Theorem seqcaoprg 10882

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 6215	. 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 541	. . . . . . 7 |- ( ( ph /\ ( ( a e. S /\ b e. S ) /\ ( c e. S /\ d e. S ) ) ) -> c e. S )
5		simprlr 540	. . . . . . 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 6218	. . . . . . 7 |- ( ( ph /\ ( c e. S /\ b e. S ) ) -> ( c .+ b ) = ( b .+ c ) )
8	3, 4, 5, 7	syl12anc 1272	. . . . . 6 |- ( ( ph /\ ( ( a e. S /\ b e. S ) /\ ( c e. S /\ d e. S ) ) ) -> ( c .+ b ) = ( b .+ c ) )
9	8	oveq1d 6073	. . . . 5 |- ( ( ph /\ ( ( a e. S /\ b e. S ) /\ ( c e. S /\ d e. S ) ) ) -> ( ( c .+ b ) .+ d ) = ( ( b .+ c ) .+ d ) )
10		simprrr 542	. . . . . 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 6221	. . . . . 6 |- ( ( ph /\ ( c e. S /\ b e. S /\ d e. S ) ) -> ( ( c .+ b ) .+ d ) = ( c .+ ( b .+ d ) ) )
13	3, 4, 5, 10, 12	syl13anc 1276	. . . . 5 |- ( ( ph /\ ( ( a e. S /\ b e. S ) /\ ( c e. S /\ d e. S ) ) ) -> ( ( c .+ b ) .+ d ) = ( c .+ ( b .+ d ) ) )
14	11	caovassg 6221	. . . . . 6 |- ( ( ph /\ ( b e. S /\ c e. S /\ d e. S ) ) -> ( ( b .+ c ) .+ d ) = ( b .+ ( c .+ d ) ) )
15	3, 5, 4, 10, 14	syl13anc 1276	. . . . 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 2275	. . . 4 |- ( ( ph /\ ( ( a e. S /\ b e. S ) /\ ( c e. S /\ d e. S ) ) ) -> ( c .+ ( b .+ d ) ) = ( b .+ ( c .+ d ) ) )
17	16	oveq2d 6074	. . 3 |- ( ( ph /\ ( ( a e. S /\ b e. S ) /\ ( c e. S /\ d e. S ) ) ) -> ( a .+ ( c .+ ( b .+ d ) ) ) = ( a .+ ( b .+ ( c .+ d ) ) ) )
18		simprll 539	. . . 4 |- ( ( ph /\ ( ( a e. S /\ b e. S ) /\ ( c e. S /\ d e. S ) ) ) -> a e. S )
19	1	caovclg 6215	. . . . 5 |- ( ( ph /\ ( b e. S /\ d e. S ) ) -> ( b .+ d ) e. S )
20	3, 5, 10, 19	syl12anc 1272	. . . 4 |- ( ( ph /\ ( ( a e. S /\ b e. S ) /\ ( c e. S /\ d e. S ) ) ) -> ( b .+ d ) e. S )
21	11	caovassg 6221	. . . 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 1276	. . 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 6215	. . . . 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 6221	. . . 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 1276	. . 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 2277	. 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 10880	1 |- ( ph -> ( seq M ( .+ , H ) ` N ) = ( ( seq M ( .+ , F ) ` N ) .+ ( seq M ( .+ , G ) ` N ) ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1005   = wceq 1398   e. wcel 2205   ` cfv 5357  (class class class)co 6058   ZZ>=cuz 9871   ...cfz 10361   seqcseq 10833

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-addass 8245  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-0id 8251  ax-rnegex 8252  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-ltadd 8259

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-iord 4492  df-on 4494  df-ilim 4495  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-frec 6635  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-inn 9255  df-n0 9514  df-z 9595  df-uz 9872  df-fz 10362  df-fzo 10499  df-seqfrec 10834

This theorem is referenced by:  gsumfzmptfidmadd  14140