Theorem seqfeq3 10987

Description: Equality of series under different addition operations which agree on an additively closed subset. (Contributed by Stefan O'Rear, 21-Mar-2015.) (Revised by Mario Carneiro, 25-Apr-2016.)

Hypotheses

Ref	Expression
seqfeq3.m	|- ( ph -> M e. ZZ )
seqfeq3.f	|- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S )
seqfeq3.cl	|- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )
seqfeq3.id	|- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) = ( x Q y ) )

Assertion

Ref	Expression
seqfeq3	|- ( ph -> seq M ( .+ , F ) = seq M ( Q , F ) )

Distinct variable groups:   ph, x, y   x, F, y   x, M, y   x, .+ , y   x, Q, y   x, S, y

Proof of Theorem seqfeq3

Dummy variable a is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2238	. . . 4 |- ( ZZ>= ` M ) = ( ZZ>= ` M )
2		seqfeq3.m	. . . 4 |- ( ph -> M e. ZZ )
3		seqfeq3.f	. . . 4 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S )
4		seqfeq3.cl	. . . 4 |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )
5	1, 2, 3, 4	seqf 10922	. . 3 |- ( ph -> seq M ( .+ , F ) : ( ZZ>= ` M ) --> S )
6	5	ffnd 5537	. 2 |- ( ph -> seq M ( .+ , F ) Fn ( ZZ>= ` M ) )
7		seqfeq3.id	. . . . 5 |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) = ( x Q y ) )
8	7, 4	eqeltrrd 2316	. . . 4 |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x Q y ) e. S )
9	1, 2, 3, 8	seqf 10922	. . 3 |- ( ph -> seq M ( Q , F ) : ( ZZ>= ` M ) --> S )
10	9	ffnd 5537	. 2 |- ( ph -> seq M ( Q , F ) Fn ( ZZ>= ` M ) )
11	5	ffvelcdmda 5846	. . . 4 |- ( ( ph /\ a e. ( ZZ>= ` M ) ) -> ( seq M ( .+ , F ) ` a ) e. S )
12		fvi 5763	. . . 4 |- ( ( seq M ( .+ , F ) ` a ) e. S -> ( _I ` ( seq M ( .+ , F ) ` a ) ) = ( seq M ( .+ , F ) ` a ) )
13	11, 12	syl 14	. . 3 |- ( ( ph /\ a e. ( ZZ>= ` M ) ) -> ( _I ` ( seq M ( .+ , F ) ` a ) ) = ( seq M ( .+ , F ) ` a ) )
14	4	adantlr 481	. . . 4 |- ( ( ( ph /\ a e. ( ZZ>= ` M ) ) /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )
15	3	adantlr 481	. . . 4 |- ( ( ( ph /\ a e. ( ZZ>= ` M ) ) /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S )
16		simpr 110	. . . 4 |- ( ( ph /\ a e. ( ZZ>= ` M ) ) -> a e. ( ZZ>= ` M ) )
17	7	adantlr 481	. . . . 5 |- ( ( ( ph /\ a e. ( ZZ>= ` M ) ) /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) = ( x Q y ) )
18		fvi 5763	. . . . . 6 |- ( ( x .+ y ) e. S -> ( _I ` ( x .+ y ) ) = ( x .+ y ) )
19	14, 18	syl 14	. . . . 5 |- ( ( ( ph /\ a e. ( ZZ>= ` M ) ) /\ ( x e. S /\ y e. S ) ) -> ( _I ` ( x .+ y ) ) = ( x .+ y ) )
20		fvi 5763	. . . . . . 7 |- ( x e. S -> ( _I ` x ) = x )
21	20	ad2antrl 494	. . . . . 6 |- ( ( ( ph /\ a e. ( ZZ>= ` M ) ) /\ ( x e. S /\ y e. S ) ) -> ( _I ` x ) = x )
22		fvi 5763	. . . . . . 7 |- ( y e. S -> ( _I ` y ) = y )
23	22	ad2antll 495	. . . . . 6 |- ( ( ( ph /\ a e. ( ZZ>= ` M ) ) /\ ( x e. S /\ y e. S ) ) -> ( _I ` y ) = y )
24	21, 23	oveq12d 6106	. . . . 5 |- ( ( ( ph /\ a e. ( ZZ>= ` M ) ) /\ ( x e. S /\ y e. S ) ) -> ( ( _I ` x ) Q ( _I ` y ) ) = ( x Q y ) )
25	17, 19, 24	3eqtr4d 2281	. . . 4 |- ( ( ( ph /\ a e. ( ZZ>= ` M ) ) /\ ( x e. S /\ y e. S ) ) -> ( _I ` ( x .+ y ) ) = ( ( _I ` x ) Q ( _I ` y ) ) )
26		fvi 5763	. . . . 5 |- ( ( F ` x ) e. S -> ( _I ` ( F ` x ) ) = ( F ` x ) )
27	15, 26	syl 14	. . . 4 |- ( ( ( ph /\ a e. ( ZZ>= ` M ) ) /\ x e. ( ZZ>= ` M ) ) -> ( _I ` ( F ` x ) ) = ( F ` x ) )
28	8	adantlr 481	. . . 4 |- ( ( ( ph /\ a e. ( ZZ>= ` M ) ) /\ ( x e. S /\ y e. S ) ) -> ( x Q y ) e. S )
29	14, 15, 16, 25, 27, 15, 28	seq3homo 10985	. . 3 |- ( ( ph /\ a e. ( ZZ>= ` M ) ) -> ( _I ` ( seq M ( .+ , F ) ` a ) ) = ( seq M ( Q , F ) ` a ) )
30	13, 29	eqtr3d 2273	. 2 |- ( ( ph /\ a e. ( ZZ>= ` M ) ) -> ( seq M ( .+ , F ) ` a ) = ( seq M ( Q , F ) ` a ) )
31	6, 10, 30	eqfnfvd 5812	1 |- ( ph -> seq M ( .+ , F ) = seq M ( Q , F ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   _I cid 4435   ` cfv 5380  (class class class)co 6088   ZZcz 9655   ZZ>=cuz 9937   seqcseq 10905

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-iinf 4737  ax-cnex 8276  ax-resscn 8277  ax-1cn 8278  ax-1re 8279  ax-icn 8280  ax-addcl 8281  ax-addrcl 8282  ax-mulcl 8283  ax-addcom 8285  ax-addass 8287  ax-distr 8289  ax-i2m1 8290  ax-0lt1 8291  ax-0id 8293  ax-rnegex 8294  ax-cnre 8296  ax-pre-ltirr 8297  ax-pre-ltwlin 8298  ax-pre-lttrn 8299  ax-pre-ltadd 8301

This proof depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-id 4440  df-iord 4513  df-on 4515  df-ilim 4516  df-suc 4518  df-iom 4740  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-recs 6580  df-frec 6666  df-pnf 8368  df-mnf 8369  df-xr 8370  df-ltxr 8371  df-le 8372  df-sub 8506  df-neg 8507  df-inn 9314  df-n0 9575  df-z 9656  df-uz 9938  df-seqfrec 10906

This theorem is used by:  mulgpropdg  14026