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Theorem seq3caopr 10469
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 6021 . 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 6024 . . . . . . 7  |-  ( (
ph  /\  ( c  e.  S  /\  b  e.  S ) )  -> 
( c  .+  b
)  =  ( b 
.+  c ) )
83, 4, 5, 7syl12anc 1236 . . . . . 6  |-  ( (
ph  /\  ( (
a  e.  S  /\  b  e.  S )  /\  ( c  e.  S  /\  d  e.  S
) ) )  -> 
( c  .+  b
)  =  ( b 
.+  c ) )
98oveq1d 5884 . . . . 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 6027 . . . . . 6  |-  ( (
ph  /\  ( c  e.  S  /\  b  e.  S  /\  d  e.  S ) )  -> 
( ( c  .+  b )  .+  d
)  =  ( c 
.+  ( b  .+  d ) ) )
133, 4, 5, 10, 12syl13anc 1240 . . . . 5  |-  ( (
ph  /\  ( (
a  e.  S  /\  b  e.  S )  /\  ( c  e.  S  /\  d  e.  S
) ) )  -> 
( ( c  .+  b )  .+  d
)  =  ( c 
.+  ( b  .+  d ) ) )
1411caovassg 6027 . . . . . 6  |-  ( (
ph  /\  ( b  e.  S  /\  c  e.  S  /\  d  e.  S ) )  -> 
( ( b  .+  c )  .+  d
)  =  ( b 
.+  ( c  .+  d ) ) )
153, 5, 4, 10, 14syl13anc 1240 . . . . 5  |-  ( (
ph  /\  ( (
a  e.  S  /\  b  e.  S )  /\  ( c  e.  S  /\  d  e.  S
) ) )  -> 
( ( b  .+  c )  .+  d
)  =  ( b 
.+  ( c  .+  d ) ) )
169, 13, 153eqtr3d 2218 . . . 4  |-  ( (
ph  /\  ( (
a  e.  S  /\  b  e.  S )  /\  ( c  e.  S  /\  d  e.  S
) ) )  -> 
( c  .+  (
b  .+  d )
)  =  ( b 
.+  ( c  .+  d ) ) )
1716oveq2d 5885 . . 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 6021 . . . . 5  |-  ( (
ph  /\  ( b  e.  S  /\  d  e.  S ) )  -> 
( b  .+  d
)  e.  S )
203, 5, 10, 19syl12anc 1236 . . . 4  |-  ( (
ph  /\  ( (
a  e.  S  /\  b  e.  S )  /\  ( c  e.  S  /\  d  e.  S
) ) )  -> 
( b  .+  d
)  e.  S )
2111caovassg 6027 . . . 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 1240 . . 3  |-  ( (
ph  /\  ( (
a  e.  S  /\  b  e.  S )  /\  ( c  e.  S  /\  d  e.  S
) ) )  -> 
( ( a  .+  c )  .+  (
b  .+  d )
)  =  ( a 
.+  ( c  .+  ( b  .+  d
) ) ) )
231caovclg 6021 . . . . 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 6027 . . . 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 1240 . . 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 2220 . 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 10468 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 978    = wceq 1353    e. wcel 2148   ` cfv 5212  (class class class)co 5869   ZZ>=cuz 9517    seqcseq 10431
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584  ax-cnex 7893  ax-resscn 7894  ax-1cn 7895  ax-1re 7896  ax-icn 7897  ax-addcl 7898  ax-addrcl 7899  ax-mulcl 7900  ax-addcom 7902  ax-addass 7904  ax-distr 7906  ax-i2m1 7907  ax-0lt1 7908  ax-0id 7910  ax-rnegex 7911  ax-cnre 7913  ax-pre-ltirr 7914  ax-pre-ltwlin 7915  ax-pre-lttrn 7916  ax-pre-ltadd 7918
This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-id 4290  df-iord 4363  df-on 4365  df-ilim 4366  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-riota 5825  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-recs 6300  df-frec 6386  df-pnf 7984  df-mnf 7985  df-xr 7986  df-ltxr 7987  df-le 7988  df-sub 8120  df-neg 8121  df-inn 8909  df-n0 9166  df-z 9243  df-uz 9518  df-fz 9996  df-fzo 10129  df-seqfrec 10432
This theorem is referenced by:  ser3add  10491  prod3fmul  11533
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