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Theorem sersub 11366
Description: The difference of two infinite series. (Contributed by NM, 17-Mar-2005.) (Revised by Mario Carneiro, 27-May-2014.)
Hypotheses
Ref Expression
sersub.1  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
sersub.2  |-  ( (
ph  /\  k  e.  ( M ... N ) )  ->  ( F `  k )  e.  CC )
sersub.3  |-  ( (
ph  /\  k  e.  ( M ... N ) )  ->  ( G `  k )  e.  CC )
sersub.4  |-  ( (
ph  /\  k  e.  ( M ... N ) )  ->  ( H `  k )  =  ( ( F `  k
)  -  ( G `
 k ) ) )
Assertion
Ref Expression
sersub  |-  ( ph  ->  (  seq  M (  +  ,  H ) `
 N )  =  ( (  seq  M
(  +  ,  F
) `  N )  -  (  seq  M (  +  ,  G ) `
 N ) ) )
Distinct variable groups:    k, F    k, G    k, M    ph, k    k, H    k, N

Proof of Theorem sersub
Dummy variables  w  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 addcl 9072 . . 3  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( x  +  y )  e.  CC )
21adantl 453 . 2  |-  ( (
ph  /\  ( x  e.  CC  /\  y  e.  CC ) )  -> 
( x  +  y )  e.  CC )
3 subcl 9305 . . 3  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( x  -  y
)  e.  CC )
43adantl 453 . 2  |-  ( (
ph  /\  ( x  e.  CC  /\  y  e.  CC ) )  -> 
( x  -  y
)  e.  CC )
5 addsub4 9344 . . . 4  |-  ( ( ( x  e.  CC  /\  y  e.  CC )  /\  ( z  e.  CC  /\  w  e.  CC ) )  -> 
( ( x  +  y )  -  (
z  +  w ) )  =  ( ( x  -  z )  +  ( y  -  w ) ) )
65eqcomd 2441 . . 3  |-  ( ( ( x  e.  CC  /\  y  e.  CC )  /\  ( z  e.  CC  /\  w  e.  CC ) )  -> 
( ( x  -  z )  +  ( y  -  w ) )  =  ( ( x  +  y )  -  ( z  +  w ) ) )
76adantl 453 . 2  |-  ( (
ph  /\  ( (
x  e.  CC  /\  y  e.  CC )  /\  ( z  e.  CC  /\  w  e.  CC ) ) )  ->  (
( x  -  z
)  +  ( y  -  w ) )  =  ( ( x  +  y )  -  ( z  +  w
) ) )
8 sersub.1 . 2  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
9 sersub.2 . 2  |-  ( (
ph  /\  k  e.  ( M ... N ) )  ->  ( F `  k )  e.  CC )
10 sersub.3 . 2  |-  ( (
ph  /\  k  e.  ( M ... N ) )  ->  ( G `  k )  e.  CC )
11 sersub.4 . 2  |-  ( (
ph  /\  k  e.  ( M ... N ) )  ->  ( H `  k )  =  ( ( F `  k
)  -  ( G `
 k ) ) )
122, 4, 7, 8, 9, 10, 11seqcaopr2 11359 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 359    = wceq 1652    e. wcel 1725   ` cfv 5454  (class class class)co 6081   CCcc 8988    + caddc 8993    - cmin 9291   ZZ>=cuz 10488   ...cfz 11043    seq cseq 11323
This theorem is referenced by:  serle  11378  cvgcmp  12595  abelthlem6  20352  atantayl  20777
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-n0 10222  df-z 10283  df-uz 10489  df-fz 11044  df-fzo 11136  df-seq 11324
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