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Theorem seqcaopr 11841
Description: The sum of two infinite series (generalized to an arbitrary commutative and associative operation). (Contributed by NM, 17-Mar-2005.) (Revised by Mario Carneiro, 30-May-2014.)
Hypotheses
Ref Expression
seqcaopr.1  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  .+  y
)  e.  S )
seqcaopr.2  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  .+  y
)  =  ( y 
.+  x ) )
seqcaopr.3  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S  /\  z  e.  S ) )  -> 
( ( x  .+  y )  .+  z
)  =  ( x 
.+  ( y  .+  z ) ) )
seqcaopr.4  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
seqcaopr.5  |-  ( (
ph  /\  k  e.  ( M ... N ) )  ->  ( F `  k )  e.  S
)
seqcaopr.6  |-  ( (
ph  /\  k  e.  ( M ... N ) )  ->  ( G `  k )  e.  S
)
seqcaopr.7  |-  ( (
ph  /\  k  e.  ( M ... N ) )  ->  ( H `  k )  =  ( ( F `  k
)  .+  ( G `  k ) ) )
Assertion
Ref Expression
seqcaopr  |-  ( ph  ->  (  seq M ( 
.+  ,  H ) `
 N )  =  ( (  seq M
(  .+  ,  F
) `  N )  .+  (  seq M ( 
.+  ,  G ) `
 N ) ) )
Distinct variable groups:    k, F    k, G    k, H    x, k, y, z, ph    k, M    .+ , k, x, y, z    S, k, x, y, z   
k, N
Allowed substitution hints:    F( x, y, z)    G( x, y, z)    H( x, y, z)    M( x, y, z)    N( x, y, z)

Proof of Theorem seqcaopr
Dummy variables  a 
b  c  d are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 seqcaopr.1 . . 3  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  .+  y
)  e.  S )
21caovclg 6253 . 2  |-  ( (
ph  /\  ( a  e.  S  /\  b  e.  S ) )  -> 
( a  .+  b
)  e.  S )
3 simpl 457 . . . . . . 7  |-  ( (
ph  /\  ( (
a  e.  S  /\  b  e.  S )  /\  ( c  e.  S  /\  d  e.  S
) ) )  ->  ph )
4 simprrl 763 . . . . . . 7  |-  ( (
ph  /\  ( (
a  e.  S  /\  b  e.  S )  /\  ( c  e.  S  /\  d  e.  S
) ) )  -> 
c  e.  S )
5 simprlr 762 . . . . . . 7  |-  ( (
ph  /\  ( (
a  e.  S  /\  b  e.  S )  /\  ( c  e.  S  /\  d  e.  S
) ) )  -> 
b  e.  S )
6 seqcaopr.2 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  .+  y
)  =  ( y 
.+  x ) )
76caovcomg 6256 . . . . . . 7  |-  ( (
ph  /\  ( c  e.  S  /\  b  e.  S ) )  -> 
( c  .+  b
)  =  ( b 
.+  c ) )
83, 4, 5, 7syl12anc 1216 . . . . . 6  |-  ( (
ph  /\  ( (
a  e.  S  /\  b  e.  S )  /\  ( c  e.  S  /\  d  e.  S
) ) )  -> 
( c  .+  b
)  =  ( b 
.+  c ) )
98oveq1d 6104 . . . . 5  |-  ( (
ph  /\  ( (
a  e.  S  /\  b  e.  S )  /\  ( c  e.  S  /\  d  e.  S
) ) )  -> 
( ( c  .+  b )  .+  d
)  =  ( ( b  .+  c ) 
.+  d ) )
10 simprrr 764 . . . . . 6  |-  ( (
ph  /\  ( (
a  e.  S  /\  b  e.  S )  /\  ( c  e.  S  /\  d  e.  S
) ) )  -> 
d  e.  S )
11 seqcaopr.3 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S  /\  z  e.  S ) )  -> 
( ( x  .+  y )  .+  z
)  =  ( x 
.+  ( y  .+  z ) ) )
1211caovassg 6259 . . . . . 6  |-  ( (
ph  /\  ( c  e.  S  /\  b  e.  S  /\  d  e.  S ) )  -> 
( ( c  .+  b )  .+  d
)  =  ( c 
.+  ( b  .+  d ) ) )
133, 4, 5, 10, 12syl13anc 1220 . . . . 5  |-  ( (
ph  /\  ( (
a  e.  S  /\  b  e.  S )  /\  ( c  e.  S  /\  d  e.  S
) ) )  -> 
( ( c  .+  b )  .+  d
)  =  ( c 
.+  ( b  .+  d ) ) )
1411caovassg 6259 . . . . . 6  |-  ( (
ph  /\  ( b  e.  S  /\  c  e.  S  /\  d  e.  S ) )  -> 
( ( b  .+  c )  .+  d
)  =  ( b 
.+  ( c  .+  d ) ) )
153, 5, 4, 10, 14syl13anc 1220 . . . . 5  |-  ( (
ph  /\  ( (
a  e.  S  /\  b  e.  S )  /\  ( c  e.  S  /\  d  e.  S
) ) )  -> 
( ( b  .+  c )  .+  d
)  =  ( b 
.+  ( c  .+  d ) ) )
169, 13, 153eqtr3d 2481 . . . 4  |-  ( (
ph  /\  ( (
a  e.  S  /\  b  e.  S )  /\  ( c  e.  S  /\  d  e.  S
) ) )  -> 
( c  .+  (
b  .+  d )
)  =  ( b 
.+  ( c  .+  d ) ) )
1716oveq2d 6105 . . 3  |-  ( (
ph  /\  ( (
a  e.  S  /\  b  e.  S )  /\  ( c  e.  S  /\  d  e.  S
) ) )  -> 
( a  .+  (
c  .+  ( b  .+  d ) ) )  =  ( a  .+  ( b  .+  (
c  .+  d )
) ) )
18 simprll 761 . . . 4  |-  ( (
ph  /\  ( (
a  e.  S  /\  b  e.  S )  /\  ( c  e.  S  /\  d  e.  S
) ) )  -> 
a  e.  S )
191caovclg 6253 . . . . 5  |-  ( (
ph  /\  ( b  e.  S  /\  d  e.  S ) )  -> 
( b  .+  d
)  e.  S )
203, 5, 10, 19syl12anc 1216 . . . 4  |-  ( (
ph  /\  ( (
a  e.  S  /\  b  e.  S )  /\  ( c  e.  S  /\  d  e.  S
) ) )  -> 
( b  .+  d
)  e.  S )
2111caovassg 6259 . . . 4  |-  ( (
ph  /\  ( a  e.  S  /\  c  e.  S  /\  (
b  .+  d )  e.  S ) )  -> 
( ( a  .+  c )  .+  (
b  .+  d )
)  =  ( a 
.+  ( c  .+  ( b  .+  d
) ) ) )
223, 18, 4, 20, 21syl13anc 1220 . . 3  |-  ( (
ph  /\  ( (
a  e.  S  /\  b  e.  S )  /\  ( c  e.  S  /\  d  e.  S
) ) )  -> 
( ( a  .+  c )  .+  (
b  .+  d )
)  =  ( a 
.+  ( c  .+  ( b  .+  d
) ) ) )
231caovclg 6253 . . . . 5  |-  ( (
ph  /\  ( c  e.  S  /\  d  e.  S ) )  -> 
( c  .+  d
)  e.  S )
2423adantrl 715 . . . 4  |-  ( (
ph  /\  ( (
a  e.  S  /\  b  e.  S )  /\  ( c  e.  S  /\  d  e.  S
) ) )  -> 
( c  .+  d
)  e.  S )
2511caovassg 6259 . . . 4  |-  ( (
ph  /\  ( a  e.  S  /\  b  e.  S  /\  (
c  .+  d )  e.  S ) )  -> 
( ( a  .+  b )  .+  (
c  .+  d )
)  =  ( a 
.+  ( b  .+  ( c  .+  d
) ) ) )
263, 18, 5, 24, 25syl13anc 1220 . . 3  |-  ( (
ph  /\  ( (
a  e.  S  /\  b  e.  S )  /\  ( c  e.  S  /\  d  e.  S
) ) )  -> 
( ( a  .+  b )  .+  (
c  .+  d )
)  =  ( a 
.+  ( b  .+  ( c  .+  d
) ) ) )
2717, 22, 263eqtr4d 2483 . 2  |-  ( (
ph  /\  ( (
a  e.  S  /\  b  e.  S )  /\  ( c  e.  S  /\  d  e.  S
) ) )  -> 
( ( a  .+  c )  .+  (
b  .+  d )
)  =  ( ( a  .+  b ) 
.+  ( c  .+  d ) ) )
28 seqcaopr.4 . 2  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
29 seqcaopr.5 . 2  |-  ( (
ph  /\  k  e.  ( M ... N ) )  ->  ( F `  k )  e.  S
)
30 seqcaopr.6 . 2  |-  ( (
ph  /\  k  e.  ( M ... N ) )  ->  ( G `  k )  e.  S
)
31 seqcaopr.7 . 2  |-  ( (
ph  /\  k  e.  ( M ... N ) )  ->  ( H `  k )  =  ( ( F `  k
)  .+  ( G `  k ) ) )
322, 2, 27, 28, 29, 30, 31seqcaopr2 11840 1  |-  ( ph  ->  (  seq M ( 
.+  ,  H ) `
 N )  =  ( (  seq M
(  .+  ,  F
) `  N )  .+  (  seq M ( 
.+  ,  G ) `
 N ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   ` cfv 5416  (class class class)co 6089   ZZ>=cuz 10859   ...cfz 11435    seqcseq 11804
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370  ax-cnex 9336  ax-resscn 9337  ax-1cn 9338  ax-icn 9339  ax-addcl 9340  ax-addrcl 9341  ax-mulcl 9342  ax-mulrcl 9343  ax-mulcom 9344  ax-addass 9345  ax-mulass 9346  ax-distr 9347  ax-i2m1 9348  ax-1ne0 9349  ax-1rid 9350  ax-rnegex 9351  ax-rrecex 9352  ax-cnre 9353  ax-pre-lttri 9354  ax-pre-lttrn 9355  ax-pre-ltadd 9356  ax-pre-mulgt0 9357
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-pss 3342  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-tp 3880  df-op 3882  df-uni 4090  df-iun 4171  df-br 4291  df-opab 4349  df-mpt 4350  df-tr 4384  df-eprel 4630  df-id 4634  df-po 4639  df-so 4640  df-fr 4677  df-we 4679  df-ord 4720  df-on 4721  df-lim 4722  df-suc 4723  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-riota 6050  df-ov 6092  df-oprab 6093  df-mpt2 6094  df-om 6475  df-1st 6575  df-2nd 6576  df-recs 6830  df-rdg 6864  df-er 7099  df-en 7309  df-dom 7310  df-sdom 7311  df-pnf 9418  df-mnf 9419  df-xr 9420  df-ltxr 9421  df-le 9422  df-sub 9595  df-neg 9596  df-nn 10321  df-n0 10578  df-z 10645  df-uz 10860  df-fz 11436  df-fzo 11547  df-seq 11805
This theorem is referenced by:  seradd  11846  mulgnn0di  16311  lgsdir  22667  lgsdi  22669  prodfmul  27403
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