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Theorem isersub 9560
Description: The difference of two infinite series. (Contributed by NM, 17-Mar-2005.) (Revised by Mario Carneiro, 27-May-2014.)
Hypotheses
Ref Expression
isersub.1  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
isersub.2  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( F `  k )  e.  CC )
isersub.3  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( G `  k )  e.  CC )
isersub.4  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( H `  k )  =  ( ( F `  k
)  -  ( G `
 k ) ) )
Assertion
Ref Expression
isersub  |-  ( ph  ->  (  seq M (  +  ,  H ,  CC ) `  N )  =  ( (  seq M (  +  ,  F ,  CC ) `  N )  -  (  seq M (  +  ,  G ,  CC ) `  N ) ) )
Distinct variable groups:    k, F    k, G    k, H    k, M    k, N    ph, k

Proof of Theorem isersub
Dummy variables  w  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 addcl 7160 . . 3  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( x  +  y )  e.  CC )
21adantl 271 . 2  |-  ( (
ph  /\  ( x  e.  CC  /\  y  e.  CC ) )  -> 
( x  +  y )  e.  CC )
3 subcl 7374 . . 3  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( x  -  y
)  e.  CC )
43adantl 271 . 2  |-  ( (
ph  /\  ( x  e.  CC  /\  y  e.  CC ) )  -> 
( x  -  y
)  e.  CC )
5 addsub4 7418 . . . 4  |-  ( ( ( x  e.  CC  /\  y  e.  CC )  /\  ( z  e.  CC  /\  w  e.  CC ) )  -> 
( ( x  +  y )  -  (
z  +  w ) )  =  ( ( x  -  z )  +  ( y  -  w ) ) )
65eqcomd 2087 . . 3  |-  ( ( ( x  e.  CC  /\  y  e.  CC )  /\  ( z  e.  CC  /\  w  e.  CC ) )  -> 
( ( x  -  z )  +  ( y  -  w ) )  =  ( ( x  +  y )  -  ( z  +  w ) ) )
76adantl 271 . 2  |-  ( (
ph  /\  ( (
x  e.  CC  /\  y  e.  CC )  /\  ( z  e.  CC  /\  w  e.  CC ) ) )  ->  (
( x  -  z
)  +  ( y  -  w ) )  =  ( ( x  +  y )  -  ( z  +  w
) ) )
8 isersub.1 . 2  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
9 isersub.2 . 2  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( F `  k )  e.  CC )
10 isersub.3 . 2  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( G `  k )  e.  CC )
11 isersub.4 . 2  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( H `  k )  =  ( ( F `  k
)  -  ( G `
 k ) ) )
12 cnex 7159 . . 3  |-  CC  e.  _V
1312a1i 9 . 2  |-  ( ph  ->  CC  e.  _V )
142, 4, 7, 8, 9, 10, 11, 13iseqcaopr2 9557 1  |-  ( ph  ->  (  seq M (  +  ,  H ,  CC ) `  N )  =  ( (  seq M (  +  ,  F ,  CC ) `  N )  -  (  seq M (  +  ,  G ,  CC ) `  N ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1285    e. wcel 1434   _Vcvv 2602   ` cfv 4932  (class class class)co 5543   CCcc 7041    + caddc 7046    - cmin 7346   ZZ>=cuz 8700    seqcseq 9521
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-coll 3901  ax-sep 3904  ax-nul 3912  ax-pow 3956  ax-pr 3972  ax-un 4196  ax-setind 4288  ax-iinf 4337  ax-cnex 7129  ax-resscn 7130  ax-1cn 7131  ax-1re 7132  ax-icn 7133  ax-addcl 7134  ax-addrcl 7135  ax-mulcl 7136  ax-addcom 7138  ax-addass 7140  ax-distr 7142  ax-i2m1 7143  ax-0lt1 7144  ax-0id 7146  ax-rnegex 7147  ax-cnre 7149  ax-pre-ltirr 7150  ax-pre-ltwlin 7151  ax-pre-lttrn 7152  ax-pre-ltadd 7154
This theorem depends on definitions:  df-bi 115  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-nel 2341  df-ral 2354  df-rex 2355  df-reu 2356  df-rab 2358  df-v 2604  df-sbc 2817  df-csb 2910  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3259  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-int 3645  df-iun 3688  df-br 3794  df-opab 3848  df-mpt 3849  df-tr 3884  df-id 4056  df-iord 4129  df-on 4131  df-ilim 4132  df-suc 4134  df-iom 4340  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-res 4383  df-ima 4384  df-iota 4897  df-fun 4934  df-fn 4935  df-f 4936  df-f1 4937  df-fo 4938  df-f1o 4939  df-fv 4940  df-riota 5499  df-ov 5546  df-oprab 5547  df-mpt2 5548  df-1st 5798  df-2nd 5799  df-recs 5954  df-frec 6040  df-pnf 7217  df-mnf 7218  df-xr 7219  df-ltxr 7220  df-le 7221  df-sub 7348  df-neg 7349  df-inn 8107  df-n0 8356  df-z 8433  df-uz 8701  df-fz 9106  df-fzo 9230  df-iseq 9522
This theorem is referenced by:  serile  9571
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