Theorem seqdistr 12852

Description: The distributive property for series. (Contributed by Mario Carneiro, 28-Jul-2013.) (Revised by Mario Carneiro, 27-May-2014.)

Hypotheses

Ref	Expression
seqdistr.1	|- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )
seqdistr.2	|- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( C T ( x .+ y ) ) = ( ( C T x ) .+ ( C T y ) ) )
seqdistr.3	|- ( ph -> N e. ( ZZ>= ` M ) )
seqdistr.4	|- ( ( ph /\ x e. ( M ... N ) ) -> ( G ` x ) e. S )
seqdistr.5	|- ( ( ph /\ x e. ( M ... N ) ) -> ( F ` x ) = ( C T ( G ` x ) ) )

Assertion

Ref	Expression
seqdistr	|- ( ph -> ( seq M ( .+ , F ) ` N ) = ( C T ( seq M ( .+ , G ) ` N ) ) )

Distinct variable groups:   x, y, C   x, G, y   x, M, y   x, N, y   x, .+ , y   x, F   ph, x, y   x, S, y   x, T, y

Allowed substitution hint:   F( y)

Proof of Theorem seqdistr

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		seqdistr.1	. . 3 |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )
2		seqdistr.4	. . 3 |- ( ( ph /\ x e. ( M ... N ) ) -> ( G ` x ) e. S )
3		seqdistr.3	. . 3 |- ( ph -> N e. ( ZZ>= ` M ) )
4		seqdistr.2	. . . 4 |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( C T ( x .+ y ) ) = ( ( C T x ) .+ ( C T y ) ) )
5		oveq2 6658	. . . . . 6 |- ( z = ( x .+ y ) -> ( C T z ) = ( C T ( x .+ y ) ) )
6		eqid 2622	. . . . . 6 |- ( z e. S |-> ( C T z ) ) = ( z e. S |-> ( C T z ) )
7		ovex 6678	. . . . . 6 |- ( C T ( x .+ y ) ) e. _V
8	5, 6, 7	fvmpt 6282	. . . . 5 |- ( ( x .+ y ) e. S -> ( ( z e. S |-> ( C T z ) ) ` ( x .+ y ) ) = ( C T ( x .+ y ) ) )
9	1, 8	syl 17	. . . 4 |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( ( z e. S |-> ( C T z ) ) ` ( x .+ y ) ) = ( C T ( x .+ y ) ) )
10		oveq2 6658	. . . . . . 7 |- ( z = x -> ( C T z ) = ( C T x ) )
11		ovex 6678	. . . . . . 7 |- ( C T x ) e. _V
12	10, 6, 11	fvmpt 6282	. . . . . 6 |- ( x e. S -> ( ( z e. S |-> ( C T z ) ) ` x ) = ( C T x ) )
13	12	ad2antrl 764	. . . . 5 |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( ( z e. S |-> ( C T z ) ) ` x ) = ( C T x ) )
14		oveq2 6658	. . . . . . 7 |- ( z = y -> ( C T z ) = ( C T y ) )
15		ovex 6678	. . . . . . 7 |- ( C T y ) e. _V
16	14, 6, 15	fvmpt 6282	. . . . . 6 |- ( y e. S -> ( ( z e. S |-> ( C T z ) ) ` y ) = ( C T y ) )
17	16	ad2antll 765	. . . . 5 |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( ( z e. S |-> ( C T z ) ) ` y ) = ( C T y ) )
18	13, 17	oveq12d 6668	. . . 4 |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( ( ( z e. S |-> ( C T z ) ) ` x ) .+ ( ( z e. S |-> ( C T z ) ) ` y ) ) = ( ( C T x ) .+ ( C T y ) ) )
19	4, 9, 18	3eqtr4d 2666	. . 3 |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( ( z e. S |-> ( C T z ) ) ` ( x .+ y ) ) = ( ( ( z e. S |-> ( C T z ) ) ` x ) .+ ( ( z e. S |-> ( C T z ) ) ` y ) ) )
20		oveq2 6658	. . . . . 6 |- ( z = ( G ` x ) -> ( C T z ) = ( C T ( G ` x ) ) )
21		ovex 6678	. . . . . 6 |- ( C T ( G ` x ) ) e. _V
22	20, 6, 21	fvmpt 6282	. . . . 5 |- ( ( G ` x ) e. S -> ( ( z e. S |-> ( C T z ) ) ` ( G ` x ) ) = ( C T ( G ` x ) ) )
23	2, 22	syl 17	. . . 4 |- ( ( ph /\ x e. ( M ... N ) ) -> ( ( z e. S |-> ( C T z ) ) ` ( G ` x ) ) = ( C T ( G ` x ) ) )
24		seqdistr.5	. . . 4 |- ( ( ph /\ x e. ( M ... N ) ) -> ( F ` x ) = ( C T ( G ` x ) ) )
25	23, 24	eqtr4d 2659	. . 3 |- ( ( ph /\ x e. ( M ... N ) ) -> ( ( z e. S |-> ( C T z ) ) ` ( G ` x ) ) = ( F ` x ) )
26	1, 2, 3, 19, 25	seqhomo 12848	. 2 |- ( ph -> ( ( z e. S |-> ( C T z ) ) ` ( seq M ( .+ , G ) ` N ) ) = ( seq M ( .+ , F ) ` N ) )
27	3, 2, 1	seqcl 12821	. . 3 |- ( ph -> ( seq M ( .+ , G ) ` N ) e. S )
28		oveq2 6658	. . . 4 |- ( z = ( seq M ( .+ , G ) ` N ) -> ( C T z ) = ( C T ( seq M ( .+ , G ) ` N ) ) )
29		ovex 6678	. . . 4 |- ( C T ( seq M ( .+ , G ) ` N ) ) e. _V
30	28, 6, 29	fvmpt 6282	. . 3 |- ( ( seq M ( .+ , G ) ` N ) e. S -> ( ( z e. S |-> ( C T z ) ) ` ( seq M ( .+ , G ) ` N ) ) = ( C T ( seq M ( .+ , G ) ` N ) ) )
31	27, 30	syl 17	. 2 |- ( ph -> ( ( z e. S |-> ( C T z ) ) ` ( seq M ( .+ , G ) ` N ) ) = ( C T ( seq M ( .+ , G ) ` N ) ) )
32	26, 31	eqtr3d 2658	1 |- ( ph -> ( seq M ( .+ , F ) ` N ) = ( C T ( seq M ( .+ , G ) ` N ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   ZZ>=cuz 11687   ...cfz 12326   seqcseq 12801

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-seq 12802

This theorem is referenced by:  isermulc2  14388  fsummulc2  14516  stirlinglem7  40297