Theorem ser3sub 10962

Description: The difference of two infinite series. (Contributed by NM, 17-Mar-2005.) (Revised by Jim Kingdon, 22-Apr-2023.)

Hypotheses

Ref	Expression
sersub.1	|- ( ph -> N e. ( ZZ>= ` M ) )
ser3sub.2	|- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) e. CC )
ser3sub.3	|- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( G ` k ) e. CC )
ser3sub.4	|- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( H ` k ) = ( ( F ` k ) - ( G ` k ) ) )

Assertion

Ref	Expression
ser3sub	|- ( ph -> ( seq M ( + , H ) ` N ) = ( ( seq M ( + , F ) ` N ) - ( seq M ( + , G ) ` N ) ) )

Distinct variable groups:   k, F   k, G   k, H   k, M   k, N   ph, k

Proof of Theorem ser3sub

Dummy variables w x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		addcl 8304	. . 3 |- ( ( x e. CC /\ y e. CC ) -> ( x + y ) e. CC )
2	1	adantl 277	. 2 |- ( ( ph /\ ( x e. CC /\ y e. CC ) ) -> ( x + y ) e. CC )
3		subcl 8525	. . 3 |- ( ( x e. CC /\ y e. CC ) -> ( x - y ) e. CC )
4	3	adantl 277	. 2 |- ( ( ph /\ ( x e. CC /\ y e. CC ) ) -> ( x - y ) e. CC )
5		addsub4 8569	. . . 4 |- ( ( ( x e. CC /\ y e. CC ) /\ ( z e. CC /\ w e. CC ) ) -> ( ( x + y ) - ( z + w ) ) = ( ( x - z ) + ( y - w ) ) )
6	5	eqcomd 2244	. . 3 |- ( ( ( x e. CC /\ y e. CC ) /\ ( z e. CC /\ w e. CC ) ) -> ( ( x - z ) + ( y - w ) ) = ( ( x + y ) - ( z + w ) ) )
7	6	adantl 277	. 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		ser3sub.2	. 2 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) e. CC )
10		ser3sub.3	. 2 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( G ` k ) e. CC )
11		ser3sub.4	. 2 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( H ` k ) = ( ( F ` k ) - ( G ` k ) ) )
12	2, 4, 7, 8, 9, 10, 11	seq3caopr2 10932	1 |- ( ph -> ( seq M ( + , H ) ` N ) = ( ( seq M ( + , F ) ` N ) - ( seq M ( + , G ) ` N ) ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   ` cfv 5377  (class class class)co 6085   CCcc 8177   + caddc 8182   - cmin 8497   ZZ>=cuz 9923   seqcseq 10886

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-addcom 8279  ax-addass 8281  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-0id 8287  ax-rnegex 8288  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-ltadd 8295

This proof depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-frec 6662  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-inn 9306  df-n0 9566  df-z 9647  df-uz 9924  df-fz 10414  df-fzo 10552  df-seqfrec 10887

This theorem is used by:  ser3le  10976