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Theorem seqcaopr 11099
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 6028 . 2  |-  ( (
ph  /\  ( a  e.  S  /\  b  e.  S ) )  -> 
( a  .+  b
)  e.  S )
3 simpl 443 . . . . . . 7  |-  ( (
ph  /\  ( (
a  e.  S  /\  b  e.  S )  /\  ( c  e.  S  /\  d  e.  S
) ) )  ->  ph )
4 simprrl 740 . . . . . . 7  |-  ( (
ph  /\  ( (
a  e.  S  /\  b  e.  S )  /\  ( c  e.  S  /\  d  e.  S
) ) )  -> 
c  e.  S )
5 simprlr 739 . . . . . . 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 6031 . . . . . . 7  |-  ( (
ph  /\  ( c  e.  S  /\  b  e.  S ) )  -> 
( c  .+  b
)  =  ( b 
.+  c ) )
83, 4, 5, 7syl12anc 1180 . . . . . 6  |-  ( (
ph  /\  ( (
a  e.  S  /\  b  e.  S )  /\  ( c  e.  S  /\  d  e.  S
) ) )  -> 
( c  .+  b
)  =  ( b 
.+  c ) )
98oveq1d 5889 . . . . 5  |-  ( (
ph  /\  ( (
a  e.  S  /\  b  e.  S )  /\  ( c  e.  S  /\  d  e.  S
) ) )  -> 
( ( c  .+  b )  .+  d
)  =  ( ( b  .+  c ) 
.+  d ) )
10 simprrr 741 . . . . . 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 6034 . . . . . 6  |-  ( (
ph  /\  ( c  e.  S  /\  b  e.  S  /\  d  e.  S ) )  -> 
( ( c  .+  b )  .+  d
)  =  ( c 
.+  ( b  .+  d ) ) )
133, 4, 5, 10, 12syl13anc 1184 . . . . 5  |-  ( (
ph  /\  ( (
a  e.  S  /\  b  e.  S )  /\  ( c  e.  S  /\  d  e.  S
) ) )  -> 
( ( c  .+  b )  .+  d
)  =  ( c 
.+  ( b  .+  d ) ) )
1411caovassg 6034 . . . . . 6  |-  ( (
ph  /\  ( b  e.  S  /\  c  e.  S  /\  d  e.  S ) )  -> 
( ( b  .+  c )  .+  d
)  =  ( b 
.+  ( c  .+  d ) ) )
153, 5, 4, 10, 14syl13anc 1184 . . . . 5  |-  ( (
ph  /\  ( (
a  e.  S  /\  b  e.  S )  /\  ( c  e.  S  /\  d  e.  S
) ) )  -> 
( ( b  .+  c )  .+  d
)  =  ( b 
.+  ( c  .+  d ) ) )
169, 13, 153eqtr3d 2336 . . . 4  |-  ( (
ph  /\  ( (
a  e.  S  /\  b  e.  S )  /\  ( c  e.  S  /\  d  e.  S
) ) )  -> 
( c  .+  (
b  .+  d )
)  =  ( b 
.+  ( c  .+  d ) ) )
1716oveq2d 5890 . . 3  |-  ( (
ph  /\  ( (
a  e.  S  /\  b  e.  S )  /\  ( c  e.  S  /\  d  e.  S
) ) )  -> 
( a  .+  (
c  .+  ( b  .+  d ) ) )  =  ( a  .+  ( b  .+  (
c  .+  d )
) ) )
18 simprll 738 . . . 4  |-  ( (
ph  /\  ( (
a  e.  S  /\  b  e.  S )  /\  ( c  e.  S  /\  d  e.  S
) ) )  -> 
a  e.  S )
191caovclg 6028 . . . . 5  |-  ( (
ph  /\  ( b  e.  S  /\  d  e.  S ) )  -> 
( b  .+  d
)  e.  S )
203, 5, 10, 19syl12anc 1180 . . . 4  |-  ( (
ph  /\  ( (
a  e.  S  /\  b  e.  S )  /\  ( c  e.  S  /\  d  e.  S
) ) )  -> 
( b  .+  d
)  e.  S )
2111caovassg 6034 . . . 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 1184 . . 3  |-  ( (
ph  /\  ( (
a  e.  S  /\  b  e.  S )  /\  ( c  e.  S  /\  d  e.  S
) ) )  -> 
( ( a  .+  c )  .+  (
b  .+  d )
)  =  ( a 
.+  ( c  .+  ( b  .+  d
) ) ) )
231caovclg 6028 . . . . 5  |-  ( (
ph  /\  ( c  e.  S  /\  d  e.  S ) )  -> 
( c  .+  d
)  e.  S )
2423adantrl 696 . . . 4  |-  ( (
ph  /\  ( (
a  e.  S  /\  b  e.  S )  /\  ( c  e.  S  /\  d  e.  S
) ) )  -> 
( c  .+  d
)  e.  S )
2511caovassg 6034 . . . 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 1184 . . 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 2338 . 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 11098 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 358    /\ w3a 934    = wceq 1632    e. wcel 1696   ` cfv 5271  (class class class)co 5874   ZZ>=cuz 10246   ...cfz 10798    seq cseq 11062
This theorem is referenced by:  seradd  11104  mulgnn0di  15141  lgsdir  20585  lgsdi  20587  fprodadd  25455
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-n0 9982  df-z 10041  df-uz 10247  df-fz 10799  df-fzo 10887  df-seq 11063
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