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Theorem seqcaopr 11361
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 6240 . 2  |-  ( (
ph  /\  ( a  e.  S  /\  b  e.  S ) )  -> 
( a  .+  b
)  e.  S )
3 simpl 445 . . . . . . 7  |-  ( (
ph  /\  ( (
a  e.  S  /\  b  e.  S )  /\  ( c  e.  S  /\  d  e.  S
) ) )  ->  ph )
4 simprrl 742 . . . . . . 7  |-  ( (
ph  /\  ( (
a  e.  S  /\  b  e.  S )  /\  ( c  e.  S  /\  d  e.  S
) ) )  -> 
c  e.  S )
5 simprlr 741 . . . . . . 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 6243 . . . . . . 7  |-  ( (
ph  /\  ( c  e.  S  /\  b  e.  S ) )  -> 
( c  .+  b
)  =  ( b 
.+  c ) )
83, 4, 5, 7syl12anc 1183 . . . . . 6  |-  ( (
ph  /\  ( (
a  e.  S  /\  b  e.  S )  /\  ( c  e.  S  /\  d  e.  S
) ) )  -> 
( c  .+  b
)  =  ( b 
.+  c ) )
98oveq1d 6097 . . . . 5  |-  ( (
ph  /\  ( (
a  e.  S  /\  b  e.  S )  /\  ( c  e.  S  /\  d  e.  S
) ) )  -> 
( ( c  .+  b )  .+  d
)  =  ( ( b  .+  c ) 
.+  d ) )
10 simprrr 743 . . . . . 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 6246 . . . . . 6  |-  ( (
ph  /\  ( c  e.  S  /\  b  e.  S  /\  d  e.  S ) )  -> 
( ( c  .+  b )  .+  d
)  =  ( c 
.+  ( b  .+  d ) ) )
133, 4, 5, 10, 12syl13anc 1187 . . . . 5  |-  ( (
ph  /\  ( (
a  e.  S  /\  b  e.  S )  /\  ( c  e.  S  /\  d  e.  S
) ) )  -> 
( ( c  .+  b )  .+  d
)  =  ( c 
.+  ( b  .+  d ) ) )
1411caovassg 6246 . . . . . 6  |-  ( (
ph  /\  ( b  e.  S  /\  c  e.  S  /\  d  e.  S ) )  -> 
( ( b  .+  c )  .+  d
)  =  ( b 
.+  ( c  .+  d ) ) )
153, 5, 4, 10, 14syl13anc 1187 . . . . 5  |-  ( (
ph  /\  ( (
a  e.  S  /\  b  e.  S )  /\  ( c  e.  S  /\  d  e.  S
) ) )  -> 
( ( b  .+  c )  .+  d
)  =  ( b 
.+  ( c  .+  d ) ) )
169, 13, 153eqtr3d 2477 . . . 4  |-  ( (
ph  /\  ( (
a  e.  S  /\  b  e.  S )  /\  ( c  e.  S  /\  d  e.  S
) ) )  -> 
( c  .+  (
b  .+  d )
)  =  ( b 
.+  ( c  .+  d ) ) )
1716oveq2d 6098 . . 3  |-  ( (
ph  /\  ( (
a  e.  S  /\  b  e.  S )  /\  ( c  e.  S  /\  d  e.  S
) ) )  -> 
( a  .+  (
c  .+  ( b  .+  d ) ) )  =  ( a  .+  ( b  .+  (
c  .+  d )
) ) )
18 simprll 740 . . . 4  |-  ( (
ph  /\  ( (
a  e.  S  /\  b  e.  S )  /\  ( c  e.  S  /\  d  e.  S
) ) )  -> 
a  e.  S )
191caovclg 6240 . . . . 5  |-  ( (
ph  /\  ( b  e.  S  /\  d  e.  S ) )  -> 
( b  .+  d
)  e.  S )
203, 5, 10, 19syl12anc 1183 . . . 4  |-  ( (
ph  /\  ( (
a  e.  S  /\  b  e.  S )  /\  ( c  e.  S  /\  d  e.  S
) ) )  -> 
( b  .+  d
)  e.  S )
2111caovassg 6246 . . . 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 1187 . . 3  |-  ( (
ph  /\  ( (
a  e.  S  /\  b  e.  S )  /\  ( c  e.  S  /\  d  e.  S
) ) )  -> 
( ( a  .+  c )  .+  (
b  .+  d )
)  =  ( a 
.+  ( c  .+  ( b  .+  d
) ) ) )
231caovclg 6240 . . . . 5  |-  ( (
ph  /\  ( c  e.  S  /\  d  e.  S ) )  -> 
( c  .+  d
)  e.  S )
2423adantrl 698 . . . 4  |-  ( (
ph  /\  ( (
a  e.  S  /\  b  e.  S )  /\  ( c  e.  S  /\  d  e.  S
) ) )  -> 
( c  .+  d
)  e.  S )
2511caovassg 6246 . . . 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 1187 . . 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 2479 . 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 11360 1  |-  ( ph  ->  (  seq  M ( 
.+  ,  H ) `
 N )  =  ( (  seq  M
(  .+  ,  F
) `  N )  .+  (  seq  M ( 
.+  ,  G ) `
 N ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   ` cfv 5455  (class class class)co 6082   ZZ>=cuz 10489   ...cfz 11044    seq cseq 11324
This theorem is referenced by:  seradd  11366  mulgnn0di  15449  lgsdir  21115  lgsdi  21117  prodfmul  25219
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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 2418  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702  ax-cnex 9047  ax-resscn 9048  ax-1cn 9049  ax-icn 9050  ax-addcl 9051  ax-addrcl 9052  ax-mulcl 9053  ax-mulrcl 9054  ax-mulcom 9055  ax-addass 9056  ax-mulass 9057  ax-distr 9058  ax-i2m1 9059  ax-1ne0 9060  ax-1rid 9061  ax-rnegex 9062  ax-rrecex 9063  ax-cnre 9064  ax-pre-lttri 9065  ax-pre-lttrn 9066  ax-pre-ltadd 9067  ax-pre-mulgt0 9068
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 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-nel 2603  df-ral 2711  df-rex 2712  df-reu 2713  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-pss 3337  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-tp 3823  df-op 3824  df-uni 4017  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-tr 4304  df-eprel 4495  df-id 4499  df-po 4504  df-so 4505  df-fr 4542  df-we 4544  df-ord 4585  df-on 4586  df-lim 4587  df-suc 4588  df-om 4847  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-1st 6350  df-2nd 6351  df-riota 6550  df-recs 6634  df-rdg 6669  df-er 6906  df-en 7111  df-dom 7112  df-sdom 7113  df-pnf 9123  df-mnf 9124  df-xr 9125  df-ltxr 9126  df-le 9127  df-sub 9294  df-neg 9295  df-nn 10002  df-n0 10223  df-z 10284  df-uz 10490  df-fz 11045  df-fzo 11137  df-seq 11325
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