.  S  |->  ( C T z ) ) `  (  seq M (  .+  ,  G ) `  N
) )  =  (  seq M (  .+  ,  F ) `  N
) )
273, 2, 1seqcl 12085 . . 3  |-  ( ph  ->  (  seq M ( 
.+  ,  G ) `
 N )  e.  S )
28 oveq2 6285 . . . 4  |-  ( z  =  (  seq M
(  .+  ,  G
) `  N )  ->  ( C T z )  =  ( C T (  seq M
(  .+  ,  G
) `  N )
) )
29 ovex 6302 . . . 4  |-  ( C T (  seq M
(  .+  ,  G
) `  N )
)  e.  _V
3028, 6, 29fvmpt 5943 . . 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 2505 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 1374    e. wcel 1762    |-> cmpt 4500   ` cfv 5581  (class class class)co 6277   ZZ>=cuz 11073   ...cfz 11663    seqcseq 12065
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 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-cnex 9539  ax-resscn 9540  ax-1cn 9541  ax-icn 9542  ax-addcl 9543  ax-addrcl 9544  ax-mulcl 9545  ax-mulrcl 9546  ax-mulcom 9547  ax-addass 9548  ax-mulass 9549  ax-distr 9550  ax-i2m1 9551  ax-1ne0 9552  ax-1rid 9553  ax-rnegex 9554  ax-rrecex 9555  ax-cnre 9556  ax-pre-lttri 9557  ax-pre-lttrn 9558  ax-pre-ltadd 9559  ax-pre-mulgt0 9560
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-nel 2660  df-ral 2814  df-rex 2815  df-reu 2816  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-tp 4027  df-op 4029  df-uni 4241  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-tr 4536  df-eprel 4786  df-id 4790  df-po 4795  df-so 4796  df-fr 4833  df-we 4835  df-ord 4876  df-on 4877  df-lim 4878  df-suc 4879  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6674  df-1st 6776  df-2nd 6777  df-recs 7034  df-rdg 7068  df-er 7303  df-en 7509  df-dom 7510  df-sdom 7511  df-pnf 9621  df-mnf 9622  df-xr 9623  df-ltxr 9624  df-le 9625  df-sub 9798  df-neg 9799  df-nn 10528  df-n0 10787  df-z 10856  df-uz 11074  df-fz 11664  df-seq 12066
This theorem is referenced by:  isermulc2  13431  fsummulc2  13550  stirlinglem7  31337
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