Theorem seq3caopr 10097

Description: The sum of two infinite series (generalized to an arbitrary commutative and associative operation). (Contributed by NM, 17-Mar-2005.) (Revised by Jim Kingdon, 23-Apr-2023.)

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 ) )
seq3caopr.5	|- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) e. S )
seq3caopr.6	|- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( G ` k ) e. S )
seq3caopr.7	|- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( H ` k ) = ( ( F ` k ) .+ ( G ` k ) ) )

Assertion

Ref	Expression
seq3caopr	|- ( ph -> ( seq M ( .+ , H ) ` N ) = ( ( seq M ( .+ , F ) ` N ) .+ ( seq M ( .+ , G ) ` N ) ) )

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

Allowed substitution hints:   F( x, y, z)   G( x, y, z)   H( x, y, z)   M( x, y, z)   N( x, y, z)

Proof of Theorem seq3caopr

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 5855	. 2 |- ( ( ph /\ ( a e. S /\ b e. S ) ) -> ( a .+ b ) e. S )
3		simpl 108	. . . . . . 7 |- ( ( ph /\ ( ( a e. S /\ b e. S ) /\ ( c e. S /\ d e. S ) ) ) -> ph )
4		simprrl 509	. . . . . . 7 |- ( ( ph /\ ( ( a e. S /\ b e. S ) /\ ( c e. S /\ d e. S ) ) ) -> c e. S )
5		simprlr 508	. . . . . . 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 5858	. . . . . . 7 |- ( ( ph /\ ( c e. S /\ b e. S ) ) -> ( c .+ b ) = ( b .+ c ) )
8	3, 4, 5, 7	syl12anc 1182	. . . . . 6 |- ( ( ph /\ ( ( a e. S /\ b e. S ) /\ ( c e. S /\ d e. S ) ) ) -> ( c .+ b ) = ( b .+ c ) )
9	8	oveq1d 5721	. . . . 5 |- ( ( ph /\ ( ( a e. S /\ b e. S ) /\ ( c e. S /\ d e. S ) ) ) -> ( ( c .+ b ) .+ d ) = ( ( b .+ c ) .+ d ) )
10		simprrr 510	. . . . . 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 5861	. . . . . 6 |- ( ( ph /\ ( c e. S /\ b e. S /\ d e. S ) ) -> ( ( c .+ b ) .+ d ) = ( c .+ ( b .+ d ) ) )
13	3, 4, 5, 10, 12	syl13anc 1186	. . . . 5 |- ( ( ph /\ ( ( a e. S /\ b e. S ) /\ ( c e. S /\ d e. S ) ) ) -> ( ( c .+ b ) .+ d ) = ( c .+ ( b .+ d ) ) )
14	11	caovassg 5861	. . . . . 6 |- ( ( ph /\ ( b e. S /\ c e. S /\ d e. S ) ) -> ( ( b .+ c ) .+ d ) = ( b .+ ( c .+ d ) ) )
15	3, 5, 4, 10, 14	syl13anc 1186	. . . . 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 2140	. . . 4 |- ( ( ph /\ ( ( a e. S /\ b e. S ) /\ ( c e. S /\ d e. S ) ) ) -> ( c .+ ( b .+ d ) ) = ( b .+ ( c .+ d ) ) )
17	16	oveq2d 5722	. . 3 |- ( ( ph /\ ( ( a e. S /\ b e. S ) /\ ( c e. S /\ d e. S ) ) ) -> ( a .+ ( c .+ ( b .+ d ) ) ) = ( a .+ ( b .+ ( c .+ d ) ) ) )
18		simprll 507	. . . 4 |- ( ( ph /\ ( ( a e. S /\ b e. S ) /\ ( c e. S /\ d e. S ) ) ) -> a e. S )
19	1	caovclg 5855	. . . . 5 |- ( ( ph /\ ( b e. S /\ d e. S ) ) -> ( b .+ d ) e. S )
20	3, 5, 10, 19	syl12anc 1182	. . . 4 |- ( ( ph /\ ( ( a e. S /\ b e. S ) /\ ( c e. S /\ d e. S ) ) ) -> ( b .+ d ) e. S )
21	11	caovassg 5861	. . . 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 1186	. . 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 5855	. . . . 5 |- ( ( ph /\ ( c e. S /\ d e. S ) ) -> ( c .+ d ) e. S )
24	23	adantrl 465	. . . 4 |- ( ( ph /\ ( ( a e. S /\ b e. S ) /\ ( c e. S /\ d e. S ) ) ) -> ( c .+ d ) e. S )
25	11	caovassg 5861	. . . 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 1186	. . 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 2142	. 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		seq3caopr.5	. 2 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) e. S )
30		seq3caopr.6	. 2 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( G ` k ) e. S )
31		seq3caopr.7	. 2 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( H ` k ) = ( ( F ` k ) .+ ( G ` k ) ) )
32	2, 2, 27, 28, 29, 30, 31	seq3caopr2 10096	1 |- ( ph -> ( seq M ( .+ , H ) ` N ) = ( ( seq M ( .+ , F ) ` N ) .+ ( seq M ( .+ , G ) ` N ) ) )

Syntax hints:   -> wi 4   /\ wa 103   /\ w3a 930   = wceq 1299   e. wcel 1448   ` cfv 5059  (class class class)co 5706   ZZ>=cuz 9176   seqcseq 10059

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-coll 3983  ax-sep 3986  ax-nul 3994  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-setind 4390  ax-iinf 4440  ax-cnex 7586  ax-resscn 7587  ax-1cn 7588  ax-1re 7589  ax-icn 7590  ax-addcl 7591  ax-addrcl 7592  ax-mulcl 7593  ax-addcom 7595  ax-addass 7597  ax-distr 7599  ax-i2m1 7600  ax-0lt1 7601  ax-0id 7603  ax-rnegex 7604  ax-cnre 7606  ax-pre-ltirr 7607  ax-pre-ltwlin 7608  ax-pre-lttrn 7609  ax-pre-ltadd 7611

This theorem depends on definitions:  df-bi 116  df-3or 931  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-nel 2363  df-ral 2380  df-rex 2381  df-reu 2382  df-rab 2384  df-v 2643  df-sbc 2863  df-csb 2956  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-nul 3311  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-int 3719  df-iun 3762  df-br 3876  df-opab 3930  df-mpt 3931  df-tr 3967  df-id 4153  df-iord 4226  df-on 4228  df-ilim 4229  df-suc 4231  df-iom 4443  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490  df-iota 5024  df-fun 5061  df-fn 5062  df-f 5063  df-f1 5064  df-fo 5065  df-f1o 5066  df-fv 5067  df-riota 5662  df-ov 5709  df-oprab 5710  df-mpo 5711  df-1st 5969  df-2nd 5970  df-recs 6132  df-frec 6218  df-pnf 7674  df-mnf 7675  df-xr 7676  df-ltxr 7677  df-le 7678  df-sub 7806  df-neg 7807  df-inn 8579  df-n0 8830  df-z 8907  df-uz 9177  df-fz 9632  df-fzo 9761  df-seqfrec 10060

This theorem is referenced by:  ser3add  10119