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Theorem seqdistr 11333
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 6052 . . . . . 6  |-  ( z  =  ( x  .+  y )  ->  ( C T z )  =  ( C T ( x  .+  y ) ) )
6 eqid 2408 . . . . . 6  |-  ( z  e.  S  |->  ( C T z ) )  =  ( z  e.  S  |->  ( C T z ) )
7 ovex 6069 . . . . . 6  |-  ( C T ( x  .+  y ) )  e. 
_V
85, 6, 7fvmpt 5769 . . . . 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 6052 . . . . . . 7  |-  ( z  =  x  ->  ( C T z )  =  ( C T x ) )
11 ovex 6069 . . . . . . 7  |-  ( C T x )  e. 
_V
1210, 6, 11fvmpt 5769 . . . . . 6  |-  ( x  e.  S  ->  (
( z  e.  S  |->  ( C T z ) ) `  x
)  =  ( C T x ) )
1312ad2antrl 709 . . . . 5  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( ( z  e.  S  |->  ( C T z ) ) `  x )  =  ( C T x ) )
14 oveq2 6052 . . . . . . 7  |-  ( z  =  y  ->  ( C T z )  =  ( C T y ) )
15 ovex 6069 . . . . . . 7  |-  ( C T y )  e. 
_V
1614, 6, 15fvmpt 5769 . . . . . 6  |-  ( y  e.  S  ->  (
( z  e.  S  |->  ( C T z ) ) `  y
)  =  ( C T y ) )
1716ad2antll 710 . . . . 5  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( ( z  e.  S  |->  ( C T z ) ) `  y )  =  ( C T y ) )
1813, 17oveq12d 6062 . . . 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 2450 . . 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 6052 . . . . . 6  |-  ( z  =  ( G `  x )  ->  ( C T z )  =  ( C T ( G `  x ) ) )
21 ovex 6069 . . . . . 6  |-  ( C T ( G `  x ) )  e. 
_V
2220, 6, 21fvmpt 5769 . . . . 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 2443 . . 3  |-  ( (
ph  /\  x  e.  ( M ... N ) )  ->  ( (
z  e.  S  |->  ( C T z ) ) `  ( G `
 x ) )  =  ( F `  x ) )
261, 2, 3, 19, 25seqhomo 11329 . 2  |-  ( ph  ->  ( ( z  e.  S  |->  ( C T z ) ) `  (  seq  M (  .+  ,  G ) `  N
) )  =  (  seq  M (  .+  ,  F ) `  N
) )
273, 2, 1seqcl 11302 . . 3  |-  ( ph  ->  (  seq  M ( 
.+  ,  G ) `
 N )  e.  S )
28 oveq2 6052 . . . 4  |-  ( z  =  (  seq  M
(  .+  ,  G
) `  N )  ->  ( C T z )  =  ( C T (  seq  M
(  .+  ,  G
) `  N )
) )
29 ovex 6069 . . . 4  |-  ( C T (  seq  M
(  .+  ,  G
) `  N )
)  e.  _V
3028, 6, 29fvmpt 5769 . . 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 2442 1  |-  ( ph  ->  (  seq  M ( 
.+  ,  F ) `
 N )  =  ( C T (  seq  M (  .+  ,  G ) `  N
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721    e. cmpt 4230   ` cfv 5417  (class class class)co 6044   ZZ>=cuz 10448   ...cfz 11003    seq cseq 11282
This theorem is referenced by:  isermulc2  12410  fsummulc2  12526  stirlinglem7  27700
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 2389  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664  ax-cnex 9006  ax-resscn 9007  ax-1cn 9008  ax-icn 9009  ax-addcl 9010  ax-addrcl 9011  ax-mulcl 9012  ax-mulrcl 9013  ax-mulcom 9014  ax-addass 9015  ax-mulass 9016  ax-distr 9017  ax-i2m1 9018  ax-1ne0 9019  ax-1rid 9020  ax-rnegex 9021  ax-rrecex 9022  ax-cnre 9023  ax-pre-lttri 9024  ax-pre-lttrn 9025  ax-pre-ltadd 9026  ax-pre-mulgt0 9027
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 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-nel 2574  df-ral 2675  df-rex 2676  df-reu 2677  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-pss 3300  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-tp 3786  df-op 3787  df-uni 3980  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-tr 4267  df-eprel 4458  df-id 4462  df-po 4467  df-so 4468  df-fr 4505  df-we 4507  df-ord 4548  df-on 4549  df-lim 4550  df-suc 4551  df-om 4809  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-1st 6312  df-2nd 6313  df-riota 6512  df-recs 6596  df-rdg 6631  df-er 6868  df-en 7073  df-dom 7074  df-sdom 7075  df-pnf 9082  df-mnf 9083  df-xr 9084  df-ltxr 9085  df-le 9086  df-sub 9253  df-neg 9254  df-nn 9961  df-n0 10182  df-z 10243  df-uz 10449  df-fz 11004  df-seq 11283
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