Theorem seqcaoprg 10590

Description: The sum of two infinite series (generalized to an arbitrary commutative and associative operation). (Contributed by NM, 17-Mar-2005.) (Revised by Mario Carneiro, 30-May-2014.)

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 ) )
seqcaopr.5	|- ( ( ph /\ k e. ( M ... N ) ) -> ( F ` k ) e. S )
seqcaopr.6	|- ( ( ph /\ k e. ( M ... N ) ) -> ( G ` k ) e. S )
seqcaopr.7	|- ( ( ph /\ k e. ( M ... N ) ) -> ( H ` k ) = ( ( F ` k ) .+ ( G ` k ) ) )
seqcaoprg.p	|- ( ph -> .+ e. V )
seqcaoprg.f	|- ( ph -> F e. W )
seqcaoprg.g	|- ( ph -> G e. X )
seqcaoprg.h	|- ( ph -> H e. Y )

Assertion

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

Distinct variable groups:   k, F   k, G   k, H   x, k, y, z, ph   k, M   .+ , k, x, y, z   S, k, x, y, z   k, N

Allowed substitution hints:   F( x, y, z)   G( x, y, z)   H( x, y, z)   M( x, y, z)   N( x, y, z)   V( x, y, z, k)   W( x, y, z, k)   X( x, y, z, k)   Y( x, y, z, k)

Proof of Theorem seqcaoprg

Dummy variables a b c d are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		seqcaopr.1	. . 3 |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )
2	1	caovclg 6077	. 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 ) )
7	6	caovcomg 6080	. . . . . . 7 |- ( ( ph /\ ( c e. S /\ b e. S ) ) -> ( c .+ b ) = ( b .+ c ) )
8	3, 4, 5, 7	syl12anc 1247	. . . . . 6 |- ( ( ph /\ ( ( a e. S /\ b e. S ) /\ ( c e. S /\ d e. S ) ) ) -> ( c .+ b ) = ( b .+ c ) )
9	8	oveq1d 5938	. . . . 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 ) ) )
12	11	caovassg 6083	. . . . . 6 |- ( ( ph /\ ( c e. S /\ b e. S /\ d e. S ) ) -> ( ( c .+ b ) .+ d ) = ( c .+ ( b .+ d ) ) )
13	3, 4, 5, 10, 12	syl13anc 1251	. . . . 5 |- ( ( ph /\ ( ( a e. S /\ b e. S ) /\ ( c e. S /\ d e. S ) ) ) -> ( ( c .+ b ) .+ d ) = ( c .+ ( b .+ d ) ) )
14	11	caovassg 6083	. . . . . 6 |- ( ( ph /\ ( b e. S /\ c e. S /\ d e. S ) ) -> ( ( b .+ c ) .+ d ) = ( b .+ ( c .+ d ) ) )
15	3, 5, 4, 10, 14	syl13anc 1251	. . . . 5 |- ( ( ph /\ ( ( a e. S /\ b e. S ) /\ ( c e. S /\ d e. S ) ) ) -> ( ( b .+ c ) .+ d ) = ( b .+ ( c .+ d ) ) )
16	9, 13, 15	3eqtr3d 2237	. . . 4 |- ( ( ph /\ ( ( a e. S /\ b e. S ) /\ ( c e. S /\ d e. S ) ) ) -> ( c .+ ( b .+ d ) ) = ( b .+ ( c .+ d ) ) )
17	16	oveq2d 5939	. . 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 )
19	1	caovclg 6077	. . . . 5 |- ( ( ph /\ ( b e. S /\ d e. S ) ) -> ( b .+ d ) e. S )
20	3, 5, 10, 19	syl12anc 1247	. . . 4 |- ( ( ph /\ ( ( a e. S /\ b e. S ) /\ ( c e. S /\ d e. S ) ) ) -> ( b .+ d ) e. S )
21	11	caovassg 6083	. . . 4 |- ( ( ph /\ ( a e. S /\ c e. S /\ ( b .+ d ) e. S ) ) -> ( ( a .+ c ) .+ ( b .+ d ) ) = ( a .+ ( c .+ ( b .+ d ) ) ) )
22	3, 18, 4, 20, 21	syl13anc 1251	. . 3 |- ( ( ph /\ ( ( a e. S /\ b e. S ) /\ ( c e. S /\ d e. S ) ) ) -> ( ( a .+ c ) .+ ( b .+ d ) ) = ( a .+ ( c .+ ( b .+ d ) ) ) )
23	1	caovclg 6077	. . . . 5 |- ( ( ph /\ ( c e. S /\ d e. S ) ) -> ( c .+ d ) e. S )
24	23	adantrl 478	. . . 4 |- ( ( ph /\ ( ( a e. S /\ b e. S ) /\ ( c e. S /\ d e. S ) ) ) -> ( c .+ d ) e. S )
25	11	caovassg 6083	. . . 4 |- ( ( ph /\ ( a e. S /\ b e. S /\ ( c .+ d ) e. S ) ) -> ( ( a .+ b ) .+ ( c .+ d ) ) = ( a .+ ( b .+ ( c .+ d ) ) ) )
26	3, 18, 5, 24, 25	syl13anc 1251	. . 3 |- ( ( ph /\ ( ( a e. S /\ b e. S ) /\ ( c e. S /\ d e. S ) ) ) -> ( ( a .+ b ) .+ ( c .+ d ) ) = ( a .+ ( b .+ ( c .+ d ) ) ) )
27	17, 22, 26	3eqtr4d 2239	. 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		seqcaopr.5	. 2 |- ( ( ph /\ k e. ( M ... N ) ) -> ( F ` k ) e. S )
30		seqcaopr.6	. 2 |- ( ( ph /\ k e. ( M ... N ) ) -> ( G ` k ) e. S )
31		seqcaopr.7	. 2 |- ( ( ph /\ k e. ( M ... N ) ) -> ( H ` k ) = ( ( F ` k ) .+ ( G ` k ) ) )
32		seqcaoprg.p	. 2 |- ( ph -> .+ e. V )
33		seqcaoprg.f	. 2 |- ( ph -> F e. W )
34		seqcaoprg.g	. 2 |- ( ph -> G e. X )
35		seqcaoprg.h	. 2 |- ( ph -> H e. Y )
36	2, 2, 27, 28, 29, 30, 31, 32, 33, 34, 35	seqcaopr2g 10588	1 |- ( ph -> ( seq M ( .+ , H ) ` N ) = ( ( seq M ( .+ , F ) ` N ) .+ ( seq M ( .+ , G ) ` N ) ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 980   = wceq 1364   e. wcel 2167   ` cfv 5259  (class class class)co 5923   ZZ>=cuz 9603   ...cfz 10085   seqcseq 10541

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625  ax-cnex 7972  ax-resscn 7973  ax-1cn 7974  ax-1re 7975  ax-icn 7976  ax-addcl 7977  ax-addrcl 7978  ax-mulcl 7979  ax-addcom 7981  ax-addass 7983  ax-distr 7985  ax-i2m1 7986  ax-0lt1 7987  ax-0id 7989  ax-rnegex 7990  ax-cnre 7992  ax-pre-ltirr 7993  ax-pre-ltwlin 7994  ax-pre-lttrn 7995  ax-pre-ltadd 7997

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-id 4329  df-iord 4402  df-on 4404  df-ilim 4405  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-riota 5878  df-ov 5926  df-oprab 5927  df-mpo 5928  df-1st 6199  df-2nd 6200  df-recs 6364  df-frec 6450  df-pnf 8065  df-mnf 8066  df-xr 8067  df-ltxr 8068  df-le 8069  df-sub 8201  df-neg 8202  df-inn 8993  df-n0 9252  df-z 9329  df-uz 9604  df-fz 10086  df-fzo 10220  df-seqfrec 10542

This theorem is referenced by:  gsumfzmptfidmadd  13479