Theorem ser3sub 10943

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 8298	. . 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 8519	. . 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 8563	. . . 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 10913	1 |- ( ph -> ( seq M ( + , H ) ` N ) = ( ( seq M ( + , F ) ` N ) - ( seq M ( + , G ) ` N ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   ` cfv 5375  (class class class)co 6079   CCcc 8171   + caddc 8176   - cmin 8491   ZZ>=cuz 9904   seqcseq 10867

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 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 4244  ax-sep 4247  ax-nul 4257  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-iinf 4733  ax-cnex 8264  ax-resscn 8265  ax-1cn 8266  ax-1re 8267  ax-icn 8268  ax-addcl 8269  ax-addrcl 8270  ax-mulcl 8271  ax-addcom 8273  ax-addass 8275  ax-distr 8277  ax-i2m1 8278  ax-0lt1 8279  ax-0id 8281  ax-rnegex 8282  ax-cnre 8284  ax-pre-ltirr 8285  ax-pre-ltwlin 8286  ax-pre-lttrn 8287  ax-pre-ltadd 8289

This theorem 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 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-tr 4228  df-id 4436  df-iord 4509  df-on 4511  df-ilim 4512  df-suc 4514  df-iom 4736  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-riota 6032  df-ov 6082  df-oprab 6083  df-mpo 6084  df-1st 6368  df-2nd 6369  df-recs 6570  df-frec 6656  df-pnf 8356  df-mnf 8357  df-xr 8358  df-ltxr 8359  df-le 8360  df-sub 8493  df-neg 8494  df-inn 9288  df-n0 9547  df-z 9628  df-uz 9905  df-fz 10395  df-fzo 10533  df-seqfrec 10868

This theorem is referenced by:  ser3le  10957