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Theorem seqdistr 11379
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 6092 . . . . . 6  |-  ( z  =  ( x  .+  y )  ->  ( C T z )  =  ( C T ( x  .+  y ) ) )
6 eqid 2438 . . . . . 6  |-  ( z  e.  S  |->  ( C T z ) )  =  ( z  e.  S  |->  ( C T z ) )
7 ovex 6109 . . . . . 6  |-  ( C T ( x  .+  y ) )  e. 
_V
85, 6, 7fvmpt 5809 . . . . 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 6092 . . . . . . 7  |-  ( z  =  x  ->  ( C T z )  =  ( C T x ) )
11 ovex 6109 . . . . . . 7  |-  ( C T x )  e. 
_V
1210, 6, 11fvmpt 5809 . . . . . 6  |-  ( x  e.  S  ->  (
( z  e.  S  |->  ( C T z ) ) `  x
)  =  ( C T x ) )
1312ad2antrl 710 . . . . 5  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( ( z  e.  S  |->  ( C T z ) ) `  x )  =  ( C T x ) )
14 oveq2 6092 . . . . . . 7  |-  ( z  =  y  ->  ( C T z )  =  ( C T y ) )
15 ovex 6109 . . . . . . 7  |-  ( C T y )  e. 
_V
1614, 6, 15fvmpt 5809 . . . . . 6  |-  ( y  e.  S  ->  (
( z  e.  S  |->  ( C T z ) ) `  y
)  =  ( C T y ) )
1716ad2antll 711 . . . . 5  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( ( z  e.  S  |->  ( C T z ) ) `  y )  =  ( C T y ) )
1813, 17oveq12d 6102 . . . 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 2480 . . 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 6092 . . . . . 6  |-  ( z  =  ( G `  x )  ->  ( C T z )  =  ( C T ( G `  x ) ) )
21 ovex 6109 . . . . . 6  |-  ( C T ( G `  x ) )  e. 
_V
2220, 6, 21fvmpt 5809 . . . . 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 2473 . . 3  |-  ( (
ph  /\  x  e.  ( M ... N ) )  ->  ( (
z  e.  S  |->  ( C T z ) ) `  ( G `
 x ) )  =  ( F `  x ) )
261, 2, 3, 19, 25seqhomo 11375 . 2  |-  ( ph  ->  ( ( z  e.  S  |->  ( C T z ) ) `  (  seq  M (  .+  ,  G ) `  N
) )  =  (  seq  M (  .+  ,  F ) `  N
) )
273, 2, 1seqcl 11348 . . 3  |-  ( ph  ->  (  seq  M ( 
.+  ,  G ) `
 N )  e.  S )
28 oveq2 6092 . . . 4  |-  ( z  =  (  seq  M
(  .+  ,  G
) `  N )  ->  ( C T z )  =  ( C T (  seq  M
(  .+  ,  G
) `  N )
) )
29 ovex 6109 . . . 4  |-  ( C T (  seq  M
(  .+  ,  G
) `  N )
)  e.  _V
3028, 6, 29fvmpt 5809 . . 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 2472 1  |-  ( ph  ->  (  seq  M ( 
.+  ,  F ) `
 N )  =  ( C T (  seq  M (  .+  ,  G ) `  N
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726    e. cmpt 4269   ` cfv 5457  (class class class)co 6084   ZZ>=cuz 10493   ...cfz 11048    seq cseq 11328
This theorem is referenced by:  isermulc2  12456  fsummulc2  12572  stirlinglem7  27819
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-riota 6552  df-recs 6636  df-rdg 6671  df-er 6908  df-en 7113  df-dom 7114  df-sdom 7115  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-nn 10006  df-n0 10227  df-z 10288  df-uz 10494  df-fz 11049  df-seq 11329
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