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Theorem seq3caopr 10513
Description: The sum of two infinite series (generalized to an arbitrary commutative and associative operation). (Contributed by NM, 17-Mar-2005.) (Revised by Jim Kingdon, 23-Apr-2023.)
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 ) )
seq3caopr.5  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( F `  k )  e.  S
)
seq3caopr.6  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( G `  k )  e.  S
)
seq3caopr.7  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( H `  k )  =  ( ( F `  k
)  .+  ( G `  k ) ) )
Assertion
Ref Expression
seq3caopr  |-  ( ph  ->  (  seq M ( 
.+  ,  H ) `
 N )  =  ( (  seq M
(  .+  ,  F
) `  N )  .+  (  seq M ( 
.+  ,  G ) `
 N ) ) )
Distinct variable groups:    .+ , k, x, y, z    k, F   
k, G    k, H    k, M    k, N    S, k, x, y, z    ph, k, x, y, z
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 seq3caopr
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 6048 . 2  |-  ( (
ph  /\  ( a  e.  S  /\  b  e.  S ) )  -> 
( a  .+  b
)  e.  S )
3 simpl 109 . . . . . . 7  |-  ( (
ph  /\  ( (
a  e.  S  /\  b  e.  S )  /\  ( c  e.  S  /\  d  e.  S
) ) )  ->  ph )
4 simprrl 539 . . . . . . 7  |-  ( (
ph  /\  ( (
a  e.  S  /\  b  e.  S )  /\  ( c  e.  S  /\  d  e.  S
) ) )  -> 
c  e.  S )
5 simprlr 538 . . . . . . 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 6051 . . . . . . 7  |-  ( (
ph  /\  ( c  e.  S  /\  b  e.  S ) )  -> 
( c  .+  b
)  =  ( b 
.+  c ) )
83, 4, 5, 7syl12anc 1247 . . . . . 6  |-  ( (
ph  /\  ( (
a  e.  S  /\  b  e.  S )  /\  ( c  e.  S  /\  d  e.  S
) ) )  -> 
( c  .+  b
)  =  ( b 
.+  c ) )
98oveq1d 5910 . . . . 5  |-  ( (
ph  /\  ( (
a  e.  S  /\  b  e.  S )  /\  ( c  e.  S  /\  d  e.  S
) ) )  -> 
( ( c  .+  b )  .+  d
)  =  ( ( b  .+  c ) 
.+  d ) )
10 simprrr 540 . . . . . 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 6054 . . . . . 6  |-  ( (
ph  /\  ( c  e.  S  /\  b  e.  S  /\  d  e.  S ) )  -> 
( ( c  .+  b )  .+  d
)  =  ( c 
.+  ( b  .+  d ) ) )
133, 4, 5, 10, 12syl13anc 1251 . . . . 5  |-  ( (
ph  /\  ( (
a  e.  S  /\  b  e.  S )  /\  ( c  e.  S  /\  d  e.  S
) ) )  -> 
( ( c  .+  b )  .+  d
)  =  ( c 
.+  ( b  .+  d ) ) )
1411caovassg 6054 . . . . . 6  |-  ( (
ph  /\  ( b  e.  S  /\  c  e.  S  /\  d  e.  S ) )  -> 
( ( b  .+  c )  .+  d
)  =  ( b 
.+  ( c  .+  d ) ) )
153, 5, 4, 10, 14syl13anc 1251 . . . . 5  |-  ( (
ph  /\  ( (
a  e.  S  /\  b  e.  S )  /\  ( c  e.  S  /\  d  e.  S
) ) )  -> 
( ( b  .+  c )  .+  d
)  =  ( b 
.+  ( c  .+  d ) ) )
169, 13, 153eqtr3d 2230 . . . 4  |-  ( (
ph  /\  ( (
a  e.  S  /\  b  e.  S )  /\  ( c  e.  S  /\  d  e.  S
) ) )  -> 
( c  .+  (
b  .+  d )
)  =  ( b 
.+  ( c  .+  d ) ) )
1716oveq2d 5911 . . 3  |-  ( (
ph  /\  ( (
a  e.  S  /\  b  e.  S )  /\  ( c  e.  S  /\  d  e.  S
) ) )  -> 
( a  .+  (
c  .+  ( b  .+  d ) ) )  =  ( a  .+  ( b  .+  (
c  .+  d )
) ) )
18 simprll 537 . . . 4  |-  ( (
ph  /\  ( (
a  e.  S  /\  b  e.  S )  /\  ( c  e.  S  /\  d  e.  S
) ) )  -> 
a  e.  S )
191caovclg 6048 . . . . 5  |-  ( (
ph  /\  ( b  e.  S  /\  d  e.  S ) )  -> 
( b  .+  d
)  e.  S )
203, 5, 10, 19syl12anc 1247 . . . 4  |-  ( (
ph  /\  ( (
a  e.  S  /\  b  e.  S )  /\  ( c  e.  S  /\  d  e.  S
) ) )  -> 
( b  .+  d
)  e.  S )
2111caovassg 6054 . . . 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 1251 . . 3  |-  ( (
ph  /\  ( (
a  e.  S  /\  b  e.  S )  /\  ( c  e.  S  /\  d  e.  S
) ) )  -> 
( ( a  .+  c )  .+  (
b  .+  d )
)  =  ( a 
.+  ( c  .+  ( b  .+  d
) ) ) )
231caovclg 6048 . . . . 5  |-  ( (
ph  /\  ( c  e.  S  /\  d  e.  S ) )  -> 
( c  .+  d
)  e.  S )
2423adantrl 478 . . . 4  |-  ( (
ph  /\  ( (
a  e.  S  /\  b  e.  S )  /\  ( c  e.  S  /\  d  e.  S
) ) )  -> 
( c  .+  d
)  e.  S )
2511caovassg 6054 . . . 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 1251 . . 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 2232 . 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 seq3caopr.5 . 2  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( F `  k )  e.  S
)
30 seq3caopr.6 . 2  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( G `  k )  e.  S
)
31 seq3caopr.7 . 2  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( H `  k )  =  ( ( F `  k
)  .+  ( G `  k ) ) )
322, 2, 27, 28, 29, 30, 31seq3caopr2 10512 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 104    /\ w3a 980    = wceq 1364    e. wcel 2160   ` cfv 5235  (class class class)co 5895   ZZ>=cuz 9557    seqcseq 10475
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-iinf 4605  ax-cnex 7931  ax-resscn 7932  ax-1cn 7933  ax-1re 7934  ax-icn 7935  ax-addcl 7936  ax-addrcl 7937  ax-mulcl 7938  ax-addcom 7940  ax-addass 7942  ax-distr 7944  ax-i2m1 7945  ax-0lt1 7946  ax-0id 7948  ax-rnegex 7949  ax-cnre 7951  ax-pre-ltirr 7952  ax-pre-ltwlin 7953  ax-pre-lttrn 7954  ax-pre-ltadd 7956
This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4311  df-iord 4384  df-on 4386  df-ilim 4387  df-suc 4389  df-iom 4608  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-riota 5851  df-ov 5898  df-oprab 5899  df-mpo 5900  df-1st 6164  df-2nd 6165  df-recs 6329  df-frec 6415  df-pnf 8023  df-mnf 8024  df-xr 8025  df-ltxr 8026  df-le 8027  df-sub 8159  df-neg 8160  df-inn 8949  df-n0 9206  df-z 9283  df-uz 9558  df-fz 10038  df-fzo 10172  df-seqfrec 10476
This theorem is referenced by:  ser3add  10535  prod3fmul  11580
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