Theorem seq3caopr 10451

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 6017	. 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 6020	. . . . . . 7 |- ( ( ph /\ ( c e. S /\ b e. S ) ) -> ( c .+ b ) = ( b .+ c ) )
8	3, 4, 5, 7	syl12anc 1236	. . . . . 6 |- ( ( ph /\ ( ( a e. S /\ b e. S ) /\ ( c e. S /\ d e. S ) ) ) -> ( c .+ b ) = ( b .+ c ) )
9	8	oveq1d 5880	. . . . 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 6023	. . . . . 6 |- ( ( ph /\ ( c e. S /\ b e. S /\ d e. S ) ) -> ( ( c .+ b ) .+ d ) = ( c .+ ( b .+ d ) ) )
13	3, 4, 5, 10, 12	syl13anc 1240	. . . . 5 |- ( ( ph /\ ( ( a e. S /\ b e. S ) /\ ( c e. S /\ d e. S ) ) ) -> ( ( c .+ b ) .+ d ) = ( c .+ ( b .+ d ) ) )
14	11	caovassg 6023	. . . . . 6 |- ( ( ph /\ ( b e. S /\ c e. S /\ d e. S ) ) -> ( ( b .+ c ) .+ d ) = ( b .+ ( c .+ d ) ) )
15	3, 5, 4, 10, 14	syl13anc 1240	. . . . 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 2216	. . . 4 |- ( ( ph /\ ( ( a e. S /\ b e. S ) /\ ( c e. S /\ d e. S ) ) ) -> ( c .+ ( b .+ d ) ) = ( b .+ ( c .+ d ) ) )
17	16	oveq2d 5881	. . 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 6017	. . . . 5 |- ( ( ph /\ ( b e. S /\ d e. S ) ) -> ( b .+ d ) e. S )
20	3, 5, 10, 19	syl12anc 1236	. . . 4 |- ( ( ph /\ ( ( a e. S /\ b e. S ) /\ ( c e. S /\ d e. S ) ) ) -> ( b .+ d ) e. S )
21	11	caovassg 6023	. . . 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 1240	. . 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 6017	. . . . 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 6023	. . . 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 1240	. . 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 2218	. 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 10450	1 |- ( ph -> ( seq M ( .+ , H ) ` N ) = ( ( seq M ( .+ , F ) ` N ) .+ ( seq M ( .+ , G ) ` N ) ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 978   = wceq 1353   e. wcel 2146   ` cfv 5208  (class class class)co 5865   ZZ>=cuz 9499   seqcseq 10413

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 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-coll 4113  ax-sep 4116  ax-nul 4124  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-setind 4530  ax-iinf 4581  ax-cnex 7877  ax-resscn 7878  ax-1cn 7879  ax-1re 7880  ax-icn 7881  ax-addcl 7882  ax-addrcl 7883  ax-mulcl 7884  ax-addcom 7886  ax-addass 7888  ax-distr 7890  ax-i2m1 7891  ax-0lt1 7892  ax-0id 7894  ax-rnegex 7895  ax-cnre 7897  ax-pre-ltirr 7898  ax-pre-ltwlin 7899  ax-pre-lttrn 7900  ax-pre-ltadd 7902

This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-nel 2441  df-ral 2458  df-rex 2459  df-reu 2460  df-rab 2462  df-v 2737  df-sbc 2961  df-csb 3056  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-nul 3421  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-int 3841  df-iun 3884  df-br 3999  df-opab 4060  df-mpt 4061  df-tr 4097  df-id 4287  df-iord 4360  df-on 4362  df-ilim 4363  df-suc 4365  df-iom 4584  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-f1 5213  df-fo 5214  df-f1o 5215  df-fv 5216  df-riota 5821  df-ov 5868  df-oprab 5869  df-mpo 5870  df-1st 6131  df-2nd 6132  df-recs 6296  df-frec 6382  df-pnf 7968  df-mnf 7969  df-xr 7970  df-ltxr 7971  df-le 7972  df-sub 8104  df-neg 8105  df-inn 8891  df-n0 9148  df-z 9225  df-uz 9500  df-fz 9978  df-fzo 10111  df-seqfrec 10414

This theorem is referenced by:  ser3add  10473  prod3fmul  11515